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the phenomenon of electric conduction in metals and semiconductors has been a prominent research topic ever since the early days of solid - state physics . the idea that electric currents flowing inside solid materials are effected by the transport of `` small '' charged particles ( `` charge carriers '' ) was first conceived by weber.@xcite following the discovery of the electron by thomson,@xcite weber s idea quickly found its concrete expression in attempts to understand electric ( as well as thermal ) conduction in solids as a manifestation of electron transport . the basic concept for a theoretical treatment of conduction in terms of the motion of individual carriers was outlined by riecke.@xcite relying to some extent on this concept , drude@xcite formulated his celebrated transport model , which subsequently was refined by lorentz.@xcite in drude s model , the atomistic picture of matter and the kinetic theory of gases are combined to describe conduction in terms of a homogeneous gas of non - interacting , mobile charge carriers in thermodynamic equilibrium , which are assumed to move against a background of spatially fixed , heavy atoms . when an external electric field is applied , the interplay of field - induced acceleration and subsequent thermalizing collisions with the heavy atoms gives rise to a `` drift current '' of the carriers . the magnitude of this current is determined by the `` mean free path '' , i.e. , the average distance the carriers travel between two collisions . for the drift current to be a valid concept , the mean free path must be very small as compared with typical dimensions of the sample . while originally conceived to describe carrier transport in metals , drude s model later on has been frequently used in qualitative , or semi - quantitative , analyses of transport properties of semiconductors as well .    to overcome the shortcomings of drude s model in the quantitative description of carrier transport in semiconductors , particularly in inhomogeneously doped systems , the drift current was supplemented with a diffusion current,@xcite whereby drude s model has been extended to the `` drift - diffusion model '' of transport . the latter model , while representing a substantial improvement over drude s model and serving as a benchmark of semiclassical transport theories even in modern times , is again valid in the range of very small carrier mean free paths only . in the opposite case of very large mean free paths , carrier transport in semiconductors can be described in terms of the ballistic ( `` thermionic '' ) model,@xcite in which carriers thermally emitted at the ends of a sample are assumed to traverse it without collisions with the background atoms .    until recently , no systematic attempts had been made to bridge the gap between the limiting cases of the drift - diffusion and ballistic descriptions within a unified approach . in view of this situation , we set out to develop the `` thermoballistic approach''@xcite to carrier transport in semiconductors . apart from perceiving the challenge to fill a long - standing gap in the theory of carrier transport in semiconductors , we found that recent progress in device physics , spintronics , and photovoltaics called for an extension of the theoretical framework hitherto available for analyzing experimental data in these fields . the physical concept underlying the thermoballistic approach rests on a random partitioning of the length of a semiconducting sample into `` ballistic transport intervals '' . here , `` random '' implies ( i ) an arbitrary number of intervals and ( ii ) arbitrary positions of the end - points , and thus arbitrary lengths , of the intervals . [ the drift - diffusion model may be viewed as implying a partitioning of the sample length into infinitesimally short ballistic intervals ;  in the ballistic transport model , on the other hand , the sample length constitutes a single ballistic interval . ] an individual partition defines what we call a `` ballistic configuration '' . the points linking adjacent intervals in a ballistic configuration are assumed to be points of local thermodynamic equilibrium characterized by a local chemical potential . any such point acts both as a source of , and a sink for , carriers . that is , on the one hand , carriers are thermally emitted there , with a velocity distribution determined by the local temperature ;  subsequent to their emission , the carriers are ballistically transmitted across either interval to the left and right , facing , in general , potential energy barriers arising from the internal and external potentials inside the sample . on the other hand , carriers emitted at the two equilibrium points neighboring the point under consideration on either side and transmitted to it are `` absorbed '' there , i.e. , they return to thermodynamic equilibrium instantaneously . this equilibration is assumed to result from collisions of the carriers with spatially fixed scattering centers ( `` impurities '' ) randomly distributed over the sample , a view adopted from drude s transport model .    in the manner by which the ballistic intervals have been introduced , the lengths of these intervals are stochastic variables , with associated probabilities given by the probabilities for carriers to traverse an interval without impurity scattering . these probabilities are governed by the carrier mean free path , which is allowed to have arbitrary magnitude . by averaging the ( in general , spin - dependent ) carrier currents in the individual ballistic intervals over all ballistic configurations , a position - dependent total ( i.e. , spin - summed ) thermoballistic current as well as a thermoballistic spin - polarized current are derived . these currents , in conjunction with the associated ballistic densities , represent the key elements of the thermoballistic approach . they form the point of departure for the calculation of the spin - resolved equilibrium chemical potentials . from the latter functions , in turn , transport quantities , like the current - voltage characteristic , the magnetoresistance , and ( position - dependent ) current and density spin polarizations , are obtained in terms of the potential energy profile , the mean free path , the ballistic spin relaxation length , and other parameters characterizing the sample .    the present article is devoted to a comprehensive exposition of the physical concept underlying the thermoballistic approach and its detailed implementation . we begin by describing a prototype model of thermoballistic transport ( the `` prototype model '' , for short ) . this model introduces averaging over ballistic configurations as the constitutive element of the thermoballistic concept . it is based on the simplifying assumption that the carrier current is conserved across the points of local thermodynamic equilibrium linking the ballistic transport intervals . this assumption allows the derivation of an explicit , transparent expression for the current - voltage characteristic , as well as the construction of a local chemical potential in a heuristic form . after describing the prototype model , we present a general formulation of the thermoballistic concept proper , which makes use of a local chemical potential as the essential dynamical quantity , and takes into account arbitrarily shaped , spin - split potential energy profiles as well as spin relaxation during ballistic carrier motion .    throughout this article , the formulation deviates in many respects from that in our original publications . apart from correcting flaws and inconsistencies , we introduce modifications and extensions of the formalism that make it more transparent and more comprehensible . aiming at a self - contained , unified presentation , we will develop the elements of the thermoballistic concept in considerable detail . this includes a recollection of early attempts to describe charge carrier transport in semiconductors and , at one place or another , the coherent recapitulation of background material which otherwise can be found only scattered over textbooks . while the thermoballistic approach , in its most general form , would allow a consistent treatment of three - dimensional , bipolar carrier transport in semiconducting systems , we confine ourselves here to the case of one - dimensional , unipolar transport throughout . to be specific , we consider electron transport in the conduction band of n - doped systems , with the understanding that all results for hole transport in the valence band of p - doped systems can be obtained by transcribing the results for electron transport in an obvious way . in implementing the thermoballistic approach , we aim at carrying the development to the point where we obtain explicit equations from which the relevant physical quantities can be calculated . we do not enter here into applications requiring numerical calculations . pertinent results presented in our previous publications will be briefly mentioned at various places in this article . the organization of this article is as follows . in the next section , we present an account of drude s transport model and the standard drift - diffusion and ballistic models . in sec . [ sec : gendrift ] , we begin with an outline of the probabilistic approach to carrier transport , which is prerequisite to the formulation of the thermoballistic model . this is followed by the description of the prototype model . in sec . [ sec : therconc ] , the thermoballistic concept proper is developed at length , where particular emphasis is placed on the inclusion of spin degrees of freedom . the spin - resolved ballistic currents and associated densities are introduced in terms of an `` average chemical potential '' and a `` spin accumulation function '' connected with the splitting of the spin - resolved chemical potentials . total and spin - polarized thermoballistic currents and densities are constructed by averaging the corresponding ballistic quantities over the ballistic configurations , and are evaluated in the drift - diffusion regime and in the ballistic limit . energy dissipation ( `` heat production '' ) is analyzed within the thermoballistic approach . in sec . [ sec : thersyno ] , in particular , a synopsis of the thermoballistic concept is presented , with emphasis on its physical content as well as its merits and weaknesses . the procedures devised for the implementation of the thermoballistic concept , i.e. , for the explicit calculation of the average chemical potential and of the spin accumulation function as functions of intrinsic and external physical parameters , are described in sec . [ sec : therimpl ] . expressions are derived for the current and density spin polarizations and the magnetoresistance in terms of the values of the spin accumulation function at the boundaries of a semiconducting sample . as specific examples of current interest in semiconductor and spintronics research , we treat spin - polarized transport in heterostructures formed of a nonmagnetic semiconductor and two ferromagnetic contacts , and spin - polarized transport in heterostructures involving diluted magnetic semiconductors in their paramagnetic phase . finally , in sec .  [ sec : summconc ] , we summarize the contents of this article and give an outlook towards future developments . as the thermoballistic approach is devised to bridge the gap between the drift - diffusion and ballistic descriptions of transport , we survey , in this section , the standard formulations of these two limiting cases . the drift - diffusion model is an extension of the transport model of drude , so that we begin here with an account of the latter . drude s model is far from being able to describe transport properties of semiconductors quantitatively ;  nevertheless , its exposition provides us with the opportunity to introduce and discuss the basic notions on which classical and semiclassical transport theories rely , and which will appear ubiquitously throughout this article .      in the transport model of drude,@xcite one considers a ( three - dimensional ) homogeneous classical gas of non - interacting electrons in thermodynamic equilibrium at temperature @xmath0 , which is subjected to an externally applied , constant electric field @xmath1 . the electrons collide with randomly distributed , spatially fixed scattering centers ( `` impurities '' ) . the collisions are assumed to cause the electrons to return instantaneously to complete thermodynamic equilibrium . the average vectorial velocity of the electrons emerging from this equilibration is equal to zero . their mean speed @xmath2 , i.e. , the thermal average of the magnitude of the electron velocity as derived from the three - dimensional maxwell - boltzmann velocity distribution function , is given by @xmath3 where @xmath4 is the effective electron mass , and @xmath5 , with @xmath6 the boltzmann constant . [ note that in drude s original paper,@xcite the mean electron speed was derived from boltzmann s equipartition theorem ;  the maxwell - boltzmann velocity distribution was introduced into the description of electron transport in metals by lorentz.@xcite ]    defining the _ mean free path _ ( or _ momentum relaxation length _ ) @xmath7 as the average distance ( measured along the transport direction ) travelled by the electrons between two collisions , one finds for the _ collision time _ ( or _ relaxation time _ ) @xmath8 , defined as the average time - of - flight between successive collisions , @xmath9 the mean free path @xmath7 characterizes a set of collision - free _ ballistic intervals _ of average length @xmath7 , across which electrons move ballistically under the influence of the external field @xmath1 . on average , they acquire a velocity , the _ drift velocity _ @xmath10 , given by the acceleration @xmath11 times their average time - of - flight across the ballistic intervals , i.e. , times the collision time @xmath8 , @xmath12 where @xmath13 is the electron mobility .    in the picture of drude , the charge current density @xmath14 ( current , for short ) is a _ drift current _ , @xmath15 , driven by the external electric field @xmath1 , @xmath16 where @xmath17 is the electron density . since , in drude s model , @xmath17 as well as the field @xmath1 are independent of position , @xmath18 is constant . in terms of the electron conductivity @xmath19 the current @xmath14 is expressed as @xmath20 i.e. , in the form of ohm s law .    the transport mechanism in drude s model can be elucidated@xcite by noting that eq . ( [ eq : drude003 ] ) , when written as @xmath21 states that the effect of the external force @xmath22 is balanced , on average , by that of a _ friction force _ @xmath23 , with the friction coefficient @xmath24 given by @xmath25 the friction force reflects the reset to zero of their individual velocities when , subsequent to their acceleration through the external field , the electrons are thermally equilibrated due to impurity scattering . limits on the range of validity of drude s model are set by the requirement that the perturbation of the electron gas due to the external field is sufficiently small so that the gas stays close to thermodynamic equilibrium . this condition can be met by requiring the magnitude of the drift velocity to be very small as compared with the mean speed of the electrons , @xmath26 using eqs . ( [ eq : drude001 ] ) , ( [ eq : drude003 ] ) , and ( [ eq : drude004 ] ) , condition ( [ eq : drude004c ] ) can be re - expressed in a form that restricts , for given mean free path @xmath7 , the magnitude of the electric - field vector @xmath1 , @xmath27 for a homogeneous sample of length @xmath28 subjected to an external voltage bias @xmath29 , when @xmath30 , one has @xmath31 reversely , for given voltage bias @xmath29 , the ratio of mean free path to sample length is restricted by the condition @xmath32 for room temperature , i.e. , @xmath33 ev , and values of @xmath34 of a few tens of mev , which are typical for semiconducting devices , the right - hand side of condition ( [ eq : drude004e ] ) is of order unity . hence , one may use the condition @xmath35 as a rough criterion for delimiting the range of validity of the drude model . the value of @xmath7 remains small if the density of impurities is sufficiently high . electron transport in inhomogeneous semiconductors is outside of the scope of drude s model . the nonuniformity of the donor density in inhomogeneous samples entails a position dependence of the electron density , @xmath36 , and gives rise to an @xmath37-dependent internal ( `` built - in '' ) electrostatic potential.@xcite [ throughout this article , we consider _ one - dimensional _ transport in _ three - dimensional _ , `` plane - parallel '' semiconducting samples , i.e. , samples whose parameters do not vary in the directions perpendicular to the transport direction , which is taken as the @xmath37-direction ( for a discussion of this assumption , see sec . [ sec : thersyno ] ) . further , the temperature @xmath0 is assumed to be constant across the sample . ] transport in inhomogeneous systems near thermodynamic equilibrium can be described in terms of a _ local _ thermodynamic equilibrium@xcite characterized by a _ local chemical potential _ @xmath38.@xcite disregarding spin degrees of freedom and adopting the effective - mass approximation,@xcite we have for the equilibrium electron density @xmath39 in a nondegenerate system @xmath40 here , @xmath41 is planck s constant , and we have introduced cylindrical coordinates in three - dimensional velocity space ( with cartesian coordinates @xmath42 ) such that @xmath43 with @xmath44 . the function @xmath45 is the maxwell - boltzmann energy distribution function , and @xmath46 is the total electronic energy at the equilibrium point @xmath37 , with the kinetic energy @xmath47 the potential energy profile @xmath48 comprises the conduction band edge potential ( which includes the position - dependent internal potential ) and the external electrostatic potential . in general , @xmath48 exhibits a `` multiple - barrier '' shape associated with local maxima of the profile . evaluation of the integrals in eq . ( [ eq : drude102 ] ) results in the standard form for the density , @xmath49 } , \label{eq : drude014}\ ] ] where @xmath50 is the effective density of states at the conduction band edge.@xcite    the effects of the spatial variation of the electron density and of the associated occurrence of an internal potential in inhomogeneous semiconductors can be described within an extension of drude s model , the _ drift - diffusion model_. in this model , a generalized , @xmath37-dependent drift current @xmath51 is supplemented@xcite by a _ diffusion current _ @xmath52 proportional to the density gradient along the @xmath37-axis , so that the total ( conserved ) current @xmath53 is given by @xmath54 the generalized drift current is obtained from expression ( [ eq : drude005 ] ) by replacing ( i ) the conductivity @xmath55 with the _ local conductivity _ @xmath56 given by eq . ( [ eq : drude006 ] ) , with @xmath39 in lieu of @xmath17 , and ( ii ) the constant external electric field @xmath1 with the field @xmath57/e$ ] associated with the potential energy profile @xmath48 , so that one obtains @xmath58 the diffusion current is written as @xmath59 where @xmath60 is the diffusion coefficient , which is related to the electron mobility @xmath61 via the einstein relation@xcite @xmath62 and @xmath61 is now assumed to have the form corresponding to one - dimensional transport , @xmath63 [ see eq . ( [ eq : drude004 ] ) for the three - dimensional form of @xmath61 ] . then , generalizing eq . ( [ eq : drude006 ] ) , one has @xmath65 for the total current @xmath53 , one now finds from eqs . ( [ eq : drude008f])([eq : drude007 ] ) , using eq . ( [ eq : drude006f ] ) , @xmath66 , \label{eq : drude008}\ ] ] which is the standard drift - diffusion expression for the total charge current . for the derivation of this expression from boltzmann s transport equation and for its application in device simulation , see , e.g. , ref .  . in ref .  , the expression is derived within a time - dependent tutorial treatment of diffusion in the presence of an external electric field ( `` biased - random - walk model '' ) .    substituting expression ( [ eq : drude014 ] ) for the equilibrium density @xmath39 in eq . ( [ eq : drude008 ] ) , we obtain the current @xmath53 in the form @xmath67 which shows that , in the drift - diffusion model , it is the local chemical potential which provides the driving force for electron transport . using eqs . ( [ eq : drude014 ] ) and ( [ eq : drude006f ] ) , we can rewrite eq . ( [ eq : diff001 ] ) as @xmath68 considering a sample extending from @xmath69 to @xmath70 and integrating eq . ( [ eq : diff002 ] ) over the interval @xmath71 $ ] , we can express the local chemical potential @xmath38 in the form @xmath72 an equivalent representation of @xmath38 is obtained by integrating eq . ( [ eq : diff002 ] ) over the interval @xmath73 $ ] . now , setting @xmath74 in eq . ( [ eq : diff003 ] ) , identifying the chemical potentials at the end - points @xmath75 with the potentials @xmath76 in the contacts connected to the semiconducting sample , @xmath77 and solving the resulting equation for the current @xmath53 , we then obtain , using eq .  ( [ eq : drude014 ] ) , the current - voltage characteristic of the drift - diffusion model in the form @xmath78 here , @xmath79 is the voltage bias , and @xmath80 is the maximum barrier height of the potential energy profile relative to its value @xmath81 at the left end of the sample , where @xmath82 is the overall maximum of @xmath48 in the interval @xmath83 $ ] . finally , the quantity @xmath84 } \label{eq : diff008}\ ] ] is the `` effective sample length '' . it has the appealing property of becoming equal to the sample length @xmath85 for a flat profile , i.e. , if @xmath86 , and otherwise satisfies @xmath87 . writing the current - voltage characteristic in the particular form ( [ eq : diff006 ] ) facilitates comparison with analogous expressions given below .    expression ( [ eq : diff006 ] ) shows that the current - voltage characteristic of the drift - diffusion model is controlled ( i ) by the `` barrier factor '' @xmath88 , which involves the overall maximum of the profile @xmath48 , and ( ii ) by the ratio @xmath89 or , owing to eq . ( [ eq : drude004hg ] ) , by the ratio @xmath90 , in which the effective sample length reflects , in an integral way , the shape of @xmath48 . [ in the ballistic description of transport , the barrier factor @xmath91 re - appears as the thermally averaged probability for electron transmission from @xmath69 to @xmath70 ; see eq . ( [ eq : ball005 ] ) below . ] the drift - diffusion model is based on the assumption of a continuously varying equilibrium chemical potential @xmath38 , which implies that the points of local thermodynamic equilibrium lie arbitrarily dense . then , strictly speaking , the mean free path @xmath7 must be confined to arbitrarily small values . on the other hand , @xmath7 ( or the mobility @xmath61 ) must be finite and large enough to give rise to a non - vanishing conductivity of a magnitude in the range of typical experimental values , which calls for a relaxation of the former condition .    to obtain a practical criterion for the range of validity of the drift - diffusion model , one may require @xmath7 to be so small that the effective number of points of local thermodynamic equilibrium along the length of the sample is so large that the spatial variations in the potential energy profile @xmath48 , and hence in the electron density @xmath39 , are `` resolved '' with sufficient accuracy . in terms of a local , @xmath37-dependent mean free path @xmath92 , this requirement may be expressed as @xmath93 where @xmath94 is the length of an interval , centered at the point @xmath37 , over which the relative variation of @xmath48 is very small compared to unity , i.e. , @xmath95 for constant mean free path @xmath7 , one may fulfil the above requirement in an overall way by adopting the condition @xmath96 the effective sample length @xmath97 defined by the integral ( [ eq : diff008 ] ) tends to decrease exponentially when strong variations in @xmath48 are `` switched on '' . condition ( [ eq : diff009 ] ) tightens condition ( [ eq : drude004f ] ) so as to permit the `` resolution '' of the details of the profile @xmath48 . in contrast to the drift - diffusion model , in which the points of local thermodynamic equilibrium are assumed to lie arbitrarily dense , the ballistic ( thermionic ) transport model presupposes the complete absence of such points inside the sample . then , the electrons in the sample perform a collision - free _ ballistic motion _ in the field associated with the potential energy @xmath48 . without thermal equilibration , it is meaningless to speak of a local chemical potential . only at the sample ends at @xmath75 , the electrons are forced into equilibrium , with densities @xmath98 } \label{eq : ball000}\ ] ] determined by the boundary values of the potential energy profile , @xmath99 , and by the chemical potentials @xmath76 in the contacts . here , the mean free path @xmath7 , which has been of central importance in drude s model and in the drift - diffusion model , is effectively of infinite length and does not appear in the formalism of the ballistic model . roughly speaking , one may use the condition @xmath100 to delimit the range of validity of the ballistic model .      in the ballistic model , the end - points @xmath75 of the sample are fixed points of local thermodynamic equilibrium with chemical potentials @xmath101 , out of which thermal electron currents are symmetrically emitted towards the left and right , so that only _ one half _ of each of these currents are emitted towards the inner region of the sample .    for a nondegenerate system , the classical _ electron current _ @xmath102 emitted at the left end - point @xmath69 towards the right , say , is expressed , in extension of the electron density @xmath39 given by eq . ( [ eq : drude102 ] ) , in the form @xmath103 from which we find , using eq . ( [ eq : ball000 ] ) , @xmath104 } =   v_{e } n(x_{1 } ) . \label{eq : ball001}\ ] ] here , @xmath105 is the _ emission velocity _ , which is actually equal to one half of the one - dimensional mean electron speed @xmath106 the three - dimensional analogue of which is given by eq . ( [ eq : drude001 ] ) . the electrons emitted at @xmath69 with velocity component @xmath107 move along ballistic trajectories , thereby conserving their total energy , @xmath108 where @xmath37 is any point inside the interval @xmath83 $ ] ( and is not a point of local thermodynamic equilibrium ) . the energy distribution at @xmath37 is therefore equal to that at @xmath69 . however , only electrons with total energy larger than the overall maximum @xmath82 of @xmath48 [ see eq . ( [ eq : diff007 ] ) ] are classically able to reach the right end - point of @xmath83 $ ] at @xmath70 . thus , part of the current @xmath102 emitted at @xmath69 will be reflected , and the electrons forming it are absorbed when they return to their origin at @xmath69 . the other , transmitted part @xmath109 , called the ( left ) `` ballistic current '' , is absorbed into the contact connected to the sample at @xmath70 . modifying expression ( [ eq : ball001h ] ) , we have for this part @xmath110 in the integration , the potential energy profile @xmath48 contained in the function @xmath111 drops out , and we obtain for the ( left ) ballistic current the @xmath37-independent expression @xmath112 } , \label{eq : ball004f}\ ] ] i.e. , the ballistic current is conserved ( independent of @xmath37 ) , as expected . the ( right ) ballistic current @xmath113 transmitted from the _ right _ end - point @xmath70 of the interval @xmath83 $ ] is given by @xmath114 } , \label{eq : ball006}\ ] ] in analogy to eq . ( [ eq : ball004f ] ) . the ballistic currents ( [ eq : ball004f ] ) and ( [ eq : ball006 ] ) bear a close analogy to the `` thermionic emission current '' associated with the evaporation of electrons from a heated metal ( `` richardson effect'').@xcite in semiconductor physics , `` thermionic emission '' was introduced as a mechanism of carrier transport by bethe@xcite in his treatment of electron transport across a schottky barrier . associated with the ballistic currents @xmath115 are the `` ballistic densities '' @xmath116 of the electrons making up the currents inside the ballistic interval @xmath83 $ ] . these densities will turn out to be instrumental in establishing the spin - dependent thermoballistic scheme ( see sec . [ sec : balltrans ] ) .    the density @xmath117 associated with the current @xmath109 of eq . ( [ eq : ball004 ] ) is given by @xmath118 which is evaluated to yield @xmath119 } . \label{eq : ball004ka}\end{aligned}\ ] ] here , @xmath120 } \nonumber \\ & \ & \times { \rm erfc } \bm { ( } \ { \beta [ e_{c}^{m}(x_{1 } , x_{2 } ) - e_{c}(x ) ] \}^{1/2 } \bm { ) } , \nonumber \\ \label{eq : ball004kaa}\end{aligned}\ ] ] where the function @xmath121 is the complementary error function.@xcite the ballistic density is position - dependent via the @xmath37-dependence of the function @xmath122 , i.e. , of the potential energy profile @xmath48 . the ballistic density @xmath123 associated with the current @xmath113 is obtained by replacing @xmath124 with @xmath125 in expression ( [ eq : ball004ka ] ) . the function @xmath122 determines the `` ballistic velocities '' @xmath126 which have the same magnitude for the currents transmitted from the left and right . for constant potential energy profile , one has @xmath127 , and the electrons move with speed @xmath128 , i.e , with the mean electron speed @xmath2 given by eq . ( [ eq : ball002hvl ] ) . for position - dependent profiles , when @xmath129 , the magnitude of the ballistic velocities is larger than @xmath2 .    the _ net ballistic current _ @xmath130 in the interval @xmath131 $ ] , @xmath132 equals the ( conserved ) total current @xmath133 , which we can express , using eqs . ( [ eq : ball004f ] ) and ( [ eq : ball006 ] ) , as @xmath134 this can be rewritten , using eqs . ( [ eq : diff004 ] ) and ( [ eq : ball000 ] ) , in the form @xmath135 here , @xmath136 with @xmath137 given by eq . ( [ eq : diff007 ] ) , is the thermal average of the classical transmission probability @xmath138 for electrons emitted at @xmath69 with total energy @xmath139 to be transmitted to the point @xmath70 . if , in particular , the potential energy profile is constant across the interval @xmath83 $ ] , or if its maximum lies at the emission point @xmath69 itself , then @xmath140 in eq .  ( [ eq : diff007 ] ) , and hence @xmath141 . relation ( [ eq : ball009 ] ) is the current - voltage characteristic of the ( classical ) ballistic transport model . in contrast to the characteristic ( [ eq : diff006 ] ) of the drift - diffusion model , which involves the mean free path @xmath7 and the potential energy profile @xmath48 [ via the effective sample length @xmath97 ] , the characteristic ( [ eq : ball009 ] ) is controlled by one `` material parameter '' only , _ viz . _ , the thermally averaged transmission probability @xmath143 .    for the _ joint ballistic density _ @xmath144 , @xmath145 we have @xmath146 in analogy to expression ( [ eq : ball008 ] ) for the net ballistic current .      the ballistic transport model is straightforwardly extended so as to include _ electron tunneling _ by replacing the classical transmission probability @xmath147 of eq . ( [ eq : ball005a ] ) with the corresponding _ quantal _ probability @xmath148 . the thermally averaged quantal transmission probability is then , in generalization of expression ( [ eq : ball005 ] ) , given by @xmath149 where @xmath150 the probability @xmath148 is obtained by solving the stationary schrdinger equation with the potential energy function @xmath48 . [ owing to time reversal invariance , the probability for transmission from the left equals that for transmission from the right . ] the integration in eq . ( [ eq : ball010 ] ) starts at the total energy @xmath151 , so that scattering boundary conditions can be imposed on the wavefunction both in the ranges @xmath152 and @xmath153 . in wkb approximation,@xcite the transmission probability @xmath154 to be used in eq . ( [ eq : ball010 ] ) is composed of the classical (  over - barrier " ) part @xmath147 given by eq . ( [ eq : ball005a ] ) and the remaining quantal ( `` sub - barrier '' ) part @xmath155 , @xmath156 the sub - barrier contribution has the form @xmath157 where @xmath158 with @xmath159 \}^{1/2 } \theta \bm { ( } e_{c}(x ) - e \bm { ) } , \label{eq : ball012g}\ ] ] is the barrier penetration factor .    in writing @xmath160 in the form ( [ eq : ball012 ] ) , we disregard resonance effects that may occur when @xmath48 exhibits two or more local maxima in the interval @xmath83 $ ] , with a corresponding number of one or more minima in between . then , when the energy @xmath161 is located below the second - highest maximum and above the lowest minimum , there is at least one `` valley '' in @xmath48 , across which the eletron motion is classically allowed , so that resonance formation due to quantum coherence becomes possible . in semiconductor physics , a concrete realization of this situation occurs in resonant tunneling in multiple - barrier quantum - well structures.@xcite for this case , the full wkb tunneling probability for double - barrier and triple - barrier structures , respectively , has been presented in ref .  .    from eq . ( [ eq : ball011 ] ) , we now find for the wkb form of the thermally averaged transmission probability @xmath162 of eq . ( [ eq : ball010 ] ) @xmath163 here , we have @xmath164 with @xmath165 for the over - barrier contribution , and @xmath166 for the sub - barrier contribution . the thermally averaged quantal probability @xmath167 for transmission from the left end - point at @xmath69 can be expressed in terms of @xmath168 , using eqs . ( [ eq : diff007 ] ) and ( [ eq : ball003a ] ) , as @xmath169 } . \label{eq : ball006b}\ ] ] in analogy to eq . ( [ eq : ball009 ] ) for the classical case , we have @xmath170 for the current - voltage characteristic of tunneling - enhanced ballistic transport .      in the degenerate case , when the electron system obeys fermi - dirac statistics , we write the ballistic current @xmath171 in the form [ see eqs . ( [ eq : ball001h ] ) and ( [ eq : ball004 ] ) for the nondegenerate case ] @xmath172 where @xmath173 is the fermi - dirac energy distribution function , @xmath174 expression ( [ eq : degen040 ] ) for the current @xmath175 formally agrees with the expression for the current of evaporated electrons encountered in the degenerate treatment of the richardson effect.@xcite then , following the procedure of ref . , we can reduce the threefold integration in eq . ( [ eq : degen040 ] ) to a single integration over the kinetic energy @xmath176 , obtaining @xmath177 the ballistic current @xmath113 transmitted from the right end - point of the sample at @xmath70 is the negative of expression ( [ eq : degen010 ] ) , with @xmath125 substituted for @xmath124 . the total current @xmath133 [ see eq . ( [ eq : ball007 ] ) ] is thus obtained as @xmath178 \nonumber \\ & \ & \hspace{3.0 cm } \times \theta \bm { ( } \epsilon - e_{c}^{m } ( x_{1 } , x_{2 } ) \bm { ) } . \label{eq : degen011e}\end{aligned}\ ] ] here , we have expressed the factor preceding the integral in eq . ( [ eq : degen010 ] ) in terms of the emission velocity , @xmath179 , and the effective density of states , @xmath180 , which are nondegenerate quantities given by eqs . ( [ eq : ball002 ] ) and ( [ eq : diff001a ] ) , respectively . expression ( [ eq : degen011e ] ) for the total currrent @xmath133 is the degenerate analogue to the nondegenerate current - voltage characteristic ( [ eq : ball008 ] ) .    for zero bias , the chemical potentials at the sample ends , @xmath124 and @xmath125 , differ only by an infinitesimal @xmath181 , @xmath182 then , expanding the right - hand side of eq . ( [ eq : degen011e ] ) to first order in @xmath181 , we find @xmath183_{\mu = \mu_{1 } } \nonumber \\ & = & v_{e } n_{c } \beta \delta \ln \bm { ( } 1 + e^{-\beta[e_{c}^{m}(x_{1 } , x_{2 } ) - \mu_{1 } ] } \bm { ) } . \label{eq : degen001}\end{aligned}\ ] ] setting @xmath184 , we now have for the zero - bias conductance per unit area in the ballistic transport model @xmath185 } \bm { ) } . \label{eq : degen009}\end{aligned}\ ] ] in highly doped , degenerate semiconductors , we may have @xmath186 ( see , e.g. , ref .  , where grain - boundary - limited transport in polycrystalline materials is considered ) . then , if @xmath187 \gg 1 ,   \label{eq : degen009ofd}\ ] ] we find from eq . ( [ eq : degen009 ] ) @xmath188 , \label{eq : degen009ze}\ ] ] which equals the conductance of a ballistic point contact.@xcite    in closing this subsection , we note that the ballistic transport model does not provide information on where the resistance causing the voltage drop is located along the sample . evidently , it can not be inside the collision - free sample . in the quantal description of ballistic electron transport in mesoscopic systems as formulated by landauer,@xcite the resistance is made up solely of the _ interface resistances _ arising from the abrupt change in the density of states ( `` transverse modes '' ) that the electrons encounter when they move across the interfaces separating the contacts ( with infinitely many modes ) from the sample ( with a few modes only ) . the voltage drop is located , therefore , in the immediate vicinity of the interfaces , so that , when a chemical potential is introduced _ ad hoc _ , this must be constant inside the sample and discontinuous at the interfaces . prior to the work of landauer , the importance of interface resistances in ballistic transport had been emphasized by sharvin.@xcite    anticipating the later development , we remark at this point that in the thermoballistic approach , i.e. , for _ finite _ magnitude of the mean free path , the ( local ) equilibrium chemical potential is a constitutive element of the transport mechanism ;  it is defined , and can be explicitly calculated , all along a semiconducting sample . this potential has discontinuities at the contact - sample interfaces , whose magnitude increases from near - zero in the small-@xmath7 , drift - diffusion regime to the sharvin value in the large-@xmath7 , ballistic regime . for more details , see sec . [ sec : sharvin ] . in the drift - diffusion and ballistic transport models , the parameter of central importance is the electron mean free path @xmath7 or , equivalently , the collision time @xmath8 originally introduced in drude s model . the collision time represents the average time - of - flight that elapses between successive electron collisions with the randomly distributed scattering centers in the sample . the drift - diffusion and ballistic transport mechanisms are limiting cases associated with very small and very large ( effectively , infinite ) collision times , respectively . in the present section , we proceed to the `` prototype thermoballistic model'',@xcite which represents an attempt to unify the drift - diffusion and ballistic models within a stationary ( time - independent ) description in which the mean free path is not limited in magnitude and enters as a parameter that controls collision _ probabilities_. [ in our original paper,@xcite we have called this model the `` generalized drude model '' ; the new name appears to give a better description of its features . ] we begin this section by introducing the probabilistic definition of @xmath7 ( or @xmath8 ) .      in the time - dependent probabilistic approach , one introduces the probability @xmath189 that an electron undergoes a collision , i.e. , is equilibrized with its surroundings , within unit time . that is , @xmath190 is the probability for the collision to occur within an infinitesimally short time interval @xmath191 . then , assuming the collision time @xmath8 to be constant , the probability @xmath192 that an electron moves without collision over a finite time interval @xmath193 is given@xcite by @xmath194 and the conditional probability @xmath195 that an electron undergoes a collision within the time interval @xmath191 after a collision - free flight over a time interval @xmath193 , by @xmath196 the mean time - of - flight between successive collisions now becomes @xmath197 identifying @xmath198 with the collision time @xmath8 of the drude model , one sees that the inverse of this quantity is just equal to the collision probability per unit time , @xmath199 , of the probabilistic approach . we recall that the term `` collision '' used here is that proffered by drude , meaning a collision leading to instantaneous , complete equilibration of the momentum of an electron with its surroundings . in that picture , @xmath8 is the average time of collision - free flight between points of equilibration , hence the appellation . however , @xmath189 is , more generally , the probability for a _ complete equilibration _ to occur in unit time ;  this may happen as a consequence of a `` complete collision '' ( in the drude sense ) after the average time @xmath8 , or by `` incomplete collisions '' ( leading to incomplete equilibration ) in a sequence of shorter collision times , which add up , on average , to the time @xmath8 .    within the time - dependent probabilistic picture , one can set up@xcite an expression for the net electron current @xmath200 at position @xmath37 and time @xmath193 , which embodies features that are relevant for devising the thermoballistic concept ( see sec . [ sec : thermcurr ] ) . assuming a sample with position - dependent electron density @xmath39 , one writes @xmath201 considering for the moment electrons with _ arbitrary , constant _ velocity @xmath202 , the currents @xmath203 are expressed , using eqs . ( [ eq : prob001 ] ) and ( [ eq : prob002 ] ) , as @xmath204 the contributions to these currents for fixed time @xmath193 arise from electrons that are transmitted , subsequent to their emission at the points @xmath205 [ lying to the left ( right ) of the point @xmath37 ] , without collision from @xmath207 @xmath208 to the point @xmath37 . the intervals @xmath209 $ ] and @xmath210 $ ] define `` ballistic intervals '' ;  the currents transmitted across these intervals contribute to the total current with weight @xmath211 . note that , in contrast to the situation considered above , where ballistic motion occurs _ before _ equilibration in the time interval @xmath191 ( with probability @xmath212 ) , here the ballistic motion occurs _ after _ the electrons have been equilibrated in the time interval @xmath213 ( with probability @xmath214 ) .    when the collision time @xmath8 is sufficiently small , such that @xmath215 is much smaller than the average length over which the density @xmath39 changes appreciably , one has , by expanding in expressions ( [ eq : prob004 ] ) @xmath39 to first order , @xmath216 taking into account that only electrons with @xmath217 ( @xmath218 ) contribute to the current arriving at the point @xmath37 from the left ( right ) , _ thermal averaging _ of @xmath219 now yields @xmath220 so that one obtains the local expression @xmath221 for the thermally averaged electron current . the charge current @xmath222 is then found , in view of expression ( [ eq : drude011 ] ) for the diffusion coefficient @xmath60 , to agree with relation ( [ eq : drude007 ] ) for the diffusion ( charge ) current @xmath52 of the drift - diffusion model . this result may thus be regarded as a proof of the einstein relation ( [ eq : drude011 ] ) . when @xmath223 , only electrons emitted at the left and right end , respectively , of the sample contribute to the currents @xmath203 . then , the ( thermally averaged ) net electron current @xmath200 essentially reduces to the ballistic current given by expression ( [ eq : ball008 ] ) . in the intermediate regime , i.e. , for nonzero , finite values of @xmath8 , @xmath200 combines elements of both drift - diffusion and ballistic transport .    in analogy to the time - dependent probability @xmath192 given by eq . ( [ eq : prob001 ] ) , one can introduce the probability @xmath224 for an electron to travel without collision over a finite distance @xmath37 , @xmath225 with a constant electron mean free path @xmath7 . the conditional probability for an electron to undergo a collision in the infinitesimal collision interval @xmath226 after a collision - free flight over the distance @xmath37 reads @xmath227 then one finds @xmath228 for the mean distance @xmath229 between successive electron collisions .    in the following , the mean free path @xmath7 will be assumed to take on any finite value , while , as before , the collisions are assumed to lead instantaneously to a complete equilibration of the electron momenta , and are to be interpreted as a simulation of the real situation where the distances between successive collisions are short ( of atomic dimensions ) , while the equilibration during a collision is ( usually ) far less than complete . the quantity @xmath7 , in parallel to the collision time @xmath8 , is simply to be regarded as a parameter which determines the probability ( [ eq : prob001a ] ) of collision - free ( complete or not ) traversal by an electron of the distance @xmath37 . it is noted here that when treating inhomogeneous systems , one should , strictly speaking , allow for a position dependence of the collision time , @xmath230 , and of the mean free path , @xmath231 . in that case , expressions ( [ eq : prob001 ] ) and ( [ eq : prob001a ] ) for the probabilities @xmath192 and @xmath224 , respectively , must be replaced with the more general forms @xmath232 and @xmath233 respectively . then , of course , the relation @xmath234 [ see eq . ( [ eq : drude002 ] ) ] ceases to be valid . the unification of the drift - diffusion and ballistic transport models within the prototype thermoballistic model@xcite is based on the introduction of configurations of ballistic transport intervals , which cover the length of the sample . the individual ballistic intervals in a configuration are linked by points of local thermodynamic equilibrium , and their lengths are stochastic variables occurring with probabilities determined by the probabilities for collision - free electron motion across the intervals , where the electron mean free path is allowed to have _ arbitrary _ magnitude . the electron current is , of course , conserved in each ballistic interval . in addition , it is assumed that _ the current is conserved also from one interval to the next_. averaging over the ballistic configurations then leads to a transparent and intuitively appealing expression for the current voltage - characteristic in terms of an effective transport length , in which the detailed shape of the potential energy profile is taken into account . moreover , a local chemical potential can be constructed in a heuristic way .      here , as well as in sec . [ sec : averball ] , we confine ourselves to ( one - dimensional ) classical transport in nondegenerate systems . the effects of tunneling and degeneracy will be considered in secs . [ sec : quantum ] and [ sec : degener1 ] , respectively .     exhibiting a single barrier with maximum at @xmath235 . the partitioning of the sample length @xmath28 into @xmath236 ballistic intervals @xmath237 $ ] ( @xmath238 ) is indicated , where @xmath239 is the maximum of @xmath48 in interval @xmath240 . for the definition of the mirror point @xmath241 associated with a point @xmath37 , see eq . ( [ eq : gendr009uvy ] ) below . , scaledwidth=45.0% ]    for given @xmath236 ( @xmath242 ) , the length @xmath28 of the sample is randomly partitioned into @xmath236 `` ballistic intervals '' labeled @xmath240 ( @xmath243 ) , in which the electrons move ballistically across the potential energy profile @xmath48 [ see fig . [ fig:1 ] ] . these intervals are linked by points of local thermodynamic equilibrium @xmath244 and @xmath245 ( @xmath246 ) which , in addition , include the two ( true ) equilibrium points at the interfaces with the left and right contacts at @xmath247 and @xmath248 , respectively . [ in this section , in order to avoid confusion with the labeling of the equilibrium points inside , we denote the end - point coordinates of the sample by @xmath249 and @xmath28 , instead of @xmath69 and @xmath70 , as used elsewhere . ] at @xmath244 and @xmath245 , electrons are thermally emitted towards the right and left , respectively , into the interval @xmath237 $ ] . any set @xmath250 of @xmath251 equilibrium points linking @xmath236 ballistic intervals characterizes a _ ballistic configuration_.    introducing an equilibrium chemical potential @xmath38 all across the sample , in terms of which the values @xmath252 and @xmath253 are defined , the net ballistic current @xmath254 transmitted across the interval @xmath240 in a partition with @xmath236 intervals is given , in analogy to expression ( [ eq : ball008 ] ) for the current in the ballistic transport model , by @xmath255 , \label{eq : gendr001}\ ] ] where @xmath256 denotes the absolute maximum of the potential energy profile @xmath48 in the interval @xmath240 . in terms of the ( classical ) thermally averaged transmission probability @xmath257 } \label{eq : gendr001x}\ ] ] defined in analogy to expression ( [ eq : ball005d ] ) , we can write @xmath258 in the form @xmath259 , \nonumber \\ \label{eq : gendr001y}\end{aligned}\ ] ] where @xmath260 the current @xmath258 is , of course , conserved across each interval @xmath240 .    in its general form , expression ( [ eq : gendr001y ] ) obviously does not lend itself for the calculation of transport properties in terms of external physical quantities . therefore , we make the simplifying assumption that the electron current is conserved also across the points of local thermodynamic equilibrium linking one ballistic interval to its adjacent intervals ( and hence is constant all along the length of the sample ) , so that in eq . ( [ eq : gendr001y ] ) @xmath261 for all @xmath240 and @xmath236 , where @xmath133 is the _ physical current_. we can then iterate eq .  ( [ eq : gendr001y ] ) with respect to @xmath240 ( for fixed @xmath236 ) , with the result @xmath262 here , @xmath263 denotes the overall maximum of the profile @xmath48 in the interval @xmath264 $ ] , and the function @xmath265 , defined as @xmath266 } \nonumber   \\ & \equiv & \sum_{i=1}^{n } e^{-\beta [ e_{c}^{m}(0,s ) - e_{c}^{m}(i ) ] } , \label{eq : gendr002a}\end{aligned}\ ] ] can be formally viewed as the partition function corresponding to the `` energy spectrum '' @xmath267 . it depends on the equilibrium points @xmath268 in the ballistic configuration @xmath269 via eq . ( [ eq : gendr001 m ] ) .      in order to arrive at results in terms of physical quantities , we must average eq . ( [ eq : gendr002 ] ) over the complete set of ballistic configurations @xmath270 @xmath271 . in doing so , we identify the average of the left - hand side of eq . ( [ eq : gendr002 ] ) with the `` bias term '' @xmath272 , where @xmath273 and @xmath274 are the values of the chemical potential at the contact side of the left and right contact - semiconductor interface , respectively , and obtain @xmath275 here , @xmath276 denotes the average of the partition functions @xmath265 over all ballistic configurations @xmath270 . from eq . ( [ eq : gendr004 ] ) , the current - voltage characteristic of the prototype thermoballistic model is now obtained in the general form @xmath277 } \frac{1}{\cal r }   ( 1 - e^{-\beta e v } ) , \label{eq : gendr004a}\ ] ] with the voltage bias @xmath29 given by @xmath278 aside from the barrier factor involving the absolute maximum , @xmath263 , of the potential energy profile , the current @xmath133 is controlled here by the parameter @xmath279 , which alone comprises the detailed dependence of the current - voltage characteristic on the shape of the profile and on the material parameters characterizing the sample . as it appears in eq . ( [ eq : gendr004a ] ) , @xmath279 can be viewed as a ( dimensionless ) _ reduced resistance_.    in the averaging process implied in eq . ( [ eq : gendr003 ] ) , the contribution of each ballistic interval @xmath240 in a partition with @xmath236 intervals is to be weighted with the conditional probability @xmath280 for an electron to make a collision in the infinitesimal interval @xmath281 after having traveled freely across the interval @xmath240 of length @xmath282 , into which it was emitted after a collision in the interval @xmath283 . the collision probability is connected with the _ two _ ends of each ballistic interval , and the collision interval @xmath281 is the _ equilibration interval _ for the ballistic interval @xmath240 , but it is simultaneously the _ emission interval _ for the ballistic interval @xmath284 . this `` sharing '' of collision intervals @xmath281 between the ballistic intervals @xmath240 and @xmath284 implies that effectively the density of points of equilibration is only _ one - half _ of that given originally . therefore , the conditional probability @xmath280 is , in conformance with eqs . ( [ eq : prob001a ] ) and ( [ eq : prob002a ] ) , expressed as @xmath285 with the _ effective mean free path _ @xmath286 given by @xmath287 here , the meaning of the mean free path @xmath7 is the original one introduced within the drude model . the conditional probability @xmath288 for an electron emitted at @xmath289 to undergo @xmath290 collisions at the equilibrium points @xmath245 @xmath291 , and finally be absorbed with unit probability at @xmath292 , is then found as @xmath293 \nonumber \\ & \ & \times e^{-(s - x^{n}_{n-1})/\ell } \label{eq : gendr007}\end{aligned}\ ] ] for @xmath294 , while @xmath295 in the product in eq . ( [ eq : gendr007 ] ) , the exponentials cancel out except for the factor @xmath296 , so that @xmath297 and @xmath298 [ the symbol @xmath299 is meant to imply @xmath300-fold integration over @xmath301 from @xmath249 to @xmath28 ] , i.e. , the total probability is unity . the reduced resistance @xmath279 defined by eq . ( [ eq : gendr003 ] ) can now be expressed as @xmath303 where the operators @xmath304 acting on the partition functions @xmath265 are given by @xmath305 and @xmath306 for @xmath294 . it can be shown that , owing to the separable form of @xmath307 [ see eqs . ( [ eq : gendr002a ] ) and ( [ eq : gendr002c ] ) ] , the multi - dimensional integrals in eq . ( [ eq : gendr009 ] ) can always be reduced to one - dimensional and two - dimensional integrals . the remaining infinite series can be summed up in terms of exponentials . for the classical partition function @xmath265 of eq . ( [ eq : gendr002a ] ) , the reduced resistance @xmath279 is thus obtained in the form @xmath308 \nonumber \\ & + & \hspace{-0.1 cm } \int_{0}^{s } \frac{dx'}{\ell } \int_{x'}^{s } \frac{dx''}{\ell } e^{-(x''-x')/\ell } \bar{r}(x',x '' ) , \label{eq : gendr009a}\end{aligned}\ ] ] where the ( dimensionless ) function @xmath309 is given by @xmath310 } \nonumber \\ & = & e^{- \beta [ e_{c}^{m}(0,s ) - e_{c}^{m}(x',x '' ) ] } . \label{eq : gendr009b}\end{aligned}\ ] ] here , the thermally averaged classical transmission probability @xmath311 is defined in analogy to eqs . ( [ eq : ball005d ] ) and ( [ eq : gendr001x ] ) , @xmath312 is the absolute maximum of the potential energy profile @xmath48 in the interval @xmath313 $ ] , and @xmath314 in general , we have @xmath315 and , in particular , @xmath316 .    for a flat potential energy profile , @xmath317 , when @xmath318 , expression ( [ eq : gendr009a ] ) can be immediately evaluated with the result @xmath319 so that in the general case , owing to eq . ( [ eq : gendr009uhd ] ) , @xmath320 i.e. , @xmath279 _ decreases _ when a position - dependent potential energy profile is introduced . in the current - voltage characteristic ( [ eq : gendr004a ] ) , this decrease of @xmath279 is overcompensated by the effect of the barrier factor @xmath321}$ ] , so that the current @xmath133 decreases as well .    by reducing expression ( [ eq : gendr009 ] ) for @xmath279 to the form ( [ eq : gendr009a ] ) , we have transcribed the prototype thermoballistic model , which , according to its original concept , relies on a discrete partitioning of the sample length into ballistic intervals , into a pure continuum model . expression ( [ eq : gendr009a ] ) is composed of three contributions . the first corresponds to ballistic transport all across the sample length @xmath28 , with associated probability @xmath322 . the second reflects the integrated effect of ballistic transport across intervals @xmath323 $ ] and @xmath324 $ ] , respectively , with probabilities @xmath325 and @xmath326 . the third , finally , corresponds to the integrated effect of ballistic transport across intervals @xmath327 $ ] , with probabilities @xmath328 . the central task in implementing the prototype thermoballistic model is to evaluate the reduced resistance @xmath279 from eq . ( [ eq : gendr009a ] ) [ or from the analogous equation derived from the quantal partition function ( [ eq : gendr002c ] ) ] for given potential energy profile @xmath48 . in general , the computation of @xmath48 is accomplished ( see , e.g. , ref .  ) by solving , for given distribution of the space - fixed charges in the sample , a nonlinear poisson equation .      in the ballistic transport model ( see sec . [ sec : balltunn ] ) , we have taken into account the effect of electron tunneling by introducing quantal probabilities to describe electron transmission across the whole sample length . in the prototype model , we can include tunneling effects by using quantal transmission probabilities in expression ( [ eq : gendr001y ] ) for the current @xmath258 in the individual ballistic intervals @xmath329 $ ] . replacing in eq . ( [ eq : gendr001y ] ) the probability @xmath330 with the thermally averaged quantal transmission probability @xmath331 defined in analogy to expression ( [ eq : ball010 ] ) , with @xmath75 replaced with @xmath332 , the partition function @xmath265 in eq . ( [ eq : gendr002 ] ) acquires the form @xmath333 } \nonumber \\ & \equiv & \sum_{i=1}^{n } \frac{\bar{t}(i)}{\bar{\cal t}(i ) } e^{-\beta [ e_{c}^{m}(0,s ) - e_{c}^{m}(i ) ] } , \label{eq : gendr002c}\end{aligned}\ ] ] where eq . ( [ eq : gendr001x ] ) for @xmath330 has been used ..    in generalization of expression ( [ eq : gendr009a ] ) , the reduced resistance @xmath334 including tunneling effects is obtained from the quantal partition function @xmath335 as @xmath336 \nonumber \\ & + & \int_{0}^{s } \frac{dx'}{\ell } \int_{x'}^{s } \frac{dx''}{\ell } e^{-(x''-x')/\ell } \bar{\cal r}(x',x '' ) , \label{eq : gendr009e}\end{aligned}\ ] ] where @xmath337 } \nonumber \\ & \equiv &   \frac{\bar{t}(x',x'')}{\bar{\cal t}(x',x '' ) } \bar{r}(x ' , x '' ) . \label{eq : ball003y}\end{aligned}\ ] ] here , the thermally averaged quantal transmission probability @xmath338 is defined in analogy to expression ( [ eq : ball010 ] ) , and @xmath339 is given by eq . ( [ eq : gendr009b ] ) . in view of eq . ( [ eq : ball003y ] ) , the inequality ( [ eq : gendr009y ] ) providing an upper limit on the values of the classical function @xmath279 holds also for the quantal function @xmath334 . we now adopt the wkb approximation , for which we have , using eq . ( [ eq : ball010xa ] ) , @xmath340 with @xmath341 inserting expression ( [ eq : ball003za ] ) in eq . ( [ eq : gendr009e ] ) , we find that @xmath334 separates in the form @xmath342 where the classical contribution @xmath343 is given by eq . ( [ eq : gendr009a ] ) , and the tunneling ( sub - barrier ) contribution @xmath344 by eq . ( [ eq : gendr009e ] ) , with @xmath345 replaced with @xmath346 . evidently , @xmath347 and @xmath348 .    according to its definition in analogy to eq . ( [ eq : ball010x ] ) , the thermally averaged sub - barrier transmission probability @xmath349 is nonzero only if the potential energy profile @xmath48 exhibits at least one local maximum in the range @xmath350 such that @xmath351 . in the evaluation of the sub - barrier function @xmath344 from eq . ( [ eq : gendr009e ] ) , ballistic intervals for which @xmath48 does not meet this requirement can , therefore , be excluded from the outset .      recalling the procedure developed in the nondegenerate case ( see secs . [ sec : ballconf ] and [ sec : averball ] ) , in which expression ( [ eq : gendr001 ] ) for the net ballistic current @xmath258 forms the starting point of the derivation of the current - voltage characteristic ( [ eq : gendr004a ] ) , we find that an analogous procedure can not be set up , in general , in the degenerate case . the reason is that for the argument leading to eq . ( [ eq : gendr004a ] ) , @xmath258 must factorize into a term depending on the potential energy profile @xmath48 , and terms depending on the chemical potential @xmath352 . in the degenerate case , the current analogous to @xmath258 is the current @xmath353 given by eqs . ( [ eq : ball007 ] ) and ( [ eq : degen011e ] ) , with @xmath69 and @xmath70 , respectively , replaced with @xmath354 and @xmath355 . this current does not factorize if the bias is nonzero . in the zero - bias case , however , the chemical potentials at the sample ends , @xmath356 and @xmath357 , differ only infinitesimally , and the increment of @xmath38 across a ballistic interval @xmath240 , @xmath358 is infinitesimally small as well . then , we find in analogy to eq . ( [ eq : degen001 ] ) @xmath359_{\mu = \mu(x_{i-1}^{n } ) } \nonumber \\ & = & v_{e } n_{c } \beta \delta_{i } \ln \bm { ( } 1 + e^{-\beta[e_{c}^{m}(i ) - \mu(0 ) ] } \bm { ) } , \label{eq : degen001y}\end{aligned}\ ] ] where we have replaced , in the second equation , @xmath360 with @xmath361 . expression ( [ eq : degen001y ] ) has the factorization property alluded to above , so that , following the procedure leading to eq . ( [ eq : gendr002 ] ) in the nondegenerate case , we here find @xmath362 =   \frac{j}{v_{e } n_{c } }   \frac{{\cal z}^{n}_{d } } { \ln \bm { ( } 1 + e^{- \beta [ e_{c}^{m}(0,s ) - \mu(0 ) ] }   \bm { ) } }   , \label{eq : degen003}\ ] ] where the function @xmath363 } \bm{)}}{\ln \bm { ( } 1 + e^{- \beta [ e_{c}^{m}(i ) -   \mu(0 ) ] } \bm { ) } } \nonumber \\ \label{eq : degen003a}\ ] ] generalizes the partition function @xmath265 given by eq . ( [ eq : gendr002a ] ) to the degenerate case . averaging eq . ( [ eq : degen003 ] ) over the ballistic configurations @xmath364 yields , in analogy to eq . ( [ eq : gendr004 ] ) , @xmath365 }   \bm { ) } }   , \label{eq : degen005}\ ] ] where @xmath366 is the reduced resistance for the degenerate case .      within the prototype model , we can construct , in a heuristic way , a unique chemical potential @xmath38 all along the sample length by modifying relation ( [ eq : gendr004 ] ) that connects the values of the chemical potential at the contact - semiconductor interfaces , @xmath273 and @xmath274 . we proceed as follows .    on the one hand , by replacing the fixed position @xmath28 with a variable position @xmath37 inside the sample , we turn @xmath274 into a chemical - potential function @xmath367 and , simultaneously , reduce the range of averaging over the ballistic configurations in eq . ( [ eq : gendr003 ] ) for the reduced resistance @xmath279 from @xmath368 $ ] to @xmath369 $ ] . we then obtain @xmath370 @xmath371 , where we have introduced the `` resistance function '' @xmath372 as @xmath373 } ,   \label{eq : prochem03}\ ] ] which , in turn , can be expressed as @xmath374 . \label{eq : prochem03gcx}\ ] ] on the other hand , by replacing in eq . ( [ eq : gendr004 ] ) the fixed position @xmath249 with the variable @xmath37 , so that @xmath273 turns into a chemical - potential function @xmath375 and the range of averaging is reduced to @xmath376 $ ] , we have @xmath377 @xmath371 , with the resistance function @xmath378 defined by @xmath379 } ,   \label{eq : prochem04}\ ] ] so that @xmath380 . \label{eq : prochem04ldp}\ ] ] from the definitions ( [ eq : gendr003 ] ) , ( [ eq : prochem03 ] ) , and ( [ eq : prochem04 ] ) , it follows that @xmath381 the resistance functions @xmath372 and @xmath378 are obtained in explicit form from expression ( [ eq : gendr009a ] ) [ or from expression ( [ eq : gendr009e ] ) , which includes tunneling effects ] by transcribing it so as to correspond to samples with end - point coordinates @xmath382 and @xmath383 , respectively .    from the potentials @xmath384 , we now construct a unique chemical potential @xmath352 by setting @xmath385   \label{eq : prochem05}\ ] ] for @xmath386 , while @xmath387 using eqs . ( [ eq : prochem01 ] ) and ( [ eq : prochem02 ] ) , along with eq . ( [ eq : gendr004 ] ) in eq . ( [ eq : prochem05 ] ) , we obtain @xmath388 where the function @xmath389 is defined as @xmath390 . \nonumber \\ \label{eq : prochem06gcl}\end{aligned}\ ] ] with the resistance functions @xmath391 and the reduced resistance @xmath279 calculated from eq . ( [ eq : gendr009a ] ) [ or eq .  ( [ eq : gendr009e ] ) ] , the equilibrium chemical potential in the prototype model can now be explicitly evaluated in terms of the potential energy profile @xmath48 and the momentum relaxation length @xmath7 . a noteworthy feature of the chemical potential @xmath38 is the occurrence of discontinuities at @xmath392 and @xmath393 . using eqs . ( [ eq : gendr004 ] ) , ( [ eq : gendr009a ] ) , ( [ eq : gendr009b ] ) , and ( [ eq : prochem04zwe ] ) , we find from eq . ( [ eq : prochem06 ] ) @xmath394 } - 1 = - \frac{j}{2 v_{e } n_{c } } e^{\beta   [ e_{c}(0 ) - \mu_{0 } ] }   \label{eq : prochem06amf}\ ] ] for @xmath392 , and @xmath395 } - 1 =   \frac{j}{2 v_{e } n_{c } } e^{\beta   [ e_{c}(s ) - \mu_{s } ] }    \label{eq : prochem06pky}\ ] ] for @xmath393 .    in the fully thermoballistic approach ( see sec .  [ sec : equichem ] below ) , we will derive an expression for the average chemical potential which closely resembles expression ( [ eq : prochem06 ] ) and which exhibits discontinuities analogous to those expressed by eqs . ( [ eq : prochem06amf ] ) and ( [ eq : prochem06pky ] ) , but relies on different assumptions . the present subsection is devoted to the application of the prototype thermoballistic model to specific examples . with a view to inhomogeneous semiconductors and heterostructures , in which barriers in the potential energy profile arise , e.g. , at heterojunctions , grain boundaries , and metal - semiconductor contacts ( `` schottky barriers '' ) , we deal with the evaluation of the reduced resistance @xmath279 for profiles @xmath48 exhibiting an arbitrary number of barriers and interjacent valleys . as a prelude to the general case , we treat the cases of a single barrier and of two barriers enclosing a valley separately . another example considered here is that of field - driven transport in homogeneous semoconductors , for which we evaluate the position dependence of the chemical potential .      here , we consider the case of a single barrier in the profile @xmath48 , with its maximum located at some position @xmath396 $ ] ( see fig . [ fig:1 ] ) . then , we have @xmath397 . if , in particular , @xmath398 or @xmath399 , the profile is monotonic .    in the calculation of the corresponding reduced resistance @xmath279 from expression ( [ eq : gendr009a ] ) , one must distinguish three cases . if the ballistic interval @xmath313 $ ] contains @xmath400 , we have @xmath401 ;  if it lies to the left or right of @xmath400 , we have @xmath402 or @xmath403 , respectively . we then find @xmath404 where @xmath97 is the effective sample length , @xmath405 } . \label{eq : gendr012}\ ] ] this expression is formally equal to the effective sample length ( [ eq : diff008 ] ) introduced in the context of the drift - diffusion model , but it differs in physical origin . while expression ( [ eq : gendr012 ] ) results from averaging over ballistic configurations involving intervals of arbitrary , finite length , expression ( [ eq : diff008 ] ) arises from the assumption of arbitrarily short ballistic intervals , on which the drift - diffusion model is implicitly based .    in expression ( [ eq : gendr011 ] ) for @xmath279 , the unit term corresponds to ballistic transmission all across the sample , governed by the barrier maximum at @xmath400 , while the effective sample length @xmath97 represents the effect of the potential energy profile in an integral way . the ( classical ) current - voltage characteristic ( [ eq : gendr004a ] ) for the case of a single potential energy barrier now reads @xmath406 } \frac{\ell}{\ell + \tilde{s } } ( 1 - e^{-\beta e v } ) . \label{eq : gendr004c}\ ] ] if the effective mean free path is much longer than the effective sample length , @xmath407 , the characteristic ( [ eq : gendr004c ] ) reduces to that of the ballistic transport model , eq . ( [ eq : ball009 ] ) . in the opposite case , @xmath408 , using the relation @xmath409 for the electron mobility @xmath61 [ see eqs . ( [ eq : drude004hg ] ) and ( [ eq : ball002 ] ) ] , the characteristic of the drift - diffusion model , eq . ( [ eq : diff006 ] ) , is retrieved . expression ( [ eq : gendr004c ] ) exemplifies the unification of the ballistic and drift - diffusion transport mechanisms in the prototype thermoballistic model . we note that a heuristic attempt to unify drift - diffusion and ballistic transport has been made , for the special case of transport across a schottky barrier , by crowell and sze@xcite ( see also ref .  ) , who assumed ballistic transport to prevail in the vicinity of the barrier maximum , and drift - diffusion transport elsewhere . they obtained an expression for the current - voltage characteristic equivalent to ( [ eq : gendr004c ] ) , @xmath410 } \frac{v_{e } v_{di}}{v_{e } + v_{di } } ( 1 - e^{-\beta e v } ) , \label{eq : gendr004f}\ ] ] where @xmath411 is an effective diffusion velocity , with the electron mobility @xmath61 given by eq . ( [ eq : ballim20a ] ) . a picture similar to that of ref .   has been developed and applied by evans and nelson@xcite in a study of transport across a single grain boundary barrier . other studies elucidating the transition region between drift - diffusion and ballistic transport were presented by de jong@xcite and prins _ et al._@xcite    considering now , for the single barrier , the effect of electron tunneling , we obtain , following sec . [ sec : quantum ] , @xmath412 for the sub - barrier part @xmath344 of the reduced resistance @xmath334 , while the classical part @xmath343 is given by eq . ( [ eq : gendr011 ] ) . since @xmath413 , we have from eq . ( [ eq : gendr009s ] ) @xmath414 since @xmath415 , the transmission probabilities @xmath311 and @xmath416 , and consequently the function @xmath346 , depend on @xmath417 and @xmath418 only via the function @xmath419 [ see eqs . ( [ eq : ball005d ] ) and ( [ eq : ball010x ] ) ] . then , @xmath346 is a function of one coordinate only , determined by the potential energy profile @xmath48 , @xmath420 as @xmath418 increases from @xmath400 to @xmath28 , for fixed @xmath421 $ ] , @xmath422 changes to @xmath423 when @xmath424 or , equivalently , @xmath425 ;  here , the `` mirror point '' @xmath241 is defined as that position to the right ( left ) of a maximum or minimum of the potential energy profile where it has the same value as at the point @xmath37 to the left ( right ) , i.e. , @xmath426 ( see figs .  [ fig:1 ] and [ fig:2 ] ) . the double integral in expression ( [ eq : gendr009s ] ) then reduces to two single integrals which , together with the single integrals preceding it , combine to just one single integral , so that @xmath344 attains the simple form @xmath427 this expression bears a close formal similarity to the shape term appearing in the classical treatment of the double - barrier case [ see eq .  ( [ eq : vall002a ] ) below ] . next , we consider a potential energy profile of the type shown in fig .  [ fig:2 ] , exhibiting two barriers with maxima at @xmath428 and @xmath429 , respectively ( without loss of generality , the maximum at @xmath430 is assumed to be the higher one ) , and a valley with minimum at @xmath431 in between .    in the evaluation of the reduced resistance @xmath279 , a number of cases are to be distinguished when calculating the function @xmath432 in dependence on the location of the ballistic interval @xmath313 $ ] with respect to the different ranges delimited by the points @xmath392 , @xmath430 , @xmath433 , @xmath434 , @xmath435 , and @xmath28 . from expression ( [ eq : gendr009a ] ) , we obtain @xmath436 where the effective sample length @xmath97 is given by eq . ( [ eq : gendr012 ] ) with @xmath437 . the `` shape term '' @xmath438 has the form @xmath439 . \label{eq : vall002a}\end{aligned}\ ] ] in writing @xmath438 in this compact form , we have used the identity @xmath440 where @xmath441 is the width of the valley ( see fig .  [ fig:2 ] ) . the position of the minimum of the valley , @xmath434 , enters @xmath279 implicitly via the shape of @xmath48 in the interval @xmath442 $ ] .     exhibiting two barriers with maxima at @xmath428 and @xmath443 , respectively , which enclose a valley with minimum at @xmath431 and width @xmath444 . for the definition of the mirror point @xmath241 associated with a point @xmath37 , see eq . ( [ eq : gendr009uvy ] ) . , scaledwidth=45.0% ]    comparing in eq . ( [ eq : vall002a ] ) the magnitude of the integral involving the term @xmath445 in the brackets to that of the integral involving @xmath446 , it is seen that for @xmath447 , where @xmath448 are the widths of the two barriers centered about @xmath449 , the second integral can be neglected . hence , @xmath450 } , \label{eq : vall004}\ ] ] i.e. , the two barriers contribute independently to the reduced resistance ( first and third term ) .    when @xmath451 ( ballistic limit ) , we have @xmath452 } < s , \label{eq : vall005}\ ] ] so that both the terms @xmath453 and @xmath454 in eq . ( [ eq : vall002 ] ) can be neglected . then , @xmath279 reduces to the unit term , which reflects ballistic transmission across the higher barrier maximum at @xmath430 ;  this maximum `` eclipses '' the lower maximum at @xmath435 .      in the foregoing cases of a single barrier and of a valley in - between two barriers , we have obtained explicit expressions for the reduced resistance @xmath279 in terms of the potential energy profile @xmath48 and the effective momentum relaxation length @xmath286 . if the profile contains an arbitrary combination of barriers and valleys , @xmath279 must be evaluated , in general , for each case anew , starting from the basic formula ( [ eq : gendr009a ] ) . however , an explicit expression for @xmath279 can still be obtained in the special case where the _ height _ of the barriers decreases or increases _ monotonically _ along the sample length .     exhibiting @xmath455 barriers with maxima at @xmath456 ( @xmath457 ) , and heights decreasing monotonically with increasing @xmath37 . the valleys enclosed by two adjacent barriers have minima located at @xmath458 ( @xmath459 ) and widths @xmath460 . , scaledwidth=45.0% ]    assuming , without loss of generality , the barrier height to decrease monotonically when @xmath37 increases from 0 to @xmath28 ( which includes the case of equal height of all barriers ) , we consider @xmath455 barriers , with maxima at @xmath456 @xmath461 , enclosing @xmath462 valleys , with minima at @xmath458 ( @xmath459 ) and widths @xmath460 ( see fig .  [ fig:3 ] ) . then , using eq . ( [ eq : gendr009a ] ) and proceeding as in the case of a single valley , we obtain @xmath279 in the general form given by eq . ( [ eq : vall002 ] ) , with @xmath97 again given by eq . ( [ eq : gendr012 ] ) with @xmath437 , where @xmath430 is now the position of the maximum of the heighest ( left - most ) barrier of the @xmath455 barriers considered . the shape term @xmath438 entering expression ( [ eq : vall002 ] ) , @xmath463 ,   \nonumber \\ \label{eq : vall006a}\end{aligned}\ ] ] generalizes expression ( [ eq : vall002a ] ) to the case of @xmath462 valleys ;  it appears as a sum over separate contributions from the different valleys . the contribution of valley @xmath464 , with the maximum of the adjoining lower barrier located at @xmath465 , consists of an integral which extends over the width of the valley from @xmath466 to @xmath465 .    when the barrier heights in the potential energy profile do not behave monotonically , inspection of formula ( [ eq : gendr009a ] ) shows that the reduced resistance @xmath279 can always be expressed as a sum of a term formally identical to expression ( [ eq : vall002 ] ) [ with @xmath467 in @xmath97 and @xmath438 replaced with @xmath468 , where @xmath469 is the position of the maximum of the heighest barrier ] plus additional terms arising from the combined effect of barriers and valleys on the electron transport in the ballistic intervals . introducing now the effective transport length as @xmath470 we have from eqs . ( [ eq : gendr009x ] ) , ( [ eq : gendr009y ] ) , and ( [ eq : vall002 ] ) @xmath471 from eqs . ( [ eq : gendr004a ] ) and ( [ eq : vall007a ] ) , we then obtain the current - voltage characteristic for a potential energy profile of the type depicted in fig . [ fig:3 ] in the form @xmath472 } \frac{\ell}{l } ( 1- e^{-\beta ev } ) . \label{eq : vall008}\ ] ] this expression is the principal result of the prototype thermoballistic model . it has been derived here in the framework of classical transport in nondegenerate systems , where its interpretation is most transparent . the properties of the characteristic ( [ eq : vall008 ] ) are determined by the barrier factor @xmath473}$ ] and by the ratio @xmath474 . in the effective transport length @xmath475 , the effective mean free path @xmath286 represents the ballistic contribution to the current , which is associated with the maximum @xmath467 of the highest barrier in the potential energy profile . the remaining terms give a quantitative measure of the influence of that part of the electron motion which is not purely ballistic . their contribution amounts to at most the sample length @xmath28 . the effective sample length @xmath97 given by eq . ( [ eq : gendr012 ] ) with @xmath437 represents a contribution that characterizes the potential energy profile @xmath476 in an integral way ; it does not manifestly depend on @xmath286 , only implicitly so via the profile ( an indirect relationship between profile and mean free path arises from their common dependence on the donor density ) . the shape term @xmath438 given by eq . ( [ eq : vall006a ] ) , on the other hand , depends on the detailed structure of the profile as well as explicitly on the mean free path , and thus represents the interplay of ballistic and drift - diffusion transport . this term is a distinctive feature of expression ( [ eq : vall008 ] ) , and therefore of the prototype thermoballistic model . tunneling effects in the current - voltage characteristic ( [ eq : vall008 ] ) can be taken into account by generalizing the effective transport length @xmath475 of eq . ( [ eq : vall007a ] ) so as to include the sub - barrier contribution @xmath344 to the reduced resistance . using eq . ( [ eq : ball003p ] ) , we have @xmath477 here , as exemplified by expression ( [ eq : gendr009st ] ) for the single - barrier case , @xmath344 depends explicitly on the effective mean free path @xmath286 . from expression ( [ eq : gendr009r ] ) , the inclusion of tunneling is seen to _ lower _ the _ ballistic _ contribution to the effective transport length [ see eq . ( [ eq : vall007a ] ) ] . this behavior is in line with the role of @xmath334 as a resistance , which ought to decrease when the barriers become `` transparent '' to ballistic electron transport . the ( classical ) reduced resistance in the degenerate case , @xmath478 [ see eqs . ( [ eq : degen003a ] ) and ( [ eq : degen004 ] ) ] , is expressed , in analogy to eqs . ( [ eq : vall007a ] ) and ( [ eq : vall007 ] ) for the nondegenerate case , in terms of a ( classical ) effective transport length @xmath479 given by @xmath480 where , in generalization of eq . ( [ eq : gendr012 ] ) , @xmath481 } \bm{)}}{\ln \bm { ( } 1 + e^{- \beta [ e_{c}(x ) - \mu_{0 } ] } \bm { ) } } \label{eq : degen007}\ ] ] is the effective sample length for the degenerate case , and @xmath482 } \bm { ) } } { \ln \bm { ( } 1 + e^{- \beta [ e_{c}(x_{v } ) - \mu_{0 } ] } \bm { ) } } - \frac { \ln \bm { ( } 1 + e^{- \beta [ e_{c}(x_{0 } ) - \mu_{0 } ] } \bm { ) } } { \ln \bm { ( } 1 + e^ { -\beta [ e_{c}(x ) - \mu_{0}]}\bm { ) } } \right\ } \nonumber \\ \label{eq : degen008}\end{aligned}\ ] ] generalizes the shape term ( [ eq : vall006a ] ) . the inclusion of tunneling effects is , in the degenerate case , a highly intricate task and will not be considered here .    setting @xmath483 in eq . ( [ eq : degen005 ] ) , we now have for the zero - bias conductance per unit area in the prototype thermoballistic model @xmath484 in generalization of expression ( [ eq : degen009 ] ) for the conductance in the ballistic transport model .    previously , we have applied@xcite the prototype model in calculations of transport properties of poly-  and microcrystalline semiconducting materials , in particular , of materials relevant to photovoltaics . the occurence of grain boundaries in this kind of materials gives rise to a multi - barrier structure of the band edge profile . adopting the trapping model@xcite to describe the grain boundaries , we have solved@xcite the corresponding nonlinear poisson equation to obtain zero - bias potential energy profiles for chains of grains . then , taking into account tunneling corrections and using a phenomenological relation@xcite to express the momentum relaxation length @xmath7 in terms of the donor density , we have calculated zero - bias conductivities and electron mobilities as a function of @xmath7 , and of the number and lengths of the grains . it turns out that neither the ballistic ( thermionic ) model nor the drift - diffusion model can provide an adequate description of electron transport in poly-  and microcrystalline materials . the application of the prototype model in the analysis of experimental data@xcite has led to promising results . the chemical potential @xmath352 given by eq . ( [ eq : prochem06 ] ) can be expressed in closed form for the case of electron transport in a homogeneous semiconductor ( no space charges ) , driven by a constant external electric field of magnitude @xmath485 .    with the field assumed to be directed antiparallel to the @xmath37-axis , the corresponding potential energy profile reads @xmath486 where @xmath487 . using this in the properly transcribed expression ( [ eq : gendr009a ] ) [ or , alternatively , in eq . ( [ eq : gendr011 ] ) , considering the profile ( [ eq : potfi01 ] ) a particular case of a barrier ] , we obtain the resistance functions @xmath391 in the form @xmath488 and @xmath489 \equiv { \cal r}_{1}(s - x ) . \label{eq : potfi03}\ ] ] from eq . ( [ eq : prochem04zwe ] ) , we then have @xmath490 and from eq . ( [ eq : prochem06gcl ] ) , @xmath491 inserting expressions ( [ eq : potfi04a ] ) and ( [ eq : potfi04 ] ) in eq . ( [ eq : prochem06 ] ) gives @xmath38 in closed form .    for zero bias , when @xmath492 , we have @xmath493 and @xmath494 i.e. , @xmath495 varies linearly with position . the prototype thermoballistic model developed in the preceding section has been based on the random partitioning of the length of a semiconducting sample into ballistic transport intervals linked by points of local thermodynamic equilibrium , which make up a ballistic configuration . electrons thermally emitted at either end - point of a ballistic interval , and subsequently transmitted across the potential energy profile in the sample , form the net ballistic electron current in the interval . this current is conserved across an individual ballistic interval . a distinctive assumption of the prototype model has been that the current is conserved also across the points of local thermodynamic equilibrium linking the ballistic intervals , and equals the physical current . by averaging over all ballistic configurations , the current - voltage characteristic can then be expressed , without requiring knowledge of the equilibrium chemical potential inside the sample , essentially in terms of a reduced resistance that comprises the effect of the sample parameters . the position dependence of the chemical potential has been constructed in a heuristic way only .    while adhering to the idea of introducing ballistic configurations and averaging thereover , the thermoballistic concept proper refines the prototype model in that it abandons the assumption of current conservation across the points of local thermodynamic equilibrium . [ a simple example contradicting this assumption is provided by the case of the ballistic currents in a homogeneous semiconductor at zero bias considered in sec . [ sec : homsem ] below . ] position - dependent total and spin - polarized thermoballistic currents as well as the associated densities are defined in terms of an average chemical potential and a spin accumulation function related to the splitting of the spin - resolved chemical potentials . by imposing appropriately chosen physical conditions on these dynamical functions , procedures for their explicit determination are devised . we have developed the thermoballistic approach to semiclassical carrier transport@xcite in a series of papers . in ref .  , the concept of a thermoballistic current was introduced and implemented without regard to spin degrees of freedom . the extension of this concept to spin - polarized electron transport across a spin - degenerate potential energy profile was presented in ref .  , in which spin injection out of ferromagnetic contacts into a nonmagnetic semiconducting sample was treated in detail . in ref . , we have generalized the thermoballistic concept to the case of arbitrary spin splitting of the profile , thereby covering , in particular , spin - polarized transport in diluted magnetic semiconductors in their paramagnetic phase . in the present section , we formulate the thermoballistic concept within the frame set by ref .  . classical transport in nondegenerate systems is considered throughout . effects of electron tunneling and degeneracy can be included , in principle , by resorting to the corresponding developments in secs . [ sec : ballmech ] and [ sec : gendrude ] .    for comprehensive surveys of the fundamentals of spin physics in semiconductors and their application in the field of spintronics , we refer the reader to refs .   and . enclosed between two metal contacts . the spin splitting @xmath496 of the electrostatic potential energy profile @xmath48 gives rise to the spin - dependent potential energy profiles @xmath497 . the coordinates @xmath417 and @xmath418 denote points of local thermodynamic equilibrium which delimit a ballistic transport interval @xmath327 $ ] . electrons thermally emitted at @xmath498 toward the right ( left ) move ballistically across the profiles @xmath497 to reach the end - point @xmath499 of the ballistic interval , where they are absorbed ( equilibrated ) . two net electron currents are indicated :  the ( conserved ) net total ballistic current @xmath500 determined by the values @xmath501 and @xmath502 of the average chemical potential @xmath503 and the values @xmath504 and @xmath505 of the spin accumulation function @xmath506 according to eq . ( [ eq : ball004jvx ] ) [ with the mean spin function @xmath507 expressed in terms of @xmath503 via eq . ( [ eq : eqden03zk ] ) ] , and the net relaxing ballistic spin - polarized current @xmath508 determined by @xmath504 and @xmath505 according to eq . ( [ eq : netbal02 ] ) . the quantities @xmath76 and @xmath509 are the values of the equilibrium chemical potential and the spin accumulation function , respectively , at the contact side of the contact - semiconductor interfaces [ see eqs . ( [ eq : ballim05 ] ) and ( [ eq : ballim05s ] ) ] . , scaledwidth=45.0% ]    in sec . [ sec : gendrude ] , we have introduced ballistic transport intervals with end - point coordinates @xmath510 characterized by discrete labels @xmath511 . when the ballistic configurations made up of these intervals are averaged over , the description in terms of discrete coordinates turns into one in terms of the _ continuous _ coordinates @xmath512 in expressions ( [ eq : gendr009a ] ) and ( [ eq : gendr009e ] ) for the reduced resistance @xmath279 . transferring this feature into the formulation of the thermoballistic concept , we work here with ballistic intervals @xmath313 $ ] with continuous end - point coordinates @xmath417 and @xmath418 representing points of local thermodynamic equilibrium ( see also sec . [ sec : probappr ] ) . for notational convenience , we will henceforth label the end - points of a semiconducting sample by @xmath69 and @xmath70 , respectively , so that we have @xmath513 and the sample length @xmath28 is given by @xmath85 .    in fig . [ fig:4 ] , a paramagnetic semiconducting sample enclosed between two metal contacts is depicted in a schematic diagram . various physical quantities appearing in the thermoballistic description are indicated . we write the spin - dependent potential energy profiles @xmath514 corresponding to spin - up @xmath515 and spin - down @xmath516 conduction band electrons , respectively , in the form @xmath517 here , the spin - independent part @xmath518 is assumed to comprise the conduction band edge potential and the external electrostatic potential , and @xmath519 is the spin splitting of the conduction band . having in mind electron transport in diluted magnetic semiconductors in their paramagnetic phase , we identify this splitting with the ( giant ) zeeman splitting due to an external magnetic field @xcite [ we restrict ourselves to considering a single landau level whose energy is assumed to be included in @xmath518 ] . in presenting the general formalism , we assume both @xmath518 and @xmath519 , and hence @xmath521 , to be continuous functions of @xmath417 in the interval @xmath83 $ ] . the case of abrupt changes in one or the other of these functions , which occur at the interfaces in heterostructures , may be described , in a simplified picture , in terms of discontinuous functions ( see sec . [ sec : dmsnmshe ] below ) .    in terms of the boltzmann factors @xmath522 , the ( static ) local spin polarization of the conduction band electrons , @xmath523 , is given by @xmath524 where @xmath525 so that we have @xmath526 the function @xmath527 defined by @xmath528^{2 } \right\}^{1/2 } \nonumber \\   & = & \frac{1}{\cosh \bm { ( } \beta \delta(x')/2 \bm { ) } }    \label{eq : spfra17c}\end{aligned}\ ] ] will be frequently used below . for the spin - resolved equilibrium electron densities @xmath529 , we have , in generalization of expression ( [ eq : drude014 ] ) , @xmath530 } . \label{eq : eqden02}\ ] ] here , @xmath531 are the spin - resolved chemical potentials associated with the local thermodynamic equilibrium at @xmath417 , and @xmath532 , with @xmath180 given by eq . ( [ eq : diff001a ] ) , is the effective density of states of either spin at the conduction band edge ( for simplicity , the effective electron mass @xmath4 entering @xmath180 is assumed here to be independent of position and of the external magnetic field ) . the spin - resolved chemical potentials @xmath533 and @xmath534 are independent dynamical functions whose position dependence is to be determined within the thermoballistic approach and from which , subsequently , all transport properties are to be derived . however , in implementing the thermoballistic approach , we will work not with @xmath533 and @xmath534 , but with suitably defined combinations of these functions :  ( i ) an `` average chemical potential '' , and ( ii ) a `` spin accumulation function '' related to the splitting of the potentials @xmath533 and @xmath534 . we define the `` spin functions '' @xmath535 as @xmath536 and the `` mean spin function '' @xmath507 as @xmath537 where @xmath538 we can express @xmath539 in the form @xmath540 where @xmath541 \label{eq : spfra15xxx}\ ] ] is the mean value of @xmath533 and @xmath534 [ mean chemical potential ] , and @xmath542 their splitting . [ in the following , the @xmath543 notation introduced in eqs . ( [ eq : eqden03oh ] ) and([eq : spfra15x ] ) , respectively , will be frequently used _ mutatis mutandis _ to denote spin - summed and spin - polarized quantities . ] further , writing @xmath544 we introduce the _ `` average chemical potential '' _ @xmath545 . the _ `` spin accumulation function '' _ @xmath546 is defined as @xmath547 [ this function agrees with the `` spin transport function '' @xmath504 of ref .  , but differs by a factor of two from the identically named function introduced in ref .  . ] the name here chosen for @xmath546 derives from the fact that in the limit @xmath548 , this function becomes proportional to the splitting @xmath549 , which is the dynamical quantity used in the drift - diffusion approach to spin - polarized transport , and which is commonly@xcite referred to as the  spin accumulation  there .    using eq . ( [ eq : eqden03nn ] ) , we can rewrite @xmath546 in the form @xmath550 introducing the `` reduced '' spin accumulation function @xmath551 as @xmath552 we then obtain , using eqs . ( [ eq : eqden03zk])([eq : spfra01fv ] ) , the relations @xmath553 expressing the difference between average and mean chemical potential in terms of the splitting @xmath549 and of @xmath551 , respectively . the spin functions @xmath535 are proportional to the spin - resolved electron densities @xmath554 [ see eqs . ( [ eq : eqden02 ] ) ] , so that their use results in a formulation in terms of _ linear _ equations , instead of the nonlinear description ensuing from using the spin - resolved chemical potentials @xmath531 themselves . [ this aspect has been emphasized previously @xcite within a study , based on the standard drift - diffusion approach , of electric - field effects on spin - polarized transport in nondegenerate semiconductors . ]    from eqs . ( [ eq : eqden02 ] ) , the total ( i.e. , spin - summed ) equilibrium density , @xmath555 , and the spin - polarized equilibrium density , @xmath556 , are now obtained in terms of the functions @xmath557 as @xmath558 , \label{eq : ball004jux}\ ] ] where eqs . ( [ eq : eqden60 ] ) and ( [ eq : eqden61 ] ) have been used .      in this subsection , the spin - resolved electron currents transmitted across a ballistic transport interval as well as the associated densities ( called `` ballistic currents '' and `` ballistic densities '' , for short ) are constructed by closely following the development in the ballistic ( see sec .  [ sec : ballmech ] ) and the prototype thermoballistic ( see sec .  [ sec : ballconf ] ) transport models , in which spin degrees of freedom have been disregarded . introducing spin relaxation during the ballistic electron motion , we obtain the ballistic spin - polarized currents and densities , whose dynamics are determined from a balance equation . the left end - point , @xmath417 , as well as the right end - point , @xmath418 , of the ballistic interval @xmath313 $ ] are points of local thermodynamic equilibrium but are , in general , _ not _ points of spin equilibrium . we first assume that the electrons thermally emitted at @xmath417 towards the right are _ not _ affected by spin relaxation during their motion across the interval @xmath313 $ ] . the spin - resolved densities @xmath554 [ see eq . ( [ eq : eqden02 ] ) ] then give rise to conserved ballistic spin - resolved currents ( i.e. , currents independent of the position @xmath559 $ ] ) in that interval , which have the form @xmath560 ( see sec . [ sec : ballnonc ] ) , where the thermally averaged ( classical ) transmission probabilities @xmath561 are given [ see eqs . ( [ eq : ball005 ] ) and([eq : ball005a ] ) ] by @xmath562 } , \label{eq : ball004iii}\ ] ] with @xmath563 the overall maxima of the potential energy profiles @xmath564 in @xmath313 $ ] . in terms of the spin functions @xmath535 given by eqs . ( [ eq : eqden03 mm ] ) , we can rewrite the currents @xmath565 in the form @xmath566 we then have for the total ( i.e. , spin - summed ) ballistic current , @xmath567 , and the ballistic spin - polarized current , @xmath568 , @xmath569 , \nonumber \\ \label{eq : ball004ju}\end{aligned}\ ] ] where @xmath570 and @xmath571 which are nonlocal extensions of expressions ( [ eq : eqden60 ] ) and ( [ eq : eqden06 ] ) for the functions @xmath572 and the local polarization @xmath523 , respectively . we now consider the effect of spin relaxation on the ballistic currents inside the interval @xmath313 $ ] . we introduce functions @xmath573 that depend on the position @xmath574 $ ] and are required to satisfy the initial conditions @xmath575 using these functions , we generalize the definitions of the currents @xmath576 given by eqs . ( [ eq : ball004ju ] ) to @xmath577 . \label{eq : spirel08}\end{aligned}\ ] ] now , also in the presence of spin relaxation , the _ total _ ballistic current must be conserved , @xmath578 where , from eq . ( [ eq : ball004ju ] ) , @xmath579 . \label{eq : ball004juhu}\end{aligned}\ ] ] equating this to expression ( [ eq : spirel08 ] ) for @xmath580 , we obtain the relation @xmath581 . \label{eq : balden09}\end{aligned}\ ] ] therefore , the spin - polarized current can be written as @xmath582 where @xmath583^{2 } , \label{eq : spirel15}\ ] ] and @xmath584^{2 } \right\}^{1/2 } \label{eq : emtra14}\ ] ] is the nonlocal extension of expression ( [ eq : spfra17c ] ) for the function @xmath527 . the ballistic spin - polarized current of eq . ( [ eq : spirel11 ] ) has the form @xmath585 where the first ( @xmath37-independent , nonrelaxing ) term is the `` persistent '' ballistic spin - polarized current , @xmath586 while the second ( @xmath37-dependent ) term is the `` relaxing '' ballistic spin - polarized current , @xmath587 the latter describes the spin dynamics in the current @xmath588 via the `` spin relaxation function '' @xmath589 . the @xmath37-dependence of @xmath589 will be determined explicitly in sec . [ sec : balance ] , while the procedure for calculating the spin accumulation function @xmath590 will be described in sec . [ sec : accufunc ] . the persistent ballistic spin - polarized current @xmath591 depends , via the total ballistic current @xmath567 , on the spin accumulation function @xmath546 . when @xmath546 is calculated within the thermoballistic approach , spin relaxation in all ballistic intervals ( including the interval @xmath313 $ ] under consideration ) is taken into account . therefore , while being not _ directly _ affected by spin relaxation _ inside _ the interval @xmath313 $ ] , the persistent current @xmath591 depends , via the dependence of the factor @xmath592 on @xmath546 [ see eqs . ( [ eq : ball004juhu ] ) and ( [ eq : spirel12 ] ) ] , in an _ indirect _ way on spin relaxation . the ballistic densities @xmath593 associated with the ballistic currents @xmath594 of eqs . ( [ eq : spirel08 ] ) are given by @xmath595 \nonumber \\ \label{eq : ball004m}\end{aligned}\ ] ] [ see eq . ( [ eq : ball004ka ] ) ] . here , we have taken into account that the ballistic densities correspond to _ one half _ of the thermal currents emitted symmetrically at a point of local thermodynamic equilibrium [ see , e.g. , eq .  ( [ eq : ball001h ] ) ] . the functions @xmath596 and @xmath597 are defined as @xmath598 where @xmath599 } \nonumber \\ & \ & \times { \rm erfc } \bm { ( } \ { \beta [ e_{\uparrow , \downarrow}^{m}(x ' , x '' ) - e_{\uparrow , \downarrow}(x ) ] \}^{1/2 } \bm { ) }   \nonumber \\ \label{eq : ball004n}\end{aligned}\ ] ] [ see eq . ( [ eq : ball004kaa ] ) ] , and @xmath600 in generalization of expression ( [ eq : emtra09 ] ) for the function @xmath601 .    now , inserting expression ( [ eq : balden09 ] ) for @xmath602 in eqs . ( [ eq : ball004 m ] ) , we find that the total ballistic density , @xmath603 , and the ballistic spin - polarized density , @xmath604 , have the form @xmath605 here , the first ( @xmath37-dependent , but nonrelaxing ) term is the persistent part , @xmath606    \nonumber \\ \label{eq : balden13}\end{aligned}\ ] ] and the second term is the relaxing part , @xmath607 where @xmath608 in contrast to the @xmath37-independent currents @xmath567 and @xmath591 , the persistent parts of the ballistic densities , @xmath609 , depend on position via the @xmath37-dependence of the potential energy profiles @xmath564 . for zero spin splitting of the conduction band , @xmath610 , when @xmath611 , the persistent part of the spin - polarized density as well as the relaxing part of the total density vanish , @xmath612 .      in a time - dependent formulation , spin relaxation in a system described by spin - resolved electron densities @xmath613 and currents @xmath614 is governed by the local coupled spin balance equations@xcite @xmath615 and @xmath616 where @xmath617 @xmath618 is the rate for spin - flip scattering from spin - up ( spin - down ) to spin - down ( spin - up ) states . in the stationary case , when @xmath619 , this leads to the balance equation @xmath620 connecting the relaxing part of the spin - polarized current , @xmath621 , to the relaxing parts of the spin - polarized , @xmath622 , and of the total density , @xmath623 . here , @xmath624 , defined as @xmath625 is the spin relaxation time . we apply eq . ( [ eq : spirel01ac ] ) to spin relaxation during ballistic transport . in doing this , we will disregard the term involving @xmath626 for two reasons :  ( i ) as it is preceded by the _ difference _ of the two relaxation rates @xmath617 and @xmath627 , which are estimated to be of comparable magnitude , the term may generally be considered small in comparison with the term involving @xmath628 . ( ii ) since both @xmath628 and @xmath626 are proportional to the spin relaxation function @xmath629 [ see eq . ( [ eq : balden1 ] ) ] , we can account for @xmath626 by combining its prefactors with those of @xmath628 in an _ effective _ spin relaxation time ( still denoted by @xmath624 ) depending , in general , on the potential energy profiles . [ note that we must not assume @xmath630 from the outset by adopting the arguments leading to eq . ( 2.7 ) of ref .  . the density @xmath631 appearing in that equation is the deviation of the total density from its spin equilibrium value . by contrast , the relaxing total density @xmath626 , when used in the thermoballistic description , gives the deviation of the total density from the persistent total density . in the latter , a spin non - equilibrium part enters via the spin accumulation function @xmath632 [ see eq . ( [ eq : balden13 ] ) ] . ]    in the thermoballistic approach , it is assumed that the thermally emitted electrons spend only an infinitesimally short time span at the emission point , and it is only during their motion across the ballistic interval that they can undergo spin relaxation . spin relaxation in ballistic transport is commonly described@xcite in terms of a ( ballistic ) spin relaxation length @xmath633 given by @xmath634 [ as in the case of the effective electron mass @xmath4 , we assume @xmath624 , and hence @xmath633 , to be independent of position and of the external magnetic field ;  we consider @xmath633 here an _ effective _ quantity , in line with the interpretation of @xmath624 . ] in our description of spin - polarized electron transport , spin relaxation is thus completely separated from momentum relaxation at the points of local thermodynamic equilibrium and , in this respect , is similar to the dyakonov - perel relaxation mechanism.@xcite    in terms of the relaxing ballistic spin - polarized current @xmath635 and the corresponding density @xmath636 , the balance equation governing spin relaxation during ballistic transport now reads @xmath637 where the spin relaxation length @xmath633 is given by eq . ( [ eq : spirel01 ] ) .    inserting in eq . ( [ eq : baleq01 ] ) the expressions for @xmath635 and @xmath636 from eqs . ( [ eq : spirel14 ] ) and ( [ eq : balden1 ] ) , respectively , we obtain a first - order differential equation for the spin relaxation function @xmath629 , @xmath638 where @xmath639 the solution of eq . ( [ eq : baleq02 ] ) obeying the initial condition ( [ eq : spirel05 ] ) is @xmath640 where @xmath641 with @xmath642 . now , inserting expression ( [ eq : baleq05 ] ) in eqs . ( [ eq : spirel14 ] ) and ( [ eq : balden1 ] ) , respectively , we obtain the relaxing ballistic spin - polarized current and density explicitly in terms of the spin accumulation function @xmath546 , @xmath643 and @xmath644 the @xmath37-dependence of the relaxing spin - polarized current and density in the ballistic interval @xmath327 $ ] is hence governed by the factor @xmath645 . it departs from a purely exponential behavior unless the potential energy profiles @xmath497 are constant over the interval , in which case @xmath646 . so far , we have only considered thermal emission at the _ left _ end - point , @xmath417 , of the ballistic interval @xmath327 $ ] , obtaining a variety of ballistic currents and densities summarily denoted here by @xmath647 and @xmath648 , respectively . the analogous ballistic currents and densities @xmath649 and @xmath650 corresponding to emission at the _ right _ end - point @xmath418 can be expressed in terms of those emitted at @xmath417 as @xmath651 and @xmath652 [ note that , owing to the symmetry of the functions @xmath653 , the functions @xmath654 , @xmath655 , @xmath656 , and @xmath657 entering the expressions for @xmath647 and @xmath648 are symmetric under the exchange of @xmath417 and @xmath418 . ] we then have @xmath658 and @xmath659 for the _ net _ ballistic currents and _ joint _ ballistic densities summarily denoted by @xmath660 and @xmath661 , respectively .    for the ( conserved ) net _ total _ ballistic current @xmath500 inside the ballistic interval , we now find , using eq . ( [ eq : ball004juhu ] ) , @xmath662 \nonumber \\ & \ & + { \mbox{$\frac{1}{2}$ } } p^{m}(x ' , x'')[a(x ' ) -a(x '' ) ]   \ } . \label{eq : ball004jvx}\end{aligned}\ ] ] here , the function @xmath663 has been replaced with @xmath664 [ see eq . ( [ eq : eqden03 ] ) ] , and the subscript attached to the spin accumulation function @xmath546 has been omitted , i.e. , we have set @xmath665 the current @xmath500 is seen to be dynamically determined , in general , both by the average chemical potential @xmath545 [ via the mean spin function@xmath539 ] and the spin accumulation function @xmath504 . the same then holds for the net _ persistent _ ballistic _ spin - polarized _ current @xmath667 , for which we have , using eq . ( [ eq : spirel12 ] ) , @xmath668 for zero spin splitting , when @xmath669 and hence @xmath670 , the dependence of @xmath500 on @xmath506 drops out and , further , @xmath671 .    for the net _ relaxing _ ballistic _ spin - polarized _ current , we have , using eq . ( [ eq : baleq07 ] ) , @xmath672 , \label{eq : netbal02}\end{aligned}\ ] ] which is dynamically determined by the spin accumulation function @xmath506 alone .    for the joint _ total _ ballistic density , @xmath673 , and the joint ballistic _ spin - polarized _ density , @xmath674 , respectively , we have from eq . ( [ eq : spirel11kx ] ) @xmath675 with _ persistent _ parts @xmath676 and _ relaxing _ parts @xmath677 given by @xmath678 \nonumber \\ & \ & + { \mbox{$\frac{1}{2}$ } } p^{m}(x ' , x'')[a(x ' ) + a(x '' ) ] \ } \nonumber \\ \label{eq : ball004juy}\end{aligned}\ ] ] [ see eq . ( [ eq : balden13 ] ) ] and @xmath679 \nonumber \\ \label{eq : netbal02xy}\end{aligned}\ ] ] [ see eqs . ( [ eq : balden1 ] ) and ( [ eq : baleq08 ] ) ] , in analogy to eqs . ( [ eq : ball004jvx ] ) , ( [ eq : netbal06 ] ) , and ( [ eq : netbal02 ] ) for the net currents .    from expressions ( [ eq : netbal02 ] ) and ( [ eq : netbal02xy ] ) , we now derive , using eqs . ( [ eq : baleq03 ] ) and ( [ eq : baleq06 ] ) , the balance equation @xmath680 connecting the net relaxing ballistic spin - polarized current @xmath508 and the associated joint density @xmath681 . this equation can be obtained , of course , simply by adding the balance equation eq . ( [ eq : baleq01 ] ) and the corresponding equation for emission at @xmath418 .      in a significant advance over its prototype presented in sec . [ sec : gendrift ] , where overall current conservation was introduced via the condition ( [ eq : gendr001qal ] ) , the thermoballistic concept proper rests on the introduction of a `` reference coordinate '' @xmath37 that characterizes an arbitrary point inside the semiconducting sample extending from @xmath69 to @xmath70 , as shown in fig . [ fig:4 ] . singling out this coordinate , we consider the net ballistic currents and joint ballistic densities within the ensemble of all ballistic intervals @xmath327 $ ] enclosing @xmath37 . these currents and densities form the building blocks for establishing , at @xmath37 , the corresponding thermoballistic quantities . the point labeled by the reference coordinate @xmath37 is _ not _ a point of local thermodynamic equilibrium ( we may call it a `` ballistic point '' ) . however , the `` equilibrium points '' @xmath512 may come infinitesimally close to @xmath37 .      the thermoballistic currents and densities at the point @xmath37 are constructed by performing weighted summations of the corresponding net ballistic currents and joint ballistic densities over all ballistic intervals @xmath327 $ ] subjected to the condition @xmath682 just as in the prototype thermoballistic model ( see sec .  [ sec : averball ] ) , we adopt here the probabilistic picture outlined in sec .  [ sec : probappr ] :   the contributions from the interval @xmath327 $ ] are weighted with the probability @xmath683 ( one - dimensional transport is assumed here ;  see the remark at the beginning of sec .  [ sec : diffmech ] ) that the electrons traverse the interval without collisions , multiplied by the probability @xmath684 ( @xmath685 ) that they are absorbed or emitted in an interval @xmath686 ( @xmath687 ) around the end - point @xmath417 ( @xmath418 ) . at the ends of the semiconducting sample at @xmath75 , absorption and emission occur with unit probability . like the effective electron mass @xmath4 and the spin relaxation length @xmath633 , the momentum relaxation length @xmath7 is assumed here to be independent of position and of the external magnetic field .    representing the net ballistic currents and joint ballistic densities of eqs . ( [ eq : ball004jvx])([eq : netbal02xy ] ) summarily by a function @xmath688 , and the corresponding thermoballistic currents and densities by @xmath689 , we write @xmath690 in the form @xmath691 where @xmath692 , and the infinitesimal @xmath181 has been introduced in accordance with condition ( [ eq : thebal00 ] ) . further , we set @xmath693 in parallel to eq . ( [ eq : gendr008x ] ) for the total probability in the prototype model , we have @xmath694 for @xmath695 .     according to eq . ( [ eq : thebal01 ] ) . , scaledwidth=45.0% ]    specifically , the total thermoballistic current @xmath696 is given by expression ( [ eq : thebal01 ] ) with @xmath500 of eq . ( [ eq : ball004jvx ] ) substituted for @xmath697 . the persistent thermoballistic spin - polarized current @xmath698 follows by identifying @xmath688 with @xmath667 of eq . ( [ eq : netbal06 ] ) . the relaxing thermoballistic spin - polarized current @xmath699 is obtained by replacing @xmath688 with @xmath700 of eq . ( [ eq : netbal02 ] ) . further , the thermoballistic densities @xmath701 , @xmath702 , and @xmath703 corresponding to the currents @xmath696 , @xmath698 , and @xmath699 follow by substituting the respective joint ballistic densities [ see eqs . ( [ eq : netbal03ax])([eq : netbal02xy ] ) ] for @xmath688 in eq . ( [ eq : thebal01 ] ) .    in expression ( [ eq : thebal01 ] ) when specialized to the total thermoballistic current @xmath696 , the first term on the right - hand side represents the net electron current passing through the point @xmath37 while being ballistically transmitted between @xmath69 and @xmath70 [ which occurs with probability @xmath704 , the second term refers to ballistic electron motion between any point @xmath417 @xmath705 and @xmath70 [ probability @xmath706 , the third term to ballistic electron motion between @xmath69 and any point @xmath418 @xmath707 [ probability @xmath708 , etc . the four types of ballistic current appearing in expression ( [ eq : thebal01 ] ) are illustrated by the double arrows labeled i to iv in fig . [ fig:5 ] . the derivatives with respect to @xmath37 of the various thermoballistic quantities can be written in the general form @xmath709 where the first term on the right - hand side comprises the contributions arising from differentiating the functions @xmath710 in the integrands of expression ( [ eq : thebal01 ] ) , and the second , those from differentiating the limits of integration:@xmath711 ) - \mathbb{f}_{2}(x ; [ f ] ) \ } . \label{eq : thebal06jb}\end{aligned}\ ] ] here , the functionals @xmath712)$ ] and @xmath713)$ ] , defined by @xmath714 ) & = & e^{- ( x - x_{1})/l } f(x_{1 } , x^{+ } ; x ) \nonumber \\ & \ &   + \int_{x_{1}}^{x^{- } } \frac{dx'}{l } e^{- ( x - x')/l } f(x ' , x^{+ } ; x ) \nonumber \\ \label{eq : thec00ohx}\end{aligned}\ ] ] and @xmath715 ) & = & e^{- ( x_{2 } - x)/l } f(x^{- } , x_{2 } ; x )   \nonumber \\ & \ &   + \int_{x^{+}}^{x_{2 } } \frac{dx''}{l } e^{- ( x '' - x)/l } f(x^{- } , x '' ; x )   , \nonumber \\ \label{eq : thec00gma}\end{aligned}\ ] ] respectively , represent the contributions of the function @xmath710 arising from the ranges to the left and right of the point @xmath37 .    for the total thermoballistic current @xmath696 constructed from the @xmath37-independent ballistic current @xmath500 , the first term on the right - hand side of eq . ( [ eq : thebal01dz ] ) vanishes , and we have @xmath716 with @xmath717 given by expression ( [ eq : thebal06jb ] ) for @xmath718 . we note that @xmath719 is not , in general , equal to zero , so that the total thermoballistic current @xmath696 is not conserved . the quantity @xmath720 is the increment of @xmath696 across the infinitesimal interval @xmath226 at position @xmath37 of the sample . the four types of ballistic current appearing in expression ( [ eq : thebal06jb ] ) , which contribute to this increment , are depicted by the double arrows labeled i to iv in fig . [ fig:6 ] , with the arrows i and ii representing the term @xmath712)$ ] , and iii and iv the term @xmath713)$ ] .     of the total thermoballistic current @xmath696 according to eq . ( [ eq : thebal06jb ] ) . , scaledwidth=45.0% ]    if we set @xmath721 in expressions ( [ eq : thebal01 ] ) and ( [ eq : thebal06jb ] ) , we find from eq . ( [ eq : thebal01dz ] ) for the relaxing thermoballistic spin - polarized current @xmath722 in the first term on the right - hand side of this equation , we have introduced the relaxing thermoballistic spin - polarized density @xmath703 by using the balance equation ( [ eq : baleq01zs ] ) to replace the derivatives @xmath723 which we encounter when differentiating the integrals in expression ( [ eq : thebal01 ] ) .    the thermoballistic currents and densities , eq .  ( [ eq : thebal01 ] ) , are expressed , via the corresponding ballistic currents and densities , eqs . ( [ eq : ball004jvx])([eq : netbal02xy ] ) , in terms of two dynamical functions , _ viz . _ , the mean spin function @xmath507 and the spin accumulation function @xmath506 , or , equivalently , the average chemical potential @xmath503 and the splitting of the spin - resolved chemical potentials , @xmath724 [ see eqs .  ( [ eq : eqden03zk ] ) , ( [ eq : spfra01fv ] ) , and ( [ eq : ball004jhm ] ) ] . to implement the thermoballistic concept , we must establish algorithms for calculating @xmath507 and @xmath506 in terms of the intrinsic parameters of the semiconducting system , like momentum and spin relaxation lengths , as well as of the external parameters , like applied voltage and spin polarization in the external leads .    before establishing these algorithms within the thermoballistic approach , i.e. , for arbitrary momentum relaxation length @xmath7 , we directly evaluate the thermoballistic currents and densities in the drift - diffusion regime ( small @xmath7 ) and for the ballistic case ( @xmath725 ) . the results will turn out to be equal to the respective standard physical expressions summarized in sec . [ sec : dibatran ] , which demonstrates that the thermoballistic description indeed bridges the gap between the drift - diffusion and ballistic descriptions of carrier transport .      in the _ drift - diffusion regime _ , when @xmath726 and @xmath727 , nonzero contributions to the thermoballistic currents and densities defined by eq . ( [ eq : thebal01 ] ) arise only from very short ballistic intervals @xmath313 $ ] enclosing the point @xmath37 , so that @xmath728 and @xmath729 are infinitesimals and only the double integral over @xmath417 and @xmath418 contributes . then , from eq . ( [ eq : ball004juy ] ) , the persistent part of the total thermoballistic density , @xmath730 , is seen to reduce to the total equilibrium density @xmath731 [ see eq . ( [ eq : ball004jux ] ) ] , @xmath732 with @xmath731 expressed in terms of the functions @xmath507 and @xmath506 , @xmath733 . \label{eq : ball004jum}\ ] ] further , since @xmath734 in the drift - diffusion regime , the relaxing part of the joint total ballistic density , @xmath735 [ see eq . ( [ eq : netbal02xy ] ) ] , vanishes . consequently , we have @xmath736 for the relaxing part of the total thermoballistic density , @xmath737 , and hence @xmath738 for the total thermoballistic density @xmath701 in the drift - diffusion limit . the persistent part of the thermoballistic spin - polarized density , @xmath702 , is obtained from eq . ( [ eq : ball004juy ] ) as @xmath739 , \label{eq : ball004juz}\ ] ] and the relaxing part of the thermoballistic spin - polarized density , @xmath703 , from eq . ( [ eq : netbal02xy ] ) as @xmath740 a(x ) \nonumber \\ & = &   \frac{n_{c}}{4 } b_{+}(x ) q^{2}(x ) a(x ) , \label{eq : ball004jua}\end{aligned}\ ] ] with @xmath741 given by eq .  ( [ eq : spfra17c ] ) . then , re - expressing the spin - polarized equilibrium density @xmath742 [ see eq . ( [ eq : ball004jux ] ) ] in the form @xmath743 , \label{eq : ball004jvz}\ ] ] we have @xmath744 for the thermoballistic spin - polarized density @xmath745 in the drift - diffusion limit . the net total ballistic current @xmath746 is found from eq . ( [ eq : ball004jvx ] ) , by expanding to first order in @xmath747 and @xmath748 , in the form @xmath749 where @xmath750 now , evaluating the double integral over @xmath417 and @xmath418 in eq . ( [ eq : thebal01 ] ) for @xmath751 and taking the limit @xmath752 at fixed @xmath753 , we are left with a factor @xmath754 , so that the drift - diffusion limit of the total thermoballistic current @xmath696 is given by @xmath755 here , we have used relation ( [ eq : ballim20a ] ) to introduce the electron mobility @xmath61 , and we have identified the constant , total thermoballistic current in the drift - diffusion regime with the ( conserved ) total physical current @xmath133 . then , using eqs . ( [ eq : ball004jum ] ) and ( [ eq : ball004jvz ] ) , we can express @xmath133 in terms of the equilibrium densities @xmath756 [ see eqs . ( [ eq : ball004jup ] ) and ( [ eq : ball004jut ] ) ] in the form @xmath757 . \label{eq : ballim20cc}\end{aligned}\ ] ] for zero spin splitting , @xmath758 , this expression becomes equivalent to expression ( [ eq : drude008 ] ) for the total current in the standard drift - diffusion model . however , by contrast with the latter model , we have obtained eq . ( [ eq : ballim20cc ] ) without invoking the einstein relation ( [ eq : drude011 ] ) . this feature can be traced back to the probabilistic description underlying the thermoballistic approach ( see sec .  [ sec : probappr ] ) , which allows the diffusion current to be directly expressed in terms of the collision time ( and hence of the momentum relaxation length @xmath7 ) [ see eq . ( [ eq : prob006 ] ) ] .    to obtain an explicit expression for the mean spin function @xmath507 , and hence for the average chemical potential @xmath503 , we observe that in the drift - diffusion regime the total thermoballistic density @xmath701 is equal to the total equilibrium density @xmath731 [ see eq . ( [ eq : ball004jup ] ) ] , which , in turn , is related to @xmath507 via eq . ( [ eq : ball004jum ] ) . using eq . ( [ eq : ballim30aha ] ) to solve eq . ( [ eq : ballim20 ] ) for @xmath759 , integrating over the interval @xmath71 $ ] , and using eqs .  ( [ eq : eqden60])([eq : spfra17c ] ) to simplify the integrals , we can express @xmath503 in the form @xmath760 \nonumber \\ & - & \frac{\beta}{4 } \int_{x_{1}}^{x } dx ' \frac{d \delta ( x')}{dx ' } q^{2}(x ' ) a(x ' ) , \nonumber \\ \label{eq : ballim20kf}\end{aligned}\ ] ] which generalizes expression ( [ eq : diff003 ] ) . integration over the interval @xmath73 $ ] leads to another expression for @xmath503 , which is different in form , but numerically equal to expression ( [ eq : ballim20kf ] ) . the drift - diffusion form of the spin accumulation function @xmath506 in expression ( [ eq : ballim20kf ] ) is determined by a differential equation [ see eq . ( [ eq : ballim26xx ] ) below ] . we now set @xmath761 in eq . ( [ eq : ballim20kf ] ) and identify the boundary values of @xmath503 at the interface positions @xmath75 with the values , @xmath76 , of the equilibrium chemical potential at the contact side of the contact - semiconductor interfaces , @xmath762 [ see eq . ( [ eq : diff005 ] ) ] , so that from eq . ( [ eq : eqden03zk ] ) @xmath763 similarly , we identify the boundary values of @xmath506 with external values @xmath509 , @xmath764 we then obtain the drift - diffusion form of the current - voltage characteristic , which , for simplicity , is written down here for the case of constant spin splitting , when @xmath765 , @xmath766 , and @xmath767 , @xmath768 } \nonumber \\ & \ & \times [ 1 - e^{- \beta e v } + { \mbox{$\frac{1}{2}$ } } e^{- \beta \mu_{1 } } p ( a_{1 } - a_{2 } ) ]    . \label{eq : diff006hsa}\end{aligned}\ ] ] here , we have used eqs . ( [ eq : diff004 ] ) and ( [ eq : diff008 ] ) , respectively , to introduce the voltage bias @xmath29 and the effective sample length @xmath97 .    for zero spin splitting , when @xmath769 and @xmath770 , expressions ( [ eq : ballim20kf ] ) and ( [ eq : diff006hsa ] ) become equivalent to the expressions ( [ eq : diff003 ] ) and ( [ eq : diff006 ] ) , respectively , in the standard drift - diffusion model .    the drift - diffusion limit of the persistent thermoballistic spin - polarized current @xmath698 immediately follows from eqs . ( [ eq : netbal06 ] ) and ( [ eq : ballim20 ] ) as @xmath771 so that @xmath772 for zero spin splitting . proceeding in analogy to the derivation of @xmath696 , we find the drift - diffusion limit of the relaxing thermoballistic spin - polarized current @xmath699 from eqs . ( [ eq : netbal02 ] ) and ( [ eq : thebal01 ] ) in the form @xmath773 using eq . ( [ eq : ball004jua ] ) , we can express @xmath699 in terms of the analogous thermoballistic density @xmath703 , @xmath774 , \label{eq : ballim2010}\ ] ] where @xmath775 so that @xmath776 for zero spin splitting . inserting expressions ( [ eq : ballim21 ] ) and ( [ eq : ball004jua ] ) for @xmath699 and @xmath703 , respectively , in the general balance equation ( [ eq : ballim24new ] ) , we obtain a homogeneous second - order differential equation for the spin accumulation function @xmath506 , @xmath777 where @xmath778 is the spin diffusion length . equation ( [ eq : ballim26xx ] ) can be converted into an analogous equation for the density @xmath703 , @xmath779 - \frac{1}{l^{2}_{s } } \check{\mathfrak{n}}_{-}(x ) = 0 , \nonumber \\ \label{eq : ballim25}\end{aligned}\ ] ] where eq . ( [ eq : ball004jua ] ) has been used . for zero spin splitting , this equation generalizes , by including arbitrarily shaped potential energy profiles , the drift - diffusion equation commonly used to describe electric - field effects in spin - polarized transport in semiconductors [ see eq .  ( 2.8 ) of ref .  ;  in that equation , the spin diffusion length appears as @xmath780 , with @xmath60 an effective diffusion coefficient and @xmath624 the spin relaxation time ] . note that , in contrast to eq . ( [ eq : ballim26xx ] ) for @xmath506 , eq . ( [ eq : ballim25 ] ) for @xmath781 contains terms proportional to the first and second derivatives of @xmath48 [ see eq . ( [ eq : ballim24xy ] ) ] .      in the strictly _ ballistic limit _ , when @xmath725 , there are no points of local thermodynamic equilibrium in the interval @xmath83 $ ] , and the average chemical potential @xmath782 is not defined inside this interval . expression ( [ eq : thebal01 ] ) reduces to @xmath783 i.e. , the thermoballistic currents and densities are given by the corresponding expressions for the net ballistic currents and joint ballistic densities ( see sec .  [ sec : netball ] ) , evaluated at @xmath784 and @xmath785 . in expressions ( [ eq : ball004jvx ] ) , ( [ eq : netbal02 ] ) , ( [ eq : ball004juy ] ) , and ( [ eq : netbal02xy ] ) , the boundary values of the mean spin function , @xmath786 , and those of the spin accumulation function , @xmath787 , are to be identified with the corresponding values in the contacts [ see eqs . ( [ eq : ballim05ups ] ) and ( [ eq : ballim05s ] ) , respectively ] .    from eq . ( [ eq : ball004jvx ] ) , we then obtain the total thermoballistic current @xmath696 in the form @xmath788 , \nonumber \\ \label{eq : ballim02}\end{aligned}\ ] ] where @xmath789 with @xmath790 defined by eq . ( [ eq : ballim05ups ] ) , and @xmath791 for zero spin splitting , expression ( [ eq : ballim02 ] ) becomes equivalent to expression ( [ eq : ball008 ] ) for the total current in the ballistic transport model . the ballistic limit of the persistent thermoballistic spin - polarized current @xmath698 is obtained from @xmath792 by replacing in expression ( [ eq : ballim02 ] ) the quantity @xmath793 with @xmath794 [ see eqs . ( [ eq : emtra09 ] ) and ( [ eq : netbal06 ] ) ] .    for the total thermoballistic density @xmath701 , we have , defining @xmath795 and using eqs . ( [ eq : netbal03ax])([eq : netbal02xy ] ) , @xmath796 \nonumber \\ & \ & + { \mbox{$\frac{1}{2}$ } } { \cal d}^{m}_{+}(x_{1},x_{2};x ) { \cal a}^{+}_{12}(x_{1 } , x_{2 } ; x)\ } . \label{eq : ballim03}\end{aligned}\ ] ] for zero spin splitting , this becomes equivalent to expression ( [ eq : ball007za ] ) . further , using eq . ( [ eq : netbal02 ] ) and ( [ eq : netbal02xy ] ) , we express the relaxing thermoballistic spin - polarized current @xmath699 and the corresponding density @xmath703 as @xmath797 and @xmath798 respectively . in view of eq . ( [ eq : baleq01zs ] ) , the current @xmath699 and density @xmath703 in the ballistic limit are trivially connected by the general balance equation ( [ eq : ballim24new ] ) . the stochastic equilibration of the electrons that occurs during their motion across the sample is associated with the dissipation of energy , i.e. , the net transfer of energy out of the ensemble of conduction band electrons into a reservoir ( `` heat bath '' ) of electrons in thermodynamic equilibrium .    to describe this transfer within the thermoballistic concept , we introduce the ( conserved ) ballistic _ energy _ currents @xmath799 generated by thermal electron emission at the end - points @xmath417 and @xmath418 , respectively , of the ballistic interval @xmath313 $ ] . neglecting spin degrees of freedom , we have for the current @xmath800 , by an obvious modification of expression ( [ eq : ball004 ] ) for the electron current @xmath801 , writing @xmath802 , @xmath803 } \nonumber \\ & \ & \times \theta \bm { ( } \epsilon -   e_{b}^{l}(x ' , x '' ) \bm { ) } ,   \label{eq : thec01}\end{aligned}\ ] ] where , in analogy to eq . ( [ eq : diff007 ] ) , @xmath804 is the maximum barrier height of the potential energy profile @xmath48 relative to its value at @xmath417 . evaluating the integral in expression ( [ eq : thec01 ] ) , we obtain , in extension of eq . ( [ eq : ball004f ] ) , @xmath805   e^{- \beta [ e_{c}^{m}(x ' , x '' ) - \tilde{\mu}(x ' ) ] }   . \label{eq : thec03}\end{aligned}\ ] ] for the ballistic energy current @xmath806 , we have @xmath807 in line with eq . ( [ eq : ball004rfd ] ) for the corresponding ballistic electron current , and for the net ballistic energy current in the interval @xmath313 $ ] , @xmath808 in line with eq . ( [ eq : ball004zgx ] ) for the net ballistic electron current . now , we again introduce a `` reference coordinate '' @xmath37 inside the sample , whose meaning , however , differs from that of the coordinate @xmath37 in the definition ( [ eq : thebal01 ] ) of the thermoballistic currents and densities . while in that definition , @xmath37 is a coordinate _ inside ballistic intervals _ , we here consider @xmath37 a coordinate characterizing the position of a _ point of local thermodynamic equilibrium_. more precisely , we consider a collection of such points , with density @xmath809 , in an interval @xmath226 centered about @xmath37 , in which ballistic energy currents are absorbed from , and emitted towards , either side . the equilibrium point @xmath37 , where incoming electron currents are completely equilibrated and the outgoing currents are solely determined by the parameters of the reservoir ( i.e. , by the chemical potential ) , dynamically separates the two sample partitions to the left and right of @xmath37 . we denote by @xmath810 the ( kinetic ) energy transferred per unit volume and unit time _ into _ the reservoir by absorption of electrons at the point @xmath37 , and correspondingly by @xmath811 the change in energy of the reservoir due to thermal electron emission _ out of _ it . the _ net _ energy @xmath812 , i.e. , the energy dissipated locally at the point @xmath37 , is then given by @xmath813 the energies @xmath810 and @xmath811 are each composed of two parts ( see also fig . [ fig:6 ] ) . one part of @xmath810 is given by the weighted sum of the energy currents @xmath814 over all ballistic intervals @xmath313 $ ] lying to the left of @xmath37 and having their right end at @xmath815 , where the currents are absorbed . the other part is expressed analogously in terms of the currents @xmath816 in ballistic intervals to the right of the absorption point @xmath37 . for the energy @xmath817 dissipated in an interval @xmath226 centered around @xmath37 , we then have @xmath818 ) + \mathbb{f}_{2}(x ; [ \mathbb{e}^{r } ] ) \ } . \label{eq : thec00kcg}\ ] ] here , the functionals @xmath819)$ ] and @xmath820)$ ] are given , for an arbitrary ballistic current @xmath821 , by eqs . ( [ eq : thec00ohx ] ) and ( [ eq : thec00gma ] ) , respectively . the energy current @xmath822 is obtained from eq . ( [ eq : thec00kcg ] ) by interchanging the role of @xmath814 and @xmath823 , @xmath824 ) + \mathbb{f}_{2}(x ; [ \mathbb{e}^{l } ] ) \ } , \label{eq : thec00tuz}\ ] ] which comprises all ballistic energy currents emitted at the point @xmath37 towards either side . for the dissipated energy @xmath812 , we then have @xmath825 ) + \mathbb{f}_{2}(x ; [ \mathbb{e } ] ) \ } , \label{eq : thec00phy}\ ] ] where the net ballistic energy current @xmath826 is given by eq . ( [ eq : thec00gjb ] ) . we note that in obtaining expression ( [ eq : thec00phy ] ) for @xmath812 we have _ not _ drawn on the thermoballistic energy current @xmath827 that results from identifying in eq . ( [ eq : thebal01 ] ) the function @xmath710 with net ballistic energy current @xmath826 . naively , one might expect that the dissipated energy @xmath812 can be represented by the derivative @xmath828 given by eq . ( [ eq : thebal06jb ] ) . however , comparing expressions ( [ eq : thec00phy ] ) and ( [ eq : thebal06jb ] ) , one observes that the contributions of the ballistic energy currents from the two sides of @xmath37 add up in the former , and are subtracted from one another in the latter . the result ( [ eq : thec00phy ] ) can be obtained from the thermoballistic energy current @xmath827 in the form @xmath829 at variance with expression ( [ eq : thebal06jb ] ) . we do not write down here the general expression for @xmath812 in terms of the average chemical potential @xmath782 obtained by using expressions ( [ eq : thec03])([eq : thec00gjb ] ) in eq . ( [ eq : thec00phy ] ) , and confine ourselves to considering the ballistic limit and the drift - diffusion regime . owing to the overall factor @xmath830 in the right - hand side of eq . ( [ eq : thec00phy ] ) , @xmath812 vanishes in the _ ballistic limit _ @xmath725 , which reflects the complete absence of equilibration in this limit . in the _ drift - diffusion regime _ , on the other hand , when @xmath831 , only the integral terms contribute . we evaluate these terms for the special case of a homogeneous sample subjected to a constant external electric field of magnitude @xmath485 directed antiparallel to the @xmath37-axis , so that the potential energy profile has the form @xmath832 where @xmath833 for this profile , the average chemical potential @xmath503 is found from eqs . ( [ eq : diff003 ] ) and ( [ eq : diff006 ] ) , using the relation @xmath834 , to run parallel to the profile , @xmath835 + e_{c}(x ) . \label{eq : the03b}\ ] ] therefore , the total equilibrium electron density is constant , @xmath836 . the net ballistic energy current @xmath826 can now be expressed as @xmath837 } \left [ 1 + \epsilon ( x '' - x ' ) \right ]   \ }   \nonumber \\ & \approx & \frac{v_{e}}{2 \beta } n(x_{1 } )   \epsilon^{2 } ( x''- x')^{2 } , \label{eq : the03c}\end{aligned}\ ] ] where the second , approximate equation holds in the zero - bias limit @xmath838 . inserting the approximate representation of @xmath826 in the integral terms of expression ( [ eq : thec00phy ] ) , we find for the locally dissipated energy @xmath812 in the drift - diffusion regime @xmath839 where we have introduced the conductivity @xmath55 via eqs . ( [ eq : drude006 ] ) and ( [ eq : ballim20a ] ) . for the total energy @xmath840 dissipated per unit time ( `` heat production '' ) in a sample with cross - sectional area @xmath841 and resistance @xmath842 , we then recover the ohmic expression @xmath843 . we conclude this section with a synopsis of the concept underlying the thermoballistic description of charge carrier transport in semiconductors , in which we briefly comment on its basic ingredients and elucidate the physical content of its formal structure . this will be followed by an assessment of its merits as well as its weaknesses . the basic ingredients of the ( semiclassical ) thermoballistic concept are the ballistic carrier currents and densities which are constructed within the following framework . ( i ) thermal emission of carriers occurs at points of local thermodynamic equilibrium randomly distributed over the sample . ( ii ) the equilibrium points link `` ballistic transport intervals '' across which the emitted carriers move ballistically under the influence of potential energy profiles arising from internal and external electrostatic potentials . during their ballistic motion , the carriers undergo spin relaxation controlled by a ballistic spin relaxation length @xmath633 . ( iii ) at the end - points of the ballistic intervals , instantaneous `` point - like '' thermalization ( `` absorption '' ) of the carriers takes place . here , we invoke the picture of `` reflectionless contacts''@xcite , according to which the ballistic carriers that enter a contact ( representing a `` bath '' with an effectively infinite number of transverse modes ) are completely absorbed there , after having been emitted from a similar contact at the opposite end of the sample . in the thermoballistic concept , the emission and absorption of carriers are treated in this way at _ all _ points of local thermodynamic equilibrium inside the sample . at these points , the absorption of carriers into the bath is complete ( `` reflectionless '' ) , but at the same instant carriers are emitted out of the bath into the ballistic intervals on either side.the opposing collision - free currents emitted at either end into the ballistic interval combine to form a _ net ballistic current _ inside the interval , with an associated _ joint ballistic density_.    the random distribution of points of local thermodynamic equilibrium is mirrored in a random partitioning of the length of the sample into ballistic intervals , where each partition defines a `` ballistic configuration '' , a central notion of the concept . the ballistic carrier currents and densities in these configurations are assembled to form the corresponding _ thermoballistic currents and densities_. these are constructed , at a reference position @xmath37 located arbitrarily inside the sample , by performing weighted summations , with weights controlled by a momentum relaxation length @xmath7 , over the net ballistic currents and joint ballistic densities in all ballistic intervals containing the point @xmath37 . the thermoballistic currents and densities constitute the key element of the thermoballistic concept . the physical content of the thermoballistic transport mechanism can be exhibited by analyzing the underlying formalism with regard to the intertwined effects of thermal electron emission and ballistic motion . let us consider expressions ( [ eq : ball004jvx])([eq : netbal02xy ] ) for the net ballistic currents and joint ballistic densities , which are essentially composed of two factors each . on the one hand , they contain the nonlocal barrier factors @xmath844 and @xmath845 in the expressions for the total and persistent spin - polarized currents and the relaxing spin - polarized current , respectively , and the factors @xmath846 and @xmath847 in the analogous expressions for the densities . these factors describe the collision - free motion of the electrons across the ballistic interval @xmath313 $ ] , which is essentially determined by the potential energy profiles @xmath848 inside the interval . they represent the _ ballistic attribute _ of thermoballistic transport . the factors in brackets , on the other hand , contain terms depending on the average chemical potentials @xmath545 , @xmath849 and the spin accumulation functions @xmath504 , @xmath505 , which are directly related to the spin - resolved chemical potentials @xmath850 , @xmath851 at the end - points @xmath512 of the ballistic interval . these factors , which describe the thermal emission ( `` thermal activation '' ) of the ballistic currents at the points of local thermodynamic equilibrium at @xmath417 and @xmath418 , respectively , represent the _ thermal attribute _ of thermoballistic transport . the term in brackets is the `` activation term '' . [ the joint appearance of ballistic and thermal attributes shows that the term `` thermoballistic '' indeed provides an appropriate characterization of our approach . ]    the contributions of the ballistic currents and densities ( [ eq : ball004jvx])([eq : netbal02xy ] ) , summarily denoted by @xmath710 , to the corresponding thermoballistic currents and densities @xmath689 are to be read from eq . ( [ eq : thebal01 ] ) :  they are given by the current ( or density ) @xmath710 in the interval @xmath313 $ ] , multiplied by the probability @xmath852 that the electrons traverse this interval ballistically . the momentum relaxation length @xmath7 controls the magnitude of the ballistic contribution to the entire transport process . at the same time , it determines the average number of collisions , @xmath853 , in a sample of length @xmath28 . in the _ ballistic limit _ , when @xmath725 and there is no point of local thermodynamic equilibrium inside the sample , the transport is purely ballistic between the end - points @xmath69 and @xmath70 , and only the first term on the right - hand side of eq . ( [ eq : thebal01 ] ) contributes . in the opposite limit , when @xmath854 , the points of local thermodynamic equilibrium at which the electrons are equilibrated , lie infinitesimally close to one another . then only the double integral in eq . ( [ eq : thebal01 ] ) survives , and we arrive at @xmath855 for the total thermoballistic current @xmath856 [ see eq . ( [ eq : ballim20 ] ) ] . this current is essentially given in terms of equilibrium quantities , a property that characterizes the _ drift - diffusion limit_. expression ( [ eq : syn02 ] ) has the form of the current in the standard drift - diffusion approach [ see eqs . ( [ eq : drude006f ] ) and ( [ eq : diff001 ] ) ] , @xmath857 where @xmath61 is the electron mobility given by eq . ( [ eq : ballim20a ] ) , and @xmath731 the equilibrium electron density given by eq . ( [ eq : ball004jum ] ) . in that approach , the momentum relaxation length @xmath7 is nonzero ( so that @xmath61 remains nonzero ) , but small compared with the length scales over which the other parameters vary appreciably . the activation term reduces to a derivative , and the relation between the total thermoballistic current and the average chemical potential becomes a local one . the principal merit of the thermoballistic concept is that it allows to establish a consistent and transparent formalism for bridging , within the semiclassical approximation , the gap between the standard drift - diffusion and ballistic descriptions of charge carrier transport in semiconductors . while incorporating basic features of these descriptions , the thermoballistic concept consistently unifies and generalizes them by introducing random partitionings of the sample length into ballistic configurations . the concept is transparent in a twofold way . first , as shown above , a lucid interpretation of its physical content can be given in terms of ballistic and thermal attributes . second , owing to its semiclassical character,@xcite the concept allows the effects of the different parameters describing a semiconducting system to be clearly distinguished . in the implementation of this concept , explicit equations for various transport quantities can be derived , and simple solutions can be obtained in important special cases . the merit of transparency of the thermoballistic concept carries with it some simplifications and weak points which , however , in many cases can be remedied , albeit at the cost of increased complexity :  they are not detrimental to the concept as a whole . while the formulation of the full thermoballistic concept given here describes semiclassical transport in nondegenerate semiconducting systems , we have demonstrated within the prototype model how effects of electron tunneling and degeneracy can be taken into account . quantum interference effects in the electron motion are not treated explicitly , but they may be assumed to be implicitly incorporated via an extended interpretation of the mean free path ( or momentum relaxation length ) @xmath7 , which from the outset has been taken as a phenomenological parameter . by treating it formally as the average distance that the carriers travel without collision between points of _ complete _ thermodynamic equilibrium , as in the relaxation time approximation,@xcite it simulates the effects of incomplete equilibration due to elastic or inelastic impurity scattering , of dimensionality , and , in the extreme , of quantal phase correlations . indeed , in this work the momentum relaxation length @xmath7 is the determining parameter in which a great diversity of detail is subsumed . it is introduced as a constant , so it must include in an average way the effect of spatial variations in the internal and external parameters characterizing the semiconducting system;in particular , this constant is chosen to be independent of the potential energy profile , to which , however , it should be related in a self - consistent way . moreover , the choice of the momentum relaxation length @xmath7 , rather than the relaxation time @xmath8 , is also merely one of convenience for the stationary treatment in this work . we may work with position - dependent momentum relaxation lengths , but this would increase the complexity of the formalism and obscure the general line of argument . the spin relaxation mechanism in terms of the ballistic spin relaxation length @xmath633 is again a phenomenological one , having certain similarities with the dyakonov - perel mechanism.@xcite we assume spin relaxation to occur only during the ballistic electron motion ;  however , simultaneous spin and momentum relaxation could be taken into account by introducing additional terms in the spin balace equation .    in principle , the thermoballistic concept allows a fully three - dimensional treatment of bipolar carrier transport to be implemented . however , in the present paper , in order to keep the formalism manageable , we work within a narrowed framework . first , we confine ourselves to unipolar transport , dealing specifically with electron transport in a spin - split conduction band . second , we consider three - dimensional `` plane - parallel '' samples whose parameters ( in particular , the average density of the scattering centers associated with impurities ) do not vary in the directions perpendicular to the transport direction ( the @xmath37-direction ) . nevertheless , electron transport in this kind of sample depends on the number of dimensions , @xmath17 , via `` no - scattering probabilities '' @xmath858 [ see sec . ii of ref .  ] . here , in order to be able to write down physically transparent formulae , we use one - dimensional no - scattering probabilities of the form @xmath859 ( see sec . [ sec : probappr ] ) . having presented , in the preceding section , the concept underlying the thermoballistic approach , we now turn to the implementation of this concept . we begin by establishing the physical conditions from which the algorithms for calculating the dynamical functions , _ viz . _ , the average chemical potential @xmath503 [ via the mean spin function @xmath507 ] and the spin accumulation function @xmath506 , are developed . thereafter , these algorithms will be described in detail . we call _ physical conditions _ ( i ) a relation introduced to connect the total thermoballistic current @xmath696 with the cognate total physical current @xmath133 , which allows us to calculate the function @xmath507 , and ( ii ) an assumption concerning the detailed spin relaxation mechanism , which leads to the determination of the function @xmath506 . the current @xmath861 , owing to its construction in terms of ballistic currents averaged over random ballistic configurations ( with weights controlled by the momentum relaxation length @xmath7 ) , is to be interpreted as an `` ensemble average '' of the electron current at the point @xmath37 . this average is spatially varying , and , therefore , it is the _ spatial average _ of the ensemble average over the length of the sample which is to be identified with the ( constant ) total physical current @xmath862 inside the sample . by current conservation at @xmath69 and @xmath70 , this current is equal to the total current @xmath133 in the left and right leads , so that we have @xmath863 in this condition , the current @xmath864 is defined at a ballistic point @xmath37 located inside an ensemble of ballistic intervals @xmath313 $ ] , where @xmath417 and @xmath418 , unlike @xmath37 , are points of local thermodynamic equilibrium [ see eqs . ( [ eq : thebal00 ] ) and ( [ eq : thebal01 ] ) ] . the integration over @xmath37 starts at the fixed equilibrium point @xmath69 and ends at the fixed equilibrium point @xmath70 . now , just as we have introduced the _ ballistic _ reference point @xmath37 , we consider here an _ point of reference , i.e. , a point of local thermodynamic equilibrium anywhere inside the sample , which , again , is labeled by the coordinate @xmath37 @xmath865 . such a point acts in the same way as the fixed equilibrium points @xmath69 and @xmath70 . the ballistic current entering @xmath37 , say , from the left , is completely absorbed , whereupon a thermal current is instantaneously emitted to either side of @xmath37 . the same happens to the current entering the equilibrium point @xmath866 from the right . however , in contrast to the `` true '' , externally controlled equilibrium points @xmath75 , which are located at the contact side of the contact - semiconductor interfaces , no sharvin - type interface resistance ( see sec . [ sec : sharvin ] below ) appears at @xmath37 .    within this scheme , we introduce the thermoballistic current @xmath867 , with @xmath868 , where @xmath37 is an equilibrium point , and @xmath869 a ballistic point . in conformance with eq . ( [ eq : thebal04abc ] ) , the spatial average of this current over the range @xmath71 $ ] is again , by current conservation at @xmath69 , equal to the physical current in the left lead , so that @xmath870 analogously , we have for the thermoballistic current @xmath871 in the range @xmath73 $ ] , using similar arguments as above , @xmath872 ( @xmath873 ) . equations ( [ eq : thebal04ohg ] ) and ( [ eq : thebal04zul ] ) , when expressed in terms of the function @xmath507 via eqs . ( [ eq : ball004jvx ] ) and ( [ eq : thebal01 ] ) , lead to two different volterra - type integral equations with solutions @xmath874 and @xmath875 , respectively . trivially , by satisfying eqs . ( [ eq : thebal04ohg ] ) and ( [ eq : thebal04zul ] ) , these solutions also satisfy condition ( [ eq : thebal04abc ] ) . they define , for given spin accumulation function @xmath506 , two different total equilibrium densities @xmath876 and @xmath877 , respectively , via eq . ( [ eq : ball004jum ] ) . the total equilibrium density at @xmath37 is obtained as the mean value of these , @xmath878 , \label{eq : thebal05lg}\ ] ] so that , using eq . ( [ eq : ball004jum ] ) again , we have for the unique ( thermoballistic ) mean spin function @xmath879 . \label{eq : thebal05pk}\ ] ] the physical conditions ( [ eq : thebal04ohg ] ) and ( [ eq : thebal04zul ] ) , together with eq . ( [ eq : thebal05lg ] ) , determine the procedure for calculating the average chemical potential @xmath503 . using the mean value ( [ eq : thebal05lg ] ) as the point of departure for constructing @xmath503 reflects the fact that this function expresses an intrinsic property of the semiconducting sample , with no preference for one or the other of the sample ends at @xmath69 and @xmath70 . with @xmath507 given by eq . ( [ eq : thebal05pk ] ) , it follows from eqs . ( [ eq : ball004jvx ] ) and ( [ eq : thebal01 ] ) that the unique thermoballistic current @xmath864 is given by @xmath880 , \nonumber \\ \label{eq : thebal04lfs}\end{aligned}\ ] ] a symmetric combination as in eq . ( [ eq : thebal05pk ] ) . we note that in ref . we have applied a different procedure for constructing the average chemical potential @xmath503 . there , the combination of the functions @xmath874 and @xmath875 was not chosen to be symmetric as in eq . ( [ eq : thebal05pk ] ) , but was determined by the requirement that the values of the unique total thermoballistic current at the two sample ends be equal , @xmath881 this condition derives from postulating that the position dependence of the total thermoballistic current is compensated by that of a `` background current'',@xcite such that these currents add up to the ( conserved ) total physical current . the background current is assumed to be fed by sources and sinks whose effect averages out to zero when the current is integrated over the length of the sample . in the present work , this hypothesis has been abandoned for being unphysical .      in the thermoballistic transport mechanism , we assume that spin relaxation takes place only _ inside _ the ballistic intervals ( see sec .  [ sec : balance ] ) , and that an infinitesimal shift of the end - points of the ballistic intervals does not affect the current @xmath699 . accordingly , we set the term arising from differentiating the limits of integration in expression ( [ eq : thebal01 ] ) for @xmath699 equal to zero , @xmath882 from eq . ( [ eq : thebal06 ] ) , the relaxing thermoballistic spin - polarized current and density are then seen to be connected by the balance equation @xmath883 which is of the same form as the balance equation ( [ eq : baleq01zs ] ) connecting the relaxing spin - polarized current and density in the individual ballistic intervals . the _ spin accumulation function _ @xmath506 , and hence the relaxing thermoballistic spin - polarized current @xmath699 , are determined by condition ( [ eq : thebal06rt ] ) . when written in terms of @xmath506 by using expression ( [ eq : netbal02 ] ) in eq . ( [ eq : thebal06jb ] ) , this condition turns into a linear , inhomogeneous fredholm - type integral equation of the second kind@xcite for @xmath506 . the explicit form of this equation and its conversion , in a specific case of particular importance , into a differential equation will be the subject of sec . [ sec : accufunc ] . the average chemical potential @xmath782 is determined , via expression ( [ eq : thebal05pk ] ) for the mean spin function @xmath507 , by the solutions @xmath884 of the integral equations ( [ eq : thebal04ohg ] ) and ( [ eq : thebal04zul ] ) , respectively , which are conveniently solved in terms of `` resistance functions '' . we begin by considering eq . ( [ eq : thebal04ohg ] ) . to solve it for the function @xmath874 , we define functions @xmath885 and @xmath886 via @xmath887 and @xmath888 where the spin accumulation function @xmath506 is assumed to be given . furthermore , we introduce the `` resistance function '' @xmath889 [ the choice of this name will be substantiated in sec . [ sec : currvolt ] below ] as @xmath890 along with the function @xmath891   \label{eq : inteq02ax}\ ] ] [ for given @xmath506 , @xmath892 is a given function as well ] .    now , expressing the net total ballistic current @xmath746 given by eq . ( [ eq : ball004jvx ] ) in terms of the functions @xmath889 and @xmath892 , we obtain from eq . ( [ eq : thebal01 ] ) the total thermoballistic current @xmath893 to be inserted in eq . ( [ eq : thebal04ohg ] ) in the form @xmath894 here , @xmath895 \nonumber \\ & \ & \hspace{-2.2 cm }   + \int_{\xi^{+}}^{x } \frac{dx''}{l }   w_{+}(x_{1 } , x '' ; l ) [ \tilde{\mathfrak{r}}_{1}(x '' ) -   \tilde{\mathfrak{r}}_{1}(x_{1 } ) ]   \nonumber \\ & \ & \hspace{-2.2 cm } + \left . \int_{x_{1}}^{\xi^{- } } \frac{dx'}{l } \int_{\xi^{+}}^{x } \frac{dx''}{l }   w_{+}(x ' , x '' ; l ) [ \tilde{\mathfrak{r}}_{1}(x '' ) -   \tilde{\mathfrak{r}}_{1}(x ' ) ] \right\ } , \nonumber \\ \label{eq : thebal01tzg}\end{aligned}\ ] ] while the term @xmath896 is obtained from expression ( [ eq : thebal01tzg ] ) by replacing @xmath889 with @xmath892 throughout , and @xmath897 with @xmath898 , where @xmath899 to evaluate condition ( [ eq : thebal04ohg ] ) , we use the relation @xmath900 to carry out the integration over @xmath869 , so that the condition can be expressed in the explicit form @xmath901 here , the inhomogeneity @xmath902 is given by @xmath903 and the integral kernels @xmath904 are defined as @xmath905 with @xmath906 equation ( [ eq : inteq04ftx ] ) is a linear , inhomogeneous , volterra - type integral equation of the second kind@xcite in the range @xmath907 .    the kernels @xmath908 depend on the potential energy profiles @xmath909 solely via the quantities @xmath654 , and are independent of the total physical current @xmath133 . nonetheless , for nonzero spin splitting , a @xmath133-dependence of the resistance function @xmath889 can arise , via the function @xmath892 , from a nonlinear @xmath133-dependence of the spin accumulation function @xmath506 [ see secs . [ sec : accufunc ] and [ sec : currden ] below ] . in the zero - bias limit , when @xmath506 is proportional to @xmath133 , the function @xmath892 , and hence @xmath889 , become independent of @xmath133 . for zero spin splitting , when @xmath910 , we have @xmath911 and @xmath889 does not depend on @xmath506 . the resistance function @xmath889 is discontinuous at @xmath912 :  we have @xmath913 from eq . ( [ eq : inteq02 ] ) , whereas we obtain @xmath914 by expanding eq . ( [ eq : inteq04ftx ] ) to first order in @xmath915 ( we assume the potential energy profiles @xmath916 to be continuous in the interval @xmath83 $ ] ) .    for arbitrary functions @xmath48 and @xmath917 , the calculation of the resistance function @xmath889 for a chosen parameter set consists of _ three , consecutive _ steps :  ( i ) calculation of the spin accumulation function @xmath506 as solution of the integral equation ( [ eq : spiacc02 ] ) [ see sec . [ sec : accufunc ] below ] , using the boundary conditions ( [ eq : ballim05s ] ) with given values @xmath509 at the contact - semiconductor interfaces ;  ( ii ) with @xmath506 as input in expression ( [ eq : inteq02ax ] ) , calculation of the inhomogeneity @xmath918 from eq . ( [ eq : inteq03 ] ) ; ( iii ) using @xmath902 in the volterra equation ( [ eq : inteq04ftx ] ) , calculation of @xmath889 by , in general , numerical methods .    turning now to the function @xmath919 , we proceed as for @xmath885 . in analogy to eq . ( [ eq : inteq04ftx ] ) for the resistance function @xmath889 , we introduce a resistance function @xmath920 as @xmath921 for which we obtain from eq . ( [ eq : thebal04zul ] ) a volterra - type integral equation in the range @xmath922 , @xmath923 with the inhomogeneity @xmath924 expressed in terms of the function @xmath925   \label{eq : inteq02im}\ ] ] as @xmath926 the kernels @xmath927 are obtained by replacing in expression ( [ eq : inteq06opa ] ) for @xmath928 the pair of arguments @xmath929 with @xmath930 . the resistance function @xmath931 is discontinuous at @xmath932 , @xmath933 @xmath934 in analogy to the discontinuity of the function @xmath889 at @xmath935 [ see eqs . ( [ eq : inteq08 ] ) and ( [ eq : inteq09 ] ) ] .    as to the calculation of @xmath931 from eq . ( [ eq : inteq04iz ] ) , the foregoing discussion regarding the calculation of @xmath889 from eq . ( [ eq : inteq04ftx ] ) applies _ mutatis mutandis_. alternatively , one may calculate @xmath931 by using the relation @xmath936 where @xmath937 is the solution of eq . ( [ eq : inteq04ftx ] ) corresponding to spatially reversed potential energy profiles , @xmath938 using as input a spin accumulation function @xmath506 calculated from eq . ( [ eq : spiacc02 ] ) below with the reversed profiles and with the boundary values @xmath939 and @xmath940 interchanged .    the calculation of the resistance functions @xmath889 and @xmath931 simplifies considerably if the spin splitting is constant over the sample , @xmath941 [ while @xmath48 still may be arbitrary ] . with @xmath942 we have @xmath943 and hence for the functions @xmath654 from eqs . ( [ eq : ball004u ] ) and ( [ eq : emtra09 ] ) , using eqs . ( [ eq : eqden06])([eq : spfra17c ] ) , @xmath944 and @xmath945 here , we have defined @xmath946 @xmath947 and @xmath948 is the static spin polarization .    considering , for instance , the calculation of @xmath889 , we then find from eqs . ( [ eq : inteq06opa ] ) for the kernels @xmath904 , using expressions ( [ eq:66x ] ) and ( [ eq:66y ] ) in eq . ( [ eq : thebal02 ] ) , @xmath949 and @xmath950 where the reduced kernel @xmath951 corresponds to @xmath952 . now , inserting expressions ( [ eq : inteq06wuf ] ) and ( [ eq : inteq06waf ] ) in eqs . ( [ eq : inteq04ftx ] ) and ( [ eq : inteq03 ] ) , respectively , we can rewrite the integral equation ( [ eq : inteq04ftx ] ) in the form @xmath953 @xmath954 , where @xmath955 , \label{eq : inteq09wu}\ ] ] and the function @xmath892 is independent of @xmath956 ( see sec .  [ sec : intequat ] below ) . as a solution of eq . ( [ eq : inteq04aha ] ) , the function @xmath957 is a universal function that describes an intrinsic property of the semiconducting sample ;  it is determined by the potential energy profile @xmath48 and the momentum relaxation length @xmath7 , and does not depend on the spin splitting @xmath956 and the spin accumulation function @xmath506 . the resistance function @xmath889 for constant spin splitting can now be expressed as @xmath958 the calculation here separates into _ two , independent _ steps :  ( i ) solution of eq . ( [ eq : inteq04aha ] ) for @xmath957 ;  ( ii ) calculation of @xmath506 as solution of the integral equation ( [ eq : spiacc02 ] ) and determination of @xmath892 via eqs .  ( [ eq : inteq04lil ] ) and ( [ eq : inteq02ax ] ) . in particular , for zero spin splitting , we have @xmath959      introducing the mean value @xmath960 of the functions @xmath961 , which both satisfy condition ( [ eq : thebal04abc ] ) , @xmath962   \label{eq : inteq09plb}\ ] ] [ see eq . ( [ eq : thebal05pk ] ) ] , we observe that @xmath960 satisfies condition ( [ eq : thebal04abc ] ) as well . using eqs . ( [ eq : inteq02 ] ) and ( [ eq : spiche02 ] ) , we can express @xmath960 in the form @xmath963 - { \mbox{$\frac{1}{2}$ } } [ \tilde{\mathfrak{r}}_{1}(x ) - \tilde{\mathfrak{r}}_{2}(x ) ] . \label{eq : spiche08gs}\end{aligned}\ ] ] now , writing down this expression for @xmath935 and @xmath761 , respectively , and using eqs . ( [ eq : inteq04scp ] ) , ( [ eq : inteq08 ] ) , and ( [ eq : spiche05 ] ) together with the boundary conditions ( [ eq : ballim05ups ] ) , we add the resulting two expressions to obtain for the quantity @xmath964 defined by eq . ( [ eq : ballim04 ] ) , @xmath965 \ } . \label{eq : spiche16}\end{aligned}\ ] ] on the other hand , subtracting the two expressions , we have @xmath966 with the parameter @xmath967 defined as @xmath968 . \label{eq : spiche15}\ ] ] then , using eqs . ( [ eq : spiche16 ] ) and ( [ eq : spiche14 ] ) , we can eliminate from expression ( [ eq : spiche08gs ] ) the dependence on the boundary values @xmath969 and @xmath970 of the functions @xmath961 . introducing the function @xmath971 , \nonumber \\ \label{eq : spiche18}\end{aligned}\ ] ] so that , in view of eqs . ( [ eq : inteq08 ] ) and ( [ eq : spiche05 ] ) , @xmath972 we find for the mean spin function @xmath507 , using eq . ( [ eq : inteq04scp ] ) , @xmath973 for @xmath974 , and @xmath975 equivalently , we write @xmath976 for @xmath974 , and @xmath977 expression ( [ eq : spiche17zg ] ) represents the final , general expression for the ( local ) thermoballistic average chemical potential @xmath503 inside the sample [ henceforth , we drop the attribute `` thermoballistic '' when referring to @xmath978 . it is seen to be uniquely determined by the values of the _ external _ parameters at the contact sides of the contact - semiconductor interfaces , _ viz . , _ the values @xmath76 of the equilibrium chemical potential and the values @xmath509 of the spin accumulation function [ which enter via the resistance functions @xmath889 and @xmath931 ] . comparing expression ( [ eq : spiche17zg ] ) to expression ( [ eq : prochem06 ] ) for the chemical potential @xmath38 in the prototype thermoballistic model , we observe a formally identical structure , but differences in the explicit forms of the functions @xmath389 and @xmath980 and of the reduced resistances @xmath279 and @xmath981 . an explicit expression for @xmath503 can be obtained if @xmath982 ( `` ballistic regime '' ) , in which case the probability for an electron to suffer a collision when traversing the sample of length @xmath28 , @xmath983 , is vanishingly small . then , keeping terms of order @xmath853 , we have from eq . ( [ eq : inteq04ftx ] ) @xmath984 and similarly from eq . ( [ eq : inteq04iz ] ) @xmath985 since from eqs . ( [ eq : ballim05s ] ) , ( [ eq : inteq04scp ] ) , ( [ eq : inteq02ax ] ) , and ( [ eq : inteq02im ] ) @xmath986 we now find from eq . ( [ eq : spiche15 ] ) @xmath987 for the function @xmath988 , we have from eq . ( [ eq : spiche18 ] ) , using eq . ( [ eq : inteq53cb ] ) , @xmath989 with expressions ( [ eq : inteq53pl ] ) and ( [ eq : spiche18fv ] ) inserted in eq . ( [ eq : spiche17zg ] ) , we obtain the average chemical potential @xmath503 in explicit form . in the particular case of zero spin splitting , when @xmath990 and @xmath991 , we have @xmath992 . \label{eq : spiche17ld}\ ] ] this result , here obtained in the ballistic regime , @xmath982 , contrasts with the ( strict ) ballistic limit , @xmath725 , considered in sec . [ sec : ballim ] , when there is absolutely no point of local thermodynamic equilibrium inside the sample . in this extreme case , no meaning can be attached to an average chemical potential , and there is nothing a calculation of such a quantity can be based upon .    in ref .  , we have presented and discussed numerical results for @xmath503 , calculated with the potential energy profile ( [ eq : difeq01 ] ) for values of the ratio @xmath993 ranging between @xmath994 to @xmath995 . if calculated with the procedure adopted in the present work ( see the remarks at the end of sec . [ sec : detmsf ] ) , the results for @xmath503 would differ quantitatively from the former ones , but would agree with those in all qualitative respects . the discontinuities in the resistance functions @xmath889 and @xmath931 at the points of local thermodynamic equilibrium @xmath935 and @xmath761 , respectively , give rise to discontinuities in the function @xmath988 both at @xmath996 and @xmath761 . from eq . ( [ eq : spiche18 ] ) , we find , using eqs . ( [ eq : inteq08 ] ) and ( [ eq : inteq09 ] ) , @xmath997 and similarly , using eqs . ( [ eq : spiche05 ] ) and ( [ eq : spiche06 ] ) , @xmath998 therefore , according to eq . ( [ eq : spiche17zg ] ) , the average chemical potential @xmath503 exhibits discontinuities at these points as well ( `` sharvin effect '' ;  see ref .  ) , @xmath999 } - 1 & = & - \frac{\eta_{12}^{- } e^{- \beta \mu_{1 } } } { 2 } \frac{\tilde{\mathfrak{r}}_{1 } ( x_{1}^{+})}{\tilde{\mathfrak{r } } } \nonumber \\ & = & - { \mbox{$\frac{1}{2}$ } } \beta e^{2 }   j \tilde{\rho}_{1 } ,   \label{eq : sharv03}\end{aligned}\ ] ] @xmath1000 } - 1 & = & \frac{\eta_{12}^{- } e^{- \beta \mu_{2 } } } { 2 } \frac{\tilde{\mathfrak{r}}_{2 } ( x_{2}^{-})}{\tilde{\mathfrak{r } } } \nonumber \\ & = &   { \mbox{$\frac{1}{2}$ } } \beta e^{2 } j \tilde{\rho}_{2 } ,   \label{eq : sharv04}\end{aligned}\ ] ] where we have used eqs . ( [ eq : spiche14 ] ) and ( [ eq : spiche15 ] ) to introduce the total physical current @xmath133 . the quantities @xmath1001 are the _ sharvin interface resistances_,@xcite @xmath1002 @xmath1003 for zero spin splitting , when @xmath1004 , so that @xmath1005 and @xmath1006 , we can express @xmath1001 in the form @xmath1007 where @xmath1008 } \label{eq : sharv05tf}\end{aligned}\ ] ] are the total equilibrium electron densities at the semiconductor side of the contact - semiconductor interfaces [ see eqs . ( [ eq : ball004jux ] ) and ( [ eq : ball004jum ] ) ] . equations ( [ eq : sharv03 ] ) and ( [ eq : sharv04 ] ) , respectively , are analogous to eqs . ( [ eq : prochem06amf ] ) and ( [ eq : prochem06pky ] ) for the discontinuities of the chemical potential @xmath352 in the prototype thermoballistic model . the _ current - voltage characteristic _ of the thermoballistic transport model is obtained from eq . ( [ eq : spiche14 ] ) in the form @xmath1009 where @xmath1010 is the voltage bias between the metal contacts ( we assume the voltage drop across the contacts to be negligibly small ) . in the zero - bias limit , when @xmath1011 , we have @xmath1012 for the resistance times cross - sectional area of the sample . comparing the current - voltage characteristic ( [ eq : sharv06 ] ) to that of the ( spinless ) prototype thermoballistic model , eq . ( [ eq : gendr004a ] ) , one sees that the ( dimensionless ) parameter @xmath967 directly corresponds to the reduced resistance @xmath279 of the prototype model , eq . ( [ eq : gendr003 ] ) . therefore , @xmath967 is here also called the _ reduced resistance _ of the sample , and the functions @xmath889 and @xmath931 from which it is derived , the _ resistance functions_.    the reduced resistance @xmath967 is a central element of the thermoballistic description of electron transport in semiconductors . it comprises the effect of the detailed shape of the potential energy profiles as well as that of the momentum relaxation length . moreover , via its dependence on the spin accumulation function @xmath506 , it takes into account the effect of spin relaxation . as exemplified by expression ( [ eq : inteq53pl ] ) for @xmath967 , which holds in the ballistic regime , the reduced resistance depends on the total current @xmath133 unless @xmath506 is proportional to @xmath133 . the explicit form of @xmath967 in the drift - diffusion regime can be obtained by rewriting eq . ( [ eq : ballim20kf ] ) for @xmath761 in the form of a current - voltage characteristic and comparing it to the characteristic ( [ eq : sharv06 ] ) . defining now the _ relative magnetoresistance _ @xmath1013 of the semiconducting sample@xcite as @xmath1014 where @xmath1015 is the resistance at zero external magnetic field , i.e. , at zero spin splitting , we have from eq .  ( [ eq : sharv07tf ] ) @xmath1016 the reduced resistance at zero spin splitting , @xmath1017 , is obtained by solving eq . ( [ eq : inteq04aha ] ) for the function @xmath957 and the analogous equation for @xmath1018 and using eq . ( [ eq : spiche15 ] ) , @xmath1019 . \label{eq : spiche15axa}\ ] ] for _ constant _ splitting , we find from eq . ( [ eq : spiche15 ] ) , using eq . ( [ eq : inteq09wq ] ) and the analogous equation for @xmath931 as well as eq . ( [ eq : inteq53cb ] ) , @xmath1020 and hence @xmath1021 for the magnetoresistance . the net relaxing ballistic spin - polarized current @xmath508 [ see eq . ( [ eq : netbal02 ] ) ] and the joint relaxing ballistic spin - polarized density @xmath681 [ see eq . ( [ eq : netbal02xy ] ) ] , and thus also the corresponding thermoballistic current @xmath699 and density @xmath703 evaluated from eq . ( [ eq : thebal01 ] ) , are dynamically determined solely by the spin accumulation function @xmath506 . the calculation of this function is , in general , prerequisite to a complete determination of the average chemical potential @xmath782 [ see sec . [ sec : resisfunc ] ] . following sec . [ sec : thermcurr ] , we invoke condition ( [ eq : thebal06rt ] ) , which is equivalent to the spin balance equation ( [ eq : ballim24new ] ) , and insert expression ( [ eq : netbal02 ] ) for the quantity @xmath710 in eq . ( [ eq : thebal06jb ] ) to obtain for the spin accumulation function @xmath506 an integral equation of the form @xmath1022 where @xmath1023 and @xmath1024 here , the values @xmath787 of the spin accumulation function at the interface positions @xmath75 have been set equal to the external values , @xmath509 , at the contact side of the contact - semiconductor interfaces [ see eq . ( [ eq : ballim05s ] ) ] . the function @xmath1025 is defined as @xmath1026 [ see eq . ( [ eq : baleq06 ] ) ] , and @xmath1027 with @xmath655 given by eq . ( [ eq : spirel15 ] ) .    with the first two terms in eq . ( [ eq : spiacc02 ] ) acting as an inhomogeneity , this equation is a linear , inhomogeneous , fredholm - type integral equation of the second kind@xcite for the spin accumulation function @xmath506 . the corresponding homogeneous equation is solved by @xmath1028 only , so that the solution @xmath506 of eq . ( [ eq : spiacc02 ] ) for @xmath1029 is linear and homogeneous in @xmath939 and @xmath940 . just like the average chemical potential @xmath503 , the spin accumulation function @xmath506 exhibits the sharvin effect , i.e. , it is not , in general , continuous at the interfaces with the contacts , @xmath1030 . this will be demonstrated in sec . [ sec : diffequa ] below by way of a specific example . the solution of eq . ( [ eq : spiacc02 ] ) is found , in general , numerically by applying matrix methods after discretization .    for constant spin splitting , @xmath1031 , when the functions @xmath1032 are given by eq . ( [ eq:66ixy ] ) , the function @xmath655 defined by eq . ( [ eq : spirel15 ] ) reduces to @xmath1033 with @xmath1034 given by eq . ( [ eq:66asa ] ) , and @xmath1035 by eq . ( [ eq:66blg ] ) . further , using eqs . ( [ eq:66hjc ] ) and ( [ eq:66ixy ] ) in expression ( [ eq : ball004n ] ) , we see that @xmath956 cancels out in the functions @xmath1036 , @xmath1037 inserting this in eq . ( [ eq : ball004p ] ) , we then find , via eqs . ( [ eq : balden15 ] ) , ( [ eq : baleq03 ] ) , and ( [ eq : baleq06 ] ) , the function @xmath1025 defined by eq . ( [ eq : spiacc06 ] ) to be independent of @xmath956 , @xmath1038 with this result , in conjunction with eqs .  ( [ eq : thebal03 ] ) and ( [ eq : thebal03ihi ] ) , used in eqs . ( [ eq : spiacc03 ] ) and ( [ eq : spiacc04 ] ) , the spin splitting @xmath956 is seen to drop out from the integral equation ( [ eq : spiacc02 ] ) , i.e. , its solution @xmath506 does not depend on @xmath956 .      in _ homogeneous _ semiconductors without space charge , the conduction band edge potential is constant . in this case , the integral equation for @xmath506 can be converted into a differential equation . assuming the electrons to be driven by an external electric field of magnitude @xmath485 directed antiparallel to the @xmath37-axis , the potential energy profile ( [ eq : eqden01 ] ) for zero spin splitting , @xmath1039 , is given by eq . ( [ eq : difeq01 ] ) . [ according to the above discussion , the inclusion of a nonzero , constant spin splitting would not alter the results . ] for the profile ( [ eq : difeq01 ] ) , the function @xmath1040 defined by eq . ( [ eq : baleq03 ] ) takes the form @xmath1041^{1/2}\bm { ) }   \nonumber \\ \label{eq : difeq02}\end{aligned}\ ] ] @xmath1042 , as can be shown by using eqs . ( [ eq : ball004p ] ) , ( [ eq : ball004n ] ) , and ( [ eq : balden15 ] ) .    using expression ( [ eq : difeq02 ] ) , via eq . ( [ eq : baleq06 ] ) , in eq . ( [ eq : spiacc06 ] ) , the integral equation ( [ eq : spiacc02 ] ) can now be reduced to @xmath1043 where @xmath1044 x } , \label{eq : difeq05}\ ] ] @xmath1045 x } , \label{eq : difeq06}\ ] ] and @xmath1046 \ } . \label{eq : difeq07}\ ] ] the function @xmath1047 defined by @xmath1048 obeys the relations @xmath1049 , @xmath1050 for @xmath1051 , and @xmath1052 for @xmath1053 . we can convert the inhomogeneous integral equation ( [ eq : difeq04 ] ) into a homogeneous integrodifferential equation for @xmath506 by supplementing eq . ( [ eq : difeq04 ] ) with the equations obtained by forming its first and second derivative with respect to @xmath37 , and subsequently eliminating from this set of three equations the boundary values @xmath939 and @xmath940 . [ in principle , a similar procedure could also be applied to the general equation ( [ eq : spiacc02 ] ) , but this does not seem to lead to any advantage . ] if we replace in the integral equation ( [ eq : difeq04 ] ) the function @xmath1047 with a position - independent average value @xmath1054 , the functions @xmath1055 and @xmath1056 become pure exponentials , and the corresponding integrodifferential equation reduces to a homogeneous second - order differential equation of the form @xmath1057 here , @xmath1058 , \label{eq : difeq10}\ ] ] @xmath1059 , \label{eq : difeq11}\ ] ] and @xmath1060 \ } , \label{eq : difeq12}\ ] ] where @xmath1061 and @xmath1062 with the ballistic spin relaxation length @xmath633 given by eq . ( [ eq : spirel01 ] ) . [ note that expression ( [ eq : difeq12 ] ) for the function @xmath1063 differs from the corresponding expression ( 3.48 ) of ref .  , which was derived by introducing the average value @xmath1054 in the integrodifferential equation , rather than in the original integral equation . ] since , owing to the presence of the factor @xmath1064 in the functions @xmath1055 and @xmath1056 , only the values of @xmath1065 in the range @xmath1066 contribute appreciably , we may choose @xmath1054 as the average of @xmath1065 over an @xmath37-interval of length equal to the momentum relaxation length @xmath7 , @xmath1067 in the right - hand integral in this equation , the range of small @xmath1068 , when @xmath1069 , is emphasized because of the weight factor @xmath1070 .    for large @xmath1071 ( i.e. , in the ballistic regime and/or for strong fields ) , the variation of @xmath1065 with @xmath37 becomes essential . however , for the purpose of demonstrating the principal effects of the transport mechanism , the approximation in terms of a constant average value @xmath1054 of the function @xmath1072 , in conjunction with the choice ( [ eq : difeq15 ] ) for @xmath1054 , appears to be sufficiently accurate .    in the drift - diffusion regime , when @xmath726 , @xmath727 , and @xmath1073 , we have @xmath1074 and hence @xmath1075 , where @xmath1076 equation ( [ eq : difeq09 ] ) then reduces to @xmath1077 where @xmath1078 is the spin diffusion length given by eq . ( [ eq : ballim26 ] ) . setting @xmath1079 in eq . ( [ eq : ballim26xx ] ) , we find eq . ( [ eq : difeq16 ] ) to agree with the former equation , which was obtained by directly evaluating the thermoballistic current @xmath699 and density @xmath703 in the drift - diffusion limit , and which leads to the standard form ( [ eq : ballim25 ] ) of the drift - diffusion equation for spin - dependent transport .    in the zero - bias limit @xmath1080 , the integral equation ( [ eq : difeq04 ] ) reduces to @xmath1081 from which we obtain the differential equation @xmath1082 here , @xmath1083 is the generalization of the spin diffusion length @xmath1078 given by eq . ( [ eq : ballim26 ] ) . the length @xmath475 becomes equal to the latter length , @xmath1084 , in the drift - diffusion regime , when @xmath727 , and to the ballistic spin relaxation length , @xmath1085 , in the ballistic case , when @xmath1086 and hence @xmath1087 .    for @xmath974 , the general solution of eq . ( [ eq : difeq18 ] ) reads @xmath1088 inserting this expression for @xmath506 in eq . ( [ eq : difeq17 ] ) and setting @xmath935 and @xmath761 , we obtain , using eq . ( [ eq : ballim05s ] ) , two linear equations for the coefficients , with the solution @xmath1089 , \label{eq : difeq21}\ ] ] @xmath1090 . \label{eq : difeq22}\ ] ] here , @xmath1091 , \label{eq : difeq23}\ ] ] with @xmath1092 it then follows from eqs . ( [ eq : difeq20])([eq : difeq24 ] ) that @xmath506 is discontinuous at @xmath935 and @xmath761 , @xmath1093 @xmath1094 where @xmath1095 \leq 1 , \label{eq : difeq27}\ ] ] with @xmath1096 in the drift - diffusion regime , when @xmath1084 and @xmath1097 , we have @xmath1098 and in the ballistic case , when @xmath1085 and @xmath1099 , @xmath1100 . \label{eq : difeq30}\ ] ] at @xmath935 , for example , the discontinuity of @xmath506 is @xmath1101 in the drift - diffusion regime , and @xmath1102 $ ] in the ballistic case . the position dependence of the current and density spin polarizations as well as the magnetoresistance are the physical quantities of principal interest in the study of spin - polarized electron transport in paramagnetic semiconducting systems .      in the thermoballistic approach , we define the _ persistent _ current spin polarization in the semiconducting sample , @xmath1103 , in terms of the persistent thermoballistic spin - polarized current @xmath698 and the total physical current @xmath133 as @xmath1104 @xmath1105 . analogously , the _ relaxing _ current spin polarization @xmath1106 is defined in terms of the relaxing thermoballistic spin - polarized current @xmath699 and the total current @xmath133 as @xmath1107 for the _ total _ current spin polarization @xmath1108 , we then have @xmath1109 [ note that in refs .   and , we have defined the current spin polarization in terms of the total _ thermoballistic _ current @xmath696 , rather than in terms of the total _ physical _ current @xmath133 . ]    the _ persistent _ density spin polarization @xmath1110 and the _ relaxing _ density spin polarization @xmath1111 are introduced by replacing in expressions ( [ eq:65 ] ) and ( [ eq:65ax ] ) the thermoballistic currents @xmath698 and @xmath699 with the respective densities @xmath702 and @xmath703 , and the current @xmath133 in the denominator of those expressions with the thermoballistic joint total density @xmath701 . hence , we obtain @xmath1112 for the _ total _ density spin polarization @xmath1113 .    in expression ( [ eq:91xx ] ) for the total current spin polarization inside the semiconductor , the current @xmath698 depends on the reduced spin accumulation function @xmath1114 , and hence linearly on the boundary values @xmath1115 , via the dependence of the total ballistic current @xmath500 on @xmath1114 , @xmath1116 [ see eqs . ( [ eq : ball004jvx ] ) and ( [ eq : netbal06 ] ) ;  here , we have assumed @xmath500 to be expressed in terms of @xmath1114 , rather than @xmath506 , using eq . ( [ eq : spfra01fv ] ) ] . the relaxing thermoballistic spin - polarized current @xmath699 is determined by the ballistic current @xmath508 given by eq . ( [ eq : netbal02 ] ) , which is a linear - homogeneous functional of @xmath1114 . hence , @xmath699 is linear and _ homogeneous _ in the boundary values @xmath1115 , so that we can write @xmath1117 , \nonumber \\ \label{eq:91x}\end{aligned}\ ] ] with ( dimensionless ) `` formfactors '' @xmath1118 and @xmath1119 . then , we can express the values of the total current spin polarization at the semiconductor side of the interfaces at @xmath75 as @xmath1120 . \nonumber \\ \label{eq:66c}\end{aligned}\ ] ] the boundary values @xmath1115 of the reduced spin accumulation function , which are undetermined as yet , are to be expressed in terms of the current @xmath133 and the parameters of the system by matching the current spin polarization in the semiconductor to the corresponding polarization in the contacts ( see sec . [ sec : match ] below ) .      in order to keep the formulation sufficiently general , we assume the semiconducting sample to be connected to ( semi - infinite ) _ ferromagnetic _ metal contacts , which are treated as fully degenerate fermi systems .    in the left contact located in the range @xmath1121 , the spin - resolved chemical potentials @xmath1122 have the form@xcite @xmath1123 here , @xmath1124 is the spin diffusion length , and @xmath1125 are the ( @xmath37-independent ) spin - up and spin - down bulk conductivities , respectively . setting @xmath1126 in eqs . ( [ eq:50 ] ) , we obtain @xmath1127 with the spin - resolved currents @xmath1128 given by @xmath1129 we then find for the current spin polarization in the left contact , using @xmath1130 , @xmath1131 here , @xmath1132 is the bulk ( current or density ) spin polarization , and the parameter @xmath1133 which has the dimension of interface conductance , characterizes the spin - dependent transport in the ferromagnet .    for the current spin polarization in the right contact located in the range @xmath1134 , we have @xmath1135 which follows by replacing in eq . ( [ eq:50 ] ) the coordinate @xmath37 with @xmath1136 , and the labels `` l '' attached to the parameters in eqs . ( [ eq:50])([eq:57xyz ] ) with `` r '' .    when spin - selective interface resistances are absent , the chemical - potential splitting @xmath724 is continuous at the interfaces , @xmath1137 @xmath1138 where @xmath1139 are its values at the contact side of the interfaces . the latter values are to be identified with the corresponding values @xmath1140 at the semiconductor side of the interface , @xmath1141 where the right - hand equations have been obtained by solving eq . ( [ eq : spfra01hl ] ) for @xmath549 , setting @xmath1142 , and defining @xmath1143 for the boundary values of the reduced spin accumulation function @xmath551 . with eqs . ( [ eq:57zxl])([eq:68 ] ) used in eqs . ( [ eq:57lala ] ) and ( [ eq:57zzz ] ) , respectively , we now have for the current spin polarizations at the contact side of the interfaces @xmath1144 @xmath1145 which are to be identified with the corresponding polarizations @xmath1146 at the semiconductor side of the interfaces [ see eq . ( [ eq:66zara ] ) below ] . spin - selective interface resistances @xmath1147 give rise to discontinuities of the spin - resolved chemical potentials on the contact sides of the interfaces.@xcite at @xmath935 , for example , the discontinuity has the form @xmath1148 the interface resistances @xmath1149 are located between @xmath1126 and @xmath69 in the ferromagnetic contact , adjacent to the sharvin interface resistance @xmath1150 [ see eq . ( [ eq : sharv05 ] ) ] between @xmath935 and @xmath1151 in the semiconductor . the quantity @xmath1152 to be substituted in eq . ( [ eq:57lala ] ) is obtained from eqs . ( [ eq:50 ] ) , ( [ eq : zeropol01 ] ) , and ( [ eq:57xyz ] ) as @xmath1153 \right\ } . \label{eq : karol}\end{aligned}\ ] ] the same procedure applies _ mutatis mutandis _ to the interface at @xmath74 . the connection of @xmath1139 with the interface values of the reduced spin accumulation function , @xmath1115 , is again given by eqs . ( [ eq:68 ] ) . to evaluate the current and density spin polarizations all across the contact - semiconductor system , we have to relate the boundary values of the reduced spin accumulation function , @xmath1115 , to the total physical current @xmath133 and the internal parameters characterizing the system . to this end , we exploit the continuity of the total current spin polarization @xmath1108 across the contact - semiconductor interfaces at @xmath75 , @xmath1154 using expressions ( [ eq:57angi ] ) and ( [ eq:57guido ] ) for @xmath1155 and expressions ( [ eq:66c ] ) for @xmath1146 , we obtain a pair of coupled nonlinear equations for @xmath1115 , which we will not write down here in their general form .    in the zero - bias limit , when @xmath1156 , we have @xmath1157 [ see eq . ( [ eq : spfra01hl ] ) ] , so that @xmath1158 and @xmath1159 become linear in @xmath1160 and @xmath1161 , respectively . further , to first order , the term involving the reduced spin accumulation function @xmath1114 in the persistent ballistic spin - polarized current @xmath667 [ see eq . ( [ eq : netbal06 ] ) , with @xmath746 expressed in terms of @xmath1114 via eqs.([eq : spfra01hl ] ) and ( [ eq : ball004jhm ] ) ] can be neglected , so that this current becomes independent of @xmath1115 . the same then holds for the corresponding thermoballistic current , @xmath698 , as well as for the persistent current spin polarization @xmath1103 derived therefrom . the polarizations @xmath1146 are linear in @xmath1115 , and the coupled equations for @xmath1115 become linear . since @xmath1162 in the zero - bias limit , they can be expressed , using eq . ( [ eq : ball004jux ] ) , in the form @xmath1163 \breve{a}_{1 } +   v_{e } n_{+}(x_{1 } ) { \cal f}_{2}(x_{1 } ) \breve{a}_{2 }   \nonumber \\ & \ & \hspace{-4.0 cm } = p_{l } j -   \overstar{\mathfrak{j}}_{-}(x_{1 } ) , \label{eq:66e}\end{aligned}\ ] ] @xmath1164 \breve{a}_{2 }   \nonumber \\ & \ & \hspace{-4.0 cm } = p_{r } j -   \overstar{\mathfrak{j}}_{-}(x_{2 } ) . \label{eq:66ezg}\end{aligned}\ ] ] the current @xmath698 is proportional to @xmath133 , so that the quantities @xmath1115 are , as solutions of eqs . ( [ eq:66e ] ) and ( [ eq:66ezg ] ) , proportional to @xmath133 as well . hence , the relaxing thermoballistic spin - polarized current @xmath699 is also proportional to @xmath133 , so that the total current spin polarization @xmath1108 calculated from eq . ( [ eq:91xx ] ) is , in the zero - bias limit , independent of the total physical current @xmath133 . summing up , we obtain the total current spin polarization along the entire heterostructure , @xmath1108 , as follows . in the metal contacts , it is given by expressions ( [ eq:57lala ] ) and ( [ eq:57zzz ] ) , respectively , with @xmath1152 and @xmath1165 expressed in terms of @xmath1115 via eqs . ( [ eq:57zxl])([eq:68 ] ) . in the semiconductor , it is given by expression ( [ eq:91xx ] ) in terms of the persistent , @xmath698 , and relaxing , @xmath699 , thermoballistic spin - polarized currents , where the formfactors @xmath1166 entering the latter current are determined by the solution of the integral equation ( [ eq : spiacc02 ] ) for the spin accumulation function @xmath506 . the total density spin polarization @xmath1113 is obtained in an analogous way . this section deals with the application of the thermoballistic approach to various specific examples . the emphasis is on spin - polarized transport in heterostructures including nonmagnetic and magnetic semiconductors .      in considering the case of a homogeneous semiconductor at zero bias , we illustrate the thermoballistic formalism for a particularly simple example and , moreover , collect results needed for treating heterostructures involving homogeneous layers of nonmagnetic and magnetic semiconductors ( see secs . [ sec : fmnmshet ] and [ sec : dmsnmshe ] below ) .    for a homogeneous semiconducting layer at zero bias , the potential energy profiles @xmath497 are given by @xmath1167 @xmath1105 . specializing the development presented in sec . [ sec : resf ] for constant spin splitting to the case @xmath1168 , we find from eq . ( [ eq : inteq06opa ] ) the reduced @xmath1169 kernel @xmath1170 in the explicit form @xmath1171 . \label{eq : inteq06woft}\ ] ] from the corresponding solution @xmath957 of eq . ( [ eq : inteq04aha ] ) and the solution of the analogous equation for the function @xmath1018 , we then obtain for @xmath1172 and @xmath1173 , respectively , using eq . ( [ eq : inteq09wq ] ) , @xmath1174 for the resistance functions @xmath1175 of the homogeneous semiconductor ;  at @xmath935 and @xmath761 , on the other hand , we have @xmath1176 ;  see eqs . ( [ eq : inteq08 ] ) and ( [ eq : spiche05 ] ) .    for the reduced resistance @xmath967 , we then find from eq . ( [ eq : spiche15 ] ) , using eq . ( [ eq : inteq53cb ] ) , @xmath1177 and for the function @xmath988 from eq . ( [ eq : spiche18 ] ) , using eqs . ( [ eq : ballim04st ] ) , ( [ eq : inteq04lil ] ) , ( [ eq : inteq02ax ] ) , and ( [ eq : inteq02im ] ) , @xmath1178   \label{eq:66zora}\end{aligned}\ ] ] for @xmath1029 , whereas from eq . ( [ eq : spiche15ug ] ) , @xmath1179 . the spin accumulation function @xmath506 to be inserted in this expression is given by eqs . ( [ eq : ballim05s ] ) and ( [ eq : difeq20 ] ) .    inserting expressions ( [ eq:66zick ] ) and ( [ eq:66zora ] ) in eq . ( [ eq : spiche17zg ] ) and using eq . ( [ eq : spiche14 ] ) , we find from eq . ( [ eq : ball004jvx ] ) for the net total ballistic current in a homogeneous semiconductor at zero bias @xmath1180 @xmath1181 for @xmath1182 , @xmath1183 for @xmath1184 , and @xmath1185 for @xmath1186 , in which the dependence on @xmath506 present in @xmath967 and @xmath988 has cancelled out completely . at this point , it is appropriate to note that the dependence of the ballistic current @xmath746 on the end - point coordinates @xmath512 exhibited here contradicts the basic assumption of the prototype thermoballistic model as expressed by eq . ( [ eq : gendr001qal ] ) . now , using expressions ( [ eq:66jafz])([eq:66juhu ] ) in eq . ( [ eq : thebal01 ] ) , we find @xmath1187 i.e. , the total thermoballistic current is equal to the ( constant ) total physical current . further , using eq . ( [ eq : netbal06 ] ) , we have @xmath1188 for the persistent thermoballistic spin - polarized current , and hence from eq . ( [ eq:65 ] ) for the persistent current spin polarization @xmath1189 thereby retrieving the static spin polarization . turning now to the determination of the relaxing thermoballistic spin - polarized current @xmath699 in a homogeneous semiconductor at zero bias , we have from eq . ( [ eq : netbal02 ] ) for the corresponding ballistic current @xmath1190 \nonumber \\ \label{eq : netbal02ihk}\end{aligned}\ ] ] @xmath1191 . inserting this expression in eq . ( [ eq : thebal01 ] ) and subsequently using the derivative of eq . ( [ eq : difeq17 ] ) with respect to @xmath37 , we obtain @xmath699 in the form @xmath1192 [ see eq . ( 4.11 ) of ref .  ; the factor 2 appearing in the right - hand side of the latter equation again reflects the fact that the normalization of @xmath506 used there differs from that used in the present article ] . from eqs . ( [ eq:91xx ] ) , ( [ eq:66zyxv ] ) , and ( [ eq : netbal02fln ] ) , we now have for the total current spin polarization @xmath1193 [ note that owing to its normalization to the total thermoballistic current , the expression for @xmath1108 given by eq . ( 136 ) of ref .   differs from expression ( [ eq:66ze ] ) ;  see the remark following eq . ( [ eq:91xx ] ) above . ]    as to the density spin polarization in a homogeneous semiconductor , we have from eq . ( [ eq : ball004juy ] ) for the persistent part , @xmath1194 , of the joint total ballistic density @xmath1195 @xmath1196 for @xmath1182 , @xmath1197 for @xmath1184 , and @xmath1198 for @xmath1186 . for the relaxing part of the joint total ballistic density , @xmath1199 , we find from eq . ( [ eq : netbal02xy ] ) @xmath1200 so that @xmath1201 observing eq . ( [ eq:66trfc ] ) , we now use expressions ( [ eq:66kohj])([eq:66fsnb ] ) in eq . ( [ eq : thebal01 ] ) to obtain for the thermoballistic joint total density @xmath1202 with @xmath1203 given explicitly by eq . ( [ eq:66kohj ] ) . since @xmath1204 from eq . ( [ eq : ball004juy ] ) , we have for the persistent part of the joint ballistic spin - polarized density @xmath1205 and hence for the persistent density spin polarization @xmath1206 in agreement with the corresponding current spin polarization .    by inserting expression ( [ eq : netbal02fln ] ) for the corresponding current @xmath699 in the balance equation ( [ eq : ballim24new ] ) and using eq . ( [ eq : difeq18 ] ) for the spin accumulation function @xmath506 , the relaxing thermoballistic spin - polarized density @xmath1207 for a homogeneous semiconductor is readily obtained in the form @xmath1208 so that we have for the total density spin polarization @xmath1209 with @xmath701 given by eq . ( [ eq:66gvya ] ) . now , omitting in expression ( [ eq:66gvya ] ) for @xmath701 the term proportional to @xmath133 , so that @xmath1210 , we can use expression ( [ eq:66gxky ] ) for @xmath1113 in eq . ( [ eq:66ze ] ) to express the current spin polarization @xmath1108 in terms of @xmath1113 in the form @xmath1211 differentiating this equation and using eqs . ( [ eq : difeq18 ] ) and ( [ eq:66gxky ] ) , we then obtain @xmath1212 for the density spin polarization @xmath1113 in terms of @xmath1108 .    for the magnetoresistance of a homogeneous semiconductor at zero bias , we find @xmath1213 by inserting the zero - field limit of expression ( [ eq:66zick ] ) , @xmath1214 for @xmath1017 in eq . ( [ eq : sharv07lz ] ) . we now consider heterostructures formed of a _ homogeneous , nonmagnetic _ semiconducting ( nms ) layer and two _ ferromagnetic _ metal ( fm ) contacts . the study of spin - polarized transport in this kind of structure , both experimentally and theoretically , marked the beginning of semiconductor spintronics@xcite ( for recent surveys of the physics of semiconductor - based spintronic devices , see refs .  ) . for zero bias , we evaluate the position dependence of the current and density spin polarizations using the results of secs . [ sec : pollead ] , [ sec : match ] , and [ sec : homsem ] . for nonzero bias , we obtain the spin polarizations `` injected '' from the contacts into the semiconductor , i.e. , the polarizations generated inside the semiconductor in the vicinity of either fm / nms interface regardless of the influence of the opposite interface .      inside the semiconducting layer , we have from eq . ( [ eq:66ze ] ) with @xmath1215 and @xmath1216 , using expression ( [ eq : difeq20 ] ) for the spin accumulation function @xmath506 , @xmath1217 , \nonumber \\ \label{eq:40c}\end{aligned}\ ] ] where @xmath24 is given by eq . ( [ eq : difeq24 ] ) . the coefficients @xmath1218 can be expressed via eqs . ( [ eq : difeq21 ] ) and ( [ eq : difeq22 ] ) , and using eq . ( [ eq : spfra01hl ] ) , in terms of the values @xmath1115 of the reduced spin accumulation function @xmath1114 on the contact sides of the interfaces . in order to determine the quantities @xmath1115 , we evaluate expression ( [ eq:40c ] ) on the semiconductor sides of the interfaces , obtaining @xmath1219 @xmath1220 here , we have introduced the total equilibrium density for zero spin splitting and zero bias , @xmath1221 which is constant inside the semiconducting layer . the coefficients @xmath1222 and @xmath41 in expressions ( [ eq:61a ] ) and ( [ eq:61abc ] ) are given by eqs . ( [ eq : difeq27 ] ) and ( [ eq : difeq28 ] ) , respectively .    using the continuity of @xmath1108 at the interfaces , eq . ( [ eq:66zara ] ) , we now equate expressions ( [ eq:57angi ] ) and ( [ eq:61a ] ) , and expressions ( [ eq:57guido ] ) and ( [ eq:61abc ] ) . [ when spin - selective interface resistances are included , expression ( [ eq:57angi ] ) for @xmath1158 is to be replaced with the more general expression obtained from using expression ( [ eq : karol ] ) for @xmath1152 in eq . ( [ eq:57lala ] ) , and analogously for @xmath1159 . ] for zero bias , when @xmath1157 , this results in a system of coupled linear equations for @xmath1115 [ see eqs . ( [ eq:66e ] ) and ( [ eq:66ezg ] ) ] , @xmath1223 \breve{a}_{1 } -   h \breve{a}_{2 } = \frac{2 j}{v_{e } n_{+}^{(0 ) } } p_{l }   , \label{eq:69}\ ] ] @xmath1224 \breve{a}_{2 } =   \frac{2   j}{v_{e } n_{+}^{(0 ) } } p_{r }   , \label{eq:69abc}\ ] ] where @xmath1225 the solutions of eqs . ( [ eq:69 ] ) and ( [ eq:69abc ] ) are found to be @xmath1226 p_{l } - h p_{r } \ }   , \label{eq:72abc}\ ] ] @xmath1227 p_{r } \ }   , \label{eq:72}\ ] ] where @xmath1228 [ g +   \tilde{g}_{r}^{(0 ) } ] - h^{2 }    . \label{eq:72ijk}\ ] ] expressions ( [ eq:72abc ] ) and ( [ eq:72 ] ) determine the values of the reduced spin accumulation function , @xmath1160 and @xmath1161 in terms of the current @xmath133 , of the polarizations @xmath1229 and @xmath1230 in the left and right ferromagnetic contact , respectively , and of material parameters , such as the conductivities @xmath1231 and the spin diffusion lengths @xmath1232 of the contacts [ via @xmath1233 $ ] and the momentum relaxation length @xmath7 and the spin relaxation length @xmath633 in the semiconducting sample as well as the sample length @xmath28 ( via @xmath1222 and @xmath41 ) and the ( constant ) total equilibrium electron density @xmath1234 . since the quantities @xmath1115 are proportional to the total current @xmath133 , the current spin polarization @xmath1108 is independent of @xmath133 , while the density spin polarization @xmath1113 is proportional to @xmath133 . the spin polarizations along the entire heterostructure are now obtained as follows . inside the semiconductor , the current spin polarization @xmath1108 is given by expression ( [ eq:40c ] ) , with @xmath1218 calculated from the solutions @xmath1115 of eqs . ( [ eq:72abc ] ) and ( [ eq:72 ] ) . the corresponding density spin polarization @xmath1113 is then readily obtained from eq . ( [ eq:66pgmq ] ) . the expressions for @xmath1108 in the ferromagnetic contacts are provided by eqs . ( [ eq:57lala ] ) and ( [ eq:57zzz ] ) , respectively , where the quantities @xmath1152 and @xmath1165 are calculated from eq . ( [ eq : karol ] ) and from its analogue for @xmath1165 , respectively . we do not write down the density spin polarizations in the ferromagnets , but only mention that they do not , in general , match the polarizations @xmath1235 and @xmath1236 on the semiconductor sides of the interfaces .    in refs . and , we have presented results of detailed calculations for @xmath1108 at zero bias for typical parameter values of fm / nms / fm heterostructures , emphasizing the dependence of the polarization on the momentum and spin relaxation lengths , @xmath7 and @xmath633 , and on the spin - selective interface resistances @xmath1237 ( see sec .  [ sec : pollead ] ) . we define the `` injected spin polarization '' as the polarization inside the semiconductor in the vicinity of the interface , e.g. , at @xmath912 , generated by the bulk polarization @xmath1229 of the left ferromagnet regardless of the influence of the right ferromagnet . more explicitly , we define the injected current and density spin polarizations as the polarizations @xmath1238 and @xmath1235 , respectively , in the limit of infinite sample length , @xmath1239 . the injected spin polarizations at @xmath1240 provide the initial values of the left - generated polarizations in the semiconductor , which propagate into the region @xmath1172 while being degraded by the effect of spin relaxation . we now consider the injected spin polarizations for electron transport in a homogenenous semiconductor , driven by an external electric field , i.e. , for a ( spin - degenerate ) potential energy profile of the form ( [ eq : difeq01 ] ) , with the parameter @xmath1241 defined by eq . ( [ eq : difeq01erd ] ) . in order to obtain the spin accumulation function @xmath506 for this case , one has to solve eq . ( [ eq : difeq09 ] ) [ in general , numerically ] under the asymptotic condition @xmath1242 to determine the decay length @xmath1243 , we solve eq . ( [ eq : difeq09 ] ) in the range @xmath1244 in which the @xmath37-dependence of the coefficient functions @xmath1245 , @xmath1246 , and @xmath1063 arising from the function @xmath1247 can be disregarded . this yields @xmath1248^{1/2 }   . \label{eq:80}\ ] ] since @xmath1249 , @xmath1243 is a real number . the injected current spin polarization is obtained from eq . ( [ eq:91xx ] ) with @xmath772 . we calculate the relaxing thermoballistic spin - polarized current at the interface , @xmath1250 , from the relation @xmath1251 , \label{eq:80ijhb}\ ] ] where the function @xmath1252 is given by eq . ( [ eq : spiacc04 ] ) . this relation can be shown to hold , for arbitrary sample length @xmath28 and potential energy profiles @xmath564 , by ( i ) evaluating @xmath1250 from eq . ( [ eq : thebal01 ] ) with @xmath1253 [ see eq . ( [ eq : netbal02 ] ) ] and ( ii ) eliminating from the resulting expression the boundary value @xmath940 by using the integral equation ( [ eq : spiacc02 ] ) for @xmath935 . for the case considered here , we write down the integral equation ( [ eq : difeq04 ] ) [ with @xmath1254 for @xmath935 and take the limit @xmath1255 to obtain the relation @xmath1256 where @xmath1257 relation ( [ eq:80lcvs ] ) fixes the normalization of the spin accumulation function @xmath506 in terms of the boundary value @xmath939 . then , using this relation in eq . ( [ eq:80ijhb ] ) with @xmath1258 , we can express the current @xmath1250 in the form @xmath1259 where the quantity @xmath1260 is independent of the normalization of @xmath506 , and @xmath1261 } . \label{eq:82jpfc}\ ] ] in obtaining eq . ( [ eq:81 ] ) , we have used eq . ( [ eq : ball004jum ] ) , with @xmath769 , as well as eq . ( [ eq : spfra01hl ] ) .    for the injected current spin polarization , we now have from eq . ( [ eq:91xx ] ) @xmath1262 which , by continuity , is equal to the polarization @xmath1263 at the contact side of the interface . then , equating the right - hand sides of eqs . ( [ eq:57angi ] ) and ( [ eq:81ijk ] ) , we obtain @xmath1264 this nonlinear equation for @xmath1160 is to be solved for given values of the parameters @xmath1241 , @xmath1229 , @xmath1265 , @xmath1266 , @xmath7 , and @xmath633 . we now turn to the calculation of the injected density spin polarization from eq . ( [ eq:91olsb ] ) , where @xmath1267 and @xmath1268 in the present case . proceeding similarly as for the current @xmath1250 , we obtain the relaxing thermoballistic spin - polarized density at the interface , @xmath1269 , in the form @xmath1270 where @xmath1271 for the persistent total thermoballistic density at the interface , @xmath1272 , we find by evaluating eq . ( [ eq : thebal01 ] ) with @xmath1273 [ see eq . ( [ eq : ball004juy ] ) ] and taking the limit @xmath1239 @xmath1274 where the quantity @xmath1275 }    \label{eq:82erkg}\ ] ] has been expressed in terms of the average chemical potential @xmath503 using eq . ( [ eq : eqden03zk ] ) . the injected density spin polarization now follows as @xmath1276 where the boundary value @xmath1160 of the reduced spin accumulation function is again to be determined by solving eq . ( [ eq:69rst ] ) .    the dependence of the injected current spin polarization , @xmath1238 , on the electric - field parameter @xmath1241 , for @xmath7-values ranging from the drift - diffusion to the ballistic regime and for zero as well as nonzero interface resistances @xmath1277 , has been studied numerically in ref .  . in the drift - diffusion regime , when @xmath727 and @xmath1278 , the spin accumulation function @xmath506 is given by the exponentially decreasing solution of eq . ( [ eq : difeq16 ] ) , @xmath1279 @xmath1280 , with the field - dependent spin diffusion length @xmath1281 given by @xmath1282 where @xmath1078 is the spin diffusion length defined by eq . ( [ eq : ballim26 ] ) . from eqs . ( [ eq:82 ] ) and ( [ eq:82pqr ] ) , respectively , we then have @xmath1283 and @xmath1284 and hence from eqs . ( [ eq:81ijk ] ) and ( [ eq:81zyx ] ) , respectively , @xmath1285 and @xmath1286 for the injected current and density spin polarizations in the drift - diffusion regime . in deriving the right - hand equation ( [ eq:81jslc ] ) , we have used eq . ( [ eq : diff006hsa ] ) , with @xmath1287 , @xmath1288 , and @xmath1289 , as well as eq . ( [ eq:82jpfc ] ) , to express the total current @xmath133 in the form @xmath1290 where @xmath1291 is the ( spin - summed ) conductivity of the semiconductor [ see eq . ( [ eq : drude006f ] ) ] . the right - hand equation ( [ eq:81pfqn ] ) has been obtained by using eq . ( [ eq : ballim20kf ] ) in eq . ( [ eq:82erkg ] ) . the boundary value @xmath1160 is determined by the equation @xmath1292 which follows from eq . ( [ eq:69rst ] ) by replacing its right - hand side with the right - hand side of eq . ( [ eq:81jslc ] ) and inserting expression ( [ eq:81kalo ] ) for @xmath133 . the effect of external electric fields on the current spin polarization injected at fm / nms interfaces has been studied within the standard drift - diffusion approach by yu and flatt@xcite . comparing our description to that of these authors , we find that the field - dependent spin diffusion length @xmath1281 given by eq . ( [ eq:80hui ] ) agrees with the `` up - stream '' spin diffusion length @xmath1293 given by eq . ( 2.23b ) of ref .  , provided the `` intrinsic '' spin diffusion length @xmath475 of that reference is identified with the spin diffusion length @xmath1294 of the present work . then , identifying in eq . ( 3.5 ) of ref . ( with the interface resistances set equal to zero ) the spin - summed conductivity of the semiconductor , @xmath1295 , with @xmath1296 , and the spin injection efficiency at the interface , @xmath1297 , with @xmath1298 , and using eq . ( [ eq:57xyz ] ) , we observe the equivalence of the former equation with eq . ( [ eq:69hlbv ] ) . consequently , the injected current and density spin polarizations of either work are actually identical .    for arbitrary @xmath7 , we now consider the injected current spin polarization in the zero - bias limit , when @xmath1299 . here , the spin accumulation function @xmath506 is given by the exponentially decreasing solution of eq . ( [ eq : difeq18 ] ) , @xmath1300 @xmath1280 , where the generalized spin diffusion length @xmath475 is given by eq . ( [ eq : difeq19 ] ) . from eq . ( [ eq:82 ] ) , we then have @xmath1301 with @xmath24 given by eq . ( [ eq : difeq24 ] ) , and hence from the zero - bias limit of eq . ( [ eq:69rst ] ) @xmath1302 where @xmath1303 is the sharvin interface conductance [ see eq . ( [ eq : sharv05kl ] ) ] . inserting this in eq . ( [ eq:81ijk ] ) , we find @xmath1304 i.e. , for zero bias , the injected current spin polarization is independent of the total current @xmath133 .    in the drift - diffusion regime , when @xmath727 , we have @xmath1305 so that @xmath1306 reduces to @xmath1307 where @xmath1308 with @xmath1296 defined by eq . ( [ eq:75zqsp ] ) , and @xmath1078 by eq . ( [ eq : ballim26 ] ) . the quantity @xmath1309 is seen to be the semiconductor analogue of the `` interface conductance '' @xmath1265 characterizing the ferromagnet [ see eq . ( [ eq:57xyz ] ) ] . for typical values of the parameters of the fm / nms / fm system ( see , e.g. , ref .  ) , the values of @xmath1265 exceed those of @xmath1309 by several orders of magnitude . this indicates a `` conductance mismatch '' between ferromagnet and semiconductor,@xcite which gives rise to very low values of the injected spin polarizations at fm / nms interfaces . [ to remedy this mismatch , one may introduce spin - selective interface resistances , for example , in the form of tunneling barriers.@xcite ]    in the ballistic regime , when @xmath1310 , we have @xmath1311 so that @xmath1312 here , the sharvin interface conductance @xmath1313 takes the place of the quantity @xmath1309 in eq . ( [ eq:75 ] ) . as @xmath1313 is proportional to the density @xmath1266 , eq . ( [ eq:79y ] ) yields values for @xmath1238 close to the bulk spin polarization of the feromagnet , @xmath1229 , if @xmath1266 is sufficiently high . for typical parameter values,@xcite however , the high donor densities needed to obtain the required high electron densities implies very small values of the momentum relaxation length @xmath7 , such that ballistic transport is excluded as the dominating transport mechanism . on the other hand , for densities so low that ballistic transport prevails , the injected spin polarization is confined to very small values . this result corroborates previous estimates@xcite according to which spin injection is suppressed even in the ballistic regime unless spin - selective interface resistances are introduced .      as an example of particular interest from the point of view of applications , we now consider spin - polarized electron transport in heterostructures involving diluted magnetic semiconductors ( dms ) in their paramagnetic phase ( see , e.g. , refs .   and ) . in structures of this kind , a nonmagnetic semiconducting ( nms ) layer is sandwiched between two dms layers which , in turn , are enclosed between nonmagnetic metal contacts ( see fig .  [ fig:7 ] ) .    in a complete thermoballistic description , the semiconductor part of a dms / nms / dms structure should be treated as a _ single _ sample . then , the ballistic intervals @xmath327 $ ] covering the sample may contain one or both of the dms / nms interfaces , at which the potential energy profiles must be expected to change abruptly . the same holds for the material parameters , in particular the momentum and spin relaxation lengths , so that a description in terms of position - dependent parameters would become necessary . here , we adopt a simplified treatment by assuming the different layers in a dms / nms / dms heterostructure to be _ homogeneous _ and requiring the interfaces to act as _ fixed points _ of local thermodynamic equilibrium . this allows us to apply the thermoballistic description separately to the different layers ( the potential energy profiles are then , in general , _ discontinuous _ at the interfaces ) . for each layer , we evaluate the spin accumulation function and the zero - bias current spin polarization for a homogeneous semiconductor ( see sec . [ sec : homsem ] ) , and subsequently match the current spin polarization at the interfaces to obtain its full position dependence as well as the magnetoresistance . heterostructures of the kind depicted schematically in fig . [ fig:7 ] are parametrized here by attaching labels @xmath1314 to quantities referring to the left dms layer , the nms layer , and the right dms layer , respectively . the positions of the interfaces ( including the interfaces between the dms layers and the contacts ) are denoted by @xmath1315 @xmath1316 ; this notation deviates from the one used in the preceding sections .    inside the left and right metal contacts , the position dependence of the current spin polarization is given by eqs . ( [ eq:57lala ] ) and ( [ eq:57zzz ] ) , respectively [ with @xmath70 in the latter equation replaced with @xmath1317 , with @xmath1318 [ see eq . ( [ eq : spfra01hl ] ) for @xmath1319 and @xmath1320 , @xmath1321 for @xmath1121 and @xmath1322 .    inside the semiconducting part of the heterostructure , @xmath1108 is found in terms of the static spin polarizations @xmath1323 of the three layers , and of the boundary values of the spin accumulation function @xmath506 at the interface positions @xmath1315 , @xmath1324 , which are all points of local thermodynamic equilibrium . [ the interface positions @xmath1325 are points of local thermodynamic equilibrium in the same sense as are the interface positions @xmath1326 , except that there the interface is between metal contact and dms , while here it is between dms and nms . ] in each layer @xmath53 , the function @xmath506 has the sharvin discontinuities @xmath1327 and @xmath1328 at the interface positions @xmath1329 and @xmath1330 , respectively [ see eqs . ( [ eq : difeq25 ] ) and ( [ eq : difeq26 ] ) ] . using now , for zero bias , eq . ( [ eq:66ze ] ) together with eq . ( [ eq : difeq20 ] ) separately in each layer , we have @xmath1331   \nonumber \\ \label{eq : a2}\end{aligned}\ ] ] for @xmath1332 @xmath1333 . here , @xmath1334 with @xmath1335 and @xmath1336 defined by eqs . ( [ eq : difeq14c ] ) and ( [ eq : difeq19 ] ) in terms of the momentum relaxation length , @xmath1337 , and spin relaxation length , @xmath1338 , of layer @xmath53 . further , @xmath1339 \label{eq : a4}\ ] ] and @xmath1340 , \label{eq : a5}\ ] ] where @xmath1341 , \label{eq : a7}\ ] ] and @xmath1342 is the thickness of layer @xmath53 . [ owing to a difference in normalization , expression ( [ eq : a2 ] ) for @xmath1108 differs from the corresponding expression given by eq . ( 141 ) of ref . ;  see the remark following eq . ( [ eq:66ze ] ) above . ]    in order to relate the quantities @xmath1343 to the total physical current @xmath133 and the parameters of the contact - semiconductor system , we invoke the continuity of the current spin polarization @xmath1108 at all interfaces , setting @xmath1344 ( @xmath1345 ) in eq . ( [ eq : a2 ] ) . from eqs . ( [ eq : a1 ] ) and ( [ eq : a2 ] ) , we then obtain a set of four coupled linear equations for @xmath1343 , @xmath1346 @xmath1347 @xmath1348 @xmath1349 here , @xmath1350 where @xmath1351 }   \label{eq : a853}\ ] ] is the ( constant ) total equilibrium electron density in layer @xmath53 , and @xmath1352 are the band offsets . further , @xmath1353 , \label{eq : a12}\ ] ] with @xmath1354 the system of equations ( [ eq : a8])([eq : a11 ] ) can be easily solved , so that the complete position dependence of the zero - bias current spin polarization is obtained in explicit form . we now specialize to the case of a _ symmetric _ heterostructure , for which the parameters of the right dms layer and metal contact are identical to those of the left dms layer and metal contact . assuming arbitrarily high conductivities of the contacts , @xmath1355 , so that @xmath1356 , we have from eqs .  ( [ eq : a1 ] ) and ( [ eq : a987 ] ) @xmath1357 such that @xmath1358 remains nonzero and finite . then , from eqs . ( [ eq : a9 ] ) and ( [ eq : a10 ] ) with @xmath1359 and @xmath1360 , @xmath1361 where @xmath1362 hence , from eqs . ( [ eq : a8 ] ) and ( [ eq : a11 ] ) , @xmath1363 here , the dms parameters have been labeled by d " , and the nms parameters by thus , in the symmetric case , the zero - bias current spin polarization @xmath1108 is completely determined by the quantity @xmath1161 . explicitly , we obtain , setting @xmath1364 ( where the superscript `` @xmath1365 '' refers to the metal contacts ) in eq . ( [ eq : a1 ] ) and using eqs . ( [ eq : a987 ] ) and ( [ eq : a16 ] ) , @xmath1366 if @xmath1121 and @xmath1322 , respectively . further , from eq . ( [ eq : a2 ] ) , we have , using eqs . ( [ eq : a6])([eq : a7 ] ) along with eqs . ( [ eq : spfra01hl ] ) , ( [ eq:61ujo ] ) , and ( [ eq : a15 ] ) , @xmath1367 \rule{0mm}{5 mm } \right\ } ,   \label{eq : aa2}\end{aligned}\ ] ] if @xmath1368 and @xmath1369 , respectively , and @xmath1370   \nonumber \\ & \ & \times \cosh \bm { ( } [ x - ( x_{2 } + x_{3})/2]/l_{n } \bm { ) }   ,   \label{eq : aa3}\end{aligned}\ ] ] if @xmath1371 .    in refs .   and , we have presented numerical results for @xmath1108 for values of the momentum relaxation length @xmath7 ranging from the drift - diffusion regime to the ballistic regime .      the relative magnetoresistance @xmath1013 of the dms / nms / dms heterostructure follows by inserting in eq . ( [ eq : sharv07xbs ] ) the reduced resistance @xmath967 and the corresponding zero - field resistance @xmath1017 obtained by summing up the respective contributions @xmath1372 and @xmath1373 of the ( homogeneous ) layers @xmath53 . these contributions are obtained from the ( appropriately labeled ) expressions ( [ eq:66zick ] ) and ( [ eq:66tfk ] ) by multiplication with a factor @xmath1374 , such that @xmath967 and @xmath1017 are normalized in conformance with the definition ( [ eq : inteq04scp ] ) for a single sample extending from @xmath69 to @xmath1375 . specializing immediately to the case of the symmetric structure considered above , we have , defining @xmath1376 @xmath1377 and using eqs . ( [ eq : a14 ] ) , @xmath1378 ( we have omitted here the overall factor @xmath1379 which drops out when @xmath1013is formed ) , from which @xmath1017 follows by setting @xmath1215 and @xmath1216 . then , using eq . ( [ eq : a15 ] ) , we obtain @xmath1380 here , we note that in ref . , in contrast to its definition in terms of the zero - field resistance @xmath1017 via eq . ( [ eq : sharv07xbs ] ) , the relative magnetoresistance @xmath1013 has been defined with respect to the _ spin - equilibrium _ resistance obtained by setting @xmath1381 in expression ( [ eq:66fdlv ] ) . the latter definition does not seem to correspond to a genuine magnetoresistance , and so one should not attach quantitative significance to the numerical results for @xmath1013 shown in fig .  4 of ref .  .    in the limit of low external magnetic field , when the static spin polarization @xmath1382 depends linearly on the field strength , we have from eq . ( [ eq : a17 ] ) , keeping terms of order @xmath1383 , @xmath1384   -   { \cal s}_{d}/2 \ } p^{2}}{{\cal s}_{d } + e^{- \beta \delta_{dn } }   { \cal s}_{n}/2 } . \label{eq : a17pdfg}\ ] ] while based on assumptions that differ from , and are more general , than those underlying the  two - band model  for the ( transverse ) magnetoresistance,@xcite expression ( [ eq : a17pdfg ] ) exhibits the quadratic dependence on the field strength characterizing the latter model in the low - field limit . considering first the thermoballistic description of spin - polarized transport in dms / nms / dms heterostructures in the drift - diffusion regime , we can compare our results to those of ref . obtained within the standard drift - diffusion approach .    in the drift - diffusion regime , when @xmath1385 and @xmath1386 , we have @xmath1387 and @xmath1388 [ see eqs . ( [ eq : ballim26 ] ) , ( [ eq : difeq14c ] ) , ( [ eq : difeq19 ] ) , and ( [ eq : a6 ] ) ] . hence , from eqs . ( [ eq : a12 ] ) and ( [ eq : a13 ] ) , @xmath1389 and @xmath1390 respectively . then , specializing to the symmetric heterostructure , we find for the quantity @xmath1391 defined by eq . ( [ eq : a4711 ] ) @xmath1392   . \label{eq : a23}\end{aligned}\ ] ] introducing the ( spin - summed ) conductivity @xmath1393 for layer @xmath53 @xmath1394 , @xmath1395 [ see eq . ( [ eq : drude006f ] ) ] , with @xmath1396 @xmath1397 , we can rewrite expression ( [ eq : a23 ] ) in the form @xmath1398   . \label{eq : a23cayk}\end{aligned}\ ] ] now , inserting expression ( [ eq : a23cayk ] ) in eq . ( [ eq : a15 ] ) and using the resulting expression for @xmath1161 in eq . ( [ eq : a2 ] ) , we find that the values of the zero - bias current spin polarization @xmath1108 at the positions @xmath1326 , @xmath1325 , and @xmath1399 , respectively , agree with those given by eqs . ( 17 ) , ( 16 ) , and(18 ) of ref .  , if we identify in the latter equations the spin - flip lengths @xmath1400 and @xmath1401 with @xmath1402 and @xmath1403 , respectively , and the layer thicknesses @xmath1404 and @xmath1405 with @xmath1406 and @xmath1407 , respectively .    for the magnetoresistance in the drift - diffusion regime , we obtain from eq . ( [ eq : a17 ] ) , setting @xmath1408 and inserting expression ( [ eq : a23cayk ] ) for @xmath1391 , @xmath1409 + p^{2}/\mathfrak{q}_{dn } } { s_{d}/ \sigma_{+}^{(0;d ) } + s_{n}/2   \sigma_{+}^{(0;n ) } } , \label{eq : a17zglv}\ ] ] where @xmath1410 is the conductivity for zero spin splitting , which is given by eq . ( [ eq : a16a ] ) with @xmath1411 , and @xmath1412 then , identifying the parameters as above , we find that @xmath1013 agrees with expression ( 13 ) of ref .  .    turning now to the ballistic regime , when @xmath1413 and @xmath1414 , we have @xmath1415 and @xmath1416 [ see eqs . ( [ eq : ballim26 ] ) , ( [ eq : difeq14c ] ) , ( [ eq : difeq19 ] ) , and ( [ eq : a6 ] ) ] , so that @xmath1417 and @xmath1418 for the symmetric heterostructure , we then find from eq . ( [ eq : a4711 ] ) @xmath1419 e^{\beta \delta_{dn } } . \label{eq : a21a}\ ] ] with this used in eq . ( [ eq : a15 ] ) for @xmath1161 as well as in eq . ( [ eq : a16 ] ) for @xmath1420 , we obtain the position dependence of the zero - bias current spin polarization from eqs . ( [ eq : a1 ] ) and ( [ eq : a2 ] ) .    for the magnetoresistance in the ballistic regime , we have from eqs . ( [ eq : a17 ] ) , setting @xmath1421 and inserting expression ( [ eq : a21a ] ) for @xmath1391 , @xmath1422   e^{\beta \delta_{dn } } \ } } { 1 + e^{- \beta \delta_{dn}}/2 } .   \label{eq:22}\ ] ] depending on the values assigned to the different parameters in this expression , the ballistic ( relative ) magnetoresistance is positive or negative . in this article , we have presented a comprehensive survey of the thermoballistic approach to charge carrier transport in semiconductors . the principal aim has been to develop the basic physical concept underlying this approach in detail , and to give a coherent exposition of the ensuing formalism , which unifies , and partly modifies , generalizes , and corrects , the formal developments presented in our previous publications .    to make the presentation self - contained and easy to follow , we have proceeded step by step , starting with an account of drude s model as the origin of all semiclassical transport models . we then reviewed the standard drift - diffusion and ballistic transport models , basic features of which have been adopted to shape the thermoballistic description of carrier transport in terms of averages over random configurations of ballistic transport intervals . the contributions of the individual ballistic intervals to the total carrier current are governed by collision probabilities involving the carrier mean free path , or momentum relaxation length , as the determining parameter of the thermoballistic concept . this concept finds a first concrete expression in the prototype thermoballistic model , which is based on the simplifying assumption of current conservation across the points of local thermodynamic equilibrium linking ballistic intervals . the implementation of the prototype model results in a current - voltage characteristic containing a reduced resistance , which can be expressed explicitly in terms of the parameters of the semiconducting system .    in the full thermoballistic concept , current conservation across the equilibrium points is abandoned , and position - dependent , total and spin - polarized thermoballistic currents and densities are introduced in terms of an average chemical - potential function and a spin accumulation function related to the spin splitting of the chemical potential . the algorithms for determining these dynamical functions follow from two physical conditions . first , the average of the total thermoballistic current over the length of the semiconducting sample ( as well as over the range between either end and an arbitrary point of local thermodynamic equilibrium ) is required to equal the conserved physical current , whereby one is able to set up a scheme for obtaining the average chemical potential . the explicit calculation of this function is implemented in terms of resistance functions , for which volterra - type integral equations are derived . second , spin relaxation is assumed to act only inside the ballistic transport intervals . as a result , the thermoballistic spin - polarized current and density are connected by a spin balance equation , from which one obtains an inhomogeneous fredholm - type integral equation for the spin accumulation function . in the general case of arbitrarily shaped potential energy profiles , considerable numerical effort is needed for solving the integral equations for the resistance functions and the spin accumulation function . in a number of important special cases , however , these equations reduce to a form which greatly facilitates their solution . examples are the equations for the resistance functions when the spin splitting of the band edge profile is independent of position , or the equation for the spin accumulation function for a homogeneous sample in an external electric field . in the latter case , the integral equation can be converted into an easily tractable second - order differential equation . for homogeneous semiconductors at zero bias , the solutions of the equations for the dynamical functions can be obtained in closed form throughout .    for the purpose of demonstrating the potentialities of the thermoballistic approach in present - day semiconductor and spintronics research , we have summarized in this article the treatment of a number of specific examples . the prototype model is employed to describe electron transport across potential energy profiles exhibiting an arbitrary number of barriers , where effects of tunneling and degeneracy are included . this approach proves to be of particular relevance to the description of grain boundary effects in electron transport in polycrystalline semiconductors and has already been applied with promising results in the analysis of experimental data . the full thermoballistic approach is used in the treatment of spin - polarized transport in heterostructures . for the prototype problem of semiconductor spintronics , _ viz . _ , the injection of spin polarization from ferromagnetic contacts into a semiconducting sample , the thermoballistic description extends the standard drift - diffusion description so as to allow for arbitrary values of the momentum relaxation length . the same holds for transport in heterostructures composed of layers of diluted magnetic and nonmagnetic semiconductors at zero bias , where the position dependence of the current spin polarization as well as the magnetoresistance are obtained in closed form .    in our previous publications , we have presented results of explicit calculations for a variety of specific cases . these calculations were mostly of exploratory character , with the aim to reveal qualitative trends in the parameter dependence of the relevant transport properties . in future work , emphasis should be placed on the application of the thermoballistic approach in quantitative studies , as demanded for the analysis of specific experimental results . for these studies to become successful , the careful evaluation of the potential energy profiles is prerequisite , and advanced numerical techniques for solving the integral equations for the dynamical functions are to be employed . with these goals achieved , the thermoballistic approach will certainly prove useful as a practical tool , beyond its basic theoretical relevance as the bridge between the drift - diffusion and ballistic descriptions of charge carrier transport in semiconductors . we are indebted to the board of directors and the staff of the helmholtz - zentrum berlin fr materialien und energie for generously granting access to its premises and facilities beyond the date of our retirement , thereby allowing us to continue and complete our research work on charge carrier transport in semiconductors . p.  drude , ann . ( leipzig ) * 306 * , 566 ( 1900 ) [ reprinted in _ ostwalds klassiker der exakten wissenschaften _ , vol .  298 , edited by h.  t.  grahn and d.  hoffmann ( verlag harri deutsch , frankfurt / m . , 2006 ) ] ;  * 308 * , 369 ( 1900 ) . a.  sommerfeld and h.  bethe , in _ handbuch der physik _ , vol .  24/2 , second edition , edited by h.  geiger and k.  scheel ( springer , berlin , 1933 ) [ reprinted in _ heidelberger taschenbcher _ , vol .  19 ( springer , berlin , 1967 ) ] . t.  weis , r.  lipperheide , and u.  wille , in _ proc . 2nd world conf . photovoltaic solar energy conversion _ , edited by j.  schmid , h.  ossenbrink , p.  helm , h.  ehmann , and e.  d.  dunlop ( european comm . joint res . centre , ispra , 1998 ) , p.  1438 . we use the term `` semiclassical '' essentially as defined in chap . 12 of ref .  , where the band structure of the unperturbed system is assumed to be generated from quantum - mechanical calculations , whereas the effect of an externally applied electric field is treated classically ( the effect of a static external magnetic field is assumed here to be included in the band edge potential ) . we go beyond this scope when considering corrections to the ballistic currents due to tunneling and degeneracy .
a comprehensive survey is given of the thermoballistic approach to charge carrier transport in semiconductors . 
 this semiclassical approach bridges the gap between the drift - diffusion and ballistic ( `` thermionic '' ) models of carrier transport , whose validity is limited to the range of very small and very large values , respectively , of the carrier mean free path . 
 the physical concept underlying the thermoballistic approach , while incorporating basic features of the drift - diffusion and ballistic descriptions , constitutes a novel , unifying scheme . 
 it is based on the introduction of `` ballistic configurations '' defined by a random partitioning of the length of a semiconducting sample into ballistic transport intervals . 
 the points linking adjacent ballistic intervals are assumed to be points of local thermodynamic equilibrium characterized by a local chemical potential . 
 carriers thermally emitted at any such point are ballistically transmitted across either interval , while at the same time carriers transmitted from the equilibrium points next to it are `` absorbed '' at that point , i.e. , they are assumed to be instantaneously equilibrated there . during their transmission , 
 the carriers face , in general , potential energy barriers arising from internal and external electrostatic potentials in the sample . 
 the lengths of the ballistic intervals are stochastic variables , with associated probabilities given by the probabilities for carriers to traverse an interval without collisions with the scattering centers randomly distributed over the sample . 
 these probabilities are controlled by the carrier mean free path , whose magnitude is arbitrary . by averaging the ballistic carrier currents over all ballistic configurations , 
 a position - dependent thermoballistic current is derived , which is the key element of the thermoballistic concept and forms the point of departure for the calculation of various transport properties . 
 the present article starts out with a preparatory account of the standard drift - diffusion and ballistic transport models which form the cornerstones of the thermoballistic concept , and of a prototype model which paves the way for the fully developed form of that concept . in the main body of the article , a coherent exposition of the thermoballistic approach 
 is given within a general formulation that takes into account arbitrarily shaped , spin - split potential energy profiles and spin relaxation during the carrier motion across ballistic intervals . 
 the calculational procedures devised for implementing the thermoballistic concept are described . 
 specific examples relevant to present - day semiconductor and spintronics research are considered .