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nonlinear charge transport in weakly coupled , undoped , photo - excited , type - i semiconductor superlattices ( sls ) is well described by spatially discrete drift - diffusion equations.@xcite as in the much better known case of doped sls , nonlinear phenomena include formation and dynamics of electric - field domains , self - sustained oscillations of the current through voltage - biased sls , chaos , etc . experimentally , the formation of static electric - field domains in undoped , photo - excited sls was already reported many years ago.@xcite the first experimental observation of dynamical aspects of domain formation in undoped , photo - excited sls such as self - sustained oscillations of the photo - current were reported by kwok _ _ et al.__@xcite due to the excitation condition , the oscillations were damped . subsequently , undamped self - sustained oscillations of the photo - current in undoped sls were observed for a type - ii gaas / alas@xcite and for a direct - gap gaas / alas sl.@xcite tomlinson _ _ et al.__@xcite reported the detection of undamped photo - current oscillations in an undoped gaas / al@xmath0ga@xmath1as sl , where the transport is governed by resonant tunnelling between @xmath2 states . the evolution from a static state at low carrier densities to an oscillating state at higher carrier densities was demonstrated in an undoped , photo - excited sl by increasing the photo - excitation intensity.@xcite an investigation of the bifurcation diagrams for undoped , photo - excited sls showed the existence of a transition between periodic and chaotic oscillations.@xcite for a detailed review of the nonlinear static and dynamical properties of doped and undoped superlattices , see ref .  . previous theoretical studies of undoped photo - excited sls , including studies of bifurcation and phase diagrams,@xcite are based on a discrete drift model having a constant recombination coefficient.@xcite so far , there have been no reports on considering field - dependent recombination or the fact that the time scale of the electron - hole dynamics depends strongly on the optical excitation intensity . however , the consequences of including these effects for the dynamics of electric - field domains can be striking . in this work , we incorporate into the previously studied discrete model the dependence of the electron - hole recombination on the electric field and on the photo - excitation intensity using a straightforward model that takes into account the overlap integral between the electron and hole wave functions . the field - dependent recombination coefficient decreases with increasing electric field , which has far reaching consequences .    at high photo - excitation intensities , it is possible to find only one stable electric - field domain , not two as in the case of a constant recombination coefficient.@xcite in this case , self - sustained oscillations of the current ( ssoc ) may appear under dc voltage bias . the field profile during ssoc can exhibit nucleation of dipole waves inside the sample , the splitting of one wave into two , and the motion of the resulting waves in opposite directions . these dipole waves resemble the pulses in excitable reaction - diffusion systems such as the fitzhugh - nagumo model for nerve conduction @xcite and are quite different from field profiles for a constant recombination coefficient.@xcite in an excitable dynamical system , an appropriate finite disturbance of the unique stable fixed point may cause a large excursion in phase space before returning to the stable state . when diffusion is added , the resulting reaction - diffusion system may support wave fronts , pulses , and wave trains . it is also possible to find sl configurations at high photo - excitation intensities for which there exist no stable electric - field domains . in these cases , there are ssoc , whose corresponding field profiles are wave trains , comprising a periodic succession of dipole waves . these cases are similar to wave trains in oscillatory media such as those appearing in the fitzhugh - nagumo model in the presence of a sufficiently large external current.@xcite the equations governing nonlinear charge transport in weakly coupled , undoped , photo - excited , type - i sls are @xmath3 in these equations , the tunneling current densities between the quantum wells ( qws ) as well as between the sl and the contact regions are @xmath4 the voltage bias condition is @xmath5 here @xmath6 , @xmath7 , @xmath8 , @xmath9 , @xmath10 , @xmath11 and @xmath12 denote the electron charge , the average permittivity , the conductivity of the injecting contact , the doping density of the collecting contact , the average electric field , as well as the two - dimensional electron and hole densities of the @xmath13th period of the sl , respectively . equation  ( [ 1 ] ) corresponds to the averaged poisson equation . equation  ( [ 2 ] ) denotes ampre s law : the total current density @xmath14 equals the sum of the displacement current density and @xmath15 , the electron tunneling current density across the @xmath13th barrier that separates the quantum wells @xmath13 and @xmath16 . charge continuity is obtained by differentiating eq . ( [ 1 ] ) with respect to time and using eq .  ( [ 2 ] ) in the result . tunneling of holes is neglected so that only photogeneration and recombination of holes with electrons enter into eq . ( [ 3 ] ) . for high temperatures , i.e. @xmath17/m^*$ ] , where @xmath18 denotes boltzmann s constant , @xmath19 the temperature , @xmath20 the effective electron mass , and @xmath21 $ ] the order of magnitude of the electron density , the tunneling current is given by eq . ( [ 4 ] ) , in which @xmath22 and @xmath23 are functions of the electric field given in ref . and @xmath24 is the length of one sl period ( @xmath25 and @xmath26 denote the individual widths of the quantum well and barrier , respectively ) . even for lower temperatures , the qualitative behavior of the solutions of the discrete drift - diffusion model is similar to one of the more general tunneling current models described in ref .  . for @xmath27 and a constant recombination coefficient @xmath28 , eqs . ( [ 1])([4 ] ) describe the well known discrete drift model introduced in ref .  . the indices @xmath29 and @xmath30 represent the sl injecting and collecting contacts , and eq . ( [ 2 ] ) holds for them with the phenomenological currents given by eq . ( [ 5 ] ) . the total current density follows from the voltage bias condition in eq . ( [ 6 ] ) @xmath31 photogeneration and recombination are given by @xmath32 and @xmath33 respectively . here @xmath34 , @xmath35 , @xmath36 , @xmath37 , and @xmath38 denote the photo - excitation intensity , the frequency of the exciting photon , the refractive index , the intrinsic carrier density , and the speed of light , respectively . @xmath39 and @xmath40 correspond to the two- ( 2d ) and three - dimensional ( 3d ) absorption coefficients . the 2d absorption coefficient is proportional to the square of the modulus of the electron - hole overlap integral for a constant electric field @xmath41 ( cf . @xmath42 where @xmath43 denotes the energy of the bandgap at the @xmath2 point and @xmath44 as well as @xmath45 solve the stationary schrdinger equation inside one sl period , @xmath46 $ ] , for the electrons and holes , respectively . in this equation , the electric field @xmath41 is considered to be constant and @xmath47 , where the penetration length @xmath48 solves the cubic equation @xmath49}= 1\ ; , \nonumber\ ] ] and therefore @xmath48 depends self - consistently on the eigenvalue @xmath50 . for a fixed value of @xmath34 , the recombination coefficient decreases with increasing electric - field strength @xmath41 , as depicted in fig . [ fig1 ] .     vs electric field @xmath41 for the four configurations listed in table [ t1 ] . ] to study the model described by eqs . ( [ 1])([7 ] ) , it is convenient to render them dimensionless . for a fixed value of the photo - excitation intensity @xmath34 and a constant electric field @xmath51 , the stationary solution of eq . ( [ 3 ] ) is @xmath52 . we use the maximum values @xmath21=[p]=\sqrt{\gamma(i)/r(0,i)}$ ] to define typical values for @xmath11 and @xmath12 . the field @xmath53 at the maximum of the drift velocity is a typical value of the field when there are ssoc . therefore , we adopt it as the field unit @xmath54=f_{m}$ ] . similarly , @xmath55=v_{m}$ ] and , therefore , @xmath56=e[n]v_{m}/l$ ] and @xmath57=lv_{m}$ ] . there are two possible time scales , the first one being @xmath58 $ ] , which balances maxwell s displacement current with the current density in eq . ( [ 2 ] ) , and the second one @xmath59/\gamma(i)=1/\sqrt{\gamma(i ) r(0,i)}$ ] , which balances both sides of eq .  ( [ 3 ] ) . it is reasonable to choose the time unit as the longer of the two times @xmath60 and @xmath61 . we have chosen four representative gaas / al@xmath62ga@xmath63as sl configurations with 10  nm wells and 4  nm barriers . the configurations are i : @xmath64 and @xmath65 kw/@xmath66 ; ii : @xmath64 and @xmath67 kw/@xmath66 ; iii : @xmath68 and @xmath69 kw/@xmath66 ; and iv : @xmath70 and @xmath71 kw/@xmath66 . we assume a circular cross section with a diameter of 160  @xmath72 m , a photo - excitation intensity of 60 mw , a wavelength of 413  nm , and four different beam diameters yielding the previously listed values of the laser intensity . for most of the cases listed in table [ t1 ] , @xmath73 , and therefore we choose @xmath74=t_{n}$ ] . all scaled variables are listed in table [ t1 ] . .units used to achieve a set of equations with dimensionless variables . [ cols="^,^,^,^,^,^,^,^,^,^,^",options="header " , ]      vs electric field @xmath41 . for a fixed value of the total current density @xmath75 , there may be one or three zeros of @xmath76-@xmath75 depending on the al content @xmath77 . ] it is interesting to depict the phase plane corresponding to spatially uniform solutions of eqs . ( [ 10])([12 ] ) with @xmath78 , @xmath51 @xmath79 generically and for a fixed value of @xmath75 , the nullclines @xmath80 and @xmath81 intersect in one or three fixed points , depending on the al content @xmath77 in the barriers . at these fixed points , @xmath82 the function @xmath76 is depicted in fig . [ fig2 ] for the sl configurations listed in table [ t1 ] . the calculations for arbitrary values of aluminum content @xmath77 show that for @xmath83 , there are three fixed points of the system in eq . ( [ 17 ] ) , one on each of the three branches of @xmath84 , and the ratio of @xmath85 to the average current @xmath86 is sufficiently large . as we shall see later , some nonlinear phenomena occurring in these sls are quite similar to the ones observed in doped sls : static electric - field domains with domain walls joining the stable branches of @xmath84 , ssoc due to the recycling of pulses formed by two moving domain walls having a high - field region between them , etc . for @xmath87 , @xmath76 is either increasing for positive @xmath41 ( for @xmath88 ) or the ratio of @xmath85 to the average current @xmath89 is small ( for @xmath90 ) . for @xmath88 , there is a unique fixed point at @xmath91 , which , for an appropriate value of @xmath75 , may be located on any of the three branches of @xmath84 . if the fixed point is located on one of the two stable branches of @xmath84 for which @xmath92 has a positive slope , the dynamical system of eq . ( [ 17 ] ) is excitable , whereas it is oscillatory if the fixed point is located on the second branch of @xmath84 with @xmath93 . in an excitable dynamical system , an appropriate finite disturbance of the unique stable fixed point may cause a large excursion in phase space before returning to the stable state . when diffusion is added , the resulting excitable reaction - diffusion system may support a variety of wave fronts , pulses and wave trains.@xcite a dynamical system having an unstable fixed point and a stable limit cycle around it is oscillatory . again in the presence of diffusion , oscillatory systems may support different spatio - temporal patterns.@xcite an undoped photo - excited sl with excitable or oscillatory dynamics exhibits quite unusual phenomena . under dc current bias , it is possible to have pulses moving to the right or to the left and periodic wave trains . under dc voltage bias , these pulses and wave trains may give rise to a variety of sscos . similar phenomena are observed in the case @xmath90 for which @xmath76 has a shallow valley for a narrow interval of current densities . the behavior of a dc voltage - biased sl is quite different depending on its al content and photo - excitation intensity . for an al content smaller than 45% , @xmath76-@xmath75 in eq . ( [ j - f ] ) has a single zero for any value of @xmath75 , and the only stable states of the sl are stationary ones unless the photo - excitation intensity is sufficiently large ( cf . next section ) . let us assume that the al content is larger than 45% so that @xmath76-@xmath75 in eq . ( [ j - f ] ) may have three zeros for an appropriate range of @xmath75 values . in this case , the undoped sl behaves similarly to an @xmath94-doped sl.@xcite the most interesting limit is that of small photo - excitation intensity , i.  e. , @xmath95 .     and @xmath96 , respectively . the other parameter values correspond to configuration iv in table [ t1 ] and are @xmath97 , @xmath98 , @xmath99 , @xmath100 , and al content @xmath101.,title="fig : " ]   and @xmath96 , respectively . the other parameter values correspond to configuration iv in table [ t1 ] and are @xmath97 , @xmath98 , @xmath99 , @xmath100 , and al content @xmath101.,title="fig : " ]    for the time scale @xmath102 , eqs . ( [ 11 ] ) and ( [ 12 ] ) can be rewritten as @xmath103 in the limit of large photo - excitation intensities , @xmath104 , eq . ( [ 50 ] ) indicates that the @xmath105 do not depend on @xmath106 . in this case , eq . ( [ 49 ] ) corresponds to a sl doped with a density @xmath105 , and we may expect phenomena similar to the ones observed in an @xmath94-doped sl . in the opposite limit of small photo - excitation intensities , @xmath107 , @xmath108 is a slow scale . the @xmath12 are functions of @xmath109 determined by solving eq . ( [ 50 ] ) with a zero left hand side @xmath110 the resulting equation for @xmath111 is then obtained by inserting eq . ( [ 51 ] ) into eq . ( [ 49 ] ) @xmath112d(f_{i})\;. \label{52}\end{aligned}\ ] ] this equation is similar to the one describing the electric field in a doped sl , but now there are drift and diffusion terms which are nonlinear in the differences @xmath109 . under dc voltage bias , there are ssoc mediated by pulses of the electric field . the current density varies on a slow time scale , whereas the electric - field profile consists of a varying number of wave fronts joining the stable constant solutions of eq . ( [ 52 ] ) at the instantaneous value of the current.@xcite depending on the conductivity of the injecting contact @xmath8 , there are different types of ssoc due to the periodic generation of dipoles or monopoles at the injecting contact . if the curve @xmath113 intersects the bulk current - field characteristic curve @xmath76 before its maximum ( cf . [ fig2 ] ) , sscos due to recycling and motion of charge monopole waves ( moving charge accumulation layers ) appear as shown in fig . [ fig3 ] . if @xmath113 intersects @xmath76 after its maximum ( cf . [ fig2 ] ) , sscos due to recycling and motion of dipole waves are obtained , as depicted in fig . we have indicated in figs . [ fig3 ] and [ fig4 ] whether there are one or three uniform and time - independent ( constant ) solutions of eq . ( [ 17 ] ) , solving @xmath114 and @xmath81 for the instantaneous value of the current density @xmath115 during the ssoc .     and @xmath116 , respectively . the other parameter values are the same as the ones used for fig . [ fig3 ] corresponding to configuration iv in table [ t1 ] . , title="fig : " ]   and @xmath116 , respectively . the other parameter values are the same as the ones used for fig . [ fig3 ] corresponding to configuration iv in table [ t1 ] . , title="fig : " ] if @xmath87 , the phase plane in eq .  ( [ 17 ] ) may have only one fixed point located on any branch of the nullcline @xmath84 . for small photo - excitation intensities , the only stable state is a stationary one . however , for sufficiently large photo - excitation intensities , an infinite , dc current - biased sl may exhibit pulses moving downstream or upstream and also wave trains moving downstream.@xcite the counterpart of these stable solutions for a dc voltage - biased sl is very interesting and different from anything observed in an @xmath94-doped sl . our simulations correspond to sls with different al contents , whose current - field characteristics @xmath76 and injecting contact curve @xmath117 are shown in fig . [ fig2 ] . in all cases , sscos appear for an average bias roughly in the region of negative differential resistance ( ndr ) , where @xmath118 ( e.g. , @xmath8=0.5625 in fig .  [ fig2 ] ) .     and @xmath64 ( cf . [ fig2 ] ) . these oscillations correspond to having a finite wave train . the other parameter values correspond to configuration ii in table [ t1 ] and are @xmath97 , @xmath119 , @xmath120 , @xmath121 , and @xmath122.,title="fig : " ]   and @xmath64 ( cf . [ fig2 ] ) . these oscillations correspond to having a finite wave train . the other parameter values correspond to configuration ii in table [ t1 ] and are @xmath97 , @xmath119 , @xmath120 , @xmath121 , and @xmath122.,title="fig : " ]     and the other parameters correspond to configuration i in table [ t1 ] . , title="fig : " ]   and the other parameters correspond to configuration i in table [ t1 ] . , title="fig : " ]    when the conductivity of the injecting contact is such that @xmath113 intersects @xmath76 near the maximum thereof , it is possible to have sscos that are quite different from the ones appearing in @xmath94-doped sls . for an injecting contact conductivity @xmath123 ( cf . fig . [ fig2 ] ) , pulses ( charge dipoles ) may be triggered at the injecting contact , move toward the receiving contact , and cause sscos as shown in fig . for all the instantaneous values of @xmath14 during these sscos , there is only one constant solution of eq . ( [ 17 ] ) : most of the times this solution is on the second ndr branch of @xmath76 . only when @xmath14 is near its maximum value , the constant solution is on the third branch of @xmath76 . for a constant current density , the constant solution of eq . ( [ 17 ] ) is on the ndr branch , which implies that the system is oscillatory and that periodic wave trains are possible . the realization of wave trains for a long dc - biased sl are displayed in fig . [ fig6 ] : at any time during ssoc , there are only two fully developed pulses present in this sl . these pulses experience variations in their shape and velocity when they are generated or arrive at the contacts , but longer sls allow for realizations of wave trains , in which more pulses exist simultaneously inside the sl . [ fig2 ] ) . the other parameter values correspond to configuration iii in table [ t1 ] and are @xmath124 , @xmath125 , @xmath126 , @xmath127 , and @xmath128.,title="fig : " ]   ( cf . [ fig2 ] ) . the other parameter values correspond to configuration iii in table [ t1 ] and are @xmath124 , @xmath125 , @xmath126 , @xmath127 , and @xmath128.,title="fig : " ]    in the previous example , pulses always move downstream , from left to right . for slightly larger conductivity of the injecting contact ( @xmath129 in fig .  [ fig2 ] ) , fig . [ fig7 ] shows that two pulses are formed inside the sl and move with opposite velocities toward the contacts . these pulses are similar to the ones constructed above for the case of dc current bias ( with positive or negative velocity ) , except that the current changes slowly with time during the self - oscillation and the pulses accommodate their form to the instantaneous value of the current . note that the pulses are triggered inside the sl , not at the injecting contact . a pulse moving with positive speed has a long trailing region , in which the field increases as we move away from the pulse . however , there is a depletion layer near the injecting contact , in which the field decreases as we move away from the injecting contact . in a long , but finite sl , a local maximum is formed inside the sl , when the decreasing field near the contact meets the increasing field in the trailing region of the exiting pulse with positive speed . the current increases as the exiting pulse is absorbed by the receiving contact , until it surpasses a critical value . in this case , the local maximum of the field profile inside the sl is split , and two new pulses are created . the pulse closer to the injecting contact moves toward it with negative speed whereas the other pulse moves toward the receiving contact with positive speed . the upward moving pulse reaches the injecting contact and is absorbed there before the downward moving pulse arrives at the other contact . in this case , the field profile close to the injecting contact is quasi - stationary , and a local maximum of the field is formed when we match this region with the trailing region of the downward moving pulse . after the critical value of the current is reached , another pulse pair is nucleated , and the same process is periodically repeated . it is interesting to evaluate in some detail the process of nucleation and disappearance of pulses during these sscos . in the current vs time diagram in fig . [ fig7 ] , we distinguish different regions depending on the number of constant solutions of eq . ( [ 17 ] ) that exist for the corresponding instantaneous value of @xmath14 . in region a , there is only one constant solution on the third branch of @xmath76 . in region b , there is one constant solution on the third branch and two on the second branch of @xmath76 . in region c , there are three constant solutions , one on each branch of @xmath76 , whereas two of these solutions are on the second branch and one of the first branch of @xmath76 , if @xmath14 is in region d. there is only one constant solution located on the first branch of @xmath76 , if @xmath14 is in region e. for a constant voltage bias , a pulse moving upstream may be generated only if @xmath14 surpasses a critical value ( 1.007454 ) , which is located in region c and marked by an arrow . once generated , the upstream moving pulses persist for any instantaneous value of the current density . these ssoc are apparently weakly chaotic : we have calculated the corresponding lyapunov exponents and found that there is a single positive exponent with a rather small value of @xmath130@xmath131@xmath132 . [ fig2 ] ) . the other parameter values correspond to configuration i in table [ t1 ] and are @xmath133 , @xmath134 , @xmath135 , @xmath136 , and @xmath137.,title="fig : " ]   ( cf . [ fig2 ] ) . the other parameter values correspond to configuration i in table [ t1 ] and are @xmath133 , @xmath134 , @xmath135 , @xmath136 , and @xmath137.,title="fig : " ]    we have also observed sscos mediated by dipole waves that nucleate alternatively at two different qws of the sl as shown in fig .  [ fig8 ] . during these sscos , there is only one constant solution of eq . ( [ 17 ] ) for any instantaneous value of @xmath14 . this solution is either on the first or the second branch of @xmath76 . at about @xmath138 , where @xmath14 reaches its global maximum , two dipole waves are nucleated at two different qws . they become fully developed pulses and move with positive speed toward the receiving contact . when the first one arrives there , the current increases so that the corresponding constant solution of eq . ( [ 17 ] ) is on the second branch of @xmath76 . in this case , a small dipole wave is nucleated at the local field maximum , where the tail of the first dipole meets the depletion layer near the injecting contact . the small dipole wave never grows into a fully developed field pulse , and it continues advancing , until the old large dipole wave disappears at the receiving contact . at this time ( about 4296 ) , a large current spike appears , and a new dipole wave is formed closer to the injecting contact than the small dipole . @xmath14 decreases abruptly , while the small dipole disappears , and the newly created dipole reaches a large size and moves toward the receiving contact . shortly afterward , a new current spike marks the creation of another small dipole . this small dipole travels toward the receiving contact , and it grows only when the only existing large pulse reaches the receiving contact and disappears . a small dipole formed closer to the injecting contact does not grow , until the large pulse reaches the receiving contact and disappears without triggering a new dipole wave . in this case , the corresponding pulse is close to the receiving contact . when it reaches the contact and disappears into it , two new dipoles are simultaneously triggered and become fully developed . a scenario similar to the one previously described follows , marked again by a large current spike . the situation is not repeated exactly : there are small differences in the qws at which pulses are nucleated , differences in the size and lifetimes of the small dipoles , etc . these oscillations also seem to be weakly chaotic , in which a single lyapunov exponent is small and positive .    if the injecting contact conductivity is smaller so that the contact current @xmath113 intersects @xmath76 on the ndr branch thereof , there appear standard sscos due to repeated dipole pulse nucleation at the injecting contact and motion toward the receiving contact . we have calculated the recombination coefficient as a function of the applied electric field for undoped , photo - excited , weakly coupled gaas / al@xmath62ga@xmath63as superlattices . depending on the al content @xmath77 , the superlattice may have only one static domain for small @xmath77 or two stable differentiated static domains for @xmath139 . in the latter case , a dc voltage biased sl under weak photoexcitation may exhibit self - sustained oscillations of the current due to repeated nucleation of a charge monopole or dipole waves at the injecting contact and their motion toward the collector . for high photo - excitation intensities , the walls separating electric field domains are mostly pinned , and self - sustained oscillations of the current occur only in narrow voltage intervals . for small @xmath77 among other unusual phenomena , there may appear weakly chaotic ssoc due to dipole dynamics in dc - voltage - biased sls for high photo - excitation intensities , during which nucleated dipole waves can split in two oppositely moving dipoles . these dipoles are pulses of the electric field with shapes and behavior similar to pulses in excitable media , where a sufficiently large disturbance about the unique stable domain may induce them . for other parameter values , the unique static domain is unstable , and the underlying dynamics is oscillatory so that wave trains formed by succession of pulses give rise to self - sustained oscillations of the current . the work was financially supported in part by the spanish ministry of science and innovation under grant fis2008 - 04921-c02 - 01 .
a model for charge transport in undoped , photo - excited semiconductor superlattices , which includes the dependence of the electron - hole recombination on the electric field and on the photo - excitation intensity through the field - dependent recombination coefficient , is proposed and analyzed . under dc voltage bias and high photo - excitation intensities 
 , there appear self - sustained oscillations of the current due to a repeated homogeneous nucleation of a number of charge dipole waves inside the superlattice . 
 in contrast to the case of a constant recombination coefficient , nucleated dipole waves can split for a field - dependent recombination coefficient in two oppositely moving dipoles . 
 the key for understanding these unusual properties is that these superlattices have a unique static electric - field domain . at the same time , 
 their dynamical behavior is akin to the one of an extended excitable system : an appropriate finite disturbance of the unique stable fixed point may cause a large excursion in phase space before returning to the stable state and trigger pulses and wave trains . 
 the voltage bias constraint causes new waves to be nucleated when old ones reach the contact .