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spintronics , which exploits electron spin rather than charge to develop a new generation of electronic devices , has emerged as an active field in condensed matters because of both the underlying fundamental physics and its potential impact on the information industry.prinz98science,wolf01science,awschalom02 one key issue in spintronics is the generation and efficient control of spin current . spin - orbit interaction of electrons exists extensively in metals and semiconductors and mix spin states . it provides an efficient way to control the coherent motion of electron spins . recently it is proposed theoretically that an electric field may generate a spin current in hole - doped semiconductors and in two - dimensional electron gases ( 2deg ) in heterostructures with spin - orbit coupling due to the spin helicity and the noncollinearity of the velocity of the single particle wave function.@xcite studies of this intrinsic spin hall effect has evolved into a subject of intense research . shen03xxx , sinitsyn03xxx , culcer03xxx , schliemann04prb , hu03xxx , rashba03prb , hulb04xxx the spin hall effect in a paramagnetic metal with magnetic impurities has also been discussed , in which a transverse spin imbalance will be generated when a charge current circulates.dyakonov71,hirsch99,zhang00prl,hu03prb we also note that the spin chirality in systems with strong spin - orbit interaction may induce a pure spin current@xcite    over the past two decades , remarkable phenomena have been observed in the 2deg , most notably , the discovery of integer and fractional quantum hall effect.@xcite research in spin transports provides a good opportunity to explore spin physics in the 2deg with spin - orbit couplings . the spin - orbit coupling leads to a zero - field spin splitting , and it competes with the zeeman spin splitting when a perpendicular magnetic field is applied . the result can be detected as beating in shubnikov - de haas oscillations.@xcite    very recently we have studied the spin hall effect in the 2deg with spin - orbit coupling in a strong perpendicular magnetic field , and predicted a resonant spin hall effect caused by the landau level crossing near the fermi energy.@xcite in this paper we present detailed calculations of the problem . we analyze symmetries in systems with the rashba and/or dresselhaus couplings . by using linear response theory , we calculate the spin hall conductance @xmath2 , including its magnetic field and temperature dependences for realistic parameters of ingags / ingaalgs . the non - linearity in the electric field of the spin hall current near resonance is also studied beyond the linear response theory . the resonance is a low temperature property , which shows up at a characteristic temperature of the order of the zeeman energy @xmath3 . the peak of the resonance diverges as @xmath4 ( @xmath5 : the magnetic length ) , and its weight diverges as @xmath6 at low @xmath7 and at @xmath8 . near the resonant magnetic field @xmath9 , @xmath10 . the resonance arises from the fermi level degeneracy of the zeeman - split landau levels in the presence of the spin - orbit coupling . among the two types of the spin couplings we consider , the rashba coupling reduces the zeeman splitting and is the interaction responsible for the resonance . the dresselhaus coupling further separates the zeeman splitting and suppresses the resonance . the resonant condition in the presence of both rashba and dresselhaus couplings is derived within a perturbation theory , which is accurate for small ratio of the zeeman energy to the cyclotron frequency . the paper is organized as follows . in section ii we introduce the hamiltonian of the system under consideration and analyze its symmetries . in section iii , we study the spin hall current for systems with only rashba or only dresselhaus coupling . in section iv , we consider systems with both rashba and dresselhaus couplings . by treating the couplings as small parameters , we develop a perturbation method to derive the resonance condition . the paper is concluded with a summary and discussions in section v. as an introduction , we start with the three - dimensional ( 3d ) spin - orbit interaction known for iii - v compounds such as gaas and inas , which is of the form@xcite @xmath11where @xmath12 ( @xmath13 ) are the pauli matrices for spin of electrons , @xmath14 is the momentum of the charge carrier , and @xmath15 in eq.(1 ) , the first term is the dresselhaus coupling which originates from the lack of bulk inversion symmetry,@xcite while the second term is the rashba coupling which arises from the lack of structure inversion symmetry.@xcite the effective field @xmath16 is induced by the asymmetry of the external voltage to the system . in quantum wells , by neglecting the weak interband mixing and retaining the linear contribution of @xmath14 parallel to the @xmath17 plane , the spin - orbit interaction in 3d is reduced to an effective one in 2d ,    @xmath18    where @xmath19 and @xmath20 , with the average taken over the lowest energy band of the quasi-2d quantum well . the rashba coupling can be modulated up to fifty percent by a gate voltage perpendicular to the plane@xcite . in some quantum wells such as gaas the two terms are usually of the same order of magnitude , while in narrow gap compounds like inas the rashba coupling dominates.@xcite experimentally the relative strength of the rashba and dresselhaus couplings can be extracted from photocurrent measurements . @xcite     and @xmath21 are the charge and spin hall currents.,width=321 ]    in this paper we consider a spin-1/2 particle of charge @xmath22 and effective mass @xmath23 confined by a semiconductor quantum well to a 2d @xmath17 plane of length @xmath24 and width @xmath25 . @xcite the particle is subjected to a spin - orbit interaction @xmath26 . a perpendicular magnetic field @xmath27 and an electric field @xmath28 along the y - axis are applied as shown in fig . 1 . both electron - electron interaction and impurities will be neglected in our study . the hamiltonian reads    @xmath29    where @xmath30 is the lande g - factor , and @xmath31 is the bohr magneton . in @xmath32 the momentum @xmath14 is replaced by the canonical momentum , @xmath33 . we choose the landau gauge @xmath34 and consider periodic boundary condition in the @xmath35 direction , hence @xmath36 is a good quantum number . below we rewrite the hamiltonian in terms of lowering and raising operators . for each @xmath37 , we introduce the lowering operator @xmath38\]]and the corresponding raising operator @xmath39 , with the magnetic length @xmath40 . @xmath41 and @xmath42 satisfy the commutations @xmath43 = \delta _ { kk^{\prime } } $ ] , and @xmath44 = 0 $ ] . in terms of @xmath45 and @xmath46 we have @xmath47where @xmath48 is the cyclotron frequency , @xmath49 , and @xmath50 is twice of the ratio of the zeeman energy to the cyclotron frequency ( @xmath51 : free electron mass ) . @xmath52 and @xmath53 , both inversely proportional to @xmath54 are the dimensionless rashba and dresselhaus coupling respectively . the velocity operator plays an important role in study of transport properties including the spin hall conductance . the velocity operator of a single particle is @xmath55/i\hbar $ ] ( @xmath56 ) , from which we obtain    @xmath57 , \\ v_{y } & = & \frac{i\hbar } { \sqrt{2}ml_{b}}\left [ a_{k}^{+}-a_{k}+i\sqrt{2}\eta _ { d}\sigma _ { y}+i\sqrt{2}\eta _ { r}\sigma _ { x}\right ] .\end{aligned}\ ] ]    comparing this with the standard expression of velocity for a charged particle in a magnetic field , @xmath58 the spin - orbit coupling effectively induces a spin - dependent vector potential . we analyze three symmetries of the hamiltonian in this subsection , which we will use in our calculations . _ interchange symmetry of the two couplings . _ under the unitary transformation , @xmath59 @xmath60 @xmath61 the rashba and dresselhaus couplings are interchanged@xcite ,    @xmath62    therefore a system with rashba coupling @xmath63 , dresselhaus coupling @xmath64 , and lande g - factor @xmath30 is mapped on to a system with rashba coupling @xmath63 , dresselhaus coupling @xmath65 , and lande g - factor @xmath66 . in particular , a system with only dresselhaus coupling can be mapped on to a system with only rashba coupling and an opposite sign in @xmath67 . this symmetry will be used in section iii . at the symmetric point @xmath68 , @xmath26 is invariant under the transformation . @xmath69 is another symmetric point under the transformation , @xmath70 @xmath71 @xmath72 for physical parameters , we will always consider @xmath73 .    _ signs of the couplings . _ under the transformation , @xmath74 @xmath75 @xmath76we have @xmath77 and @xmath78 . the eigenenergy spectrum is invariant under the simultaneous sign changes of the two couplings . the eigenenergy spectrum is even in @xmath79 if @xmath80 and is even in @xmath81 if @xmath82 .    _ charge conjugation . _ under the charge conjugation transformation , @xmath83 , the magnetic moment of the carrier also changes its sign , or effectively @xmath84 in eq . this transformation is equivalent to the flip of the external magnetic field @xmath85 . therefore , a system of hole carriers has the same physical properties as the corresponding electron system except for possible directional changes in the observables . @xmath86 can be solved analytically in the systems with only rashba or only dresselhaus coupling . an analytical solution is currently not available for @xmath87 with both couplings . @xcite in the next section , we shall discuss the charge and spin hall conductance of the electron system with a pure rashba coupling . the results can be mapped easily onto the system with a pure dresselhaus coupling and to the hole system in semiconductors by using the symmetries discussed above . in this section we focus on systems with the rashba coupling only . after a brief review of the single particle solution in the absence of an electric field , we will discuss the spin hall conductance by using linear response theory in iiib , and its non - linear effect and scaling behavior near the resonance in iiic . we will use the interchange symmetry to comment on the dresshaus coupling system . the single particle problem of @xmath86 with @xmath80 can be solved  rashba60 . the rashba coupling hybridizes a spin down state in the @xmath88 landau level with a spin - up state in the @xmath89 landau level , and the eigenenergies are given by    @xmath90    with @xmath91 for positive integer @xmath92 , and @xmath93 @xmath94 . there is a large degeneracy @xmath95 to each eigenergy . the corresponding eigenstates are given by @xmath96 where @xmath97 is the eigenstate of the n@xmath98 landau level with @xmath99 in the absence of the spin - orbit coupling . @xmath100 and @xmath101 for @xmath102 , with @xmath103 . the eigenenergies for the system with dresselhaus coupling only can be obtained by replacing @xmath79 by @xmath81 and @xmath104 by @xmath105 , @xmath106 the energy spectra versus @xmath79 or @xmath81 are plotted in fig . 2 . in the absence of the spin - orbit coupling , the zeeman energy splits the two degenerate @xmath107 landau levels of spin - up and spin - down electron states into two nearby ones with the lower level for spin - up and the higher level for spin - down . as @xmath79 increases from zero , the energy of the @xmath88 landau level state of spin - down is lowered because of its hybridization with the spin - up state at the @xmath89 landau level due to the rashba coupling . the rashba interaction competes with the zeeman energy and there is an energy crossing at certain values of @xmath79 or the magnetic fields as we can see in fig . the spin hall resonance we examine is closely related to this level crossing . the energy level diagram in fig . 2(b ) for the dresselhaus coupling has different features . in that case , a spin - up state , which is at the lower level due to the zeeman splitting , mixes with a spin - down state at a higher landau level , which separate further the zeeman splitting , thus there is no resonance in the spin hall current .     as a function of the dimensionless rashba coupling @xmath108 . the parameters are @xmath109 evm , @xmath110m@xmath111 , @xmath112 and @xmath113 , taken from reference @xcite for the inversion heterostructures in @xmath114ga@xmath115as / in@xmath116al@xmath117as . ( b ) same as in ( a ) , but for the dresselhaus coupling @xmath118 . , width=321 ]      we consider the charge and spin hall currents along the @xmath119axis induced by an electric field along the @xmath120axis as in fig . 1 . in terms of the velocity operator , the charge and spin-@xmath121 component current operators are defined by @xmath122respectively . we refer readers to ref . @xcite for the discussions on the other spin components . the symmeterized form of the spin current operator guarantees that it is hermitian . each single particle state @xmath123 carries a current @xmath124 . the average current density carried by the 2deg is then given by @xmath125where @xmath126 is the fermi - dirac distribution function . note that since spin is not a conserved quantity in the presence of spin - orbit couplings , the spin current defined above and the spin density do not satisfy a continuity equation . nevertheless , the expectation values of the spin density and the spin current are well defined . the charge and spin hall conductance are then given by @xmath127unlike a free electron in an uniform magnetic field , the single particle problem with the spin - orbit coupling in the presence of an electric field @xmath128 is not analytically solvable , since the landau levels mixing no longer truncates . after a replacement of @xmath129 in the operator @xmath130 by @xmath131 the hamiltonian of the system in the presence of the electric field reads , apart from a constant , @xmath132where @xmath133 is the one in eq . ( [ hh ] ) by replacing @xmath130 by @xmath134 and @xmath135 . we now consider @xmath136 as a perturbative hamiltonian to study the charge and spin hall currents . up to the first order in @xmath137 , we obtain @xmath138where the superscript refers to the 0@xmath98 order and 1@xmath139 order in the perturbation in @xmath136 , and @xmath140 in the above equations , the summation is over the complete and orthogonal set of eigenstates @xmath141 corresponding to the spectra @xmath142 in eq . ( 8) , and @xmath143 since the matrix elements vanish for other values of @xmath144 . the charge hall conductance is found to be independent of the spin - orbit coupling , @xmath145 , with @xmath146 being the filling factor . within the perturbation theory , the spin hall conductance @xmath2 can be divided into two parts . the part arising from the @xmath147 order in @xmath148 is found to be the product of the spin polarization @xmath149 per electron and the hall conductance @xmath150 , divided by the electron charge ( @xmath22 ) , @xmath151the expectation value of the spin polarization per electron is , @xmath152@xmath153 at @xmath154 is plotted in fig . the oscillation is due to the alternate filling by electrons of the energy levels with mainly spin - up and spin - down . a jump is visible at @xmath155 ( correspondi ng to inverse magnetic field @xmath156t@xmath157 ) because of the energy crossing . the second part in g@xmath158 arises from the first order in @xmath136 , @xmath159at @xmath154 , if the two degenerate energy levels ( crossing point in fig . 2(a ) ) are partially occupied , @xmath160 may become divergent . mathematically , the resonance is given by the condition @xmath161 for the electron filling factor @xmath162 , with @xmath92 an integer satisfying the equation @xmath163from the above condition , for a system with any @xmath164 , and @xmath73 , there is a unique resonant magnetic field @xmath9 such that the resonant condition is satisfied . by symmetry , we obtain the resonance condition for the system with a pure dresselhaus coupling , which is given by the solution for @xmath92 of the equation , @xmath165unlike the pure rashba coupling case , there is no solution for any @xmath73 in the pure dresselhaus coupling system . this is because the energy levels @xmath166 and @xmath167 with @xmath168 do not cross over , so the pairs of the crossing levels in fig . 2(b ) correspond to @xmath169 and do not contribute to the spin hall conductance . we have calculated the spin hall conductance numerically . @xmath160 at @xmath170 is shown in fig . in addition to the oscillation in @xmath171 similar to that of @xmath172 , there is a pronounced resonant peak at the filling @xmath155 . in fig . 4 , we show @xmath160 at several temperatures . the height of the resonance peak increases drastically as the temperature decreases below a few kelvin . in fig . 5 , we show the @xmath7-dependence of the height of the resonant peak and the two nearby side peaks . the characteristic temperature for the occurrence of the peak can be estimated to be the zeeman energy @xmath3 , which is about @xmath173 at the resonant field for the parameters in fig . more explicit derivation of this will be given in the next subsection .     and spin hall conductance as a function of 1/b at @xmath154 . the parameters are the same as in fig . 2 . ( a ) and ( b ) for the rashba coupling , and ( c ) and ( d ) for the dresselhaus coupling.,width=302 ]     at several temeratures for rashba coupling systems . the parameters are the same as those in fig . 2.,width=302 ]          in this section we study the non - linear effect of the electric field to the resonant spin hall current and the scaling behavior . since the resonance originates from the interference of two degenerate levels near the fermi energy , we will focus on those two levels to examine the problem . as an example , we shall consider in@xmath114ga@xmath174as / in @xmath116ga@xmath117 as with the parameters given in fig . 2 , in which case the resonance occurs at the filling factor @xmath155 ( see fig . 3b ) and the relevant two levels are @xmath175 and @xmath176 . the energy levels below the two levels are assumed to be fully filled , and all levels above the two to be empty . this is valid if @xmath177 @xmath178 . the hamiltonian is then , up to a constant , reduced to a @xmath179 matrix , @xmath180 where @xmath181        as we can see from the reduced hamiltonian and from fig . 6 , the electric field breaks the level degeneracy and opens an energy gap @xmath182 . denoting the two eigenstates of the reduced hamiltonian by @xmath183 , the spin hall current density is given by @xmath184where the fermi - dirac distribution @xmath185/k_{b}t\right ) + 1\}^{-1}$ ] , with @xmath186 , @xmath187 the chemical potential , and @xmath188 . the electric field and temperature dependences of the spin current @xmath21 near the resonance point is plotted in fig.7 . at low temperatures the resonant spin current approaches to a constant in a weak electric field .     the filling factor at the resonance is @xmath189,width=321 ] now we analyze the scaling behavior of the spin conductance near the resonance point . for simplicity we limit our discussion to the case of @xmath190 and @xmath191 . near the resonant point , @xmath192 where @xmath193 is the zeeman energy and @xmath194 is the reduced dimensionless magnetic field . using the identity @xmath195 , \]]we obtain the singular part of the spin hall conductance to be @xmath196 .\]]where the factor @xmath197 is a slowly varying function of @xmath7 ranging from 1 at low temperatures to @xmath198 at high temperatures . at low temperatures @xmath2 is given by , @xmath199it is only a function of the reduced magnetic field and the excess part of the filling factor from @xmath200 . at the resonant magnetic field , _ i.e. _ , @xmath201 the spin hall current approaches with lowering temperature to a constant , @xmath202 as can be seen in fig . 7 . using the resonance condition in eq.([condition ] ) , @xmath203 ( @xmath204 ) and using the fact that for large @xmath92 , @xmath92 is proportional to @xmath205 the resonant magnetic field @xmath206 approximately . thus the resonant spin current is proportional to @xmath207therefore for a given filling factor , the larger the spin - orbit coupling @xmath208 is , the stronger the spin hall resonance . the resulted spin hall conductance diverges at @xmath154 as @xmath209at temperatures @xmath210 , @xmath211and the integral @xmath212this integral reflects the weight of the resonant peak of the spin hall conductance . since the method used in this subsection is beyond perturbation theory , we conclude that the resonance spin hall conductance we predict is not an artifact of the perturbation method . instead , the resonance is caused by the interference between the two degenerate energy levels at the fermi energy . in this section we briefly discuss the resonance in the spin hall conductance in systems with both rashba and dresselhaus couplings . the hamiltonian including the electric potential reads , @xmath132 with @xmath213 . in this case @xmath86 is not solvable analytically . a state @xmath214 ( in the basis of the landau levels with @xmath215 ) is coupled to @xmath216 via the rashba coupling , which is further coupled to @xmath217 due to the dresselhaus coupling . in this way , a landau level is coupled to infinite number of other landau levels , and the analytic solution is not available . the problem , however , may be approximately solved by using perturbation theory to treat @xmath218 and @xmath81 as small parameters . this is equivalent to the limit @xmath219 , since @xmath220 . for the sample we consider in the present paper given in fig . 2 , @xmath221 at the resonant field @xmath222 tesla . in the absence of the electric field , the single particle energy , up to the second order in @xmath79 and @xmath81 , is given by    [ two - e ] @xmath223 note that the mixed term of @xmath224 does not appear in the perturbation to the second order . the two levels become degenerate if the following equation is satisfied ,    @xmath225    it follows that a necessary condition for the resonant spin hall current is @xmath226 , for @xmath191 . at @xmath227 and in the limit @xmath228 , eq . ( [ perturbation ] ) is consistent with eq . ( [ condition ] ) for the resonant condition we derived for the pure rashba system . alternatively the resonant magnetic field is @xmath229 the large number @xmath230 increases with @xmath231 for a specific density of particles . thus for a certain rashba coupling the increasing of dresselhaus coupling will decrease the resonant magnetic field @xmath232 the singular part of the spin hall conductance can be studied by examining the two level system in the presence of an electric field as we described in section iiic . at the resonant point and at low temperature , @xmath233    as the dresselhaus coupling increases from zero , the resonance is shifted to lower magnetic fields , the resonance occurs at higher landau levels with a weaker resonant strength . in summary , we have studied the spin hall effect in a two - dimensional electron system with spin - orbit couplings in a strong perpendicular magnetic field . in systems with the rashba coupling dominating over the dresselhaus coupling , there is a resonant magnetic field at which the spin hall conductance diverges at low temperature and low electric field . the physics for this resonance is the energy level crossing of the two landau levels due to the competition of the zeeman splitting and the rashba coupling . for a given system , there is a unique resonant magnetic field , at which the two landau levels become degenerate at the fermi energy . in this case , some physical properties may show singularity . as studied earlier , the spin polarization will change its sign as the magnetic field is varied passing through the resonant field . namely the magnetic susceptibility is divergent . the spin hall conductance is another singular response due to this level crossing . when an infinitesimally weak d.c . electric field is applied in the plane , the two degenerate landau levels are split accordingly and a finite spin hall current is induced . the resonance is macroscopic in the sense that a huge number of the states in the same landau level are involved in the process . we have calculated the temperature and electric field dependences of the resonance . the characteristic temperature for the resonant spin hall current is of order of the zeeman energy . as the temperature decreases , the height of the resonance peak diverge like @xmath234 and the weight diverges like @xmath235 . while the spin orbit coupling has a dramatic effect on the spin hall conductance , the charge hall conductance is not affected and remains quantized . the spin hall current is non - linear with the electric field at the resonant field . at low temperatures , the spin hall current rapidly rises linearly with the electric field and saturates at higher electric fields . at @xmath154 , the spin hall conductance diverges as @xmath236 at resonance . near the resonant magnetic field @xmath9 , it is @xmath237 . contrary to the rashba coupling , the dresselhaus coupling further increases the zeeman energy splitting to suppress the effect of the rashba coupling . the strength of the rashba coupling necessary to surpass the dresselhaus coupling by in order to have the resonant spin hall current was estimated by using a perturbation method treating the couplings as small parameters . this is accurate as long as the zeeman energy is much smaller than then cyclotron frequency . we have assumed no potential disorder in our theory . the effects of disorder in 2deg with rashba coupling , especially in a strong magnetic field , is not well understood at this point.@xcite nevertheless , it seems reasonable to assume that the spin - orbit coupling does not change the effects of disorder qualitatively . this is likely to be the case in the presence of a strong magnetic field , which ensures extended states in the landau levels when the disorder is not sufficiently strong as evidenced by the experimentally observed quantization of the hall conductance . we then assume that the disorder gives rise to broadening of the landau level and localization so that the extended states in a landau levels are separate in energy from those in the next one by localized states . inspection of the spin - orbit coupling shows that laughlin s gauge argument still holdslaughlin , halperin , and each landau level with its extended states completely filled contribute @xmath238 to the charge hall conductance . thus we conclude that the quantum hall conductance remains intact with the spin - orbit interaction , except at the special degeneracy point . as the fermi energy varies across this degenerate extended state , the charge hall conductance @xmath239 is expected to change by @xmath240 , instead of @xmath238 for the other extended levels . this fact can be used experimentally to determine the rashba interaction induced degeneracy discussed in this paper . w. knap , c. skierbiszewski , a. zduniak , e. litwin  staszewska , d. bertho , f. kobbi , j. l. robert , g. e. pikus , f. g. pikus , s. v. iordanskii , v. mosser , k. zekentes , yu . b. lyanda  geller , phys . rev . b * 53 * , 3912 ( 1996 ) .                y. xiong and x. c. xie , cond - mat/0403083 ; j. inoue , g. e. w. bauer , and l. w. molenkamp , phys . b 70 , 041303 ( 2004 ) ; s. murakami , phys . b 69 , 241202 ( 2004 ) ; e. g. mishchenko , a. v. shytov , and b. i. halperin , cond - mat/0406730 ; k. nomura , j. sinova , t. jungwirth , q. niu , and a. h. macdonald , cond - mat/0407279 ; l. sheng , d. n. sheng , and c. s. ting , cond - mat/0409038 .
following a previous work [ shen , ma , xie and zhang , phys . 
 rev . 
 lett . 
 92 , 256603 ( 2004 ) ] on the resonant spin hall effect , we present detailed calculations of the spin hall conductance in two - dimensional quantum wells in a strong perpendicular magnetic field . 
 the rashba coupling , generated by spin - orbit interaction in wells lacking bulk inversion symmetry , introduces a degeneracy of zeeman - split landau levels at certain magnetic fields . 
 this degeneracy , if occuring at the fermi energy , will induce a resonance in the spin hall conductance below a characteristic temperature of order of the zeeman energy . at very low temperatures , 
 the spin hall current is highly non - ohmic . 
 the dresselhaus coupling due to the lack of structure inversion symmetry partially or completely suppresses the spin hall resonance . 
 the condition for the resonant spin hall conductance in the presence of both rashba and dresselhaus couplings is derived using a perturbation method . in the presence of disorder , 
 we argue that the resonant spin hall conductance occurs when the two zeeman split extended states near the fermi level becomes degenerate due to the rashba coupling and that the the quantized charge hall conductance changes by @xmath0 instead of @xmath1 as the magnetic field changes through the resonant field .