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_ electrical _ spin injection , transport and detection in diluted magnetic semiconductor ( dms)-nonmagnetic semiconductor ( nms ) based systems @xcite-@xcite are three crucial prerequisites for the successful realization of semiconductor spintronics . while the spin degree of freedom naturally is important , the charge degree of freedom plays a significant role in semiconductor spin transport as well , through electric field induced effects .
several experiments@xcite-@xcite have recently shown that the spin injection efficiency is strongly dependent on both applied and intrinsic electric fields , caused by , e.g. , inhomogeneous doping . this was theoretically emphasized by yu and flatte(yf)@xcite within a drift - diffusion model from which it was shown that the magnitude and sign of an electric field strongly influences the spin relaxation length according to @xmath0 where @xmath1 is a homogeneous electric field , and @xmath2 is an intrinsic ( electric field - independent ) spin diffusion length . according to eq .
( [ yf ] ) , the electric field enhances(suppresses ) the spin relaxation length @xmath3 in the direction anti - parallel(parallel ) to the direction of the electric field . however , the underlying assumption leading up to eq .
( [ yf ] ) was _ local charge neutrality _ , and thus homogeneous electric fields .
the question is how spin polarized electrons propagate in , e.g. , inhomogeneously doped semiconductors where electric fields are inhomogeneous ?    in this paper , we use a similar approach as yf as well as self - consistent numerical calculations to study the effects of inhomogeneous electric fields .
we find that , in an analogous drift - diffusion picture , a `` quasi - local '' spin relaxation length in the presence of inhomogeneous electric fields can be defined according to @xmath4^{-1}~ , \label{ldu}\ ] ] from which a strong electric field gradient dependence becomes evident .
however , in order to understand the full influence of inhomogeneous electric fields one needs to have access to the self - consistent charge density and electric field profile .
we have performed such self - consistent numerical calculations using a boltzmann - poisson equation approach for a dms - nms inhomogeneously doped semiconductor from which we demonstrate that electric field gradients indeed can play a crucial role in the propagation of spin polarized electrons .
our findings may in particular highlight a possible difficulty in obtaining efficient spin polarized propagation at semiconductor interfaces between low - to - high doping concentration regions .    in the following we will show a derivation of the drift - diffusion model leading to eq .
( 2 ) ( section 2 ) , followed by our numerical model ( section 3 ) , and a discussion of our results ( section 4 ) .
the current for spin - up and spin - down electrons can be written as @xmath5 where @xmath6 and @xmath7 are the diffusion constants and conductivities for the spin - up(down ) species . here , the conductivities refer to @xmath8 , in terms of the mobilities @xmath9 and densities @xmath10 of spin - up(down ) electrons .
the continuity equations for the two species require that @xmath11 where @xmath12 are the spin - flip scattering rates . in steady - state eqs .
( [ dd],[continuity ] ) result in @xmath13 for a nms @xmath14 , and @xmath15 .
multiplying eq .
( [ ddc ] ) with @xmath16 and substracting them from each other we arrive to the following expression for the spin density imbalance @xmath17 @xmath18 where we have used the einstein relation @xmath19 , @xmath20 is the spin relaxation time @xmath21 ( @xmath22 ) , and @xmath23 is the intrinsic spin diffusion length .    equation ( [ ddimbalance ] ) is a drift - diffusion equation for the spin density imbalance @xmath24 and contains terms depending on both the electric field , and the electric field _
gradient_. we note that in fact eq .
( [ ddimbalance ] ) resembles the corresponding equations in refs .
@xcite , but with an _ additional term proportional to _ @xmath25 .
similarly to refs .
@xcite we can use the roots of the characteristic equation for eq .
( [ ddimbalance ] ) to define up- and down - stream spin diffusion lengths from eq .
( [ ddimbalance ] ) leading up to eq .
( [ ldu ] ) .
notice that eq .
( [ ldu ] ) is only defined `` locally '' , over a region where @xmath26 can be considered constant , and where an average value of the electric field is used .
note also that the above equations are also valid for ferromagnetic semiconductors provided the mobility and diffusion constant are replaced with the ones corresponding to the minority - spin species@xcite .    a comparison between eq .
( [ ldu ] ) and refs .
@xcite results in that eq .
( [ ldu ] ) above contains an additional term @xmath27 in the square root .
naturally , this term plays an important role in inhomogeneously doped semiconductors .
we have performed self - consistent calculations and compared them with numerical fits using eq .
( [ ldu ] ) to verify its applicability .
our self - consistent approach is based on the solution of the boltzmann and poisson equations using numerical methods developed in ref .
@xcite . working in a one - dimensional model ( choosing the @xmath28 direction ) ,
the transport of spin - polarized electrons is described by two bte equations according to @xmath29 where @xmath30 is the electron distribution for the spin - up ( down ) electrons , @xmath31 is the momentum relaxation time , and @xmath32 ( @xmath33 ) is the scattering rate for spin - up ( down ) electrons . the first term on the right - hand side of eq .
( [ bte ] ) describes the relaxation of each nonequilibrium spin distribution to a local equilibrium ( spin - dependent ) , normalized electron distribution function @xmath34 , where @xmath35 k is the lattice temperature .
the last term in eq .
( [ bte ] ) describes the relaxation of the spin polarization . from the distribution function @xmath36
we calculate the local spin density according to @xmath37 , the total charge density , @xmath38 , and the spin density imbalance , @xmath39 . in order to take into account inhomogeneous charge distributions and electric fields
we solve eq .
( [ bte ] ) together with the poisson equation @xmath40 where @xmath41 is the dielectric constant and @xmath42 is the donor concentration profile , using finite difference and relaxation methods@xcite .
we have performed self - consistent , numerical calculations for an inhomogeneously doped dms - nms structure schematically depicted in fig .
[ fig:1 ] using @xmath35 k , @xmath43 , @xmath44 ps , @xmath45 ns , @xmath46 m@xmath47 , @xmath48 @xmath49 m and @xmath50 v.    at the interface between the two doping regions with concentrations @xmath51 and @xmath52 a built - in field is expected due to the diffusion of carriers from the region with higher to the region with lower doping concentration .
this is illustrated in fig .
[ fig:2 ] where we show the electric field profile , as well as the gradient of the electric field ( inset ) around the interfacial regions , for different doping concentrations @xmath53 .
the sign and amplitude of the electric field gradients naturally depend strongly on the doping concentration @xmath52 on a length scale of the order of the screening length .    while it is not possible to use eq .
( [ ldu ] ) directly for a quantitative analysis of the spin transport properties , one can make preliminary estimates of the order of the spin relaxation length around a specific region using linear approximations of the electric field .
for example , using a linear fit of @xmath1 for @xmath54 @xmath49 m for the @xmath55 curve yields @xmath56 v/@xmath49m@xmath57 .
furthermore , using a mean value of @xmath58 kv / cm and @xmath59 @xmath49 m we obtain from eq .
( [ ldu ] ) a spin relaxation length value @xmath60 @xmath49 m along the ( charge ) direction of transport .
similarly , it is found that @xmath61 @xmath49 m . in comparison ,
an evaluation of eq .
( [ yf ] ) , i.e. without the electric field gradient term , using the same value of the electric field yields @xmath62 @xmath49 m , and @xmath63 @xmath49 m .
thus , the spin relaxation length in the direction of the charge flow is predicted to be strongly influenced by the inhomogeneous electric field around the doping interface .    in fig .
[ fig:3 ] , the numerically calculated spin density imbalance , @xmath24 , and total charge density , @xmath64 , are shown for the @xmath55 structure considered in the above estimates .
the total charge density ( dashed line ) increases monotonically as expected within a screening length from the low to the high doping concentration region .
the spin density imbalance ( solid line ) follows a similar increase , but is , however , _
significantly suppressed _ such that only a maximum of 0.5@xmath52 is reached around the interface region , beyond which @xmath24 decreases exponentially . the observed suppression is a consequence of the inhomogeneous electric fields .
a crude estimate of the validity of the values for @xmath65 can be obtained by comparing our numerical calculations with @xmath66 $ ] . in fig .
[ fig:3 ] we show the resulting fits for @xmath67 and 36 @xmath49 m , the first and latter values corresponding to the previously estimated values for @xmath65 and @xmath68 obtained from the numerically calculated electric field profiles .
we see that the higher values of @xmath68 indeed yield that @xmath69 .
in contrast , the @xmath68 values 0.16 and 0.3 @xmath49 m , which are of the order of the screening length , yield better fits to the numerically calculated data .
our results and analysis shows that inhomogeneous electric fields can strongly influence the spin relaxation length , and that significant spin polarization suppression can occur as a result thereof .
we have also high - lighted a possible mechanism that may inhibit propagation of spin polarized electrons from low - to - high regions of doping concentrations , such as encountered in future semiconductor spintronic devices .
this work was supported by the indiana 21@xmath70 century research and technology fund .
numerical calculations were performed using the facilities at the center for computational nanoscience at ball state university .
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we discuss the influence of electrical effects on spin transport , and in particular the propagation and relaxation of spin polarized electrons in the presence of inhomogeneous electric fields . 
 we show that the spin relaxation length strongly depends on electric field gradients , and that significant suppression of electron spin polarization can occur as a result thereof . 
 a discussion in terms of a drift - diffusion picture , and self - consistent numerical calculations based on a boltzmann - poisson approach shows that the spin relaxation length in fact can be of the order of the charge screening length .