Thermoballistic spin-polarized electron transport in paramagnetic semiconductors

# Thermoballistic spin-polarized electron transport in paramagnetic semiconductors

R. Lipperheide    U. Wille Helmholtz-Zentrum Berlin für Materialien und Energie (formerly Hahn-Meitner-Institut Berlin), Lise-Meitner-Campus Wannsee,
Glienicker Straße 100, D-14109 Berlin, Germany
July 26, 2019
###### Abstract

Spin-polarized electron transport in diluted magnetic semiconductors (DMS) in the paramagnetic phase is described within the thermoballistic transport model. In this (semiclassical) model, the ballistic and diffusive transport mechanisms are unified in terms of a thermoballistic current in which electrons move ballistically across intervals enclosed between arbitrarily distributed points of local thermal equilibrium. The contribution of each interval to the current is governed by the momentum relaxation length. Spin relaxation is assumed to take place during the ballistic electron motion. In paramagnetic DMS exposed to an external magnetic field, the conduction band is spin-split due to the giant Zeeman effect. In order to deal with this situation, we extend our previous formulation of thermoballistic spin-polarized transport so as to take into account an arbitrary (position-dependent) spin splitting of the conduction band. The current and density spin polarizations as well as the magnetoresistance are each obtained as the sum of an equilibrium term determined by the spin-relaxed chemical potential, and an off-equilibrium contribution expressed in terms of a spin transport function that is related to the splitting of the spin-resolved chemical potentials. The procedures for the calculation of the spin-relaxed chemical potential and of the spin transport function are outlined. As an illustrative example, we apply the thermoballistic description to spin-polarized transport in DMS/NMS/DMS heterostructures formed of a nonmagnetic semiconducting sample (NMS) sandwiched between two DMS layers. We evaluate the current spin polarization and the magnetoresistance for this case and, in the limit of small momentum relaxation length, find our results to agree with those of the standard drift-diffusion approach to electron transport.

###### pacs:
72.20.Dp, 72.25.Dc, 75.50.Pp

## I Introduction

The electrical injection of spin-polarized carriers from magnetic contacts into, and their subsequent transport across, nonmagnetic semiconductors (NMS) constitute outstanding issues in the field of semiconductor spintronics. aws02 (); sch02 (); zut04 (); sch05 (); dya08 () While spin-based semiconductor devices still await realization, major progress has been achieved over the past years in the understanding of the conditions and mechanisms governing polarized-carrier injection and transport.

The low injection efficiency observed for ferromagnetic metal (FM) contacts found an explanation sch00 () in the large conductivity mismatch between the contacts and the semiconducting sample. To circumvent this obstacle, the introduction of spin-selective interface resistances was suggested fil00 (); ras00 () and analyzed theoretically in some detail. smi01 (); fer01 (); ras02 (); yuf02 (); yuf02a (); alb02 (); alb03 (); kra03 () A considerable enhancement of the injection efficiency due to externally applied electric fields was predicted. yuf02 (); yuf02a () On the other hand, the mismatch problem is mitigated from the outset if contact layers made up of magnetic semiconductors are used. zut04 (); sch05 (); roy06 () Promisingly high values of the spin injection efficiency were obtained in experiments oes99 (); fie99 (); jon00 () using paramagnetic (II,Mn)VI DMS contacts. In the latter, the conduction band is spin-split due to the giant Zeeman effect fur88 (); die94 (); cib08 () in an external magnetic field, thereby making possible contact spin polarizations close to 100%. For this kind of spin injector, a novel magnetoresistance effect was observed sch01 (); sch04 () and theoretically analyzed sch01 (); yuf02a (); kha05 () within the standard drift-diffusion theory of electron transport.

Spin-polarized transport in semiconducting structures formed of NMS layers and paramagnetic (II,Mn)VI DMS layers means electron transmission across spin-dependent potential profiles. These profiles depend on position via the position dependence of the electrostatic potential (composed of the conduction band edge potential and the external potential) and of the magnetic-field-induced spin splitting of the conduction band, which changes abruptly at the NMS/DMS interfaces. Transport across spin-dependent potential profiles corresponding to specific combinations of NMS and DMS layers has been studied theoretically in a number of cases egu98 (); guo00 (); egu01 (); cha01 () in the (quantum-coherent) ballistic limit. Recently, the transport properties of II-VI resonant tunneling devices coupled to paramagnetic DMS contacts were found san07 (); slo07 () to depend strongly on the magnitude of the applied magnetic field.

In the present work, we consider arbitrarily shaped potential profiles including internal and external electrostatic potentials and exhibiting arbitrary, position-dependent spin splitting, thereby covering any combination of NMS layers and paramagnetic DMS layers that may occur in semiconducting structures. We study spin-polarized electron transport within the (semiclassical) thermoballistic description lip03 () of carrier transport in nondegenerate semiconductors. The basic element of this description is the thermoballistic current, in which electrons move ballistically across intervals enclosed between points of local thermal equilibrium. The contribution to the current of each such ”ballistic interval” is governed by the momentum relaxation length (”mean free path”). The thermoballistic transport mechanism is intermediate between the diffusive (Drude) process, where the electrons move from one state of equilibrium to another, infinitesimally close-lying state (mean free path tending toward zero), and, on the other hand, the ballistic, collision-free motion across the electrostatic potential profile (mean free path tending toward infinity). In the thermoballistic current, diffusive and ballistic transport are thus unified in a way that allows the effect of these two aspects of the transport mechanism to be studied. In Ref. lip05, , we have introduced spin relaxation into the thermoballistic description for the case of electron transport in NMS. There, we have disregarded any spin splitting of the conduction band. Applications lip05 (); lip06 () have dealt with spin-polarized transport in heterostructures formed of an NMS layer sandwiched between two ferromagnetic metal contacts. In the present paper, we put forward the systematic extension of the thermoballistic approach to spin-polarized electron transport across spin-split potential profiles. While we are mainly concerned here with the general formulation of this extension, we also treat, within a simplified picture, spin-polarized transport in heterostructures formed of an NMS layer enclosed between two DMS layers. Applications to specific experimental configurations will be deferred to future work.

As a prerequisite to the thermoballistic description, we formulate, in the next two sections, the details of ballistic spin-polarized electron transport across spin-split potential profiles. In Sec. II, we introduce the electron densities at the points of local thermal equilibrium enclosing a ballistic interval, and construct the currents injected from these points into that interval. In Sec. III, we continue the injected currents and the densities from the points of local thermal equilibrium into the ballistic interval, and introduce the balance equation that describes spin relaxation inside the interval. The off-equilibrium ballistic spin-polarized current and density are expressed in terms of a spin transport function that is related to the splitting of the spin-resolved chemical potentials. In Sec. IV, the thermoballistic currents and densities are constructed by summing up the contributions from all ballistic intervals. The magnitude of these contributions is governed by the momentum relaxation length. The current and density spin polarizations as well as the magnetoresistance are each expressed as the sum of an equilibrium term determined by the spin-relaxed chemical potential, and an off-equilibrium contribution given in terms of the spin transport function. The spin-relaxed chemical potential is described through a resistance function. The latter, as well as the spin transport function, are each calculated from an integral equation. Section V deals with the thermoballistic description of spin-polarized electron transport in DMS/NMS/DMS heterostructures. We derive explicit expressions for the current spin polarization and the magnetoresistance and compare these, in the limit of small momentum relaxation length, to those of the standard drift-diffusion approach to electron transport. Numerical results for DMS/NMS/DMS heterostructures are presented and discussed in Sec. VI. In Sec. VII, we summarize the contents of this paper and make some concluding remarks.

## Ii Points of thermal equilibrium: Densities and injected currents

The thermoballistic description of spin-polarized electron transport in NMS lip05 () makes use, in a one-dimensional geometry, of ballistic electron currents and densities in “ballistic intervals” between two points of local thermal equilibrium, and (); see Fig. 1. The ballistic electron transport across such an interval is determined by the densities at the points of local thermal equilibrium and by the shape of the potential profile inside the interval. For the purpose of generalizing the thermoballistic approach to spin-polarized electron transport in paramagnetic DMS, we introduce the thermal-equilibrium densities and for spin-up and spin-down conduction band states, as well as the spin-dependent potential profiles

 ϵ↑,↓(x)=Ec(x)±Δ(x)/2 (1)

at position . Here, the (spin-independent) potential comprises the conduction band edge potential and the external electrostatic potential, and is the Zeeman splitting of the conduction band due to an external magnetic field fur88 () [we restrict ourselves to considering a single Landau level whose energy is assumed to be included in ]. In developing our formalism, we assume both and , and hence , to be continuous functions of in the interval . 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. This aspect will be illustrated in Sec. V below for the case of DMS/NMS/DMS heterostructures.

In the thermoballistic transport mechanism, momentum relaxation takes place exclusively and instantaneously at the points of local thermal equilibrium. Spin relaxation via spin-flip scattering processes, on the other hand, occurs during the electron motion across the ballistic intervals (since the electrons spend only an infinitesimally short time span at the points of local thermal equilibrium, it is only inside the ballistic intervals that they can experience spin relaxation). This separation of momentum and spin relaxation is in accordance with the D’yakonov-Perel’ relaxation mechanism dya71 (); dya08a (), which is known zut04 (); dya08 () to be the dominant relaxation mechanism at large donor doping levels and at high temperatures. Therefore, our approach covers a broad range of cases interesting from the point of view of experiment.

### ii.1 Electron densities

The electron motion in a ballistic interval is activated at a point of local thermal equilibrium at one end and terminated at another such point at the other end. In the semiclassical model, the points of local thermal equilibrium are characterized by a local chemical potential (”quasi-Fermi level”). In a nondegenerate system, the spin-resolved electron densities, , are given by

 n↑,↓(x′)=Nc2e−β[ϵ↑,↓(x′)−μ↑,↓(x′)], (2)

where are the spin-resolved chemical potentials, is the effective density of states of either spin at the conduction band edge, is the effective mass of the electrons (which, for simplicity, is assumed to be independent of position and of the external magnetic field), and . In the spin-relaxed state, the potentials become equal to the common ”spin-relaxed chemical potential” . Thus, the spin-relaxed densities at the point of thermal equilibrium , denoted by , are

 ~n↑,↓(x′)=Nc2e−β[ϵ↑,↓(x′)−~μ(x′)]. (3)

For the Boltzmann factors , we introduce the notation

 ϑ↑,↓(x′)=e−βϵ↑,↓(x′) (4)

and define

 ϑ±(x′)=ϑ↑(x′)±ϑ↓(x′) (5)

(the notation will be used mutatis mutandis for various other quantities appearing further below), so that we have

 ~n±(x′)=Nc2ϑ±(x′)eβ~μ(x′). (6)

The total (i.e., spin-summed) spin-relaxed density, , is

 ~n(x′)=Nc2ϑ+(x′)eβ~μ(x′), (7)

and for the “static” spin polarization of the conduction band electrons, i.e., the polarization in the spin-relaxed state at zero external electric field, we have

 P(x′)≡~n−(x′)~n(x′)=ϑ−(x′)ϑ+(x′)=−tanh(βΔ(x′)/2). (8)

We note the relation

 ϑ↑,↓(x′)=12[1±P(x′)]ϑ+(x′) (9)

expressing the Boltzmann factors in terms of .

For the dynamical description of spin-polarized transport, we now introduce the ratios of the densities (2) to their spin-relaxed counterparts (3),

 α↑,↓(x′)≡n↑,↓(x′)~n↑,↓(x′)=eβ[μ↑,↓(x′)−~μ(x′)]; (10)

working with these quantities will provide us with a description in terms of linear equations, instead of the nonlinear description that we would encounter when employing the chemical potentials themselves. This aspect has been emphasized previously yuf02a () within the diffusive approach to spin polarization in nondegenerate semiconductors. [We note that the quantity introduced in Eq. (10) is different from that of Eq. (3.8) in Ref. lip05, ; in the spin-relaxed state, we have in the present case , while in Ref. lip05, we have .] Using Eqs. (4), (7), (9), and (10), we can write the densities in the form

 n↑,↓(x′)=12~n(x′)[1±P(x′)]α↑,↓(x′), (11)

so that

 n±(x′)=12~n(x′)[α±(x′)+P(x′)α∓(x′)]. (12)

Here, the functions and appear as independent dynamical quantities.

We now require the total density, , to be unaffected by spin relaxation, i.e., we set

 n(x′)=~n(x′). (13)

This requirement is justified yuf02 () for n-doped (unipolar) systems, for which spin-flip scattering processes involving valence band states can be disregarded. Substituting Eq. (13) in the upper Eq. (12), we find and related by

 α+(x′)=2−P(x′)α−(x′), (14)

so that we can eliminate from the lower Eq. (12) to obtain the spin-polarized density, , as

 n−(x′)=n(x′){P(x′)+12[Q(x′)]2α−(x′)}, (15)

where

 [Q(x′)]2≡1−[P(x′)]2=4ϑ↑(x′)ϑ↓(x′)[ϑ+(x′)]2=1cosh2(βΔ(x′)/2). (16)

In the spin-relaxed state, we have

 α↑,↓(x′)=1, (17)

and hence

 α−(x′)=0,α+(x′)=2. (18)

Then, from Eq. (15), the equilibrium spin-polarized density at the point of thermal equilibrium , , is

 ~n−(x′)=n(x′)P(x′) (19)

[see Eq. (8)], and the off-equilibrium spin-polarized density, , is there

 ˇn−(x′)≡n−(x′)−~n−(x′)=12n(x′)[Q(x′)]2α−(x′). (20)

The off-equilibrium spin-polarized density is proportional to the quantity , i.e., to the difference of the spin-resolved densities relative to their spin-relaxed values [see Eq. (10)], but also contains, via the factor , the effect of the static polarization .

For later use, we establish the relation between the difference of the spin-resolved chemical potentials, , and the quantity which will appear as the key element determining the spin dynamics of off-equilibrium spin-polarized transport in the thermoballistic approach (see Secs. III.B and IV.C). From Eq. (13), we obtain, using Eqs. (2), (7), and (10),

 ϑ↑(x′)α↑(x′)+ϑ↓(x′)α↓(x′)=ϑ+(x′), (21)

and hence, using Eq. (9),

 α↑,↓(x′)=1±12[1∓P(x′)]α−(x′). (22)

With the help of Eq. (10), we then have

 μ−(x′)=1βln(α↑(x′)α↓(x′))=1βln⎛⎝1+12[1−P(x′)]α−(x′)1−12[1+P(x′)]α−(x′)⎞⎠, (23)

and, reversely,

 α−(x′)=2tanh(βμ−(x′)/2)1+P(x′)tanh(βμ−(x′)/2). (24)

For the mean chemical potential , we find

 ¯μ(x′) = ~μ(x′)+12βln(α↑(x′)α↓(x′)) (25) = ~μ(x′)+12βln({1+12[1−P(x′)]α−(x′)}{1−12[1+P(x′)]α−(x′)}) = ~μ(x′)+12βln(1−P(x′)α−(x′)−14[Q(x′)]2[α−(x′)]2).

Finally, we quote the relation between the spin-resolved chemical potentials and the common spin-relaxed chemical potential ,

 12[1+P(x′)]eβμ↑(x′)+%$12$[1−P(x′)]eβμ↓(x′)=eβ~μ(x′), (26)

which, like Eq. (21), follows from Eq. (13), but this time using Eqs. (2), (4), (7), and (9).

### ii.2 Current injection

In the thermoballistic description, a point of thermal equilibrium lies between two ballistic intervals (except at the ends of the sample, which will be considered further below), and electrons from the left- and right-lying intervals enter into it to be equilibrated instantaneously. At the same time, equilibrated electrons, which have no preferred direction of motion, are injected symmetrically (i.e., ”half-and-half”) into either ballistic interval, forming the spin-resolved electron current densities (electron currents, for short) , where is the emission velocity. If, however, there lies a potential barrier is the maximum of the spin-dependent potential profile in the interval between, say, the left point of current injection, , and the opposite point of current absorption, , part of the current will be reflected at this barrier (and equilibrated while it travels back and forth between and the left side of the barrier, – we may call it the “confined current”). The part that surmounts the barrier (henceforth called the “injected current”) is given by

 Jl↑,↓(x′,x′′) = Nc2(2β/πm∗)1/2∫∞0dppm∗e−β[p2/2m∗+ϵ↑,↓(x′)−μ↑,↓(x′)]Θ(p2/2m∗+ϵ↑,↓(x′)−ϵm↑,↓(x′,x′′)) (27) = ven↑,↓(x′)Tl↑,↓(x′,x′′),

where

 Tl↑,↓(x′,x′′)≡e−β[ϵm↑,↓(x′,x′′)−ϵ↑,↓(x′)]=ϑm↑,↓(x′,x′′)ϑ↑,↓(x′) (28)

is the (classical) transmission probability for the spin-resolved current injected at the left end-point of the ballistic interval to reach the opposite right end-point . Here, we have introduced the notation

 ϑm↑,↓(x′,x′′)=e−βϵm↑,↓(x′,x′′), (29)

which corresponds to the definition (4) for the Boltzmann factors , with replaced with [note that the factors and are local functions depending, for a given ballistic interval , each on the position of the maximum of the potential profiles and , respectively]. If the potential profile is constant along the interval or if its maximum lies at the injection point itself, then and .

In analogy to Eq. (11) for the spin-resolved densities, we now write the currents (27) in the form

 Jl↑,↓(x′,x′′)=12~Jl(x′,x′′)[1±Pm(x′,x′′)]α↑,↓(x′), (30)

where

 ~Jl(x′,x′′)≡~Jl+(x′,x′′)=veNc2ϑm+(x′,x′′)eβ~μ(x′) (31)

is the total spin-relaxed current injected at into the interval from the left, and

 Pm(x′,x′′)=ϑm−(x′,x′′)ϑm+(x′,x′′) (32)

is a static, “nonlocal” spin polarization depending on the (generally different) positions of the maximum of the potential profile for the spin-up state and for the spin-down state, respectively [compare with the analogous relation (8) for the local spin polarization ].

From expression (30), we obtain, in analogy to Eq. (12),

 Jl±(x′,x′′)=12~Jl(x′,x′′)[α±(x′)+Pm(x′,x′′)α∓(x′)]. (33)

In line with the condition (13) imposed on the total density , we now require the total current to equal its spin-relaxed limit,

 Jl(x′,x′′)=~Jl(x′,x′′). (34)

Following the argument leading to expression (15) for the spin-polarized density , we then find

 Jl−(x′,x′′)=Jl(x′,x′′){Pm(x′,x′′)+12[Qm(x′,x′′)]2α−(x′)}, (35)

where

 [Qm(x′,x′′)]2≡1−[Pm(x′,x′′)]2=4ϑm↑(x′,x′′)ϑm↓(x′,x′′)[ϑm+(x′,x′′)]2 (36)

(see the analogous relation (16) for ).

In the spin-relaxed state [see Eq. (18)], Eq. (33) yields for the injected equilibrium spin-polarized current

 ~Jl−(x′,x′′)=Jl(x′,x′′)Pm(x′,x′′), (37)

so that the injected off-equilibrium spin-polarized current is

 ˇJl−(x′,x′′)≡Jl−(x′,x′′)−~Jl−(x′,x′′)=12Jl(x′,x′′)[Qm(x′,x′′)]2α−(x′), (38)

in parallel to Eqs. (19) and (20), respectively.

All the preceding formulas hold also in the case where the left end-point of the ballistic interval coincides with the left end-point of the sample, , and similarly for the right end-points, , with the understanding that the chemical potentials and have fixed values given by the chemical potentials in the contacts bordering on the semiconducting sample at either end.

## Iii Spin-polarized transport across a ballistic interval

The electron currents discussed up to now are the currents injected into the ballistic interval at the point of thermal equilibrium as they leave the latter. Once they have entered this interval, they become transmitted ballistic currents, which will be considered in the following.

### iii.1 Continuing the injected currents and densities into the ballistic interval

We again consider injection at the left point of thermal equilibrium, , of the ballistic interval . If the system is in the spin-relaxed state at the injection point , we have and . Then the transmitted (in the following, we omit the attribute “transmitted”) ballistic spin-resolved current at , given by expression (30), propagates along the ballistic interval without spin relaxation, i.e., it is conserved. If, however, , the injected current will relax along the ballistic interval and hence will no longer be conserved. It is then natural to continue expression (30) into the ballistic interval by replacing with a more general quantity depending both on the end-points and and on the position , so that the ballistic spin-resolved current is obtained as

 Jl↑,↓(x′,x′′;x)=12% Jl(x′,x′′)[1±Pm(x′,x′′)]αl↑,↓(x′,x′′;x)=veNc2ϑm↑,↓(x′,x′′)eβ~μ(x′)αl↑,↓(x′,x′′;x), (39)

with the initial condition

 αl↑,↓(x′,x′′;x′)=α↑,↓(x′), (40)

which holds for all .

In line with the density ratios in Eq. (30), the generalized quantities express the deviation of the currents inside the ballistic interval from their spin-relaxed values, so that here the condition for spin equilibrium is again

 αl↑,↓(x′,x′′;x)=1 (41)

[see Eq. (17)], and hence

 αl−(x′,x′′;x)=0,αl+(x′,x′′;x)=2, (42)

as in Eq. (18).

Extending condition (34) and the procedure ensuing therefrom, we obtain for the (conserved) total ballistic current inside the ballistic interval

 Jl(x′,x′′;x)≡Jl+(x′,x′′;x)=Jl(x′,x′′)=veNc2ϑm+(x′,x′′)eβ~μ(x′) (43)

and, similarly, for the (conserved) ballistic equilibrium spin-polarized current

 ~Jl−(x′,x′′;x)=Jl(x′,x′′)Pm(x′,x′′)=veNc2ϑm−(x′,x′′)eβ~μ(x′) (44)

[see Eq. (37)], while for the ballistic off-equilibrium spin-polarized current we have

 ˇJl−(x′,x′′;x) = 12Jl(x′,x′′)[Qm(x′,x′′)]2αl−(x′,x′′;x) (45) = veNc2ϑm(x′,x′′)eβ~μ(x′)αl−(x′,x′′;x)

[see Eq. (38)], where

 ϑm(x′,x′′)=2ϑm↑(x′,x′′)ϑm↓(x′,x′′)ϑm+(x′,x′′)=12ϑm+(x′,x′′)[Qm(x′,x′′)]2. (46)

In Eq. (45), the spin dynamics of the ballistic off-equilibrium spin-polarized current is determined by the factor , whose dependence on the coordinate reflects the spin relaxation inside the ballistic interval . This dependence will be considered in Sec. III.B.

The density associated with the ballistic spin-resolved current will be called “ballistic spin-resolved density”. It is obtained from Eq. (39) by replacing with ,

 nl↑,↓(x′,x′′;x)=Nc4Dm↑,↓(x′,x′′;x)eβ~μ(x′)αl↑,↓(x′,x′′;x) (47)

(see the analogous relation between the current (2.2) and the density (2.7) in Ref. lip05, ). Here,

 Dm↑,↓(x′,x′′;x)=C↑,↓(x′,x′′;x)ϑm↑,↓(x′,x′′) (48)

and

 C↑,↓(x′,x′′;x)=1T↑,↓(x′,x′′;x)erfc([−lnT↑,↓(x′,x′′;x)]1/2) (49)

[see Eq. (2.8) of Ref. lip05, ], with

 T↑,↓(x′,x′′;x)≡e−β[ϵm↑,↓(x′,x′′)−ϵ↑,↓(x)]=ϑm↑,↓(x′,x′′)ϑ↑,↓(x); (50)

the latter quantity may be interpreted as the transmission probability corresponding to injection at the point toward the region containing the maximum of the potential profile in the interval . The ballistic velocity is given by

 v↑,↓(x′,x′′;x)=Jl↑,↓(x′,x′′;x)nl↑,↓(x′,x′′;x)=2veC↑,↓(x′,x′′;x). (51)

It is not affected by spin relaxation, since the spin-flip mechanism is assumed not to influence the kinematics of the electron motion.

From Eq. (47), the continuation of half (see the remarks at the beginning of Sec. II.B) the densities (12) into the ballistic interval now follows as

 nl±(x′,x′′;x)=12~nl(x′,x′′;x)[αl±(x′,x′′;x)+PmC(x′,x′′;x)αl∓(x′,x′′;x)], (52)

where

 ~nl(x′,x′′;x)≡~nl+(x′,x′′;x)=Nc4Dm+(x′,x′′;x)eβ~μ(x′) (53)

[see Eq. (7)] is the total ballistic spin-relaxed density, and, in generalization of expression (32) for ,

 PmC(x′,x′′;x)=Dm−(x′,x′′;x)Dm+(x′,x′′;x). (54)

Extending condition (13) into the ballistic interval by requiring

 nl(x′,x′′;x)≡nl+(x′,x′′;x)=~nl(x′,x′′;x), (55)

we find from Eqs. (52)

 αl+(x′,x′′;x)=2−PmC(x′,x′′;x)αl−(x′,x′′;x) (56)

[see Eq. (14)], and hence for the ballistic spin-polarized density inside the interval , in parallel to Eq. (15),

 nl−(x′,x′′;x)=nl(x′,x′′;x){PmC(x′,x′′;x)+12[QmC(x′,x′′,x)]2αl−(x′,x′′;x)}, (57)

where

 [QmC(x′,x′′;x)]2≡1−[PmC(x′,x′′;x)]2=4Dm↑(x′,x′′;x)Dm↓(x′,x′′;x)[Dm+(x′,x′′;x)]2 (58)

[see Eq. (36)]. Thus, we obtain for the ballistic equilibrium spin-polarized density

 ~nl−(x′,x′′;x)=Nc4Dm−(x′,x′′;x)eβ~μ(x′) (59)

[see Eq. (19)], and for the ballistic off-equilibrium spin-polarized density

 ˇnl−(x′,x′′;x)=Nc4Dm(x′,x′′;x)eβ~μ(x′)αl−(x′,x′′;x) (60)

[see Eq. (20)], where

 Dm(x′,x′′;x)=2Dm↑(x′,x′′;x)Dm↓(x′,x′′;x)Dm+(x′,x′′;x)=12Dm+(x′,x′′;x)[QmC(x′,x′′;x)]2, (61)

in parallel to Eq. (46).

We remark that for constant potential profiles, , when so that , Eq. (53) becomes . This reflects the fact that the left-hand side of this relation refers to the density associated with the ”half-sided” injected current, while is the total density at the point of thermal equilibrium .

### iii.2 Spatial behavior of the off-equilibrium spin-polarized current and density

The total ballistic current and density , as well as the ballistic equilibrium spin-polarized current and density , are determined by the quantities and [which, in turn, are completely determined by the potential profiles ], and, most importantly, by the spin-relaxed chemical potential , which is the only dynami