Out-of-plane equilibrium spin current in a quasi-two-dimensional electron gas under in-plane magnetic field

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by  E. Nakhmedov

Out-of-plane equilibrium spin current in a quasi-two-dimensional electron gas under
in-plane magnetic field
E. Nakhmedov1,2 and O. Alekperov2
1Institut fu¨r Theoretische Physik, Universita¨t Wu¨rzburg, Am Hubland, D-97074 Wu¨rzburg, Germany
2Institute of Physics, Azerbaijan National Academy of Sciences,H. Cavid str. 33, AZ1143 Baku, Azerbaijan
(Dated: January 24, 2012)
Equilibriumspin-currentiscalculatedinaquasi-two-dimensionalelectrongaswithfinitethickness
2 underin-planemagneticfieldandinthepresenceofRashba-andDresselhausspin-orbitinteractions.
1 The transverse confinement is modeled by means of a parabolic potential. An orbital effect of
0 the in-plane magnetic field is shown to mix a transverse quantized spin-up state with nearest-
2 neighboringspin-downstates. Theout-off-planecomponentoftheequilibriumspincurrentappears
tobenotzerointhepresenceofanin-planemagneticfield,providedatleasttwotransverse-quantized
n
levels are filled. In the absence of the magnetic field the obtained results coincide with the well-
a
J knownresults,yieldingcubicdependenceoftheequilibriumspin currenton thespin-orbitcoupling
constants. Thepersistent spin-current vanishes in the absence of themagnetic field if Rashba- and
3
Dresselhaus spin-orbit coefficients, α and β, are equal each other. In-plane magnetic field destroys
2
this symmetry, and accumulates a finite spin-current as α→β. Magnetic field is shown to change

stronglytheequilibriumcurrentofthein-planespincomponents,andgivesnewcontributionstothe
]
l cubic-dependenton spin-orbit constants terms. These new terms depend linearly on the spin-orbit
l
a constants.
h
- PACSnumbers: 74.78.-w,74.62.-c,74.70.Kn,74.50.+r
s
e
m
I. INTRODUCTION entofthecrystalpotentialislarge. Mostprominentsemi-
t. conductingcompoundshaveeitherzinc-blendestructure,
a
like GaAs and most of III-V compounds, or wurtzite
m Acentralgoalofspintronicsresearchisanachievement
structureinII-VIcompoundswithBIA.Lackofthebulk
- of an electron spin manipulation by means of an exter-
inversion symmetry in these compounds was shown by
d nal electric field [1] instead of a magnetic field, which
n Dresselhaus [5] to originate another macroscopic SO in-
is widely used now in semiconducting devices for en-
o teraction.
hancement of an information processing speed. An elec-
c
[ tric field controlled spin-orbital coupling is a promising Effects ofboth Rashba-and Dresselhaus-SOcouplings

tool [2] in realization of futuristic spin transport de- to the physical properties of two-dimensional (2D) elec-
1
v vices. During the last ten years, there has been im- tron gas are not trivial even in the absence of an exter-
3 pressing progress in both experimental and theoretical nal magnetic field. Existence of a SIA or a BIA in a
8 understanding ofthe spin dynamicsin quantumwells on disordered 2D system changes reversely the sign of the
6
thebaseof,particularly,narrow-gapsemiconductorswith phase-coherent localization correction to the conductiv-
4
highg-factor,andmetal-oxide-semiconductorfield-effect- ity [6–8], driving the system from a weak localization
.
1 transistor (MOSFET) structures. Although the dimen- regime into an antilocalization one. Rashba- and Dres-
0
sionless spin-orbit (SO) coupling parameter in vacuum selhaus SO interactions equally and independently con-
2
is as small as E /(m c2) 10−6, where E 1eV is tributetotheweakantilocalizationcorrection. Contribu-
1 F 0 F
∼ ∼
: the Fermi energy of an electron and m0c2 1MeV is tions of Rashba and Dresselhaus SO interactions to the
v ∼
the Diracgap,the largevalue ofthe SOcouplingenergy, D’yakonov-Perel’sspin relaxationrate [9] were shown to
i
X comparable with the Fermi energy, can be ensured by be also additive [10]. On the other hand, the anisotropic
r large potential gradient on the semiconductor/insulator contribution to the conductivity tensor [11] in the pres-
a
interface of these structures in the presence of macro- ence of both Rashba and Dresselhaus SO terms, the ab-
scopicstructuralinversionasymmetry(SIA).Indeed,the sence of spin polarization and suppression of spin accu-
gate potential applied across the substrate in MOSFET mulation especially at the equal values of the coupling
results in inhomogeneous space charge distribution near constants α = β, [12–14], restoration of the weak lo-
±
the semiconductor/insulator interface. The nonuniform calization regime back at α = β [15–17] manifest an
±
macroscopic potential, confining the electrons near the existence of the interference between Rashba and Dres-
interface, varies over a wide range, 10 1000 nm, selhaus SO interactions. Although the SO coupling gen-
∼ ÷
with largerpotential gradient,which originates so-called erallybreaksthespinrotationalsymmetry,anewtypeof
Rashba SO interaction [3, 4]. On the other hand, higher SU(2)symmetryappears[18]inthecaseofα=β,which
value of the SO coupling is achieved by choosing special renders the spin lifetime. In the presence of Rashba and
semiconductingmaterialswithabulk inversionasymme- Dresselhaus terms with equal strength, the SO interac-
try (BIA) in their crystalline structure, where the gradi- tionrotateselectronspinsaroundasinglefixedaxis. The
2
spinalongthisaxisbecomesconserved,neverthelessspin chargebalancecondition. Finitethicknessoftheelectron
alignedintheperpendiculardirectionsundergoesadeter- gas was recently suggested by Rashba and Efros [26, 27]
ministic rotation depending only on the initial and final in order to study the time-dependent gate voltage ma-
points of their trajectory. nipulation of electron spins in MOSFETs and quantum
Differentexperimentaltechniqueshavebeendeveloped wells,sincethespinresponsetoaperpendicular-toplane
recently to control a coupling of spin to the electric field electric field can be achieved due to a deviation from
[19–21]. An efficient gˆ-tensor modulation resonance, ob- strict 2D limit.
servedinaparabolicAl Ga Asquantumwell[19]with In this work, we report on our investigation of both
x 1−x
varying Al content x=x(z) acrossthe well, providedan orbital and spin effects of in-plane magnetic field in a
opportunity to manipulate electron spins by means of quasi-two-dimensional (quasi-2D) electron gas with a fi-
various electron spin resonance type techniques. An in- nite thickness on the spin precessionand splitting in the
plane magnetic field in all of these experiments seems to presence of Rashba and Dresselhaus SO interactions. In
be rather favorable for getting a pronounced spin reso- this paper we calculate persistent spin current. Note
nance. SO interactions in a 2D electronic system pro- thatthemodelhasbeenconsideredinourpreviouspaper
duce an effective in-plane field, which results in an drift- [28, 29] in order to study the energy spectrum and the
driven in-plane spin polarization [22]. An external in- Fermisurfaceunder variationsofSO couplingconstants,
plane magnetic field appears to be not always summed the gate electric field, the magnetic field and g-factor.
algebraicallywith SO induced effective field, and can re- Generation of a spin flux and its change under exter-
sult in the surprising out-of-plane spin polarization [23], nal destructive factors is still a controversial issue [30]
which has been observed in a strained n InGaAs film in spintronics. Generation of a dissipationless transverse
−
[20]. On the other hand, Hanle precession of optically spin current or a spin Hall current by a driving electric
oriented2Delectronsin GaAs [24]is welldescribedby a field E was predicted [31, 32] in a clean, infinite and ho-
total in-plane field, given as a sum of the external- and mogeneous structural inversion asymmetric 2D system.
SO effective fields. All these facts show nontrivialeffects Even an arbitrary small concentration of non-magnetic
of in-plane magnetic field on spin dynamics in quasi-2D impuritieswasshown[33–36]tosuppresstotallytheuni-
systems. Effect of an external magnetic field, aligned in versal value of the spin Hall conductivity peculiar to a
thenormaldirectiontotheelectrongas,hasbeenstudied clean system. As it was shown by Rashba [37], a spin
very well, since the problem can be solved exactly for a current in the presence of Rashba SO coupling appears
non-interactingelectrongasinthe presenceofone ofthe even at equilibrium in the absence of an external elec-
SO interaction. In the previous activities, the selective tric field, though it does not result in any accumulation
coupling of the in-plane magnetic field to the electronic of spin. A universal equilibrium spin current was shown
spindegreeoffreedominthepresenceofSOinteractions [38]toappearasadiamagneticcolorcurrentduetoare-
has been used to probe the interplay of Zeeman split- sponsetoaneffectiveYang-Millsmagneticfieldproduced
ting with the SO coupling. The in-plane magnetic field bySOinteractions,whichprovidesanexplicitrealization
in non-ideal 2D systems with a finite width, which is ofanon-AbelianLandaudiamagnetism. Theequilibrium
particularly relevant for heterojunctions and MOSFET spin current in a 2D electron gas with a slightly modu-
structures, couples also to the orbital motion, and can lated Rashba parameter was shown [39] to transfer spin
considerably modify the physics involved. It is there- from areaswhere spin is produced to areaswhere spin is
foreimportanttocharacterizethevariousphysicaleffects absorbed. Itwasrecentlyshown[40]thatanequilibrium
generated in the presence of a parallel magnetic field, in spincurrentina2DsystemwithRashbaSOinteractions
order to gain a better understanding of the influence of results in a mechanical torque on a substrate near edge
orbital magnetic effects on the physical properties of an of the medium, which provides an experimental tool to
electron gas with SO interactions. detect the equilibrium spin current. Therefore, it can be
Although the electrongas, formedon the semiconduc- concludedthat a relationof the equilibrium spin current
tor/insulator interface in the heterojunctions and MOS- to spin transport should not be ruled out.
FET structures, has a finite thickness [25], in the most Thecentralresultofthepaperisanappearanceofout-
activities concerning the SO interactions it is taken as of-plane equilibrium spin current in the quasi-2D elec-
a strictly 2D object by neglecting the finite thickness. tron gas under in-plane magnetic field in the presence
The thicknessofthe confinedelectrongasinthese struc- of the SO interactions. In the absence of the magnetic
tures is variedinthe largeintervalby the gatepotential, field, the average values of the spin currents JSx and
applied acrossthe electrongas,from10 nm in the inver- JSy are shown to coincide with the well-known results
sion regime up to 1000 nm in the depletion regime. The [37, 39, 41, 42] obtained for a strictly 2D electron gas,
charge distribution and the electron gas thickness can revealing a cubic dependence on the SO coupling con-
be experimentally measured and theoretically estimated stants. Weshowthatthemagneticfieldstronglychanges
with high accuracy by means of the self-consistent so- the in-plane spin-current components, contributing new
lutions of Schr¨odinger and Poisson equations under the terms to them. The new contributions turn to be pro-
3
portional,in additionto the magnetic field, either to the Dresselhaus term [5, 15], Hˆ , due to a BIA in the crys-
D
gateelectric fieldor to Zeemansplitting. The new terms talline structure. Dresselhaus SO interaction in bulk
depend linearlyonthe SO coupling constantsinthe lim- semiconductors with a zinc-blende crystal symmetry is
iting case if one of the SO coupling constant is vanish- proportional to the third order of the electron momen-
ingly small. Therefore, these contributions may prevail tum P
over the cubic dependent on the SO coupling constants
vtearnmissh.esTchoemopulet-toelfy-pwlainthetchoemmpoangnenetticJfiSyeld=,a{n0d, d0e,pJeznSdz}s HˆD = ~η σiPi(Pi2+1−Pi2+2),(i=x,y,z;i+3→i),
i
quadratically or linearly on the SO coupling constants. X (2)
The paper is organized as follows. In Section II of whereηisacharacteristicbulkcoefficientoftheSOsplit-
this work we describe explicitly an analytical solution ting. Sincetheaveragewavevectorinthedirectionofthe
of quantum mechanical problem of one particle, moving quantum confinement z is large, the terms involving p2
z
inaquasi-2Dsystemwithfinite thicknessunderin-plane will dominate in Dresselhaus SO coupling for a quasi-
magneticfieldandinthepresenceofRashbaandDressel- 2D electron gas with finite thickness. The expressionfor
haus SO interactions by imposing a parabolic confining SO interaction Hamiltonian in MOSFETs and quantum
potential in the transverse direction. We take into ac- wells of the width d grown along [001] crystallographic
count in this work a gate potential too, which produces axis reads
the SIA and Rashba SO interaction. In Section III we
calculate the spin current in equilibrium. Conclusions
α β
are given in Section IV. In Appendix we present some Hˆ =Hˆ +Hˆ = (σ P σ P )+ (σ P σ P ),
so R D ~ x y − y x ~ x x− y y
routine calculations of the persistent spin current. (3)
where α and β = η p2 = η(π/d)2 are the sample de-
− h zi −
pendent parameters of Rashba- and Dresselhaus-SO in-
II. ENERGY SPECTRUM IN THE PRESENCE
teractions, correspondingly. Rashba coefficient α is pro-
OF AN IN-PLANE MAGNETIC FIELD
portional to the gate electric field. Spin-orbital interac-
tion can be interpreted as an interaction of a spin with
We consider a quasi-2D gas of electrons, moving un-
randomlyoriented,in accordancewith the electronwave
deranexternalin-planemagneticfieldinthe presenceof
vector,effective magnetic field, which lies in the plane of
bothRashbaandDresselhausSOinteractionsandagate
the electron gas:
potential. Single particle Hamiltonian of the system in
the effective mass approximation can be written as
~ 2α
Hˆ = σ ΩR with ΩR = (P zˆ), (4)
P2 m∗ω2z2 1 R 2 · eff eff ~2 ×
Hˆ = 2m∗ + 20 −eEgz+Hˆso+ 2gµBσB (1) Hˆ = ~σ ΩD , with
D 2 · eff
whereP=p eAisanelectronmomentuminthepres-
ence ofa vect−orc-potentialA, m∗ ande are the electronic ΩDeff = 2~η2{Px(Py2−hPz2i),Py(hPz2i−Px2),0}. (5)
effective mass and charge, respectively; E is a strength
g
of the gate electric field. The second term in Eq. (1) is
the confining potentialin z-direction,approximatedas a Although the effective magnetic field ΩR , corre-
eff
parabola with a frequency ω0, which is a characteristic spondingtoRashbatermisperpendiculartothe2Dwave
parameter of the electron gas thickness. This potential vectorofanelectron,x-componentofΩD isinthesame
eff
does not produce a structural inversion asymmetry and, direction as p while its y-component is directed in the
x
consequently, SO interaction. Since Rashba SO interac- opposite to p direction.
y
tion in the conduction band of a semiconductor is deter-
Anexternalmagneticfieldischosentobe inthe plane
minedbytheelectricfieldinthevalencebandratherthan
of the 2D electron gas, along x axis B= B,0,0 under
by that in the conduction band [43], the parabolic con- { }
the gauge A= 0, Bz,0 . The magnetic field tends to
finementapproximationneglectsasmallinterfacecontri- { − }
polarize the electron spin in the x-direction due to the
butiontoRashbaSOcouplingconstant. Thelasttermin
Zeemaneffect,andcreatesanangularmomentumdueto
Eq. (1) is Zeeman splitting energy in the external mag-
the orbital motion.
netic field B with ω ~ = gµ B/2, where µ = e~ is
z B B 2m0 In order to solve Schr¨odinger equation i~∂Ψ(x,y,z;t) =
the Bohr magneton of a free electron with mass m0, g is ∂t
the effective Land´e factor, and σ = σ ,σ ,σ are the HˆΨ(x,y,z;t) with a spinor Ψ(x,y,z;t) = Ψ↑ one ex-
{ x y z} Ψ↓
Pauli spin matrices. presses the electron wave functions with spin-up Ψ
(cid:0) (cid:1) ↑
Spin-orbital Hamiltonian Hˆ in Eq. (1) contains and spin-down Ψ orientations as Ψ (x,y,z;t) =
so ↓ ↑,↓
Rashba [3, 4] term, HˆR, due to a macroscopic SIA and eikxx+ikyyψ↑,↓(z,t) which yields
4
∂ψ ~2 ∂2 m∗ω2z2 ~2k2
i~ ↑ = + +(k ~ω eE )z+ ψ + (α+iβ)(k +ω m∗z/~)+(β+iα)k +~ω ψ ;(6)
∂t −2m∗∂z2 2 y B− g 2m∗ ↑ y B x z ↓
(cid:26) (cid:27)
∂ψ ~2 ∂2 m∗ω2z2 ~2k2 (cid:2) (cid:3)
i~ ↓ = + +(k ~ω eE )z+ ψ + (α iβ)(k +ω m∗z/~)+(β iα)k +~ω ψ ,(7)
∂t −2m∗∂z2 2 y B− g 2m∗ ↓ − y B − x z ↑
(cid:26) (cid:27)
(cid:2) (cid:3)
where ω = eB/m∗c is the cyclotron frequency, ω = ψ (z) and ψ (z); they furthermore satisfy the condition
B ↑ ↓
ωB2 +ω02 is the effective frequency, and k = kx2+ky2 ψ↑,↓ =eiθψ↓∗,↑. Acoordinate-dependentterm,∝ωBm∗z,
in the off-diagonal part of Eqs. (6) and (7) is origi-
ips the modulus of a 2D-wave vector. In theqstation- nated from the orbital magnetic field effect, which links
ary case the wave function is chosen as Ψ(x,y,z;t) =
nth orbital of a spin-up electron with (n 1)th or-
exp( iEt/~)Ψ(x,y,z),where E is the totalenergyofan ±
− bital of a spin-down electron and vice versa. Eqs. (6)
electron. In the absence of SO interactions and Zeeman
and (7) are easily solved in the absence of this spatial-
term, Eqs. (6) and (7) are decoupled and reducedto the
dependent term. Indeed, let us replace z in the ’mix-
oscillator equation with real wave function
ing’ terms of Eqs. (6) and (7) by the coordinate of
~2 d2 m∗ω2 the magnetic orbital center z0, and seek the solution as
(cid:26)−2m∗dz2 + 2 (z−z0)2−E˜(cid:27)ψ(0)(z)=0, (8) Ψin(ng0)s(yxs,tye,mz)o=f eeqiukxaxti+oinkysyfψorn(0A)(z)anABdnn B. We get the follow-
n n
(cid:0) (cid:1)
where z is the z-coordinate of a magnetic orbit, z =
0 0
eerEggmy−∗skωpy2~eωcBtr,uamndofEt˜h=eqEua−nti~2z2mekd∗2 o+rb(ietEagl2,−mE~˜∗knωy2ω=B)~2ωi(snt+h1e/e2n)-, c(∗EA˜/~+ω−(E˜n/~−ω1/2n)An1−/2c)0BBn==00, (10)
corresponding to the nth state − 0 n − − n
ψn(0)(z)=(√π2nn!)−1/2e−(z2−az2B0)2Hn za−z0 (9) where the dimensionless coefficient c0 is given
(cid:18) B (cid:19) 1 ω2 eE ω
c = (α+iβ) k 0 + g B +(iα+β)k +ω ~ .
with Hn(z) and aB = ~/m∗ω being the Hermite poly- 0 ~ω yω2 ~ω2 x z
(cid:20) (cid:18) (cid:19) (cid:21)
nomial and the Bohr radius, correspondingly. (11)
p
SOinteractionsinEqs. (6)and(7)mixthetransverse- The energy spectrum is immediately obtained from
quantized levels, yielding the complex wave functions Eq.(10)
~2k2 (k ~ω eE )2 (k ω2+eE ω /~)2
E±(k ,k )= y B− g +ω~(n+1/2) (α2+β2) k2+ y 0 g B +
n x y 2m∗ − 2m∗ω2 ±( (cid:20) x ω4 (cid:21)
1/2
(k ω2+eE ω /~) (k ω2+eE ω /~)
+4αβk y 0 g B +ω2~2+2ω ~ α y 0 g B +βk . (12)
x ω2 z z ω2 x
(cid:20) (cid:21))
The coefficients A and B in the spinor are completely linear combinations of ψ(0)(z)
n n n
definedfromthenormalizationcondition A 2+ B 2 =
n n
1 and Eq. (10) | | | | ψ↑(z)=e−(z2−az2B0)2 ∞ an Hn z−z0 ; (14)
n=0 aB√π2nn! (cid:18) aB (cid:19)
X
A = 1 and B = |c0| . (13) ψ↓(z)=e−(z2−az2B0)2 ∞ p bn Hn z−z0 , (15)
n √2 n ±√2 c0 nX=0 aB√π2nn! (cid:18) aB (cid:19)
p
wherethecoefficientsa andb satisfythenormalization
n n
condition ∞ (a 2 + b 2) = 1. From the condition
n=0 | n| | n|
General solutions of Eqs. (6) and (7) are sought as that Eqs. (6) and (7) are complex conjugate each other,
P
5
one gets a = eiθb∗ and b = eiθa∗ with θ being a real are simplified
n n n n
phase shift. Therefore, a 2 = b 2 = 1. It is
n| n| n| n| 2 1
easy to estimate the average value of the spin operator
c = [iα(k ik )+β(k +ik )], and c =0,
components S = Sˆx,SˆyP,Sˆz = ~/2Pσx,σy,σz over the 0 ω0~ x− y x y 1
{ } { } (21)
stationary states given by Eqs. (14) and (15)
and, as a result, a mixing between the transverse-
~ quantized levels is left off (see, Eqs. (18) and (19)). A
hSˆxik = 2 (a∗nbn+b∗nan), simple exact expression for the energy spectrum in the
Xn absence of the magnetic field is obtained
~
hSˆyik =−i2 (a∗nbn−b∗nan), 1 ~2k2 e2E2
~ Xn En± =~ω0(n+ 2)+ 2m∗ − 2m∗ωg2
Sˆz k = (an 2 bn 2)=0, 0
h i 2 | | −| |
Xn ± (α2+β2)k2+4αβkxky, (22)
hSˆ+ik =~ a∗nbn, and hSˆ−ik =~ b∗nan. (16) q
which is a particular form of Eq. (12) written at B =0,
n n
X X
since Eq. (12) is exact in this limit.
So a spin precesses around the normal to the plane, and
By expressing b in Eq. (19) through a ,a and
Sˆz k averagesouttozero,whereasin-planecomponents substituting into Enq. (18) we get an equatnionn±fo1r the
h i
of the spin take finite values. The averageposition of an
vector a = a ,a ,a ,... . An equation for the vector
0 1 2
electron in the confining potential z can be calculated { }
b = b ,b ,b ,... is obtained by the same way; finally
h i 0 1 2
by the same way, { }
we get:
a
hzik = √B2 √n+1 {(a∗n+1an+a∗nan+1)+ Nˆa=0 (23)
Xn Mˆb=0 (24)
(b∗ b +b∗b ) +z , (17)
n+1 n n n+1 } 0
Nˆ and Mˆ are square penthadiagonal matrices of infinite
which means that an overlap between the neighbor-
order with non-zero entries N = 0 (M = 0) only if
ing transverse-quantized levels shifts the center of the i,j 6 i,j 6
i j 2, and Nˆ = (Mˆ)∗. Apart from the non-zero
magnetic orbit of both spin-up and spin-down electrons
| − | ≤
equally, in addition to the magnetic- and gate electric main diagonal Nn,n, the matrix Nˆ contains the first two
fields shift z0, in z-direction. diagonals,Nn,n±1andNn,n±2,aboveandbelowit,which
Equations for the coefficients a and b with n = are given as
n n
0,1,2,3,... can be obtained by putting Eqs. (14) and
(15) into Eqs. (6) and (7)
E 1 c 2
~E˜ω −n− 21!an−c0bn−√2n c1 bn−1− Nn,n2=n(cid:18)c~ω2 −n−2(n2(cid:19)+−1)~cEω2−| 0n| − 12 −
1 1
| | | | ; (25)
− 2(n+1) c1 bn+1 =0, (18) − ~Eω −n+ 21 − ~Eω −n− 23
~E˜ω −n− 21!bn−pc∗0an−√2n c∗1 an−1− Nn,n−1 =−√2n ~Eω −c∗1nc0− 21 + ~Eω −c∗0nc1+ 12!; (26)
− 2(n+1) c∗1 an+1 =0, (19) N = 2(n+1) c∗1c0 + c∗0c1 ; (27)
where c0 is defined by Eqp. (11), and the coefficient c1 is n,n+1 −p ~Eω −n− 21 ~Eω −n− 23!
given as 2 n(n 1) c 2
1
N = − | | ; (28)
n,n−2 − E n 1
ω m∗ p~ω − − 2
B
c1 =(α+iβ)2~ωrω~. (20) Nn,n+2 =−2 (n+E 1)(nn+23) |c1|2. (29)
p ~ω − − 2
Note that a =b =0 for n<0 in Eqs. (18) and (19).
n n
It is easy to see that the approximate equations (10) The energy spectrum has to be found from the secular
canbe obtainedfromEqs.(18)and(19) byneglecting all equation, by equating the determinant of the matrix Nˆ
terms c . to zero. The infinite penthadiagonalmatrix is truncated
1
∼
In the absence of the external magnetic field, B = 0, down to the first n rows and n columns, the roots of
theexpressionsforc andc ,givenbyEqs.(11)and(20), which can be found by numeric methods, [28, 29].
0 1
6
III. SPIN CURRENT IN EQUILIBRIUM charge current
e~ nm d2k eα eβ
The equilibrium spin current in the previous activi- hJxi= m∗ (2π)2kx− ~ hσyi+ ~ hσxi=
ties [37–42] has been studied for a pure 2D electron gas n=0Z
X
in the absence of an external magnetic field. This sec- nm d2k ~k β+iα β iα
e x + a∗b + − b∗a , (33)
tion is addressedto study the spin-currentin a quasi-2D (2π)2 m∗ ~ n n ~ n n
electron gas with finite thickness in the presence of an nX=0Z (cid:26) (cid:27)
in-plane magnetic field and the gate potential. In order ~ nm d2k α β
J =e k + σ σ +ω z
to write the continuity equation for the charge density h yi m∗ (2π)2 y ~h xi− ~h yi Bh i
(cid:26) n=0Z (cid:27)
ρ = eΨ(z)2 = ψ 2 + ψ 2 and for the γ-component X
oeqfuthatei|osnpsin(6d|)enasnit|dy↑(S|7γ)=ar|(e~↓/m|2u)(ltΨip†lσieγdΨ)t,oththeeSirchcro¨omdipnlgexer- =e nm (2dπ2k)2 ~mk∗y + α+~iβa∗nbn+ α−~iβb∗nan+
n=0Z (cid:26)
conjugate components ψ∗ or ψ∗, which yields X
↑ ↓ +ω z +ω a 2(n+1) (a∗ a +b∗ b ) , (34)
B 0 B B n+1 n n+1 n
(cid:27)
∂ρ p
+ J=0
∂t ∇· J =0 (35)
z
∂S h i
γ + JSγ =G , (30)
∂t ∇· γ Note that the equations (33)-(35) for the charge cur-
rent components can be obtained according to J =
i
h i
where J and JSγ are the charge- and the spin-current, evhv=iidarsiw=elli[bHˆy,ursi]n.g the Heisenberg equationof motion
correspondingly. A violation of the spin conservation in i dt ~ i
Theaveragevaluesofthespincurrentcomponentsread
the system results in an additional source term (torque)
as
G [39]inthespin-balanceequation. Thecomponentsof
γ
the charge- and spin currents read nm d2k ~k α β a
JSj = (a∗b∗) xSˆ + ǫ + ǫ n
h x i (2π)2 n n m∗ j 2 jx 2 jy b
e~ eα nX=0Z (cid:18) (cid:19)(cid:18) n(cid:19)
J+j =eβ−Ψi†2σ˜mΨ∗(cid:0)Ψ†∇e2jΨψ−†A∇ΨjΨ, †Ψ(cid:1)− ~ Ψ†(σ×zˆ0)jΨ(31) =Z (2dπ2k)2 (cid:26)~22mk∗xhσjik,nm + α2ǫjx+ β2ǫjy(cid:27), (36)
~ j − m∗c j
nm d2k ~k ω2 eE ω
JSj = (a∗b∗) y 0 + g B Sˆ +
and h y i (2π)2 n n m∗ ω2 m∗ω2 j
n=0Z (cid:26) (cid:18) (cid:19)
X
α β a
~2 + ǫ + ǫ +ω a 2(n+1)(a∗ b∗ )Sˆ n
JjSγ =−i4m∗[Ψ†σγ∇jΨ−∇jΨ†σγΨ]− 2 jy 2 jx B B n+1 n+1 j(cid:27)(cid:18)bn(cid:19)
− α4{Ψ†[σγ(σ×zˆ0)j +(σ×zˆ0)jσγ]Ψ}+ = (2dπ2k)2 ~m2k∗y ωω022 +peEmg∗ωωB2~ hσjik,nm +
Z (cid:26)(cid:18) (cid:19)
+ β4{Ψ†[σ˜jσγ +σγσ˜j]Ψ}− 2me~∗cAj(Ψ†σγΨ). (32) + α2ǫjy + β2ǫjx+ a2BωB~hσjiokf,nfm (37)
(cid:27)
and
where σ˜ = σ , σ ,0 , zˆ is a unit vector, normal to
x y 0
{ − }
the electron gas plane, and Aj is jth component of the ~ nm d2k
vector-potential. It is easy to see that the structure of JSz =i 2(n+1)
the contributions coming from Rashba and Dresselhaus h kzi m∗aB n=0Z (2π)2 ×
X p
t(e4r)masndis(s5im).ilaTrhteomthaegnHeatimcilfiteolndiahnasfoarmcognivtreinbubtyionEqtso. (a∗n+1b∗n+1)Sˆz abn =i2m~∗2a (2dπ2k)2hσzik,nm(3,8)
the charge current as well as to the spin current, which (cid:18) n(cid:19) B Xnm Z
cttahoiennseSidsOtsbiynwtimetrheaaccntoisnotnorsfi.bHuatimonilttoonthmeetchhoadrgeincuthrreenabts[e4n4c]eobo-f whJhySezrie=j =0; ǫxi,jyisanad2DhJzSaxnitis=ymhJmzSeytiric=te0n,soarlsowihtJhxSczoim=-
ponents ǫxy = −ǫyx = 1. The expressions for hσxik,nm,
The equilibrium spin- and charge currents at T=0 are hσyik,nm, hσzik,nm and their evident momentum depen-
foundbyaveragingEqs. (31)and(32)overthestationary dences are calculated in Appendix.
states,givenbythewavefunctions(14)and(15),andby It is evident that the normal to the electron gas com-
integrating over all occupied states, which yields for the ponent of JSz in Eq. (38) arises exclusively due to the
h i
7
transverse confinement and the in-plane magnetic field. where λ for n =0,1 indicates the sub-band index with
n
Nevertheless this term does not accumulate a spin, an λ = and λ = +, and the sign shows the spin-
0 1
− ±
electrontunnelingfromnthto(n+1)thlevelisaccompa- branch index. In-plane momentum dependence of the
nied,accordingtoEq. (38),byreverseflowfrom(n+1)th energy spectrum, Eq. (42), is determined by the terms
to nth level. c 2 and (c∗c +c c∗)2
| 0| 0 1 0 1
The spin-continuityequation(30)containsthe source-
term G due to a violation of the spin conservation, the
γ
components of which are given by the following expres- c 2 = 1 (α2+β2) k ω02 + eEgωB 2+k2 +
sions | 0| ~2ω2 yω2 ~ω2 x
(cid:26) (cid:20)(cid:18) (cid:19) (cid:21)
Gj =−iα2{Ψ†[σ×(zˆ0×∇)]jΨ−[(∇×zˆ0)×σ]jΨ†Ψ}− 4αβkx kyωω022 + eE~ωgω2B +
β e (cid:18) (cid:19)
−i2{Ψ†(∇˜ ×σ)jΨ−(∇˜ ×σ)jΨ†Ψ}+βcΨ†(A×σ)Ψ+ 2ωz~ α kyωω022 + eE~gωω2B +βkx , (43)
+ 1gµ Ψ†(B σ)Ψ+αeΨ†((zˆ A) σ)Ψ, (39) (cid:20) (cid:18) (cid:19) (cid:21)(cid:27)
B 0
2 × c × ×
where ˜ = , ,0 and σ˜ = σ , σ ,0 . The
x y x y
∇ {∇ −∇ } { − }
averagingof x- and y-components of the torque over the
quantum-mechanic states gives
ω m∗ ω2 eE ω
c∗c +c c∗ = B (α2+β2) k 0 + g B +
hGxi=0, and hGyi=0, (40) 0 1 0 1 (ω~)2rω~(cid:26) (cid:18) yω2 ~ω2 (cid:19)
whichisinconsistencewitharesultinstrictly2Dsystem +2αβkx+αωz~ (44)
[39] in the absence of an external magnetic field. Nev- (cid:27)
ertheless z-component of the torque is not averaged to
zero: So, c 2 O(α2,β2), whereas (c∗c + c c∗)2
| 0| ∼ 0 1 0 1 ∼
nm d2k β O(α4,β4,α2β2) in the absence of Zeeman splitting.
hGzi= (2π)2 (αkx+βky + ~z0ωBm∗)(a∗nbn+ Therefore,expansionof Eq. (42) oversmall SO coupling
n=0Z constants up to quadratic in α,β terms yields
X (cid:8)
1 α
+b∗a ) i(αk +βk + gµ B+ z ω m∗)(a∗b
n n − y x 2 B ~ 0 B n n−
−b∗nan)−iα~aBωBm∗ 2(n+1) (a∗n+1bn−b∗n+1an)+ E±(0) ≈ ~22mk∗2 − (eEg2−m~∗ωωB2ky)2 + 21~ω±|c0|~ω; (45)
β p
+ a ω m∗ 2(n+1) (a∗ b +b∗ a ) . (41)
~ B B n+1 n n+1 n
p (cid:9)
The integration at T = 0 is taken over each momen-
tum component i = x,y and for both spin branches in ~2k2 (eE ~ω k )2 3
eachtransverse-quantizedsubbanduptotheFermilevel, E±(1) ≈ 2m∗ − g2−m∗ωB2 y + 2~ω∓|c0|~ω. (46)
Ki k Ki . Inordertocalculatetheaverage
− nm,± ≤ i ≤ nm,±
valuesofthe charge-andspincurrentcomponents,given
byEqs. (35)-(30),evidentexpressionsofa andb arere- The limit of integration over the occupied states can be
n n
quired. TheFermilevelisassumedtobesetbetweennm found by fixing the Fermi energy EF in Eq. (42) and
and n +1 subbands, so that all levels up to K ,n solving this equation for the momentum. It is neces-
m m m
are occupied with a = 0, b = 0 for n { n and} sarytonotethataninterferencebetweenthegateelectric
n n m
a = b = 0 for n > n6 . A simp6 lest case, w≤hich takes field and the orbital effect of the in-plane magnetic field
n n m
intoaccounttheinter-subbandmixingduetoaninterfer- shifts the Fermi surface along ky axis (see, Eqs. (45),
ence between the SO interactions and in-plane magnetic (46) and (43)). Furthermore, Zeeman splitting makes
field, is nm =1. In this case equations (18) and (19) are the energy spectra asymmetric along both kx and ky
simplified to the form, given by Eq. (79) in Appendix. axes. Therefore, the integrations over kx and ky have to
Theanalyticalexpressionsfortheenergyspectruminthe be taken, generally speaking, over asymmetric intervals
firstandsecondtransverse-quantizedsubbandsaregiven −Kn′xm,± ≤kx ≤Knxm,± and −Kn′ym,± ≤ky ≤Knym,±.
according to Eqs. (82)-(85)
The integrationlimit is calculatedin Appendix by fix-
~2k2 (eE ~ω k )2 ~ω ing the Fermi energy and transforming the momentum
E±(n) = 2m∗ − g2−m∗ωB2 y +~ω+λn 2 × componentsinEqs. (45)and(46)intopolarcoordinates,
k =kcosϕ, k =ksinϕ. WeexpressEq. (110)forkF
x y n,±
1+4c 2+8c 2 4 c 2+2(c∗c +c c∗)2,(42) as kF =kF δk, where
| 0| | 1| ∓ | 0| 0 1 0 1 n,± n ±
r
q
8
1 eE ω 2m∗ ω2
kF = g B sinϕ+ [E ω~(n+1/2)](cos2ϕ+ 0 sin2ϕ)+
n cos2ϕ+ ωω022 sin2ϕ(cid:26)− ~ω2 (cid:20) ~2 F − ω2
e2E2 m∗2 ω4 ω2 1/2
g + (α2+β2) cos2ϕ+ 0 sin2ϕ +4αβ 0 sinϕcosϕ , (47)
ω2~2 ~4 ω4 ω2
(cid:18) (cid:19) (cid:21) (cid:27)
(cid:2) (cid:3)
m∗ ω4 ω2
δk = (α2+β2) cos2ϕ+ 0 sin2ϕ +4αβ 0 sinϕcosϕ. (48)
~2(cos2ϕ+ ωω022 sin2ϕ)s (cid:18) ω4 (cid:19) ω2
for n=0,1. In-planemagneticfieldinducesaninter-sub-bandcou-
The average values of the Pauli spin-matrices pling terms, σ off with i = x,y, which give a con-
h iik,nm
hσjik,nm,j =x,y,zarecalculatedinAppendix. Byusing tribution to the y-components of the spin current (37).
theexpressions(11),(20),(43)and(99)inEqs. (94),(95) These terms are calculated in Appendix. By neglect-
and (98) in Appendix, one gets an explicit momentum- ing the small terms [c∗2c + c2c∗ + (c∗ + c )c 2] and
0 1 0 1 1 1 | 0|
dependent expressions for the averagedPauli matrices [c∗2c c2c∗+(c∗ c )c 2], Eqs. (97) and (97) in Ap-
0 1− 0 1 1− 1 | 0|
pendix are approximated as
ak+b
σ = λ , (49)
h xik±,n ± n√ck2+dk+e
a¯k+¯b
σ = λ i , (50)
h yik±,n ∓ n √ck2+dk+e
a˜k+˜b
σ = λ i , (51)
h zik±,n ∓ n √ck2+dk+e σ off c∗1+c1 =λ (c∗+c )=
h xik,nm ≈ 2(ǫ˜ 1) n 1 1
where k is the momentum modulus k ,k = −
x y
{ } αω m∗
kcosφ,ksinφ in spherical-polar system and B
=λ , (61)
{ } n ω~ ω~
r
ω2 c∗ c
a=α 0 sinφ+βcosφ, (52) σ off i 1− 1 = iλ (c∗ c )=
ω2 h yik,nm ≈− 2(ǫ˜ 1) − n 1− 1
−
eE ω
b=α ~gω2B +ωz~, (53) =−λnβωω~B mω~∗. (62)
ω2 r
a¯=αcosφ+β 0 sinφ, (54)
ω2
eE ω
¯b=β g B, (55)
~ω2
ω m∗
a˜= B (α2 β2)cosϕ, (56)
ω~ ω~ −
r So, the inter-subband coupling terms σ off and
˜b=−ωωBrmω~∗βωz, (57) dhσeypieoknf,ndfmondetpheensdpionn-lbyraonncthhiensduebx-.band indhexxaink,dnmdo not
ω2 2 ω2 2
c= α 0 sinφ+βcosφ + β 0 sinφ+αcosφ ,(58)
ω2 ω2
(cid:18) (cid:19) (cid:18) (cid:19)
2eE ω ω2 ω2
d=2 g B α α 0 sinφ+βcosφ +β β sinφ Inordertocalculatetheequilibriumspin-currentcom-
~ω2 ω2 ω2
(cid:18) (cid:19)(cid:20) (cid:18) (cid:19) (cid:18) ponents, Eqs. (36)- (38) are integrated firstly over the
ω2 momentummodulus k by taking into accountEqs. (49)-
+αcosφ +2ω ~ α 0 sinφ+βcosφ , (59)
z ω2 (62). Routine calculations yield under this condition the
(cid:19)(cid:21) (cid:18) (cid:19)
2 2 following results for the equilibrium spin-current JSx =
eE ω eE ω
e= α ~gω2B +ωz~ + β ~gω2B . (60) {JxSx,JySx,0}
(cid:18) (cid:19) (cid:18) (cid:19)
9
m∗2ω 1 eE ω 2 1 1
JSx = 0β(β2 α2)+ g B (α2+2β2 α2 β2 )+ (α2 β2)2
h x i 3π~4ω − 2πβ ω2~ 2 −| − | 8α2 − −
(cid:18) 0 (cid:19) (cid:26)
α2+β2 α2 β2 ω eE ω (cid:2)
α2 β2 (α2+β2) + 1 | − | + z g B α2+β2 α2 β2 +
| − | 3 − (α2 β2) 4παβ ω2 −| − |
(cid:18) − (cid:19)(cid:27) 0 (cid:26)
7α2 α2 β2(cid:3) 5ω2~2 α2 β2 α2+β2 α2 β2
1 | − | + z 1 | − | −| − | , (63)
3 − (α2 β2) 12πβ − (α2 β2) − α2
(cid:18) − (cid:19)(cid:27) (cid:26) − (cid:27)
m∗2ω 3ω2 αm∗ω2 3 1 eE ω 2
JSx = 0α (α2 β2)+ B(α2+β2) + B E ω~ g B 3α4
h y i 3π~4ω − ω2 π~2ωω F − 2 − 48παβ2 ω2~ −
0 0 (cid:18) (cid:19) (cid:18) 0 (cid:19) (cid:26)
(cid:2) (cid:3) (α4 β4) ω eE ω
17β4 62α2β2+3(3β2 α2)α2 β2 8β2 − z g B 3α4 3β4
− − − | − |− α2 β2 − 12πα2β2 ω2 − −
| − |(cid:27) 0 (cid:26)
α2 β2 5ω2~2 α2 β2 α2+β2 α2 β2
3(α2 β2)α2 β2 6α2β2| − | + z 1+ | − | −| − | , (64)
− − | − |− (α2 β2) 12πα (α2 β2) − β2
− (cid:27) (cid:26) − (cid:27)
as well as for JSy = JxSy,JySy,0
{ }
m∗2ω 1 eE ω 2
JSy = 0α(β2 α2)+ g B 3(α2+5β2)α2 β2 (α2+β2)(3α2+23β2)
h x i 3π~4ω − 48παβ2 ω2~ | − |− −
(cid:18) 0 (cid:19) (cid:26)
α2 β2 ω eE ω
8β2(α2+β2)| − | z g B (α2 β2)α2 β2 (α4 β4)+
− (α2 β2) − 24πα2β2 ω2 − | − |− −
− (cid:27) 0 (cid:26)
α2 β2 ω2~2 α2 β2 α2+β2 α2 β2
+2α2β2| − | z 1+ | − | −| − | , (65)
(α2 β2) − 6πα (α2 β2) − β2
− (cid:27) (cid:26) − (cid:27)
m∗2ω 3ω2 βm∗ω2 3 1 eE ω 2
JSy = 0β (α2 β2) B(α2+β2) B E ω~ + g B 3α4+
h y i 3π~4ω − − ω2 − π~2ωω F − 2 48πα2β ω2~
(cid:20) 0 (cid:21) 0 (cid:18) (cid:19) (cid:18) 0 (cid:19) (cid:26)
α2 β2 ω eE ω
+3β4 3(α2+β2)α2 β2 8α2(α2+β2) 1 | − | + z g B 5α2 3β2
− | − |− − (α2 β2) 24παβ ω2 − − −
(cid:18) − (cid:19)(cid:27) 0 (cid:26)
α2 β2 ω2~2 α2 β2 α2+β2 α2 β2
3α2 β2 +2(4α2 3β2)| − | z 1 | − | −| − | , (66)
− | − | − (α2 β2) − 6πβ − (α2 β2) − α2
− (cid:27) (cid:26) − (cid:27)
Thenon-zerocomponentofSz spin-current,JSz ={0,0,JzSz},inducedbythein-planemagneticfield,canbepresented
as
m∗ω (α2 β2) eE ω α2 β2 m∗βω ω
JSz = B − (α2+β2 α2 β2 ) g B +αω ~ 1 | − | + B z. (67)
h z i 4παβω~2 −| − | ~ω2 z − α2 β2 πω ~
(cid:26) (cid:18) (cid:19) (cid:18) − (cid:19)(cid:27) 0
The above expressions for the persistent spin current are simplified considerably as the gate voltage and Zeeman
splitting vanish , E =ω =0,
g z
m∗2ω m∗2ω 3ω2 αm∗ω2 3
JSx = 0β(β2 α2), 0α (α2 β2)+ B(α2+β2) + B E ω~ , 0 (68)
3π~4ω − 3π~4ω − ω2 π~2ωω F − 2
(cid:26) (cid:20) 0 (cid:21) 0 (cid:18) (cid:19) (cid:27)
m∗2ω m∗2ω 3ω2 βm∗ω2 3
JSy = 0α(β2 α2), 0β (α2 β2) B(α2+β2) B E ω~ , 0 ; (69)
3π~4ω − 3π~4ω − − ω2 − π~2ωω F − 2
(cid:26) (cid:20) 0 (cid:21) 0 (cid:18) (cid:19) (cid:27)
JSz = 0,0,0 . (70)
{ }
Although the spin-current components turn to vanish at 2D electron gas in the absence of the magnetic field
α = β in the absence of the magnetic field, the latter
m∗2 m∗2
destroys this symmetry and yields a finite spin current JSx = β(β2 α2), α(α2 β2), 0 ;(71)
3π~4 − 3π~4 −
as α β. Indeed, the expressions (63)-(67) yield the (cid:26) (cid:27)
well-k→nown results of Refs. [41, 42] for spin current in a m∗2 m∗2
JSy = α(β2 α2), β(α2 β2), 0 , (72)
3π~4 − 3π~4 −
(cid:26) (cid:27)
JSz = 0, 0, 0 .(73)
{ }
10
It is easy to check that the diagonal components of the for arbitrary α=0. Nevertheless
spincurrentvanish,hJxSxi=hJySyi=hJzSzi=0asβ →0 6
m∗2ω α3 3ω2 αm∗ω2 3 29α eE ω 2
JSx = 0 1+ B + B E ω~ + g B ; (74)
h y i 3π~4ω ω2 π~2ωω F − 2 24π ~ω2
(cid:18) (cid:19) 0 (cid:18) (cid:19) (cid:18) 0 (cid:19)
m∗2ω α3 11α eE ω 2
JSy = 0 g B , (75)
h x i − 3π~4ω − 24π ~ω2
(cid:18) 0 (cid:19)
under these conditions. On the other hand the diagonal spin-current components are nonzero for α 0 and β =0
→ 6
m∗2ω β3 11β eE ω 2
JSx = 0 + g B ; 0; 0 , (76)
h i ( 3π~4ω 24π (cid:18) ~ω02 (cid:19) )
m∗2ω β3 3ω2 βm∗ω2 3 β eE ω 2
JSy = 0; 0 1+ B B E ω~ g B ; 0 , (77)
h i ( − 3π~4ω (cid:18) ω2 (cid:19)− π~2ω0ω (cid:18) F − 2 (cid:19)− 3π (cid:18) ~ω02 (cid:19) )
m∗βω ω
JSz = 0; 0; B z . (78)
h i 2πω ~
(cid:26) 0 (cid:27)
New contributions to the pure 2D ( α3,β3) spin- mixing by means of in-plane magnetic field. Although
∼
current expressions in Eqs. (63)-(66) are caused by theequilibriumspincurrentvanishesatα=β inthe ab-
electron-transfer mechanism between nearest-neighbor sence of the magnetic field (see, Refs. [41, 42]), in- plane
transverse-quantizedsub-bands, induced by the in-plane magnetic field destroys this symmetry, yielding non-zero
magneticfield,andtheyvanish,consequently,withmag- persistent spin-current at α = β. The magnetic field
netic field. These terms are proportional either to the strongly changes JSx and JSy spin-current components,
gateelectricfieldortoZeemansplitting too. Indeed,the and contributes new terms to them.
gateelectricfieldchanges,ontheonehand,theelectronic
energy spectrum, and shifts, on the other hand, the cen-
ter of an magnetic orbit z0 along z-axis. The coefficient ACKNOWLEDGMENT
c ,whichhasaphysicalmeaningofprobabilityamplitude
0
foranelectrontransitionfromonespin-polarizedbranch
This research was supported by the DFG under grant
to other one in the same transverse-quantized subband
Op28/8-1 and by the governmental grant of Azerbaijan
n, according to Eqs. (18) and (19), parametrically de-
Republic under grant EIF-2010-1(1)-40/01-22.
pends on E and ω . The new terms in the spin-current
g z
expressionsdependlinearlyontheSOcouplingconstants
in the simplest case if one of the SO coupling constant
APPENDIX
is zero. Therefore, these terms may dominate over the
pure2Dtermsforsomevaluesofα, β, E andmagnetic
g
field. Equations(18)and(19)arereducedforn=1intothe
following form
IV. CONCLUSIONS (ǫ˜ 3/2) a c b √2c b =0
1 0 1 1 0
− − −
(ǫ˜ 3/2) b c∗ a √2c∗ a =0
Inthispapertheequilibriumspincurrentiscalculated − 1− 0 1− 1 0
(ǫ˜ 1/2) a c b √2c b =0
for a quasi-2D electron gas with finite thickness under 0 0 0 1 1
− − −
in-plane magnetic field in the presence of Rashba- and (ǫ˜ 1/2) b c∗ a √2c∗ a =0, (79)
− 0− 0 0− 1 1
Dresselhaus spin-orbit interactions. Note that the prob-
lemhasbeensolvedfor2DelectrongasinRefs. [37–42]in where ǫ˜= E˜/(~ ω), and the higher order terms a and
2
the absence of the magnetic field. Our calculations show b are neglected, since the levels n = 0 and n = 1 only
2
that the in-plane magnetic field generates out-of-plane are filled. Excluding a and b from these equations one
1 1
spin current, which appears exclusively due to sub-band gets