Kinetic investigation on extrinsic spin Hall effect induced by skew scattering

0.18 MB·English

by  J. L. Cheng

Kinetic investigation on extrinsic spin Hall effect induced by skew scattering
J. L. Cheng and M. W. Wu∗
Hefei National Laboratory for Physical Sciences at Microscale and Department of Physics,
University of Science and Technology of China, Hefei, Anhui, 230026, China
(Dated: February 5, 2008)
The kinetics of the extrinsic spin Hall conductivity induced by the skew scattering is performed
fromthefullymicroscopickineticspinBlochequationapproachin(001)GaAssymmetricquantum
8
well. In the steady state, the extrinsic spin Hall current/conductivity vanishes for the linear-k
0
dependentspin-orbitcouplingandisverysmall forthecubic-kdependentspin-orbitcoupling. The
0
spin precession inducedbytheDresselhaus/Rashba spin-orbitcoupling playsaveryimportant role
2
in the vanishment of the extrinsic spin Hall conductivity in the steady state. An in-plane spin

n polarization is induced by the skew scattering, with the help of the spin-orbit coupling. This spin
a polarization is verydifferent from thecurrent-inducedspin polarization.
J

0 PACSnumbers: 72.25.Pn,72.25.Rb,71.70.Ej,71.10.-w
1

] Generating and manipulating the spin polarization in additional spin-dependent electron-impurity2,48 or the
l
l semiconductors is one of the important prerequisites for electron-phonon16,17 skew scattering. The extrinsic SHE
a
h the realization of the new spintronic device.1 Spin Hall induced by the skew scattering alone has been widely
- effect (SHE) is considered as the convenient method to studied by using both the Kubo formula and the Boltz-
s generatespinpolarizationinadditionaltothetraditional mannequationmethod,6,7,28,29,30,34,35,36andthenonzero
e
m methods such as the external magnetic field, the cir- extrinsic SHC is obtained. Lately, Tse and Das Sarma31
cular/linear polarized laser,2,3 the spin-galvanic effect4 have proved the vanishment of the extrinsic SHC by
.
t and the spin injection from the ferromagnetic metal to considering the vortex correction of the linear SOC in
a
m semiconductor.5 The SHE is induced by intrinsic or ex- the Kubo formula. However, a fully microscopic cal-
trinsicspin-orbitcoupling(SOC)6 whichgivesrisetothe culation of the extrinsic SHC from the kinetic equation
-
d spin current vertical to the charge current without ap- approach is still missing and the vanishment of the ex-
n plying external magnetic field and/or spin accumulation trinsic SHC by the vortex correction from the Kubo ap-
o at sample edges.7,8,9 Experimentally, the spin Hall con- proach needs to be verified from the kinetic approach.
c ductivity(SHC)isestimatedindirectlybythe spinaccu- The other is the additional spin-dependent position and
[
mulation at the sample edges,10,11,12,13,14 or the charge velocityoperatorswhich bringthe correctionto the defi-

2 current with a transverse magnetic field applied in a nitionofthespincurrent,andarereferredtoastheside-
v gyrotropic systems.15,16,17 Recently, a direct electronic jumpmechanism.6 Heretwocorrectionsonthedefinition
0
measurementsoftheSHEisgivenbyValenzuelaandTin- should be specified. The first comes from the electrical
7
kham in the metallic conductor.18 All these effects are potential which gives an intrinsic-like contribution and
3
explained as the intrinsic6,19,20,21,22,23,24,25,26,27 and/or againcannotcontributetothespinaccumulationaccord-
0
. the extrinsic SHE6,7,28,29,30,31,32,33,34,35,36,37,38 theoreti- ing to Refs. 24,25,47. The second comes from the spin
6 cally by using the Kubo formula19,20,30,31 or the Boltz- dependentscatteringinwhichhighordercorrelationsbe-
0
mann equation6,7,23,34,36 with only the carrier-impurity tweendifferentwavevectorsneedtobeconsidered.6,34 It
7
scattering included. ishardtoincludethesecondcorrectionintheBoltzmann
0
: equation approach,6 though it also gives rise to the ex-
v The intrinsic SHE is induced by the intrinsic SOC
trinsic SHE, together with the skew scattering. In the
i (i.e., theDresselhaus39 and/orthe Rashba40 SOCs)with
X following, we concentrate on the extrinsic SHE induced
the applied external electric field and the resulting SHC
r by the skew scattering.
a was at first thought as dissipationless one in perfect
crystals.19,20 Later investigations proved that this kind The spin polarization is not only accumulated at the
of SHC disappears even for the infinitesimal impurity sampleedgesduetotheextrinsicSHE,butalsoobserved
density with the vertex correction22,23,41,42,43,44 in the simultaneously inside the samples which is induced by
Rashba model or the linear Dresselhaus model in quan- thechargecurrent.10,11,13,49,50,51,52 Bycomparingtheex-
tum wells,45 but remains a finite value for the cubic periments in Refs. 10,13,51, it is easy to find that both
SOC.46 However, the spin current is not an observable spin polarizations are in the same order. In theory, En-
quantity. The Laughlin’sgaugegedankenexperimentin- gelet al.49 andTrushinandSchliemann52 attributedthe
dicatesthattheintrinsicSHEcannotleadtoanyspinac- current-inducedspinpolarization(CISP)inthe homoge-
cumulation at sample edges,24,25 unless in a mesoscopic neous system as the results of the current induced effec-
system. InadditionaltotheDresselhausandtheRashba tive magnetic field (EMF) from the SOC.51 Tarasenko
SOC, the mixing between the valance and the conduc- showed that the spin-flip phonon scattering in asym-
tion bands gives rise to two corrections: One is the metric quantum wells can also induce this kind of spin
2
polarization.50 However, the spin-conserving skew scat- E
tering in two dimensional GaAs semiconductor can also k
y
induce spinpolarizationinside the sample due to the ex-
trinsicSHE.Thiseffecthasnotbeenstudiedinthe liter-
ature. Moreover,afullymicroscopickineticinvestigation
on the extrinsic SHE is also missing in the literature.
In this paper, we focus on the kinetic process of the
extrinsicSHEinducedbytheskewscatteringinsymmet- kx
ricGaAs(001)quantumwellfromthekineticspinBloch
equation(KSBE)approach.53,54 Wedemonstratetheim-
portantroleofthespinprecessioninducedbythe(intrin-
sic)Dresselhaus/RashbaSOCtothe SHEandshowthat
itisinadequatetostudytheextrinsicSHEfromtheKubo
formalism without considering the Dresselhaus/Rashba
SOC in the literature. We further show that the extrin- FIG. 1: (Color online) The schematic for the skew scatter-
sic SHE can generate spin polarizations in homogeneous ing and the intrinsic spin orbit coupling in the k-space. ⊙
system. (⊗)standsforthespin-up(-down)polarization. Thebluear-
By using the non-equilibrium Green function method rowsgivethedirectionoftheEMFfromtheDresselhausSOC
and the generalized Kadanoff-Baym Ansatz,55 we con- which rotatesthespinpolarization tothedirection indicated
struct the KSBE53,54 for electrons as follows bythe red dashed arrows.
∂ρ (t) ∂ρ ∂ρ ∂ρ ∂ρ
k k k k k
eE + + + =0. (1)
∂t − ∂k ∂t (cid:12) ∂t (cid:12) ∂t (cid:12) side. A schematic for the skew scatteringis givenin Fig.
x (cid:12)coh (cid:12)scat (cid:12)ss
(cid:12) (cid:12) (cid:12) 1 for a left moving electron.
f (cid:12)ρ (cid:12) (cid:12)
Here, ρ (t) = k↑ k↑↓ is electron density ma- We first analysisthe KSBE for some simple cases. Af-
k (cid:18)ρk↓↑ fk↓(cid:19) ter carrying out the summation over k, one gets the
trix with wave vector k at time t. The applied elec- equation of continuity for the spin density S= s =
ntreitcicfiefiledldEBisisaasslounmgedthaeloxn-gy (twheellx)-apxlaisnea.ndThtheecomhaegr-- kTr[ρkσ] as Pk k
P
ent term describes the spin precession along the mag- ∂S(t)
ΩD(k) s (t) gµ B S(t)=0 . (3)
netic/effective magnetic field and is given by ∂∂ρtk = ∂t −Xk × k − B ×
(cid:12)coh
i[1(ΩD(k)+gµ B) σ,ρ (t)],whereΩD(k)=γ(k(cid:12)(k2
2 B · k x(cid:12) y− Thesummationofthespin-conservingscatteringisnatu-
(πa)2),ky((πa)2 − kx2),0) represents the EMF from the rally zero. The summation of the skew-scattering in Eq.
Dresselhaus SOC39 with γ standing for the material- (2) for the two-dimension system is also zero because it
determinedSOCstrength56,57andabeingthewellwidth. canonlyskewthespin-upand-downelectronsseparately
Infinite-well-depth assumption is adopted here and only instead of flip them. Equation (3) is consistent with the
the lowest subband is taken into account due to small result in Ref. 44 which is obtained by using the general
well width. The spin conserving scattering ∂ρk is operator commutation relations. When only the linear-
∂t
(cid:12)scat k terms in ΩD(k) is retained, for the steady state one
givenbythespinconservingelectron-impurityscat(cid:12)tering,
(cid:12) obtains
the electron-phonon scattering and the electron-electron
Coulomb scattering which are given in detail in Ref. 58. egµ B
Jz = B y Sz . (4)
The skewscattering is givenby the third orderexpan- y γm∗((π/a)2 k2 )
sion of the electron-impurity scattering and reads:6,30,31 −h xi
Here Jz = e ~k/m∗(f f ) is the spin cur-
∂ρk = 2π2N λγ2 δ(ε ε )δ(ε ε ) rent which i−s tPheksimplest kb↑ut−thke↓ most widely-used
∂t (cid:12) − i k2 − k k1 − k definition6 without considering contribution from the
(cid:12)(cid:12)ss k1kX2;q1q2 side-jumpeffect. TheSHCisgivenasσz =Jz/E. Inthe
V(k(cid:12) k ,q )V(k k ,q )V(k k, q q ) y y
××{(k−−k11)×1(k2−1k−)·σ2,ρ2k2} . 2− − 1− 2 (2) d−emrei∗vhaktiy2oinJ,xzwaenadlsotkakkeyktx2hsezkap=pr−oxmeim∗haktx2iioJnyzPwkitkhxkhky2sx2zk/y=i,
Here λγ2 = η(2−η) is the strength of the skew scat- the average valuePof k2 . If the strain-induced SOC
2m∗Eg(3−η) x/y
tering with η = ∆+∆Eg.2 Eg and ∆ are the band gap and RHassh=baα(SkOyσCx)−isktaxkσeyn)i(nwtoithactchoeunsta,mEeq.ex(4p)recshsaionngeosfinthtoe
the spin-orbit splitting of the valance band separately.
V(q,qz) = q2+−qez22+κ2I(q2zπa) with I(x) = eiπxπsxi(n1(−πxx)2) Jz = γ((πa)2−hky2i)By −2αBx egµBSz . (5)
being the form factor and κ denoting the Debye-Hu¨cke y γ2((π)2 k2 )((π)2 k2 ) 4α2 m∗
screening constant. This term produces an asymmetric a −h xi a −h yi −
spin-conserving scattering of electrons so that the spin- Withouttheexternalmagneticfield,Eqs.(4)and(5)give
up/downelectronsprefertobescatteredtotheleft/right Jz = 0, which verifies the zero extrinsic SHC given by
y
3
Ref. [31] from the Kubo approach. However, the skew 100
1.2
scatteringdoesgeneratethe spincurrentswhenthe elec-
tric field is applied (in Fig. 1), but it is just the dynamic
10 1
one. To make clear how the spin current disappears, we − )0.3 0.1
) h
rewrite the x-component of Eq. (3) as h 2/ h)
It is hence∂eSasxy/∂tto=finmd∗tγh[a(πt/tah)e2s−pinhk-Hz2ia]Jllyzc/uerr.ent is co(6n)- 22/e− 10−2 2(10e−00..21 22/(10e 0.8 4P−x
verted to the spin polarization along the x-axis and it (10 10−3 σzy 00 2 40σx 10−
tends to zero in the steady state. This can further be zy t(ps) 0.4
understood from Fig. 1: after the spin current is excited σ
10 4
by the skew scattering, the spin polarization distributes −
( and ) anti-symmetrically at k . However the y
y
⊙ ⊗ ±
components of the EMF due to the SOC (the blue ar- 10 5 0
−
rows) have the same symmetry and rotate the spin po- 0 200 400 600 800 1000
larizations to the x direction (the red dashed arrows). t (ps)
−
Therefore, the spin Hall current is converted to the in-
plane spin polarization.
Now we show the time evolution of the SHC σz and FIG.2: TimeevolutionoftheSHCwith(solidcurve)/without
y
the spin polarization P =Sx/N calculated by numeri- (chain curve) coherent terms and the time evolution of the
x e
cally solving the KSBE with all the scattering explicitly spin polarization (dashed curve) along the x-axisat T =200
included at T =200 K in Fig. 2. The parameters in the K. Ni = Ne = 4×1011 cm−2 and the electric field E =
0.1 kV/cm. The corresponding time evolutions of the SHC
calculation are taken as following: the electron and im-
purity densities N = N = 4 1011 cm−2; a = 7.5 nm; of the first 4 ps are shown in the inset, together with the
E = 0.1 kV/cm; γe = 1i1.4 eV×˚A3; and √λγ = 2.07 ˚A. chargeconductivityσx. Itisnotedthatthescalesofthespin
· polarization/charge conductivity are on the right hand side
From the figure, the SHC increases with time first from
of the frames.
zero value to a maximum one in nearly 1 ps, then de-
creasesslowly to a very small value (in stead of zero due
to the inclusion of the Dresselhaus SOC with the cubic
Dresselhaus/Rashba SOC. To show this effect, we drop
k terms) in a characteristic time scale about 50 ps. The
thecoherenttermandplotthetimeevolutionoftheSHC
spinpolarizationis alongthe x-axisandincreasesfrom
zero to its steady state P =1−.2 10−4. by chain curves in Fig. 2. We find that the steady SHC
The above evolution isx easy t×o understand with the σyx ∼0.4×10−3σx is much closer to the one above given
bytheKuboformulaat0K.Therefore,weconcludethat
help of the schematic in Fig. 1. When the positive elec-
it is not suitable to calculate the extrinsic SHC without
tricfieldisapplied,theskewscatteringscattersthespin-
considering the spin precession. Furthermore, the side
downelectronstotheupperpanelofthek-spaceandthe
jumpmechanismgivesantime-independentcontribution
spin-up ones to the lower panel. This leads to the spin
as σz = 2λγ2e2N 2 10−3e2/h with the opposite
currents flowing along the y-direction. The strength of y − e ∼− ×
sign of the one from the skew scattering.
theskewscatteringisdeterminedbytheshiftoftheelec-
Itisfurthernotedthatalthoughthespinprecessionin-
tron distribution, so the SHC increases fast to its max-
imum value σz 2.5 10−3e2 at the time scale of the ducedbytheEMFleadstothevanishmentoftheSHC,it
y ∼ × h plays a very important role in generatingthe spin polar-
charge current (see the dashed curve in the inset of Fig.
ization. From the discussion above, it is easy to see that
2). Then the spin polarization precesses along the y-
theskewscatteringalonecannotgenerateanyspinpolar-
components of the EMF from the SOC to the x direc-
− ization due to the anti-symmetrical spin polarization at
tion to generate the in-plane spin polarization. Due to
k . DifferingfromtheCISP49wheretheSOCisusedto
the symmetry, the spin polarizations along the y and z ±proyvide a current induced EMF,10,58 here the SOC acts
axes are zero.
asananti-symmetricalprecessionfield,whichisthesame
It is interesting to see that the obtained SHC in the
as the spin polarization induced by the spin-dependent
steady state is orders of magnitude smaller than the one
phononscattering.3 We further note that the spinpolar-
obtainedfromtheKuboformulawidelyusedinthelitera-
ization induced by the skew scattering is very different
turewhereonlythelowestorderdiagramisconsidered.30
fromthe CISP:(i)Thespinpolarizationinducedbycur-
With only the impurity scattering included at zero tem-
rent induced EMF prefers the small well width which
perature, the Kube formula gives the extrinsic SHC30 as
2πm∗λγ2εFσ 3.37 10−3σ withthechargeconductiv- gives a large EMF,49 whereas the one induced by the
~2 x ∼ × x skew scattering and the spin precession prefers a large
ity σx = nem2τ. This value is orders of magnitude larger well width. The later can be seen as following: from
thanourresultσz 4.1 10−8σ ,obtainedfromthefully Fig.2,theevolutionofthe extrinsicspinHallconcurrent
microscopicKSByE∼.The×differenxce is causedby the inho- can be written as Jz(t)=Jz e−t/τs with the relaxation
y y,0
mogeneous broadening59 in spin precession due to the time τ . Jz approximatesto the maximum value which
s y,0
4
is only determined by the skew scattering. Therefore temperature with all the scattering explicitly included.
Sx = m∗τ γ((π/a)2 k2 )Jz /e. For the D’yakonov- We find the spin precession induced by the Dressel-
s −h xi y,0
Perel’ mechanism,60 τ [γ((π/a)2 k2 )]−2. Hence haus/Rashba SOC has a very important effect on the
Sx γ((π/a)2 k2 s)−∝1. (ii) When−hthxeielectric field extrinsic SHC and verify the vanishment of the SHC for
∝ −h xi lineark-dependent SOC. We also showthat the SHC in-
is along the x-axis, for the Dresselhaus SOC, the CISP
duced by the skew scattering calculated from the Kubo
is along the x direction, while the polarization induced
formula in the literature is inadequate without consider-
by the skew scattering is along the x direction. How-
− ing the spin precession. Finally we show that with the
ever,for the strain-inducedorthe RashbaSOC,the spin
joint effects from the skew scattering and the spin pre-
polarizations from both mechanisms are along the same
cession, an in-plane spin polarization can be generated
direction. (iii) The CISP decreases with the impurity
density49becausehighimpuritydensityreducestheEMF whichcanbefurtherrotatedtothez-directionbyapply-
ing an external in-plane magnetic field.
effectively. However, the skew-scattering-induced spin
polarization increases with impurity density as the skew The authors acknowledge valuable discussions with Z.
scattering is proportional to the impurity density. Y. Weng. This work was supported by the Natural Sci-
In summary, we investigate the SHC and the spin po- ence Foundation of China under Grant Nos. 10574120
larization induced by the k-asymmetric spin-conserving and 10725417, the National Basic Research Program of
skew scattering in symmetrical (001) GaAs quantum China under Grant No. 2006CB922005 and the Knowl-
well from the fully microscopic KSBE approach at high edgeInnovationProjectofChineseAcademyofSciences.
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