Spin accumulation and spin-coherent excitations in a strongly confined two-dimensional hole gas

P. Kleinert1 and V. V. Bryksin2

1Paul-Drude-Intitut für Festkörperelektronik, Hausvogteiplatz 5-7, 10117 Berlin, Germany2A. F. Ioffe Physical Technical Institute, Politekhnicheskaya 26, 194021 St. Petersburg, Russia

�Received 27 April 2007; revised manuscript received 29 May 2007; published 29 August 2007�

Based on a rigorous quantum-kinetic approach, spin-charge coupled drift-diffusion equations are derived fora strongly confined two-dimensional hole gas. An electric field leads to a coupling between the spin and chargedegrees of freedom. For weak spin-orbit interaction, this coupling gives rise to a spin accumulation at the edgesof a stripe. There exists a threshold value of the spin-orbit coupling constant that separates spin diffusion fromballistic spin transport. In the latter regime, undamped spin-coherent oscillations are observed. This result isconfirmed by an exact microscopic approach valid in the ballistic regime.

DOI: 10.1103/PhysRevB.76.073314 PACS number�s�: 72.25.�b, 72.10.�d, 72.15.Gd

The generation and manipulation of a spin polarization byexclusively electronic means in nonmagnetic semiconductorsat room temperature is a major challenge of spintronics.Among many interesting phenomena, the intrinsic spin-Halleffect1,2 �SHE� has recently attracted considerable interest.Experimental studies3–5 reveal an electric-field-induced spinaccumulation near the edges of a confined two-dimensionalelectron �hole� gas. Most theoretical interpretations of theseexperimental data rely on the notion of a spin current ori-ented transverse to the applied electric field.1,2 Interestingly,this seemingly clear physical picture still remains the subjectof serious debates.6,7 The relationship between the spin cur-rent and the induced spin polarization seems to be a verysubtle issue. The main problem underlying the debates is thenotion of a spin current itself because spin is not a conservedquantity in spin-orbit coupled systems. Consequently, anyapproach that avoids the intricate identification of a more orless suitable spin current is superior. Such an alternative ap-proach not only introduces different calculational techniques,but also suggests alternative interpretations of the effects un-der consideration. As widely anticipated, a complete physicaldescription of spin-related phenomena is provided by micro-scopic models based on the spin-density matrix or KeldyshGreen functions together with an analysis of its long-wavelength and low-frequency limit. These approaches aremore general and free from artifact associated with ambigu-ous definitions of the spin current.

In this Brief Report, we propose an alternative approachand derive spin-charge coupled drift-diffusion equations for atwo-dimensional hole gas �2DHG�, which refers to the popu-lated heavy-hole band of thin p-type quantum wells. Therelated heavy-hole Hamiltonian of the cubic Rashba modelhas the second quantized form

H = �k,�

ak�† ��k − �F�ak� − �

k,�,��

���k · �����ak�† ak��

+ u�k,k�

��

ak�† ak�� − ieE�

k,�

���ak−�/2�† ak+�/2���=0,

�1�

where ak�† �ak�� denote the creation �annihilation� operators

with in-plane quasimomentum k= �kx ,ky ,0� and spin �. The

electric field E is oriented along the x axis. Furthermore, �Fdenotes the Fermi energy, � the vector of Pauli matrices,�k=�2k2 / �2m�, and u the strength of the “white-noise” elas-tic impurity scattering, which gives rise to the momentumrelaxation time �. Contrary to a phenomenological approach,we treat elastic scattering on a full microscopic scope. Thespin-orbit coupling is given by

��k =�

2�i�k+

3 − k−3�,�k+

3 + k−3�,0� , �2�

where k±=kx± iky and ��k=�k3. Based on the Born approxi-mation for elastic impurity scattering, the Laplace-transformed kinetic equations for the four physical compo-

nents of the spin-density matrix f = �f , f� have the form7

sf −i�

m�� · k�f + i���k� · f +

eE

��kf =

1

�� f − f� + f0, �3�

sf + 2��k f� −i�

m�� · k�f + i���k�f +

eE

��kf

=1

��f − f� +

1

�

�

��kf��k −

��k

�

�

��kf + f0, �4�

where an additional frequency appears

���k� =3�

���ky

2 − kx2��y − 2kxky�x,�kx

2 − ky2��x − 2kxky�y,0� ,

�5�

which depends on �. The cross line over k-dependent quan-tities denotes an integration over the polar angle of thein-plane vector k. f0 and f0 are the initial charge and spin-density components, respectively. The quantum BoltzmannEqs. �3� and �4� are treated in the long-wavelength limit inorder to derive spin-charge coupled drift-diffusion equations.To this end, the kinetic equations are written in a matrix form

Af +E f =Bf¯

+ �, where the matrix A collects all contributionsthat are independent of the electric field E and not integratedover the angle . To calculate the matrix B on the right-handside of this equation, we assume �kF

3 ��kFand expand up to

�2. The matrix equation is iteratively solved in the case of

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weak electric field �the matrix E contains first-order deriva-

tives and f is decomposed according to f0+ f1, with f0�E0

and f1�E�. The solution of the equation is written in theform

�1 + C1−1C2�−1C1

−1 f¯

= � , �6�

where f¯

0= C1� and f¯

1= C2�. The general expressions for thetransport coefficients are very cumbersome, but simplifyconsiderably in the low-field case and under the condition�kF

3 ��kF.

As we are mainly interested in spin accumulation, let usfocus on the coupling between the spin and charge degrees offreedom. By applying the outlined schema, we obtain ourmain analytical result

�s + ivd

0�x + D0�2 f + i�z�y f z = f0, �7�

�s +1

�sz+ i

vd

0�x + Dz�

2 f z + i�0�y f = 0, �8�

with the transport coefficients

D0 =D

02 , Dz = D

02 − 12�2

� 02 + 4�2�2 , �9�

�0 = vd9�2

2�

3 02 − 4�2

� 02 + 4�2�2 ,

1

�sz=

4�2

0�, �10�

�z = vd9�2

2� 0

2 4 0�2 + 8�2 − 3 02s�

� 0s� + 4�2�� 02 + 4�2�

. �11�

The parameters introduced in these equations are calculatedat the Fermi surface and are given by vd=eE� /m, 0=s�+1, D=vF

2� /2, �=�kF� /�, and �=�kF

�=K�kF�l �withK�kF�=�mkF

2 /�2 and the mean free path l=vF��. As the con-tributions ��x do not affect our analysis, we took them tolowest order in �. Equations �7� and �8� have been derivedfor � /��1 but unrestricted values of �. In the absence ofthe electric field �E=0�, Eqs. �7� and �8� completely de-couple. This decoupling, which applies to all components ofthe spin-density matrix, represents a specialty of the cubicRashba model.8,9

The time dependence of the coupled spin-charge transportis calculated by an inverse Laplace transformation of thesolution of Eqs. �7� and �8�. Due to the complicated s depen-dence of all transport coefficients, a non-Markovian temporalevolution is expected in general.

The character of the coupled spin-charge transportstrongly depends on the strength of the spin-orbit coupling,which is expressed by the dimensionless parameter �=K�kF�l. It is the most striking feature of the drift-diffusionEqs. �7� and �8� that the behavior of spin transport changesradically with increasing coupling strength �. The appear-ance of such a crossover is due to the unusual expression forthe diffusion coefficient Dz in Eq. �9�, which has recentlybeen obtained by an alternative approach.8 Moreover, the

very same result is also obtained for the linear Rashba modelwhen the frequency �k is appropriately redefined. With in-creasing spin-orbit coupling K�kF� or relaxation time �, thediffusion coefficient Dz changes its sign. A negative diffusioncoefficient indicates an instability of the spin system. Underthis condition, spin diffusion has the tendency to strengtheninitial spin fluctuations. The competition between these self-strengthening and spin-relaxation processes results in a spa-tial oscillatory spin pattern. Going from weak ���1/12�to strong ���1/12� spin-orbit coupling, we observe atransition in the spin system from a diffusive behavior to aballistic regime.

As an illustrative example, Eqs. �7� and �8� are treated fora stripe geometry �−L0�y�L0�. The inverse Fourier trans-formation is accomplished by the replacement �y→ i� /�y,whereas along the x axis, all quantities are constant. What we

expect from the spatial variation of the charge density f�y� isan induced electric field Ey�y� oriented along the y axis. Thisfield enters Eqs. �7� and �8� via the drift term and satisfies the

Poisson equation dEy /dy=4�e� f − f0� /�, with f0 denotingthe background density and � the dielectric constant. Choos-ing appropriate boundary conditions is a difficult task10,11

that is not examined here since the qualitative behavior ofthe solution is completely independent of the boundary con-dition and solely dictated by Dz in Eq. �9�. We assume thatcharge traps give rise to an electric field at the boundaries

Ey�y= ±L0�= ±E0. The corresponding solution f z�y� is anti-symmetric. In order to catch up with simple analytical resultsused to interpret experimental data,3 we require the spin po-

larization f z�y� to vanish at the boundaries. We obtain

f z�y� = −�E0�0

4�e�sz

�12�2

2

�12 − �2

2� sinh��1y�sinh��1L�

−sinh��2y�sinh��2L�� ,

�12�

where �1,2 are positive roots of the characteristic equation

�2�0�z − ��2Dz −1

�sz��2D0 +

4�e

��f0� = 0. �13�

A numerical example for the field-induced out-of-plane spinpolarization is presented in Fig. 1. The thick line illustratesthe result for the spin polarization in the diffusive regime�Dz�0� when the spin-orbit coupling is weak ��1/12. Asin the SHE, the electric field aligned along the x axis inducesa spin polarization at the edges of the stripe. Many theoreti-cal studies of the SHE �see, for instance, Refs. 8 and 12�introduced the spin current J�

i by a symmetrized product ofspin and velocity operators �v� i+ iv�� /2. It was claimedthat, at least for the cubic Rashba model, the SHE introducedin such a manner is robust against disorder. Experimentalresults4 seem to confirm this physical picture. However, thereis a principal difficulty with such an approach. The abovementioned definition of the current is not sufficiently general.It would completely fail for any hopping transport problem,for which the eigenstates have no dispersion. This definitiononly applies whenever the Hamiltonian commutes with thedipole operator. Obviously, this is not the case for the Rashba

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Hamiltonian. Consequently, it is necessary to go back to themore general definition, which expresses the current by thetime derivative of the dipole operator.6,7 This more generaltreatment of the spin current reveals a close relationship be-tween the field-induced spin accumulation and the spin cur-rent expressed by a quasichemical potential. From a techni-cal point of view, the current that applies in its most generalform to a homogeneous system is calculated from the quan-

tity �� f�k ,�� at �=0 and not only from the density matrix

f�k�. Based on this general definition of the spin current, itwas concluded that an electric-field-induced steady-statespin-Hall current does not exist in the cubic Rashba model.13

On the other hand, a SHE was demonstrated by a recentexperiment.4 This calamity indicates that the notion of a spincurrent is not useful for studying the SHE. An alternative,which is proposed in this Brief Report, determines the spinaccumulation from quantum kinetic equations or the associ-ated spin-charge coupled drift-diffusion equations.

The character of the SHE dramatically changes instrongly spin-orbit coupled systems ���1/12 so thatDz�0�. The result is illustrated in Fig. 1 by the thin line. Aspin-coherent standing wave travels through the stripe. It isremarkable that these oscillations are not damped although afinite elastic scattering is present. The occurrence of a peri-odic spin pattern is not unusual and has been investigated inthe literature.14–17 However, here such an oscillatory spinpattern can be induced by an electric field. Whether the pre-dicted oscillations of the spin polarization show up in experi-ment depends on spin dephasing mechanisms.18 Besides in-homogeneous broadening, we mention another possiblesource of spin relaxation: The rapid variation of the out-of-plane spin polarization induces a magnetic field that leads tocirculating microscopic currents. The retroaction of thesecurrents on spin may result in a finite damping of spin oscil-lations.

In the strong-coupling limit ��=K�kF�l�1�, the oscilla-tory spin pattern changes on a length scale K−1�kF� that ismuch smaller than the mean free path l. This fact conflictswith basic assumptions of the drift-diffusion approach, which

is only applicable for diffusion lengths much smaller than themean free path. Although macroscopic transport equationswere found to be valid even when the spin-diffusion length issomewhat less than the mean free path,19 it is indispensableto treat this point in detail. Large values for � give rise tospin-relaxation times �sz, which are much smaller than theelastic scattering time �. This condition characterizes the bal-listic spin regime. Therefore, we go back to the kinetic Eqs.�3� and �4� and solve them under the condition �→� and tofirst order in the electric field. For the out-of plane spin po-larization, we obtain

� 2 + 4�k2�fz +

eE

�

�

�kxfz =

2i

��kx��y − �ky��x�f0,

�14�

where =s− i� ·v. Calculating the inverse Laplace and Fou-rier transformation and integrating over the angle , we ar-rive at the analytical solution

fz�k,r,t� =3f0

16�r

�

�k eE

�k� sin 2K�k�r

2K�k�r− cos 2K�k�r��

��y − �vt�2 − x2� for 0 � y � vt

− ��y + �vt�2 − x2� for − vt � y � 0,��15�

which describes the field-induced spin polarization that oc-curs, when initially a drop of charge carriers is injected intothe 2DHG at the position r=0. A �-like wave front of spinpolarization travels through the homogeneous 2DHG. Theamplitude of this narrow wave front oscillates as illustratedin Fig. 2. As expected, the relief is antisymmetric with re-spect to the y axis. Most interesting for our comparison withthe above drift-diffusion approach is the observation that thewavelength of the spatial spin pattern is of the order ofK−1�kF�. Therefore, this exact analytical result confirms theexistence of a field-induced long-living spin pattern instrongly spin-orbit coupled systems as predicted by the drift-diffusion equations.

−20 −10 0 10 20

−5

0

5

x 10−4

kFy

fz/f

0

FIG. 1. Out-of plane spin polarization induced by an electricfield applied parallel to the stripe of a 2DHG �for m� /�2=2 nm,kF=0.1 nm−1, E0=100 V/cm, and f0=0.11018 cm−3�. The thickand thin lines were calculated with �=0.05 ps ��=0.173� and�=0.4 ps ��=1.3836�, respectively.

−10−5

05

10 −10

−5

0

5

10

−101

x

y

F(r)

FIG. 2. The amplitude F�r�=�xd��sin x /x−cos x� /x� /dx withdependence on r, with x=2K�kF�r and K�kF�=0.5 nm−1. For thefactor �, we have �= ±1 for y�0.

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Our study of electric-field-induced spin phenomena re-vealed a close relationship between the SHE and spin-coherent oscillations. We compare this conclusion with re-cent results that claimed that the intrinsic SHE and theZitterbewegung are essentially the same kind ofphenomena.17 Therefore, the question arises whether theabove treated spin-coherent waves have to be identified withthe Zitterbewegung, which is a purely relativistic effect. In-deed, both kinds of oscillatory spin excitations exhibit com-mon features. The characteristic wavelength of both types ofoscillations amounts to about 100 nm �calculated by adopt-ing the typical values �m /�2�2 nm and kF�0.1 nm−1�.Moreover, both the Zitterbewegung20 and the spin-coherentwaves �cf. Eq. �12�� are resonantly enhanced whenever thewidth of the stripe matches a characteristic wavelength of thespin excitation. However, some features of spin-coherentwaves are not compatible with such an identification with theZitterbewegung. The dispersion relation of coupled spin-charge excitations is calculated from the vanishing determi-nant of Eqs. �7� and �8�. In general, one obtains not onlyspin-coherent solutions but also damped excitations, whereasthe transition between them could be driven by the electricfield. In addition, the spin-coherent waves that appear at

strong spin-orbit coupling are separated from the SHE by asharp threshold. For the Zitterbewegung, such a threshold isnot expected as its existence is solely due to at least twoenergy bands separated by a nonzero gap. We think that theinteresting study of the relationship between spin-coherentwaves, the Zitterbewegung, and long-living spin-coherentstates14 will continue in the near future.

The experimental observation of the field-induced spin-coherent waves should be possible by high-resolutionscanning-probe microscopy imaging techniques. The directexperimental proof of this effect would facilitate develop-ments both in spintronics and basic research.

In conclusion, we treated the spin accumulation of a2DHG by spin-charge coupled drift-diffusion equations. Itwas shown that for weak spin-orbit coupling, the electricfield induces a spin accumulation at the edges of the stripe.In the strong-coupling regime, an oscillatory spin pattern waspredicted to appear.

This work was supported by the Deutsche Forschungsge-meinschaft and the Russian Foundation of Basic Researchunder Grant No. 05-02-04004.

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