Spin-orbit coupling mediated spin torque in a single ferromagnetic layer

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Spin-orbit coupling mediated spin torque in a single ferromagnetic layer

A. Matos-Abiague

Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany

R. L. Rodríguez-Suárez

Departamento de Física, Pontifícia Universidad Católica de Chile, Casilla 306, Santiago 6904411, Chile 共Received 25 May 2009; revised manuscript received 13 August 2009; published 30 September 2009兲 By considering a linear in momentum but otherwise arbitrary spin-orbit coupling共SOC兲, we derive a simple analytical expression for the current-driven spin torque in a single ferromagnetic layer. Explicit forms of the spin torque are given for structures with different SOC fields, in dependence of strain effects, growth direction, and/or symmetry under spatial inversion. The Landau-Lifshitz-Gilbert equation including the effects of the SOC mediated spin torque on the magnetization dynamics is briefly discussed.

DOI:10.1103/PhysRevB.80.094424 PACS number共s兲: 75.60.Jk, 75.75.⫹a, 72.25.⫺b, 72.10.⫺d

I. INTRODUCTION

Since its first theoretical description^1,2the phenomenon of spin-transfer torque has attracted increasing attention due to its potential applications in spintronic devices. This phenomenon occurs in spin-valve structures composed of two ferromagnetic layers separated by a nonmagnetic one. A trans- verse共perpendicular to the layers兲charge current through the device produces a flow of spin-polarized conduction electrons from the fixed layer into the free layer. This causes a direct transfer of angular momentum from the spin-polarized flowing electrons to the local magnetization of the free layer,^1–7 resulting in a torque that may produce magnetic reversal or steady-state precessions with frequencies in the microwave range.^7–14 This spin-transfer mechanism allows nanomagnets to be manipulated without magnetic fields, and is the subject of extensive research for new applications in nonvolatile memory technology and radio-frequency oscilla- tors.

Up to now the majority of investigations on the spin transfer driven excitations have been performed on planar spin- valve nanopillar or nanocontact structures in which a noncollinear configuration of the magnetic structure is required.

However, spin torque phenomena may also be present in collinear spin valves composed of two ferromagnetic con- tacts separated by a two-dimensional electron gas with spin- orbit coupling共SOC兲.¹⁵Recent theoretical investigations^16,17 have shown that, even in a single, uniformly magnetized ferromagnetic layer, when SOC is present, an in-plane current can induce a spin torque on the magnetization of the layer without the need for noncollinear ferromagnetic configuration of the structure. In these previous studies^16,17 the authors considered a共001兲ferromagnetic layer with magnetization lying on the plane of the layer and in the presence of Bychkov-Rashba¹⁸ and Dresselhaus¹⁹SOCs. Here we investigate the current-driven SOC mediated spin torque for the case of an arbitrary magnetization orientation and a linear in momentum but otherwise arbitrary form of the SOC.

The paper is organized as follows. In Sec. IIwe present the basic assumptions and theoretical model. A general expression for the SOC mediated spin torque is derived in Sec.

IIIfor the case of a single, uniformly magnetizated ferromagnetic layer in the presence of a linear in momentum SOC

field. The modified Landau-Lifshitz-Gilbert 共LLG兲equation accounting for the SOC mediated torque is briefly discussed in Sec. IV. Specific expressions for the SOC mediated spin torque are given in Sec.Vfor the cases of共001兲,共110兲, and 共111兲ferromagnetic layers with structure inversion asymme- try共SIA兲 and/or bulk inversion asymmetry共BIA兲. The spin torque mediated by strain-induced SOC is investigated in Sec. VI. Finally, conclusions are given in Sec.VII.

II. THEORETICAL MODEL

We consider a two-dimensional uniformly magnetized ferromagnetic layer and assume a model in which the local magnetization of the ferromagnet is determined by localized d-like electrons rather than from itinerant electrons. Such a model appears to be appropriate for electrodes composed of 3d ferromagnetic metals and alloys lying on the negative slope side of the Slater-Pauling curve 共e.g., Co, CoFe, Ni, and NiFe兲.^6,20,21

The localized spins couple to the itinerant spins through the exchange interaction

Hex= −Jex

兺

i,j

共Si·sj兲, 共1兲

whereJexis the exchange coupling constant, Sidenotes the localized spin at the ith site and sj is the spin of the jth itinerant electron.

Averaging Eq. 共1兲 over the localized states, summing up the contributions of all the localized spins, and considering

M=␥0具具Si典典_loc=␥0

兺

_i ^具Sⁱ^典, ^共2兲

withM,␥0, and具Si典 as the macroscopic magnetization, gy- romagnetic factor, and local spin density at site i, respec- tively, one obtains the effective one-particle exchange Hamiltonian for an itinerant electron,

H_ex^it = −Jex

␥0

共M·s兲, 共3兲

whereMˆ =M/M_s 共withM_s as the saturation magnetization兲 is a unit vector along the magnetization direction. Equation

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共3兲can be rewritten in the more familiar form

H_ex^it = −⌬ex共Mˆ ·␴兲共4兲 by introducing the exchange splitting energy as

⌬ex=J_exបM_s 2␥0

. 共5兲

Thus, in the presence of SOC, the motion of the itinerant electrons is described by the one-particle Hamiltonian

H^it=ប²k²

2m −⌬exMˆ ·␴⁺w共k兲·␴^, 共6兲 wherekis the length of the in-plane wave vectork,mis the carrier mass, and ␴ is a vector whose components are the Pauli matrices. The last term in Eq. 共6兲 represents the SOC which is determined by the SOC field 共SOCF兲 w共k兲. The time-reversal symmetry implies that w共k兲= −w共−k兲. There- fore, in the lowest approximation, the SOCF is always linear in the wave vector. One can then express theith component of the SOCF as

w_i=

兺

l=x,y

c_i^lk_l; 共i=x,y,z兲, 共7兲 where kl are the wave-vector components and the coeffi- cients c_i^l determine the explicit form of the SOCF. Correc- tions of higher order in k may, in principle, play some role when large currents are applied and states with high values of k_储 become relevant. Furthermore, it has already been shown that up to the first order in⌬ex/wi共kF兲共withkFas the length of the Fermi wave vector兲 the cubic terms in the SOCF do not contribute to the torque.¹⁷Here we consider the case of a strong ferromagnet for which⌬exⰆw共kF兲and limit our analysis to the case of moderate currents. In such a regime, the linear approximation used in Eq.共7兲suffices.

The effective exchange Hamiltonian for the localized spins can be obtained from Eq. 共1兲 by averaging over the ensemble of itinerant electrons. The result is

H_ex^loc= −Jex共S·具具s典典兲, 共8兲 where具具s典典is the spin density of itinerant electrons共explicit calculations of this quantity are given in the following sec- tion兲.

III. SOC MEDIATED SPIN TORQUE

By definition, the spin torque T is the change of spin angular momentum per unit time. The average spin torque exerted by the itinerant electrons on the localized spins is then

T=d具具S典典loc

dt = i ប具具关Hex

loc,S兴典典loc, 共9兲 where具具. . .典典_locrefers to the ensemble average over localized spins. Working out the commutator关Hex

loc,S兴and taking into account Eqs.共2兲,共5兲, and共8兲we obtain

T=2⌬ex

ប² 共Mˆ ⫻具具s典典兲. 共10兲

We note that, since具具s典典=共ប/2兲具具␴典典, Eq.共10兲coincides共up to a factor 1/Ms兲共Ref. 22兲with Eq.共6兲in Ref.17.

When a current flows through the system, the spin density of the itinerant electrons deviates from its equilibrium value 共具具s典典0兲 by an amount 具具␦s典典, i.e., 具具s典典=具具s典典0+具具␦s典典. In equilibrium there is no preferential direction for the motion of the itinerant electrons and, in average, the SOCF, which is an odd function of the momentum, vanishes. Thus the equilibrium spin density具具s典典0储Mand does not contribute to the spin torque. On the contrary, the presence of a current defines a preferential direction for the electron motion and results in a finite SOC mediated nonequilibrium spin density 具具␦s典典, which is noncollinear with M and, in turn, induces a spin torque on the localized spins. The spin torque in Eq.共10兲can then be rewritten as

T=2⌬ex

ប² 共Mˆ ⫻具具␦s典典兲. 共11兲 Considering Eq. 共6兲 we find that the eigenenergies and wave functions of itinerant electrons are given, respectively, by

E_k^⫾=ប²k²

2m ⫾兩w−⌬exMˆ兩共12兲 and

␺k⫾= e^ik.r

冑

S共1 +A_⫾²兲

冉

^A^⫾¹^eⁱ^␰^k

冊

^, ^共¹³^兲

whereS is the area of the film,

A_⫾= 共wzMs−⌬exMz兲⫾Ms兩w−⌬exMˆ兩

冑

共wxMs−⌬exMx兲²+共wyMs−⌬exMy兲², 共14兲 and

tan␰k= −wyMs−⌬exMy

wxMs−⌬exMx

. 共15兲

In writing Eqs. 共14兲and共15兲 we took into account that兩M兩

=Ms.

The nonequilibrium spin density of itinerant electrons is determined by the relation

具具␦s典典=ប 2

兺

␴=⫾

冕

共2^d␲²^k兲²具⌿k␴兩␴兩⌿k␴典␦^f␴共k兲, 共16兲 where ␦^f␴=f_␴−f_␴^共⁰^兲 represents the deviation of the distribu- tion function f_␴ corresponding to the␴ band from its equilibrium value f_␴^共0兲.

We consider the most relevant case 共from the practical point of view兲of a strong ferromagnet in which the exchange splitting dominates over the SOC effects and EFⰇ⌬ex

Ⰷ兩w共kF兲兩, wherekF=

冑

2mEF/ប². In such a case the interband transitions can be neglected and the scattering by impurities

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can be treated within a constant relaxation-time approximation of the Boltzmann equation关for details see Refs.16,17, and23兴. One can then write²⁴

␦^f␴⬇e␶

បE·⵱kf_␴^共0兲, 共17兲 where ␶ is the relaxation time, e is the electron charge 共e

⬎0兲, and Eis the electric field.

The nonequilibrium spin density具具␦s典典 can be calculated by combining Eqs. 共16兲 and 共17兲. An analytical expression for 具具␦s典典 can be obtained in the limit of large exchange coupling 共⌬exⰇ兩w共kF兲兩兲 by expanding the spin expectation values 具⌿k␴兩␴兩⌿k␴典 in powers of wi/⌬ex共i=x,y,z兲 and keep- ing up to the first order only. After integration over the in- plane wave vector we obtain

具具␦^sx典典=q

兺

l=x,y

jl关共My

2+M_z²兲cx

l−MxMycy

l−MxMzcz l兴,

共18兲具具␦^sy典典=q

兺

l=x,y

jl关−MxMyc_x^l+共Mx

2+M_z²兲cy

l−MyMzc_z^l兴, 共19兲 and

具具␦^sz典典=q

兺

l=x,y

j_l关−M_xM_zc_x^l−M_yM_zc_y^l+共M_x²+M_y²兲c_z^l兴. 共20兲 In the equations abovej_lrepresents thelth component of the charge current density²⁵ j=共jx,jy, 0兲^T and q=mP/2e⌬exM_s², whereP is the spin polarization of the current.

By substituting Eqs. 共18兲–共20兲 into Eq. 共11兲 and considering that P⬇⌬ex/E_F we obtain the nonequilibrium spin torque,

T= − m⌬ex

eប²E_FM_s关jx共M⫻dx兲+jy共M⫻dy兲兴, 共21兲 where we have introduced the vectors

dl=c_x^lxˆ+c^l_yyˆ+c_z^lzˆ; 共l=x,y兲, 共22兲 withxˆ,yˆ, andzˆas the unit vectors along thex,y, andzaxes, respectively. Equation 共21兲 is valid for strong ferromagnets and for linear inkbut otherwise arbitrary SOCF and for any orientation of the magnetization. It reveals in an elegant and simple way how the nonequilibrium spin torque is determined by the SOCF whose properties are encoded in the vectorsdl.

We remark that the SOC mediated spin torque in Eq.共21兲 is different from the conventional torque produced by inject- ing a spin-polarized current into a ferromagnetic layer. The conventional torque requires a source 共usually an additional ferromagnetic layer兲of spin-polarized electrons with noncollinear magnetization compared to the magnetization of the free layer共where the spin torque is exerted兲and can be finite even in the absence of SOC.^1–7 In contrast, the SOC mediated spin torque vanishes in the absence of SOC 关see Eq.

共21兲兴but does not require the injection of spin-polarized cur-

rents and can, therefore, be present in structures containing a single ferromagnetic layer.

IV. LANDAU-LIFSHITZ-GILBERT EQUATION We now consider the presence of an external magnetic field and include the effects of the SOC mediated spin torque T 关see Eq. 共21兲兴 in the standard LLG equation.^26,27 The modified LLG equation describing the magnetization dynamics of a single ferromagnetic layer in the presence of SOC and subjected to an in-plane current flow and an external magnetic field reads

dM

dt = −␥0共M⫻H_eff兲+␣G

Ms

冉

^M^⫻^dMdt

冊

⁺^␥⁰^T^, ^共23兲

where␣Gis the Gilbert damping parameter and the effective field reads as

H_eff=H_ext+H_an+H_d+A_exⵜ²M. 共24兲 HereH_extis the applied external field,H_anis the anisotropy field, H_dis the demagnetizing field due to axial dipole coupling, andA_exis an exchange constant.²⁷Taking into account Eq. 共21兲, one can rewrite the modified LLG equation 关Eq.

共23兲兴as dM

dt = −␥0关M⫻共Heff+H_drive兲兴+␣G

Ms

冉

^M^⫻^dMdt

冊

^,

共25兲 with the effective driving field

H_drive= m⌬ex

eប²E_FM_s共jxdx+jydy兲. 共26兲 The SOC mediated driving field H_drive competes with H_eff 关see Eq.共25兲兴and could be useful for current-driven magnetization switching in single, uniformly magnetized ferromagnetic layers, as already discussed in Refs. 16and17. How- ever, since H_drive does not compete with the damping, the SOC mediated excitation of steady magnetization precessions in such systems is not possible.

V. SOC MEDIATED SPIN TORQUE IN SYSTEMS WITH SIA AND/OR BIA

In noncentrosymmetric materials, the lack of bulk inversion symmetry results in the BIA-induced SOC.^28,29 This kind of SOC is, in general, present if the magnetic layer consists of a noncentrosymmetric ferromagnet and/or at interfaces between ferromagnets and noncentrosymmetric materials such as zinc blende semiconductors.³⁰ The SIA- induced SOC does not need the presence of noncentrosymmetric materials; it originates from the lack of inversion symmetry of the structure itself 共e.g., a ferromagnetic layer sandwiched between two different materials兲and is determined by build-in and/or external electric fields.^28,29 For systems with both BIA and SIA the two SOC mecha- nisms coexist. That is the case, for example, of zinc-blende semicondutor/cubic ferromagnet interfaces such as 共001兲

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GaAs/Fe, where the BIA-induced SOC of the noncentrosymmetric semiconductor interferes with the SIA-like SOC resulting from the strong build-in electric field at the interface.

Such an interference leads to a net twofold symmetric SOCF which reflects the C_2v symmetry of the 共001兲 GaAs/Fe interface.^29,30

We focus now on the specific form of the nonequilibrium spin torque mediated by BIA- and SIA-induced SOCs in ferromagnetic layers grown in the directions 共001兲, 共110兲, and共111兲.

A. (001) layers with axes xˆ¸[100], yˆ¸[010], and zˆ¸[001]

The SOC containing both SIA-induced Bychkov-Rashba and BIA-induced Dresselhaus terms is given by18,19,29,31

H_SO=␣共kx␴y−ky␴x兲+␥共kx␴x−ky␴y兲, 共27兲 where␣ ^and␥ are the corresponding Bychkov-Rashba and linearized-Dresselhaus parameters, respectively. The values of the SOC parameters are material dependent. For zinc- blende semiconductors the order of these parameters ranges from 10⁻³ eV Å to 10⁻¹ eV Å.²⁹ For the case of ferromagnetic materials the values of the SOC parameters are less known. The SOCFs in an Fe/GaAs slab have recently been extracted fromab initio calculations.³² For such a structure the Bychkov-Rashba-type and Dresselhaus-type SOC parameters were found to vary from about⫾10⁻² to 10⁻¹ eV Å.³² The components of the SOCF, w, corresponding to Eq. 共27兲 are wx=␥kx−␣ky, wy=␣kx−␥ky, and wz= 0. From Eqs.共7兲and共22兲one obtains

d_x=␥^x^ˆ⁺␣^y^ˆ^; ^dy= −␣^x^ˆ⁻␥^y^ˆ^. 共28兲 The SOC mediated spin torque can be straightforwardly obtained by placing the above relation into Eq.共21兲. The result is shown in Table I关see case A兴. The angle ␾0 denotes the direction of the in-plane charge current, i.e.,jx=jcos␾0and j_y=jsin␾0. By introducing polar coordinates for the magnetization,M=Ms共sin␪cos␾, sin␪sin␾, cos␪兲^T, the components of the spin torque T=共Tx,Ty,Tz兲^Tcan be rewritten as

Tx= m⌬exj eប²EF

共␣^cos␾0cos␪⁻␥^sin␾0cos␪兲, 共29兲

T_y=m⌬exj

eប²E_F共␣^sin␾0cos␪⁻␥^cos␾0cos␪兲, 共30兲 and

Tz=m⌬exj eប²EF

sin␪关␣cos共␾⁻␾0兲−␥sin共␾⁺␾0兲兴. 共31兲 In the particular case of an in-plane magnetization 共i.e., ␪

= 90°兲we obtain,Tx=Ty= 0 and T=m⌬exj

eប²EF

关␣cos共␾⁻␾0兲−␥sin共␾⁺␾0兲兴zˆ. 共32兲 After the transformations ⌬ex→Jsd, ␣→−␣, ␥→␤, ␾→␪, and␾0→␪0so that the notations here and those used in Ref.

17 match each other, Eq.共32兲 reduces共up to a factor of 2兲共Ref. 33兲to Eq.共50兲of Ref.17.

B. (110) layers with axes xˆ¸[11¯0], yˆ¸[001], and zˆ¸[110]

Here the SOC is determined by^31,34,35

H_SO=␤^kx␴y−␣^ky␴x+␭kx␴z, 共33兲 where ␣ ^and ␤ are parameters related to the SIA-induced SOC and ␭ is the strength of the BIA-induced SOC. Note that, in general, ␣⫽␤ which is a manifestation of the re- duced symmetry of 共110兲 layers with respect to the 共001兲 structures.^31,34

The SOC mediated spin torque can be obtained by following the same procedure as before and the result is displayed as case B in Table I. In comparison with the previous case 共case A兲 the spin torque posses now an extra component in the direction zˆ⫻M.

C. (111) layers with axes xˆ¸[112¯], yˆ¸[1¯10], and zˆ¸[111]

In this case the SOC can be written as^31,35

H_SO=共␣⁺␥兲共kx␴y−ky␴x兲, 共34兲 where␣and␥are the strengths of the SIA- and BIA-induced SOCs, respectively. Equation 共34兲 is equivalent to a Bychkov-Rashba SOC with strength ␣⁺␥. Therefore, the spin torque can readily be obtained from the results in case A by performing the transformations ␥→0 and ␣→共␣⁺␥兲. The resulting spin torque is shown as case C in Table I. In the regime ␣^{= −}␥ 共this could in principle be achieved by tuning␣electrostatically兲the SOC is zero关see Eq.共34兲兴and the spin torque vanishes. This open the possibility of switching the SOC mediated spin torque by means of a bias volt- age.

VI. SPIN TORQUE MEDIATED BY STRAIN-INDUCED SOC

We now investigate the effects of strain-induced SOC on the spin torque. We consider a共001兲layer with axesxˆ储关100兴, yˆ储关010兴, and zˆ储关001兴 with a strain-induced SOC of the form³⁶

TABLE I. SOC mediated spin torque in units ofm⌬ex/eប²E_FM_s for different growth directions.

Case

Growth

direction Spin torque

A 共001兲 j关共␣sin␾0−␥cos␾0兲共M⫻xˆ兲

−共␣cos␾0−␥sin␾0兲共M⫻yˆ兲兴

B 共110兲

j兵␣sin␾0共M⫻xˆ兲− cos␾0关␤共M⫻yˆ兲 +␭共M⫻zˆ兲兴其

C 共111兲 j共␣+␥兲关sin␾0共M⫻xˆ兲− cos␾0共M⫻yˆ兲兴

(5)

H_SO=␣n关共uzxkz−uxyky兲␴x+共uxykx−uyzkz兲␴y

+共u_yzk_y−u_zxk_x兲␴z兴+␥n关k_x共u_yy−u_zz兲␴x

+ky共uzz−uxx兲␴y+kz共uxx−uyy兲␴z兴, 共35兲 whereuijare the components of the strain tensor and␣ ^and

␥are material parameters. The spin torque deduced from Eq.

共35兲is given by T= − m⌬exj

eប²E_FM_s兵␣共uyzsin␾0−uzxcos␾0兲共M⫻zˆ兲 +关␣^uxycos␾0+␥共uzz−uxx兲sin␾0兴共M⫻yˆ兲

+关␥共u_yy−u_zz兲cos␾0−␣^uxysin␾0兴共M⫻xˆ兲其. 共36兲 The parametrical dependence of the SOC mediated, strain-induced spin torque is quite rich and suggests the possibility of engineering its form by appropriately designing the strain properties. In the particular case of an in-plane uniaxial strain such that the only nonvanishing components of the strain tensor are u_xx=u_yy⫽u_xy=u_yx the spin torque

acquires a form similar to the one given in case A but with strain-renormalized SIA and BIA parameters ␣→␣uxy and

␥→␥^uxx.

VII. SUMMARY

We have derived a general expression for the current- induced spin torque in a single ferromagnetic layer in the presence of a linear in momentum SOC field. We have performed explicit calculations of the spin torque in 共001兲, 共110兲, and 共111兲 layers lacking bulk and/or structure inversion symmetry. The spin torque mediated by strain-induced SOC has also been investigated.

ACKNOWLEDGMENTS

We thank J. Fabian and M. Gmitra for valuable discus- sions. This work was supported by the Deutsche Forschungs- gemeinschaft via Grant No. SFB 689 and the FONDECYT grant共Grant No. 1085229兲.

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22In the definition of the spin torque T 关Eq. 共6兲兴in Ref. 17 the authors use the magnetizationMin place of the unit vectorMˆ

=M/M_s. This leads to wrong units forT. However, since the

calculated torque 关see, for example, Eq. 共50兲 in Ref. 17兴 has correct units, the inappropriate replacement ofMˆ byMin their Eq.共6兲seems to be a misprint.

23J. Schliemann and D. Loss, Phys. Rev. B 68, 165311共2003兲.

24An expression for ␦^f␴ which differs from Eq. 共17兲 has been reported by M. Trushin and J. Schliemann 关Phys. Rev. B 75, 155323共2007兲兴for systems with Bychkov-Rashba and Dressel- haus SOCs. We found that, up to first order in the SOCF there is no difference between using our Eq.共17兲or the expression given by Trushin and Schliemann. Similarly, the use of the velocity operator with or without SOC leads, up to first order in the SOCF, to the same results for the spin density 关Eqs.共18兲 and 共19兲兴.

25Up to first order in the SOCF the charge currents with and without SOC coincide.

26T. L. Gilbert, IEEE Trans. Magn. 40, 3443共2004兲.

27Spin Dynamics in Confined Magnetic Structures I, Topics in Ap- plied Physics, edited by B. Hillebrands and K. Ounadjela 共Springer, New York, 2002兲, Vol. 83;Spin Dynamics in Confined Magnetic Structures II, Topics in Applied Physics 共Springer, New York, 2003兲, Vol. 87.

28I. Žutić, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 共2004兲.

29J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. Žutić, Acta Phys. Slov. 57, 565共2007兲.

30J. Moser, A. Matos-Abiague, D. Schuh, W. Wegscheider, J. Fa- bian, and D. Weiss, Phys. Rev. Lett. 99, 056601 共2007兲; A.

Matos-Abiague and J. Fabian, Phys. Rev. B 79, 155303共2009兲.

31X. Cartoixà, L.-W. Wang, D. Z.-Y. Ting, and Y.-C. Chang, Phys.

Rev. B 73, 205341共2006兲.

32M. Gmitra, A. Matos-Abiague, C. Ambrosch-Draxl, and J. Fa- bian, arXiv:0907.4149共unpublished兲.

33The extra factor of 2 in Eq.共50兲of Ref.17is a consequence of the same extra factor in Eq.共47兲of that paper. A simple calcu-

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lation shows that this factor of 2 cannot be obtained when placing the wave functions关Eq.共37兲in Ref.17兴into Eq.共47兲. We suspect the authors of Ref.17may have missed the cancellation of such a factor with the contribution 1/2 from the normalization constant of the wave function.

34H. Diehl, V. A. Shalygin, V. V. Bel’kov, Ch. Hoffmann, S. N.

Danilov, T. Herrle, S. A. Tarasenko, D. Schuh, Ch. Gerl, W.

Wegschelder, W. Prettl, and S. D. Ganichev, New J. Phys. 9, 349共2007兲.

35A. Matos-Abiague, M. Gmitra, and J. Fabian, Phys. Rev. B 80, 045312共2009兲.

36G. E. Pikus and A. N. Titkov, in Optical Orientation, Modern Problems in Condensed Matter Science, edited by F. Meier and B. P. Zakharchenya共North-Holland, Amsterdam, 1984兲, Vol. 8.