Spin-orbit coupling induced by a mass gradient
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Spin-orbit coupling induced by a mass gradient
A. Matos-Abiague
Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany (Dated: October 29, 2018)The existence of a spin-orbit coupling (SOC) induced by the gradient of the effective mass inlow-dimensional heterostructures is revealed. In structurally asymmetric quasi-two-dimensionalsemiconductor heterostructures the presence of a mass gradient across the interfaces results in aSOC which competes with the SOC created by the electric field in the valence band. However, ingraded quantum wells subjected to an external electric field, the mass-gradient induced SOC canbe finite even when the electric field in the valence band vanishes.
PACS numbers: 71.70.Ej,73.61.Ey,73.20.Qt,73.21.Fg
Semiconductor spintronics is an emerging field basedon the controlled manipulation of the carrier spins fordata processing and device operations.
Most proposalsfor spintronic devices rely on the ability of manipulatingelectron spins by using the spin-orbit coupling (SOC),which is the most fundamental spin-dependent interac-tion in nonmagnetic semiconductors. In semiconductorheterostructures the SOC results from the lack of inver-sion symmetry. The bulk inversion asymmetry (BIA) ofzinc-blende semiconductors leads to the so-called Dres-selhaus SOC, while the structure inversion asymmetry(SIA) of the heterostructure itself results in the Bychkov-Rashba (BR) SOC. In this paper I focus on the investigation of the BR-type SOC and show that, in addition to the SOC gen-erated by the electric field in the valence band, thereis a mass-gradient contribution to the SIA-induced SOC.In general, the two contributions compete. However, insome specific cases the mass-gradient induced SOC dom-inates.The emergence of SOC due to the existence of a massgradient can be better understood by establishing ananalogy between the nonrelativistic limit of Dirac’s the-ory and the effective-mass Hamiltonian for conductionelectrons in semiconductor heterostructures.For simplicity and without loss of generality, I considerthe case of a time-independent system in the presenceof an electrostatic potential V . Starting with the time-independent Dirac equation the upper ψ u and lower ψ l components of the four-component spinor Ψ = ( ψ u , ψ l ) T are found to be coupled through the equations ( σ · p ) ψ l = 1 c ( ǫ − V ) ψ u , (1)( σ · p ) ψ u = 1 c ( ǫ − V + 2 m c ) ψ l , (2)where ǫ is the particle energy (measured from the restenergy m c ), σ is the vector of Pauli matrices and p , m , and c , refer to the momentum operator, the bareelectron mass, and the velocity of light, respectively. Itfollows from Eq. (2) that ψ l is smaller than ψ u by a factor ∼ v/c . Thus, in the nonrelativistic limit ( v ≪ c ) themain contribution to the four-component spinor comes from ψ u . Using Eq. (2) one can eliminate ψ l from Eq. (2)and obtain, after some algebra, an equation that involvesonly ψ u , (cid:26) p · (cid:18) µ p (cid:19) + ~ (cid:20) ∇ (cid:18) µ (cid:19) × p (cid:21) · σ + V (cid:27) ψ u = ǫψ u , (3)where I have introduced the position-dependent potential mass µ = m (cid:18) ǫ − V m c (cid:19) . (4)Note that no approximation has been made in deriv-ing Eq. (3). The two-component spinor ψ u is, how-ever, not normalized. The standard procedure to over-come this problem in the nonrelativistic limit is to in-troduce a new normalized two-component spinor ˜ ψ =(1 + p / m c ) ψ u . In doing this, however, some approx-imations has to be made and the Hamiltonian for thenormalized spinor ˜ ψ acquires additional terms. However,these additional terms are irrelevant for our discussionhere and will be omitted in our analysis.The Hamiltonian in Eq. (3) is Hermitian and resem-bles the effective-mass Hamiltonian describing the mo-tion of electrons in a solid with position-dependent ef-fective mass. Interestingly, the spin-orbit coupling seemsto originate from the gradient of the potential mass µ .In the Dirac theory the energy gap, E = 2 m c , whichseparates the energy spectra of the free particles and an-tiparticles is a position-independent constant. Therefore,the only position dependence in µ which can lead to a fi-nite SOC has to come from the electrostatic potential V .Thus, in the Dirac theory, the SOC emerges purely fromthe electric field E = ( − /e ) ∇ V (here e denotes the elec-tron charge). In the nonrelativistic approximation thedominant energy is the vacuum gap E and the inversepotential mass can be approximated as [see Eq. (4)]1 µ ≈ m + V − ǫ m c . (5)As a result the SOC reduces to the well-known form H so = ~ (cid:20) ∇ (cid:18) µ (cid:19) × p (cid:21) · σ ≈ ~ c E ( ∇ V × p ) · σ . (6)In the case of semiconductors the analogue to the po-tential mass µ is the effective mass m ∗ , while the equiva-lent to the vacuum gap E is the energy gap E g separat-ing the energy spectrum of electrons in the conductionband from the hole spectrum in the valence band. Incontrast to the vacuum, where E is a constant, in semi-conductor heterostructures the energy gap E g becomesposition dependent. Therefore, the position dependenceof the effective mass may originate from both the elec-trostatic potential V and the band gap E g . As a result,in addition to the conventional SOC produced purely bythe electric field a finite SOC contribution induced by theposition dependence of the effective mass emerges.To investigate in more details the mass gradient in-duced SOC, I consider a semiconductor heterostructuregrown in the z direction. In such a case the mass gradientinduced SOC can be related to the well-known Bychkov-Rashba (BR) SOC observed in quasi two-dimensionalsystems with structure inversion asymmetry (SIA). The effective Hamiltonian describing the motion of theconduction band electrons in the heterostructure can beobtained by using the envelope function approximation.I consider the (8 ×
8) Kane Hamiltonian which ac-counts for the Γ c , Γ v , and Γ v bands [see bands C,HH and LL, and SO bands, respectively, in Fig. 1]. Theconduction and valence band states can be decoupled byusing the folding-down (L¨owdin) technique. Neglect-ing the non-parabolicity effects, the effective Hamiltonianfor the conduction electrons is found to be H eff = p k m ∗ ( z ) − ~ ddz (cid:20) m ∗ ( z ) ddz (cid:21) + V c ( z ) + H so , (7)where1 m ∗ ( z ) = 1 m − P m (cid:20) V v ( z ) + 1 V v ( z ) − ∆ ( z ) (cid:21) (8)is the z -dependent, inverse effective mass for the conduc-tion band electrons and H so = α ( z ) ~ ( p y σ x − p x σ y ) (9)with α ( z ) = ~ m ddz " ˜ P V v ( z ) − ˜ P V v ( z ) − ∆ ( z ) (10)is the SIA-induced SOC. In the equations above p x and p y are the components of the in-plane momentum p k and ˜ P = h S | p x | P i represents the non-vanishing momen-tum matrix elements involving the s-like band edge Blochstate ( | S i ) of the conduction band and the p-like holestates ( | P i = | X i , | Y i , | Z i ). The correction to the effec-tive mass due to the interaction with remote bands canbe included by using perturbation theory. As shownin Fig. 1, ∆ ( z ) refers to the spin-orbit splitting energy,while V c ( z ) and V v ( z ) are the potential profiles of theconduction and valence band edges, respectively. A0 D B0 D B g E ( ) v V z ( ) c V z z = A B
CHHLHSO A g E FIG. 1: (color online). Schematic of the band-edge profileof an A/B semiconductor interface located at z = 0. Theconduction, heavy hole, light hole, and split-off bands arelabelled as C, HH, LH, and SO, respectively. For the case of a quantum well (QW) grown along the z -direction one can obtain an effective Hamiltonian de-scribing the in-plane motion of a tow-dimensional elec-tron gas by averaging Eq. (7) with the spin-independent z -component, f ( z ), of the wave function. This re-sults in the so-called BR SOC with α BR = h α ( z ) i c = R α ( z ) | f ( z ) | dz as the BR SOC strength.At first glance it seems that the BR spin splittingshould be proportional to the electric field which brakesthe spatial inversion symmetry. However, the fact thatfor quasi 2D systems with position-independent effectivemass the electric field along the direction of confinementmust average to zero (this follows from Ehrenfest’s the-orem which states that the average force on a bound statevanishes ) generated intensive discussions about the na-ture of the electric field causing the BR SOC. Itwas latter found that for a quantum well growth along the z direction, with z -dependent effective mass, the averageelectric field E c ∼ h p z [( ∂ z m − ) p z ] i c may not vanish. However, the estimated value for this field was found tobe too small as to explain the experimentally ob-served spin-splitting due to the SIA SOC. Actually, theSOC induced by E c represents a high-order correctionwhich is not even present in the effective Hamiltonianobtained with the standard (8 ×
8) Kane approximation.In an attempt to clarify the origin of the BR SOC,Lassnig showed that the BR spin splitting in the con-duction band is related to the electric field in the valenceband [ E v = ( − /e ) ∇ V v ] whose average (over the con-duction states) does not necessarily vanish (note that inthis case Ehrenfest’s theorem does not apply ). Below Ishow that in addition to the SOC induced purely by E v acontribution originating from the existence of a mass gra-dient appears. I remark, however, that the mass-gradientcontribution discussed here is much larger than the onegenerated by the mass-gradient induced electric field E c (see discussion in the previous paragraph) and appearsreadily in the effective Hamiltonian resulting from the(8 ×
8) Kane model. In fact, the BR SOC can be rein- ( ) c V z ( ) v V z ( ) c V z v V ext E (a) (b) FIG. 2: (color online). Schematics of the conduction- andvalence-band potentials in a linearly graded quantum wellwith infinite barriers, in the absence (a) and presence (b)of an external electric field E ext . The external electric fieldcompensates the electric field in the valence band in such away that the total electric field in the valence band vanishes(i.e., ∇ V fv = 0). terpreted as resulting from the competition between thehere proposed mass-gradient SOC and the SOC inducedpurely by E v . Interestingly, the mass gradient contri-bution (and therefore the BR SOC) can be finite evenwhen the electric field in the valence band vanishes (i.e, E v = 0).Combining Eqs. (8) and (10) one can rewrite the SOCparameter as α ( z ) = ~ ddz (cid:20) m ∗ ( z ) (cid:21) + ~ m ddz " ˜ P V v ( z ) . (11)When modelling semiconductor heterostructures, themomentum matrix element, ˜ P , is commonly consideredto be position independent. Within this approximationand taking into account that for most semiconductors V v is of the same order of the band gap (i.e., | V v − E g | ≪ E g ,one can approximate Eq. (11) as α ( z ) = ~ ddz (cid:20) m ∗ ( z ) (cid:21) − ~ ( ˜ P /m ) E g ddz V v ( z ) . (12)Thus, one can rewrite the SOC in Eq. (9) as H so = H m so + H v so , where H m so = ~ (cid:20) ∇ (cid:18) m ∗ (cid:19) × p (cid:21) · σ (13)is the mass gradient induced SOC and H v so = − ~ ( ˜ P /m ) E g ( ∇ V v × p ) · σ (14)is the SOC created by the electric field in the valenceband.Equation (14) has the same structure as Eq. (6), which confirms that, indeed, H v so corresponds to the stan-dard SOC generated, purely, by an electric field; in this case by the valence-band electric field E v = ( − /e ) ∇ V v .Consequently, H v so vanishes when E v = 0. On the con-trary, it is clear from Eqs. (8) and (13) that even in thecase of vanishing E v , the mass gradient induced SOC, H m so , remains, in general, finite.For an estimation of the strengths of the mass gradientand valence-band electric field induced SOCs, I considerthe case of an A/B abrupt interface between two III-Vsemiconductors [see Fig. 1]. In such a case the band pa-rameters, and therefore m ∗ and V v , are step-like functionsof z and the mass-gradient and valence-band electric fieldinduced SOCs in Eq. (11) reduce to H m,v so = α m,v ~ δ ( z )( p y σ x − p x σ y ) (15)with α m = ~ (cid:18) m ∗ B − m ∗ A (cid:19) , (16)and α v = ~ m ˜ P E A g − ˜ P E B g ! , (17)respectively. Here the values of the parameters in regionsA and B are indicated with the respective labels. Notealso, that for a better accuracy, the step-like positiondependence of the momentum matrix element has alsobeen considered in Eq. (17).The calculated values of α m , α v , and the total interfaceSOC strength, α int = α m + α v , for different interfaces arelisted in Table I. For all the considered interfaces, α m and α v are of the same order but with opposite signs. Thus,the competition between the mass-gradient and valence-band electric field induced SOC contributions results inthe decrease of the total interface SOC.In systems in which α m and α v are of the same order(as the ones considered above) the mass-gradient contri-bution to the SOC is masked by the SOC induced by thevalence-band electric field. Therefore the experimentalmeasurement of α m alone may be difficult in such sys-tems. To overcome this problem, I propose to measure α m in a graded, semiconductor quantum well subjectedto an external electric field.For illustration, I consider a Ga − x Al x As-based quan-tum well with high potential barriers, so that interfaceeffects play a little role and can be neglected. In a lin-early graded quantum well (i.e, with Al concentrationvarying linearly with the position) the energy gap to-gether with the band parameters become position de-pendent. For small grading, the band parameters in-terpolate linearly with the Al concentration. Conse-quently, both the potential profile of the conduction ( V c )and valence ( V v ) band edges change linearly with z [seeFig. 2(a)]. In the presence of a constant external electricfield, E ext , oriented along the growth direction the bandedge profiles are modified as V c = V c ( z ) − e E ext z and V v = V v ( z ) − e E ext z . Since V v ( z ) is a linear function TABLE I: Interface spin-orbit coupling parameters (in eV˚A ) for A/B abrupt interfaces composed of arsenides. α m and α v correspond to the contributions due to the mass gradient and valence-band electric field, respectively, while α int is the totalinterface SOC strength.A B α m α v α int AlAs InAs 138.93 -169.02 -30.09GaAs AlAs -35.88 39.07 3.19InAs GaAs -103.05 129.95 26.9 of z one can find a target electric field for which V v be-comes position independent and the total electric field inthe valence bands vanishes [see Fig. 2(b)]. Therefore, forsuch a target external field, α v ≈
0, while α m remains fi-nite. Under this condition, the SOC is determined solelyby the mass-gradient contribution.I now estimate the values of α m and the target externalfield E target for a quantum well with Al concentrationvarying linearly from x min = 0 to x max = 0 .
1, i.e., x ( z ) = x max z/d. (18)Here d = 200 ˚A is the well width. The composition de-pendence of the band parameters is evaluated accord-ing to the interpolation scheme developed in Ref. 12. Ithen expand Eq. (8) up to the linear order in x [whichfor the small concentrations considered here ( x ≤ . target ≈
30 kV/cm the valence-band elec-tric field vanishes (i.e., α v = 0), while the strength ofthe mass-gradient induced SOC is found to be α m ≈− .
74 meV ˚A. Apart from the sign, this value is of thesame order but still larger than the SIA SOC parametersexperimentally measured in GaAs-AlGaAs asymmetric quantum wells.
Beyond the case of semiconductor heterostructures,the mass-gradient induced SOC is expected to be relevantin metal/semiconductor interfaces across which the val-ues of the effective mass have large and abrupt changes.In summary, I have revealed the existence of a mass-gradient contribution to the SOC in systems with struc-ture inversion asymmetry. I have shown that for somesemiconductor heterostructures, the mass-gradient con-tribution is of the same order as the SOC generated bythe electric field in the valence band. However, in theparticular case of a linearly graded semiconductor quan-tum well subjected to a conveniently designed externalfield, the electric field in the valence band vanishes andthe remaining SOC is purely induced by the mass gradi-ent.
Acknowledgments
I am grateful to J. Fabian for fruitful discussions. Thiswork was supported by the Deutsche Forschungsgemein-schaft via SFB 689. I. ˇZuti´c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. , 323 (2004). J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, andI. ˇZuti´c, Acta. Phys. Slovaca , 565 (2007). G. Dresselhaus, Phys. Rev. , 580 (1955). Y. A. Bychkov and E. I. Rashba, J. Phys. C , 6039(1984). R. Lassnig, Phys. Rev. B , 8076 (1985). R. Winkler,
Spin-orbit coupling effects in two-dimensionalelectron and hole systems (Springer, Berlin, 2003). R. Winkler, Physica E , 450 (2004). J. J. Sakurai,
Advanced Quantum Mechanics (Addison-Wesley, Reading, Massachusetts, 1967). E. O. Kane, Lecture Notes in Physics , 13 (1980). P. L¨owdin, J. Chem. Phys. , 1396 (1951). E. A. de Andrada e Silva, G. C. La Rocca, and F. Bassani,Phys. Rev. B , 16293 (1997). I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, J.Appl. Phys. , 5815 (2001). A. D¨arr, J. P. Kotthaus, and T. Ando, in
Proceedings of the 13th International Conference of Semiconductors , editedby F. G. Fumi (North-Holland, Amsterdam, 1976), p. 774. J. J. Sakurai,
Modern Quantum Mechanics (Addison-Wesley, Reading, Massachusetts, 1994). F. J. Ohkawa and Y. Uemura, J. Phys. Soc. Jpn. , 1325(1974). F. Malcher, G. Lommer, and U. R¨ossler, Superlatt. Mi-crostruct. , 267 (1986). W. Zawadzki and P. Pfeffer, Phys. Rev. B , 235313(2001). Here Kane’s momentum matrix element ˜ P plays the roleof m c in the Dirac theory. This correspondence has beenpreviously recognized in Ref. 7. B. Jusserand, D. Richards, G. Allan, C. Priester, andB. Etienne, Phys. Rev. B , 4707 (1995). J. B. Miller, D. M. Zumb¨uhl, C. M. Marcus, Y. B. Lyanda-Geller, D. Goldhaber-Gordon, K. Campman, and A. C.Gossard, Phys. Rev. Lett.90