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Theory of electric dipole spin

resonance in a parabolic quantum well

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Citation Efros, Al. L., and E. I. Rashba. 2006. “Theory of Electric Dipole Spin Resonance in a Parabolic Quantum Well.” Physical Review B 73 (16) (April 25). doi:10.1103/physrevb.73.165325.

Published Version doi:10.1103/physrevb.73.165325

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Theory of electric dipole spin resonance in a parabolic quantum well

Al. L. Efros

Naval Research Laboratory, Washington, DC 20375, USA E. I. Rashba

Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 共Received 26 January 2006; published 25 April 2006兲

A theory of electric dipole spin resonance共EDSR兲, that is caused by various mechanisms of spin-orbit coupling, is developed as applied to free electrons in a parabolic quantum well. Choosing a parabolic shape of the well has allowed us to find explicit expressions for the EDSR intensity and its dependence on the magnetic field direction in terms of the basic parameters of the Hamiltonian. By using these expressions, we have investigated and compared the effect of specific mechanisms of spin orbit共SO兲 coupling and different polar- izations of ac electric field on the intensity of EDSR. It is our basic assumption that the SO coupling energy is small compared with all different competing energies共the confinement energy, and the cyclotron and Zeeman energies兲 that allowed us to describe all SO coupling mechanisms in the framework of the same general approach. For this purpose, we have developed an operator formalism for calculating matrix elements of the transitions between different quantum levels. To make these calculations efficient enough and to derive explicit and concise expressions for the EDSR intensity, we have established a set of remarkable identities relating the eigenfrequencies and the angles defining the spatial orientation of the quantizing magnetic field B共␪,␾兲.

Applicability of these identities is not restricted by EDSR and we expect them to be useful for the general theory of parabolic quantum wells. The angular dependences of the EDSR intensity, found for various SO coupling mechanisms, show a fine structure consisting of alternating up and down cusps originating from repopulating different quantum levels and their spin sublevels. Angular dependences of the EDSR intensity are indicative of the relative contributions of the competing mechanisms of SO coupling. Our results show that electrical manipulating electron spins in quantum wells is generally highly efficient, especially by an in-plane ac electric field.

DOI:10.1103/PhysRevB.73.165325 PACS number共s兲: 78.67.De, 71.70.Ej, 76.20.⫹q, 85.75.⫺d

I. INTRODUCTION

Efficient manipulation of electron spins by an external ac field is one of the central problems of semiconductor spintronics,1,2 quantum computing3 and information processing.4The original proposals regarding spin manipula- tion were based on using a time-dependent magnetic field H˜ 共t兲. However, there is a growing understanding of the ad- vantages of spin manipulation by a time-dependent electric field E˜ 共t兲 that allows the access to electron spins with na- nometer precision and can provide much stronger coupling to electron spins through various mechanisms of SO interaction.5,6 Different options of electrical manipulating electron spins in semiconductor nanostructures range from adiabatic pumping spin currents from quantum dots7 to ma- nipulating electron spins in quantum wells共QWs兲 at the spin resonance frequency by employing various mechanisms of SO coupling. Spin orbit interaction

SO= Hˆ

orb共kˆ,兲 + HˆZ共r,␴兲 共1兲 can be usually represented as a sum of the orbital contribu- tion Hˆ

orb共kˆ,兲 depending on the momenta kˆ and Pauli ma- trices ␴, and the Zeeman contribution HˆZ共r,␴兲 depending on the coordinates r and matrices ␴. Because both terms include orbital operators共kˆ or r兲 and matrices␴, they rep- resent different mechanisms of SO coupling.

For two-dimensional 共2D兲 electrons in QWs, two basic mechanisms of the orbital SO coupling are directly related to the QW symmetry properties. They stem from the structure inversion asymmetry 共SIA兲 mechanism described by the Rashba term,8,9 and the A3B5 compound bulk inversion asymmetry共BIA兲 mechanism described by the Dresselhaus term.10 In the principle crystal axes, the bulk Dresselhaus three-dimensional 共3D兲 spin-orbit interaction HˆD can be written as5

D=␦共␴·ˆ兲, ␬ˆx= kˆyxy− kˆzxz, 共2兲 where␬ˆyand␬ˆzcan be derived from␬ˆxby cyclic permuta- tions, and␦ is a parameter. Here kˆj共j=x,y,z兲 are the projec- tions of the momentum operator kˆ = −i⵱ +eA/បc of an elec- tron, A is the vector potential of the magnetic field B共␪,␾兲,

andare the polar angle and the azimuth of B, and −e is the electron charge. In a strong confinement limit, when car- riers are in a quasi-2D regime,Dreduces to a 2D Dressel- haus Hamiltonian. For a rectangular关001兴 QW of the width d,

D=␣D共␴xx−␴yy兲 共3兲 with␣D= −␦具kz2典=−␦共␲/ d兲2.11–15The 2D Dresselhaus Hamil- tonian possesses a discrete symmetry C2vof the关001兴 face of a cubic lattice of the Tdsymmetry typical of the zinc blende

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modification of A3B5crystals, while the Rashba SO Hamil- tonian

R=␣R共␴xy−␴yx兲 共4兲 possesses a continuous group C⬁vof the rotations about the 关001兴 axis. Experimental data of Refs. 16–18 suggest that for GaAs QWs both SIA and BIA terms are of the same order of magnitude. It is a general consensus that in narrow-gap com- pounds the SIA mechanism typically dominates,19 however, the ratio of the coupling constants of only ␣R/␣D⬇2 has been recently reported for InAs QWs of width d = 15 nm.20 The tunability of␣Rby gate voltage21–23is believed to be of critical importance for the operation of semiconductor spin- tronic devices.

The spatial dependence of Zeeman energy Hˆ

Z共r,␴兲 stems either from the inhomogeneity of the field B24 or from a position-dependent g factor.25–28 Recently Kato et al.29 achieved operating electron spins in parabolic AlxGa1−xAs QWs through the gˆ-tensor modulation technique based on the difference in the spatial dependences of the various com- ponents of gˆ = gˆ共z兲. Graded parabolic QWs were originally designed for producing a high mobility electron gas30,31 by applying the modulation doping technique.32The early work on their dynamical responses was mostly concerned with the effect of electron-electron interaction on the electron orbital dynamics and the related transition frequencies,33 and also with generalizing the Kohn theorem.34The recent success in electrical spin manipulation29shifted the interest to the spin flip transitions in such systems and the effect of various mechanisms of SO interaction.

The existence of several mechanisms of SO coupling makes it important to develop reliable experimental tech- niques for identifying them. Also, the relative efficiency of different SO coupling mechanisms strongly depends on the choice of the semiconductor materials and the shape of a QW or a heterojunction. The efficiency of gˆ-tensor modulation technique29is based on the anomalously small g-factor value in AlxGa1−xAs, 兩g兩ⱗ0.1, and is therefore specific for GaAs based devices. Developing similar techniques for narrow-gap A3B5 semiconductors with typically large g values,兩g兩ⲏ10, needs different approaches, and we show that orbital mechanisms of SO coupling can be rather efficient for them. Moreover, we prove that the efficiency of different mechanisms depends strongly on the polarization of the ac electric field E˜ .

To consider all these problems in a framework of an uni- fied approach and to derive analytical expressions for the transition probabilities, we make two basic assump- tions. First, we choose a parabolic shape for the QW. Second, we accept that SO coupling is not too strong, i.e., the SO coupling energy ESO is small compared with all differ- ent energies, including the confinement energyប␻0, the cy- clotron energy ប␻c, and the Zeeman energy ប␻s. Here ␻0

is the characteristic frequency of the parabolic potential,

c= eB / mc is the cyclotron frequency of electrons with the effective mass m, ands= gBB /ប is the spin resonance fre- quency, with␮B= eប/2m0c being the Bohr magneton. These assumptions allow one to account for HˆSOonly in the matrix

elements of spin transitions and disregard the effect of Hˆ

SO

on the position of energy levels.35In this approximation, one can take advantage of the exact solution38,39for the quanti- zation of electron levels in a magnetic field tilted with re- spect to the confinement plane.

Meanwhile, applying the exact solution found by Maan38 and Merlin39 is not straightforward when it comes to calcu- lating matrix elements of spin flip transitions. Indeed, their solution depends on an auxiliary angle␥ defined by the de- coupling condition of two normal modes, hence, all matrix elements depend on ␥. We have found an extensive set of remarkable identities relating␥, frequencies␻␰,共␪兲 of two eigenmodes, and the polar angle ␪. The symmetry of the problem underlying these identities is far from obvious, but the identities permitted us to eliminate ␥ and find all final results in an explicit form. Some of these results, without the derivation, were published in our previous papers.40,41 We expect that the technique will facilitate developing the theory of different properties of parabolic QWs which is timely be- cause of the recent progress in experimental work25,29 and the competition of the different mechanisms of SO coupling that manifests itself in various phenomena.

In this paper we develop a general theory of electric di- pole spin resonance共EDSR兲 in QWs which is caused by the standard Hamiltonians of the Hˆ

orb共kˆ,␴兲 type for two basic geometries: with the ac electric field in the QW plane and perpendicular to this plane. We solve exactly the problem of an electron confined in a parabolic QW being a subject to a tilted magnetic field B and find the EDSR intensity in the Dresselhaus and Rashba models versus the B direction. Our results show that electric dipole spin resonance is especially strong when it is excited by an in-plane electrical field. How- ever, we show that it is also strong enough in the geometry when the time-dependent potential is applied to a gate. Our results demonstrate convincingly that an efficient electrical spin manipulation can be achieved through the orbital mechanisms of spin-orbit coupling. Our results also suggest that the angular dependence of EDSR intensity is a unique characteristic of various competing mechanisms of spin-orbit coupling contributing to EDSR.

The paper is organized as follows. In Sec. II we develop a theoretical approach to an electron in a parabolic quantum well and derive operator expressions for the basic variables like the coordinates and kinetic momenta. While we apply these results for calculating the EDSR intensity, they are rather general and can be applied to different problems re- lated to parabolic quantum wells. In Sec. III we calculate the EDSR intensity for the Dresselhaus and Rashba 2D spin- orbit coupling Hamiltonians for the two basic geometries with an in-plane and perpendicular-to-plane electric field. We also investigate in detail the dependence of the EDSR inten- sity on the direction of the magnetic field and present figures that illustrate the basic mechanisms and characteristic fea- tures that should help assigning specific EDSR bands when experimental work begins. In Sec. IV we develop a theory of EDSR in a parabolic well with a 3D Dresselhaus SO inter- action and unveil the specific features of EDSR that distin- guish it from the results of the 2D model corresponding to a strong confinement limit. The discussion of obtained results

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and estimates of the EDSR intensity are given in Sec. V.

Appendix A includes a number of identities that have been highly instrumental in deriving the results of Secs. III and IV, and that we expect to be useful for future theoretical work on parabolic quantum wells. Appendix B supports the calcula- tions of Sec. IV.

II. AN ELECTRON IN A PARABOLIC WELL A SUBJECT TO A TILTED MAGNETIC FIELD

Everywhere in what follows we suppose that SO correc- tions to the energy levels are small as compared with the separation between adjacent Landau levels and Zeeman sub- levels. Therefore, we disregard the effect of SO coupling on the energy spectrum and begin with the Hamiltonian of the orbital motion in a parabolic quantum well

0= ប2

2m

− i⵱ +បce A

2+ 12m0

2z2, 共5兲

where A is the vector potential of the magnetic field B. Be- cause the Hamiltonian is quadratic in the coordinates, the problem can be solved exactly, and the energy spectrum is well known.38,39However, because for calculating matrix el- ements of EDSR we need explicit expressions for the opera- tors of coordinates and momenta, somewhat lengthly calcu- lations should be performed.

A. Diagonalization of the Hamiltonian

First, we perform a transformation U from the original coordinate system x , y , z related to the principal crystal axes to a new 共“primed”兲 reference system x, y, zwith the zaxis parallel to B and the yaxis lying in the x , y plane. The coordinates in both systems are related as

xyz

= Uˆ

xyz

,

where

Uˆ =

coscos− sincossin − sincos0 sinsincoscossin

. 共6兲

Similar equations are valid for momenta projections, i= Uˆ

iii, where i = x , y , z and summation over repeated co- ordinate indices is implied. Because y⬘does not appear in the potential energy m0

2z2/ 2, it is convenient to choose the Lan- dau gauge with Ax= 0, Ay= Bx, Az= 0. Then y⬘ is a cyclic variable, the Landau momentum k⬅kyis a c number, and

0= ប2

2m关−⳵x2⬘−⳵2z⬘+共k + x⬘/␭22兴 +1

2m0

2共− sin␪x+ cosz⬘兲2, 共7兲 where ␭=

cប/eB is the magnetic length. To decouple the

motion in two degrees of freedom in Eq.共7兲, we perform a rotation

x= cos␥␰+ sin␥␩,

z= − sin␥␰+ cos␥␩. 共8兲 Then the cancelation condition for the mixed product ␰␩

results in an equation on the auxiliary angle␥

sin 2␥=共␻0/␻c2sin关2共␪+␥兲兴 共9兲 and the coefficients at ␰2 and2, together with the kinetic energy, define the eigenfrequencies

2共␪兲 =␻c

2cos2␥+␻0

2sin2共␪+␥兲

2共␪兲 =␻c

2sin2+02cos2共␪+␥兲 共10兲 of two normal modes,␰and␩. Equations共9兲 and 共10兲 com- plete the diagonalization of the quadratic part of Hˆ

0and are in agreement with the results by Maan38and Merlin.39

Eliminating in Eq.共7兲 the term that is linear in xcan be achieved by shifting␰andby

0= − k␭2cos␥

cos2+共␻0/␻c2sin2共␪+␥兲= −␭2kcos共␪+␥兲 cos␪ ,

0= − k␭2sin␥

sin2␥+共␻0/␻c2cos2共␪+␥兲= −␭2ksin共␪+␥兲 cos␪ .

共11兲 In these equations, first expressions for␰0 and␩0 come di- rectly from calculations, while the simplified form of them can be found by using Eqs.共A13兲.

The free term in Eq.共7兲 equals E0共k兲 =2k2

2mm

2共␰022+022兲 = 0. 共12兲 Vanishing of this term follows from Eqs.共9兲–共11兲, and inde- pendence of the energy from the Landau momentum k is a consequence of the translational symmetry in the 共x,y兲 plane. The shifts␰0,␩0satisfy the identities

cos␥␰0+ sin␥␩0= −␭2k,

sin共␪+␥兲␰0− cos共␪+␥兲␩0= 0. 共13兲 Equation共10兲 allows one to find the spectrum of the sys- tem, i.e., the set of two frequencies, 兵␻共␪兲,␻共␪兲其. How- ever, because Eq. 共10兲 includes the auxiliary angle ␥ that should be found from Eq.共9兲, the shape of the curves␻2共␪兲 and␻2共␪兲 is not obvious, depends on the ratio ␻c/␻0, and identification of a specific eigenvalue as␻2共␪兲 or␻2共␪兲 is a matter of convention. In what follows, we identify␻共␪兲 and

共␪兲 as the frequencies that at ␪= 0 coincide with the cy- clotron frequency␻cand the confinement frequency ␻0, re- spectively. Then, using Eqs.共A14兲 and 共A16兲, we derive ex- plicit expressions for␻2共␪兲 and␻2共␪兲

2共␪兲 =1

2共␻02+␻c2−⍀2sign兵␻0−␻c其兲,

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2共␪兲 =1 2共␻0

2+␻c

2+⍀2sign兵␻0−␻c其兲, 共14兲

where

2共␪兲 =

04+␻c 4− 2␻0

2c

2cos 2␪. 共15兲 This spectrum is plotted in Fig. 1共a兲 as a function of

c/␻0 for three values of the polar angle ␪. For= 0, the spectrum consists of the cyclotron and confinement branches, ␰ and ␩, respectively. However, for ␪⫽0 the

branches interchange at ␻c=␻0. Left parts of the lower branches and right parts of the upper branches are described by the␻ solution and show a cyclotron-like behavior. The opposite parts of the same branches are described by the␻ solution and show a confinement-like behavior. Of course, all branches are continuous and smooth.

The same spectrum is plotted in Fig. 1共b兲 as the function of␪for three values of␻c/␻0. All branches retain their iden- tity,␰or␩, in the whole interval 0艋␪艋␲/ 2. Allbranches originate at the same frequency␻共␪= 0兲=␻0; they are upper branches for␻c⬍␻0and lower branches for␻c⬎␻0. On the contrary,␰branches are lower branches for␻c⬍␻0and up- per branches for␻c⬎␻0; they originate at the frequencies␻c

that are B dependent. Forc=␻0, the spectrum is described by a simple equation ␻±共␪兲=␻0共1±sin␪兲1/2. These curves are separatrices dividing ␰ and␩ regions, hence, neither ␰ nor␩identity can be ascribed to them.

The Hamiltonian Hˆ

0of Eq.共7兲, when written in the vari- ables␰and, reads

0=

2m22+21m202

, 共16兲

where the summation is performed over␨=,␩. After intro- ducing step operators

0=

ប/2m共a++ a兲,

= − i= i

m/2ប共a+− a兲, 共17兲 the Hamiltonian takes the standard oscillator form

0=1 2

ប␻共a+a+ aa+兲. 共18兲

B. Operator representation for coordinates and momenta Similarly to Eq. 共6兲, one can express the components j= −iⵜj+共e/បc兲Aj of the kinetic momentum in the crystal frame through its components in the primed frame. In the latter frame, the vector-potential A contributes only to the component kˆy⬘that equals

y= k + x⬘/␭2=关cos␥共␰0兲 + sin␥共␩0兲兴/␭2. 共19兲 When deriving Eq.共19兲, the upper identity of Eq. 共13兲 has been used. Using Eq.共8兲, one can express kˆxand kˆz⬘through the derivatives ⳵ and rewrite the components of the mo- menta in the crystal frame in terms of a, a+by applying Eq.

共17兲

x=

共Xa+ X*a+兲,

y=

共Ya+ Y*a+兲, FIG. 1. 共Color online兲 The energy spectrum of electrons con-

fined in a parabolic quantum well in a tilted magnetic field. Here,

0is the confinement frequency,␻c= eB / mc is the cyclotron fre- quency, ␪ is the polar angle of the field B, ␻ and ␻ are the frequencies of two eigenmodes␻,␨=␰,␩.共a兲 The dependence of

on the ratio␻c/␻0for three values of␪. Cyclotron-like modes are shown by dotted lines and confinement-like modes by full lines.

For␪=0 these modes intersect, while for ␪⫽0 they interchange at

c=␻0.共b兲 The dependence of␻on␪ for three values of ␻c/␻0. The modes retain their identity in the whole interval 0艋␪艋␲/2, and␻⬎␻ for␻c⬍␻0while␻⬍␻ for␻c⬎␻0. At the degen- eracy point,␻c=␻0, the two␻,branches have no specific identity.

They are separatrices dividing the plane into one␩共gray兲 and two

␰ 共white兲 regions.

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z=

共Za+ Z*a+兲. 共20兲 Here

X=

m2ប

− i cos共+兲cosccos␥sin

,

X=

m2ប

− i sin+兲coscsinsin

,

Y= −

m2

i cos共+兲sinccos␥cos␾

,

Y= −

m2ប

i sin共+兲sinc sincos

,

Z= i

m2បsin共␪+␥兲,

Z= − i

m2 cos共␪+␥兲. 共21兲 These coefficients depend only on the angles 共␪,␾兲 and the eigenfrequencies␻and do not depend on k,0, and␩0. Therefore, despite the fact that Eqs.共20兲 and 共21兲 have been derived in the Landau representation, the final results are gauge invariant and may be conveniently employed for find- ing operators共xˆ,yˆ,zˆ兲 of the coordinates 共x,y,z兲.

It is seen from Eq.共5兲 that the operator zˆ can be found in terms of共a, a+兲 from the difference H0−ប22/ 2m. Employ- ing Eqs. 共18兲, 共20兲, and 共21兲 and some of the identities of Appendix A, we arrive after somewhat lengthly algebra at

zˆ =

共iប/m兲Z共a+ a+兲. 共22兲

The sign of zˆ has been found from the commutation relation 关kˆz, zˆ兴=−i, and one can easily check that z˙ˆ=共i/ប兲关Hˆ0, zˆ兴

=បkˆz/ m.

For finding xˆ and yˆ, we generalize the Johnson and Lippman42procedure for the quantization of electron motion in a strong magnetic field for an electron confined in a para- bolic quantum well. Using the equations of motion for the operators of momenta and coordinates

k˙ˆ

x=␻c共byz− bzy兲, k˙ˆ

y=␻c共bzx− bxz兲, 共23兲 x˙ =បkˆx/m, y˙ =បkˆy/m, z˙ =បkˆz/m, 共24兲 where b = B / B, and eliminating the components of the mo- mentum kˆ, we arrive at the equations

x˙ˆ =共␭2k˙ˆ

y+ bxz˙ˆ兲/bz, y˙ˆ =共− ␭2k˙ˆ

x+ byz˙ˆ兲/bz. 共25兲 Equations共25兲 suggest existence of two integrals of motion

0= xˆ −共bxzˆ +2y兲/bz, 0= yˆ −共byzˆ −2x兲/bz, 共26兲 that generalize the well known guiding center coordinates.

They obey the same commutation relation

关xˆ0,yˆ0兴 = i␭2/cos␪ 共27兲 and commute with operators a. Remarkably, in a parabolic well the operators xˆ0and yˆ0involve the vertical coordinate zˆ.

Nevertheless, Eq.共27兲 ensures that the degeneracy of states is completely controlled by Bz, the component of B perpen- dicular to the QW plane, and is described by the standard Landau formula nL共␪兲=cos␪/ 2␲␭2.

Because the operators kˆx, kˆy, and zˆ are already known, Eq.

共26兲 allows one to find xˆ and yˆ. To derive explicit expres- sions for these operators, it is convenient to use identities

Y+共ic/␻兲sin␪cos␾Z=共ic/␻兲cos␪X, X+共ic/␻兲sin␪sinZ=共ic/␻兲cos␪Y 共28兲 that can be checked using Eqs. 共10兲 and 共21兲. Finally, the operators of in-plane coordinates are

xˆ =

共iប/m兲共Xa− X*a+兲 + xˆ0,

yˆ =

共iប/m兲共Ya− Y*a+兲 + yˆ0. 共29兲 Because Z*= −Z, Eqs.共29兲 are similar to Eq. 共22兲 and differ from it only by presence of the guiding center operators.

Therefore, six operators of coordinates and momenta, xj and kˆj, j = x , y , z, are expressed in terms of four Bose opera- tors 共a, a+兲 and two generalized guiding center coordinates 共xˆ0, yˆ0兲. The latter commute with all operators 共a, a+兲;

hence, their presence in Eq.共29兲 does not influence electron dynamics in homogeneous external fields that are coupled to an electron only through the operators共a, a+兲. Operators xˆj

and kˆj, defined by Eqs.共20兲, 共22兲, and 共29兲 obey the standard commutation relations

关kˆj,xˆ兴 = − ij,关kˆj,kˆ兴 = − i␭−2jᐉmBm/B, 共30兲 where⑀jᐉmis the Levi-Civita tensor.

III. INTENSITY OF EDSR

The EDSR occurs in QWs due spin-orbit Hamiltonians

SOdescribed by Eqs. 共2兲–共4兲 which mix the electron spin projections on the magnetic field direction, hence, each elec- tron state acquires admixture of the opposite spin projection that is small when Hˆ

SOis weak. As a result, a time-dependent electric field E˜ 共t兲 causes EDSR, i.e., electrically induced spin-flip transitions. Matrix elements of EDSR are comprised of two contributions, of which one comes from the perturba- tion of the wave functions of stationary states and the second from the direct coupling of the electron spin to the field E˜ 共t兲.

A convenient way for calculating EDSR matrix elements is

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based on employing a canonical transformation exp共Tˆ兲 elimi- nating Hˆ

so in the first order of perturbation theory.5,6After the transformation, the time-independent part of Hˆ conserves the electron spin projection on the magnetic field direction, and spin-flip transitions are induced only by the spin-orbit contribution e共vso· A˜ 兲/c to the time-dependent part of Hˆ, where A˜ 共t兲 is a time-dependent vector potential, and vˆso is the spin-orbit part of the velocity operator.

Because of the equation of motion vˆ =共i/ប兲关Hˆ,rˆ兴, the ma- trix elements of the operators vˆ and rˆ for spin-flip transitions are related as具vˆ典=is具rˆ典, withs being the spin-flip transi- tion frequency. Neither of the two competing contributions to vˆ includes the factors, hence, it indicates the existence of massive cancellations that tremendously complicate calcula- tions based on the operator vˆ. It is much more convenient to write the time-dependent part of the Hamiltonian as Hˆ

int共t兲

= e共rˆ·Eˆ共t兲兲. In the original representation, the coordinate op- erator rˆ is diagonal in spin indices, and Hˆ

int共t兲 produces spin- flip transitions due to the level mixing. However, after the Tˆ transformation the operator rˆ acquires a SO part rˆso=关Tˆ,rˆ兴 that drives spin-flip transitions.

The total Hamiltonian of an electron confined in a para- bolic QW is Hˆ =Hˆ0+ Hˆ

Z+ Hˆ

so+ Hˆ

int共t兲, with HˆZ=12gB共␴· B兲.

The energy levels of the Hamiltonian Hˆ

0+ Hˆ

Zare E共n,n兲 =

ប␻共␪兲共n+ 1/2兲 + ប␻s/2, 共31兲 where n␰,艌0 and the spin index ␴= ± 1. The spin-flip fre- quency␻sshould be taken algebraically;␻s⬍0 for electrons in negative g-factor semiconductors. The time-independent spin-orbit interaction Hˆ

so will be considered as a perturba- tion.

The term Hˆ

so=HD, Hˆ

D, or Hˆ

R in the time-independent part of the Hamiltonian Hˆ leads to the mixing of spin sub- levels. As we have stated above, it can be eliminated in the first order of the perturbation theory by a canonical transfor- mation exp共Tˆ兲. The operator Tˆ is nondiagonal in the orbital quantum numbers共n, n兲, and its matrix elements are

具n,n,␴⬘兩Tˆ兩n,n,␴典 =具n,n,␴⬘兩Hˆso兩n,n,␴典 E共n,n兲 − E共n,n兲. 共32兲 In terms of Tˆ, the matrix elements of spin-flip transitions diagonal in共n, n兲 are

具n,n,↑兩共E˜ · rˆso兲兩n,n,↓典

=

n,n

兵具n,n,↑兩Tˆ兩n,n⬘,↓典具n,n兩共E˜ · rˆ兲兩n,n

具n,n兩共E˜ · rˆ兲兩n,n典具n,n⬘,↑兩Tˆ兩n,n,↓典其, 共33兲 with n= n± 1 and n= n± 1. The sum in Eq. 共33兲 is re- stricted because the operator rˆ is linear in共a, a+兲, cf. Eqs.

共22兲 and 共29兲. This restriction significantly simplifies matrix

elements 具n, n,↑兩Tˆ兩n, n↓典 since their denominators do not depend on共n, n兲 and are equal to ប共±␻+␻s兲.

When the operator Hˆ

SO is linear in momenta and conse- quently in the operators共a, a+兲, Eq. 共33兲 reduces to

具n,n,↑兩共E˜ · rˆso兲兩n,n,↓典

= −␨=␰,

共E˜ · l*ប共兲具0,↑兩Hˆ−␻sso兩1,↓典 +共E˜ · l兲具1,↑兩Hˆso兩0,↓典

ប共␻+s

共34兲

with

lx= iប

mX, ly= iប

mY, lz= iប

mZ. 共35兲 Remarkably, this matrix element does not depend on共n, n兲.

For calculating the matrix element具↑兩Hˆso兩↓典 in Eq. 共34兲, we transform Pauli matrices from the crystal frame 共x,y,z兲 to the primed frame 共x, y, z兲 asˆi= Uˆ

ii⬘␴ˆi⬘, similarly to Eq.

共6兲. Substituting ␴ˆx,y into the 2D Dresselhaus and Rashba Hamiltonians of Eqs. 共3兲 and 共4兲, and using 具↑兩ˆx兩↓典=1, 具↑兩ˆy兩↓典=−i, and 具↑兩ˆz兩↓典=0, we arrive at

具↑兩Hˆso兩↓典 =

共H+ H++兲, 共36兲

with the matrix elements H±that should be found from Eqs.

共20兲 and 共21兲. Applying Eq. 共A13兲, we find for a Rashba Hamiltonian

HR,␨⫿ =␣R

m/2ប共␻/␻c± 1兲f, 共37兲 with f= cos共␪+兲 and f= sin共␪+␥兲, and for a 2D Dressel- haus Hamiltonian

H2D,␨⫿ =␣D

m2ប

i cos 2

ccos ⫿ cos

− sin 2␾

c

⫿ 1

冊 册

f. 共38兲

Substituting Eq.共36兲 into Eq. 共34兲, we find 具n,n,↑兩共E˜ · rˆso兲兩n,n,↓典 = −1

1

2−␻s2

⫻ 兵␻关H共l· E˜ 兲*+ H+共l· E˜ 兲兴 +␻s关H共l· E˜ 兲*− H+共l· E˜ 兲兴其.

共39兲

A. Electric field perpendicular to the QW plane Let us start with a time-dependent electric field perpen- dicular to the QW plane, E˜ 共t兲zˆ. In this geometry lz,␨

= lz,␨* , and the matrix element of Eq. 共39兲, Lz

=具n, n,↑兩zˆso兩n, n,↓典, equals

(8)

Lz= −1

lz

H2+ H+

−␻s 2 +␻s

H− H+

2s

2

. 共40兲

For a Rashba SO coupling, substituting Eq.共37兲 into Eq. 共40兲 results in

LRz=␣R

2ប

1 +cs

sD共22sin 2共␪+␥兲, 共41兲 with

D共␪兲 ⬅ 共␻22s兲共␻2s2

=␻02c2cos2s2共␻02+␻c2−␻s2兲; 共42兲 here, Eq.共A10兲 has been applied. By using Eq. 共A15兲, one arrives at the final result40,43

LRz = −␣Rs

2ប

共␻c+␻s兲␻c

D共␪兲 sin 2␪. 共43兲 For Dresselhaus SO coupling, a similar procedure leads to the matrix element of Ref. 40

LDz = −␣Ds

csin␪

D共␪兲 关sin 2␾cos␪共␻c−␻s

− i cos 2␾共␻ccos2␪−␻s兲兴. 共44兲 B. Electric field in the quantum well plane

For an in-plane electric field, we calculte the scalar prod- uct共l· E˜ 兲. If the field E˜共t兲 is polarized at the angleto the crystal x axis, E˜ 共t兲=E˜共t兲共cos␺, sin␺, 0兲, then 共l· E˜ 兲

= lE˜ 共t兲 with

l= lxcos␺+ lysin␺

=

2mf

cos共兲 − iccossin共␾−␺兲

.

共45兲 When deriving Eq. 共45兲, the identity of Eq. 共A7兲 has been

used. Equation共45兲 allows one to rewrite the matrix element L=具n, n,↑兩xˆsocos␺+ yˆsosin␺兩n, n,↓典 of Eq. 共34兲 as

L= −1 ប

Re兵l

H2+ H+

−␻s 2 +␻s

H− H+

2s 2

+ i

Im兵l

sH2+ H−␻s2++␻H− H+

2s2

. 共46兲

For Rashba SO coupling, using Eq.共37兲 results in LR= −␣Rcos共␾␺兲

1cf222

−␻s 2+␻s

f2

2s 2

− iRsin共␾␺兲

cos␪ប␻c

1 +cs

f222s

2. 共47兲

Applying identities of Eq.共A3兲, one finds an explicit expres- sion for the dependence of the matrix element on the mag- netic field direction

LR= −␣R

ប cos共␾−␺兲

⫻␻ccos2␪共␻0 2−␻s

2兲 +␻s共␻0 2+␻c

2sin2s 2D共␪兲

− iR

ប cos␪sin共␾␺兲共␻c+␻s兲共␻0 2−␻s

2

D共␪兲 . 共48兲

In the strong 2D limit, when␻0Ⰷ␻c, this expression simpli- fies significantly, and we arrive at the result of Ref. 41.

LR= − ␣R

ប共␻c*2−␻s2兲关cos共␾−␺兲共␻c

*cos␪+␻s+ i sin共␺兲共␻c

*+␻scos␪兲兴. 共49兲

Here␻c

*=␻ccos␪is the cyclotron frequency in a tilted mag- netic field; from now on␪ is defined as 0艋␪艋␲/ 2.

For a 2D Desselhaus SO coupling, substituting Eq. 共38兲 into Eq.共46兲 results in

LD/␣D= −cos共␾␺兲

f222

−␻s 2

i cos 2− sin 2cos

ccos␪ +

sf2

2s

2共sin 2␾− i cos 2cos␪兲

i sin共␺兲

cos␪ប␻c

f22

2

−␻s

2

s共i cos 2c− sin 2cos cos+ sin 2− i cos 2cos

. 共50兲

By using identities of Eq.共A3兲, an explicit expression for this matrix element can be found

LD/␣D= −cos共␾␺兲

ccosD共022s共i cos 2− sin 2cos兲 +s20+c2D共sin22s共sin 2− i cos 2cos

i sin共␾␺兲 ប

共␻0 2−␻s

2

D共␪兲 关␻s共i cos 2− sin 2cos␪兲 +␻ccos␪共sin 2␾− i cos 2cos␪兲兴. 共51兲

(9)

In the strong 2D limit, ␻0Ⰷ␻c, through a number of transformations this expression can be brought to the final form41

LD= ␣D

ប共␻c*2−␻s2兲关sin共␾+␺兲共␻c

*cos␪−␻s

− i cos共+␺兲共␻c

*−␻scos␪兲兴. 共52兲

In conclusion, we note that in this and the previous sub- section we considered the Hamiltonians Hˆ

D and Hˆ

R on a similar footing. However, there is a considerable difference in their ranges of applicability. The Hamiltonian Hˆ

D origi- nates from the 3D HamiltonianDand is valid only in the strong confinement limit when dⰆkF

−1,␭, where kF is the Fermi momentum. When the confinement length d becomes comparable to kF−1or the magnetic length␭, additional terms like kxky2 should be included in the 2D Hamiltonian. From this standpoint, only Eq. 共52兲 has a real physical meaning while Eq.共51兲 should be only considered as an auxiliary one.

Having those limitations in mind, Eq.共52兲 will be applied in Sec. III C 2 only to electrons confined at the ground level, n, n= 0. Equation共44兲 is subject to similar restrictions, and EDSR with n, n⫽0 electrons in a field E˜zˆ will be con- sidered in Sec. IV in the framework of a more general theory.

The applicability range of the Hamiltonian Hˆ

R is much wider because it is applicable not only to zinc blende crystals under the conditions of 2D confinement but is inherent in the wurtzite modification of A3B5 compounds as a bulk property.6,8Currently, active experimental work on SO prop- erties of microstructures including the wurtzite modifications of InAs,44InN,45GaN,46–48etc., is under way, and large SO splittings up to 9 meV have been reported.47 Following the initial calculations of the band structure of wurtzite-type compounds,49 more general models have been developed recently,50,51and it looks like the band folding in the hexago- nal direction共resulting from the different size of the elemen- tary cells in the zinc blende and wurtzite lattices兲 plays a role in developing large spin splittings.52 Therefore, the Hamil- tonian Hˆ

R will be applied for arbitrary n, n⫽0.

C. Polarization dependence of EDSR intensity: General properties

In this section, we discuss at a qualitative level the basic properties of EDSR following from the equations of Secs.

III A and III B.

In all cases, the intensity of EDSR shows a pole when one of the eigenfrequencies coincides with兩␻s兩,␻共␪pole兲=兩␻s兩. It is seen from Fig. 1共b兲 that such a pole always exists be- cause one of the frequencies ␻共␪兲 vanishes for ␪=/ 2.53 For␻0⬎␻cit happens to the␻mode while for ␻0⬍␻cto the ␻ mode. The magnitude of ␪pole depends on the rela- tive magnitudes of␻0,␻c, and␻s. For␻0Ⰷ␻c,␪poledepends mostly on the ratio␻s/␻c. It is very close to ␲/ 2 for GaAs because of its anomalously small g factor, and equals

pole⬇0.89⫻␲/ 2 for InAs and ␪pole⬇0.78⫻␲/ 2 for InSb.

Therefore, for the materials like InAs and InSb the maximum can still be achieved in the range of␪and B values where the

transformation of Eq.共32兲 remains justified. Additional de- pendence of the matrix elements on␪ comes from the nu- merators of Eqs. 共43兲 and 共44兲 and their analogs for an in- plane electric field. This dependence is SO coupling specific.

Singularities at the poles are cut off by a level width.

The ␪ dependencies of the EDSR transition matrix ele- ment for E˜zˆ and E˜ ⬜zˆ are significantly different. First, for the perpendicular-to-plane polarization, the factor sin 2␪ in Eq.共43兲 and the factor sin␪in Eq.共44兲 nullify these matrix elements at ␪= 0. There are no such factors in the matrix elements of Sec. III B. Therefore, EDSR can be observed only in a tilted magnetic field B when E˜ 共t兲zˆ, but an arbi- trary orientation of B can be used 共including B兲 for an in-plane Eˆ 共t兲.

The second difference in EDSR with different E˜ 共t兲 polar- izations concerns its intensity. This difference is essentially pronounced in the strong confinement regime, ␻0Ⰷ␻s,␻c. Comparing Eqs. 共43兲 and 共44兲 with Eqs. 共49兲 and 共52兲, one can estimate the corresponding matrix elements as

⬃␣R,Ds/ប␻02 for E˜ 共t兲zˆ and as ⬃␣R,D/ប␻c for E˜ 共t兲⬜zˆ, respectively, when the inequality ␻cⲏ兩␻s兩 is satisfied. The first estimate indicates that when E˜ 共t兲zˆ, EDSR is possible only due to the deviation of the system from the strict 2D limit. On the contrary, when E˜ 共t兲⬜zˆ, EDSR survives in the strict 2D limit. The ratio of the matrix elements is about

cs/␻02Ⰶ1, hence, in the strong confinement limit an in- plane field E˜ 共t兲 is much more efficient than a perpendicular- to-plane field.

For BIA, it is also instructive to compare the magnitudes of the EDSR transition matrix elements in a strong 2D con- finement limit, LD, described by Eq.共52兲, and in the bulk, L3D⬃␦m /2, as found by Rashba and Sheka.5 We find LD/ L⬃␻0/␻cⰇ1 using␣D⬃␦m0/ប from Eq. 共67兲 below.

This enhancement of EDSR in a QW in in-plane geometry can be attributed to a strong confinement with the square of the confinement momentum, m0/ប, large compared with

−2. Extraordinarily high efficiency of EDSR in the in-plane geometry has been emphasized in Ref. 41.

We note that despite the fact that the intensity of EDSR with E˜ 共t兲zˆ is much less than with E˜ 共t兲⬜zˆ, it is usually high enough for efficient electrically manipulating electron spins.

Specific estimates can be found in Ref. 40 and will be given in Sec. V below.

As has been discussed above, there is a similarity in some of the properties of EDSR caused by BIA and SIA. The most striking difference in the effect of these mechanisms is seen in the angular dependences of the EDSR intensity, especially in its dependence on the azimuth␾, as will be discussed in Sec. III C 2.

1. Dependence of EDSR on the polar angle

Using the equations of Sec. III A, we will provide and discuss here the dependence of the EDSR intensity on the polar angle␪as applied to InAs. We restrict ourselves to SIA because Eq. 共3兲 for 2D Dresselhaus coupling is applicable only under strong confinement conditions when␻0Ⰷ␻c; cf.

References

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