We formulate a semiclassical theory for systems with spin-orbit interactions.
Using spin coherent states, we start from the path integral in an extended
phase space, formulate the classical dynamics of the coupled orbital and spin
degrees of freedom, and calculate the ingredients of Gutzwiller's trace formula
for the density of states. For a two-dimensional quantum dot with a spin-orbit
interaction of Rashba type, we obtain satisfactory agreement with fully
quantum-mechanical calculations. The mode-conversion problem, which arose in an
earlier semiclassical approach, has hereby been overcome.Comment: LaTeX (RevTeX), 4 pages, 2 figures, accepted for Physical Review
  Letters; final version (v2) for publication with minor editorial change

A. Alscher

A. Bohr

A. Sugita

A. M. Perelomov

Ch. Amann

H. Frisk

H. Kuratsuji

H. G. Solari

M. Brack

M. Mehta

M. Miettinen

M. Oestreich

M. Pletyukhov

M. Sieber

M. Stone

M. C. Gutzwiller

R. G. Littlejohn

S. Das Sarma

S. Datta

T. Darnhofer

T. Matsuyama

V. Smirnov

English

arXiv

Crossref

Semiclassical Theory of Spin-Orbit Interactions using Spin Coherent States

We formulate a semiclassical theory for systems with spin-orbit interactions. Using spin coherent states, we start from the path integral in an extended phase space, formulate the classical dynamics of the coupled orbital and spin degrees of freedom, and calculate the ingredients of Gutzwiller's trace formula for the density of states. For a two-dimensional quantum dot with a spin-orbit interaction of Rashba type, we obtain satisfactory agreement with fully quantum-mechanical calculations. The mode-conversion problem, which arose in an earlier semiclassical approach, has hereby been overcome

Pletyukhov, Mikhail

Amann, Christian

Mehta, Mitaxi

Brack, Matthias

University of Regensburg Publication Server

arXiv:nlin.CD/0203015v2   23 Jul 2002Semiclassical theory of spin-orbit interactions using spin coherent states∗M. Pletyukhov1, Ch. Amann, M. Mehta2 and M. BrackInstitut fu¨r Theoretische Physik, Universita¨t Regensburg, D-93040 Regensburg, Germany1e-mail: mikhail.pletyukhov@physik.uni-regensburg.de2present address: Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad 211019, India(March 26, 2007)We formulate a semiclassical theory for systems with spin-orbit interactions. Using spin coherent states, we start fromthe path integral in an extended phase space, formulate theclassical dynamics of the coupled orbital and spin degrees offreedom, and calculate the ingredients of Gutzwiller’s traceformula for the density of states. For a two-dimensional quan-tum dot with a spin-orbit interaction of Rashba type, we ob-tain satisfactory agreement with fully quantum-mechanicalcalculations. The mode-conversion problem, which arose inan earlier semiclassical approach, has hereby been overcome.03.65.Sq,71.70.Ej,73.20.Dx,75.10.HkThe incorporation of spin degrees of freedom in thesemiclassical treatment of quantum systems represents aconsiderable challenge to the theorist. The periodic or-bit theory (POT), initiated by Gutzwiller through hissemiclassical trace formula [1], has been very successfulin promoting the research on quantum chaos [2] and inexplaining prominent quantum shell effects occurring infinite fermion systems in many domains of physics (see,e.g., [3]). The growing interest in “spintronics”, i.e., thespin-polarized transport and spin dynamics in variouselectronic materials [4], with the particular scope of de-veloping a spin transistor [5], make it desirable to formu-late also the semiclassical theory including spin.Littlejohn and Flynn have extended the POT for sys-tems with multi-component wavefunctions and used it forthe WKB quantization of spherical systems with spin-orbit interactions [6]. They did not give an explicit traceformula, however, the main obstacle being the problemof “mode conversion” which arises when the spin-orbitinteraction in phase space locally becomes zero. Friskand Guhr [7] have applied the method of [6] to deformedcavities and promoted the hypothesis that spin-flips oc-cur at the mode-conversion points through diabatic tran-sitions between the two adiabatic spin-polarized energysurfaces. Bolte and Keppeler [8] have derived a relativis-tic trace formula from the Dirac equation and studiednonrelativistic limits, thereby justifying some ad hoc as-sumptions made in [7]. Their approach works well in theweak-coupling limit where the spin does not affect the or-bital motion [9]. It may not be extensible [10], however,to strong spin-orbit interactions such as that observedin p-InAs or in InGaAs-InAlAs heterostructures [11], oralso that amongst the nucleons in atomic nuclei [12]. Forsuch systems, the approach presented below provides asemiclassical theory with a larger range of validity.In this letter, we report on a semiclassical approachthat allows for the explicit coupling between spin and or-bital dynamics. We use the spin coherent state methodand the path integral in an extended phase space to de-rive classical equations of motion which include the spindegrees of freedom. We then obtain the amplitudes inthe semiclassical trace formula for the density of stateswithout encountering the mode-conversion problem.The path integral for a system with spin in the SU(2)spin coherent state representation originally appeared ina paper by Klauder [13] as an integral on S2. Kuratsuji etal. [14] have represented it as an integral over paths in theextended complex plane C¯1. From the group theoreticalpoint of view, the SU(2) coherent state can be associatedwith the unitary irreducible representation of the SU(2)group [15]. To present its construction, we follow a recentpaper by Kochetov [16] and his notations. The coherentstate |z, S〉 for spin S is defined by|z;S〉 = (1 + |z|2)−S exp(zsˆ+) |S,−S〉 , (1)sˆ−|S,−S〉 = 0 , (2)where z ∈ C¯1 is a complex number, and sˆ± = sˆ1 ± isˆ2,and sˆ3 are the generators of the spin SU(2) algebra:[sˆ3, sˆ±] = ±sˆ± , [sˆ+, sˆ−] = 2sˆ3 . (3)S ∈ N/2 denotes the representation index.The irreducibility as well as the existence of the groupinvariant de Haar measure dµS ensures that the resolu-tion of unity holds in the spin coherent state basis:∫|z;S〉〈z;S| dµS(z) = I2S+1 ,dµS(z) =2S + 1pi (1 + |z|2)2 d2z , (4)which turns out to be the most important property ofspin coherent states that allows for the path integral con-struction, whereby the measure dµS takes account of thecurvature of the sphere S2. In what follows, we denote|z;S〉 simply by |z〉 and use z = u− iv with u, v ∈ R1.Let us now consider a quantum Hamiltonian with spin-orbit interactionĤ = Ĥ0(qˆ, pˆ) + κh¯ sˆ · Ĉ(qˆ, pˆ) , (5)where sˆ = (sˆ1, sˆ2, sˆ3). Hereby Ĉ = (Ĉ1, Ĉ2, Ĉ3) is a vec-tor function of the coordinate and momentum operatorsqˆ, pˆ, and the parameter κ regulates the strength of theinteraction. We make use of Smirnov’s idea [17] to write1the expression for the respective quantum propagator interms of a path integral in both the orbital variables q,pand the spin coherent state variables v, u. Imposing pe-riodic boundary conditions on the propagation and thusintegrating over closed paths, we arrive at the expressionfor the partition function (or trace of the propagator):Z(T ) =∫D[q]D[p]DµS [z] exp{iR[q,p, z;T ]/h¯}, (6)where R is Hamilton’s principal action function, calcu-lated along closed paths over a time interval TR[q,p, z;T ] =∮ T0[12(p·q˙− q·p˙) + 2Sh¯ (uv˙ − vu˙)(1+|z|2)−H(q,p, v, u)]dt , (7)and the path integration in (6) is taken over the 2(d+1)-dimensional extended phase space:D[q]D[p]DµS [z] = limN→∞N∏k=1d∏j=1dqj(tk)dpj(tk)2pih¯dµS(zk) .Hereby the time interval T is divided into N time stepstk = kT/N , and zk = z(tk). Note that the periodicboundary conditions enable the antisymmetrization ofthe orbital part of the symplectic 1-form in (7).The classical phase-space symbol H(q,p, v, u) of theHamiltonian (5), appearing in the integrand of (7), isH(q,p, v, u) = H0(q,p) + κh¯S n(v, u) ·C(q,p) , (8)where H0(q,p) and C(q,p) are the Wigner symbols ofthe operators Ĥ0 and Ĉ, and n = (n1, n2, n3) = 〈z|sˆ|z〉/Sis the unit vector of dimensionless classical spin compo-nents given in terms of u and v by n1 = 2u/(1 + |z|2),n2 = 2v/(1 + |z|2), and n3 = −(1− |z|2)/(1 + |z|2).The path integral (6) receives its largest contributionsfrom the neighborhood of the classical paths along whichthe principal functionR is stationary according to Hamil-ton’s variational principle δR = 0. The first variationhereby yields the following equations of motionv˙ =(1+|z|2)24h¯S∂H∂u= κ(1+|z|2)24∂n(v, u)∂u·C(q,p) ,u˙ = − (1+|z|2)24h¯S∂H∂v= −κ (1+|z|2)24∂n(v, u)∂v·C(q,p) ,q˙i =∂H∂pi=∂H0∂pi+ κh¯S n(v, u) · ∂C(q,p)∂pi,p˙i = −∂H∂qi= −∂H0∂qi− κh¯S n(v, u) · ∂C(q,p)∂qi, (9)whose solutions define the classical orbits in the extendedphase space. Note the appearance of h¯ multiplying S ev-erywhere in (9), which reflects the non-classical nature ofthe spin. The two equations for the spin variables v, u areequivalent to n˙ = −κn×C, with the constraint n2 = 1.For spin-boson coupling in the Jaynes-Cummings model,equations analogous to (9) have been derived in [18].Sugita [19] has recently given a re-derivation of Gutz-willer’s trace formula, starting from Z(T ) whose Fourier-Laplace transform yields the density of states:g(E) =1ih¯∫ ∞0eiET/h¯ Z(T ) dT . (10)Following the general arguments of Gutzwiller [1], onemay evaluate the integrations in (6) using the stationaryphase approximation, which becomes exact in the classi-cal limit R≫ h¯. The semiclassical approximation of thepartition function Z(T ) then turns into a sum over allclassical periodic orbits (po) with fixed period TZsc(T ) =∑poeiRpo/h¯∫D[η] exp {iR(2)po [η, T ]/h¯} , (11)where Rpo are the principal functions (7), evaluated nowalong the periodic orbits in the extended phase space.R(2)po [η, T ] are the second variationsR(2)po [η, T ] =∮ T0[12η · J η˙ −H(2)]dt , (12)where η is the following 2(d + 1)-dimensional extendedphase-space vector of small variationsη = (λ, ν,ρ, ξ) =(δq, δv2√h¯S(1 + |z|2) , δp, δu2√h¯S(1 + |z|2)),and J =(0 I−I 0)is the 2(d+ 1)-dimensional unit sym-plectic matrix. H(2) is the second variation of the classi-cal Hamiltonian, calculated along the periodic orbits:H(2) = 12∑i,j[λiλj∂2H∂qi∂qj+ 2λiρj∂2H∂qi∂pj+ ρiρj∂2H∂pi∂pj]+κ2(ν2 + ξ2) (−uC1 − v C2 + C3)+ κ√h¯S(1 + |z|2)2(ν∂n∂v+ ξ∂n∂u)·∑j(λj∂C∂qj+ ρj∂C∂pj). (13)After a stationary-phase evaluation of the Fourier integ-ral (10), one finally obtains Gutzwiller’s trace formula inthe standard form [1,19]. What is new here is that allits ingredients are obtained from the dynamics in the ex-tended phase space: the actions Spo(E) = Rpo+ETpo (atfixed energy E) and the periods Tpo = dSpo/dE of theperiodic orbits, the Maslov indices σpo for which Sugitahas given general formulae [19], and the monodromymatrix Mpo defined by η(Tpo) = Mpo η(0) in termsof the solutions of the linearized equations of motionη˙ = J ∂H(2)/∂η. We refer to a forthcoming extendedpaper [20] for the details of our calculations, where wealso show that in the weak-coupling limit we obtain thesame results as Bolte and Keppeler [8].2We shall presently illustrate our method by apply-ing it to a simple model Hamiltonian describing a two-dimensional electron gas (S = 1/2) in a semiconductorheterostructure, laterally confined to a quantum dot, in-cluding a spin-orbit interaction of Rashba type [21]Ĥ =(pˆ2x + pˆ2y)2m∗+ V (x, y) + κh¯ (σy pˆx − σxpˆy) . (14)Here m∗ is the effective mass of the electron, σi are thePauli matrices, and the lateral confining potential V (x, y)is approximated as a deformed harmonic oscillatorV (x, y) = m∗(ω2x x2 + ω2y y2)/2 . (15)We use ωx = 1.56ω0, ωy = 1.23ω0, and units such thath¯ = m∗ = ω0 = 1 and that E and κ become dimension-less. The spin-orbit coupling strength κ in (14) dependson the band structure [22]. E.g., for a InGaAs-InAlAsquantum dot with ∼ 100 confined electrons one wouldobtain a value of κ ∼ 0.25. We investigate in the fol-lowing a situation where κ is large enough for the spindynamics to affect the orbital motion. For the examplesbelow we have chosen κ = 0.67. The equations of motion(9) were solved numerically.For a large range of parameters with 0 < κ ∼< 0.75,we find the following periodic solutions of Eq. (9): 1)two pairs of adiabatic orbits A±x and A±y , librating alongthe x and y axes with fully polarized spin ny = ±1 andnx = ±1, respectively; 2) two pairs of diabatic orbits, D±x1and D±x2, oscillating around A±x with ny∼ 0; and 3) twopairs of diabatic orbits, D±y1 and D±y2, oscillating aroundA±y with nx ∼ 0. The superscripts (±) of the diabaticorbits denote their senses of rotation. The stabilities ofthese orbits will be discussed elsewhere [20]. For strongercouplings κ ∼> 0.75 and for energies E ∼< 8, new orbitsbifurcate from the A and D orbits. In these situations,the stability amplitudes will have to be regularized bysuitable uniform approximations [23].The diabatic orbits obtained in our approach reflectthe explicit coupling of the spin and orbital degrees ofmotion. Such orbits have not been discussed in earliersemiclassical methods. The adiabatic orbits with “frozenspin” correspond to those discussed in [6,7,10] where theycould not be used semiclassically, though, because of themode conversion occurring at their turning points. Werepeat that we do not encounter the mode-conversionproblem here, and that the stability amplitudes of allorbits can be readily calculated numerically.In Fig. 1 we present the (x, y) shapes of the orbits A+x,D+x1, and D+x2 (left panels), and the time dependence oftheir spin components nx, ny, and nz over one period(right panels), all evaluated at E = 60. (The orbits A±y ,D±y1, and D±y2 have analogous shapes concentrated alongthe y axis, and the behavior of their nx and ny compo-nents is reversed.) We see that along the diabatic orbitsD+x1 and D+x2, the spin rotates mainly near the (nx, nz)plane (with ny ∼ 0) in a non-uniform way. The diabatic-8 -4 0 4 8x-4-2024y0 1 2 3 4t-101nx-101ny-101nz-8 -4 0 4 8x-4-2024y0 1 2 3 4t-101nx-101ny-101nz-8 -4 0 4 8x-4-2024y0 1 2 3 4t-101nx-101ny-101nzA+xD+x1D+x2FIG. 1. Periodic orbits in the 2-dimensional harmonic oscil-lator with Rashba spin-orbit interaction (see text for param-eters). Left panels: orbits in the (x, y) plane. Right panels:Spin components nx, ny, and nz versus time. From top tobottom: Adiabatic orbit A+x along x axis with polarized spinin y direction, and diabatic orbits D+x1 and D+x2 oscillatingaround the x axis with spin rotating near the (nx, nz) plane.spin-flip hypothesis of [7] has thus been replaced here bythe more sophisticated spin dynamics obtained from thecoupled equations of motion (9).In Fig. 2 we show the oscillating part of the densityof states δg(E), obtained quantum-mechanically (solidline) using exact diagonalization of the Hamiltonian (14),and semiclassically (dashed line) using Gutzwiller’s traceformula. Since the periodic orbit sum generally doesnot converge in systems with mixed classical dynam-ics, we have convoluted it with a normalized Gaussian,exp{−(E/γ)2}/γ√pi. This brings the sum to convergence[24], whereby only the orbits with shortest periods con-tribute, and reflects the prominent gross-shell structureof the quantum spectrum. We have used γ = 0.6 whereit was sufficient to include the above twelve primitiveperiodic orbits. We observe a rather good agreement be-tween the semiclassical and quantum-mechanical results.The regular beat-like structure comes about through theinterference of the six types of classical orbits which allhave frequencies close to either ωx or ωy.310 20 30 40 50 60E-101g(E)FIG. 2. Oscillating part of the coarse-grained density of states (Gaussian averaging parameter γ = 0.6) versus energy for thesame system as in Fig. 1. Solid line: quantum-mechanical result, dashed line: semiclassical result using the 12 primitive orbits.In summary, we have presented a novel approach to include a non-adiabatic coupling of spin and orbital dynamicsinto the semiclassical theory of periodic orbits. Our approach overcomes some of the difficulties with earlier approaches,in particular the restriction to purely adiabatic spin motion and the problem of mode conversion. For the simple modelof a two-dimensional quantum dot with Rashba spin-orbit interaction, we obtain a satisfactory semiclassical descriptionof the coarse-grained density of states. To study its experimental implications, we are in the process of calculatingsemiclassically the conductance of this system in response to an external magnetic field [20].We should point out that there exists a subtle problem connected with the fact that the measure D[η] of thepath integral over the extended phase-space variations η in (11) has the proper normalization only in the large-spinlimit. For small spin, like S = 1/2, this calls for an appropriate renormalization scheme [25]. Although the differentrenormalizations needed for pure spatial motion [26] and pure spin motion [13] are known, a valid scheme for thecombined spin-orbit dynamics under investigation here is yet to be found. It might lead to extra phase corrections inthe semiclassical expressions, as discussed in [27], and thereby affect the numerical results to some extent.We acknowledge helpful and critical discussions with O. Zaitsev and encouraging comments by K. Richter.∗ Work supported by Deutsche Forschungsgemeinschaft.[1] M. C. Gutzwiller, J. Math. Phys. 12, 343 (1971).[2] M. C. Gutzwiller: Chaos in Classical and Quantum Mechanics (Springer, New York, 1990); K.-F. Berggren and S. A˚berg(Eds.): Quantum Chaos Y2K, Proceedings of Nobel Symposium 116, Physica Scripta Vol. T90 (2001).[3] M. Brack and R. K. Bhaduri: Semiclassical Physics, Frontiers in Physics Vol. 96 (Addison-Wesley, Reading, USA, 1997).[4] M. Oestreich, Nature 402, 735 (1999); S. Das Sarma, J. Fabian, X. Hu, and I.Zutic, IEEE Trans. Magn. 36, 2821 (2000).[5] S. Datta and B. Das, App. Phys. Lett. 56, 665 (1990); see also S. Das Sarma, Am. Sci. 89, 516 (2001).[6] R. G. Littlejohn and W. G. Flynn, Phys. Rev. A 44, 5239 (1991); Phys. Rev. A 45, 7697 (1992).[7] H. Frisk and T. Guhr, Ann. Phys. (N.Y.) 221, 229 (1993).[8] J. Bolte and S. Keppeler, Phys. Rev. Lett. 81, 1987 (1998); Ann. Phys. (N.Y.) 274, 125 (1999).[9] S. Keppeler and R. Winkler, Phys. Rev. Lett. 88, 046401 (2002).[10] Ch. Amann and M. Brack, J. Phys. A 35, 6009 (2002).[11] C.-M. Hu et al., Phys. Rev. B 60, 7736 (1999); T. Matsuyama et al., Phys. Rev. B 61, 15588 (2000).[12] see, e.g., A. Bohr and B. R. Mottelson: Nuclear Structure, Vol. I (W. A. Benjamin, New York, 1969).[13] J. R. Klauder, Phys. Rev. D 19, 2349 (1979).[14] H. Kuratsuji and T. Suzuki, J. Math. Phys. 21, 472 (1980); H. Kuratsuji and Y. Mizobuchi, J. Math. Phys. 22, 757 (1981).[15] A. M. Perelomov, Commun. Math. Phys. 26, 222 (1972).[16] E. Kochetov, J. Math. Phys. 36, 4667 (1995).[17] V. Smirnov, J. Phys. A 32, 1285 (1999).[18] A. Alscher and H. Grabert, Eur. Phys. J. D 14, 127 (2001).[19] A. Sugita, Ann. Phys. (N.Y.) 288, 277 (2001).[20] M. Pletyukhov, O. Zaitsev, and M. Brack, to be published.[21] Y. Bychkov and E. Rashba, J. Phys. C 17, 6039 (1984), and earlier references quoted therein.[22] see, e.g., T. Darnhofer, M Suhrke, and U. Ro¨ssler, Europhys. Lett. 35 591 (1996).[23] M. Sieber and H. Schomerus, J. Phys. A 31, 165 (1998), and earlier references.[24] M. Sieber and F. Steiner, Phys. Rev. Lett. 67, 1941 (1991).[25] M. Miettinen, J. Math. Phys. 37, 3141 (1996).[26] H.G. Solari, J. Math. Phys. 28, 1097 (1987).[27] M. Stone, K.-S. Park, and A. Garg, J. Math. Phys. 41, 8025 (2000).4

Semiclassical theory of spin-orbit interactions using spin coherent states

Semiclassical theory of spin-orbit interactions using spin coherent
  states

Semiclassical theory of spin-orbit interactions using spin coherent states

Abstract

Similar works

Full text

Available Versions

Crossref

Crossref

University of Regensburg Publication Server