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Spin-Orbit Coupling in Condensed Matter Physics

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Spin-Orbit Coupling in Condensed Matter Physics THE PAPER "PROPERTIES OF A 2D ELECTRON-GAS WITH LIFTED SPECTRAL DEGENERACY."(1984) SPIN-ORBIT COUPLING IN CONDENSED MATTER PHYSICS. Emmanuel I. Rashba1 1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA The subject of our paper [1] with Yuri A. Bychkov actively involved in physics of 2D systems for a couple of is two-fold spin splitting of the band spectrum of years. two-dimensional charge carriers, electrons and holes, in Now I turn to the rst factor that requires a more asymmetric heterostructures, and manifestation of it detailed explanation. Electrons possess charge and spin, in two independent experiments. Heterostructures are and existence of spin follows from the relativistic Dirac articial crystals carefully grown as systems of layers of theory. As a result, SO coupling is also a relativistic a few dierent semiconductors and engineered in such eect. Formally, it is weak in the ne structure constant 2 a way to achieve a desirable electron energy spectrum. αf = e =¯hc ≈ 1=137. But in crystals it is multiplied Such structures were developed for semiconductor by the atomic number Z, and in the middle of the optoelectronics, in particular, semiconductor lasers periodic system Zαf is not weak. Therefore, SO coupling beginning from 1960s. Electrons in them behave as nearly essentially modies the energy spectrum. In the late two-dimensional (2D). In early 1980s, two fundamentally 1950s, we embarked with my student Valentin Sheka on new physical phenomena were discovered in 2D systems, nding energy spectrum of uniaxial non-centrosymmetric the Integer and Fractional Quantum Hall Eects (IQHE crystals of the CdS type by group-theoretical methods, and FQHE). These discoveries deeply inuenced all and arrived at the relativistic part of the Hamiltonian following developments in the low-dimensional physics ^ of semiconductors, transforming it from an applied to a Hso = α(σ × k) · ν: (1) fundamental eld of research, and promoted qualitatively Here α is a SO coupling constant, σ is a vector new theoretical concepts. IQHE demonstrated the of Pauli matrices, k = p=¯h is the electron quasi- rst topological phase transition, and FQHE indicated momentum, and ν is a unit vector directed along that electron-electron interaction can produce exotic the symmetry axis. When added to a nonrelativistic quasiparticles that carry fractional charges (fractions of Hamiltonian H = ¯h2k2=2m, H^ splits the spin- the fundamental electron charge e) and obey a fractional 0 so degenerate spectrum into two branches. The lower branch statistics (that diers from the textbook Fermi and Bose reaches its minimum at a circle in the k -space with statistics). 2 a radius kso = mα=¯h rather than in a single point, I attribute the success of our paper to four factors. as distinct from all energy spectra known by then. The First, it stood at a rm theoretical ground. Second, it branches intersect, at the energy ϵ = 0, in a conical was timely. Third, it included interesting physics which point known as the Dirac point, and isoenergy surfaces had impact on the future developments. And, last but are toroidal at ϵ < 0 and spherical at ϵ > 0. We were not the least, it was published in JETP Letters, with a excited by the result but unlucky with the paper. A follow-up paper [2] in J. Phys. C: Solid State Phys., two short version of it was rejected by JETP as being of journals that were easily available and widely read then. no general interest, and a full version was published I begin with the second factor. Our paper was a fast by Solid State Physics (in 1959) in a "Collection of response to two papers [3, 4] that were published one next Papers"issue that has never been translated into English. to the other in the same issue of the Physical Review Meantime, we found that the Hamiltonian H^so has a Letters and reported manifestation of spin-orbit (SO) dramatic eect on spin resonance [5] As distinct from coupling in spin and cyclotron resonances in two dierent the textbook concept, spin resonance of free electrons 2D systems. It is worth mentioning that both papers is predominantly excited by the electrical, rather than came from prominent experimental groups, and Klaus by magnetic, component of electromagnetic eld, and von Klitzing and Horst Stormer who would become Nobel under proper conditions this statement is also true for Laureates for the discovery of the IQHE and FQHE, localized electrons. This prediction was initially met respectively, were among the authors. In our paper, we with scepticism, and my talk at the Landau Seminar proposed an unied approach for estimating SO coupling was interrupted by Vitaly L. Ginzburg who exclaimed: constants from both sets the experimental data. Our "Because the nal result is denitely wrong, there should fast response was only possible because we were already be some error there". Landau' immediate response: 2 "Vitya, don't you see what kind of Hamiltonian he crystals with strong SO coupling, mostly from high-Z has?"settled the problem. Currently, this resonance is compounds. Among them is BiTeI with α = 3:85 eVA. I known as Electric Dipole Spin Resonance (EDSR); it admire experiments of the Tokyo group which managed, was observed and investigated in numerous systems [6]. by moving the Fermi level µ across the Dirac point, Therefore, using the Hamiltonian H^so for describing to observe a topological transition from toroidal to two SO coupling in asymmetrical heterostructures was well spherical Fermi surfaces, and the related paramagnet- justied. to-diamagnet phase transition [17]; we predicted it back Illuminating the third factor is most challenging in 1960 by analytical calculations [18]. A closely related because of the multitude of directions in which spin subject are low-dimensional topological superconductors physics developed during two last decades. The concept that host Majorana fermions: particles that constitute of spintronics is focused on applications and is based their own antiparticles and are prospective candidates on the perception that the Si based electronics is for topologically protected quantum computation [19]. approaching its fundamental limits. Spintronics relies on The simplest construction includes a low-dimensional adding spin degree of freedom to charge based devices semiconductor with a Hamiltonian H = H0 + H^so that is and electrical control of electron spin [7] The spin a subject to a moderate magnetic eld and turned into transistor proposed by Datta and Das is its paradigmatic superconductor by the proximity eect [20]. It eliminates device [8]. This interference device is based on spin the Dirac point, creates a gap in the internal spectrum precession in a SO coupled conductor connecting two branch, and renders the system become spinless"when ferromagnetic electrodes, and electric current across it µ is inside the gap. While intensive search for Majoranas is controlled by the gate-potential-dependent parameter is underway, a number of currently available fast qubits α. As a result of huge eorts, a proof of the principle InAs (elements of prospective quantum computers) are driven device has been recently demonstrated [9]. Modications by EDSR. And seemingly unrelated but conceptually of it, as well as spin injection devices based on SO closely connected systems are ultracold atoms in atomic and Aharonov- Bohm controlled interference, have been traps with laser-produced articial SO coupling [21]. also proposed. Creating perfect crystal surfaces covered High density of states near the spectrum bottom by submonolayers of high-Z elements allowed producing of the Hamiltonian H results in enhanced attraction 2D electron gases with giant α's that were directly between particles, and experiments with atomic traps measured by the Spin Angular Resolved Photo-Emission provide unique perspective onto the eect of strong SO Spectroscopy (SARPES). In Pt nanowires grown on coupling on the competition between the BCS (Bardeen, Si substrate α is of atomic scale, α = 1:36 eVA[10]. Cooper, Schrieer) and BEC (Bose-Einstein condensate) Such systems are considered as prospective candidates mechanisms of the transition into superconducting state. for creating spin devices with nanoscale interference Finally, I mention emergence of SO metamaterials for spin-controlled photonics [22]. They bridge spin physics lengths `so ≈ 1=kso. Meantime, there was a remarkable progress in the magnetic memory technology based on of electrons and light. the discovery of giant magnetoresistance celebrated by In conclusion, physics of spin-orbit coupling penetrated the Nobel Prizes for Albert Fert and Peter Grunberg. As into numerous branches of condensed matter physics a result, fast transport of magnetic domains by electric and unied them into an extensive, interconnected, and current rather than by mechanical motion emerged as an exciting eld including both fundamental problems and urgent problem, and the torques exerted onto domain practical applications. I guess that our paper became successful because it was one of the rst, and timely, steps walls by SO coupling of the H^so type and the spin- Hall eect predicted by D'yakonov and Perel' [11] are of this journey. considered as relevant mechanisms [12]. Recently, 2D electron gases with strong SO coupling were discovered in a new class of systems, conducting LaAlO3/SrTiO3 interfaces separating two insulating ceramics [13]. These [1] Yu. A. Bychkov and E. I. Rashba, Sov. Phys. - JETP interfaces demonstrate coexistence of superconductivity Lett. 39, 78 (1984). and magnetism, and their exotic properties are ascribed [2] Yu. A. Bychkov and E. I. Rashba, J. Phys. C: Solid State to SO coupling; they are considered as candidates for new Phys. 17, 6039 (1984). devices. One more eld emerged due to the discovery [3] D. Stein, K. v. Klitzing, and G. Weimann, Phys. Rev. of non-centrosymmetric superconductors. In them, SO Lett. 51, 130 (1983). coupling admixtures a triplet component to singlet [4] H.
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