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BOUND STATES IN ELECTRON SYSTEMS INDUCED BY THE SPIN-ORBIT INTERACTION Magarill L.I. in collaboration with Chaplic A.V. A shallow and narrow potential well = 3D: No bound states 2D, axially symmetric well: one bound s-state, | E |~ exp( 1 / ) 1D, symmetric potential: 2 one bound state, | E |~ U 0 Valid with SOI neglected Hamiltonian of 2D electrons with SOI in the Bychkov-Rashba form interacting with an axially symmetric potential well Dispersion relation for 2D with SOI 0.75 0.5 0.25 1 0 + 0.6 - 0.4 0.2 0 -1 0,0 me 2 2 p -0.5 0 0.5 p0 1 p0 loop of extrema p0=me The lower branch of the dispersion law of 2D electrons has a form and corresponds to a 1D particle at least in the sense of density of states. Formally the particle has anisotropic effective mass: radial component is me , azimuthal component = (the dispersion law is independent of the angle in the p-plane). p-representation of the Schrodinger equation: U(p) dreiprU (r ) 2 drrU (r ) J 0 ( pr ) Cylindrical harmonics of the spinor wave functions: J 0 (| p p ' |) J k 0 ( pr ) J 0 ( p ' r ) cos(k ) For m-th harmonic: s-state 2 ln(2me|E|R ) 2 2 E exp(4 / ) 2 me R E 2 2 32 me 2 0 meR=0.1 -10 -20 meR=0.01 =0 2meU0R2 -30 -40 0 2 4 6 8 10 12 14 16 18 1/ 20 p-state 2meER 2 No bound states for zero SOI at < c =x12; J0(x1)=0 0,0 meR = 0.1 -2,0x10-7 2meER 2 lower p-state upper p-state 0,0 -4,0x10 -7 -6,0x10 -7 -0,2 -0,4 -8,0x10 j=1/2 -0,6 j=3/2 -0,8 c -7 -1,0 4,0 4,5 5,0 5,5 6,0 6,5 7,0 7,5 8,0 -1,0x10-6 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 Effect of the magnetic field ground state (s-level) Direct Zeemann contribution neglected (g=0), only SOI induced effect ^ Splitting 3 me eB 64 | E | me c 2 Liquid He-4 Roton dispersion relation: E ( p p ) / 2M 2 2 0 2 l fold degeneracy 2D electrons with B-R SOI in 1D short-range potential y U (x) x 0.5U00 =0.5U =U =U00 There exists pc and at |py| > pc «+»-state becomes delocalized. py =0.5 meU0 py =1.5 meU0 Narrow quantum well and 3D electrons z U=-U0d(z) Dresselhaus SOI VSO=g[(sypysxpx)pz2 sz (px2-py2) pzpx py(sxpy-sypx)] Localization or delocalization of an electron in z-direction depends on the orientation of longitudinal momentum p||: [110] - lower subband: localization for all p|| upper subband: termination point p (m U 2 / 2g )1/3 ||c e 0 [100] – both subbands for all values of p|| relate to the localized states Small longitudinal momenta Two independent equations for two components of the wave function: d 2 dz 2 2m ( E U ( z )) 0 1 1 (1 d ) m me ( 1) d 2meg p 1 Asymmetric well 1 U2 U1 1 GaxAl1-xAs/A3B5/GayAl1-yAs Two identical wells a U0 U0 meU0a (1 d ) < 1 < meU0a (1 d ) For 1 two localized states, for 1 only one. Conclusion We have shown that 2D electrons interact with impurities in a very special way if one takes into account SO coupling: because of the loop of extrema, the system behaves as a 1D one for negative energies close to the bottom of continuum. This results in the infinite number of bound states even for a short-range potential. 1D potential well in 2DEG and 3DEG for proper values of characteristic parameters form bound states for only one spin state of electrons. The ground state in a short-range 2D potential well possesses the anomalously large effective g-factor.