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Section 11. Physics
Section 11. Physics
Rasulov Voxob Rustamovich, Ph D. Senior Lecturer of Fergana State University E-mail: [email protected] Rasulov Rustam Yavkachovich, Professor of Fergana State University Sultonov R. R.,
Researcher of Kokand State Pedagogical Institute
Ro'ziboyev V. U., Researcher of Fergana State University
SPIN-DEPENDENT DIMENSIONAL QUANTIZATION IN A SEMICONDUCTOR HETEROSTRUCTURE
Abstract. A theory ofspin-dependent dimensional quantization is contributed in a semiconductor heterostructure, where spin-orbit splitting is taken into account by introducing the Dresselhaus term into the effective Hamiltonian.
It is shown that the energy spectrum of electrons in the direction of size quantization consists of a set of non-equidistant discrete levels, depending on the wave vector directed along the interface of the heterostructure.
Keywords: energy spectrum, quantum well, wave function, dimensional quantization.
In connection with the growing interest in spin-depen- wave vector of electrons is perpendicular (parallel) to the axis dent phenomena, the kinetics of spin-polarized electrons in Oz along which tunneling can occur. In the absence of exter-semiconductors of various symmetries attracts great interest nal influence, the state of the electrons can be described by in order to create spin detectors. For the first time, the pos- the Hamiltonian:
sibility of creating a spin filter based on tunneling through an asymmetric barrier in semiconductors is indicated in [1; 2], where the influence of spin on motion is taken into account by introducing the Rashba term [3] into the effective Hamiltonian of current carriers. The ability to obtain carrier polarization in a heterostructure, where asymmetry is created by doping, was analyzed in [4]. A theoretical model of a spin injector based on a symmetric barrier, where the Dresselhaus spin-orbit interaction was taken into account, was built in [5].
In this work, the size quantization of the energy spectrum of spin-polarized electrons in the quantum well of a piezoelectric semiconductor is theoretically considered. We assume that the structure is grown so that the normal to the interfaces of the structure is directed along one of the main crystal-lographic axes. For example, Al GaxSb has a lattice of zinc blende type and it does not have a center of symmetry. Therefore, in the effective Hamiltonian of electrons, the terms are cubic in the wave vector. For definiteness, we assume that the axes Ox, Oy and Oz are directed along the crystallographic axes [100], [010] and [001], respectively. Let k±(kz) is the
H =-
h2
2mf dx2 2mf dx
h k\
+ V, (x) + H(
(D)
(1)
Here m( is the effective electron mass, V, (x) is the potential, which depend on the layer number I of the structure, where the electron energy is measured from the bottom of the conduction band in the emitter [6].
In piezoelectric semiconductors of the type GaAs of spinorbital interaction of an electron with a lattice field, a cubic in the wave vector term of the Dresselhaus term HD appears [7]:
HD) =Y X °X(kl -K)> (2)
where y, is the Dresselhaus constant in the layer I of the heterostructure, oa (a = x, y, z) is the Pauli matrix. For example, for Al03Ga07Sb, it takes the values 76 meV • nm-1 [8], for GaSb it takes 187 meV • nm-1 [5]. We will research the tunneling of electrons with kinetic energies (in the emitter and collector) substantially smaller than the depth of the well and the height of the barrier, which allows us to leave only terms linear in the wave vectors kx, ky and write the Dresselhaus term in (1) in the form:
SPIN-DEPENDENT DIMENSIONAL QUANTIZATION IN A SEMICONDUCTOR HETEROSTRUCTURE
Hr =7, ()),HT =Y () H?D) =y(*)) (3)
Let an electric field be applied to the heterostructure. We will consider it sufficiently small, so that the distortion of the heterostructure caused by it is much less than the height of the barriers and the depth of the well. This allows us to consider it as a small perturbation and neglect the Rashba effect [7; 8], as will be shown later. So, the Hamilto-nian for electrons in a layer with a number l, with which you can analyze dimensional quantization and tunneling, we write in the form:
»2 (Z )2
H (z)=-
h2 d2 2 m* dxZ
2m,
-v, (x
;V>
-HfD\
(4)
where Z = 1,2,3 correspond to the interfaces yz, xz and xy
respectively, (jfcf )2 = % + k2, (fc® )2 = ^ + k], ((?> )2 = k\ + k2y, Xi — x, X] — y, X3 — z .
The wave function of an electron with a wave vector kf>, following [7], will be sought in the form:
wie±)=ziVS±)(x (5)
and the spinor x(±) is chosen so that it diagonalizes the Dres-selhaus Hamiltonian H(D1) i
X;^ =
in (3): 1 i 1
(6)
where ty is the polar angle of the vector components ïf) describing the electron motion in the interface plane of the structure.
Then the Dresselhaus Hamiltonian takes the form:
H(ZD) = — k(V -
(7)
and the Schrodinger equation for the components of the wave
(xz) or
function u (z ) has the form H(
h2 d2
(z)u(z±)(
2mZ dx2z
+Y x )
Here = mP
(
1 +
2yl mi*
(xz ) = E (xC ) = E;
= E(z±)u (z±)
(z±)
= Ez)u(Z±)(xf ),
(8)
and it depends kp.
Next, we solve equations (8), where we assume that the potential in each layer is constant, i.e. Y* (z ) — Y*. Then the solution of equation (8) well be sought in the form
Uz)(xz ) = x, (9)
or
uf±Hxz ) = C{*z±) cos(kP>xz ) + sin(kPxz ), (10)
where kP =
^mf(p) , values A(i±),B( ±
or
c(z±\ d\z±) are determined from the boundary conditions of the problem. For example, from the condition u( ±^(xf = 0) = 0 we have cf^ = 0, so
uf±)(x( ) = sin (4%), from which when
uf+)(x^ = a) = 0
we
, a ( EP= Y, +
= 1,2,...j, from where
n2 h2
2m* a '
)2
1+
2^ mPk(P ^
(11) have
(12)
Thus, the energy spectrum of electrons in the direction of size quantization consists of a set of non-equidistant discrete levels, depending on the wave vector kp).
The work is partially financed by the grant OT-F2-66.
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