Spin–orbit interaction and energy states in semiconductor
quantum dots
O. Voskoboynikov
a;∗, C.P. Lee
a, O. Tretyak
baDepartment of Electronics Engineering and Institute of Electronics, National Chiao Tung University, 1001 Ta Hsueh Rd.,
Hsinchu 30010, Taiwan, ROC
bKiev Taras Shevchenko University, 64 Volodymirska St., 252030 Kiev, Ukraine
Abstract
We present a theoretical study of the impact of the spin–orbit interaction on electron energy states in small cylindrical quantum dots. In our calculations, we use the e,ective one electronic band Hamiltonian and the spin dependent boundary conditions. It has been found that the spin–orbit interaction can modify the energy spectrum of narrow gap semiconductor quantum dots. The modi.cation consists of the energy state spin splitting that strongly depends on the dot size. The spin splitting demonstrates a non-monotonic dependence on the dot size and can provide a situation when only the lower spin split states with angular quantum number |l| = 1 are bound in the dot. c 2001 Elsevier Science B.V. All rights reserved. PACS: 71.70.Ej; 85.30.Vw
Keywords: Spin–orbit interaction; Quantum dots; Energy states
The electron spin plays an important role in the de-sign of quantum dot electron energy levels and can signi.cantly alter the properties of the electron energy states [1–3]. A new branch of semiconductor electron-ics (so called spintronelectron-ics [4–7]) has produced much interest in the spin-dependent energy structure of semi-conductor quantum dots. In semisemi-conductor spintronics devices, the carriers generation, recombination, and ∗Corresponding author. Tel.: +886-3-5612121, ext. 54174; fax: +886-3-5724361.
E-mail address: [email protected] (O. Voskoboynikov).
transport will be controlled by electron spin polariza-tion as well as the electron charge. A study of the spin dependent electron con.nement in semiconduc-tor quantum dots can be an essential part of semicon-ductor spintronics development.
There is an additional reason to investigate the spin dependence of the electron energy state in III–V semi-conductor quantum dots and nanocrystals. When the potential for electrons in III–V semiconductors is in-version asymmetric, the spin–orbit (SO) interaction removes the spin degeneracy and considerably a,ects the electron energy states [8,9]. It has been found that
1386-9477/01/$ - see front matter c 2001 Elsevier Science B.V. All rights reserved.
the Rashba SO interaction [9] can critically e,ect the energy states and electronic properties of III–V semi-conductor quantum structures [10–22]. The Rashba spin–orbit interaction term is used successfully to in-terpret experimental results in various quantum well and quantum wire structures [10,11,16–22]. In this paper, we investigate the e,ect of the SO interac-tion on narrow gap semiconductor dependent quan-tum dot energy states. We use the e,ective one band spin-dependent Hamiltonian and the spin-dependent boundary conditions [12,13]. It will be demonstrated that SO interaction can change the positions of elec-tron energy states of the quantum dots.
We consider a cylindrical quantum dot with a hard-wall con.nement potential that is induced by the discontinuity of the conduction band edge of the system. This model is commonly used in calculations of the electron energy states of quantum dots embed-ded into a di,erent material matrix [23]. The model allows us to solve the three-dimensional SchrEodinger equation with a small number of additional approxi-mations. It must be emphasized that basic parameters of the materials for semiconductor quantum dot struc-tures (such as energetic gaps, e,ective masses, band o,sets etc.) are a,ected by di,erent factors (strain, for instance) and are poorly known. In modern lit-erature of this topic, values of the parameters vary within a wide range [24–26]. In our calculations we adjust the material parameters in accordance to the literature data.
In three-dimensional semiconductor quantum struc-tures the approximate one electron band e,ective Hamiltonian is given in the form
ˆ H = ˆH0+ ˆVso; (1) where ˆ H0= −˝ 2 2∇r 1 m(E; r)∇r+ V (r)
is the Hamiltonian without SO interaction, ∇r stands
for the spatial gradient, m(E; r) is the energy and po-sition dependent electron e,ective mass
1 m(E; r) = P2 ˝2 2 E + Eg(r) − V (r) + E + E 1 g(r) − V (r) + G(r) ;
V (r) is the con.nement potential, Eg(r) and G(r)
stand, respectively, for the position dependent band gap and the spin–orbit splitting in the valence band, P is the momentum matrix element. The SO interaction Vso(r) for conducting band electrons is described by
[12,13,27] ˆVso(r) = i∇r (E; r) · [×∇r]; (2) where (E; r) = P22 1 E + Eg(r) − V (r) −E + E 1 g(r) − V (r) + G(r) ; and = {x; y; z}
is the vector of the Pauli matrices.
For systems with a sharp discontinuity of the con-duction band edge between the quantum dot (material 1) and the crystal matrix (material 2), the hard-wall con.nement potential can be presented as
V (r) =
0; r ∈ 1 V0; r ∈ 2:
From integration of the SchrEodinger equation with Hamiltonian (1) along the direction perpendicular to the interface (rn) we obtain the spin dependent Ben
Daniel–Duke boundary conditions for the electron wave function (r) 1(rs) = 2(rs); ˝2 2m(E; r)∇r− i (E; r)[×∇r] n ×(rs) = const:; (3)
where rsdenotes position of the system interface.
When the quantum dot has a disk shape of radius 0
and thickness z0we treat the problem with cylindrical
coordinates (; ; z). The origin of the system lies in the center of the disk and the z-axis is chosen along the rotation axis. Because of the system cylindrical symmetry, the wave function can be represented as
where l = 0; ±1; ±2; : : : is the orbital quantum num-ber. To derive the equation for (; z) we will use the adiabatic approximation [23,28], which represents an approximate solution in the form
(; z) ≈ R()Z(z):
First we consider the ground state of z-direction electron motion, and solve the one-dimensional quan-tum well problem. The wave function of this ground state has the form
Z(z) = A cos(kz); |z| ¡ z0=2; B exp(−|z|); |z| ¿ z0=2; (5) where k(E; Ez) = 2m1(E)Ez=˝; =2m2(E)(V0− Ez)=˝;
mi(E) is the energy dependent electron e,ective mass
inside (i = 1) and outside (i = 2) the dot, E = E+ Ez
is the total electron energy that consists of the and z direction motion e,ective energies. From the Ben Daniel–Duke boundary conditions [29] in z-direction we can obtain a transcendental equation.
tan[k(E; Ez)z0=2] =mm1(E)(E; Ez)
2(E)k(E; Ez): (6)
Eq. (6) gives the Ez(E) dependence in an implicit
form. With the wave function (5) (after proper nor-malization), we substitute in the three-dimensional SchrEodinger equation and integrate out the z coordi-nate by taking the average
ˆ H=
dz Z∗(z) ˆHZ(z):
When we neglect the kinetic energy contribution from the z-dependent part for ¿ 0 [28] the
quasi one-dimensional SchrEodinger equation in the -direction is given by the form
−2 ˜m˝2 1 d2 d2 + d d − l2 2 R1() = ER1(); ¡ 0; −2m˝2 2 d2 d2 + d d − l2 2 R2() = (V0− E)R0(); ¿ 0; (7)
with the spin-dependent boundary conditions R1(0) = R2(0); 1 ˜m1 dR1 d 0 − m1 2 dR2 d 0 +2( ˜ 1− ˜ 2) ˝20 R1(0) = 0: (8) In Eqs. (7) and (8) 1 ˜m1(E; Ez) = 1 m1(E) |z|¡z0=2 dz|Z(z)|2 +m1 2(E) |z| ¿ z0=2 dz|Z(z)|2; ˜ 1(E; Ez) = 1(E) |z|¡z0=2 dz|Z(z)|2; ˜ 2(E; Ez) = 2(E) |z| ¿ z0=2 dz|Z(z)|2;
and refers to the spin polarization along the z-axis. Eq. (7) with the boundary conditions (8) are used to obtain the solution of the problem. A formal solution of (7) is well known as follows:
R1() ∼ J|l|(p);
R2() ∼ K|l|(g);
where Jnand Knare, respectively, the Bessel function
and modi.ed Bessel function, p(E; Ez) = 2 ˜m1(E; Ez)E=˝; g(E; Ez) = 2 ˜m1(E; Ez)(V0− E)=˝:
On using the boundary conditions (7), the equation that gives the energy states is found to be
p ˜m1 |l| p0J|l|(p0) − J|l|+1(p0) K|l|(g0) − g m2 |l| g0K|l|(g0) − K|l|+1(g0) J|l|(p0) + 2l( ˜ 1− ˜ 2) ˝20 J|l|(p0)K|l|(g0) = 0: (9)
We need the equation above, in addition to Eq. (6), to solve the problem. The total energy E = Ez+ Eis a
complicated function of the dot parameters, the elec-tron angular momentum, and spin. The energy states system consists of discrete levels numerated by a set of numbers {n; l; }, where n denotes nth solution of (9) with .xed l and . States with the same n and parallel (antiparallel) orbital momentum and spin (the same sign for l and ) remain two-fold degenerate (the Kramers degeneracy). But levels with the same n and antiparallel orbital momentum and spin are sepa-rated from those with parallel orbital momentum and spin. The spin–orbit interaction removes the four-fold degeneracy of the energy states in the cylindrical quantum dots.
The energy of the con.ned electron states is found by numerically solving of the system of Eqs. (6) and (9). For cylindrical quantum dots, we use conven-tional notation for electron energy states: nL, where
n is the main quantum number of the -plane motion, L = S; P; D; : : : ; denotes the absolute value of l, and refers to the electron spin direction in respect to the angular momentum direction (+1 if the directions are parallel and −1 otherwise). For all calculations we choose the lowest energy state in the z-direction.
In our calculations of the energy states for InAs quantum dots in a GaAs matrix we tuned the band parameters to take into account e,ects of strain in small InAs quantum dots. In accordance to results of Ref. [26], we use band structure parameters of InAs: energy gap is E1g= 0:52 eV, spin–orbit
split-ting is #1= 0:48 eV, the value of the nonparabolicity
parameter is E1p= 3m0P12=˝2= 22:2 eV; m0 refers to
the free electron e,ective mass. For GaAs we choose: E2g= 1:52 eV; #2= 0:34 eV, E2p= 24:2 eV. The band
o,set parameter is taken as V0= 0:55 eV. It should
be noted, that the larger E1g makes the spin–orbit
ef-fect lower but at the same time more realistic for the strained quantum dots.
The spin splitting e,ect is zero for the lowest state energy 1S±1 follows from Eq. (9). The size
depen-dence of the 1P level splitting GE1P= E1P+1− E1P−1is
shown in Fig. 1. The theory demonstrates noticeable spin splitting for relatively small quantum dots. The splitting is strongly dependent on the dot size and de-creases rapidly when the size inde-creases. But for dots with very small height the spin splitting is reduced. This is a result of electron wave function penetration into the barrier along the z direction. The averaged ef-fective mass ˜m1 and spin–orbit interaction parameter
Fig. 1. Spin splitting of |l| = 1 levels for InAs quantum dots of di,erent sizes.
˜ 1are closer to those of the GaAs matrix. This makes the di,erence between ˜ 1 and ˜ 2 smaller. When z0
increases the di,erence also increases and becomes z-independent after z0about 50 MA. This is a reason for
the GE1P dependence on 0 with .xed z0 to
demon-strate the nonmonotonic behavior [28].
An interesting consequence of the spin–orbit inter-action is the possibility to .nd quantum dot param-eters, when one set of the spin split states is located below the dot top energy V0 and another one is
lo-cated above V0. Basically, we can obtain a dot of
a critical (maximal) size {zc0; c0} when only the
1P−1 electron states are bound. For InAs quantum
dots we have calculated the following critical sizes:
{15 MA; 46 MA}; {20 MA; 39 MA}; {30 MA; 36 MA}; {40 MA; 32 MA}, {50 MA; 30 MA}. When we reach the critical size and
continue to reduce {z0; 0}, the 1P+1 levels become
unbound very soon when {z0; 0} are about few
angstroms less than {zc0; c0}. Thus, the range of
InAs dot sizes when we have only the 1P−1 bound
states is very narrow. In an experimental situation, it can be a random number of quantum dots with only the 1P−1 bound states. Certainly, the spin splitting
impact becomes stronger for states with |l| ¿ 1. It should be noted that the critical sizes mentioned above are found to be out of the adiabatic approximation validity region [28]. A detailed study of the critical dot sizes could only be performed with a more powerful numerical technique like the .nite element method. Another possibility is to consider InSb quantum dots with a stronger SO interaction.
In short conclusion, we have studied theoretically the e,ect of the spin–orbit interaction on the energy spectrum of quantum dots. The calculation is based on a simple e,ective one electronic band Hamiltonian and spin-dependent boundary conditions. Our results show that the spin–orbit interaction can noticeably modify the energy states of narrow gap semiconduc-tor InAs cylindrical quantum dots. The modi.cation is strongly dependent on the dot sizes. The spin split-ting in cylindrical quantum dots demonstrates a non-monotonic dependence on the dot size. The splitting can provide a situation when only the lower spin split electron energy states are bound in the dot.
Acknowledgements
This work was supported by the National Sci-ence Council under contract number NSC89-2215-E009-013.
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