CC BY
7
http://dx.doi.org/10.35596/1729-7648-2021-19-8-20-25
Original paper
UDC 530.145:538.915:538.958
ENERGY LEVELS OF AN ELECTRON IN A CIRCULAR QUANTUM DOT IN THE PRESENCE OF SPIN-ORBIT INTERACTIONS
ALEXANDR V. BARAN, VLADIMIR V. KUDRYASHOV
B.I. Stepanov Institute of Physics of the National Academy of Science of Belarus (Minsk, Republic of Belarus)
Submitted 17 November 2021
© Belarusian State University of Informatics and Radioelectronics, 2021
Abstract. The two-dimensional circular quantum dot in a double semiconductor heterostructure is simulated by a new axially symmetric smooth potential of finite depth and width. The presence of additional potential parameters in this model allows us to describe the individual properties of different kinds of quantum dots. The influence of the Rashba and Dresselhaus spin-orbit interactions on electron states in quantum dot is investigated. The total Hamiltonian of the problem is written as a sum of unperturbed part and perturbation. First, the exact solution of the unperturbed Schrodinger equation was constructed. Each energy level of the unperturbed Hamiltonian was doubly degenerated. Further, the analytical approximate expression for energy splitting was obtained within the framework of perturbation theory, when the strengths of two spin-orbit interactions are close. The numerical results show the dependence of energy levels on potential parameters.
Keywords: circular quantum dots, confinement potential, spin-orbit interactions, energy levels.
Conflict of interests. The authors declare no conflict of interests.
For citation. Baran A.V., Kudryashov V.V. Energy levels of an electron in a circular quantum dot in the presence of spin-orbit interactions. Doklady BGUIR. 2021; 19(8): 20-25.
Introduction
The motion of an electron in an inner layer of a double semiconductor heterostructure is usually treated as two-dimensional in the (x,y) plane. In addition, the planar motion is also restricted if an electron is placed in a quantum dot localized in the middle layer of heterostructure. The Rashba VR [1] and Dresselhaus VD [2] interactions are presented by the formulas
vr = a r Kpypx)/K Vd = a D (axpx-vypy)/k (1)
where ox and oy are the standard Pauli spin-matrices. The strengths of these interactions depend
on the materials used. The contributions of two spin-orbit interactions can be measured within various experimental methods [3, 4]. In the general case the whole spin-orbit interaction has the form VR + VD . At the same time, considerable attention is paid to the special case [3, 5, 6], when the spin-orbit interactions of Rashba and Dresselhaus have equal strength aR = aD. It can be experimentally
achieved due to the fact that the Rashba interaction strength can be controlled by an external electric field, and the Dresselhaus interaction strength can be varied by changing the width of quantum well along the z axis [3, 7].
As a rule, circular quantum dots are simulated with the help of axially symmetric confinement
potentials V(x,y) = V(p), where p = ^Jx2 + y2 . In [8, 9], a simple but sufficiently adequate
rectangular potential of finite depth was proposed. This model with a discontinuous potential describes the main properties of circular quantum dots but without taking into account the individual characteristics. In [10], the smooth confinement potential of a new type which has finite depth and width was applied in the case of equal strengths aR = a D. The presence of additional potential
parameters allows us to simulate different kinds of circular quantum dots. In the actual paper, we use this potential in order to calculate the energy levels of electron for unequal but close strengths aR ^ aD.
Methods and results
The circular quantum dot of radius p0 is described by means of the confinement potential V(p) = V0 v(r), where V0 is the depth of the potential well. The function v(r) depends on ratio r = p / p0 in the following way
0, 0 < r < g, v1 (r), g < r < s,
v(r ) =
(2)
v2(r), s < r <1, w
1, r >1.
The functions v1(r) and v2(r) have the following forms:
f „2 12 , n , 2
v1(r) = d r - ^ , 4=^ (3)
r /
1 (1+s2)
2(1 - g2)(s2 - g2)'
v2(r ) = 1 - d2 f r --1 , d2=- (g 2 + ^ )2. (4)
2 f r y 2 2(1 - g2)(1 - s2) W
The parameters g and s change within ranges 0 < g <1 and g < s < 1. The function v(r) and its first derivative are continuous in the inflection points r = g , r = s and r = 1.
The total Hamiltonian of the problem can be written as a sum H = H0 + H1, where
H P2 + ^ + (aR +aD )( )( + ) + V( )
H 0= +-—-K y )(P + Py ) + V (p), (5)
2M~ 2n
(a +a ) a -a
H1= Y-^-K + y)(P - P), Y =-■-, (6)
2n a +a„ v/
R D
Mf is the effective electron mass which characterizes the motion in a semiconductor.
We shall solve the full Schrodinger equation H¥ = E¥ in two stages. First, we obtain an exact solution of the unperturbed Schrodinger equation H0¥0 = E0¥0 and then we shall take into account the perturbation H1 within the framework of the perturbation theory.
By analogy with [10] it is easy to show that the required solutions of the unperturbed Schrodinger equation admit a factorization
1
f 1 1 f_ .(a r +a „ )Me# (x + y)
¥0(x"") = V? U-" — r^' (7)
where m = 0, ±1, ±2,... is the angular momentum quantum number. Here we use the polar coordinates p,^ (x = p cos y = p sin ^).
Introducing dimensionless quantities
2M „p2 2M f p2 M f p
eo =-fZ^ vo =-V0, a = —f-Ha R + aD ),
h2 0 h we get the radial equation
h2
i2w 1 dw m2w / 2 .
- +--:---— +(eo + a - vov(r))w = 0.
dr2 r dr r
(8)
(9)
It is seen that the wave function depends only on the combination e0 + a2.
In the region 0 < r < g, the finite at r ^ 0 solution of radial equation is expressed via
the Bessel function [11] by means of the formula w1 (r) = Jm e0 + a2 r j.
In the region g < r < s, it is simple to obtain two solutions in terms of the confluent hypergeometric functions [11]:
w2 (r) = rB exp
w3(r) = rB exp
where
M ( A,1 + B,Jd~vj2 ),
U ( A,1 + B,JdlVj2 ),
A =1 + B e0 + a2 + 2di g 2Vo
B = V m2 + di g 4Vo.
2 ^VdvVT
In the region s < r < 1, it is easy to show that two solutions are w4(r)= w+ (r ) + (r ), w5(r)= w+(r) " w-(r)
2
2i
f
w± (r) = r ± exp where
_\j d2Vo
+
2 o r2
M ( A± ,1 + B±, ±4-dJj2 ),
A± =
1 + B± . v - e - a2 + 2d2v„
2
, B± = ±y[m2—d~v~o.
(10)
(11)
(12)
(13)
(14)
(15)
Note that the functions w4(r) and w5(r) are real if d2v0 > m2.
In the region r > 1, the decreasing solution is expressed via the modified Bessel function [11] with the help of the formula w6(r) = Km ^Jv0 - e0 - a2 r j. Thus, we obtain the radial wave function
c1w1 (r), 0 < r < g,
c2w2(r) + c3w3(r), g < r < s, c4w4(r) + c5w5(r), s < r <1,
w(r ) =
(16)
c6 w6(r X
r >1.
The coefficients ci are found from the continuity condition for function w(r) and its first derivative w (r) at three inflection points r = g , r = s , and r = 1. The fulfilment of this condition and the continuity of the potential and its first derivative guarantee the continuity of the second and the third derivative of the wave function.
1vo ..2
r
2
2
r
2
2
Six coefficients c. satisfy six linear algebraic equations
T (g, s, y,, m, a, «0) X = 0,
where X = {c1, c2, c3, c4, c5, c6} and matrix T has the form
(17)
T (g, s, v0, m, a, «0) =
f W1( g) -w^ g) -w3( g) 0 0 0
wj (g) -w2( g) -w3( g) 0 0 0
0 -w2(s) -w3(s) w4(s) w5( s) 0
0 -w2(s) -w3 (s) w4(s) w5( s) 0
0 0 0 W4 (1) W5 (1) -w6(1)
0 0 0 w4(1) w5 (1) -w6(1)
A
(18)
V /
Then the dependence of dimensionless energy e0(g, s,v0, m,a) on three dimensionless potential parameters g , s , and v0 is determined by the transcendental equation
k (g, s, v0, m, a, e0) = det T (g, s, v0, m, a, e0) = 0. (19)
This equation is solved numerically. Each level of energy is degenerate with two eigenfunctions ¥ + (x, y) and ¥ - (x, y).
When the exact values of e0(g,s,v0,m,a) are found it is not hard to obtain the values of coefficients c. from the system (17) and the standard normalization condition.
So, the exact solution of the unperturbed Schrodinger equation is constructed for an electron in a circular quantum dot which is simulated by the smooth potential (2).
We introduce the dimensionless perturbation h = 2Mef p02H^n2 and consider the contribution
of h with the help of the perturbation theory in the degenerate case for the small value of y .
In the basis of the eigenvectors | ¥ +) and | ¥-) of the unperturbed Hamiltonian we have the following equalities (¥± | h | ¥±) = 0 for the diagonal matrix elements. Off-diagonal matrix elements are given by <¥; \ \\ ¥-) = <¥ 0- | \\ ¥;) = Y5(m, v, a), where
i« 2 / f « 2 0 J^2ar)w (r)dr/ J0 w (r)rdr.
(20)
Then we get splitting e± = e0 ±y5 for the energy levels. Normalized eigenfunctions in zero-order approximation, which correspond to the eigenvalues e±, are described by the formula The distinctive feature of the used approximation is zero correction for zero
angular momentum (m = 0).
Now we present some numerical illustrations in addition to the analytical results. If we choose the value of effective electron mass Mf = 0.067Me related to GaAs, where Me is the electron mass in
vacuo, and assume p0 =30 nm, then the following correspondences a=1^(aR +aD )/2=18.9579 meV nm,
e = 1 ^ E = 0.631933 meV between the dimensionless and dimensional quantities are obtained.
Tab. 1, 2 demonstrate the dependence of energy levels on potential parameters at the following angular quantum numbers m = 0, ±1, ±2 . First of all we emphasize that the number of discrete levels is finite. This number increases if the parameters v0 and g grow and decreases if m grows. The energy level decreases if the parameter s grows. The ratio 5 /e0 decreases if e0 grows.
Doklady BGUIR
TT. 19, №8 (2021)_V. 19, No.8 (2021
Table 1. The dependence of e0 and 5 on potential parameters for a = 1 and v0 = 100
I m | eo, (S)
g = 0.1 g = 0.9
s = 0.325 s = 0.775 s = 0.925 s = 0.975
0 37.8202 (0.00000) 21.1503 (0.00000) 4.36674 (0.00000) 4.20242 (0.00000)
98.3459 (0.00000) 66.4228 (0.00000) 27.0087 (0.00000) 26.158 (0.00000)
— - 66.1516 (0.00000) 64.1934 (0.00000)
1 78.2272 (1.87550) 43.1724 (1.83448) 12.5861 (1.66374) 12.1710 (1.65410)
— 87.9681 (1.61613) 43.8652 (1.65402) 42.5166 (1.64521)
— - 89.2195 (1.54573) 86.8404 (1.55437)
2 — 66.3115 (3.50209) 23.3249 (3.17680) 22.5838 (3.15368)
- - 62.8864 (3.22045) 61.0017 (3.20327)
Table 2. The dependence of e0 and 5 on potential parameters for a = 1 and v0 = 400
1 m 1 e0 , (S)
g = 0.1 g = 0.9
s = 0.325 s = 0.775 s = 0.925 s = 0.975
0 68.4065 (0.00000) 37.9282 (0.00000) 4.97095 (0.00000) 4.78748 (0.00000)
248.296 (0.00000) 130.485 (0.00000) 30.4112 (0.00000) 29.4418 (0.00000)
374.034 (0.00000) 223.558 (0.00000) 75.9617 (0.00000) 73.569 (0.00000)
- 315.804 (0.00000) 141.174 (0.00000) 136.714 (0.00000)
- 394.955 (0.00000) 225.085 (0.00000) 217.952 (0.00000)
- - 325.146 (0.00000) 315.041 (0.00000)
1 154.175 (1.95605) 82.8206 (1.91536) 14.1512 (1.69837) 13.6850 (1.68944)
324.261 (1.87921) 176.327 (1.83530) 49.6767 (1.70195) 48.1080 (1.69318)
398.235 (1.16738) 269.601 (1.75627) 105.147 (1.69960) 101.834 (1.69064)
- 359.397 (1.64424) 179.906 (1.69342) 174.211 (1.68430)
- - 272.482 (1.67878) 263.873 (1.67029)
- - 377.369 (1.59580) 366.451 (1.61028)
2 244.117 (3.86453) 129.076 (3.74964) 26.2025 (3.26501) 25.3643 (3.24341)
380.222 (3.53219) 222.626 (3.59204) 71.8550 (3.35158) 69.5931 (3.33240)
- 315.315 (3.42475) 137.158 (3.36809) 132.829 (3.34919)
- 396.324 (2.76619) 221.216 (3.36065) 214.206 (3.34202)
- - 321.637 (3.31530) 311.605 (3.30183)
Conclusion
The confinement model potential for a quantum dot considered in the present paper is smooth, has finite depth and width and permits the exact solutions of the separated unperturbed Schrodinger equation for electron states in the presence of the spin-orbit interaction of Rashba and Dresselhaus. The contribution of perturbation is really small in comparison with the unperturbed energy e0 if the strength aR is sufficiently close to the strength aD (y ^ 1). Further, we intend to construct higher-oder corrections to the energy levels.
References
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Authors' contribution
Both authors equally contributed to the writing of the article.
Information about the authors
Baran A.V., PhD., Senior Researcher at the B.I. Stepanov Institute of Physics of the National Academy of Sciences of Belarus.
Kudryashov V.V., PhD., Leading Researcher at the B.I. Stepanov Institute of Physics of the National Academy of Sciences of Belarus.
Address for correspondence
220072, Republic of Belarus,
Minsk, Nezavisimosti Ave., 68-2,
B.I. Stepanov Institute of Physics
of the National Academy of Sciences of Belarus;
tel. 8-029-568-98-11;
e-mail: a.baran@dragon.bas-net.by
Baran Aleksandr Valer'evich
