Energy spectrum of carriers in a Kane-type cylindrical cavity

ARTICLE IN PRESS

Physica E 28 (2005) 37–42 www.elsevier.com/locate/physe

Energy spectrum of carriers in a Kane-type cylindrical cavity A.M. Babayeva,b,, E. Artunca, S. Cakmaka, S. Cakmaktepea a

Institute of Physics, Azerbaijan Academy of Sciences, Baku 370143, Azerbaijan Department of Physics, University of Suleyman Demirel, Isparta 32260, Turkey

b

Received 6 January 2005; accepted 10 January 2005 Available online 26 April 2005

Abstract The electronic states of a Kane-type cylindrical cavity with and without magnetic field are theoretically investigated. The eigenvalues and eigenstates of the Kane’s Hamiltonian are obtained. Calculations are performed for a hard-wall confinement potential and electronic states are obtained as a function of the magnetic field applied along the cavity cylinder axis. The size dependence of the effective g-value in bare InSb cavity cylinder for electrons are calculated. It has been seen that the effective g-value of the electrons are decreased with a decrease is the cavity cylinder’s radius. r 2005 Elsevier B.V. All rights reserved. PACS: 73.20.D; 71.70.E; 71.18 Keywords: Nanostructures; Spin-orbit coupling; g-factor

1. Introduction In the past few decades, there have been many studies about the optical properties of quantum nanostructures such as quantum dots, quantum wires, quantum wells and others, as a result of technological advancement in two-dimensional nanostructure fabrication techniques [1,2]. It is known that the magnetic field application could provide additional information about the properCorresponding author. Institute of Physics, Azerbaijan Academy of Sciences, Baku 370143, Azerbaijan. Tel.: +24 62111340; fax: +24 62371106. E-mail address: [email protected] (A.M. Babayev).

ties of carriers in solids as well in nanostructures. The energy spectrum of carriers in quantum dots and quantum wires was investigated in Refs. [3–6] theoretically. The electron energy states in a uniform magnetic field directed along the quantum wire were studied using a free electron model [3]. In Ref. [4] the energy spectrum of an electron confined to a quantum wire by an infinite square potential by using effective-mass approximation both with and without axial magnetic field was calculated. The energy levels of an electron confined in barrier-like multilayer quantum wire structure with perpendicular magnetic field along the z-direction in cylindrical coordinates are derived in Ref. [5]. Magneto-optical properties of

1386-9477/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2005.01.019

ARTICLE IN PRESS 38

A.M. Babayev et al. / Physica E 28 (2005) 37–42

quantum dots in semiconductors have been considered for the model of hard-wall confinement taking into account the real band structure of InSb-type materials (narrow energy gap and strong spin–orbital interaction) [6]. The spin–orbital interaction and the contribution of the electrons to the g factor were detailed in Refs. [7,8]. For calculating the electron g-values in Ref. [7] the three-band Kane’s model where the nonparabolicity of the electron and light-hole bands and the complex structure of the valence bands were taken into account simultaneously was used. This model describes the energy band structure around the G point of the Brillouin zone very well. In Masale [9] the energy spectrum of a single electron confined near a cylindrical cavity is determined within the effective-mass approximation as a function of the parallelly applied magnetic field. In Ref. [9] the energy spectrum change as a function of magnetic field explained that for a smaller radius, all the mp
3 states fall in the infinitely degenerate ground states, where the energy eigenvalues for a larger radius showed a somewhat parabolic variation with the magnetic field. In Ref. [10], a confined electronic system is modeled as an electron moving outside a cylinder of radius a. The magnetic field configuration for its Landau states and for the Aharonov–Bohm effect on such states and then the respective Schrodinger equations were successfully formulated and solved. They named the confined electronic system an antidot, or perhaps quantum antiwire seems a better name. The experimental advantages of using narrow-gap semiconductors for the reduced dimensionality make it necessary to take into account the real band structure of materials. While considering the nonparabolicity of the electron dispersion in narrow and medium gap semiconductors the coupling of conduction and valance bands should be taken into account. This is the purpose of our work. In this work, using three-band Kane’s model including the conduction band, light and spin–orbital hole bands, the electron spectrum with and without magnetic field and electron effective gfactor of cavity cylinder are calculated. The system considered here might simply be a very long cylindrical cavity of radius R etched out of a bulk

composite semiconductor material. We take the potential of the cavity cylinder to be infinitive at the boundary and zero elsewhere. We omit the free-electron term in the diagonal part and the Pauli spin term, as they give only small contributions to the effective mass and the spin g value of electrons in InSb. The Kane problem, i.e. determination of the energy spectrum taking into account the interaction of three bands, conduction band, light hole band and spin–orbital splitting band, in a magnetic field was considered in a paper of Bowers and Yafed [11]. In the three-band Kane’s Hamiltonian, the valence and conduction bands interaction is taken into account via the only matrix element P (socalled Kane’s parameter). The system of Kane equations including the nondispersional heavy hole bands have, from [12,13] rffiffiffi Pk
2 Pkþ
EC 1
pffiffiffi C 3 þ Pkz C 4 þ pffiffiffi C 5 3 2 6 Pkz Pkþ þ pffiffiffi C 7 þ pffiffiffi C 8 ¼ 0, ð1Þ 3 3 rffiffiffi Pk
2 Pkþ p ffiffi ffi
EC 2
Pkz C 5 þ pffiffiffi C 6 C4 þ 3 6 2 Pk
Pkz þ pffiffiffi C 7
pffiffiffi C 8 ¼ 0, 3 3

ð2Þ

Pkþ
pffiffiffi C 1
ðE þ E g ÞC 3 ¼ 0, 2

(3)

rffiffiffi 2 Pkþ Pkz C 1
pffiffiffi C 2
ðE þ E g ÞC 4 ¼ 0, 3 6

(4)

rffiffiffi 2 Pk
Pkz C 2 þ pffiffiffi C 1
ðE þ E g ÞC 5 ¼ 0, 3 6

(5)

Pk
pffiffiffi C 2
ðE þ E g ÞC 6 ¼ 0, 2

(6)

Pkz Pkþ pffiffiffi C 1 þ pffiffiffi C 2
ðD þ E þ E g ÞC 7 ¼ 0, 3 3

(7)

ARTICLE IN PRESS A.M. Babayev et al. / Physica E 28 (2005) 37–42

Pk
Pkz pffiffiffi  C 1
pffiffiffi C 2
ðD þ E þ E g ÞC 8 ¼ 0. 3 3

Ip

(8)

39

IIp

0.8

2. Zero magnetic field

(9) where D3 is the three-dimensional Laplacien. In cylindrical coordinates the eigenfunction is

ð11Þ

(12)

where w2 ¼

3 EðE þ E g ÞðE þ E g þ DÞ
k2z . ð3E þ 3E g þ 2DÞ P2

(13)

Eq. (12) is Bessel’s differential equation [14], the solution of Eq. (12) that is well behaved at infinity is R1;2 ðrÞ ¼ CY m ðwrÞ,

(14)

where Y m ðwrÞ is the second-order Bessel function. For an infinite wall of radius R, the boundary condition is R1;2 ðRÞ ¼ 0, so the eigenvalue equation is Y m ðwRÞ ¼ 0.

200

400

600

800

1000

° R(A) Fig. 1. The energy spectrum of electrons calculated as a function of radius in cavity cylinder for InSb. I is for the first root of Bessel differential equation, II is for the second root of Bessel differential equation. (p) electrons with parabolic dispersion law, (np) electrons with Kane’s dispersion law (nonparabolic).

(10)

where A is a normalization factor, kz is the axial wave number and m the azimuthal quantum number. The radial function RðrÞ is found to satisfy the following differential equation;  2  P 2 1 þ 3 E þ Eg E þ Eg þ D  2   d 1 d m2 2
þ
k

þ E R1;2 ðrÞ ¼ 0. z dr2 r dr r2 Eq. (11) can be rewritten in the form:  2  d 1 d m2 2
þ þ w R1;2 ðrÞ ¼ 0, dr2 r dr r2

np 0.4 0.2

Substituting expressions (3)–(8) into formulas (1) and (2) we obtain     P2 2 1
E
þ D3 C 1;2 ¼ 0, 3 E þ Eg E þ Eg þ D

C 1;2 ¼ A expðimf þ ikz zÞR1;2 ðrÞ,

np 0.6 E / Eg

Here P is the Kane parameter, Eg is the band gap energy, D is the value of spin–orbital splitting and ~ k ¼ kx  iky ; k~ ¼
ir.

(15)

Eqs. (13) and (15) together show that the radial eigenvalue spectrum is ! 2 z2mp EðE þ E g ÞðE þ E g þ DÞ 2 P ¼ , (16) þ k z 3E þ 3E g þ 2D 3 R2 where zmp is the pth root of the mth Bessel function Y ðzÞ. Eq. (16) determines the energies of electrons, light holes, and the spin–orbit split-off band of holes in a cavity cylinder. In Fig. 1, the dependencies of E(R) for two cases are presented: (a) electrons with parabolic dispersion law, (b) electrons with Kane’s dispersion law for InSb cavity cylinder according to both first and second roots of Bessel’s differential equation. As seen Fig. 1, with the increase of R, the electron energy levels in parabolic and nonparabolic cases become close to each other. At rather small sizes of R, the variance electron dispersion laws becomes more important and, therefore, the curves for E(R) keep away from each other strongly in II with respect to I. The following band parameters have been used for InSb: Eg¼ 0.2368 eV, D ¼ 0:810 eV, 2 0 E P ¼ 23:41 eV, E P ¼ 2m and m0 is the freeP _2 electron mass [15].

ARTICLE IN PRESS A.M. Babayev et al. / Physica E 28 (2005) 37–42

40

3. Applied magnetic field

for cavity cylinder, where

For a uniform magnetic field, H directed along the z-axis, the vector potential may be chosen in the form   Hy Hx ~ ; ;0 (17) A¼
2 2

a1;2 ¼

and k have the forms

o0 ð¼ eH=m0 cÞ is the cyclotron frequency for free qffiffiffiffiffi _c electrons, l H ð¼ eH Þ is the magnetic length and

k ! k  i12 lH r ,

(18)

b ¼ jmj þ 1

where r ¼ x  iy;

lH ¼

eH . _c

(19)

Substituting expressions (3)–(8) into formulas (1) and (2), and using relations (19) and (20) we obtain 0

1   P2 2 1 1 2 2 2 B
E þ 3 E þ E þ E þ E þ D
r þ lH Lz þ 4 lH r C g g B C B C  B P2 l  1 C 1 @ A H
 E þ Eg E þ Eg þ D 3

C 1;2 ¼ 0, ð20Þ where Lz is the z component of angular momentum operator L, r2 ¼ x2 þ y2 . To seek the solution of Eq. (21) in cylindrical coordinates in the form   x jmj C 1;2 ¼ A1;2 expðimj þ ikz zÞ exp
x 2 F1;2 ðxÞ 2 (21) we obtain the following equation for the radial function F(x): x

1 m jmj 1 2 2 þ þ þ kz l H 2 2 2 2 3E ðE þ E g ÞðE þ E g þ DÞ
_o0 ð3E þ 3E g þ 2DÞE p D  2ð3E þ 3E g þ 2DÞ

d2 F dF
a1;2 F ¼ 0. þ ðjmj þ 1
xÞ 2 dx dx

(22)

Eq. (22) is the canonical form of Kummer’s equation for the confluent hypergeometric function [16]. In Eq. (21) x ¼ r2 =2l 2H is the dimensionless variable and l is the radial wave number. The solution of Eq. (22) is   FðxÞ ¼ U a1;2 ; b; x (23)

ð24Þ

(25)

are the parameters of the Kummer function in standard notation. The boundary conditions which correspond to the infinite potential at r ¼ R are C 1;2 ¼ 0. These lead to the eigenvalue equations ! R2 U a1;2 ; b; 2 ¼ 0. (26) 2l H Eq. (26) is just the eigenvalue problem solved by C akmak et al. [17] for the case of a Kane-type quantum wire, except that here the function Uða1;2 ; b; xÞ replaces Mða1;2 ; b; xÞ. We can not only find the energy spectrum of electrons EðR; m; l H ; kz Þ but also the energy spectrum of light holes and spin–orbit splitting hole bands, from Eq. (24). For this it is necessary to find a1 and a2 from Eq. (26) for a given R, azimuthal quantum number m and l H , and then substitute them into Eq. (24). The energy spectrum of the cavity cylinder is labeled by the azimuthal quantum number m, the radial quantum number n, and two spin sublevels s ¼ 12; ðE nms Þ. Figs. 2 and 3 show examples of the magnetic field dependence of the lowest order subbands for l ¼ 0, kz ¼ 0 electron energy eigenvalues in an InSb cavity cylinder of radii R ¼ 50 A˚ and R ¼ 300 A˚, respectively, with infinite potential. In Fig. 2 the values of m are given by m ¼ 0; 1; 2; 3;
4;
5;
6. Note that the near-linearity variation of these energy subbands with the magnetic field for smaller radius in Fig. 2, and also that all mp
2 states fall in the infinitely degenerate ground state. The energy eigenvalues for a larger radius in Fig. 3 show a somewhat parabolic variation with

ARTICLE IN PRESS A.M. Babayev et al. / Physica E 28 (2005) 37–42

41

° R=50A m=3 m=3

0.2

m=2 m=2

E / Eg

0.15

m=1

0.1

m=1 m=0 m=-1 m=-2 m=0 m=-1 m=-6,..-2

0.05

0.5

1

1.5

2

H (Magnetic Field) Fig. 2. Some of the lowest order l z ¼ 0, kz ¼ 0 electron energy eigenvalues as a function of axial field H for R ¼ 50 A˚, in an InSb-type cavity cylinder. The values m are m ¼ 0;
1;
2;
3;
4;
5.

° R=300 A

− 40

m=0

0.3

− 41

m=-1

− 42

0.25

g

E / Eg

m=-2

0.2

m=-3

0.15

m=-4 m=-5

0.1

− 43 − 44 − 45 − 46

0.05

400 1

2 3 H (Magnetic Field)

4

5

Fig. 3. Some of the lowest order l z ¼ 0, kz ¼ 0 electron energy eigenvalues as a function of axial field H for R ¼ 300 A˚ in an InSb-type cavity cylinder with the energy of the m ¼ 0;
1;
2;
3;
4;
5, s ¼
12 states.

the magnetic field and a more pronounced separation in energy of the mp0, s ¼
12 states. This is the same as in the energy of the mp0, s ¼ þ12 states. These results are in good agreement with the study of Masale [9]. The effective g factor can be determined from the Zeeman splitting of subbands: gðEÞ ¼

E"
E# . mB H

(27)

600 ° R(A)

800

1000

Fig. 4. The electron g factor change versus radius in cavity cylinder for InSb.

Here E " and E # are the electron energy for spin +z and – z directions, respectively. Fig. 4 shows g factor dependence on R for the fixed magnetic value H ¼ 1 T. As seen in Fig. 4, with the decrease in R values, g factor approaches its asymptotic bulk value.

4. Conclusion In the present work which uses the three-band Kane’s model, the electron spectrum with and

ARTICLE IN PRESS 42

A.M. Babayev et al. / Physica E 28 (2005) 37–42

without magnetic field and electron effective g factor for cavitiy cylinder are calculated. The nonparabolicity of the spectrum of electrons, light holes and spin–orbit splitting valence bands were taken into account. The effective g factor was studied for the ground state as a function of radius. The differences between the parabolic and nonparabolic band model in zero magnetic field are represented. References [1] T. Chakraborty, Quantum Dots, Elsevier, Amsterdam, 1999. [2] D. Bimberg, M. Grundman, M. Ledentsov, Quantum Dot Heterostructures, Wiley, New York, 2001. [3] N.C. Constantinou, M. Masale, D.R. Tiley, J. Phys.: Condens. Matter 4 (1992) 4499. [4] N.C. Constantinou, M. Masale, D.R. Tiley, Phys. Rev. B 46 (1992) 15432. [5] Der-San Chuu, Chi-Suan Wang, Physica B 202 (1994) 118.

[6] W. Zawadski, M. Kubisa, Semicond. Sci. Technol. 8 (1993) S240. [7] A.A. Kiselev, E.L. Ivchenko, Pismo JETF 67 (1) (1996) 41. [8] A.A. Kiselev, E.L. Ivchenko, U. Rossler, Phys. Rev. B 58 (24) (1998) 16353. [9] M. Masale, Physica B 291 (2000) 256. [10] N. Aquino, E. Castano, E. Ley-Koo, Chinese J. Phys. 41 (3) (2003) 276. [11] Bowers, Yafed, Phys. Rev. 115 (1959) 1165. [12] B.M. Askerov, Kinetic Effects in Semiconductors, Nauka, Leningrad, 1970. [13] A.I. Anselm, Introduction to Semiconductor Theory, Nauka, Moscow, 1978; A.I. Anselm, Introduction to Semiconductor Theory, Prentice-Hall, Englewood Cliffs, NJ, 1978. [14] V.M. Galitskii, B.M. Karnakov, V.I. Kogan, A Collection of Problems Quantum Mechanics Science, Nauka, Moscow, 1981 (in Russian). [15] Al.L. Efros, Rosen, Phys. Rev. B 58 (11) (1998) 7120. [16] M. Abramovich, A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965, p. 504. [17] S. C - akmak, A.M. Babayev, E. Artunc- , A. Ko¨kce, S- . C akmaktepe, Physica E 18 (2003) 365.