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A quantum dot in topological insulator nanofilm

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 115302 (http://iopscience.iop.org/0953-8984/26/11/115302) View the table of contents for this issue, or go to the journal homepage for more

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 115302 (7pp)

doi:10.1088/0953-8984/26/11/115302

A quantum dot in topological insulator nanofilm Thakshila M Herath1 , Prabath Hewageegana2 and Vadym Apalkov1 1 2

Department of Physics and Astronomy, Georgia State University, Atlanta, GA 30303, USA Department of Physics, University of Kelaniya, Kelaniya 11600, Sri Lanka

E-mail: [email protected] Received 11 November 2013, revised 8 January 2014 Accepted for publication 16 January 2014 Published 3 March 2014

Abstract

We introduce a quantum dot in topological insulator nanofilm as a bump at the surface of the nanofilm. Such a quantum dot can localize an electron if the size of the dot is large enough, &5 nm. The quantum dot in topological insulator nanofilm has states of two types, which belong to two (‘conduction’ and ‘valence’) bands of the topological insulator nanofilm. We study the energy spectra of such defined quantum dots. We also consider intraband and interband optical transitions within the dot. The optical transitions of the two types have the same selection rules. While the interband absorption spectra have multi-peak structure, each of the intraband spectra has one strong peak and a few weak high frequency satellites. Keywords: quantum dot, topological insulator, absorption spectrum (Some figures may appear in colour only in the online journal)

1. Introduction

confinement potential with smooth boundaries. Such nonconventional behavior of electrons in graphene is determined by their unique low energy dispersion, which is gapless and relativistic, while the corresponding states are chiral [5, 6]. The electrons in graphene behave as massless Dirac fermions. Another system showing a dispersion law similar to that of graphene and correspondingly similar localization properties is that of the 3D topological insulators (TI) [7–13], which were first predicted theoretically and then observed experimentally in Bix Sb1−x , Bi2 Te3 , Sb2 Te3 , and Bi2 Se3 materials. The unique features of 3D TIs are gapless surface states with a low energy dispersion; its law is similar to the dispersion law of a massless Dirac fermion. Such relativistic dispersion laws have been observed by different experimental techniques [9, 10, 12–21]. Like for graphene, the conventional quantum dots, which can localize an electron, cannot be realized in TIs through an electrostatic confinement potential. To introduce a quantum dot in TI, one can consider a TI of finite nanoscale size [22, 23] or introduce a gap in the dispersion law of the surface states. Such a gap in the energy dispersion is opened for TI nanofilm [24–26]. Due to the finite extension of the surface states into the bulk, the surface states at two

Unique properties of semiconductor quantum dots, or ‘artificial atoms’, are determined by their discrete energy spectrum, which can be tuned externally through the nature and the strength of the confinement potential [1]. Such zero-dimensional systems show both specific electron transport with nonlinear features and controllable optical properties. The main interest in the quantum dots is related to their potential for applications, ranging from those in novel lasers and photodetectors to those in quantum information processing. In conventional semiconductor systems, quantum dots are introduced either by placing one nanosized material into another material, e.g. by the Stranski–Krastanov growth technique, or by applying a specially designed electrostatic confinement potential to low-dimensional systems. In both cases the confinement potential is introduced, which results in electron localization within the quantum dot region. Recently, quantum dots of a new type, graphene quantum dots [2–4] with electrostatic confinement potential, were considered. In such quantum dots, due to the Klein paradox, the electrons cannot be localized, and can only be trapped for a long enough time [3]. The longest trapping time is realized in a 0953-8984/14/115302+07$33.00

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surfaces of TI nanofilm are coupled. Such coupling introduces a gap in the energy dispersion law. Below we consider only TIs for which the gap is opened for the nanofilm system. For example, our results are applicable to Bi2 Se3 and we will use the parameters of this TI for numerical analysis. We introduce a quantum dot in such TI nanofilm as a finite region of TI nanofilm with a larger thickness. Due to the gapped structure of the energy dispersion in TI nanofilm, such a quantum dot can localize an electron. We consider only a single-electron problem and for a single electron calculate both the energy spectra and the optical transitions within the TI quantum dot. The paper is organized as follows. In section 2, we introduce a quantum dot in TI nanofilm and describe a method of calculating the energy and optical spectra of such a quantum dot. In section 3, we present the results of calculations. In section 4 we present the concluding discussion of the results obtained. 2. The model and main equations

Figure 1. (a) A schematic illustration of a TI quantum dot of

cylindrical symmetry. The TI nanofilm has a thickness L 2 . The quantum dot is introduced as a region of nanofilm with thickness L 1 and is characterized by its height h = L 1 − L 2 and radius R. (b) A band edge profile of a TI quantum dot is shown schematically for conduction and valence bands of the system. The bandgaps 1g,1 and 1g,2 are the bandgaps of TI nanofilm with thicknesses L 1 and L 2 , respectively.

We employ a TI nanofilm, which is characterized by nanoscale thickness L, to design a TI quantum dot. A quantum dot in such nanofilm is introduced as a region of the nanofilm of larger thickness. We assume that the quantum dot has cylindrical symmetry and is characterized by its height h and radius R—see figure 1. Thus the quantum dot can be described as topological insulator nanofilm with thickness L at ρ > R and thickness L + h at ρ < R, where ρ is the 2D distance from the center of the quantum dot. Below we consider only one set of thicknesses of the system, i.e., L = 20 Åand h = 20 Å, and we vary the radius R of the quantum dot. To find the energy spectrum of the quantum dot system and the corresponding optical transitions, we introduce a 4 × 4 matrix Hamiltonian which represents the effective low energy model of topological insulator nanofilm. The Hamiltonian describes the 2D in-plane electron dynamics in TI nanofilm and has the following form [25]: H+ 0 H= 0 H

For homogeneous TI nanofilm, the electron states are characterized by the 2D wavevector, (k x , k y ) = ( px , p y )/h¯ , and the corresponding energy spectrum, π2 E ± (k, L) = D1 2 + D2 k 2 L s 2 π2 (2) ± B1 2 + B2 k 2 + A22 k 2 , L has two bands (branches): (i) the ‘conduction’ band with energy E + and positive effective mass at p = 0 and (ii) the ‘valence’ band with energy E − and negative effective mass. The band edges for TI nanofilm of thickness L are at E c (L) = 2 (D1 + B1 ) πL 2 for the conduction band and at E v (L) = D1 −

−

π2

pˆ 2

A2 σˆ x pˆ y − σˆ y pˆ x h¯ ¯ π2 pˆ 2 + B1 2 + B2 2 τˆz ⊗ σˆ z L h¯

= D1

L2

+ D2

h2

+

B1 πL 2 for the valence band. The corresponding bandgap is 2

(1)

1g = 2B1 πL 2 . Thus the variable thickness of TI nanofilm (figure 1) introduces variation of the conduction and valence band edges, which generates confinement potential for both conduction and valence bands—see figure 1(b). To find the energy spectrum of a quantum dot we determine at a given energy E the general solution of the Schr¨odinger equation with Hamiltonian (1) in two regions, ρ < R and ρ > R. Then continuity condition at the boundary ρ = R determines the discrete energy spectrum of the quantum dot. From the eigenvalue equation, derived from matrix Hamiltonian (1), one can obtain that for a given energy E there are four values of wavevectors, corresponding to decaying and 2

where σˆ i (i = x, y, z) are Pauli matrices corresponding to spin degrees of freedom, and τˆz is the Pauli matrix corresponding to electron pseudospin, which describes the block diagonal Hamiltonian matrix with H+ (for τz = 1) and H− (for τz = −1). Here pˆ i = −h¯ ∂/∂ri (i = x, y) is the electron 2D momentum, L is the thickness of the nanofilm, and A2 , B1 , B2 , D1 , D2 are parameters of the 3D model of the TI. Below we consider Bi2 Se3 TI, for which these parameters are [27] A2 = 4.1 eV Å, B1 = 10 eV Å2 , B2 = 56.6 eV Å2 , D1 = 1.3 eV Å2 , D2 = 19.6 eV Å2 . 2

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propagating waves in the nanofilm of a given thickness. These wavevectors are s p β + β 2 + 4αγ k1,± (E, L) = ± , (3) 2α s p β − β 2 + 4αγ , (4) k2,± (E, L) = ± 2α 2 where α = B22 − D22 , β = B1 B2 − D1 D2 2π + 2E D2 + L2 4 2 + (D12 − B12 ) πL 4 . If E > E + or A22 , and γ = E E − D1 2π L2 E < E − , i.e. the energy is not in the energy gap of the TI nanofilm, then k1,± are real and correspond to two propagating waves, whereas k2,± are purely imaginary and determine decaying and growing modes. If E + > E > E − , i.e. the energy is in the bandgap, then all solutions, k1,± and k2,± , are imaginary and describe either decaying or growing modes. The localized modes of a quantum dot exist only in the following energy ranges: E − (L + h) > E > E − (L) (valence band states) and E + (L) > E > E + (L + h) (conduction band states). In these energy ranges, the wavevector k1,± (E, L + h) is real and corresponds to a propagating mode inside a quantum dot, while k2,± (E, L + h), k1,± (E, L), and k2,± (E, L) are imaginary and correspond to decaying modes both inside and outside the quantum dot. A specific feature of a TI quantum dot is that inside the quantum dot there are both propagating and decaying modes, simultaneously. Because of the double pseudospin degeneracy of the electronic states, below we calculate the energy spectra of the TI quantum dot for one component, e.g., τz = 1, of the pseudospin only. In two regions (ρ < R and ρ > R) the electron wavefunction with a z-component of angular momentum m has the following form: [a1 Q 1 Jm (κρ) + a2 M1 Im (qρ)]eimϕ ψ(ρ) = (5) a1 Q 2 Jm+1 (κρ) + a2 M2 Im+1 (qρ) ei(m+1)ϕ

In equations (5) and (6), Jm is the Bessel function of the first kind, and Im and K m are modified Bessel functions of the first and the second kind, respectively. The energy spectrum is determined by the boundary conditions, which are the continuity of the wavefunction and that of its derivative at the boundary of the quantum dot, i.e., ψ|ρ=R+0 = ψ|ρ=R−0 , ∂ψ ∂ψ = . ∂ρ ρ=R+0 ∂ρ ρ=R−0

(7) (8)

From these conditions we obtain the matrix equation, M(E) a = 0, on coefficients a j , j = 1, . . . , 4, where M(E) is a 4 × 4 matrix, the elements of which depend on the energy E. Then the energy spectrum is determined from numerical solution of the nonlinear equation det M(E) = 0. The optical transitions between the localized states of the quantum dot are determined by the velocity operator, vi ∝ ∂ H/∂ pi , which, for example for y-polarized light, has the form −2N k −A v y = −A1 y −2N 2k , (9) 2

2 y

where we consider only one component of the pseudospin, τz . Then intensity of the optical transitions between the initial state i and the final state f is Ii f ∝ |hψi |v y |ψ f i|2 .

(10)

This expression determines the intensities of both interband and intraband optical transitions. The above expressions (9) and (10) also define the selection rule for optical transitions, which is 1m = m f − m i = ±1,

(11)

where m i is the angular momentum of an electron in the initial state and m f is the angular momentum of an electron in the final state. The selection rule (11) is the same for both interband and intraband transitions.

for ρ < R and [a3 31 K m (q1 ρ) + a4 F1 K m (q2 ρ)]eimϕ ψ(ρ) = a3 32 K m+1 (q1 ρ) + a4 F2 K m+1 (q2 ρ) ei(m+1)ϕ (6)

3. Results and discussion

for ρ > R, where the constants a j , j = 1, . . . , 4, are determined from the continuity of the wavefunction and its derivative and from the normalization condition. Here κ = k1,+ (E, L + h), q = k2,+ (E, L + h)/i, q1 = k1,+ (E, L)/i, and q2 = k2,+ (E, L)/i. We also introduced the following notation:

3.1. The energy spectrum

Below we consider only the single-particle energy spectrum. For a many-electron system in the quantum dot this means that we disregard the inter-electron interactions. At a given radius R, a TI quantum dot has a finite number of localized energy levels in the conduction and valence bands. The number of levels in the quantum dot increases with radius R. In figure 2 the energy levels with angular momentum m = 0, 1, and 2 are shown as functions of the quantum dot radius. The energy levels follow the general tendency: with increasing R the absolute value of the level’s energy decreases. The interlevel energy separation also decreases with increasing R. For a given angular momentum m there is a critical size (cr) of the quantum dot Rm , so the energy levels with angular momentum m exist only if the radius of the dot is larger than (cr) the critical one, R > Rm . For the conduction band, the critical

Q 1 = − A2 κ, Q 2 = (D1 + B1 )π 2 /(L + h)2 + (D2 + B2 )κ 2 − E, M1 = − A2 q, M2 = (D1 + B1 )π 2 /(L + h)2 − (D2 + B2 )q 2 − E, 31 = A2 q1 , 32 = (D1 + B1 )π 2 /L 2 − (D2 + B2 )q12 − E, F1 = A2 q2 , F2 = (D1 + B1 )π 2 /L 2 − (D2 + B2 )q22 − E. 3

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to the lowest energy state at momentum m f = m i + 1 in the conduction band. The number of peaks is equal to the number of different angular momenta in the valence band. For example, at R = 100 nm, the states in the valence band exist at only two different angular momenta, m i = 0 and 1. In this case there are two main peaks B1 and B2 in the interband absorption spectra. Similarly, at R = 200 nm, there are three main peaks B1 , B2 , and B3 . There are also peaks B4 and B5 , which have almost the same frequency as B1 and B2 but smaller intensity. The intensities of the peaks of the multi-peak absorption spectra decrease with increasing angular momentum of the initial state. In figure 5 the intraband optical spectra are shown for different quantum dot sizes. The optical spectra each consist of one main peak A1 , which corresponds to the transition from the lowest state in the conduction band with angular momentum m i = 0 to the lowest state in the conduction band with angular momentum m f = 1. There is also a higher energy satellite optical line A2 with low intensity, which corresponds to the transition from the lowest state with m i = 1 to the lowest state with m f = 2. Thus, the intraband optical spectra have mainly a single-peak structure with the frequency, which varies with the radius of the quantum dot and is in the range of 30–70 meV.

Figure 2. The energy levels of the conduction band (positive energies) and valence band (negative energies) are shown as functions of the quantum dot radius R. Only the levels with angular momentum m = 0, 1, and 2 are shown. (cr)

radii for angular momentum m = 0, 1, and 2 are R0 = 15 AA, (cr) (cr) R1 = 55 Å, and R2 = 88.5 Å. For the valence band, the (cr) (cr) corresponding critical radii are R0 = 53 Å, R1 = 84 Å, (cr) and R2 = 111.5 Å. The data shown in figure 2 illustrate that the quantum dot system does not have symmetry between the conduction and valence band energy spectra. The critical radii for the conduction and valence bands are also different. In figure 3 the whole energy spectra are shown for four radii of the TI quantum dot: 50, 100, 150 and 200 Å. For a small quantum dot, R = 50 Å, there is only one energy level. This level belongs to the conduction band and has angular momentum m = 0, while no energy levels exist in the valence band. Upon increasing the size of the dot, more levels appear both in the conduction band and in the valence band.

3.3. The electron density

At a given value of the pseudospin, for example, τz = 1, the electron wavefunctions in a TI quantum dot have two components, ψ(ρ) = [φ1 (ρ), φ2 (ρ)]—see equations (5) and (6). These two components correspond to two directions of the electron spin, e.g., spin up and spin down. In this case we can introduce two characteristics of the wavefunction: spatial distributions of the charge density and spin density. They are defined through the following expressions: 4charge (ρ) = |ψ1 (ρ)|2 + |ψ2 (ρ)|2

(12)

for the electron charge density and 3.2. Optical transitions

4spin (ρ) = |ψ1 (ρ)|2 − |ψ2 (ρ)|2

We consider optical transitions of two types: (i) interband transitions between the states of valence and conduction bands and (ii) intraband optical transitions between the states of the conduction band. In both cases we consider a non-interacting electron system, i.e., the TI quantum dot is occupied by a single electron, which can be in different initial (valence of conduction band) states. For interband and intraband optical transitions the selection rules, which are given by equation (11), are the same. The main optical transitions with the largest intensities are shown by dashed lines in figure 3 for TI quantum dots of different radii. The intraband transitions are labeled ‘A’, while interband transitions are labeled ‘B’. The optical transitions mainly occur between the lowest energy states at a given angular momentum. Although other optical transitions are allowed by the selection rule, their intensities are small. In figure 4 interband optical spectra are shown for different sizes of the TI quantum dot. The spectra show multi-peak structure. The main transitions occur from the highest energy states at a given angular momentum, m i , in the valence band

(13)

for the electron spin density. The electron charge density for a quantum dot of radius R = 150 Åis shown in figure 6 for the states with angular momentum m = 0 and 1. For each angular momentum the charge density is shown for the two lowest energy states of the conduction band. The density profile shows the general tendency expected for localized modes of semiconductor quantum dots. The density is finite at ρ = 0 only for the states with m = 0, while for m > 0 the electron density is zero at ρ = 0. With increasing energy of the state the electron density shows more oscillations and an electron becomes more localized near the center (ρ = 0) of the quantum dot. The specific feature of the TI quantum dot is the existence in region ρ < R of two waves: a propagating wave with real wavevector and decaying waves with imaginary wavevector. Similarly, in the region ρ > R there are also two waves; both of them are decaying waves. For reflection of the electron wave from a step-like defect at the surface of the TI nanofilm, the existence of waves of two types results in enhancement of the 4

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T M Herath et al

Figure 3. The energy spectra of the TI quantum dot are shown as a function of the angular momentum m for different radii R of the quantum

dot: (a) 50 Å, (b) 100 Å, (c) 150 Å, and (d) 200 Å. At R = 50 Åthere is only one energy level in the conduction band. The dotted lines with arrows show the main intraband (labeled ‘A’) and interband (labeled ‘B’) optical transitions.

Figure 4. The absorption interband optical spectra are shown for

Figure 5. The absorption intraband optical spectra are shown for

different radii R of the quantum dot: (a) 100 Å, (b) 150 Å, and (c) 200 Å. The labels of the optical lines correspond to the labels of the transitions in figure 3. The optical spectra have multi-peak structure.

different radii R of the quantum dot: (a) 100 Å, (b) 150 Å, and (c) 200 Å. The labels of the optical lines correspond to the labels of transitions in figure 3. The optical spectra have one strong line with small satellites.

5

J. Phys.: Condens. Matter 26 (2014) 115302

T M Herath et al

Figure 6. The electron charge density distribution is shown for a quantum dot of radius 150 Å. The density is shown for conduction band states with angular momentum (a) m = 0 and (b) m = 1. The numbers ‘1’ and ‘2’ next to the lines correspond to the lowest energy state and the first excited state with a given angular momentum, respectively.

Figure 7. The electron density distribution is shown for a quantum

dot of radius 150 Åfor two states with angular momentum m = 0. The numbers ‘1’ and ‘2’ next to the lines correspond to the lowest energy state and the first excited state, respectively. (a) The density of electrons with spin up, |ψ1 (ρ)|2 ; (b) the density of electrons with spin down, |ψ2 (ρ)|2 ; (c) the electron spin density, 4spin (ρ).

electron density near the step-like defect [26]. For the localized modes in the TI quantum dots, as follows from figure 6, there is no enhancement or singularity near the boundary of the quantum dot. In regular semiconductor quantum dots without spin–orbit interaction the charge and spin densities are equal to each other, 4charge (ρ) = 4spin (ρ). With spin–orbit interaction the charge and spin densities are different [28, 29]. In topological insulators, it is a strong spin–orbit interaction that results in such unique gapless surface states. Therefore, in TI quantum dots, the spin–orbit interaction, which is already included implicitly in effective Hamiltonian (1), is expected to be strong. In this case the electron charge density and spin density should be strongly different. In figure 7 the electron spin density is shown for a quantum dot of radius R = 150 Åfor the two lowest energy levels of the conduction band with angular momentum m = 0. In figures 7(a) and (b) the electron spin density is shown for spin up and spin down, respectively. The data show that the electron density with spin down is almost 6 times smaller than the corresponding density with spin up. The spin density, which is shown in figure 7(c), is positive at almost all ρ points. Thus, in a TI quantum dot the spin–orbit interaction does not affect the electron spin strongly and the electron has almost the same orientation of spin at all points within a quantum

dot. The spin density is also close to the charge density (see figure 6(a)). The reason for such a weak effect of the spin–orbit interaction on the localized electron states in the TI quantum dot is related to the following property of TI nanofilm. For homogeneous TI nanofilm with uniform thickness, the mixing due to spin–orbit interaction of the states with different spins depends on the electron wavevector. For the TI nanofilm of thickness 40 Å, the strongest mixing of ≈15% is realized at the wavevector 0.03 Å−1 . For other wavevectors the mixing is smaller. For example, for the wavevector 0.01 Å−1 the mixing is ≈7%. The localized states of the quantum dot are combinations of the plane waves, where the main contribution comes from the 1/R ≈ 0.01 Å−1 wavevector. For such a wavevector, the mixing of the state with different spins is small (≈7%), which is illustrated in figure 7. For example, for the function number ‘1’, shown in figure 7, the ratio of coefficients Q 1 and Q 2 (see equation (5)), which determines the mixing of the states with different spins, is Q 1 /Q 2 ≈ 3. As a result, the electron density with spin down is an order of magnitude smaller than the electron density with spin up (see also figure 7). 6

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4. Concluding remarks

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The gap in the energy dispersion of the surface states of TI can be opened for a TI nanofilm. The coupling of the surface states at opposite boundaries of TI nanofilm introduces conduction and valence bands in the energy spectrum of the TI nanofilm. The bands are separated by a bandgap, which is proportional to 1/L 2 . In this case the quantum dot can be realized as a region of nanofilm of larger thickness. Such a quantum dot is characterized by its height and radius. The quantum dot can be grown experimentally by the droplet epitaxy method [30], which allows one to design nanostructures of different types. We calculated, within an effective model of TI nanofilm, an energy spectrum and optical transitions in a single-electron system of TI quantum dots. The electron states are characterized by the z-component of the angular momentum, m. For each value of m, there is a critical size (radius) of the quantum dot, which depends on the height of the quantum dot, such that the states with angular momentum m exist only in quantum dots with radius larger than the critical radius, R (cr) . The selection rule for the optical transitions is 1m = ±1; it is the same for interband and intraband transitions. The interband optical absorption occurs mainly between the lowest energy states of the valence and conduction bands, increasing the angular momentum by 1. The interband absorption spectra have multi-peak structure with the number of peaks equal to the number of different angular momentum at which the states in the valence band exist. The intraband absorption spectra each consist of one main peak and small high frequency satellites. The main peak corresponds to an optical transition between the lowest energy state with angular momentum m = 0 and the lowest energy state with angular momentum m = 1. The charge and spin density distributions for electron states in a TI quantum dot show that the electron spin has almost the same direction at all points inside the quantum dot. Thus the spin density almost coincides with the charge density. References [1] Chakraborty T 1999 Quantum Dots (Amsterdam: Elsevier) Chakraborty T 1992 Comments Condens. Matter Phys. 16 35 [2] Bunch J S, Yaish Y, Brink M, Bolotin K and McEuen P L 2005 Nano Lett. 5 287

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