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Thermoelectric properties of an ultra-thin topological insulator
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 165303 (http://iopscience.iop.org/0953-8984/26/16/165303) 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) 165303 (7pp)
doi:10.1088/0953-8984/26/16/165303
Thermoelectric properties of an ultra-thin topological insulator SK Firoz Islam and T K Ghosh Department of Physics, Indian Institute of Technology-Kanpur, Kanpur-208 016, India E-mail: [email protected] Received 17 December 2013 Accepted for publication 17 February 2014 Published 3 April 2014 Abstract
Thermoelectric coefficients of an ultra-thin topological insulator are presented here. The hybridization between top and bottom surface states of a topological insulator plays a significant role. In the absence of a magnetic field, the thermopower increases and thermal conductivity decreases with an increase in the hybridization energy. In the presence of a magnetic field perpendicular to the ultra-thin topological insulator, thermoelectric coefficients exhibit quantum oscillations with inverse magnetic field, whose frequency is strongly modified by the Zeeman energy and whose phase factor is governed by the product of the Landé g-factor and the hybridization energy. In addition to the numerical results, the low-temperature approximate analytical results for the thermoelectric coefficients are also provided. It is also observed that for a given magnetic field these transport coefficients oscillate with hybridization energy, at a frequency that depends on the Landé g-factor. Keywords: thermoelectric, thin topological insulator, quantum transport (Some figures may appear in colour only in the online journal)
1. Introduction
Though there have been several experimental works on the surface states of TIs, one of the major obstacles in studying the transport properties of the surface is the unavoidable contribution of the bulk. One of the best methods to minimize this problem is to grow TI samples in the form of ultra-thin films, in which the bulk contribution becomes very small in comparison to the surface contribution [13–15]. The transition from 3D to 2D TIs leads to several effects, which have been studied for different thickness [15, 16]. The ultra-thin TI not only reduces the bulk contribution, but also possesses some new phenomena such as possible excitonic superfluidity [17], unique magneto-optical response [18–20] and better thermoelectric performances [21]. Moreover, the small thickness leads to the overlap of the wave functions between top and bottom surfaces, which introduces a new degree of freedom hybridization [22, 23]. However, it happens to a certain thickness of five to ten quintuple layers [15, 24], i.e. of the order of 10 nm. The oscillating exponential decay of hybridization induced band gap with reducing thickness in Bi2Te3 has been also reported theoretically [25]. The formation of Landau levels has been confirmed by several experiments [26, 27] in thin TIs. Moreover, several theoretical studies on
Recently a new class of material, called a topological insulator, has had much attention from condensed matter physicists [1–6]. A topological insulator (TI) shows the conduction of electrons on the surface of 3D materials but otherwise behaves as an insulator. This is due to the time-reversal symmetry possessed by materials such as Bi2Se3, Sb2Te3 and Bi2Te3 [6]. The conducting surface states of these material show a single Dirac cone, in which spin is always locked perpendicular to its momentum. Angle resolved photoemission spectroscopy [7–9] or scanning tunneling microscopy [10] has been used to realize the single Dirac cone in TIs. In two-dimensional electron systems, under the presence of a perpendicular magnetic field, electrons conduct along the boundary due to the circular orbits bouncing off the edges, leading to skipping orbits. However, in 3D materials, even in the absence of magnetic field electron conduction takes place on the surface. Here, strong Rashba spin–orbit coupling (RSOC) plays the role of magnetic field. The RSOC originates from the lack of structural inversion symmetry of the sample [11, 12]. 0953-8984/14/165303+7$33.00
1
© 2014 IOP Publishing Ltd Printed in the UK
S K Firoz Islam and T K Ghosh
J. Phys.: Condens. Matter 26 (2014) 165303
with h(k) = Δhσz + vF(pyσx − pxσy). Here p is the two- dimensional momentum operator, vF is the Fermi velocity of the Dirac fermion, σ = (σx, σy, σz) are the Pauli spin matrices and Δh is the hybridization matrix element between the states of the top and bottom surfaces of the TI. A typical value of Δh varies from 0 to 102 meV depending on the thickness of the 3D TI [15]. Because of the block-diagonal nature, the above Hamiltonian can be written as
low-temperature transport properties have been reported in a series of works [24, 28–31]. Thermoelectric properties of materials [32] have always been interesting topic for providing an additional way to explore the details of an electronic system. When a temperature gradient is applied across the two ends of the electronic system, the migration of electrons from the hotter to the cooler side leads to the development of a voltage gradient across these ends. This voltage difference per unit temperature gradient is known as longitudinal thermopower. In addition to this temperature gradient, if a perpendicular magnetic field is applied to the system, due to Lorentz force a transverse electric field is also established and gives transverse thermopower. In a conventional 2D electronic system, Landau level induced quantum oscillation (Shubnikov–de Haas) in thermoelectric coefficients has been reported theoretically as well as experimentally in a series of works [33–37]. In 3D TIs, improvement of thermoelectric performance without magnetic field has been predicted theoretically in a series of paper [38–41]. In the newly emerged relativistic-like 2D electron system graphene, thermoelectric effects have been also studied [42–45]. In this paper, we study the effect of hybridization on the thermopower and the thermal conductivity of ultra-thin TIs in the absence/presence of magnetic field. We find that thermopower increases and thermal conductivity decreases with increase of the hybridization energy when magnetic field is absent. In the presence of perpendicular magnetic field, thermoelectric coefficients oscillate with inverse magnetic field. The frequency of the quantum oscillations is strongly modified by the Zeeman energy, and phase factor is determined by the product of the Landé g-factor and the hybridization energy. The analytical expressions of the thermoelectric coefficients are also obtained. It is also shown that these transport coefficients oscillate with frequency depending on hybridization energy and Landé g-factor. This paper has the following structure. Section 2 briefly discusses the energy spectrum and the density of states of an ultra-thin TI in the absence and presence of magnetic field. In section 3, we study how the hybridization affects the thermoelectric coefficients in zero magnetic field. In section 4, a complete analysis of thermoelectric coefficients in the presence of magnetic field is provided with numerical and analytical results. We provide a summary and conclusion of our work in section 5.
H = v F(σxpy − τzσypx ) + Δh σz,
where τz = ± denotes symmetric and anti-symmetric surface states, respectively. The energy spectrum of the Dirac electron is given by
E = λ (ℏv Fk )2 + Δ2h .
D0 ( E ) =
2E . π ℏ2v F2
(4)
2.2 Non-zero magnetic field case
In the presence of magnetic field perpendicular to the surface, the Hamiltonian for a Dirac electron with hybridization is H = v F(σxΠy − τzσyΠx ) + (τzΔz + Δh )σz, (5) where Π = p + eA is the two-dimensional canonical momentum operator. Using Landau gauge A ⃗ = (0, Bx , 0), exact Landau levels can be obtained very easily [28, 31]. For n = 0, there is only one energy level, which is given as E0τz = − ( Δz + τzΔz ). When integer n ⩾ 1, there are two energy bands denoted by + corresponding to the electron and − corresponding to the hole with energy Enτ,zλ = λ 2n(ℏωc )2 + (Δz + τzΔh )2 ,
(6) where ωc = vF/l is the cyclotron frequency with l = ℏ / ( eB ) the magnetic length, and Δz = gμBB/2 with g the Landé g-factor. The corresponding eigenstates for a symmetric surface state are Ψn+ ( r ) =
⎛ ⎞ eik yy ⎜ c1ϕn − 1 ( x + x 0 ) ⎟ , L y ⎜⎝ c2ϕn ( x + x 0 ) ⎟⎠
Ψn− ( r ) =
⎛ ⎞ eik yy ⎜ c2ϕn − 1 ( x + x 0 ) ⎟ , L y ⎜⎝ − c1ϕn ( x + x 0 ) ⎟⎠
2.1 Zero magnetic field case
Let us consider a surface of an ultra-thin TI in the xy-plane with Lx × Ly dimensions, where the carriers are Dirac fermions occupying the top and bottom surfaces of the TI. The quantum tunneling between top and bottom surfaces gives rise to hybridization, and consequently the Hamiltonian can be written as the symmetric and anti-symmetric combination of both surface states as [22]
(3)
Here λ = ± stands for electron and hole bands. The density of states is given by
2. Energy spectrum and density of states
⎡ h (k ) 0 ⎤ H=⎢ ⎥, ⎣ 0 h*(k ) ⎦
(2)
(7)
(8) 2 2 where ϕn ( x ) = (1 / π 2nn ! l ) e−x /2l Hn ( x / l ) is the normalized harmonic oscillator wave function, x0 = −kyl2, c1 = cos( θτz / 2) and c2 = sin( θτz / 2) with θτz = tan−1[ n ℏωc / ( Δz + τzΔh )]. The anti-
symmetric surface state can be obtained by exchanging n and n − 1. We have derived an approximate analytical form of the density of states for n > 1 by using the Green's function technique, which is given as (see the appendix)
(1)
2
S K Firoz Islam and T K Ghosh
J. Phys.: Condens. Matter 26 (2014) 165303
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
−0.7
(9)
where Δτz = Δz + τzΔh and Γ0 is the impurity induced Landau level broadening.
κ (nW/K)
0.8
j=4 j=3 j=2
10
20 ∆h (meV)
30
40
The classical Boltzmann transport equation can be used to calculate the zero magnetic field electrical conductivity, which is given as [49] σij ( E ) = e 2τ ( E )
(10)
2
∫ (2dπk)2 δ [ E − E ( k )] vi ( k ) v j ( k ),
where i, j = x, y. For an isotropic vx2 = vy2 = (1 / 2)( vx2 + vy2 ) = (1 / 2) v 2. In our case, v2 =
(15)
system
v F2 ⎡ ⎛ Δh ⎞2 ⎤ ⎟ ⎥. ⎢1−⎜ 2 ⎣ ⎝ E ⎠ ⎦
(16) Using these in equation (15), the energy dependent conductivity becomes E ⎡ ⎛ Δ ⎞2 ⎤ σ ( E ) = e 2τ ( E ) 2 ⎢ 1 − ⎜ h ⎟ ⎥ . (17) πℏ ⎣ ⎝ E ⎠ ⎦ Assuming the energy dependent scattering time to be τ = τ0( E / EF )m, where m is a constant depending on the scat2 + Δ2 tering mechanism, EF = EF0 h is the Fermi energy with EF0 = ℏv FkF0. Here, the Fermi vector kF0 = 2πne . Substituting equation (17) into equation (13), the diffusion thermopower is obtained as
⎤
∫ d E ⎢⎣ − ∂ f∂(EE ) ⎥⎦ ( E − η )(r ) σ ( E ),
(12) where r = 0, 1, 2 and f(E) = 1/[1 + exp(E − η)β] is the Fermi–Dirac distribution function with η the chemical potential and β = ( kBT )−1. Here, σ(E) is the energy-dependent electrical conductivity. When the circuit is open, i.e. for J = 0, the thermopower can be defined as S = Q12/Q11. By using the Sommerfeld expansion in the low temperature regime, the diffusive thermopower S and thermal conductivity κ can be obtained from Mott's relation and the Wiedemann–Franz law as
and
1.0
j=5
Figure 1. Plots of the thermopower versus hybridization constant for m = 1 and for various carrier densities.
(11) Jq = Q 21E + Q 22 ( −∇ T ), where E is the electric field, ∇T is the temperature gradient and Qij (i, j = 1, 2) are the phenomenological transport coefficients. The above equations describe the response of the electronic system under the combined effects of thermal and potential gradients. Moreover, Qij can be expressed in terms of an integral I(r): Q11 = I(0), Q21 = TQ12 = −I(1)/e, Q22 = I(2)/ (e2T) with
⎡ d ⎤ lnσ ( E ) ⎥ S = − L 0eT ⎢ ⎣ dE ⎦ E = EF
j=2
ne = j × 1015/m2
0.4 0
In this sub-section, the effect of hybridization on thermopower and thermal conductivity is presented. We follow the most conventional approach in the low temperature regime. The electrical current density J and the thermal current density Jq for Dirac electrons can be expressed under the linear response regime as
⎡
−1.6
0.6
3.1 Zero-magnetic field case
I (r ) =
j=3
1.2
In this section, we shall calculate thermoelectric coefficients of an ultra-thin TI in zero and non-zero magnetic fields.
and
j=4
−1.3
−1.9
3. Thermoelectric coefficients
J = Q11E + Q12 ( −∇ T )
j=5
−1.0 S (µeV/K)
∞ ⎡ ⎧ ⎛ ⎞2 ⎫ ⎢ 1 + 2 ∑ exp ⎨ − s⎜ 2π Γ0E ⎟ ⎬ 2 2 ⎢⎣ ⎩ ⎝ ℏ ωc ⎠ ⎭ s=1 ⎫⎤ ⎧ × cos ⎨ πs ( E2 −Δ2τz ) / ( ℏωc )2 ⎬ ⎥ , ⎩ ⎭ ⎥⎦
D (E) Dτz ( E ) ≃ 0 2
S = −L0
⎛ Δ ⎞2 ⎤ ⎛ Δ ⎞2 eT ⎡ ⎢ ( m + 1) + 2⎜ h ⎟ ⎥ / 1 + ⎜ h ⎟ . ⎝ EF0 ⎠ ⎥⎦ ⎝ EF0 ⎠ (18) EF0 ⎢⎣
We plot thermopower versus hybridization for different carrier densities in the upper panel of figure 1. It shows that thermopower increases with increasing hybridization for a particular carrier density. However, for higher carrier density, this rate of enhancement with hybridization becomes very slow. Thermal conductivity can be directly obtained from the Wiedemann–Franz law given in equation (14), where the electrical conductivity σ ( EF ) is given as
(13)
κ = L 0Tσ (EF ) . (14) 2 2 2 −8 −2 Here, L 0 = ( π kB ) / (3e ) = 2.44 × 10 WΩK is the Lorentz number and σ (EF ) is the electrical conductivity at the Fermi energy.
3
⎛ Δ ⎞2 σ = σ0 / 1 + ⎜ h ⎟ . ⎝ EF0 ⎠
(19)
S K Firoz Islam and T K Ghosh
J. Phys.: Condens. Matter 26 (2014) 165303
Sxy
25
∆h=0 meV
0.4
∆h=40 meV
0.2 20
0 −0.2
Sxy
15
∆h=10 meV
0.2
10
0 −0.2
0 meV
5
∆h=20 meV
0.4 Sxy
20 meV
κxx
0.4
0
0.2
4
6
8 1/B (1/T)
0
10
12
−0.2 −0.4
2
5
6
7
8
9
Figure 3. Plots of the thermal conductivity in units of kB / h versus
10
inverse magnetic field.
1/B (1/T)
Figure 2. Plots of the thermopower in units of kB/e versus inverse magnetic field for different values of the hybridization constant.
gradient, thermal transport coefficients can be expressed as ℒ (xxr) = ℒ (xxr)col = ℒ (yyr)col and ℒ (yyr) = ℒ (yyr)dif + ℒ (yyr)col = ℒ (yyr)col . In [31], the exact form of the finite-temperature collisional conductivity has been calculated for the screened impurity potential U (k) = 2πe 2 / (ϵ k 2 + k s2 ) ≃ 2πe 2 / (ϵk s ) = U0 under the limit of small |k| ≪ ks, with ks and ϵ being the inverse screening length and dielectric constant of the material, respectively. In this limit, one can use τ02 ≈ πl 2ℏ2 / NI U02, where τ0 is the relaxation time, U0 is the strength of the screened impurity potential and NI is the two-dimensional impurity density. The exact form of the finite-temperature conductivity [31] can be reduced to the zero-temperature energy-dependent electrical conductivity as
Here, σ0 = e2τ0EF0/(πℏ2) is the Drude conductivity without the hybridization constant for the Dirac system. The thermal conductivity is plotted in the lower panel of figure 1. We note that the thermal conductivity is diminished with increasing hybridization. However, unlike the case of thermopower, here thermal conductivity increases with carrier density. 3.2 Non-zero magnetic field case
In the presence of magnetic field, the classical approach can not explain the phenomenon depending on energy quantization. In this sub-section we follow a quantum mechanical approach, based on linear response theory, to study thermal transport coefficients. Thermoelectric coefficients for a twodimensional electron system in the presence of magnetic field were derived by modifying the Kubo formula in [46, 47]. These phenomenological transport coefficients are σμν = ℒ (0) μν
where
(21)
κμν =
1 (1) [ ℒ (2) μν − eT ( ℒ S )μν ], e 2T
(22)
r) ℒ (μν =
⎡
e 2 NI U02 ∑ Inτz, h πΓ0l 2 τ
(24) z where Inτz = [n{1 + cos2(θτz )} − cos(θτz )]. Here we have used −∂f / ∂E = δ[E − Enτz ]. Using equation (23), the finite-temperature diagonal ( ℒ (xxr) ) and off-diagonal coefficients ( ℒ (yxr) ) can be written as
(20)
1 [( ℒ (0) )−1 ℒ (1) ]μν eT
Sμν =
σxx ( E ) =
ℒ (xxr) =
and
⎡ ⎛ ∂f (E) ⎞ ⎤ e 2 NI U02 ⎟⎥ I τz ( E − η )r ⎜ − 2∑ n⎢ ⎝ h πΓ0l n, τ ⎣ ∂ E ⎠ ⎦ E = Enτz (25) z
sin2θ e2 1 ℒ (yxr) = ∑ 2 τz h 2 n, τ Δn z
⎤
∫ dE ⎢⎣ − ∂ f∂(EE ) ⎥⎦ ( E − η )r σμν ( E ).
(23) Here, μ, ν = x, y. Also, σμν(E), Sμν and κμν are the zero-temperature energy-dependent conductivity, thermopower and thermal conductivity tensors, respectively. Generally, diffusive and collisional mechanisms play a major role in electron conduction. The quantized energy spectrum of electrons reveals itself through Shubnikov–de Haas oscillation by the collisional mechanism. In our case, electron transport is mainly due to the collisional instead of the diffusive mechanism. The zero drift velocity of the electrons does not allow us to have a diffusive contribution. In the presence of a temperature
Here,
En + 1, τz
∫E ,τ
n z
⎛ ∂f (E) ⎞ ⎟ dE . (26) ( E − η )r ⎜ − ⎝ ∂E ⎠
⎛ Δ τ ⎞2 ⎛ Δ τ ⎞2 Δn = 2n + ⎜ z ⎟ − 2( n + 1) + ⎜ z ⎟ ⎝ ℏωc ⎠ ⎝ ℏωc ⎠ .
4. Numerical results and discussion In our numerical calculations, the following parameters are used: carrier concentration ne = 2 × 1015 m−2, g = 60, vF = 4 × 105 ms−1 and T = 0.7 K. These numerical parameters are consistent with [15, 30, 31]. 4
S K Firoz Islam and T K Ghosh
J. Phys.: Condens. Matter 26 (2014) 165303
1
0.2
0.8
κxx /10
0.6
0
κxx and Sxy
Sxy
0.1
−0.1
0.4 0.2 Sxy
10
0
κxx
8
−0.2
6 4
−0.4 0
2 5
6
7
1/B (1/T)
8
9
Sxy = − Syx =
⎤ 1 ⎡ σxx (1) ℒ (1) yx ⎢ ℒ xx + ⎥ eT ⎣ S0 σyx ⎦
2
⎧ ⎛ Γ E ⎞2 ⎫ Ω D = exp ⎨ − ⎜ 2π 20 F2 ⎟ ⎬ ⎩ ⎝ ℏ ωc ⎠ ⎭ ⎪
and ℒ (2) xx ≃
Sxy
z
z
z
(32)
σ0T ∑ UτzFτz ( ωcτ0 )2 τz (33) × [1 − 6Ω DG ′′ ( x )cos(2πf / B − τzϕ )],
where the frequency f is given as f =
1 ( EF2 0 −Δ2z ) 2eℏv F2
(34) and the phase factor ϕ = πgμB Δh / ( eℏv F2 ). Therefore, the thermopower and the thermal conductivity oscillate with the same frequency f, which is independent of Δh, which can also be shown from numerical results. The oscillation frequency is strongly reduced by the Zeeman energy Δz. On the other hand, the phase factor φ is related to the product of the Landé g-factor and Δh, and it vanishes if either of them is zero. Although the frequency and the phase of Sxx and κxx are the same, the damping factor and amplitude are different.
z
τz
1/2 ⎛ Δh ⎞2 ⎤ kB 1 16π ⎡ ⎢ ⎥ ≃ Ω DG ′ ( x ) 1+⎜ ⎟ ⎝ EF0 ⎠ ⎥⎦ e kF0l ωcτ0 ⎢⎣ Uτ × ∑ 2 z Fτzsin(2πf / B − τzϕ ) sin ( θτz ) τz
κxx ≃ L 0
∫
∑ Uτ Fτ [1 − 3ΩDG ′′ ( x )cos(2πFτ )],
⎪
and
(28)
4π e 2 Γ0EF ΩDG ′ ( x ) ∑ UτzFτzsin(2πFτz ) (29) β h ( ℏωc )2 τ
L 0e 2T 2σ0 ( ωcτ0 )2
⎪
(31) and the temperature dependent damping factor is the derivative of the function G(x) with G(x) = x/ sinh (x). Here, x = T/Tc, where Tc = ( ℏωc )2 / (2π 2kBEF ) is the critical temperature, which depends on the strength of hybridization through the Fermi energy. Note that G(x) is the temperature dependent damping factor for the electrical conductivity tensor. The off-diagonal thermopower Sxy and the diagonal thermal conductivity κxx are obtained as given by ⎪
Here, S0 = σxxσyy − σxyσyx. The dominating term in the above two equations is the last term. The analytical form of κxx and Sxy can be obtained directly by deriving the analytical form of the phenomenological transport coefficients. The analytical form of the density of states given in equation (9) allows us to obtain asymptotic expressions of Sxy and κxx. This is done by replacing the summation over discrete quantum numbers n by the integration, i.e. ∑ → 2πl 2 Dτz ( E )dE; then we get ℒ (1) xx ≃
100
where Fτz = ( EF2 −Δ2τz ) / ( 2 ℏωc ) , Uτz = [1 + cos2 ( θτz )], the impurity induced damping factor is
(27)
⎤ ℒ (1) 1 ⎡ σxx xx ⎢ ⎥. ( −ℒ (1) xy ) + eT ⎣ S0 σyx ⎦
n
75
Figure 5. Plots of the thermopower and thermal conductivity versus the hybridization constant (Δh). Here, B = 0.2 T is taken.
In figure 2, the off-diagonal thermopower, Sxy, is shown as a function of the inverse magnetic field for different values of the hybridization constant. Similarly, the thermal conductivity, κxx, is shown versus inverse magnetic field for different values of the hybridization constant in figure 3. Careful observation of these two figures clearly shows that Sxy and κxx oscillate with the same frequency, which does not depend on the hybridization energy. The hybridization energy only influences the phase of oscillations. To determine the frequency and the phase of the quantum oscillations in the thermoelectric coefficients, we shall derive analytical expressions of these coefficients. The components of the thermopower are given by
and
50 ∆h (meV)
10
Figure 4. Plots of the numerical and analytical results of the thermoelectric coefficients versus inverse magnetic field for Δh = 20 meV. Solid and dashed lines stand for numerical and analytical results, respectively.
Sxx = Syy =
25
(30) 5
S K Firoz Islam and T K Ghosh
J. Phys.: Condens. Matter 26 (2014) 165303
Appendix
Now we compare the numerical and analytical results for sxy and κxx in figure 4. For better visualization, we have taken weak Landau level broadening Γ0 = 0.01 meV for Sxy and κxx. The analytical results, in particular the frequency f, match very well with the numerical results. We must mention here that for different values of Γ0 analytical results may differ from numerical ones in the amplitude, but the frequency and phase are always in good agreement. It is interesting to see from the analytical expressions of the thermopower and the thermal conductivity that these transport coefficients possess weak periodic oscillation with the hybridization energy for a given magnetic field. This oscillation is shown in figure 5. The frequency and phase factor of these oscillations for a fixed B are ν = ϕ / (2π ) = gμB / (2eℏv F2 ) and Φ = 2πf/B, respectively. In the presence of the magnetic field, these approximate analytical expressions of the thermopower and thermal conductivity can also be used for monolayer graphene by putting Δh = 0. There are several experimental results [42, 48] for thermoelectric properties of a graphene monolayer, but analytical expressions are not available in the literature.
The derivation of asymptotic analytical expression of density of states of a two-dimensional electronic system in the presence of impurities can be done by calculating the self-energy [50, 51], which is given as Σ −(E ) = Γ20 ∑
n
E−
Enτz
1 . − Σ −(E )
(A.1)
The imaginary part of the self-energy is related to the den⎡ Σ -( E ) ⎤ sity of states as D ( E ) = Im ⎢ 2 2 2 ⎥. ⎣ π l Γ0 ⎦ By using the residue theorem, we calculate the summation 2π Γ20E in equation (A.1), which gives Σ − ( E ) ≃ cot( πn 0 ), ( ℏωc )2 where n0 is the pole and is given as ⎡ ⎤ 1 { E − Σ − ( E )}2 − ( Δz + τzΔh )2 ⎥ . (A.2) 2 ⎢ ⎦ 2( ℏωc ) ⎣ By writing the self-energy as the sum of the real and imaginary parts, it can be further simplified as n0 =
5. Conclusion
Δ+i
⎡ ( u − iv ) ⎤ 2π Γ20E Γ ≃ cot ⎢ 2 ⎣ 2 ⎥⎦ 2 ( ℏωc )
u=
⎤ π ⎡ 2 E − ( Δz + τzΔh )2 ⎥ 2⎢ ⎦ ( ℏωc ) ⎣
Here,
We have presented a theoretical study of the thermoelectric coefficients of ultra-thin topological insulators in the presence/absence of magnetic field. In the absence of magnetic field, the thermopower and the thermal conductivity are modified due to the hybridization between top and bottom surface states. The thermopower is enhanced and the thermal conductivity is diminished due to the hybridization. The quantum oscillations in the thermopower and the thermal conductivity for different values of the hybridization constant are also studied numerically. In addition to the numerical results, we obtained the analytical expressions for the thermopower (Sxy) and the thermal conductivity (κxx). The analytical results match very well with the numerical results. We have also provided analytical expressions for the oscillation frequency and phase. The oscillation frequency is the same for both the thermopower and the thermal conductivity. It is independent of the hybridization constant but strongly suppressed by the Zeeman energy. On the other hand, the hybridization constant plays a very significant role in the phase as well as in the amplitude of the oscillations. From the analytical results, critical temperature (Tc) is found to be reduced with increasing hybridization constant. Thermoelectric coefficients also show a very low-frequency oscillation with the hybridization constant for a given magnetic field. Moreover, our analytical expressions of the thermopower and thermal conductivity are also applicable for a graphene monolayer by setting Δh = 0.
(A.3)
(A.4)
and v = π ΓE/(ℏωc)2. The imaginary part is Γ 2π Γ20E sinh v . Now, this can be re-written by = 2 ( ℏωc )2 cosh v − cos u using the following standard relation: ∞
sinh v = 1 + 2 ∑ e−svcos( su ). cosh v − cos u s=1
(A.5)
Here, the most dominant term is for s = 1 only. We can write
∞ ⎤ Γ 2π Γ20E ⎡⎢ −sπ ΓE /(ℏωc )2 ⎥. + e cos( ) = 1 2 u ∑ ⎥⎦ 2 ( ℏωc )2 ⎢⎣ s=1
(A.6)
In the limit of πΓ ≫ ℏωc, after the first iteration, we have Γ / 2 = 2πΓ02E / ( ℏωc )2. Substituting this in the earlier expression, we get ∞ ⎧ ⎛ ⎞2 ⎫ 2π Γ20E ⎡ Γ ⎢ 1 + 2 ∑ exp ⎨ − s⎜ 2π Γ0E ⎟ ⎬ = 2 2 2 ( ℏωc )2 ⎢⎣ ⎩ ⎝ ℏ ωc ⎠ ⎭ s=1
⎪
⎪
⎪
⎪
⎧ ⎫⎤ cos ⎨ sπ ( E2 −Δ2τz ) / ( ℏωc )2 ⎬ ⎥ . ⎩ ⎭ ⎥⎦ ⎪
⎪
⎪
⎪
(A.7)
Here, Δτz = Δz + τzΔh. Finally, the density of states for the two branches can be obtained as ∞ ⎧ ⎛ 2π Γ E ⎞2 ⎫ D0 ( E ) ⎡⎢ 1 + 2 ∑ exp ⎨ − s⎜ 2 0 2 ⎟ ⎬ Dτz ( E ) = 2 ⎢⎣ ⎩ ⎝ ℏ ωc ⎠ ⎭ s=1
Acknowledgments This work is financially supported by the CSIR, Government of India, under grant CSIR-SRF-09/092(0687) 2009/EMR F-O746.
6
⎪
⎪
⎪
⎪
⎧ ⎫⎤ cos ⎨ sπ ( E2 −Δ2τz ) / ( ℏωc )2 ⎬ ⎥ . ⎩ ⎭ ⎥⎦ ⎪
⎪
⎪
⎪
(A.8)
S K Firoz Islam and T K Ghosh
J. Phys.: Condens. Matter 26 (2014) 165303
[25] Linder J, Yokoyama T and Sudbo A 2009 Phys. Rev. B 80 205401 [26] Cheng P et al 2010 Phys. Rev. Lett. 105 076801 [27] Jiang Y, Wang Y, Chen M, Li Z, Song C, He K, Wang L, Chen X, Ma X and Xue Q K 2012 Phys. Rev. Lett. 108 016401 [28] Zyuzin A A and Barkov A A 2011 Phys. Rev. B 83 195413 [29] Wang J, DaSilva A M, Chang C Z, He K, Jain J K, Samarth N, Ma X C, Xue Q K and Chan M H W 2011 Phys. Rev. B 83 245438 [30] Tahir M, Sabeeh K and Schwingenschlogl U 2013 Sci. Rep. 3 1261 [31] Tahir M, Sabeeh K and Schwingenschlogl U 2013 J. Appl. Phys. 113 043720 [32] Nolas G S, Sharp J and Goldsmid H J 2001 Thermoelectrics (Berlin: Springer) [33] Fletcher R, Maan J C, Ploog K and Weimann G 1986 Phys. Rev. B 33 7122 [34] Fletcher R, Coleridge P T and Feng Y 1995 Phys. Rev. B 52 2823 [35] Fletcher R 1999 Semicond. Sci. Technol. 14 R1 [36] Maximov S, Gbordzoe M, Buhmann H, Molenkamp L W and Reuter D 2004 Phys. Rev. B 70 121308(R) [37] Goswami S, Siegert C, Pepper M, Farrer I, Ritchie D A and Ghosh A 2011 Phys. Rev. B 83 073302 [38] Tretiakov O A, Abanov Ar, Murakami S and Sinova J 2010 Appl. Phys. Lett. 97 073108 [39] Tretiakov O A, Abanov Ar and Sinova J 2011 Appl. Phys. Lett. 99 113110 [40] Tretiakov O A, Abanov Ar and Sinova J 2012 J. Appl. Phys. 111 07E319 [41] Takahashi R and Murakami S 2012 Semicond. Sci. Technol. 27 124005 [42] Zuev Y M, Chang W and Kim P 2009 Phys. Rev. Lett. 102 096807 [43] Wei P, Bao W, Pu Y, Lau C N and Shi J 2009 Phys. Rev. Lett. 102 166808 [44] Hwang E H, Rossi E and Das S 2009 Phys. Rev. B 80 235415 [45] Patel A A and Mukerjee S 2012 Phys. Rev. B 86 075411 [46] Smreka L and Streda P 1977 J. Phys. C: Solid State Phys. 10 2153 [47] Oji H 1984 J. Phys. C: Solid State Phys. 17 3059 [48] Wei P, Bao W, Pu Y, Lau C N and Shi J 2009 Phys. Rev. Lett. 102 166808 [49] Ashcroft N W and Mermin N D 2010 Solid State Physics (Stamford, CT: Cengage Learning) [50] Ando T, Fowler A B and Stern F 1982 Rev. Mod. Phys. 54 437 [51] Zhang C and Gerhardts R R 1990 Phys. Rev. B 41 12850
References [1] Kane C L and Mele E J 2005 Phys. Rev. Lett. 95 146802 [2] Bernevig B A, Hughes T L and Zhang S C 2006 Science 314 1757 [3] Konig M, Wiedmann S, Brune C, Roth A, Buhmann H, Molenkamp L W, Qi X L and Zhang S C 2007 Science 318 766 [4] Moore J E and Balents L 2007 Phys. Rev. B 75 121306 [5] Fu L, Kane C L and Mele E J 2007 Phys. Rev. Lett. 98 106803 [6] Zhang H, Liu C X, Qi X L, Dai X, Fang Z and Zhang S C 2009 Nat. Phys. 5 438 [7] Hsieh D, Qian D, Wray L, Xia Y, Hor Y S, Cava R J and Hasan M Z 2008 Nature 452 970 [8] Hsieh D et al 2009 Science 323 919 [9] Xia Y et al 2009 Nat. Phys. 5 398 [10] Roushan P, Seo J, Parker C V, Hor Y S, Hsieh D, Qian D, Richardella A, Hasan M Z, Cava R J and Yazdani A 2009 Nature 460 1106 [11] Rashba E I and Sheka V I 1959 Fizika Tverdogo Tela; Collected Papers 2 (Moscow and Leningrad: Academy of Sciences of the USSR) p 162 Rashba E I 1960 Sov. Phys. Solid State 2 1109 [12] Bychkov Y A and Rashba E I 1984 J. Phys. C: Solid State 17 6039 [13] Zhang G, Qin H, Teng J, Guo J, Guo Q, Dai X, Fang Z and Wu K 2009 Appl. Phys. Lett. 95 053114 [14] Peng H, Lai K, Kong D, Meister S, Chen Y, Qi X L, Zhang S C, Shen Z X and Cui Y 2010 Nat. Mater. 9 225 [15] Zhang Y et al 2010 Nat. Phys. 6 584 [16] Liu C X, Zhang H, Yan B, Qi X L, Frauenheim T, Dai X, Fang Z and Zhang S C 2010 Phys. Rev. B 81 041307 [17] Seradjeh B, Moore J E and Franz M 2010 Phys. Rev. Lett. 103 066402 [18] Tse W K and MacDonald A H 2010 Phys. Rev. Lett. 105 057401 [19] Tse W K and MacDonald A H 2010 Phys. Rev. B 82 161104 [20] Maciejko J, Qi X L and Zhang S C 2010 Phys. Rev. Lett. 104 166803 [21] Ghaemi P, Mong R S K and Moore J E 2010 Phys. Rev. Lett. 105 166603 [22] Yu R, Zhang W, Zhang H J, Zhang S C, Dai X and Fang Z 2010 Science 329 61 [23] Lu H Z, Shan W Y, Yao W, Niu Q and Shen S Q 2010 Phys. Rev. B 81 115407 [24] Cao H, Tian J, Miotkowski I, Shen T, Hu J, Qiao S and Chen Y P 2012 Phys. Rev. Lett. 108 216803
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Thermoelectric properties of an ultra-thin topological insulator
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 165303 (http://iopscience.iop.org/0953-8984/26/16/165303) 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) 165303 (7pp)
doi:10.1088/0953-8984/26/16/165303
Thermoelectric properties of an ultra-thin topological insulator SK Firoz Islam and T K Ghosh Department of Physics, Indian Institute of Technology-Kanpur, Kanpur-208 016, India E-mail: [email protected] Received 17 December 2013 Accepted for publication 17 February 2014 Published 3 April 2014 Abstract
Thermoelectric coefficients of an ultra-thin topological insulator are presented here. The hybridization between top and bottom surface states of a topological insulator plays a significant role. In the absence of a magnetic field, the thermopower increases and thermal conductivity decreases with an increase in the hybridization energy. In the presence of a magnetic field perpendicular to the ultra-thin topological insulator, thermoelectric coefficients exhibit quantum oscillations with inverse magnetic field, whose frequency is strongly modified by the Zeeman energy and whose phase factor is governed by the product of the Landé g-factor and the hybridization energy. In addition to the numerical results, the low-temperature approximate analytical results for the thermoelectric coefficients are also provided. It is also observed that for a given magnetic field these transport coefficients oscillate with hybridization energy, at a frequency that depends on the Landé g-factor. Keywords: thermoelectric, thin topological insulator, quantum transport (Some figures may appear in colour only in the online journal)
1. Introduction
Though there have been several experimental works on the surface states of TIs, one of the major obstacles in studying the transport properties of the surface is the unavoidable contribution of the bulk. One of the best methods to minimize this problem is to grow TI samples in the form of ultra-thin films, in which the bulk contribution becomes very small in comparison to the surface contribution [13–15]. The transition from 3D to 2D TIs leads to several effects, which have been studied for different thickness [15, 16]. The ultra-thin TI not only reduces the bulk contribution, but also possesses some new phenomena such as possible excitonic superfluidity [17], unique magneto-optical response [18–20] and better thermoelectric performances [21]. Moreover, the small thickness leads to the overlap of the wave functions between top and bottom surfaces, which introduces a new degree of freedom hybridization [22, 23]. However, it happens to a certain thickness of five to ten quintuple layers [15, 24], i.e. of the order of 10 nm. The oscillating exponential decay of hybridization induced band gap with reducing thickness in Bi2Te3 has been also reported theoretically [25]. The formation of Landau levels has been confirmed by several experiments [26, 27] in thin TIs. Moreover, several theoretical studies on
Recently a new class of material, called a topological insulator, has had much attention from condensed matter physicists [1–6]. A topological insulator (TI) shows the conduction of electrons on the surface of 3D materials but otherwise behaves as an insulator. This is due to the time-reversal symmetry possessed by materials such as Bi2Se3, Sb2Te3 and Bi2Te3 [6]. The conducting surface states of these material show a single Dirac cone, in which spin is always locked perpendicular to its momentum. Angle resolved photoemission spectroscopy [7–9] or scanning tunneling microscopy [10] has been used to realize the single Dirac cone in TIs. In two-dimensional electron systems, under the presence of a perpendicular magnetic field, electrons conduct along the boundary due to the circular orbits bouncing off the edges, leading to skipping orbits. However, in 3D materials, even in the absence of magnetic field electron conduction takes place on the surface. Here, strong Rashba spin–orbit coupling (RSOC) plays the role of magnetic field. The RSOC originates from the lack of structural inversion symmetry of the sample [11, 12]. 0953-8984/14/165303+7$33.00
1
© 2014 IOP Publishing Ltd Printed in the UK
S K Firoz Islam and T K Ghosh
J. Phys.: Condens. Matter 26 (2014) 165303
with h(k) = Δhσz + vF(pyσx − pxσy). Here p is the two- dimensional momentum operator, vF is the Fermi velocity of the Dirac fermion, σ = (σx, σy, σz) are the Pauli spin matrices and Δh is the hybridization matrix element between the states of the top and bottom surfaces of the TI. A typical value of Δh varies from 0 to 102 meV depending on the thickness of the 3D TI [15]. Because of the block-diagonal nature, the above Hamiltonian can be written as
low-temperature transport properties have been reported in a series of works [24, 28–31]. Thermoelectric properties of materials [32] have always been interesting topic for providing an additional way to explore the details of an electronic system. When a temperature gradient is applied across the two ends of the electronic system, the migration of electrons from the hotter to the cooler side leads to the development of a voltage gradient across these ends. This voltage difference per unit temperature gradient is known as longitudinal thermopower. In addition to this temperature gradient, if a perpendicular magnetic field is applied to the system, due to Lorentz force a transverse electric field is also established and gives transverse thermopower. In a conventional 2D electronic system, Landau level induced quantum oscillation (Shubnikov–de Haas) in thermoelectric coefficients has been reported theoretically as well as experimentally in a series of works [33–37]. In 3D TIs, improvement of thermoelectric performance without magnetic field has been predicted theoretically in a series of paper [38–41]. In the newly emerged relativistic-like 2D electron system graphene, thermoelectric effects have been also studied [42–45]. In this paper, we study the effect of hybridization on the thermopower and the thermal conductivity of ultra-thin TIs in the absence/presence of magnetic field. We find that thermopower increases and thermal conductivity decreases with increase of the hybridization energy when magnetic field is absent. In the presence of perpendicular magnetic field, thermoelectric coefficients oscillate with inverse magnetic field. The frequency of the quantum oscillations is strongly modified by the Zeeman energy, and phase factor is determined by the product of the Landé g-factor and the hybridization energy. The analytical expressions of the thermoelectric coefficients are also obtained. It is also shown that these transport coefficients oscillate with frequency depending on hybridization energy and Landé g-factor. This paper has the following structure. Section 2 briefly discusses the energy spectrum and the density of states of an ultra-thin TI in the absence and presence of magnetic field. In section 3, we study how the hybridization affects the thermoelectric coefficients in zero magnetic field. In section 4, a complete analysis of thermoelectric coefficients in the presence of magnetic field is provided with numerical and analytical results. We provide a summary and conclusion of our work in section 5.
H = v F(σxpy − τzσypx ) + Δh σz,
where τz = ± denotes symmetric and anti-symmetric surface states, respectively. The energy spectrum of the Dirac electron is given by
E = λ (ℏv Fk )2 + Δ2h .
D0 ( E ) =
2E . π ℏ2v F2
(4)
2.2 Non-zero magnetic field case
In the presence of magnetic field perpendicular to the surface, the Hamiltonian for a Dirac electron with hybridization is H = v F(σxΠy − τzσyΠx ) + (τzΔz + Δh )σz, (5) where Π = p + eA is the two-dimensional canonical momentum operator. Using Landau gauge A ⃗ = (0, Bx , 0), exact Landau levels can be obtained very easily [28, 31]. For n = 0, there is only one energy level, which is given as E0τz = − ( Δz + τzΔz ). When integer n ⩾ 1, there are two energy bands denoted by + corresponding to the electron and − corresponding to the hole with energy Enτ,zλ = λ 2n(ℏωc )2 + (Δz + τzΔh )2 ,
(6) where ωc = vF/l is the cyclotron frequency with l = ℏ / ( eB ) the magnetic length, and Δz = gμBB/2 with g the Landé g-factor. The corresponding eigenstates for a symmetric surface state are Ψn+ ( r ) =
⎛ ⎞ eik yy ⎜ c1ϕn − 1 ( x + x 0 ) ⎟ , L y ⎜⎝ c2ϕn ( x + x 0 ) ⎟⎠
Ψn− ( r ) =
⎛ ⎞ eik yy ⎜ c2ϕn − 1 ( x + x 0 ) ⎟ , L y ⎜⎝ − c1ϕn ( x + x 0 ) ⎟⎠
2.1 Zero magnetic field case
Let us consider a surface of an ultra-thin TI in the xy-plane with Lx × Ly dimensions, where the carriers are Dirac fermions occupying the top and bottom surfaces of the TI. The quantum tunneling between top and bottom surfaces gives rise to hybridization, and consequently the Hamiltonian can be written as the symmetric and anti-symmetric combination of both surface states as [22]
(3)
Here λ = ± stands for electron and hole bands. The density of states is given by
2. Energy spectrum and density of states
⎡ h (k ) 0 ⎤ H=⎢ ⎥, ⎣ 0 h*(k ) ⎦
(2)
(7)
(8) 2 2 where ϕn ( x ) = (1 / π 2nn ! l ) e−x /2l Hn ( x / l ) is the normalized harmonic oscillator wave function, x0 = −kyl2, c1 = cos( θτz / 2) and c2 = sin( θτz / 2) with θτz = tan−1[ n ℏωc / ( Δz + τzΔh )]. The anti-
symmetric surface state can be obtained by exchanging n and n − 1. We have derived an approximate analytical form of the density of states for n > 1 by using the Green's function technique, which is given as (see the appendix)
(1)
2
S K Firoz Islam and T K Ghosh
J. Phys.: Condens. Matter 26 (2014) 165303
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
−0.7
(9)
where Δτz = Δz + τzΔh and Γ0 is the impurity induced Landau level broadening.
κ (nW/K)
0.8
j=4 j=3 j=2
10
20 ∆h (meV)
30
40
The classical Boltzmann transport equation can be used to calculate the zero magnetic field electrical conductivity, which is given as [49] σij ( E ) = e 2τ ( E )
(10)
2
∫ (2dπk)2 δ [ E − E ( k )] vi ( k ) v j ( k ),
where i, j = x, y. For an isotropic vx2 = vy2 = (1 / 2)( vx2 + vy2 ) = (1 / 2) v 2. In our case, v2 =
(15)
system
v F2 ⎡ ⎛ Δh ⎞2 ⎤ ⎟ ⎥. ⎢1−⎜ 2 ⎣ ⎝ E ⎠ ⎦
(16) Using these in equation (15), the energy dependent conductivity becomes E ⎡ ⎛ Δ ⎞2 ⎤ σ ( E ) = e 2τ ( E ) 2 ⎢ 1 − ⎜ h ⎟ ⎥ . (17) πℏ ⎣ ⎝ E ⎠ ⎦ Assuming the energy dependent scattering time to be τ = τ0( E / EF )m, where m is a constant depending on the scat2 + Δ2 tering mechanism, EF = EF0 h is the Fermi energy with EF0 = ℏv FkF0. Here, the Fermi vector kF0 = 2πne . Substituting equation (17) into equation (13), the diffusion thermopower is obtained as
⎤
∫ d E ⎢⎣ − ∂ f∂(EE ) ⎥⎦ ( E − η )(r ) σ ( E ),
(12) where r = 0, 1, 2 and f(E) = 1/[1 + exp(E − η)β] is the Fermi–Dirac distribution function with η the chemical potential and β = ( kBT )−1. Here, σ(E) is the energy-dependent electrical conductivity. When the circuit is open, i.e. for J = 0, the thermopower can be defined as S = Q12/Q11. By using the Sommerfeld expansion in the low temperature regime, the diffusive thermopower S and thermal conductivity κ can be obtained from Mott's relation and the Wiedemann–Franz law as
and
1.0
j=5
Figure 1. Plots of the thermopower versus hybridization constant for m = 1 and for various carrier densities.
(11) Jq = Q 21E + Q 22 ( −∇ T ), where E is the electric field, ∇T is the temperature gradient and Qij (i, j = 1, 2) are the phenomenological transport coefficients. The above equations describe the response of the electronic system under the combined effects of thermal and potential gradients. Moreover, Qij can be expressed in terms of an integral I(r): Q11 = I(0), Q21 = TQ12 = −I(1)/e, Q22 = I(2)/ (e2T) with
⎡ d ⎤ lnσ ( E ) ⎥ S = − L 0eT ⎢ ⎣ dE ⎦ E = EF
j=2
ne = j × 1015/m2
0.4 0
In this sub-section, the effect of hybridization on thermopower and thermal conductivity is presented. We follow the most conventional approach in the low temperature regime. The electrical current density J and the thermal current density Jq for Dirac electrons can be expressed under the linear response regime as
⎡
−1.6
0.6
3.1 Zero-magnetic field case
I (r ) =
j=3
1.2
In this section, we shall calculate thermoelectric coefficients of an ultra-thin TI in zero and non-zero magnetic fields.
and
j=4
−1.3
−1.9
3. Thermoelectric coefficients
J = Q11E + Q12 ( −∇ T )
j=5
−1.0 S (µeV/K)
∞ ⎡ ⎧ ⎛ ⎞2 ⎫ ⎢ 1 + 2 ∑ exp ⎨ − s⎜ 2π Γ0E ⎟ ⎬ 2 2 ⎢⎣ ⎩ ⎝ ℏ ωc ⎠ ⎭ s=1 ⎫⎤ ⎧ × cos ⎨ πs ( E2 −Δ2τz ) / ( ℏωc )2 ⎬ ⎥ , ⎩ ⎭ ⎥⎦
D (E) Dτz ( E ) ≃ 0 2
S = −L0
⎛ Δ ⎞2 ⎤ ⎛ Δ ⎞2 eT ⎡ ⎢ ( m + 1) + 2⎜ h ⎟ ⎥ / 1 + ⎜ h ⎟ . ⎝ EF0 ⎠ ⎥⎦ ⎝ EF0 ⎠ (18) EF0 ⎢⎣
We plot thermopower versus hybridization for different carrier densities in the upper panel of figure 1. It shows that thermopower increases with increasing hybridization for a particular carrier density. However, for higher carrier density, this rate of enhancement with hybridization becomes very slow. Thermal conductivity can be directly obtained from the Wiedemann–Franz law given in equation (14), where the electrical conductivity σ ( EF ) is given as
(13)
κ = L 0Tσ (EF ) . (14) 2 2 2 −8 −2 Here, L 0 = ( π kB ) / (3e ) = 2.44 × 10 WΩK is the Lorentz number and σ (EF ) is the electrical conductivity at the Fermi energy.
3
⎛ Δ ⎞2 σ = σ0 / 1 + ⎜ h ⎟ . ⎝ EF0 ⎠
(19)
S K Firoz Islam and T K Ghosh
J. Phys.: Condens. Matter 26 (2014) 165303
Sxy
25
∆h=0 meV
0.4
∆h=40 meV
0.2 20
0 −0.2
Sxy
15
∆h=10 meV
0.2
10
0 −0.2
0 meV
5
∆h=20 meV
0.4 Sxy
20 meV
κxx
0.4
0
0.2
4
6
8 1/B (1/T)
0
10
12
−0.2 −0.4
2
5
6
7
8
9
Figure 3. Plots of the thermal conductivity in units of kB / h versus
10
inverse magnetic field.
1/B (1/T)
Figure 2. Plots of the thermopower in units of kB/e versus inverse magnetic field for different values of the hybridization constant.
gradient, thermal transport coefficients can be expressed as ℒ (xxr) = ℒ (xxr)col = ℒ (yyr)col and ℒ (yyr) = ℒ (yyr)dif + ℒ (yyr)col = ℒ (yyr)col . In [31], the exact form of the finite-temperature collisional conductivity has been calculated for the screened impurity potential U (k) = 2πe 2 / (ϵ k 2 + k s2 ) ≃ 2πe 2 / (ϵk s ) = U0 under the limit of small |k| ≪ ks, with ks and ϵ being the inverse screening length and dielectric constant of the material, respectively. In this limit, one can use τ02 ≈ πl 2ℏ2 / NI U02, where τ0 is the relaxation time, U0 is the strength of the screened impurity potential and NI is the two-dimensional impurity density. The exact form of the finite-temperature conductivity [31] can be reduced to the zero-temperature energy-dependent electrical conductivity as
Here, σ0 = e2τ0EF0/(πℏ2) is the Drude conductivity without the hybridization constant for the Dirac system. The thermal conductivity is plotted in the lower panel of figure 1. We note that the thermal conductivity is diminished with increasing hybridization. However, unlike the case of thermopower, here thermal conductivity increases with carrier density. 3.2 Non-zero magnetic field case
In the presence of magnetic field, the classical approach can not explain the phenomenon depending on energy quantization. In this sub-section we follow a quantum mechanical approach, based on linear response theory, to study thermal transport coefficients. Thermoelectric coefficients for a twodimensional electron system in the presence of magnetic field were derived by modifying the Kubo formula in [46, 47]. These phenomenological transport coefficients are σμν = ℒ (0) μν
where
(21)
κμν =
1 (1) [ ℒ (2) μν − eT ( ℒ S )μν ], e 2T
(22)
r) ℒ (μν =
⎡
e 2 NI U02 ∑ Inτz, h πΓ0l 2 τ
(24) z where Inτz = [n{1 + cos2(θτz )} − cos(θτz )]. Here we have used −∂f / ∂E = δ[E − Enτz ]. Using equation (23), the finite-temperature diagonal ( ℒ (xxr) ) and off-diagonal coefficients ( ℒ (yxr) ) can be written as
(20)
1 [( ℒ (0) )−1 ℒ (1) ]μν eT
Sμν =
σxx ( E ) =
ℒ (xxr) =
and
⎡ ⎛ ∂f (E) ⎞ ⎤ e 2 NI U02 ⎟⎥ I τz ( E − η )r ⎜ − 2∑ n⎢ ⎝ h πΓ0l n, τ ⎣ ∂ E ⎠ ⎦ E = Enτz (25) z
sin2θ e2 1 ℒ (yxr) = ∑ 2 τz h 2 n, τ Δn z
⎤
∫ dE ⎢⎣ − ∂ f∂(EE ) ⎥⎦ ( E − η )r σμν ( E ).
(23) Here, μ, ν = x, y. Also, σμν(E), Sμν and κμν are the zero-temperature energy-dependent conductivity, thermopower and thermal conductivity tensors, respectively. Generally, diffusive and collisional mechanisms play a major role in electron conduction. The quantized energy spectrum of electrons reveals itself through Shubnikov–de Haas oscillation by the collisional mechanism. In our case, electron transport is mainly due to the collisional instead of the diffusive mechanism. The zero drift velocity of the electrons does not allow us to have a diffusive contribution. In the presence of a temperature
Here,
En + 1, τz
∫E ,τ
n z
⎛ ∂f (E) ⎞ ⎟ dE . (26) ( E − η )r ⎜ − ⎝ ∂E ⎠
⎛ Δ τ ⎞2 ⎛ Δ τ ⎞2 Δn = 2n + ⎜ z ⎟ − 2( n + 1) + ⎜ z ⎟ ⎝ ℏωc ⎠ ⎝ ℏωc ⎠ .
4. Numerical results and discussion In our numerical calculations, the following parameters are used: carrier concentration ne = 2 × 1015 m−2, g = 60, vF = 4 × 105 ms−1 and T = 0.7 K. These numerical parameters are consistent with [15, 30, 31]. 4
S K Firoz Islam and T K Ghosh
J. Phys.: Condens. Matter 26 (2014) 165303
1
0.2
0.8
κxx /10
0.6
0
κxx and Sxy
Sxy
0.1
−0.1
0.4 0.2 Sxy
10
0
κxx
8
−0.2
6 4
−0.4 0
2 5
6
7
1/B (1/T)
8
9
Sxy = − Syx =
⎤ 1 ⎡ σxx (1) ℒ (1) yx ⎢ ℒ xx + ⎥ eT ⎣ S0 σyx ⎦
2
⎧ ⎛ Γ E ⎞2 ⎫ Ω D = exp ⎨ − ⎜ 2π 20 F2 ⎟ ⎬ ⎩ ⎝ ℏ ωc ⎠ ⎭ ⎪
and ℒ (2) xx ≃
Sxy
z
z
z
(32)
σ0T ∑ UτzFτz ( ωcτ0 )2 τz (33) × [1 − 6Ω DG ′′ ( x )cos(2πf / B − τzϕ )],
where the frequency f is given as f =
1 ( EF2 0 −Δ2z ) 2eℏv F2
(34) and the phase factor ϕ = πgμB Δh / ( eℏv F2 ). Therefore, the thermopower and the thermal conductivity oscillate with the same frequency f, which is independent of Δh, which can also be shown from numerical results. The oscillation frequency is strongly reduced by the Zeeman energy Δz. On the other hand, the phase factor φ is related to the product of the Landé g-factor and Δh, and it vanishes if either of them is zero. Although the frequency and the phase of Sxx and κxx are the same, the damping factor and amplitude are different.
z
τz
1/2 ⎛ Δh ⎞2 ⎤ kB 1 16π ⎡ ⎢ ⎥ ≃ Ω DG ′ ( x ) 1+⎜ ⎟ ⎝ EF0 ⎠ ⎥⎦ e kF0l ωcτ0 ⎢⎣ Uτ × ∑ 2 z Fτzsin(2πf / B − τzϕ ) sin ( θτz ) τz
κxx ≃ L 0
∫
∑ Uτ Fτ [1 − 3ΩDG ′′ ( x )cos(2πFτ )],
⎪
and
(28)
4π e 2 Γ0EF ΩDG ′ ( x ) ∑ UτzFτzsin(2πFτz ) (29) β h ( ℏωc )2 τ
L 0e 2T 2σ0 ( ωcτ0 )2
⎪
(31) and the temperature dependent damping factor is the derivative of the function G(x) with G(x) = x/ sinh (x). Here, x = T/Tc, where Tc = ( ℏωc )2 / (2π 2kBEF ) is the critical temperature, which depends on the strength of hybridization through the Fermi energy. Note that G(x) is the temperature dependent damping factor for the electrical conductivity tensor. The off-diagonal thermopower Sxy and the diagonal thermal conductivity κxx are obtained as given by ⎪
Here, S0 = σxxσyy − σxyσyx. The dominating term in the above two equations is the last term. The analytical form of κxx and Sxy can be obtained directly by deriving the analytical form of the phenomenological transport coefficients. The analytical form of the density of states given in equation (9) allows us to obtain asymptotic expressions of Sxy and κxx. This is done by replacing the summation over discrete quantum numbers n by the integration, i.e. ∑ → 2πl 2 Dτz ( E )dE; then we get ℒ (1) xx ≃
100
where Fτz = ( EF2 −Δ2τz ) / ( 2 ℏωc ) , Uτz = [1 + cos2 ( θτz )], the impurity induced damping factor is
(27)
⎤ ℒ (1) 1 ⎡ σxx xx ⎢ ⎥. ( −ℒ (1) xy ) + eT ⎣ S0 σyx ⎦
n
75
Figure 5. Plots of the thermopower and thermal conductivity versus the hybridization constant (Δh). Here, B = 0.2 T is taken.
In figure 2, the off-diagonal thermopower, Sxy, is shown as a function of the inverse magnetic field for different values of the hybridization constant. Similarly, the thermal conductivity, κxx, is shown versus inverse magnetic field for different values of the hybridization constant in figure 3. Careful observation of these two figures clearly shows that Sxy and κxx oscillate with the same frequency, which does not depend on the hybridization energy. The hybridization energy only influences the phase of oscillations. To determine the frequency and the phase of the quantum oscillations in the thermoelectric coefficients, we shall derive analytical expressions of these coefficients. The components of the thermopower are given by
and
50 ∆h (meV)
10
Figure 4. Plots of the numerical and analytical results of the thermoelectric coefficients versus inverse magnetic field for Δh = 20 meV. Solid and dashed lines stand for numerical and analytical results, respectively.
Sxx = Syy =
25
(30) 5
S K Firoz Islam and T K Ghosh
J. Phys.: Condens. Matter 26 (2014) 165303
Appendix
Now we compare the numerical and analytical results for sxy and κxx in figure 4. For better visualization, we have taken weak Landau level broadening Γ0 = 0.01 meV for Sxy and κxx. The analytical results, in particular the frequency f, match very well with the numerical results. We must mention here that for different values of Γ0 analytical results may differ from numerical ones in the amplitude, but the frequency and phase are always in good agreement. It is interesting to see from the analytical expressions of the thermopower and the thermal conductivity that these transport coefficients possess weak periodic oscillation with the hybridization energy for a given magnetic field. This oscillation is shown in figure 5. The frequency and phase factor of these oscillations for a fixed B are ν = ϕ / (2π ) = gμB / (2eℏv F2 ) and Φ = 2πf/B, respectively. In the presence of the magnetic field, these approximate analytical expressions of the thermopower and thermal conductivity can also be used for monolayer graphene by putting Δh = 0. There are several experimental results [42, 48] for thermoelectric properties of a graphene monolayer, but analytical expressions are not available in the literature.
The derivation of asymptotic analytical expression of density of states of a two-dimensional electronic system in the presence of impurities can be done by calculating the self-energy [50, 51], which is given as Σ −(E ) = Γ20 ∑
n
E−
Enτz
1 . − Σ −(E )
(A.1)
The imaginary part of the self-energy is related to the den⎡ Σ -( E ) ⎤ sity of states as D ( E ) = Im ⎢ 2 2 2 ⎥. ⎣ π l Γ0 ⎦ By using the residue theorem, we calculate the summation 2π Γ20E in equation (A.1), which gives Σ − ( E ) ≃ cot( πn 0 ), ( ℏωc )2 where n0 is the pole and is given as ⎡ ⎤ 1 { E − Σ − ( E )}2 − ( Δz + τzΔh )2 ⎥ . (A.2) 2 ⎢ ⎦ 2( ℏωc ) ⎣ By writing the self-energy as the sum of the real and imaginary parts, it can be further simplified as n0 =
5. Conclusion
Δ+i
⎡ ( u − iv ) ⎤ 2π Γ20E Γ ≃ cot ⎢ 2 ⎣ 2 ⎥⎦ 2 ( ℏωc )
u=
⎤ π ⎡ 2 E − ( Δz + τzΔh )2 ⎥ 2⎢ ⎦ ( ℏωc ) ⎣
Here,
We have presented a theoretical study of the thermoelectric coefficients of ultra-thin topological insulators in the presence/absence of magnetic field. In the absence of magnetic field, the thermopower and the thermal conductivity are modified due to the hybridization between top and bottom surface states. The thermopower is enhanced and the thermal conductivity is diminished due to the hybridization. The quantum oscillations in the thermopower and the thermal conductivity for different values of the hybridization constant are also studied numerically. In addition to the numerical results, we obtained the analytical expressions for the thermopower (Sxy) and the thermal conductivity (κxx). The analytical results match very well with the numerical results. We have also provided analytical expressions for the oscillation frequency and phase. The oscillation frequency is the same for both the thermopower and the thermal conductivity. It is independent of the hybridization constant but strongly suppressed by the Zeeman energy. On the other hand, the hybridization constant plays a very significant role in the phase as well as in the amplitude of the oscillations. From the analytical results, critical temperature (Tc) is found to be reduced with increasing hybridization constant. Thermoelectric coefficients also show a very low-frequency oscillation with the hybridization constant for a given magnetic field. Moreover, our analytical expressions of the thermopower and thermal conductivity are also applicable for a graphene monolayer by setting Δh = 0.
(A.3)
(A.4)
and v = π ΓE/(ℏωc)2. The imaginary part is Γ 2π Γ20E sinh v . Now, this can be re-written by = 2 ( ℏωc )2 cosh v − cos u using the following standard relation: ∞
sinh v = 1 + 2 ∑ e−svcos( su ). cosh v − cos u s=1
(A.5)
Here, the most dominant term is for s = 1 only. We can write
∞ ⎤ Γ 2π Γ20E ⎡⎢ −sπ ΓE /(ℏωc )2 ⎥. + e cos( ) = 1 2 u ∑ ⎥⎦ 2 ( ℏωc )2 ⎢⎣ s=1
(A.6)
In the limit of πΓ ≫ ℏωc, after the first iteration, we have Γ / 2 = 2πΓ02E / ( ℏωc )2. Substituting this in the earlier expression, we get ∞ ⎧ ⎛ ⎞2 ⎫ 2π Γ20E ⎡ Γ ⎢ 1 + 2 ∑ exp ⎨ − s⎜ 2π Γ0E ⎟ ⎬ = 2 2 2 ( ℏωc )2 ⎢⎣ ⎩ ⎝ ℏ ωc ⎠ ⎭ s=1
⎪
⎪
⎪
⎪
⎧ ⎫⎤ cos ⎨ sπ ( E2 −Δ2τz ) / ( ℏωc )2 ⎬ ⎥ . ⎩ ⎭ ⎥⎦ ⎪
⎪
⎪
⎪
(A.7)
Here, Δτz = Δz + τzΔh. Finally, the density of states for the two branches can be obtained as ∞ ⎧ ⎛ 2π Γ E ⎞2 ⎫ D0 ( E ) ⎡⎢ 1 + 2 ∑ exp ⎨ − s⎜ 2 0 2 ⎟ ⎬ Dτz ( E ) = 2 ⎢⎣ ⎩ ⎝ ℏ ωc ⎠ ⎭ s=1
Acknowledgments This work is financially supported by the CSIR, Government of India, under grant CSIR-SRF-09/092(0687) 2009/EMR F-O746.
6
⎪
⎪
⎪
⎪
⎧ ⎫⎤ cos ⎨ sπ ( E2 −Δ2τz ) / ( ℏωc )2 ⎬ ⎥ . ⎩ ⎭ ⎥⎦ ⎪
⎪
⎪
⎪
(A.8)
S K Firoz Islam and T K Ghosh
J. Phys.: Condens. Matter 26 (2014) 165303
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