Thermoelectric quantum transport in 3D topological insulators

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Mathematical Physical and Natural Sciences

Master’s Degree in Physics

Thermoelectric quantum transport in

3D topological insulators

Author:

Francesco Guarino

Supervisor:

Fabio Taddei

Academic year: 2015/2016

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Chapter 1 Introduction

The need of providing a sustainable energy to the world population is becoming increasingly important. It is likely that in the following decades the efforts of the scientific community will be focused on this direction and in particular to the heat-to-work conversion processes. An important issue under investigation is the thermoelectric power generation and refrigeration. Thermoelectricity refers to the phenomenon in which temperature differences are directly con-verted into electric voltages (and vice-versa) in solid state systems. Thanks to thermoelectric effects, it is possible to realize small solid state thermal ma-chines, which exploit incoming heat fluxes to produce usable work in a very reliable way and without polluting emissions. However, those devices suffer from low efficiency in heat-to-work conversion with respect to their mechani-cal counterparts.

The energy conversion efficiency of thermoelectric devices is quantified by

a dimensionless quantity known as the figure of merit ZT = (σS2_{T )/(κ}

e+ κl),

where σ is the electric conductivity, S the thermopower which quantifies the

impact of the Seebeck effect, and κe,l the electronic/lattice thermal

conduc-tivity. When ZT → ∞ then the thermodynamic efficiency, defined as the ratio between the output power and the heat currents absorbed, reaches its maximum, that is the Carnot efficiency. Therefore, it is crucial to maximize the figure of merit in order to obtain very efficient thermoelectric devices. However, the coefficients composing the figure of merit are not completely in-dependent, but one needs to find a compromise among them to achieve high efficiencies. The state of the art for the best thermoelectric materials includes

alloys as Bi2Te3, Sb2Te3, Bi2Se3, etc., which reach ZT values of ∼ 1 at room

temperature. However, it is generally accepted that ZT ∼ 3 is a target value for efficient competing thermoelectric technology and, so far, no clear paths exist which may lead to reach that target [1].

A very promising solution to the efficiency problem of solid state thermal devices is to exploit quantum phenomena arising in nanoscopic devices. For instance, resonant systems which realize narrow energy filters are expected to exhibit large heat-to-work conversion efficiency, due to the suppressed thermal conductance. Moreover, the study of thermoelectric properties of nanodevices

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CHAPTER 1. INTRODUCTION 2 is of fundamental importance in many applications, as for example the im-provement of microprocessors operations. When chips are too small (under 100 nm), they begin to get too hot and the heat generated is hard to get rid of [2]. One possibility is to use thermoelectric materials for the refrigeration of such devices.

In this thesis we investigate the potential impact in thermoelectricity of topological insulators. This type of materials represents a new state of matter, in which the bulk behaves as an insulator, while the boundaries as conduct-ing states. In 2D systems they are one dimensional edge states and in 3D they are two dimensional surface states. These boundary states are charac-terized by a linear dispersion in energy (Dirac-like) appearing inside the bulk gap. Moreover, such boundary states have the peculiar characteristic of being topologically protected, which means that they are insensitive to smooth de-formations of the Hamiltonian, so that they can not be destroyed by disorder, for example due to the presence of impurities. The presence of disorder lowers the lattice thermal conductivity, while the topological protection of the surface states against disorder implies that the electronic thermoelectric properties of such states remain unchanged with respect to a clean system. Interestingly, materials that have typically good thermoelectric properties, such as the ones mentioned before, have been discovered to be also topological insulators. Despite this interesting properties, there is very little literature about thermo-electric transport in topological insulators, and in most cases only a semiclas-sical picture is considered [3, 4, 5, 6, 7]. In this thesis we present a theoretical study of thermoelectric quantum transport in 3D topological insulators, in or-der to better unor-derstand this phenomena at the nano-scale and to optimize the efficiency of such systems.

We present in the following a brief outline of the thesis. In Chapter 2 we review the most important theoretical aspects of topological insulators. We first introduce them in an heuristic and intuitive way, then we follow the standard derivation of their most important peculiarity within the topological band theory. In the final two subsections, we show how to derive the effective models Hamiltonians for the time reversal symmetric 2D and 3D topological insulators. The final Hamiltonian of the effective model for the 3D topological

insulators of the Bi2Se3 family crystals, will be the one used in next chapters.

In Chapter 3 we describe first the basic concepts of non-equilibrium ther-modynamics. In the linear response we show how to derive the Onsager matrix, whose elements are related to the thermoelectric coefficients. Then, we intro-duce quantum coherent transport, which is described by the Landauer-Büttiker formalism. We focus on two-terminal systems, composed of a conductor at-tached to two electronic reservoirs, kept at different temperatures and chemical potentials. Through the Landauer-Büttiker formalism we express the currents in terms of the transmission probabilities of electrons through the conductor and show how they are related to the thermoelectric coefficients.

In Chapter 4 we consider a single surface of 3D topological insulator and explore its thermoelectric properties. We first calculate analytically the

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trans-CHAPTER 1. INTRODUCTION 3 mission probability of the surface states in the presence of one potential barrier and study the thermoelectric quantities varying the parameters of the problem. Specifically we focus on the way to obtain a high efficiency of such a system. Then, we introduce a second barrier to see what happens to the transport coefficients and compare them with the one barrier case.

In Chapter 5 we use a more refined full 3D model and calculate numer-ically the transport quantities on a clean slab of 3D topological insulator in order to go beyond our analytical results. Starting from the continuum model Hamiltonian, we derive a tight binding one through a discretization procedure. With such tight binding Hamiltonian we are able to calculate numerically the scattering matrix by using a numerical toolbox which matches the wave func-tions of the leads and scatterer. With this model we simulate a realistic slab of topological insulator. Finally, we introduce disorder into the system and ver-ify the topological protection of surface states, in order to support the results obtained in the case of a clean slab.

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Chapter 2 Topological Insulators

2.1 A brief introduction to topological

insula-tors

A recurring theme in condensed matter physics has been the discovery and classification of distinctive phases of matter. Often, phases can be understood using Landau’s approach, which characterizes states in terms of underlying symmetries that are spontaneously broken. Over the past 30 years, the study of the quantum Hall (QH) effect has led to a different classification paradigm. In the QH effect, the bulk of a two dimensional (2D) sample is insulating, but at the edge of the sample there is a non-zero electric current, which avoids dissipation and give rise to a quantized effect. The QH state provided the first example of a quantum state which is topologically distinct from all states of matter known before. The quantization of the Hall conductivity is explained by the fact that it is a topological invariant, which can only take integer values

in units of e2_{/h, where e is the electron charge and h the Planck constant,}

inde-pendent of material details [8]. The integer value assumed by the conductivity is called the Chern number. This is the most simple example of topological insulator (TI).

Mathematicians have introduced the concept of topological invariance in order to classify broadly geometrical objects, as, for example, 2D surfaces la-beled by the number of holes in them, or genus. The surface of a sphere is topologically equivalent to the surface of an ellipsoid, since both does not have any hole. A coffee cup is equivalent to a donut, because both contain a single hole. The key concept which tie all this arguments is that of smooth deforma-tion. Two objects are topological equivalent if it is possible to deform one into another without creating any hole. In this sense, the surface of a sphere will be never topological equivalent to a donut, or a cup of coffee. The topological invariants do not change under a smooth deformation, and thus represent a topological way of classification. In physics this arguments are directly con-nected to the smooth deformation of a manifold into the Hilbert space, in which is defined the wave function of a many-particle system. Practically, the

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CHAPTER 2. TOPOLOGICAL INSULATORS 5 topological arguments are bound to the Hamiltonian. However, how can we define a smooth deformation of the Hamiltonian? It can be defined as a change in the Hamiltonian which does not close the bulk gap. This definition can be applied to insulators and superconductors with a full energy gap, but not to gapless states as metals. Therefore, one gapped state cannot be deformed into another gapped state in a different topological state class, unless a quantum phase transition occurs. Such a transition becomes evident for instance, at the interface between the QH state and the vacuum, which is a trivial insulator, where gapless edge states appear. Indeed, in order for the topological invariant to change, at the interface the gap must close. In general, if we state that two insulators are topologically equivalent if it is possible to deform their Hamil-tonians smoothly into each other without closing the gap, then when we try to connect by a smooth deformation two topologically inequivalent insulators a gapless state must appear during the transition, otherwise they would be equivalent.

The QH state it is not the only topological class. There exists another large family of materials which belong to another topology class which preserve the time reversal symmetry (TRS). In two dimensions, they are labeled by a dif-ferent topological invariant from that of the QH state, which distinguish all 2D TIs into two distinct classes: the trivial ones, with a simple gap between the conducting and valence states, and the non-trivial ones, with the presence of edge states inside the bulk gap, with the peculiar characteristic of having a linear dispersion of the energy with respect to the wave vector. The concept of TR symmetric TI has been extended also to three dimensional (3D) materials, which have the characteristic presence of surface states. The peculiarity of both 2D and 3D TIs which preserve TR, is that the edge and surface states are helical, in the sense that at a given edge or surface, two states with oppo-site spin counter-propagate. For this reason 2D TIs are also called quantum spin Hall (QSH) insulators. Moreover, the topological non-trivial state pro-tects the robustness of edge/surface states from disorder sources which do not destroy TRS. Actually, this is a specific and unique characteristic of all TIs, which is directly bound to their topological nature. Indeed, the disorder in a non-trivial topological system, if it is not too strong, represents a deformation which affects the bulk states, but cannot affects the gapless ones, because the topological invariant can not change.

A 2D TI has been theoretically predicted [9] and experimentally observed [10] in the CdTe/HgTe/CdTe quantum well (QW). When the thickness of the quantum well is less than a critical value a trivial TI is realized, meanwhile when the thickness is above the critical value a 2D TI is obtained, with opposite spins edge state propagating in opposite directions. Transport measurements in HgTe/CdTe QW [10] show that, among the bulk gap energies, the resistance along the edges directions, in the QSH state, has quantized values in units of

e2/h, confirming the existence of gapless edge states, despite the trivial state

in which has very large values.

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CHAPTER 2. TOPOLOGICAL INSULATORS 6 x [11]. Angled resolved photo-emission spectroscopy (ARPES) measurements has confirmed the existence of topological non-trivial surface states [12].

Sim-pler versions of 3D TIs, has been demonstrated in Bi2Se3, Bi2Te3 and Sb2Te3

compounds with large bulk gaps, and a gapless surface state consisting in a single Dirac cone [13]. ARPES measurements has demonstrated the linear dis-persion of such surface states [14] [15].

In the following sections the features of the topological invariants are dis-cussed more quantitatively through the so called topological band theory, in which electrons are assumed non-interacting, and effective models for 2D and 3D TIs are introduced.

2.2 Topological band theory

We will follow Ando [16] and Kane [17] in order to explain the basics of this theory and find the most important results, the topological nature of the

quan-tum Hall state and the Z2 topological invariant in 2D time reversal invariant

systems and its extension to 3D systems.

2.2.1 Quantum Hall state

Are all electronic states with an energy gap equivalent to the vacuum? The answer is no and the reasons are of topological matter. The simplest example is the QH state, which occurs when the electrons confined in two dimensions are placed in a strong magnetic field B. The quantization of the electrons’

circular orbits with cyclotron frequency ωc leads to quantized Landau levels

with energy m = ¯hωc(m + 1₂), with m integer. If N Landau levels are filled

and the rest are empty, then an energy gap separates the occupied and empty states just like in an insulator [17]. However we want to view the Landau levels as a band structure. In the presence of a magnetic field the momentum is not a

good quantum number, but if we define a unit cell with area _eBhc enclosing a flux

quantum, where c is the light velocity in the vacuum, then lattice translations do commute and a 2D crystal is simulated, and is possible to label the states with the crystal momentum k. This leads to a band structure similar to that of an ordinary insulator. Consider a 2D electron system of size L × L along x and y axes, with a magnetic field B and electric field E along the z and y axes respectively. Our convention is to take the minus sign of the electron charge into e. By treating the effect of the electric field E as a perturbation potential V = −eEy, one may use the perturbation theory to approximate at first order

the n-th eigenstate |niE:

|niE = |ni + X m6=n hm|(−eEy)|ni En− Em |mi, (2.1)

where En and Em are the n-th and m-th eigenvalues. The current density

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where vx is the electron velocity along the x-direction and f (En) is the Fermi

distribution function. The Heisenberg equation of motion vy = _i¯1_h[y, H] leads

to

hm|vy|ni =

1

i¯h(En− Em)hm|y|ni, (2.3)

which allows one to evaluate the Hall conductivity as

σxy = hjxi E = −i¯he 2 L2 X n f (En) X m6=n

hn|vx|mihm|vy|ni − hn|vy|mihm|vx|ni

(En− Em)2

,

(2.4)

where the sum is all over the n occupied bands. We now consider the system in a periodic potential and its Bloch states as the eigenstates, the identity

humk0|v_i|u_nki = 1 ¯ h(Enk− Emk0)humk0| ∂ ∂ki unki, (2.5)

allows us to write Eq. (2.4) as

σxy = − ie2 ¯ hL2 X k X n6=m f (Enk)× × ∂ ∂kx hunk| ∂ ∂ky unki − ∂ ∂ky hunk| ∂ ∂kx unki . (2.6)

let us define the Berry connection for the unk state in the following convenient

form

an(k) = −ihunk|

∂

∂k|unki (2.7)

the Hall conductivity reduces to, for T → 0

σxy = ν

e2

h (2.8)

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CHAPTER 2. TOPOLOGICAL INSULATORS 8 ν = X n Z BZ d2_k 2π ∂a_ny ∂kx − ∂anx ∂ky , (2.9)

where BZ denotes the first Brillouin zone. If we now consider the n-th contri-bution to ν we find νn= Z BZ d2k 2π ∂a_ny ∂kx − ∂anx ∂ky = 1 2π I ∂BZ dk · an(k) = 1 2πγn[∂BZ], (2.10)

where ∂BZ indicates the border of the BZ, while γn[∂BZ] is the phase

ac-quired by the electron after encircling the BZ. Because of the single-valued nature of the wave function, its change in the phase factor after encircling the

Brillouin zone boundary can only be an integer multiple of 2π. ν = P

nνn is

called the Chern number, and represents a topological invariant, which means that it can not change under a smooth deformation of the Hamiltonian of the system which does not close the gap. A fundamental consequence of the topological classification of gapped band structures is the existence of gapless conducting states at interfaces where the topological invariants changes. They may be understood in terms of the skipping motion electrons execute as their cyclotron orbits bounce off the edge [17] (Fig. 2.1). However, more in general they can be understood if one imagines an interface in which a trivial insulator become a topological insulator. Somewhere along the way the energy gap has to vanish in order to allow the topological insulator to change.

2.2.2 Time-reversal symmetry and Bloch Hamiltonian

It is well known that in a periodic system the Bloch’s theorem leads to the form of the wave function

|ψnki = eik·r|unki, (2.11)

where |unki is the cell-periodic eigenstate and satisfy the equation

H(k)|unki = Enk|unki. (2.12)

It is important to note that when H(k) preserves time-reversal symmetry, [H(k), Θ] = 0 and

H(−k) = ΘH(k)Θ−1, (2.13)

where Θ is the time reversal operator. This last identity means that at a given energy there are two distinct states at ±k. They are called Kramers pairs.

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CHAPTER 2. TOPOLOGICAL INSULATORS 9

Figure 2.1: One way to understand the edge states in the quantum Hall effect, through the skipping motion of the electrons when their orbits bounce off the edge.

Due to the periodicity of the BZ there exist TR-invariant momentum (TRIM) where the Kramers pairs become degenerate and +k becomes equivalent to −k. It is convenient to define a matrix ω(k), whose elements are

ωαβ(k) = huα,−k|Θ|uβ,ki, (2.14)

where α and β are band indices. The two Bloch states are though related as

|uα,−ki =

X

β

ω∗_αβ(k)Θ|uβ,ki. (2.15)

The matrix ω(k) is unitary and has the following property:

ωβα(−k) = −ωαβ(k). (2.16)

This equation means that at a TRIM point Λ the ω matrix becomes antisym-metric:

ωβα(Λ) = −ωαβ(Λ). (2.17)

Now because of the two Kramers pairs we have to define the Berry connection matrix a(k) whose elements are

aαβ(k) = −ihuαk|∇k|uβki, (2.18)

which is a generalization of the Berry connection. Using the anti-linear prop-erty of Θ and using Eq. (2.15) it can be shown that

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Noticing that tr[a] = tr[a∗] (because aβα = a∗αβ), and that ω∇ω† = −(∇ω)ω†

(because ω is unitary) then we obtain

tr[a(k)] = tr[a(−k)] + itr[ω†(k)∇kω(k)]. (2.20)

These properties are important in the calculation of the Z2 topological

invari-ant.

2.2.3 Z

2

topological invariant

In order to derive the Z2 topological invariant for a 2D electron system

preserv-ing TRS, we will consider first a TR-invariant 1D periodic system, with length L, lattice constant a = 1, periodic boundary conditions and two Kramers pairs

|u1(k)i and |u2(k)i. For simplicity we assume that there are no degeneracies

other than those required by TRS, and that the band parameters change with time and return to the original values at t = T . Then, the Hamiltonian satisfies the following properties:

H[t + T ] = H[t] (2.21)

H[−t] = ΘH[t]Θ−1. (2.22)

The charge polarization Pρ (in units of e) is defined, up to a lattice constant,

by integrating the Berry connection of the occupied states over the BZ

Pρ=

Z π

−π

dk

2πA(k), (2.23)

where A(k) = tr[a]. For the boundary conditions this integral is a loop over the BZ and we can see that the space (k, t) on which is defined the wave function is a torus. If after one period T we take the difference of the two polarizations calculated in t = T and t we obtain:

P(T ) − P(t) = 1 2π hI cT dkA(t, k) − I ct dkA(t, k)i, (2.24)

where cT (t) is the loop in the BZ at time T (t). It is possible to write the

integrals in terms of the Berry curvature

B(k, t) = i X

i=1,2

h∇tui(k, t)|∇kui(k, t)i + c.c., (2.25)

over the surface τ of the torus spanned by k and t as

Pρ(T ) − Pρ(t) = 1 2π Z τ dtdkB(k, t). (2.26)

This integral over the torus surface is the Chern number of the torus and takes an integer value, which is equal to zero. Physically can be understood

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by the periodic boundary conditions which require that Pρ(T ) − Pρ(t) = 0.

Thus apparently such an insulator seems to be trivial, but actually there exist another topological invariant. The contribution from each band to the total

polarization Pρ is Pi = Z π −π dk 2πaii(k) (i = 1, 2). (2.27)

The time-reversal polarization is defined by

Pθ = P1− P2 = 2P1− Pρ, (2.28)

which intuitively gives the difference in charge polarization between spin-up and spin-down bands, if the spin projection on z is a good quantum number.

We are interested in the explicit calculation of Pθ in the TR-invariant points,

which for the periodic boundary conditions are at k = 0, π and t = 0, T /2. The system is TR symmetric and for any k, at a fixed time and up to a phase term, we have for the Kramers theorem

Θ|u2(k)i = −e−iχ(k)|u1(−k)i (2.29)

Θ|u1(k)i = e−iχ(−k)|u2(−k)i. (2.30)

The second equation follows from the first, along with the property Θ2 = −1.

Then the ω matrix becomes ω(k) = 0 e−iχ(k) −e−iχ(−k) ₀ . (2.31)

Substituting the wave functions written in Eqs. (2.29), (2.30) in the expressions

for P1 ( Eq. (2.27) with i = 1) and Pρ we obtain for Pθ

Pθ = Z π 0 dk 2π[A(k) − A(−k)] − i πlog ω12(π) ω12(0) . (2.32)

Using Eq. (2.20), we finally obtain

Pθ = 1 iπlog pω₁₂(0)2 ω12(0) · ω12(π) pω12(π)2 . (2.33)

We can see that the log argument can be ±1. Since log(±1) = iπm, with

m even/odd, we can see that Pθ can be expressed only in mod 2. Physically,

the two values corresponds to two different polarizations states of the system which occurs at time t. Because the polarization is defined up to a lattice constant, we are more interested in the variation of the polarization which is well defined. Therefore, at the TR-invariant times t = 0, T /2 we calculate the difference in the half cycle 0, T /2

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∆ = Pθ(T /2) − Pθ(0). (2.34)

where ∆ denotes a Z2 topological invariant specified in mod 2. When ∆ = 0,

Pθ does not change in the half cycle, and consequently in the all cycle for the

periodic boundary conditions on t, the system remains in the insulating state

in which P1− P2 is equal to an even number. On the contrary, with ∆ = 1, the

time reversal polarization changes in the half cycle, leading to an insulating

state in which P1 − P2 is equal to an odd number. The two insulating states

are not equivalent because the change in the polarization of an odd number leads to the removal of Kramers degeneracy during the cycle in certain TRI points, which one can see as the presence of an unpaired spin, if the spin is a good quantum number, at the end of the 1D system (because far from the edges the unpaired spin would couple with another flipped). Consequently, it is not possible to deform the Hamiltonian smoothly preserving TRS and

contemporaneously changing the polarization Pθ. In this sense the two states

are topologically different. From Eq. (2.33) is possible to find

(−1)∆= 4 Y i=1 ω12(Λi) pω12(Λi)2 , (2.35)

where Λ1 = (0, 0), Λ2 = (π, 0), Λ3 = (0, T /2) and Λ4 = (π, T /2) as in Fig. 2.2.

In general if we have 2N occupied bands then there will be N Kramers pairs and the ω will be an antisymmetric 2N × 2N matrix. The previous formula for ∆ become (−1)∆= 4 Y i=1 Pf[ω(Λi)] pdet[ω(Λi)] , (2.36)

where the Pfaffian is defined for an antisymmetric matrix A and is related to the determinant as:

Pf[A]2 = det[A]. (2.37)

By reinterpreting the periodic space (k, t) as the 2D Brillouin zone (kx, ky),

this theory provides a Z2 topological classification of 2D TR-invariant

insula-tors with 2N occupied bands. The direct consequence of passing from a trivial

insulator to a non-trivial one characterized by the Z2 invariant, is the

appear-ance of an odd number of edge states, whose propagation is determined by the spin, or the pseudospin, if the spin is not a good quantum number.

2.2.4 Extension to 3D systems

3D systems are characterized by four Z2 topological invariants. Their physical

origin might be understood easily. For simplicity consider a cubic system with lattice constant a = 1 (Fig. 2.3). In the BZ there are eight TRIMs denoted

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Figure 2.2: The TRIMs for a TR-symmetric 1D system.

six plane in x = 0, x = ±π, y = 0, y = ±π, z = 0 and z = ±π which have

a Z2 invariant. But if we see Eq. (2.36), only four invariants are independent

because the products ∆(x=0)∆(x=π), ∆(y=0)∆(y=π), ∆(z=0)∆(z=π) are redundant.

For each TRIMs Λi we define [16]

δ(Λi) ≡

Pf[ω(Λi)]

pdet[ω(Λi)]

, (2.38)

and then we define the four Z2 invariants as

(−1)ν0 ₌ Y nj=0,π δ(Λn1,n2,n3,) (2.39) (−1)νi ₌ Y nj6=i=0,π;ni=π δ(Λn1,n2,n3,) (i = 1, 2, 3). (2.40)

The invariant ν0is unique for a 3D system and specifies whether the topological

insulator is strong (ν0 = 1) or weak (ν0 = 0). The other three invariants can

be seen as Miller indices to specify the direction of vector Λi in the reciprocal

space. In the transition between a trivial and non-trivial insulator, the gap is closed by an odd number of surface states, in the case of a strong TI, or even number, in the case of a weak one. The difference with the 2D TI , in which an even number of edge states indicates a trivial state, is due to the fact that the 3D TIs are characterized by four indices, which determines a more specific classification.

2.3 Effective models

Now we want to discuss more in detail the topological insulators’ properties. In this section we will see how it is possible to find effective Hamiltonians de-scribing 2D and 3D TRS systems in the topological non trivial phase, showing the existence of topologically protected edge and surface states.

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Figure 2.3: The TRIMs for a TR-symmetric 3D system.

2.3.1 Quantum spin Hall state

It has been shown that the topological non trivial phase in a 2D system pre-serving time reversal symmetry is possible when the p-orbital like and s-orbital like bands of the normal insulator are switched near the Γ point. This is well described by the BHZ model [9] in which is considered a CdTe-HgTe-CdTe

quantum well (QW). If the QW thickness is beyond a critical thickness dc,

the heterostructure bands are inverted and a non trivial topological insulator is obtained. More in details, the HgTe band structure is inverted due to the strong spin-orbit coupling (SOC) interaction, while in CdTe remains normal (Fig. 2.4). When the thickness is under a certain value the confinement is strong and the heterostructure behaves similarly to CdTe, while for thickness larger than the critical value the confinement is weak and the heterostructure behaves like HgTe.

In the following we will follow Qi and Zhang [8], for the derivation of the effective model for this system. Under our assumption of TR symmetry the two bands must split into Kramers pairs, thus we will express the effective

Hamiltonian in the basis {|E+i, |H+i, |E−i, |H−i}, where H and E are the

va-lence and the conduction subbands in the normal order. The states |E±i and

|H±i have opposite parity; hence a Hamiltonian matrix element that connect

them must be odd under parity. Thus to lowest order in k (|E+(−)i, |H+(−)i)

will each be coupled by a linear term in k. In order to preserve rotational symmetry around the growth axes, diagonal terms must be proportional to

even powers of k± = kx± iky. The others non diagonal terms must be zero

in order to protect TR-symmetry and not to induce higher order processes coupling different ± states and splitting degeneracy in the TRIM point. Then

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Figure 2.4: Bulk band structure of HgTe and CdTe.

the final effective Hamiltonian will be:

H =h(k) 0

0 h∗(−k)

(2.41)

h(k) = (k) + da(k)σa, (2.42)

where σa _{are the Pauli matrices and}

(k) = C − D(k_x2+ k2_y),

d(k) = [Akx, −Aky, M(k)],

M(k) = M − B(k2

x+ ky2),

(2.43)

where A, B, C, D and M are material’s parameters that depend on the quan-tum well geometry. We choose the zero energy to be the valence band edge of HgTe at k = 0 (C = 0). The energy bulk spectrum is given by

E± = (k) ± p dada = (k) ± q A2_(k2 x+ ky2) + M2(k). (2.44)

In this last equation is evident the different mass terms, with M being the Dirac mass and B being the kinetic classic term, which is negative in this case, in order to have a positive kinetic energy. Considering the z-axes along the

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CHAPTER 2. TOPOLOGICAL INSULATORS 16 growth direction, we calculate the explicit solution of the Hamiltonian (2.42) in the half space x > 0, in the xy plane. Firstly we can see that the Hamiltonian

is divided in two parts, one kx-dependent and the other ky-dependent,

H = H0(kx) + H1(ky). (2.45)

In this system, translational symmetry along the y-direction is preserved, so

ky is a good quantum number. For ky = 0 we have H1 = 0, and replacing kx

with −i∂x we have the following eigenvalue equation

H0(kx → −i∂x)Ψ(x) = EΨ(x). (2.46)

Since the Hamiltonian is block diagonal, the eigenstates have the form

Ψ↑(x) = ψ₀ 0 , Ψ↓(x) = 0 ψ0 , (2.47)

where 0 and ψ0 are two-components vectors, and Ψ↑(x) is related to Ψ↓(x) by

TR. Now all we have to do is to calculate explicitly ψ0 through the equation

" ˜ (−i∂x) + _˜ M (−i∂x) −iA∂x −iA∂x − ˜M (−i∂x) # ψ0(x) = Eψ0(x), (2.48)

where ˜(−i∂x) = C − D∂x2 and ˜M (−i∂x) = M − B∂x2. In order to show the

existence of the edge states we will neglect ˜ for simplicity (which is equivalent

to set C = D = 0), but the same result is also achievable by not neglecting it.

Without ˜ the wave equation has particle-hole symmetry, therefore we expect

an edge state with E = 0 can exist. With the wave function ansatz ψ0 = φeλx

the equation becomes

[(M + Bλ2)σyφ = −Aλφ (2.49)

where φ is a two-components spinor and must be an eigenstate of σy. Defining

φ± by σyφ± = ±φ± Eq. (2.49) is simplified to a quadratic equation for λ. If

λ is a solution for φ+, then −λ is a solution for φ−. Consequently, the general

solution is given by ψ0(x) = (aeλ1x+ beλ2x)φ−+ (ce−λ1x+ de−λ2x)φ+ (2.50) where λ1,2= 1 2B A ± √ A2_{− 4M B.} _(2.51)

The coefficients a, b, c and d are determined by imposing the open boundary

condition ψ0(0) = 0. Together with the normalization of the wave function

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<(λ1,2) < 0 (c = d = 0) or <(λ1,2) > 0 (a = b = 0). It can be seen that this

conditions are achievable only in the inverted regime when M/B > 0. Thus, for the edge states at the Γ point we have the solutions

ψ0(x) = ( a(eλ1x− eλ2x_)φ − A/B < 0, c(e−λ1x− e−λ2x_)φ + A/B > 0. (2.52) The sign of A/B determines the spin polarization of the edge states and

lc = max{_|<(λ1_1,2_)|} determines the decaying length of the edge state. The

effective model can be obtained by projecting the bulk Hamiltonian onto the

edge states Ψ↑ and Ψ↓. This simple procedure define an effective 2×2 effective

Hamiltonian for the edge states H_edgeα,β(ky) = hΨα| ˆH0+ ˆH1|Ψβi. In the leading

order in ky we obtain

Hedge = ˜Akyσz. (2.53)

where ˜A ' 3.6 eV Å for HgTe QWs, with a Dirac velocity for the edge state

v = ˜A/h ' 5.5 × 105 m/s [8]. ˜A represents the expectation value of A on the

edge states. From this calculations we can see that there exist at energy E = 0

and for small ky massless Dirac edge states which are protected by TRS. From

Eq. (2.52) we can see that the propagation of the edge states is completely determined by the spin (pseudo-spin), which gives them the name of helical. This is the reason why the 2D TI which preserves TRS is also called QSH state. In realistic materials all the parameters must be considered, but the

form of ψ0 remains valid.

2.3.2 Effective model of a 3D system

In this section we will give some brief details about the derivation of the

ef-fective Hamiltonian valid for the 3D TIs Bi2Se3, Bi2Te3 and Sb2Te3. These

material are insulators and all belong to the same rhombohedral crystal

struc-ture with space group D5_3d and five atoms per unit cell. In order to get a

physical picture of the band structure of these materials, we will focus on

Bi2Se3 crystal structure (Fig. 2.5) and follow Liu [18]. The crystal has a

lay-ered structure along the z direction with five atoms (two Bi and three Se) in

one unit cell, including two equivalent Se atoms (Se1 and Se10), two equivalent

Bi atoms (Bi1 and Bi10), and one Se atom (Se2) which is inequivalent to the

Se1 and Se10 atoms. Therefore five atomic layers can be viewed as one unit,

which is usually called a quintuple layer. The coordinate is set as the follow-ing: the origin is set at the Se2 site; the z direction is set perpendicular to the atomic layer, the x direction is taken along the binary axis with twofold rotation symmetry, and the y direction is taken along the bisectrix axis, which is the crossing line of the reflection plane and the Se2 atomic layer plane.

In order to get a physical picture of the band structure, we start from

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Figure 2.5: Bi2Se3 crystal structure. ~t1,2,3 are the primitive lattice vector, given by

~t1 = (

√

3a/3, 0, c/3), ~t2 = (−

√

3a/6, a/2, c/3) and ~t3 = (−

√

3a/6, −a/2, c/3), where a is the xy plane lattice constant and c is the z lattice constant.

and that of Se is 4s2_4p4_{. The outermost shells for both Bi and Se are p}

or-bitals, thus it seems natural to consider only these orbitals and neglect the others. The chemical bonding is very strong within one quintuple layer but two neighboring quintuple layers are only coupled by the van der Waals force. Therefore it is reasonable to first focus on one quintuple layer. Within one quintuple layer there are fifteen orbitals, due to the fact that each atom has

three p orbitals (px, py, pz). All the Se layers are separated by Bi layers, thus

the strongest coupling in this system is between the Se layer and the Bi layer. Such coupling causes level repulsion, so that the Bi energy levels are pushed

up and form new hybridized states |Bi and |B0i, while the Se energy levels

are pushed down and yield three states |Sαi, |S

0

αi and |S0αi as shown in Fig

2.6 (I), with α = px, py and pz. Since the system has inversion symmetry, it

is convenient to define the bonding and anti-bonding states with the definite parity |P 1±, αi = √1 2(|Bαi ∓ |B 0 αi) |P 2±, αi = √1 2(|Sαi ∓ |S 0 αi) (2.54)

They are split when coupling between Bi and Se atoms is taken in account, with the antibonding state having higher energy (Fig. 2.6 (II)). Now we focus

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CHAPTER 2. TOPOLOGICAL INSULATORS 19 Bi Se Sx,y,z S_x,y,z0 S0x,y,z Bx,y,z B_x,y,z0 P 2+_x,y,z P 2−_x,y,z P 1−_x,y,z P 1+ x,y,z P 2−_z P 2−_x,y P 1+_z P 1+ x,y P 2−_x+iy,↑, P 2−_x−iy,↓ P 2−_x+iy,↓, P 2−_x−iy,↑ P 1+_x+iy,↑, P 2+_x−iy,↓ P 1+_x+iy,↓, P 2+_x−iy,↑ P 2−_z,↑, P 2−_z,↓ P 1+_z,↑, P 1+_z,↓ I II III IV

Figure 2.6: Schematic picture of the evolution of Bi and Se p orbitals in the four stages cited in the text

our attention on the states nearer to the Fermi surface, |P 1+, αi and |P 2−, αi

and neglect the other states. The z direction is different from the x or y

directions in the atomic plane. Thus there is an energy splitting between pz

and px,y orbitals. It has been found that for antibonding state pz has higher

energy, while for bonding state px,y has higher energy. Thus the conduction

band mainly consists of |P 1+, pzi orbital, while the valence band is dominated

by the |P 2−, pzi orbital (Fig. 2.6 (III)). Finally we consider the SOC effect,

which gives a further splitting. As shown in Fig. 2.6 (IV) SOC introduces a repulsion between z orbitals and x, y orbitals. If the SOC constant is larger than a critical value [8] the order of z orbitals is inverted, and this indicates that the material is in the non trivial topological phase, similar to the case of

HgTe QW. Bi2Se3is a strong topological insulator and its indices are (1, 0, 0, 0).

This means that a topologically protected surface state exists as we will see later in this chapter.

Generally, any 4 × 4 Hamiltonian can be expanded with Dirac Γ matrices as ˆ Hef f = (k)I + X i di(k)Γi+ X ij dij(k)Γij (2.55)

where the five Dirac matrices Γi(i = 1, 2, 3, 4, 5) satisfy {Γi, Γj} = δij and the

ten commutators are defined by Γij = [Γi, Γj]/2i. (k), di(k) and dij(k) can be

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is written in the basis {|P 1+_z,↑i, |P 2−_z,↑i, |P 1+_z,↓i, |P 2−_z,↓i}, basically because the

most relevant physics of this TIs family is collected in these four states. In this basis we can express the symmetries of the system as following.

1. Time reversal : T = ΘK, where Θ = iσ2 ⊗ 1 and K is the conjugate

operator,

2. Threefold rotation around the z direction: R3 = e

i

2(σ3⊗1)θ with θ = 2π/3,

3. Twofold rotation around x direction: R2 = iσ1⊗ τ3,

4. Inversion: P = 1 ⊗ τ3.

In the above ~σ acts on the spin basis, while ~τ on the orbital basis. Now,

defining Γi matrices as Γ1 = σ1⊗ τ1, Γ2 = σ2⊗ τ1, Γ3 = σ3⊗ τ1, Γ4 = 1 ⊗ τ2, Γ5 = 1 ⊗ τ3, (2.56)

it is possible to obtain the space group irreducible representation of each Γ matrix through the former transformations. Then, due to the commutation between the Hamiltonian and the symmetries operators previously defined, the

function di(k) [dij(k)] must have the same behavior to the corresponding Γi

[Γij] matrix under symmetry operations, which means they should belong to

the same space group representation. More details are given by Liu [18]. The final result is, up to the second order,

Hef f = (k) + M(k)Γ5+ B0Γ4kz+ A0(Γ1ky − Γ2kx), (2.57)

where

(k) = C0+ C1k2z+ C2k2k, (2.58)

M(k) = M0+ M1kz2+ M2k2k, (2.59)

with kk = k2x + ky2. The coefficients in the last two equations are material’s

parameters which can be changed in order to represent the band structure

of Bi2Se3, Bi2Te3 and Sb2Te3. The topological phase is evident when surface

states appear at the interface with the vacuum. Consider the Hamiltonian (2.57) in the half space given by z > 0. We can divide our Hamiltonian in two parts

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CHAPTER 2. TOPOLOGICAL INSULATORS 21 In the xy plane the system is infinite and invariant under translation

transfor-mations, thus kx and ky are good quantum numbers. This does not happen

in the z direction, therefore we replace kz by −i∂z and obtain the eigenvalue

equation

H0(−i∂z)Ψ(z) = EΨ(z). (2.61)

Since the Hamiltonian H0 is block diagonal, we are in the same situation of

Eq. (2.46), with the only difference that now the spinor composing ψ0 is a τ1

eigenvalue. Thus, the solutions for surface states, neglecting , are of the same form of Eq. (2.52) and can be written as

ψ0(z) = ( a(eλ1z _{− e}λ2z_)φ + B0/M1 > 0, c(e−λ1z_{− e}−λ2z_)φ − B0/M1 < 0. (2.62)

with a, c being some constants, lc = max{_|<(λ1_1,2_)|} being the decaying length

of the surface states and

λ1,2 = 1 2M1 − B0± q 4M0M1+ B02 , (2.63)

where it is clear that only if M0 < 0 surface state solutions are possible, which

means that the bands near the Fermi energy are inverted and the system is in the non-trivial topological phase. Using the following relations

hΨ|Γ1,2|Ψi = α1σx,y,

hΨ|Γ5|Ψi = α3,

(2.64)

where α1 = hψ0|τ1|ψ0i and α3 = hψ0|τ3|ψ0i, the effective Hamiltonian for the

surface states defined as H_surfα,β (kx, ky) = hΨα|H0+ ˆH1|Ψβi leads to

Hsurf = α3M2k2k+ α1A0(σxky − σykx). (2.65)

In the limit k → 0 the linear term dominates, and this is the feature of the Dirac surface state in 3D topological insulators. As said in Subsection 2.3.1, although this is a qualitative way to understand the surface states, the form of

ψ0 remains valid. In this work we have used this final Hamiltonian to study

important features and thermoelectric properties of 3D topological insulators belonging to this family materials.

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Chapter 3 Non-Equilibrium thermodynamics

and quantum coherent transport

In this chapter we illustrate the formalism of non-equilibrium thermodynamic and quantum coherent transport. Non-equilibrium thermodynamics describes the behavior of systems at non-equilibrium thermodynamical states. With quantum coherent transport instead, it is possible to describe the transport at the nano-scale, when the conductor’s length is too small to satisfy an ohmic behavior. In this case the important reference length is the mean free path, which represents the length under which the electron momentum is preserved by the interaction with impurities, phonons or other electrons. Another refer-ence length is the phase relaxation length, which is the distance beyond which the electron’s phase is randomized. We assume to be under these lengths and that the electrons do not undergo interactions with phonons.

3.1 Non-Equilibrium Thermodynamics

3.1.1 Basic concepts

For the description of non-equilibrium thermodynamics we will follow Ref. [1]. The basic idea of non-equilibrium thermodynamics is that we can describe a system in terms of the forces which take it out of equilibrium, and the fluxes, which are established in order to compensate the perturbation done by the forces. Since the entropy is maximum at the equilibrium, we can define for

every extensive quantity dependent on time ηi(t), the related fluxes Ji and

forces Xi as Ji = dηi dt , (3.1) Xi = ∂S ∂ηi , (3.2) 22

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CHAPTER 3. QUANTUM COHERENT TRANSPORT 23

where S(η1, η2, ..., ηn) is the entropy. Differentiating the entropy with respect

to the time we obtain the entropy production rate which reads ∂S ∂t = X i dηi dt ∂S ∂ηi =X i JiXi. (3.3)

Consider the system shown in Fig. 3.1, connected to two reservoirs at

temper-ature TL, TR and chemical potentials µL, µR, with TL> TR and µL> µR. The

work W done by the system against an external force F is W = −F x, where

x is the conjugate variable. The heat coming from the hottest reservoir is QL,

while the heat absorbed by the coldest reservoir is −QR. Since we are

consid-ering steady states with no variation of the internal energy the first principle reads

W + QL− QR= 0. (3.4)

The entropy production rate is ˙ S = Q˙L TL −Q˙R TR , (3.5)

where the dot indicates the time derivative. Using the first principle we

elim-inate Q2 obtaining ˙ S = Q˙L TL − Q˙L TR − W˙ TR . (3.6)

Using Eqs. (3.1) and (3.2) we write the force X1 and the associated flux J1 as

X1 =

F

TR

, (3.7)

J1 = ˙x. (3.8)

In the system in Fig. 3.1 the electromotive force is F = ∆V = (µL− µR)/e,

and the flux J1 is the charge flux transfered from the reservoirs with the highest

chemical potential. In analogy with the force and flux X1, J1, we can define

the force X2 and the associated heat flow J2 as

X2 = 1 TR − 1 TL , (3.9) J2 = ˙QL (3.10)

3.1.2 Linear response and Onsager matrix

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TL, µL S TR, µR

Figure 3.1: A system S is in contact with two reservoirs at temperatures TL, TR

and chemical potentials µ_L, µ_R. We assume T_L > TR and µL > µR. The voltage

difference is hence, ∆V = (µ_L− µ_R)/e.

Ji ≡ Ji(X1, X2, ..., Xn, λ), (3.11)

where λ denotes state variables as temperature, pressure, chemical potential, etc.. Onsager proposed in [19] to expand the fluxes around the forces at the equilibrium, which for definition are zero as the fluxes. Thus at the first order, the fluxes are written as

Ji ∼ X j ∂Ji ∂Xj Xj=0 Xj. (3.12) The coefficients Lij = _∂X∂Ji j Xj=0

are defined as the Onsager coefficients, which form a n × n matrix called the Onsager matrix. We also stress that we focus on stationary and steady state situations, where all forces and currents are independent of time, on average, apart from fluctuations.

For what concern the system in Fig. 3.1, the linear response is justified when only small perturbations are done by the forces. Quantitatively, we consider a

small perturbation as µL = µ, µR ∼ µ and TL = T , TR ∼ T . Thus the forces

read X1 = ∆µ eT = ∆V T , (3.13) X2 = ∆T T2 . (3.14)

where ∆µ = µL− µR, ∆µ/µ 1 and ∆T = (TL− TR), ∆T /T 1. Hence,

the relation between the forces and the fluxes in the linear response are (

J1 = L11X1+ L12X2,

J2 = L21X1+ L22X2.

(3.15) Imposing positive entropy production rate for the second principle of thermo-dynamics, we have

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CHAPTER 3. QUANTUM COHERENT TRANSPORT 25 ˙ S = X1J1+ X2J2 = = (L11X1+ L12X2)X1+ (L21X1+ L22X2)X2 = = L11X12+ L22X22+ 2L12L21X1X2 ≥ 0. (3.16)

and we obtain the following conditions for the Onsager coefficients:

L11≥ 0, (3.17) L22≥ 0, (3.18) L11L22− 1 4(L12+ L21) 2 _{≥ 0.} (3.19) If time-reversal symmetry is assumed, Onsager [19] derived fundamental rela-tions known as the Onsager reciprocal relarela-tions for the cross coefficients of the Onsager matrix:

Li,j = Lj,i. (3.20)

So that, Eq. (3.19) becomes det[L] ≥ 0. The Onsager coefficients are related to the familiar transport coefficients. In the case of thermoelectricity we have

G = J1 ∆V ∆T =0 = L11 T , (3.21) Ξ = J2 ∆T J1=0 = det[L] T2_L 11 , (3.22) S = −∆V ∆T J1=0 = L12 T L11 , (3.23) Π = J₂ J1 ∆T =0 = L21 L11 , (3.24)

where G is the (isothermal) electrical conductance, Ξ the thermal conductance, S the thermopower (or Seebeck coefficient), and Π the Peltier coefficient.

3.1.3 Thermoelectric efficiencies and figure of merit

If the power P = W = −T X˙ 1J1 and the heat flux ˙Q1 = J2, are positive

quantities, within the linear response we define the efficiency of steady state heat to work conversion as

η = W˙ ˙ Q1 = −T X1J1 J2 = −T X1(L11X1+ L12X2) L21X1+ L22X2 . (3.25)

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CHAPTER 3. QUANTUM COHERENT TRANSPORT 26 X₁max = L22 L21 s det[L] L11L22 − 1 ! X2. (3.26)

For systems with time-reversal symmetry L12 = L21, substituting Eq. (3.26)

into Eq. (3.25), we have

ηmax = ηC

√

ZT + 1 − 1 √

ZT + 1 + 1, (3.27)

where ηC = X2T = ∆T_T is the Carnot efficiency, and the figure of merit

ZT = L

2 12

det[L] (3.28)

is a dimensionless parameter. We can see the meaning of the figure of merit

from Eq. (3.27): when ZT → 0, ηmax → 0, while when ZT → ∞, ηmax → ηC.

As function of G, Ξ and S the figure of merit reads

ZT = GS

2

Ξ T. (3.29)

For the positiveness of the entropy production rate, ZT ≥ 0, and ηmax is a

monotonous growing function of ZT . To reach the Carnot efficiency, ZT must diverge. This happens only when the Onsager matrix is ill-conditioned, that is for det[L] = 0. Under this condition and time-reversal symmetry, using Eq.

(3.15), the equation for J2 becomes

J2 = X1L21+ X2 L2₁₂ L11 = = X1L21+ L12 L11 (J1− X1L11) = = L12 L11 J1 = ΠJ1. (3.30)

The maximum efficiency is achieved only when the energy flow is proportional to the charge flow through the Peltier coefficient. This is also known as the tight coupling condition, which means that all the heat produced is converted in work.

The output power

P = −T X1J1 = −T X1(L11X1+ L12X2) (3.31) is maximal when X₁0 = − L12 2L11 X2 (3.32) and is given by

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CHAPTER 3. QUANTUM COHERENT TRANSPORT 27 Pmax = T 4 L2 12 L11 X₂2 = ηC 4 L2 12 L11 X2. (3.33)

Using Eqs. (3.21) and (3.23) we can also write

Pmax=

1

4S

2_{G(∆T )}2_. _(3.34)

We can see from this last equation that the maximum power is directly set by

the combination S2_{G, at fixed ∆T , known for this reason as the power factor}

Q. Note that P reaches its maximum value when X1 is equal to half of the

stopping force,

X₁stop = −L12

L11

X2, (3.35)

that is, to the value for which J1 = 0. For system with time reversal symmetry

the efficiency at maximum power is

η(Pmax) = Pmax Q1(X10) = = ηC 4 L2 12 L11 X2 1 L2 12 2L11X2+ L22X2 = ηC 2 L2 12 L22L11+ detL = ηC 2 ZT ZT + 2. (3.36)

We can see from this last equation that for ZT → 0, η(Pmax) → 0, while

when ZT → ∞, η(Pmax) → ηC/2. In Fig. 3.2 we can appreciate the different

behaviors of ηmax and η(Pmax). For ZT ∼ 0 ηmax ∼ η(Pmax), while as ZT

increases its value, the difference between the two efficiencies becomes greater. The figure of merit is a parameter of great interest in the thermoelectricity because is completely determined by the transport coefficients. It allows to predict the behaviors of the maximum efficiency achievable and the efficiency at the maximum output power. The latter is of crucial importance since for low ZT , it is very near to the maximum efficiency achievable, and one in principle could set the system in order to reach the maximum output power and at the same time the maximum efficiency. In Chapters 4 and 5 we will calculate in a two-terminal system the figure of merit analytically for a single surface of

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Figure 3.2: ηmax and η(Pmax) vs ZT . For small values of ZT , ηmax ∼ η(Pmax),

while for ZT → ∞, ηmax → ηC, and η(Pmax) → ηC/2

3.2 Quantum coherent transport

3.2.1 Introduction

The idea of quantum coherent transport is to relate the transport properties of the system to the scattering properties, which are assumed to be known from a quantum-mechanical calculation. In this work we consider only two-terminal systems, as that in Fig. 3.3 composed by a mesoscopic sample connected to two reservoirs through two leads, to be referred to as ’left” (L) and ”right” (R). It is assumed that the reservoirs are so large that they are at the

equilib-rium and can be characterized by a temperature TL,Rand a chemical potential

µL,R. The electrons in each reservoirs obey to the Fermi distribution functions

fα, with α = L, R. We must underline that although there are no inelastic

processes in the sample, a strict equilibrium in the reservoirs can be estab-lished only by inelastic processes. However, we consider the reservoirs to be wide compared to the typical cross section of the mesoscopic conductor, which in turn represents only a small perturbation of the system. In this way it is justified to describe the transport between the two reservoirs in terms of an equilibrium state. [20, 21, 22, 23]

Far from the sample, we can assume, without loss of generality, that trans-verse and longitudinal motion of the electrons are separable. We take the x axis as the longitudinal direction. It is advantageous to separate incoming and outgoing states, and introduce the transverse energy labeled by the discrete

index n which denotes the quantized transverse modes with energy En. These

states are in the following referred to as transverse (quantum) channels or

modes. The total energy of the system is thus E = En+ El, where El indicates

the longitudinal energy. In the following section we will see how the incoming and outgoing states are related.

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CHAPTER 3. QUANTUM COHERENT TRANSPORT 29 RESERVOIR L µL, TL LEAD L aL,n bL,n aR,n bR,n SAMPLE RESERVOIR R µR, TR LEAD R

Figure 3.3: A sample connected to two reservoirs at chemical potentials µ_L, µR and

temperatures TL, TR.

3.2.2 The S-matrix

We now introduce the creation and annihilation operators of electrons (fermions

more in general) in the incoming states, a†_α,n(E) and aα,n(E) respectively, with

total energy E, in channel n and lead α. The same can be done defining b†_α,n(E)

and bα,n(E) for the outgoing states. They obey anticommutation relations

{a†_α,n(E), aβ,n0(E0)} = δ_α,βδ_n,n0δ(E − E0), (3.37)

{aα,n(E), aβ,n0(E0)} = 0, (3.38)

{a†_α,n(E), a†_β,n0(E0)} = 0. (3.39)

(3.40) The operators a and b are related via the scattering matrix (S-matrix), defined as          bL,1 .. . bL,NL bR,NR .. . bR,NR          = S          aL,1 .. . aL,NL aR,NR .. . aR,NR          , (3.41)

with NLand NRindicating the number of channels in left and right lead

respec-tively, which in general depends on the total energy E. The creation operators

a†and b†, are related by the same relation with the hermitian conjugate matrix

S†. The S-matrix has dimensions (NL+ NR) × (NL+ NR), and depends on

the total energy E. It has a block structure of the type

r t0

t t

. (3.42)

The matrices r (r0) of size NL× NL (NR× NR), describe electron reflection

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CHAPTER 3. QUANTUM COHERENT TRANSPORT 30 a1 b1 a2 b2 a3 b3 SAMPLE 2 SAMPLE 1

Figure 3.4: The case of two samples in series with a single channel everywhere.

size NR× NL (NL× NR) is responsible for the electron transmission through

the sample. The current conservation in the scattering process implies that the

S-matrix must be unitary. In the presence of time reversal symmetry r = r0

and t = t0.

3.2.3 Composition of S-matrices

We want to discuss how the S-matrices are composed in the presence of two samples in series. We will see how this calculation will reconnect with the transmission probability for two potential barriers calculated in Chapter 4. We simplify the treatment considering one channel incoming from the left reservoir, going through two samples and outgoing into the right reservoir (Fig. 3.4). We omit the explicit dependence on energy E, in order to have less heavy notations. In this way the S-matrix blocks of the single sample are simply complex numbers. We write the S-matrices as

b1 b2 = s(1)a1 a2 =r1 t 0 1 t1 r10 a1 a2 , (3.43) and a2 b3 = s(2) b2 a3 =r2 t 0 2 t2 r20 b2 a3 . (3.44)

The final result must be of the form

b1 b3 =r t 0 t r0 a1 a3 , (3.45)

where r, t, r0, t0 are the total reflection and transmission amplitudes. In order

to find their expression we write Eqs. (3.43) and (3.44) as a system:          r1a1+ t01a2 = b1, t1a1+ r01a2 = b2, r2b2+ t02a3 = a2, t2b2+ r20a3 = b3. (3.46)

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CHAPTER 3. QUANTUM COHERENT TRANSPORT 31 b2 = t1a1+ r01t 0 2a3 1 − r0 1r2 , (3.47) a2 = r2 1 − r0₁r2 [t1a1+ r01t 0 2a3] + t02a3. (3.48)

Then, substituting these expressions into the equations for b1 and b3, we

ob-tain the final S-matrix and the expression for the reflection and transmission amplitudes: r = r1+ r2t1t01 1 − r0₁r2 , (3.49) t = t1t2 1 − r₁0r2 , (3.50) r0 = r0₂+ r 0 1t2t02 1 − r0₁r2 , (3.51) t0 = t 0 1t 0 2 1 − r₁0r2 . (3.52)

Squaring the amplitudes, we have the final transmission and reflection

proba-bilities. Writing r0₁ and r2 as

r0₁ = |r₁0|eiδ01+θ01_, _(3.53)

r2 = |r2|eiδ2+θ2, (3.54)

where δ₁0, δ2 are the intrinsic phases and θ10, θ2 the phases acquired after two

reflections of r0₁,r2, Eq. (3.50) becomes

t = t1t2

1 − |r0

1||r2|ei(δ

0

1+δ2+θ01+θ2), (3.55)

and the same happens for t0. We can note that the expressions for the total

transmission amplitudes are the result of a geometric series as in a Fabry-Perot after infinite reflections:

t = ∞ X p=1 t1t2(|r10||r2|)p−1e2i(p−1)(δ 0 1+δ2+θ01+θ2) = t1t2 1 − |r0₁||r2|ei(δ 0 1+δ2+θ01+θ2) (3.56)

The physical meaning is that the total transmission is obtained with the sum of multiple reflections of the electron inside the scatterer. Squaring Eq. 3.55 we have the total transmission probability:

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CHAPTER 3. QUANTUM COHERENT TRANSPORT 32 T = T1T2 1 + R0₁R2− 2pR01R2cos (δ01+ δ2+ θ10 + θ2) , (3.57) where T1,2 = t∗1,2t1,2, R01 = r 0∗ 1r 0

1 and R2 = r2∗r2. In the case of the presence

of more channels, the generalization with the block matrices r, r0, t and t0 is

straightforward and the form of equations remains the same for the amplitudes, but transmission probabilities have to be defined in a different way as we will see in the next section.

3.2.4 Current operator

In the following we follow Ref. [20]. We define the field operators Ψα and Ψ†α,

with α = L, R as Ψα(r, t) = Z dEe−iEt¯h NL(E) X n=1 χn(r⊥) p2πhvL,n(E)

aL,neikL,n(E)x+ bL,ne−ikL,n(E)x,

(3.58) Ψ†_α(r, t) = Z dEeiEt¯h NL(E) X n=1 χ∗_n(r⊥) p2πhvL,n(E) a† L,ne −ikL,n(E)x_{+ b}† L,ne ikL,n(E)x. (3.59)

where r⊥is the transverse coordinate(s) and x is the coordinate along the leads

(measured form left to right). χn are the transverse wave functions, and kL,n is

the longitudinal wave vector, where the summation includes only channels with

real kL,n.The electric current operator in the left lead (far from the sample) is

expressed as (J1)L(z, t) = ¯ he 2im Z dr⊥ Ψ†_L(r, t) ∂ ∂xΨL(r, t) − ∂ ∂xΨ † L(r, t) ΨL(r, t) . (3.60) Substituting the field operators, after some algebra, we obtain

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CHAPTER 3. QUANTUM COHERENT TRANSPORT 33 (J1)L(z, t) = e 4π¯h X n Z dEdE0ei(E−E0)t¯h 1 pvL,n(E)vL,n(E0) × ×nvL,n(E) + vL,n(E0) h a†_L,n(E)aL,n(E0)ei[kL,n(E 0_)−k L,n(E)]x

− b†_L,n(E)bL,n(E0)ei[kL,n(E)−kL,n(E

0_)]xi

+vL,n(E) − vL,n(E0)

h

a†_L,n(E)bL,n(E0)e−i[kL,n(E)+kL,n(E

0_)]x

− b†_L,n(E)aL,n(E0)ei[kL,n(E)+kL,n(E

0_)]xio .

(3.61)

The result is too complicated. However, we can simplify it by considering that

for the observables the energies E, E0 are close to each others, and that the

electrons which contribute to the current are all those with velocity near to

the Fermi velocity vF. Thus we can take v(E) ∼ v(E0) ∼ vF and write

(J1)L(z, t) = e 2π¯h X n Z dEdE0ei(E−E0)t¯h a† L,n(E)aL,n(E0) − b † L,n(E)bL,n(E0). (3.62)

Now we can express the current in terms of the only a and a† operators using

the scattering matrix as

(J1)L(t) = e 2π¯h X αβ X mn Z

dEdE0ei(E−E0)th¯ aαm(E)Amn

αβ(L; E, E

0

)aβn(E0),

(3.63) where α and β label the reservoirs. The matrix A is defined as

Amn_αβ(L; E, E0) = δmnδαLδβL− X k S_Lα;mk† SLβ;kn= =X k S_α,L;mk† Sβ,L;kn− X k S_Lα;mk† SLβ;kn , (3.64)

where we have used the unitarity of S. Now we consider that each reservoir is at the thermodynamic equilibrium, thus

ha†_α,m(E)aβ,n(E0)i = δα,βδmnδ(E − E0)fα(E), (3.65)

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h(J1)L(t)i =

e h

Z

Tr[t†(E)t(E)](fL(E) − fR(E))dE. (3.66)

This is the Landauer-Büttiker formula for the current in a coherent conductor

between two terminals. The matrix t†(E)t(E) is hermitian, thus is

diagonaliz-able and has a set of real eigenvalues Tn(E), called the transmission

probabili-ties of an electron in a channel n of lead L to be transmitted into any channel of lead R. They can assume values between zero and one. In this basis, called the eigen channels basis, Eq. 3.66 becomes

h(J1)L(t)i = e h X n Z

Tn(E)(fL(E) − fR(E))dE. (3.67)

One may asks what happens to the heat current. At fixed temperature T , the heat current is due only to the entropy variation T dS = dU − µdN , with U the inner energy, µ the chemical potential and N the number of particles. The density heat current thus reads

j = jE− µjN. (3.68)

Redefining the energy and particle current operators, from the left lead we have (J2)L,E(z, t) = ¯ hE 2im Z dr⊥ Ψ†_L(r, t) ∂ ∂xΨL(r, t) − ∂ ∂xΨ † L(r, t) ΨL(r, t) , (3.69) (J2)L,N(z, t) = ¯ hµL 2im Z dr⊥ Ψ†_L(r, t) ∂ ∂xΨL(r, t) − ∂ ∂xΨ † L(r, t) ΨL(r, t) . (3.70) Hence, following the same procedure as for the electric current we finally obtain for the average heat current in the eigen channels basis

h(J2)L(t)i = 1 h X n Z

Tn(E)(E − µL)(fL(E) − fR(E))dE. (3.71)

We have deduced these results, starting from a parabolic dispersion of the electron energy. However it can be demonstrated as well that the Landauer-Büttiker formula is also valid for an arbitrary dispersion of the electron energy [23].

3.2.5 Linear response and relations with the Onsager

co-efficients

We consider now the two terminal system within the linear response, in which the forces associated with the fluxes are small enough. With two small

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pertur-CHAPTER 3. QUANTUM COHERENT TRANSPORT 35

bations δµ = (µL− µR)/2 and δT = (TL− TR)/2, we can expand fL(E) and

fR(E) at the first order in δµ/µ and δT /T , as

fL(E) = f0(E, µ + δµ, T + δT ) = f0+ ∂f0 ∂µ δµ + ∂f0 ∂β δβ, (3.72) fR(E) = f0(E, µ − δµ, T − δT ) = f0− ∂f0 ∂µ δµ − ∂f0 ∂β δβ, (3.73)

where f0 denotes the Fermi distribution calculated in T ,µ and β = (kBT )−1,

with kB being the Boltzmann constant. When we make the substitution T →

T + δT we have for β the following relation

β → 1 kB(T + δT ) = 1 kBT (1 + δT_T ) ∼ 1 kBT 1 −δT T , (3.74) therefore, δβ = −_kδT

BT2. We can write, though, at first order

fL(E) − fR(E) = 2 ∂f0 ∂µ δµ + 2 ∂f0 ∂β δβ = = ∂f0 ∂µ (µL− µR) + ∂f0 ∂β − 2 δT kBT2 = = − ∂f0 ∂E e∆V + − ∂f0 ∂E (E − µ)∆T T , (3.75)

where ∆V = (µL − µR)/e is the voltage difference, and ∆T = TL − TR.

Substituting Eq. (3.75) into Eqs. (3.67), (3.71), focusing on only one channel, we have        J1 = e 2 h R dET (E) −∂f0 ∂E ∆V + e_hR dET (E)−∂f0 ∂E ∆T T , J2 = _heR dE(E − µ)T (E) − ∂f0 ∂E ∆V + _h1R dE(E − µ)2_{T (E)}₋ ∂f0 ∂E ∆T T . (3.76) where we have omitted L, because we consider only currents from the left. Defining the integrals

Im = 1 h Z dE(E − µ)mT (E)−∂f0 ∂E , (3.77)

whit m integer, we obtain      J1 = e2T I0∆V_T + eT I1∆T_T2, J2 = eT I1∆V_T + T I2∆T_T2. (3.78)

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CHAPTER 3. QUANTUM COHERENT TRANSPORT 36 The Onsager coefficients are

L11 = e2T I0, (3.79)

L12 = L21 = eT I1, (3.80)

L22 = T I2, (3.81)

where we can see that the system described is time-reversal symmetric.

Con-sequently the thermoelectric coefficients are in terms of the integrals Im

G = e2I0, (3.82) Ξ = 1 T I2− I2 1 I0 , (3.83) S = 1 eT I1 I0 , (3.84) ZT = GS 2 Ξ T = I2 1 I0I2− I12 . (3.85)

These final formulas are the most important results of the Chapter. Now, we are able to calculate the thermodynamic coefficients in a quantum coherent conductor, simply knowing the one electron probability to transmit from one lead to another. The generalization with more channels is straightforward by

simply considering the integrals Im redefined as

Im = 1 h X n Z dE(E − µ)mTn(E) − ∂f0 ∂E . (3.86)

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Chapter 4 Transport and thermoelectric

properties in a single surface of

topological insulator

In this chapter we show the most important analytical results of this thesis. We analyze the thermoelectric quantum properties of a single surface of a 3D

topological insulator belonging to the Bi2Se3 family crystals, in a two-terminal

set up. In order to do this calculation we use the effective model Hamiltonian (2.65) for the surface states. We first focus on the transport properties of such states when a single potential barrier is inserted into the system, and then we add a second barrier, in order to exploit a resonant form of the transmission for the maximization of the performances.

4.1 Transport with a single potential barrier

4.1.1 Transmission probability

We start with the analytical calculation of transport quantities of one surface, in a two-terminal system, in the case of a single potential barrier. We use the

surface Hamiltonian (2.65), with Bi2Se3 coefficients, which at the first order in

k reads

Hsurf = ¯hvF(σxky − σykx), (4.1)

where ¯hvF = ˜A0, vF is the Fermi velocity of the surface states, and we have

omitted ˜C0 1 which is only a little shift in energy not determinant in the

whole problem. Now we include a potential barrier of height V0 and width D

on the x direction as in Fig. 4.1. In order to calculate the thermoelectric coef-ficients within the quantum coherent transport formalism, we have to calculate first the transmission probability of the electron in the surface state. We solve the Schrödinger equation in region I (III) and then the solution in region II

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CHAPTER 4. TRANSPORT IN A SINGLE SURFACE OF TI 38 x 0 k E I V0 D II III

Figure 4.1: A potential barrier of height V0 and width D. The incoming electron has

energy E and momentum k. We are in the hypothesis that the energy dispersion E(k) is linear also inside the barrier.

will be simply given by the substitution E → E − V0. Thus, the Schrödinger

equation in region I (III) is

HsurfΨ(x, y) = EΨ(x, y), (4.2)

where Ψ(x, y) is a two-component spinor. Explicitly we have

0 ¯hvF(ky + ikx) ¯ hvF(ky − ikx) 0 ψ1(x, y) ψ2(x, y) = Eψ1(x, y) ψ2(x, y) . (4.3)

Therefore the equations for the two spinors’ components are ( ¯ hvF(ky + ikx)ψ2(x, y) = Eψ1(x, y) ¯ hvF(ky − ikx)ψ1(x, y) = Eψ2(x, y). (4.4)

Solving the second equation for ψ2 and substituting into the first we obtain

for ψ1

¯

h2v2_F(k_y2+ k2_x)ψ1(x, y) = E2ψ1(x, y). (4.5)

Because on the x direction there is a broken translational symmetry we make

the replacement kx → −i∂x. The general solution is the sum of one incoming

wave and one reflected

ψ1(x, y) = (eikxx+ re−ikxx)eikyy, (4.6)

where r is the reflection coefficient and

kx = s E2 (¯hvF)2 − k2 y. (4.7)

Calling in general ψinc

i the incoming wave and ψouti the outgoing one, with

Thermoelectric quantum transport in 3D topological insulators

Mathematical Physical and Natural Sciences

Master’s Degree in Physics