In general, any kind of transport in a material can be understood as the response of this carriers to any external perturbations. The external stimuli can be electric or magnetic field, temperature, etc. The transport theory explains all possible ways of response of the carriers to the stimuli. Two transport theories are well established, one is Boltzmann transport theory[120] and the other is Green Kubo theory[121]. The thermoelectric properties of investigated compounds in the present thesis are performed using semi-classical Boltzmann transport theory implemented in BoltzTraP code[122]. Here we would like to discuss the semi-classical Boltzmann theory and few assumptions included in the same.
Boltzmann transport equation starts with the definition of the electron distribution function f(r, k, t)drdk (here the carriers are considered to be electrons), which is the number of electrons at point r with wave number k in the small phase space volume drdk. The core of this problem is the time evolution of the electron distribution functionf(r, k, t). The total number of electrons in the system can be calculated by integrating the electron distribution function inkandrspace. To
understand the variation of the distribution function in time, the knowledge of the stimuli is needed, in general external field, temperature gradient and scattering are considered as the cause of carrier transport, which cause diffusion, drift and collision. After incorporating all these perturbations, the variation off(r, k, t) in time can be seen as
f(r, k, t)
∂t = f(r, k, t)
∂t dif f usion+f(r, k, t)
∂t drif t+f(r, k, t)
∂t collision (2.41)
Each term in the above equations can be understood separately, To understand the diffusion term, let us fix the ’k’ point, and see how the carriers vary as function of ’r’, and ’t’. If vk is the velocity, the particles in a small volumedratrand at time tare considered to be same as were at the pointr−vkdt and at timet−dt. Then the variation of the distribution function in time due to diffusion can be written as
f(r, k, t)
∂t dif f usion=−vk.∂f(r)
∂r (2.42)
In similar way, at a fixed ’r’ electrons with wave numberk were same with wave number k − dtdk/dt, the change of distribution function due to drift
f(r, k, t)
∂t drif t=−k.∂f(k)
∂k (2.43)
The challenge here is to address the collision term, in general collision depends on all the scatter- ing mechanism and the form will be very complicated. Here we have used the simplest case, where the change in distribution function due to collision is considered as
f(r, k, t)
∂t collision =−f(k)−f0(k)
τ(k) (2.44)
After substituting the exact form of all the terms, the Boltzmann transport equation can be written as
f(r, k, t)
∂t =−vk.∂f(r)
∂r −k.∂f(k)
∂k −f(k)−f0(k)
τ(k) (2.45)
At the steady state, the time derivative of the distribution function will be zero, and then
−vk.∂f(r)
∂r −k.∂f(k)
∂k −f(k)−f0(k)
τ(k) = 0 (2.46)
The knowledge of the electron distribution function helps to compute several transport properties.
The transport coefficients can be obtained by solving the current density equation, the equation for electric current density is
Je= 2e 8π3
Z
v(k)f(k)dk (2.47)
Further heat current density due to electrons can be written as JQ= 2
8π3 Z
v(k)[E−µ]f(k)dk (2.48)
where µ is the chemical potential. After substituting the electron distribution function obtained
from Boltzmann transport theory, the above two equations become Je= 2e
8π3 Z
v(k)v(k)τ(k)(−∂f0
∂E )[eǫ− ∇µ+E−µ
T (−∇T)]dk (2.49)
Then
JQ = 2 8π3
Z
v(k)v(k)τ(k)(−∂f0
∂E )[eǫ− ∇µ+E−µ
T (−∇T)][E−µ]dk (2.50) To minimize the complexity, response function has been defined
In = 1 4π3
Z
v(k)v(k)τ(k)( −∂f0
∂E)[E−µ]dk(2.51) Both the current densities can be expressed in terms of response function
Je=e2I0ǫ+eI1
T(−∇T) (2.52)
JQ=eI1ǫ+I2
T(−∇T) (2.53)
When the temperature gradient is zero, Je = σǫ where σ is the electrical conductivity. The electrical conductivity can be calculated usingσ= e2I0Similarly, by switching off the electric filed and allow a temperature gradient, the equation for JQ becomes JQ = k(-∇T) where k represents the thermal conductivity due to electrons. ‘k’ can be calculated using
k= 1
T[I2−I12 I0
] (2.54)
The equation for Seebeck coefficient, where the temperature gradient is non-zero and in absence of electric current,
S= I1
eT I0 (2.55)
Using Boltzmann transport theory, one can find electrical conductivity, Seebeck coefficient (ther- mopower) and electronic part of thermal conductivity. The calculation of these transport properties in the present thesis was performed using BoltzTraP code, which is based on two assumptions, which are discussed here. Constant Scattering Time Approximation (CSTA), which assumes that the relaxation time is a constant. Another approximation is rigid band approximation, and in this approximation, the small change in the number of valence electrons does not effect the electronic structure profile, and bands are considered to be rigid, and are explained below.
2.7.1 Constant Relaxation Time Approximation
In the previous section, we have discussed the three main perturbation which cause carrier trans- port, among that collision involves several scattering processes such as electron-electron scattering, electron-phonon scattering, scattering with boundary spaces. To simplify the collision term in BTE, constant scattering time approximation (CSTA) was introduced. After imposing this approximation
one can write the change in electron distribution function due to collision as f(r, k, t)
∂t collision =−f(k)−f0(k)
τ(k) (2.56)
where ’τ(k)’ is defined as the relaxation time. The physical meaning of the relaxation time can be explained as follows. Consider at time t =0, both external field and temperature gradient perturbations are switched off, the change in electron distribution function will be only due to collision, and the relaxation time is defined as the characteristic time for a system to set back to its equilibrium state.
2.7.2 Rigid band Approximation
The aim of the present thesis is to explore the thermoelectric properties of few prospective materials within the frame work of first principles calculations, where one should understand the TE properties as a function of hole and electron doping. To perform the hole and electron doping calculations, large super cells are needed which is computationally very expensive. A feasible approximation known as rigid band approximation (RBA) is introduced to overcome the above issue. Considering the number of valence electron in the system as ’N0’, and change in the valence electron after doping as ’δn’, the total number of electrons can be written as
Ne= Z
N(E)f(T, µ)dE (2.57)
where ‘µ’ represents the chemical potential for the doped material with carrier concentration Ne= N0 +δn. It is assumed that, the electronic structure properties such as band profile and density of states remains rigid for small change in valence electrons. In the present thesis, the thermoelectric properties have been calculated based on RBA. Several studies proved that doping with the elements Sb and Bi does not effect the band structure[123, 124]. The validity of RBA has been verified for optimum doping range and significant number of TE materials are successfully predicted using this approximation [125, 126].