Thermal Transport

Abstract

The electrons in solids not only conduct electricity but also conduct heat, as they transfer energy from a hot junction to a cold junction . Just as the electrical conductivity characterizes the response of a material to an applied voltage, the thermal conductivity likewise characterizes the material with regard to heat flow in the presence of a temperature gradient. In fact, the electrical conductivity and thermal conductivity are coupled, since thermal conduction also transports charge and electrical conduction also transports energy .

Problems

For problems 8.1–8.7 below, assume the following properties for Bi$_2$Te$_3$

• Effective masses: m$_x = 0.02$, m$_y = 0.08$, m$_z = 0.32$

• Mobility: $\mu _x$ = 1200 cm$^2$V$^{-1}$

• Lattice thermal conductivity: $\kappa _L = 1.5$  Wm$^{-1}$K$^{-1}$

Perform all calculations at room temperature, T = 300 K.

8.1

Write a function in Matlab or Mathematica (or other software of your choice) that calculates the dimensionless Fermi integral:

$$\begin{aligned} F_i = F_i(\zeta ^*) = \int ^\infty _0 \frac{x^idx}{e^{(x-\zeta ^*)}+1} \end{aligned}$$

(8.112)

where

$$\begin{aligned} \zeta ^* = \frac{E_F}{k_BT} \end{aligned}$$

(8.113)

8.2

The transport coefficients of a bulk (3D) material in the constant relaxation time approximation are given by the following Boltzmann transport equations:

$$\begin{aligned} \sigma = \frac{e}{3\pi ^2}\left( \frac{2k_BT}{\hbar ^2}\right) ^{\frac{3}{2}}(m_xm_ym_z)^{\frac{1}{2}}\mu _x\left( \frac{3}{2}F_{1/2}\right) \end{aligned}$$

(8.114)

$$\begin{aligned} \mathscr {S} = -\frac{k_B}{e}\left( \frac{5F_{3/2}}{3F_{1/2}}-\zeta ^*\right) \end{aligned}$$

(8.115)

$$\begin{aligned} \kappa _e = \frac{k_B^2T}{3\pi ^2e}\left( \frac{2k_BT}{\hbar ^2}\right) ^{\frac{3}{2}}(m_xm_ym_z)^{\frac{1}{2}}\mu _x\left( \frac{7}{2}F_{5/2}-\frac{25F^2_{3/2}}{6F_{1/2}}\right) \end{aligned}$$

(8.116)

Using these equations, write functions in MatLab for the following quantities:

• (a) Electrical conductivity ($\sigma$)

• (b) Seebeck coefficient ($\mathscr {S}$) in units of $\mu$V/K

• (c) Electrical component of the thermal conductivity ($\kappa _e$).

Plot these coefficients as a function of Fermi level E$_F$ (or chemical potential).

8.3

The transport coefficients of a 2D material in the constant relaxation time approximation are given by the following Boltzmann transport equations:

$$\begin{aligned} \sigma = \frac{e}{2\pi a}\left( \frac{2k_BT}{\hbar ^2}\right) (m_xm_y)^{\frac{1}{2}}\mu _x\left( F_0\right) \end{aligned}$$

(8.117)

$$\begin{aligned} \mathscr {S} = -\frac{k_B}{e}\left( \frac{2F_1}{F_0}-\zeta ^*\right) \end{aligned}$$

(8.118)

$$\begin{aligned} \kappa _e = \frac{k_B^2T}{2\pi ae}\left( \frac{2k_BT}{\hbar ^2}\right) (m_xm_y)^{\frac{1}{2}}\mu _x\left( 3F_2-\frac{4F^2_1}{F_0}\right) \end{aligned}$$

(8.119)

For a 5 nm quantum well (a = 5 nm), write functions in MatLab for the following quantities:

• (a) Electrical conductivity ($\sigma$)

• (b) Seebeck coefficient ($\mathscr {S}$) in units of $\mu$V/K

• (c) Electrical component of the thermal conductivity ($\kappa _e$).

Plot these coefficients as a function of Fermi level E$_F$ (or chemical potential) .

8.4

In 1D, the transport coefficients are given by

$$\begin{aligned} \sigma = \frac{2e}{\pi a^2}\left( \frac{2k_BT}{\hbar ^2}\right) ^\frac{1}{2}m_x^{\frac{1}{2}}\mu _x\left( \frac{1}{2}F_{-\frac{1}{2}}\right) \end{aligned}$$

(8.120)

$$\begin{aligned} \mathscr {S} = -\frac{k_B}{e}\left( \frac{3F_{1/2}}{3F_{-1/2}}-\zeta ^*\right) \end{aligned}$$

(8.121)

$$\begin{aligned} \kappa _e = \frac{2k_B^2T}{\pi ^2 ae}\left( \frac{k_BT}{\hbar ^2}\right) ^\frac{1}{2}m_x^{\frac{1}{2}}\mu _x\left( \frac{5}{2}F_{3/2}-\frac{9F^2_{1/2}}{2F_{-1/2}}\right) \end{aligned}$$

(8.122)

For a nanowire with a 5 nm $\times$ 5 nm square cross section, write functions in Matlab for the following quantities

• (a) Electrical conductivity ($\sigma$)

• (b) Seebeck coefficient ($\mathscr {S}$) in units of $\mu$V/K

• (c) Electrical component of the thermal conductivity ($\kappa _e$).

Plot these coefficients as a function of Fermi level E$_F$ (or chemical potential).

8.5

The thermoelectric figure of merit is given by

$$\begin{aligned} Z = \frac{\mathscr {S}^2\sigma }{\kappa _e+\kappa _L} \end{aligned}$$

(8.123)

Using the functions derived in the problems above, plot the dimensionless figure of merit ZT (at room temperature) as a function of the Fermi energy for the three cases

• (a) 3D bulk

• (b) 2D (5 nm width)

• (c) 1D (5 nm $\times$ 5 nm cross section)

8.6

Find the optimum value for ZT (with respect to the Fermi energy) for each of the 3 cases above. Plot the optimum ZT value as a function of quantum well width and nanowire width for 1, 2, 3, 4 and 5 nm.

8.7

The Wiedemann-Franz Law states that

$$\begin{aligned} \mathbf {K} = \mathbf {\sigma }T \frac{\pi ^2k_B^2}{3e^2} \end{aligned}$$

(8.124)

On this basis, explain how it is possible for sapphire to be both an excellent thermal conductor and an excellent electrical insulator.

8.8

Suppose that you measure the thermal conductivity of a sample at 100$^\circ$C. How would you estimate the fraction of the heat that is carried by electron (or hole) carriers?

8.9

Suppose that Si is doped with an isoelectronic impurity in column IV of the periodic table (such as Ge or Sn), will the doping effect be greater on the electrical conductivity or on the thermal conductivity, and why?

8.10

Could this thermoelectric cooler (see figure below) be modified to become a thermoelectric heater? If so, explain how it can be done.