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## 物理代写|固体物理代写Solid-state physics代考|Thermal transport

We finally discuss the thermal conduction in metals. The basic assumption in this case is that most of the heat current is carried by the gas of conduction electrons, consistently with the empirical observation that metals are as good electrical conductors as they are good thermal conductors: it is therefore natural to look at electrons as the primary microscopic carriers for both transport phenomena. More specifically, we understand that electrons are able to transport heat since those coming from the hotter region of the sample carry a larger amount of thermal energy than the electrons moving from the colder region.

The approximation to consider only electrons as heat carriers implies that in the Drude theory we formally assume $\kappa_{\mathrm{tot}}=\kappa_{\mathrm{e}}+\kappa_1 \sim \kappa_{\mathrm{e}}$, where $\kappa_{\mathrm{e}}$ and $\kappa_1$ are the separate contributions due to the electronic and ionic degrees of freedom, respectively. In order to calculate the actual form of $\kappa_e$, we can proceed by analogy ${ }^9$ with the lattice case discussed in section $4.3$ by setting
$$\kappa_{\mathrm{e}}^{\text {Drude }}=\frac{1}{3} \tau_{\mathrm{e}}\left\langle\left(v_{\mathrm{e}}^{\text {th }}\right)^2\right\rangle c_V^{\mathrm{e}}(T),$$
where $c_V^e(T)$ is the constant-volume specific heat of the electron gas. It is important to remark that in this equation we used the same symbol $\tau_{\mathrm{e}}$ as before, but with a subtly different meaning: in equation $(7.15)$ it is understood as the relaxation time occurring in the heat current phenomenon, while in equation (7.7) it was associated with the charge current one. This abuse of notation suggests that within the present free electron theory we are assuming the charge current and heat current relaxation times to be just the same: indeed a useful approximation, which however will be revised in section 7.3.4. By describing the conduction gas classically, we calculate ${ }^{10}$ $\left\langle\left(v_{\mathrm{e}}^{\mathrm{th}}\right)^2\right\rangle=3 k_{\mathrm{B}} T / m_{\mathrm{e}}$ and $c_V^e(T)=3 n_{\mathrm{e}} k_{\mathrm{B}} / 2$ and, by means of equation (7.7), we predict the ratio between its thermal and electrical conductivity to be
$$\frac{\kappa_{\mathrm{e}}^{\text {Drude }}}{\sigma_{\mathrm{e}}}=\frac{3}{2}\left(\frac{k_{\mathrm{B}}}{e}\right)^2 T,$$
which is in remarkable good agreement with the experimental Wiedemann-Franz law stating that in most metals the ratio between the thermal and electric conductivities of the conduction gas is proportional to $T$ at sufficiently high temperatures (room temperature belongs to this range). To a very good extent, the proportionality constant is found to be about the same for all metals. Equation (7.16) not only accurately predicts the high temperature trend of the $\kappa_{\mathrm{e}} / \sigma_{\mathrm{e}}$ ratio, but it also provides the estimation of the constant
$$\frac{\kappa_e^{\text {Drude }}}{\sigma_{\mathrm{e}} T}=\frac{3}{2}\left(\frac{k_{\mathrm{B}}}{e}\right)^2=1.11 \cdot 10^{-8} \mathrm{~W} \Omega \mathrm{K}^{-2},$$
which is known as the Lorenz number. Finally, we remark that this result critically depends on the assumption that the relaxation times limiting charge and heat currents are the same.

## 物理代写|固体物理代写Solid-state physics代考|Failures of the Drude theory

From the discussion developed so far, one could draw the conclusion that Drude theory is basically sufficient to explain the main physical features of electron gas in a metal, given its apparent success in all topics where we have applied it. Unfortunately, this is not true.

First of all, we observe that the predicted value for the Lorenz number is (roughly) only half of the measured one $[2,3]$. Second, the ability to explain the phenomenological Wiedemann-Franz law is just casual since it is based on two large numerical errors pointing in opposite directions: (i) the actual specific heat of an electron gas at the typical metallic density is much smaller than predicted classically and (ii) the typical electron velocities are much larger than calculated by equipartition. By chance these two rather inaccurate estimations almost perfectly compensate for each other, thus giving an illusory impression of robustness to the Drude theory. In addition, the Lorenz number is not a constant at low temperatures [7], mainly because it is found that $\kappa_e$ is a function of temperature. As a matter of fact, only a full quantum theory is able to correct such discrepancies so as to match the experimental evidences.

In addition to the above failures, the Drude theory is also unsuccessful in describing the electron gas under the action of both an electric and magnetic field, as found when investigating the Hall effect $[2,3]$. Let us consider the situation shown in figure $7.3$ where a constant and uniform magnetic field $\mathbf{B}=(0,0, B)$ is applied normal to the current density $\mathbf{j}$ generated by a constant and uniform electric field $\mathbf{E}=\left(E_x, E_y, 0\right)$. As before, the equation of motion for the electrons of the conducing gas are Newton-like
$$-e \mathbf{E}-e \mathbf{v}{\mathbf{d}} \times \mathbf{B}=m_e \dot{\mathbf{v}}{\mathbf{d}}+\frac{m_e}{\tau_{\mathrm{e}}} \mathbf{v}{\mathrm{d}},$$ where the left-hand side is now given by the Lorentz force $-e\left(\mathbf{E}+\mathbf{v}{\mathrm{d}} \times \mathbf{B}\right)$. If we consider a steady-state regime, we have $\dot{\mathrm{v}}{\mathrm{d}}=0$ and the equation of motion leads to $v{\mathrm{d}, x}=-\frac{e \tau_{\mathrm{e}}}{m_{\mathrm{e}}} E_x-\frac{e \tau_{\mathrm{e}}}{m_{\mathrm{e}}} B \quad v_{\mathrm{d}, y} \quad$ and $\quad v_{\mathrm{d}, y}=-\frac{e \tau_{\mathrm{e}}}{m_{\mathrm{e}}} E_y+\frac{e \tau_{\mathrm{e}}}{m_{\mathrm{e}}} B v_{\mathrm{d}, x}$,

By imposing the condition that no transverse current flows in the $y$ direction (an open-circuit configuration corresponding to the condition $v_y=0$ ) we get
$$\frac{E_y}{E_x}=-\frac{e \tau_{\mathrm{e}}}{m_{\mathrm{e}}} B,$$
which allows us to define the Hall coefficient $R_{\mathrm{H}}$ as
$$R_{\mathrm{H}}=\frac{E_y}{j_x B}=\frac{E_y}{\sigma_{\mathrm{e}} E_x B}=-\frac{1}{n_{\mathrm{e}} e},$$
where $j_x$ is the current density along the $x$ direction and equation (7.7) has been used for the conductivity $\sigma_{\mathrm{e}}$. Therefore, Drude theory predicts that the Hall coefficient is independent of the applied magnetic field and, in any case, negative. Unfortunately, both conclusions are wrong!

# 固体物理代写

## 物理代写|固体物理代写Solid-state physics代考|热输运

$$\kappa_{\mathrm{e}}^{\text {Drude }}=\frac{1}{3} \tau_{\mathrm{e}}\left\langle\left(v_{\mathrm{e}}^{\text {th }}\right)^2\right\rangle c_V^{\mathrm{e}}(T),$$

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