# 物理代写|固体物理代写Solid-state physics代考|KYA322

## 物理代写|固体物理代写Solid-state physics代考|The conceptual framework

In developing an atomistic and quantum picture of the solid state we had to admit that it is necessary to adopt approximations, even very drastic ones as thoroughly reported in section 1.3. As far as electrons are concerned, this has led to equation (1.15) which defines the constitutive eigenvalue problem for the total crystalline electron wavefunction ${ }^1 \Psi_{\mathrm{e}}^{(\mathbf{R})}(\mathbf{r})$ : by solving it, we ultimately define the electronic structure of the crystal.

Despite the remarkable number of adopted simplifications already introduced ${ }^2$, equation (1.15) still represents a formidably complicated many-body quantum problem. The search for general methods able to solve it, either analytically or numerically, represents an advanced topic in theoretical condensed matter physics, ranging from single-particle theories to many-body formalisms to quantum field theories [1-6]. These methods are currently the topic of active research and mostly fall well beyond the level (and the scope) of this introductory textbook. In our attempt to elaborate a more elementary theory we will follow a simplified approach, which basically consists in two logical steps: first, we will root our theory in the single-particle approximation; second, we will take profit from some materials-specific characteristics so as to further reduce the mathematical complexity of the physical problem.

The first step was already introduced in section 1.4.1: electron-ion and electronelectron interactions are replaced by the crystal field potential $V_{\mathrm{cfp}}(\mathbf{r})$, namely a local one-electron potential with the same periodicity of the underlying crystal lattice and effectively describing the most relevant many-body features. Hereafter, we will assume that such a potential is known, for instance through a self-consistent calculation, as outlined in section 1.4.1. Its practical implementation is an advanced topic of computational solid state theory [1-3].

## 物理代写|固体物理代写Solid-state physics代考|The Fermi–Dirac distribution function

Let us consider a gas of $N$ electrons, confined within a volume $V$ in equilibrium at temperature $T$. Quantum mechanics provides the energy spectrum of this system, as we will extensively discuss in chapters 7 and 8 ; we label by $E_i$ with $i=1,2,3, \ldots$ the energies of the single-electron levels ${ }^8$. Hereafter in this section we assume that such energies are known and ask ourselves how likely each level is to be occupied.

Let us assume that two energy levels $E_1$ e $E_2$ are occupied with probability $n_{\mathrm{FD}}\left(E_1, T\right)$ and $n_{\mathrm{FD}}\left(E_2, T\right)$, respectively. Since the system is in equilibrium, on average the number of electrons undergoing the transition $E_1 \rightarrow E_2$ in the unit time equals the number of electrons undergoing the inverse transition $E_2 \rightarrow E_1$. This statement represents a specific formulation of the more general principle of microreversibility ${ }^9$. Such transitions could be generated by any kind of mechanism, like electron-electron or electron-defect scattering.

The number of electrons undergoing the transition $E_1 \rightarrow E_2$ per unit time is calculated as the product between the number of electrons initially occupying the level $E_1$ and the rate $R_{1 \rightarrow 2}$ of occurrence of such transition. The initial number of electrons is in turn given by the product between the total number of particles $N$ and the probability that the initial energy level is occupied. Similar definitions hold for the inverse transition. If electrons were classical particles, the occupation probability of any given level with energy $E$ would be proportional to the Boltzmann factor $\exp \left(-E / k_{\mathrm{B}} T\right)[11]$ and, therefore, we could write the classical version of microreversibility as
$$\exp \left(-E_1 / k_{\mathrm{B}} T\right) R_{1 \rightarrow 2}^{\text {classical }}=\exp \left(-E_2 / k_{\mathrm{B}} T\right) R_{2 \rightarrow 1}^{\text {classical }},$$
where $R_{1 \rightarrow 2}^{\text {classical }}$ and $R_{1 \rightarrow 2}^{\text {classical }}$ are the transition rates calculated according to classical physics. Since, however, electrons are identical and indistinguishable quantum particles they must obey the Pauli principle (see appendix E): two electrons cannot have the same set of quantum numbers. This suggests the most general way to switch from classical to quantum transition rates
$$R_{1 \rightarrow 2}^{\text {quantum }}=R_{1 \rightarrow 2}^{\text {classical }}\left[1-n_{\mathrm{FD}}\left(E_2, T\right)\right] \text { and } R_{2 \rightarrow 1}^{\text {quantum }}=R_{2 \rightarrow 1}^{\text {classical }}\left[1-n_{\mathrm{FD}}\left(E_1, T\right)\right],(6.2)$$
where the terms in square parenthesis $[\cdots]$ define the probability that the final state is not previously occupied. Quantum microreversibility is straightforwardly cast in the form
$$n_{\mathrm{FD}}\left(E_1, T\right) R_{1 \rightarrow 2}^{\text {quantum }}=n_{\mathrm{FD}}\left(E_2, T\right) R_{2 \rightarrow 1}^{\text {quantum }},$$
which, by inserting equation (6.1), leads to
$$\frac{R_{1 \rightarrow 2}^{\text {classical }}}{R_{2 \rightarrow 1}^{\text {cassical }}}=\frac{\exp \left(-E_2 / k_{\mathrm{B}} T\right)}{\exp \left(-E_1 / k_{\mathrm{B}} T\right)}=\frac{n_{\mathrm{FD}}\left(E_2, T\right)}{n_{\mathrm{FD}}\left(E_1, T\right)} \frac{1-n_{\mathrm{FD}}\left(E_1, T\right)}{1-n_{\mathrm{FD}}\left(E_2, T\right)},$$
or equivalently
$$\frac{1-n_{\mathrm{FD}}\left(E_1, T\right)}{n_{\mathrm{FD}}\left(E_1, T\right)} \exp \left(-E_1 / k_{\mathrm{B}} T\right)=\frac{1-n_{\mathrm{FD}}\left(E_2, T\right)}{n_{\mathrm{FD}}\left(E_2, T\right)} \exp \left(-E_2 / k_{\mathrm{B}} T\right)$$

## 物理代写|固体物理代写Solid-state physics代考|The Fermi–Dirac distribution function

$$\exp \left(-E_1 / k_{\mathrm{B}} T\right) R_{1 \rightarrow 2}^{\text {classical }}=\exp \left(-E_2 / k_{\mathrm{B}} T\right) R_{2 \rightarrow 1}^{\text {classical }}$$

$$R_{1 \rightarrow 2}^{\text {quantum }}=R_{1 \rightarrow 2}^{\text {classical }}\left[1-n_{\mathrm{FD}}\left(E_2, T\right)\right] \text { and } R_{2 \rightarrow 1}^{\text {quantum }}=R_{2 \rightarrow 1}^{\text {classical }}\left[1-n_{\mathrm{FD}}\left(E_1, T\right)\right],(6.2)$$

$$\frac{R_{1 \rightarrow 2}^{\text {classical }}}{R_{2 \rightarrow 1}^{\text {cassical }}}=\frac{\exp \left(-E_2 / k_{\mathrm{B}} T\right)}{\exp \left(-E_1 / k_{\mathrm{B}} T\right)}=\frac{n_{\mathrm{FD}}\left(E_2, T\right)}{n_{\mathrm{FD}}\left(E_1, T\right)} \frac{1-n_{\mathrm{FD}}\left(E_1, T\right)}{1-n_{\mathrm{FD}}\left(E_2, T\right)},$$

$$\frac{1-n_{\mathrm{FD}}\left(E_1, T\right)}{n_{\mathrm{FD}}\left(E_1, T\right)} \exp \left(-E_1 / k_{\mathrm{B}} T\right)=\frac{1-n_{\mathrm{FD}}\left(E_2, T\right)}{n_{\mathrm{FD}}\left(E_2, T\right)} \exp \left(-E_2 / k_{\mathrm{B}} T\right)$$

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