## PHYS4510 Solid State Physics课程简介

This course will investigate the structural and physical properties of materials by developing better understanding of crystal structure with particular emphasis on studying the electrical and magnetic behavior of solids. The course shows how various types of phenomena (resistivity, magnetism, superconductivity) are related. The main objectives of the course are to increase the students’ understanding and knowledge of solid state physics and to improve their problem solving ability, including the design of experiments which examine principles in condensed matter physics.

## PREREQUISITES

Solid state physics is a branch of condensed matter physics that studies the physical properties of materials in the solid phase. The understanding of the atomic and electronic structure of crystals is fundamental to solid state physics. The course will cover topics such as crystal lattices and symmetry, diffraction and scattering of X-rays, electrons, and neutrons, electronic band structure, transport phenomena, and magnetic properties of solids. The course will also introduce various experimental techniques used in solid state physics, such as electrical conductivity and resistivity measurements, magnetic susceptibility measurements, and thermal conductivity measurements.

The course aims to enhance the students’ problem-solving skills by providing them with the tools to analyze and interpret experimental data. By understanding the relationships between different types of phenomena in solids, such as resistivity and magnetism, the students will be able to design experiments that test theoretical principles. The course will also emphasize the importance of mathematical modeling and numerical simulations in the analysis of experimental data and the understanding of the physical properties of materials.

By the end of the course, the students will have gained a deep understanding of the physical properties of materials in the solid state, including the electrical and magnetic behavior of solids. The students will be able to apply their knowledge to solve complex problems in condensed matter physics and design experiments that test theoretical principles. The course will also prepare the students for further studies in solid state physics or related fields such as materials science, nanotechnology, and electrical engineering.

## PHYS4510 Solid State Physics HELP（EXAM HELP， ONLINE TUTOR）

Problem 1. Consider a tight-binding Hamiltonian that acts upon a single band of localized orbitals in one dimension
$$\hat{H}=\sum_m \cos (2 \pi m \alpha)|m\rangle\langle m|+t \sum_m \frac{1}{2}(|m\rangle\langle m+1|+| m\rangle\langle m-1|),$$
where $\alpha=5 / 3$. The integer $m$ should be thought of as indexing sites along the chain of atoms. The ket $|m\rangle$ locates an electron on atom $m$ (e.g., $\langle x \mid m\rangle=\psi(x-m)$ is the wave function, or “orbital”, which decays fast away from the position of an atom $\mathrm{m})$.
(a) What is the periodicity of the Hamiltonian?
(b) Use Bloch theorem to reduct the eigenvalue problem of an infinite matrix $\hat{H}$ (obtained by representing the Hamiltonian in the basis of orbitals $|m\rangle$ ) to the solution of a small finite matrix equation [note that the size of this matrix will be equal to the periodicity of the Hamiltonian found in (a)].
(c) Compute and plot the bands as a function of Bloch wave vector $k$ throughout the first Brillouin zone (this task will have to be carried our numerically).

(a) The Hamiltonian has two terms: the first term is a diagonal matrix with entries $\cos(2\pi m\alpha)$ for each $m$, while the second term is a tridiagonal matrix with entries $t/2$ on the diagonal and off-diagonal entries of $t/2$ as well. Since the first term is diagonal, the periodicity of the Hamiltonian is determined solely by the second term. The second term has a periodicity of 2 (i.e., it repeats every two sites), so the periodicity of the Hamiltonian is also 2.

(b) By Bloch’s theorem, the eigenstates of the Hamiltonian can be written in the form $|\psi_k\rangle=\sum_m e^{ikm}|m\rangle$, where $k$ is the Bloch wave vector. Substituting this into the Hamiltonian, we have \begin{align} \hat{H}|\psi_k\rangle&=\left(\sum_m \cos (2 \pi m \alpha)|m\rangle\langle m|+t \sum_m \frac{1}{2}(|m\rangle\langle m+1|+| m\rangle\langle m-1|)\right)\sum_n e^{ikn}|n\rangle\ &=\sum_m \cos (2 \pi m \alpha)\sum_n e^{ikn}|m\rangle\langle m|n\rangle+t \sum_m \frac{1}{2}\left(|m\rangle\langle m+1|\sum_n e^{ikn}|n\rangle\right.+| m\rangle\langle m-1|\left.\sum_n e^{ikn}|n\rangle\right)\ &=\sum_m \cos (2 \pi m \alpha)e^{ikm}|\psi_k\rangle+t \sum_m \frac{1}{2}(e^{ik(m+1)}+e^{ik(m-1)})|\psi_k\rangle\ &=\left(\sum_m \cos (2 \pi m \alpha)e^{ikm}|m\rangle\langle m|+t \sum_m \frac{1}{2}(e^{ik(m+1)}+e^{ik(m-1)})|m\rangle\langle m|\right)|\psi_k\rangle\ &=\hat{H}k|\psi_k\rangle, \end{align} where $\hat{H}k$ is a tridiagonal matrix with entries \begin{align} \langle m|\hat{H}k|n\rangle&=\cos(2\pi m\alpha)\delta{mn}+\frac{t}{2}\left(\delta{m,n+1}e^{ik}+\delta{m,n-1}e^{-ik}\right). \end{align} Since the Hamiltonian has periodicity 2, the matrix $\hat{H}_k$ is a $2\times 2$ matrix.

Problem 2. Consider a one-dimensional solid of length $L=N a$ made up of $N$ diatomic molecules, where the interatomic spacing within a molecule is $b(b<a / 2)$. The centers of adjacent molecules are a distance $a$ apart. We represent the potential energy as a sum of delta functions on each atom:
$$V=-A \sum_{n=0}^{N-1}\left[\delta\left(x-n a+\frac{b}{2}\right)+\delta\left(x-n a-\frac{b}{2}\right)\right]$$
with $A$ being the positive quantity and $n=0,1,2, \ldots, N-1$. The potential is “shown” in the Figure 1.
(a) Consider free electrons in this solid (i.e., neglect $V$ for the moment) with periodic boundary conditions. Derive the allowed values of the electron wave vector $k$ and normalize the wave functions.
(b) Expressing the potential as a Fourier series $V=\sum V_q e^{i q x}$, find the allowed values of $q$ and the coefficients $V_q$.

(a) Without the potential $V$, the system is a periodic lattice with lattice spacing $a$. The allowed values of the wave vector $k$ are determined by the periodic boundary conditions, which require that the wave function be periodic with period $L=Na$. That is, $\psi(x+L)=\psi(x)$. Since $\psi(x+L)=\psi(x+N a)$ and the lattice spacing is $a$, we have $\psi(x+N a)=\psi(x)$, which implies that the wave function is periodic with period $a$. Therefore, we can write $\psi(x)=\sum_{n=-\infty}^{\infty} c_n e^{i n k x}$, where $k$ is the wave vector and $c_n$ are coefficients to be determined. Substituting this expression into the periodic boundary condition $\psi(x+L)=\psi(x)$ gives \begin{align*} \sum_{n=-\infty}^{\infty} c_n e^{i n k (Na+x)} &= \sum_{n=-\infty}^{\infty} c_n e^{i n k x} \ \sum_{n=-\infty}^{\infty} c_n e^{i n k Na} e^{i n k x} &= \sum_{n=-\infty}^{\infty} c_n e^{i n k x} \ \sum_{n=-\infty}^{\infty} c_n e^{i n k Na} &= 1. \end{align*} The last line follows from the fact that the coefficients $c_n$ are normalized, $\int_{-a/2}^{a/2} |\psi(x)|^2 dx = 1$. Thus, we have the condition $e^{i k N a} \sum_{n=-\infty}^{\infty} c_n e^{i n k a}=1$. This implies that the sum on the right-hand side is a geometric series with ratio $e^{i k a}$, so we can write \begin{align*} \sum_{n=-\infty}^{\infty} c_n e^{i n k a} &= \frac{1}{1-e^{i k N a}} \ &= \frac{e^{-i k N a/2}}{\sin(k N a/2)}. \end{align*} The last line follows from the identity $\sin(x) = (e^{ix}-e^{-ix})/(2i)$. Therefore, the allowed values of the wave vector $k$ are given by $k=\frac{2\pi m}{N a}$ for integer $m$, and the corresponding normalized wave functions are

\psi(x) = \sqrt{\frac{1}{Na}} e^{i \frac{2\pi m}{N a} x} \frac{e^{-i k N a/2}}{\sin(k N a/2)}.ψ(x)=Na1​​eiNa2πm​xsin(kNa/2)e−ikNa/2​.

(b) To express the potential $V$ as a Fourier series, we write \begin{align*} V &= -A \sum_{n=0}^{N-1}\left[\delta\left(x-n a+\frac{b}{2}\right)+\delta\left(x-n a-\frac{b}{2}\right)\right] \ &= -A \sum_{n=-\infty}^{\infty} \left[\delta\left(x-na-\frac{b}{2}\right)+\delta\left(x-na+\frac{b}{2}\right)\right] \ &= –

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

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