## 物理代写|固体物理代写Solid-state physics代考|PH581

statistics-lab™ 为您的留学生涯保驾护航 在代写固体物理Solid-state physics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写固体物理Solid-state physics代写方面经验极为丰富，各种代写固体物理Solid-state physics相关的作业也就用不着说。

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

In a typical semiconductor with a band gap of $1-2 \mathrm{eV}$, very few electrons will be thermally excited enough to jump from the valence band to the conduction band. It is possible to get free carriers in another way, however, by selective doping of the material.

As discussed in Section 1.11.2, typical semiconductors are made from II-VI, III-V or Group IV elements, which have eight electrons per unit cell filling eight molecular bonding states. Suppose that in a III-V semiconductor, we replace a single atom with one from Group VI. This atom will have an extra electron and an extra positive charge on its nucleus. It makes sense that the extra electron will tend to stay near the nucleus with the extra charge, but if the energy of the extra electron is high enough, it may leave the dopant atom and migrate through the crystal as a free quasiparticle.

As in the case of excitons, if the dielectric constant of the material is high, then the bound states of carriers at an impurity can be viewed as hydrogen-like states of a carrier orbiting a charged particle. If a dopant nucleus has one extra positive charge relative to the atom it replaces in the lattice, then a negative electron will see a single positive charge sitting in the background “vacuum” of the normal band structure of the material. It can then orbit this positive charge in a hydrogenic state with a binding energy given by (2.3.5). The reduced mass in this case will just be the effective mass of the electron, because the impurity cannot move, and therefore it effectively has infinite mass.

In the same way, a dopant nucleus with one less charge than the one it replaces in the lattice will look like a single negative charge to the quasiparticles. Therefore, a hole can orbit this atom in a hydrogenic bound state. Again, the negative impurity has effectively infinite mass because it cannot move, and so the binding energy of the hole will simply be proportional to the mass of the hole, according to (2.3.5) and (2.3.6).

At high temperature, that is, when $k_B T$ is much greater than the binding energy of a carrier orbiting an impurity, the carriers will no longer be bound to the impurities and will move freely through the crystal. Therefore, an atom with more positive charge on its nucleus than the one it replaces contributes an electron to the conduction band. This type of impurity is therefore called a charge donor, and a semiconductor doped with this type of atom is called $\boldsymbol{n}$-type (for “negative”). In the same way, an atom with a nucleus with more negative charge than the one it replaces contributes a hole to the valence band, which is the same as accepting an electron from the valence band. This type of impurity is therefore called an electron acceptor, and a semiconductor doped with this type of atom is called “p-type” (for “positive”).

## 物理代写|固体物理代写Solid-state physics代考|EquilibriumPopulations in Doped Semiconductors

At $T=0$, each additional electron and hole will be in its ground state, orbiting the impurity atom from which it came. At higher temperature, however, these quasiparticles can leave their bound states and move freely through the material. The probability of this happening depends on the binding energy of the carrier on the impurity. As we did in Section 2.5.1, we can write a mass action equation for the equilibrium concentration of the quasiparticles. For example, for electrons from donor atoms, at low temperature we write
$$N_e N_h=N_Q^{(e)} N_D e^{-\left(E_c-E_D\right) / k_B T},$$
where $N_D$ is the number of donor states available to the holes. At high temperature, however, this equation will break down, because the assumption that $k_B T \ll\left(E_C-E_D\right)$ made in the derivation of (2.5.8) will no longer be true. As discussed at the end of Section 2.5.1, if the number of donor states is much less than the number of states in the conduction band, as is typically the case in doped semiconductors, the chemical potential will be pushed down toward the donor state energy. In this case, we must use the Fermi-Dirac occupation number for the holes in the donor states,
$$N_h=N_D \frac{1}{e^{\left(\mu-E_D\right) / k_B T}+1} .$$
This can be rewritten as
\begin{aligned} N_h & =N_D\left(1-\frac{1}{e^{\left(\mu-E_D\right) / k_B T}+1}\right) e^{-\left(\mu-E_D\right) / k_B T} \ & =N_D\left(1-f_h\left(E_D\right)\right) e^{-\left(\mu-E_D\right) / k_B T} \ & =\left(N_D-N_h\right) e^{-\left(\mu-E_D\right) / k_B T} . \end{aligned}

# 固体物理代写

## 物理代写|固体物理代写Solid-state physics代考|EquilibriumPopulations in Doped Semiconductors

$$N_e N_h=N_Q^{(e)} N_D e^{-\left(E_c-E_D\right) / k_B T},$$

$$N_h=N_D \frac{1}{e^{\left(\mu-E_D\right) / k_B T}+1} .$$

\begin{aligned} N_h & =N_D\left(1-\frac{1}{e^{\left(\mu-E_D\right) / k_B T}+1}\right) e^{-\left(\mu-E_D\right) / k_B T} \ & =N_D\left(1-f_h\left(E_D\right)\right) e^{-\left(\mu-E_D\right) / k_B T} \ & =\left(N_D-N_h\right) e^{-\left(\mu-E_D\right) / k_B T} . \end{aligned}

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写固体物理Solid-state physics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写固体物理Solid-state physics代写方面经验极为丰富，各种代写固体物理Solid-state physics相关的作业也就用不着说。

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

As discussed in Section 1.8, at zone center and at every critical point on a zone boundary, there is a maximum or minimum of the electron energy bands, and away from these points, the energy varies as $\left(\vec{k}-\vec{k}{\text {crit }}\right)^2$. This is the same form of dependence as expected for free particles in vacuum, for speeds much less than the speed of light, $$E=\frac{\hbar^2 k^2}{2 m},$$ where $m$ is the mass of the electron in vacuum. As we saw in the discussion of $k \cdot p$ theory in Section 1.9.4, in the case of isotropic bands, the curvature of the bands in solids near a band minimum takes the form $$E=E_0+\frac{\hbar^2 k^2}{2 m{\mathrm{eff}}},$$
where $m_{\mathrm{eff}}$ is an effective mass which can be quite different from the vacuum electron mass. Once we have taken into account this effective mass due to the band structure, a free electron near the conduction band minimum will behave exactly like a free particle in vacuum.

Because the Bloch states of the crystal are eigenstates, a free electron in a perfect crystal moves without scatteringin a straight line through the crystal, behaving just like a particle in a vacuum with mass $m_{\text {eff }}$, despite the presence of the $10^{23}$ or so closely packed atoms of the crystal. It is important to remember that we are talking about a quasiparticle that does this. Of course, the underlying electrons interact with each other and the atoms constantly, but all of these interactions are taken into account in the band energies that give rise to the effective mass. The quasiparticle itself does not scatter unless it interacts with other quasiparticles or with an imperfection in the crystal. In the latter case, we treat the imperfection (which can be a single atom defect, or a large number of atoms out of place, known as a dislocation, discussed in Section 1.11.4) as an independent object sitting in the “vacuum” of a perfect crystal.

## 物理代写|固体物理代写Solid-state physics代考|Metals and the Fermi Gas

Suppose that in some solid, there is a partially filled band as illustrated in Figure 2.3. At low temperature, the electrons will fall into the lowest energy states, but because of the Pauli exclusion principle, only one electron can occupy each state. At $T=0$, the electrons will fill up all the states below some energy $E_F$, which is called the Fermi level and all the states above this level will be empty. This is called the Fermi sea As discussed in Section 1.11.2, this system will conduct electricity at low temperatures, because electrons at the top of the Fermi sea can be accelerated by an electric field into nearby, empty states with slightly higher energy, as illustrated in Figure 2.3.

At first glance, there seems to be an inconsistency. We have written down the energy of the electrons as simply the free-particle energy $E=\hbar^2 k^2 / 2 m$, where $m$ is the effective mass of the band, but what about the energy due to all the repulsive Coulomb interactions of the negatively charged electrons? In a gas of electrons of substantial density, we would expect a strong effect due to the electron charge.

The answer is that there is indeed a contribution of the electron-electron Coulomb interaction to the energy of the electrons, as well as the Coulomb interaction of the electrons with the positively charged nuclei in the solid, but this energy is already taken into account in the shape of the band. As discussed in Section 1.9.5, a proper calculation of the band structure of a material must include the effects of the electrons on each other self-consistently. Once we have a given band, almost the entire effect of the Coulomb interactions is accounted for in the value of the effective mass and the band gaps. (There will be a small, additional effect of electron-electron correlations, as will be discussed in Section 8.15.) Therefore, in our model of the electrons, the effect of Coulomb repulsion of the electrons in their ground state is ignored. We don’t ignore the Coulomb interaction of electrons and holes in an exciton, as discussed in Section 2.3, because that is an excited state. Similarly, when a metal is at finite temperature, there can be effects of the Coulomb interactions of the electrons, which will be discussed in Chapter 8 . But as we will see in Section 2.4.2, in many metals the zero-temperature approximation works well even at room temperature.

# 固体物理代写

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|固体物理代写Solid-state physics代考|PHYS460

statistics-lab™ 为您的留学生涯保驾护航 在代写固体物理Solid-state physics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写固体物理Solid-state physics代写方面经验极为丰富，各种代写固体物理Solid-state physics相关的作业也就用不着说。

## 物理代写|固体物理代写Solid-state physics代考|Dangling Bonds and Defect States

In a perfectly periodic, infinite crystal, we can imagine that every atomic orbital is involved in a bond of the type discussed above. However, in any real crystal there will be some orbitals that are not. One reason is disorder, which always occurs at some level. We have already discussed the case of long-range disorder in Section 1.8.2. In the short-range limit, we can speak of point defects at single lattice sites in a crystal. These can can consist either of missing atoms in the lattice (“vacancies”), extra atoms where they should not be (“interstitials”), atoms of a different type, giving the wrong stoichiometry for a crystal (“impurities”), and shifts of one part of the lattice relative to another part (“dislocations”). Figure 1.37 illustrates some of these point defects, which are listed in Table 1.2. Defects and dislocations play a major role in many aspects of solid state physics, as we will see in the coming chapters.

Defect states tend to have “dangling bonds,” that is, orbitals that do not substantially overlap with other atomic orbitals in the crystal. Because of this, there will be defect states with energies that fall inside the band gaps of the crystal. We can understand this by realizing that for an orbital with no overlap with a neighboring orbital, there will be no symmetric-antisymmetric energy splitting. Since the appearance of bands and band gaps is deeply connected to the overlap integrals that give the symmetric-antisymmetric splitting, orbitals with little or no overlap will look very much like the original atomic orbitals.

When there are just a few of these defects compared to the number of atoms in the whole crystal, these defect states will be mostly isolated from each other. Since they are localized to small regions, the defect states will have discrete energies, like the confined states in a square well. Thus, in addition to the Urbach tail discussed in Section 1.8.2 which describes long-range disorder, isolated defects can give sharp lines in the density of states corresponding to particular sets of defect states.

Defect states are closely related to surface states which we will examine in Section 1.12. Like defects, atoms on the surface of a crystal have orbitals that stick out into space and do not overlap substantially with other atomic orbitals. This leads to surface states that fall within the energy gaps of the bulk crystal.

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

As discussed in Section 1.3, Bloch’s theorem is based on the assumption of invariance under a given set of translations; that is, it assumes that the properties of the system are the same if we observe a location that is moved from the present point by any translation that belongs to the lattice. But every real crystal is finite; there are boundaries on the outsides. In Section 1.7, we looked at two ways to treat a finite crystal: either to assume fictional, periodic (Born-von Karman) boundary conditions, so that our imaginary crystal effectively has no surfaces, or to create standing waves with nodes at the surfaces of the crystal, as the sum of two Bloch waves with $k$ and $-k$ in opposite directions.

Kronig-Penney model of surface stateshere is another way to satisfy the boundary conditions at the surfaces. Let us return to the Kronig-Penney model we looked at in Section 1.2. The solutions were found to satisfy (1.2.5), that is,
$$\frac{\left(\kappa^2-K^2\right)}{2 \kappa K} \sinh (\kappa b) \sin (K a)+\cosh (\kappa b) \cos (K a)=\cos (k(a+b)),$$
where both $K$ and $\kappa$ depend on the energy $E$. In Section 1.2, we treated $k$ as a free parameter which we picked, and then we solved for $E$ to get the electron bands.

Suppose, instead, that we pick the energy $E$, and solve for $k$. Clearly, if the left side of (1.12.1) is greater than 1 , then $k$ cannot be real. This condition corresponds to energies inside the band gap. In that case, the inverse cosine function will give us a value of $k$ that is complex. Figure 1.38 shows the real and imaginary parts of $k$ as a function of $E$ found using (1.12.1). When $k$ is complex, the wave will have the form $\psi(x) \sim e^{i k_R x} e^{-k_I x}$, where $k_R=\operatorname{Re} k$ and $k_I=\operatorname{Im} k$. This means that the wave has a decaying part. It therefore cannot be a solution for an infinite periodic system, but it can be a solution if there is a boundary. In this case, the solution will be nonzero near the boundary and decay exponentially into the bulk. Positive $k_I$ corresponds to decay from boundary on the left, while negative $k_I$ corresponds to a state decaying from a boundary on the right. This is another way of deriving the existence of surface states, which we have already encountered in Section 1.11.4.

We cannot pick $E$ to be any value, however. For surface states, we have the additional constraint of the boundary condition that the wave function must satisfy. Suppose that there is an infinite barrier at $x=x_0$. Then we have, for the Kronig-Penney wave function of Section 1.2,
$$\psi_1\left(x_0\right)=A_1 e^{i K x_0}+B_1 e^{-i K x_0}=0,$$
where $A_1$ and $B_1$ depend on $E$ and $k$ through the matrix equation (1.2.4). We thus have two equations which we can solve for the two unknowns, $E$ and $k$. For the KronigPenney model, there is just one solution within each band gap. Figure 1.39 gives an example of a surface state for the Kronig-Penney model that satisfies this boundary condition.

# 固体物理代写

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

$$\frac{\left(\kappa^2-K^2\right)}{2 \kappa K} \sinh (\kappa b) \sin (K a)+\cosh (\kappa b) \cos (K a)=\cos (k(a+b)),$$

$$\psi_1\left(x_0\right)=A_1 e^{i K x_0}+B_1 e^{-i K x_0}=0,$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|固体物理代写Solid-state physics代考|Angle-Resolved Photoemission Spectroscopy

statistics-lab™ 为您的留学生涯保驾护航 在代写固体物理Solid-state physics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写固体物理Solid-state physics代写方面经验极为丰富，各种代写固体物理Solid-state physics相关的作业也就用不着说。

## 物理代写|固体物理代写Solid-state physics代考|Angle-Resolved Photoemission Spectroscopy

One of the most powerful tools for determining the band structure of a material is the photoemission process, by which an incoming photon kicks an electron out of the solid.

In vacuum, the electron will travel ballistically with the momentum and energy it had when it left the material. A current of electrons ejected in this way from the material can then be analyzed for their direction of motion and kinetic energy. This measurement is known as angle-resolved photoemission spectroscopy(ARPES).

Typically, the momentum of the photon is negligible compared to the momentum of the electron. The absorption of the photon can therefore be viewed as a “vertical” process, in which the electron moves to higher energy while staying at nearly the same $k$-vector. The high-energy electron can then have enough energy to overcome the work function of the material and leave the crystal.

In thinking of the process by which the electron leaves the solid, the question immediately arises of what conservation rules to apply. We have already seen that $\hbar k$ is not the true momentum of an electron; this is given by $(1.6 .10)$,
$$\left\langle\psi_{\vec{k}}|\vec{p}| \psi_{\vec{k}}\right\rangle=\hbar \vec{k}-i \hbar \int d^3 r u_{n \vec{k}}^* \nabla u_{n \vec{k}} .$$
When the electron crosses the boundary of the solid, do we conserve momentum, or do we conserve $\hbar k$ ? The answer is that we conserve $\hbar k$ in the direction parallel to the surface, not the total electron momentum. This can be understood as a consequence of the wave nature of the electrons, in analogy with Snell’s law, which is discussed in detail in Chapter 3. We write $\vec{k}=\vec{k}{|}+k{\perp} \hat{z}$, where $\vec{k}{|}$is the wave vector component parallel to the surface and $k{\perp}$ is the component perpendicular to the surface. The spacing of the wave fronts along a direction $\vec{x}$ on the surface is given by the condition $\vec{k} \cdot \vec{x}=k_{|} x=2 \pi n$, where $n$ is an integer. The distance between points of phase $2 \pi$ is therefore $\Delta x=2 \pi / k_{|}$. This spacing must be the same for the wave both inside and outside of the solid, a condition generally known as phase matching Although $\vec{k}_{|}$is conserved, the total momentum of the electron is in general not conserved. Therefore, the crystal must recoil slightly, taking up the difference

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

In general, electronic bands and molecular bonds are intimately related. Harrison (1980) gives an excellent discussion of the relation of chemical bonds and bands in detail.

As discussed in Section 1.1.2, when two electron orbitals overlap, two new states are created that can be approximated as symmetric and antisymmetric linear combinations of the original orbitals. As mentioned in Section 1.1.2, these are called the bonding and antibonding states, respectively.

Figure 1.34 shows how bonding occurs in the case of two atoms with overlapping $s$-orbitals, in the LCAO approximation. If there is one electron per atom, then the electrons from both atoms can fall into the lowest state, thus reducing the total energy of the pair of atoms. This is why we say they are bonded. Separating the two atoms would increase the total energy of the system, that is, would require work.

If the two $s$-orbitals were filled, in other words, if each original orbital had two electrons, then the energy splitting due to the wave function overlap would not lead to bonding. Two of the electrons would fall into the lower, bonding state, while the remaining two would have to go into the higher, antibonding state, because of the Pauli exclusion principle. Since, according to (1.1.12), the average energy of the two states remains the same, there is no decrease of the total energy of the atoms. This is why, for example, helium atoms do not form homo atomic molecules.

Figure 1.35 shows the case of two atoms with partially filled, overlapping $s$ – and p-orbitals. Without knowing the exact location of the atoms, we cannot say how the orbitals will split when the atomic states overlap, but we can say that, in general, there will be an equal number of states shifting upward and downward by the same amounts. This follows from the general mathematical theorem that the sum of the eigenvalues of a matrix is equal to the trace of the matrix, no matter how large the off-diagonal elements are. In the LCAO approximation, we construct a square matrix as in (1.1.11), in which the diagonal elements are the unperturbed atomic state energies, and the off-diagonal elements are the coupling integrals. If these are nonzero, the energy eigenvalues will shift, but the sum of all the shifted energies will remain the same. This means that if some states are shifted to lower energy, other states must shift upward by the same amount.

If the total number of electrons in the atomic states is less than half of the total number of states, then all of the electrons can lower their energy when the atoms get near to each other. This leads to stable bonding. As illustrated in Figure 1.35, if the total number of electrons in the two types of orbitals is equal to eight, then the lower, bonding orbitals will be completely full and the upper, antibonding orbitals will be completely empty. This is known as a “full shell” in chemistry terminology.

# 固体物理代写

## 物理代写|固体物理代写Solid-state physics代考|Angle-Resolved Photoemission Spectroscopy

$$\left\langle\psi_{\vec{k}}|\vec{p}| \psi_{\vec{k}}\right\rangle=\hbar \vec{k}-i \hbar \int d^3 r u_{n \vec{k}}^* \nabla u_{n \vec{k}} .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|固体物理代写Solid-state physics代考|Electron Band Calculationsin Three Dimensions

statistics-lab™ 为您的留学生涯保驾护航 在代写固体物理Solid-state physics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写固体物理Solid-state physics代写方面经验极为丰富，各种代写固体物理Solid-state physics相关的作业也就用不着说。

## 物理代写|固体物理代写Solid-state physics代考|Electron Band Calculationsin Three Dimensions

So far we have looked at the wave function of one electron in fixed external potential. In solids, however, the potential energy felt by an electron arises from the interaction of each electron with all the nuclei and all the other electrons. The potential energy felt by an electron is not only given by the positively charged nuclei, but also by the average negative charge of all the other electrons.

If a crystal has a certain periodicity, then the potential created from all these particles will have that periodicity, so all of the above theorems for periodic potentials still apply. Determining the exact nature of the electron bands, however, is a difficult task. We cannot simply solve for the eigenstate of a single electron in a fixed potential; we must solve for the eigenstates of the whole set of electrons, taking into account exchange between the identical electrons, which must be treated according to Fermi statistics. Methods of treating the Fermi statistics of electrons will be discussed in Chapter 8 .

In general, the calculation of band structure is not an exact science. Typically, determining the band structure of a given solid involves interaction between experiment and theory – the crystal symmetry is determined by x-ray scattering, a band structure is calculated, this is corrected by other experiments such as optical absorption and reflectivity, etc. Many band structure calculations use experimental inputs from chemistry such as the electronegativity of ions, bond lengths, and so on. Calculating band structures from first principles, using nothing but the charge and masses of the nuclei, is still an area of frontier research, involving high-level math and supercomputers.

In this book, we will typically treat the electron bands as known functions for a given solid. Band diagrams have been published for many solids and tabulated in books, for example, Madelung (1996). At the same time, there are several useful approximation methods which allow us to write simple mathematical formulas for the bands without needing to go through all of the calculations to generate a band structure. The value of these approximations is not so much to predict the actual band structures quantitatively; rather, these models help to give us physical intuition about the nature of electron bands.

## 物理代写|固体物理代写Solid-state physics代考|How to Read a Band Diagram

In a three-dimensional crystal, the full calculation of all the band energies involves finding the energy of each band at every point in the three-dimensional Brillouin zone. This is a large amount of information, which we need to present in a simple fashion for it to be useful.

As discussed in Section 1.6, there are certain critical points in the Brillouin zone which correspond to the points on the surfaces of the zone that are half way between the origin and another reciprocal lattice vector. Figure 1.25 gives the standard labeling of these critical points for common lattice structures. Typically, band structure calculations give the band energies along lines from the center of the Brillouin zone to one of these points, or from one of these points to another one. Figure 1.26(a) shows a typical band structure plot for silicon, a cubic crystal. The critical points are labeled according to the drawing shown in Figure $1.25(\mathrm{~b})$. Note that the diagram is not symmetric about the $\Gamma$ point because two different paths away from this point are plotted. Note also that the $\mathrm{U} / \mathrm{K}$ point is not the midpoint between two reciprocal lattice points, and therefore the slope of the bands is not zero in every direction there; in particular, the slope is not zero along lines that are not normal to the zone boundary. The $\mathrm{L}$ and $\mathrm{X}$ points are critical points, and therefore if the bands are plotted with enough detail, one will see that they have zero slope at those points.

Figure 1.26(b) shows the density of states for the same crystal. As seen in this figure, van Hove singularities (discontinuities in the slope) correspond to critical points in the band structure. When bands overlap in energy, the density of states is just the sum of the density of states of the two bands. Notice that there is a gap in the density of states which corresponds to the energy gap in one band at the $\Gamma$ point (zone center) and the minimum of the next higher band at the $X$ point (the zone boundary in the [100] direction).

# 固体物理代写

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|固体物理代写Solid-state physics代考|Electron Band Calculationsin Three Dimensions

statistics-lab™ 为您的留学生涯保驾护航 在代写固体物理Solid-state physics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写固体物理Solid-state physics代写方面经验极为丰富，各种代写固体物理Solid-state physics相关的作业也就用不着说。

## 物理代写|固体物理代写Solid-state physics代考|Electron Band Calculationsin Three Dimensions

So far we have looked at the wave function of one electron in fixed external potential. In solids, however, the potential energy felt by an electron arises from the interaction of each electron with all the nuclei and all the other electrons. The potential energy felt by an electron is not only given by the positively charged nuclei, but also by the average negative charge of all the other electrons.

If a crystal has a certain periodicity, then the potential created from all these particles will have that periodicity, so all of the above theorems for periodic potentials still apply. Determining the exact nature of the electron bands, however, is a difficult task. We cannot simply solve for the eigenstate of a single electron in a fixed potential; we must solve for the eigenstates of the whole set of electrons, taking into account exchange between the identical electrons, which must be treated according to Fermi statistics. Methods of treating the Fermi statistics of electrons will be discussed in Chapter 8 .

In general, the calculation of band structure is not an exact science. Typically, determining the band structure of a given solid involves interaction between experiment and theory – the crystal symmetry is determined by x-ray scattering, a band structure is calculated, this is corrected by other experiments such as optical absorption and reflectivity, etc. Many band structure calculations use experimental inputs from chemistry such as the electronegativity of ions, bond lengths, and so on. Calculating band structures from first principles, using nothing but the charge and masses of the nuclei, is still an area of frontier research, involving high-level math and supercomputers.

In this book, we will typically treat the electron bands as known functions for a given solid. Band diagrams have been published for many solids and tabulated in books, for example, Madelung (1996). At the same time, there are several useful approximation methods which allow us to write simple mathematical formulas for the bands without needing to go through all of the calculations to generate a band structure. The value of these approximations is not so much to predict the actual band structures quantitatively; rather, these models help to give us physical intuition about the nature of electron bands.

## 物理代写|固体物理代写Solid-state physics代考|How to Read a Band Diagram

In a three-dimensional crystal, the full calculation of all the band energies involves finding the energy of each band at every point in the three-dimensional Brillouin zone. This is a large amount of information, which we need to present in a simple fashion for it to be useful.

As discussed in Section 1.6, there are certain critical points in the Brillouin zone which correspond to the points on the surfaces of the zone that are half way between the origin and another reciprocal lattice vector. Figure 1.25 gives the standard labeling of these critical points for common lattice structures. Typically, band structure calculations give the band energies along lines from the center of the Brillouin zone to one of these points, or from one of these points to another one. Figure 1.26(a) shows a typical band structure plot for silicon, a cubic crystal. The critical points are labeled according to the drawing shown in Figure $1.25(\mathrm{~b})$. Note that the diagram is not symmetric about the $\Gamma$ point because two different paths away from this point are plotted. Note also that the $\mathrm{U} / \mathrm{K}$ point is not the midpoint between two reciprocal lattice points, and therefore the slope of the bands is not zero in every direction there; in particular, the slope is not zero along lines that are not normal to the zone boundary. The $\mathrm{L}$ and $\mathrm{X}$ points are critical points, and therefore if the bands are plotted with enough detail, one will see that they have zero slope at those points.

Figure 1.26(b) shows the density of states for the same crystal. As seen in this figure, van Hove singularities (discontinuities in the slope) correspond to critical points in the band structure. When bands overlap in energy, the density of states is just the sum of the density of states of the two bands. Notice that there is a gap in the density of states which corresponds to the energy gap in one band at the $\Gamma$ point (zone center) and the minimum of the next higher band at the $X$ point (the zone boundary in the [100] direction).

# 固体物理代写

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写固体物理Solid-state physics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写固体物理Solid-state physics代写方面经验极为丰富，各种代写固体物理Solid-state physics相关的作业也就用不着说。

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

The reciprocal lattice has a natural connection to $\mathrm{x}$-ray scattering. Suppose a plane wave with wave vector $\vec{k}0$ impinges on a crystal, as shown in Figure 1.17. We write this plane wave as $$A{\text {in }}=e^{i\left(\vec{k}_0 \cdot \vec{r}-\omega t\right)}$$
Atoms in the crystal will lead to scattering of the incoming wave. In the Fraunhofer limit, a scattered wave far away can also be approximated by a plane wave with wave vector $\vec{k}$. We define the scattering vectoras the difference between the incoming and outgoing (scattered) wave vectors:
$$\vec{s}=\vec{k}-\vec{k}_0 .$$

If $\vec{a}$ is the vector from one atom to another, then the phase difference between the scattered waves from these two atoms will be
$$\delta=\vec{k} \cdot \vec{a}-\overrightarrow{k_0} \cdot \vec{a}=\left(\vec{k}-\overrightarrow{k_0}\right) \cdot \vec{a}=\vec{s} \cdot \vec{a} .$$
The amplitude of the scattered wave from these two atoms will be proportional to
$$A_{\mathrm{sum}}=\left(e^{i(0)}+e^{i \vec{s} \cdot \vec{a}}\right) e^{-i \omega t}$$
and therefore the intensity will be proportional to
\begin{aligned} I=A_{\mathrm{sum}}^* A_{\mathrm{sum}} & =\left(1+e^{-i \vec{s} \cdot \vec{a}}\right)\left(1+e^{i \vec{s} \cdot \vec{a}}\right) \ & =2(1+\cos \vec{s} \cdot \vec{a}) . \end{aligned}

## 物理代写|固体物理代写Solid-state physics代考|General Properties of Bloch Functions

Even without knowing anything about the periodic potential in a particular crystal, Bloch’s theorem allows us to make several general statements about the eigenstates of the system.

Bloch theorem as a Fourier serißse can use Fourier transform theory to express the cell function $u_{n \vec{k}}(\vec{r})$ in terms of the reciprocal lattice vectors. Since this function has the same periodicity as the lattice, we can use the Fourier transform formula
\begin{aligned} F_u(\vec{q}) & =\int_{-\infty}^{\infty} d^3 r u_{n \vec{k}}(\vec{r}) e^{i \vec{q} \cdot \vec{r}} \ & =\sum_{\vec{R}} e^{i \vec{q} \cdot \vec{R}}\left(\int_{\text {cell }} d^3 r_b u_{n \vec{k}}\left(\vec{r}_b\right) e^{i \vec{q} \cdot \vec{r}_b}\right), \end{aligned}
where in the second line we have written $\vec{r}=\vec{R}+\vec{r}_b$, and have broken the integral over all space into a sum over all Bravais lattice positions $\vec{R}$ and an integral over the relative coordinate $\vec{r}_b$ within each primitive cell. We have introduced a new reciprocal-space variable $\vec{q}$ because we are leaving $\vec{k}$ constant.

As discussed in Section 1.4, the Fourier transform (1.6.1) has nonzero values only when $\vec{q}$ is equal to a reciprocal lattice vector $\vec{Q}$. The cell function $u_{n \vec{k}}(\vec{r})$, which is the inverse Fourier transform of $F_u$, can therefore be written as a sum over the full set of reciprocal lattice vectors,
$$u_{n \vec{k}}(\vec{r})=\sum_{\vec{Q}} C_{n \vec{k}}(\vec{Q}) e^{-i \vec{Q} \cdot \vec{r}},$$
where $C_{n \vec{k}}(\vec{Q})$ is a weight factor. ${ }^2$ The dependence of $C_{n \vec{k}}(\vec{Q})$ on $\vec{k}$ gives an overall multiplier for the whole set of weight factors.
The full Bloch functions are then given by
\begin{aligned} \psi_{n \vec{k}}(\vec{r}) & =\frac{1}{\sqrt{V}} u_{n \vec{k}}(\vec{r}) e^{i \vec{k} \cdot \vec{r}} \ & =\frac{1}{\sqrt{V}} \sum_{\vec{Q}} C_{n \vec{k}}(\vec{Q}) e^{i(\vec{k}-\vec{Q}) \cdot \vec{r}} \end{aligned}

# 固体物理代写

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

$$\vec{s}=\vec{k}-\vec{k}_0 .$$

$$\delta=\vec{k} \cdot \vec{a}-\overrightarrow{k_0} \cdot \vec{a}=\left(\vec{k}-\overrightarrow{k_0}\right) \cdot \vec{a}=\vec{s} \cdot \vec{a} .$$

$$A_{\mathrm{sum}}=\left(e^{i(0)}+e^{i \vec{s} \cdot \vec{a}}\right) e^{-i \omega t}$$

\begin{aligned} I=A_{\mathrm{sum}}^* A_{\mathrm{sum}} & =\left(1+e^{-i \vec{s} \cdot \vec{a}}\right)\left(1+e^{i \vec{s} \cdot \vec{a}}\right) \ & =2(1+\cos \vec{s} \cdot \vec{a}) . \end{aligned}

## 物理代写|固体物理代写Solid-state physics代考|General Properties of Bloch Functions

\begin{aligned} F_u(\vec{q}) & =\int_{-\infty}^{\infty} d^3 r u_{n \vec{k}}(\vec{r}) e^{i \vec{q} \cdot \vec{r}} \ & =\sum_{\vec{R}} e^{i \vec{q} \cdot \vec{R}}\left(\int_{\text {cell }} d^3 r_b u_{n \vec{k}}\left(\vec{r}_b\right) e^{i \vec{q} \cdot \vec{r}_b}\right), \end{aligned}

$$u_{n \vec{k}}(\vec{r})=\sum_{\vec{Q}} C_{n \vec{k}}(\vec{Q}) e^{-i \vec{Q} \cdot \vec{r}},$$

## 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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