Posted on Categories:固体物理, 物理代写

## 物理代写|固体物理代写Solid Physics代考|k · p Theory

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## 物理代写|固体物理代写SOLID PHYSICS代考|k · p Theory

Another approximation method, which is very useful for understanding interactions between bands, uses a perturbation expansion of a different type. This method takes note of the fact that the critical points of the Brillouin zone have well-defined properties. If the energies at these critical points are known, then we can treat the band energy at a nearby point in the Brillouin zone as the sum of the energy at the critical point plus a small perturbation.
We begin by writing the Schrödinger equation in terms of the Bloch functions,
$$\left(\frac{p^2}{2 m}+U(\vec{r})\right) u_{n \vec{k}}(\vec{r}) e^{i \vec{k} \cdot \vec{r}}=E_n(\vec{k}) u_{n \vec{k}}(\vec{r}) e^{i \vec{k} \cdot \vec{r}},$$

where $\vec{p}=-i \hbar \nabla$. Since the derivative of $e^{i \vec{k} \cdot \vec{r}}$ is known, we can rewrite this as
$$\left(\frac{1}{2 m}|\vec{p}+\hbar \vec{k}|^2+U(\vec{r})\right) u_{n \vec{k}}(\vec{r})=E_n(\vec{k}) u_{n \vec{k}}(\vec{r}) .$$
We can then write this as the sum of three terms,
$$\left(H_0+H_1+H_2\right) u_{n \vec{k}}(\vec{r})=E_n(\vec{k}) u_{n \vec{k}}(\vec{r})$$
where
\begin{aligned} H_0 & =\frac{p^2}{2 m}+U(\vec{r}) \ H_1 & =\frac{\hbar}{m} \vec{k} \cdot \vec{p} \ H_2 & =\frac{\hbar^2 k^2}{2 m} . \end{aligned}

## 物理代写|固体物理代写SOLID PHYSICS代考|Other Methods ofCalculating Band Structure

We have already seen in Section 1.6 that the Bloch states of different bands are orthogonal. Since the core electrons are nearly the same as the atomic states, which have slow variation near the atomic nucleus, this means that the electron wave functions for higher levels will tend to have strong spatial oscillations near a nucleus, so that the overlap integral $\int \psi_n^* \psi_m d^3 r$ will vanish. This leads to problems for numerical calculations.

One way to solve for the higher band states without using rapidly oscillating wave functions is the pseudopotential method. In this method, instead of using just the potential $U(\vec{r})$ of the bare nucleus, a new $U(\vec{r})$ is used which includes the effects of the Coulomb repulsion and Pauli exclusion of the core electrons, to repel the electrons in higher states from the core region.

Using this new $U(\vec{r})$, the upper electron states can be calculated using the nearly free electron approximation; the inner, core electron states are assumed to remain nearly the same as the atomic core states. This strong distinction between the two types of states is one of the major assumptions of this method.

There is no exact way of calculating the potential $U(\vec{r})$; in this method one simply starts with a guess and then improves $U(\vec{r})$ by iteration. This can be done either by comparing the calculated band structure to experimental data or by adjusting $U(\vec{r})$ to give self-consistency. Once the valence electron states are calculated, the local charge density due to these electrons can be calculated, which is proportional to $\rho(\vec{r})=\psi^*(\vec{r}) \psi(\vec{r})$. The Coulomb repulsion from this charge density then gives an adjustment to $U(\vec{r})$. Eventually, the adjusted $U(\vec{r})$ will not change upon iteration, when it is consistent with the charge density of the valence states.

The band structure of silicon in Figure 1.26(a) was calculated using a pseudopotential method. Notice how the bands have the character of nearly free electrons – for example, the lowest energy band is nearly parabolic and the next energy band has a maximum at zone center, as in Figure 1.30. In general, pseudopotential methods give reasonable predictions of many band structure parameters, but still require some experimental input for realistic calculations.

## 物理代写|固体物理代写SOLID PHYSICS代考|k · p Theory

$$\left(\frac{p^2}{2 m}+U(\vec{r})\right) u_{n \vec{k}}(\vec{r}) e^{i \vec{k} \cdot \vec{r}}=E_n(\vec{k}) u_{n \vec{k}}(\vec{r}) e^{i \vec{k} \cdot \vec{r}},$$

$$\left(\frac{1}{2 m}|\vec{p}+\hbar \vec{k}|^2+U(\vec{r})\right) u_{n \vec{k}}(\vec{r})=E_n(\vec{k}) u_{n \vec{k}}(\vec{r}) .$$

$$\left(H_0+H_1+H_2\right) u_{n \vec{k}}(\vec{r})=E_n(\vec{k}) u_{n \vec{k}}(\vec{r})$$

\begin{aligned} H_0 & =\frac{p^2}{2 m}+U(\vec{r}) \ H_1 & =\frac{\hbar}{m} \vec{k} \cdot \vec{p} \ H_2 & =\frac{\hbar^2 k^2}{2 m} . \end{aligned}

## 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。

Posted on Categories:固体物理, 物理代写

## 物理代写|固体物理代写Solid Physics代考|Density ofStatesatCritical Points

avatest固体物理Solid Physics代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。avatest™， 最高质量的固体物理Solid Physics作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于统计Statistics作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此固体物理Solid Physics作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

avatest™ 为您的留学生涯保驾护航 在物理Physics代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的物理Physics代写服务。我们的专家在固体物理Solid Physics代写方面经验极为丰富，各种固体物理Solid Physics相关的作业也就用不着 说。

## 物理代写|固体物理代写SOLID PHYSICS代考|Density ofStatesatCritical Points

As we saw in Section 1.6, $\nabla_{\vec{k}} E$ vanishes at zone center and at the boundaries of the Brillouin zone. This means that the density of states will have special properties at these points. It might seem that the density of states diverges at these points, but this is not always the case. For example, in the case of isotropic bands, the density of states formula (1.8.4) can be simplified to
$$\mathcal{D}(E) d E=\frac{V}{(2 \pi)^3} 4 \pi d E \frac{1}{\left|\nabla_{\vec{k}} E\right|} k^2(E)$$

Since the band at zone center must be at a minimum or maximum, we can expand the energy in powers of $k$ as
$$E(k)=E_0+\left.\frac{1}{2} \frac{\partial^2 E}{\partial k^2}\right|{k=0} k^2+\cdots$$ The leading order of the gradient of $E$ is therefore linear in $k$, which means that the density of states is proportional to $k$, which implies $$\mathcal{D}(E) d E \propto \sqrt{\left(E-E_0\right)} d E$$ The same thing occurs at the critical points on the zone boundaries where $\nabla{\vec{k}} E$ vanishes, discussed in Section 1.6. In general, the band minimum or maximum at the critical point can be expanded in powers of $k$ as
$$E(k)=E_0+\frac{1}{2} \sum_{i, j} \frac{\partial^2 E}{\partial k_i \partial k_j}\left(k_i-k_i^{\mathrm{crit}}\right)\left(k_j-k_j^{\mathrm{crit}}\right)+\cdots$$

## 物理代写|固体物理代写SOLID PHYSICS代考|Disorderand Density ofStates

Density-of-states plots give us a natural way to look at the effect of disorder, that is, what happens to the electron bands when a crystal is not perfectly periodic. As discussed in Section 1.1 , bands and band gaps appear whenever there is overlap of atomic orbitals, regardless of periodicity.

In the long wavelength limit (when the characteristic length of the disorder is much longer than the atomic lattice spacing), we can model disorder as regions with slightly larger or smaller spacing between atoms. We can then approximate the effect of the disorder by recalculating the band energy for a larger or smaller lattice spacing in each region. Larger spacing corresponds to less orbital overlap of adjacent atoms, which means less bonding-antibonding splitting. This corresponds to a smaller band gap; in other words, the upper, antibonding states will have lower energy and the lower, bonding states will have higher energy. This means that in a region of larger lattice spacing, there will be electron states inside the nominal energy gap.

In the absence of any other information, we can assume that the disorder is distributed randomly. In the long wavelength limit, we can view the disordered crystal as a set of perfectly ordered crystals with band gaps that are distributed according to a Gaussian distribution, according to the central limit theorem,
$$P\left(E_g\right)=\frac{1}{\sqrt{2 \pi}(\Delta E)} e^{-\left(E_g(0)-E_g\right)^2 / 2(\Delta E)^2},$$
where $E_g$ is the band gap for a perfectly ordered crystal and $\Delta E_g$ is a characteristic range of energy fluctuations. The total density of states of the crystal will then be given by the convolution of this distribution with the density of states for a periodic structure,
$$\mathcal{D}(E)=\int d E_g \mathcal{D}\left(E-E_g\right) P\left(E_g\right)$$

The effect of the convolution is to smear out the band gaps of a solid. Disorder does not necessarily eliminate the existence of bands and band gaps, however. Figure 1.23(b) illustrates how a small degree of disorder smears the bands, while leaving them still much the same. In general, every real crystal has some degree of band smearing because there is always some degree of disorder.

## 物理代写|固体物理代写SOLID PHYSICS代考|Bloch’sTheorem

$$\psi_{n \vec{k}}(\vec{r}+\vec{R})=\psi_{n \vec{k}}(\vec{r}) e^{i \vec{k} \cdot \vec{R}}$$

\begin{aligned} & \psi_{n \vec{k}}(\vec{r}+\vec{R}) e^{-i \vec{k} \cdot \vec{r}}=\psi_{n \vec{k}}(\vec{r}) e^{i \vec{k} \cdot \vec{R}} e^{-i \vec{k} \cdot \vec{r}} \ & \psi_{n \vec{k}}(\vec{r}+\vec{R}) e^{-i \vec{k} \cdot(\vec{R}+\vec{r})}=\psi_{n \vec{k}}(\vec{r}) e^{-i \vec{k} \cdot \vec{r}} . \end{aligned}

$$\psi_{n \vec{k}}(\vec{r})=\frac{1}{\sqrt{V}} u_{n \vec{k}}(\vec{r}) e^{i \vec{k} \cdot \vec{r}},$$

## 物理代写|固体物理代写SOLID PHYSICS代考|BravaisLattices and ReciprocalSpace

$$\vec{R}=N_1 \vec{a}_1+N_2 \vec{a}_2+N_3 \vec{a}_3$$

## 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。