# 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|Some preliminary concepts

In our presentation of elementary solid state physics, semiconductors play a twofold pedagogical and applicative role: on the one hand, they represent a paradigmatic playground for developing a microscopic theory of charge transport in non-metallic systems (that is in systems where the free electron model cannot be used as effectively as in metals); on the other hand, they have a fundamental role in modern information and communication (nano)technologies. For these reasons they deserve the careful treatment here developed where both issues are addressed, together with a statistical treatment of charge carrier populations either in equilibrium condition or out of equilibrium.

Throughout this chapter we assume that the band structure of a semiconductor is known, as obtained by applying anyone of the methods developed in chapter 8 . Therefore, concepts like the valence (VB) and conduction $(\mathrm{CB})$ band, the energy gap and the effective mass of charge carriers will be extensively used.

We will distinguish between intrinsic and extrinsic semiconductors, provided that a pristine sample or a doped one (see section 2.5.1) is, respectively, considered. Basically, they differ in that only thermally excited carriers from the VB to the CB are available in pristine materials (hereafter referred to as intrinsic carriers), while additional electrons in $\mathrm{CB}$ and holes in $\mathrm{VB}$ are added in doped semiconductors (hereafter referred to as extrinsic carriers). This is of course the result of doping, that is, the alteration of the chemistry of the pristine material by insertion of donor or acceptor impurities: in the first case, the impurity atoms carry an excess of electrons with respect to the pristine material, while in the second case they lack of electrons or, equivalently, they carry an excess of holes with respect to the intrinsic population. In engineering applications doped semiconductors are mainly used.
Given the fundamental role played in semiconductor physics by doping, a preliminary question should be addressed, namely: what is the effect of doping on the underlying band structure of the material? The answer to this question is not at all trivial, since it requires the extensive use of the quantum mechanical perturbation theory: the presence of a dopant defect is described as a perturbation to the ideal case of perfect crystal and the way such a perturbation affects the band structure is accordingly calculated ${ }^1$. The graphical rendering of this concept is reported in figure $9.1$, which will soon be explained in full detail.

## 物理代写|固体物理代写Solid-state physics代考|Density of states for the conduction and valence bands

For further convenience we elaborate an explicit expression of the eDOS (previously introduced in section 7.3.1 only for metals) for a specific semiconductor band $n$. To this aim, we will combine the parabolic bands approximation outlined in section 8.3.1 with the concept of effective mass introduced in section 8.3.5.

Let us consider a generic band $E_n(\mathbf{k})$ and indicate by $G_n(E) d E$ the corresponding number of electron states with energy in the interval $[E, E+d E]$. If we know the single-electron energies, then we can calculate the eDOS $G_n(E)$ as
$$G_n(E)=\sum_{\mathbf{k} \in \mathrm{1BZ}} \delta\left(E-E_n(\mathbf{k})\right)=\frac{V}{(2 \pi)^3} \int_{\mathbf{k} \in 1 \mathrm{BZ}} \delta\left(E-E_n(\mathbf{k})\right) d \mathbf{k},$$
where the discrete sum takes into account that only a set of discrete $\mathbf{k}$-points are rigorously allowed in the $1 \mathrm{BZ}$, while the integral expression is a good approximation valid in the limit of a very large crystal when we can treat the wavevector as a continuous variable ${ }^3$. It is convenient to normalise the eDOS with respect to the crystal volume, so as to obtain an expression independent of geometrical factors. To this aim, we introduce the density of states per unit volume $g_n(E)$ for the nth band as
$$g_n(E)=\frac{G_n(E)}{V}=\frac{1}{(2 \pi)^3} \int_{\mathbf{k} \in 1 \mathrm{BZ}} \delta\left(E-E_n(\mathbf{k})\right) d \mathbf{k},$$
which allows us to directly calculate the total density of states (per unit volume) as $g_{\text {tot }}(E)=\sum_n g_n(E)$.

While equation (9.2) represents the most accurate way to calculate the eDOS for each band, in the following we can take profit from the twofold fact that: (i) most of the semiconductor physics is ruled over by carriers in the proximity of the forbidden gap; and (ii) the valence and the conduction bands are to a very good approximation parabolic near such a gap.

## 物理代写|固体物理代写固态物理学代考|传导和价带的态密度

$$G_n(E)=\sum_{\mathbf{k} \in \mathrm{1BZ}} \delta\left(E-E_n(\mathbf{k})\right)=\frac{V}{(2 \pi)^3} \int_{\mathbf{k} \in 1 \mathrm{BZ}} \delta\left(E-E_n(\mathbf{k})\right) d \mathbf{k},$$
，其中离散和考虑到$1 \mathrm{BZ}$中严格允许只有一组离散的$\mathbf{k}$ -点，而积分表达式是一个很好的近似，在一个非常大的晶体的极限下有效，当我们可以将波矢器视为一个连续变量${ }^3$。方便的是将eDOS与晶体体积归一化，从而得到与几何因素无关的表达式。为此，我们引入第n波段的单位体积态密度$g_n(E)$为
$$g_n(E)=\frac{G_n(E)}{V}=\frac{1}{(2 \pi)^3} \int_{\mathbf{k} \in 1 \mathrm{BZ}} \delta\left(E-E_n(\mathbf{k})\right) d \mathbf{k},$$
，这使我们可以直接计算出(单位体积态密度)的总态密度$g_{\text {tot }}(E)=\sum_n g_n(E)$ .

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部: