## 物理代写|固体物理代写Solid-state physics代考|PHYSICS3544

2022年7月27日

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## 物理代写|固体物理代写Solid-state physics代考|The Brillouin zone

The conventional way to generate the reciprocal primitive unit cell is by following the Wigner-Seitz construction: the resulting cell is referred to as the first Brillouin zone. By construction, it contains all the wavevectors that are not linked by a reciprocal translational vector $\mathbf{G}$. Its volume is $(2 \pi)^{3} / V_{\mathrm{c}}$, where $V_{\mathrm{c}}$ is the volume of the primitive unit cell of the corresponding direct lattice defined in equation (2.2). We remark that the use of the adjective ‘first’ will be clear when discussing the vibrational and electronic properties of crystalline solids. We will hereafter make use of the acronym $1 \mathrm{BZ}$ to indicate the first Brillouin zone.

The boundaries of the $1 \mathrm{BZ}$ are given by planes which, as explained in the previous section, are in turn defined by means of reciprocal lattice vectors. The general principle is that the $1 B Z$ is the smallest volume in the reciprocal space which is enclosed by planes normally bisecting reciprocal lattice vectors drawn for the origin. With reference to table $2.2$, we can calculate that the boundary planes of the $1 \mathrm{BZ}$ for the three cubic lattices are defined as follows (the $1 \mathrm{BZ}$ of the hexagonal lattice is added for completeness):

• simple cubic lattice: take the six planes normal to the vectors $\pm 2 \pi \hat{\mathrm{i}} / a, \pm 2 \pi \hat{\mathrm{j}} / a$, and $\pm 2 \pi \hat{\mathrm{k}} / a$ at their midpoints: they define a cubic volume;
• body-centred cubic lattice: take the 12 planes normal to the vectors $2 \pi(\pm \hat{j} \pm \mathrm{k}) / a, 2 \pi(\pm \hat{k} \pm \hat{\hat{i}}) / a$, and $2 \pi(\pm \hat{\mathrm{i}} \pm \hat{\mathrm{j}}) / a$ at their midpoints: they define a rhombic dodecahedron volume;
• face-centred cubic lattice: take the eight planes normal to the vectors $2 \pi(\pm \hat{i} \pm \hat{j} \pm \hat{k}) / a$ at their midpoints and further cut them by another set of six planes bisecting the reciprocal lattice vectors $\pm 4 \pi \hat{\mathrm{i}} / a, \pm 4 \pi \hat{\mathrm{j}} / a$, and $\pm 4 \pi \hat{\mathrm{k}} / a$ : the resulting volume is a truncated octahedron.

A number of high symmetry lines and points can be identified as shown in figure $2.16$ by red lines and black dots, respectively. High-symmetry points of the $1 \mathrm{BZ}$ typically lie at the centre of the zone, edges, and faces, as well as at the corner points. They play an important role in solid state physics: whenever we need to visualise a crystalline physical property depending upon a wavevector ${ }^{11}$, the conventional choice is to follow the path marked in red colour in figure 2.16, corresponding to the edges of the so called irreducible part of the $1 \mathrm{BZ}$. For further reference, we report a standard labelling used for fcc crystals to indicate some highsymmetry directions: the three directions connecting the $\Gamma$ zone-centre to the $X, K$, and $L$ point are indicated as $\Delta, \Sigma$, and $\Lambda$, respectively.

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

The property of translational invariance extensively discussed in the previous sections generates ideally perfect crystalline structures. This is valid if we either consider an infinite lattice or apply Born-von Karman periodic boundary conditions. While useful in many circumstances to develop the constitutive ideas of solid state physics, this idealised situation is surely a strong approximation to reality: in fact, perfect crystals do not exist since at any finite temperature a solid state system does contains defects, i.c. lattice imperfections, that locally break the translational invariance. Their role is key in affecting many physical properties like, for instance, the transport of electric charge or thermal energy.

We will prove the unavoidable presence of defects in crystals by applying a simple thermodynamical argument to a monoatomic Bravais lattice containing $N$ atoms and kept at constant non-vanishing temperature $T$ and pressure $P$. Its energy content is provided by the Gibbs free energy $\mathcal{G}=\mathcal{U}-T S+P V=\mathcal{H}-T S$, where $\mathcal{U}$ and $\mathcal{H}=\mathcal{U}+P V$ are the internal energy and enthalpy, respectively (in appendix $\mathrm{C}$, reference is made to the thermodynamic potentials used in this demonstration). The generation of a lattice defect (we mean: the local alteration of the crystal structure) requires a work
$$\Delta \mathcal{H}{\mathrm{f}}=\mathcal{H}-\mathcal{H}{0} \text {, }$$
known as the formation energy of the defect. In this equation $\mathcal{H}_{0}$ represents the enthalpy of the pristine ideal crystal. In order to make physical concepts clear, we consider the actual case of a lattice vacancy and a self-interstitial defect: in the first case, a single atom is removed and taken far away from the crystal, while in the former case an extra atom of the same chemical nature is added to the crystal in a position not corresponding to any lattice point ${ }^{12}$. These defects are named native, since the chemistry of the crystal is unaffected by their existence. While these are specific (but realistic) situations, the reasoning developed below will lead to conclusions of general validity. We will further assume that the crystal is always in thermodynamical equilibrium, even after defects have been generated in it. A more thorough discussion on the formation of crystal defects is found elsewhere $[16,17]$.

# 固体物理代写

## 物理代写|固体物理代写Solid-state physics代考|The Brillouin zone

• 简单立方晶格: 取垂直于向量的六个平面 $\pm 2 \pi \hat{\mathrm{i}} / a, \pm 2 \pi \hat{\mathrm{j}} / a ，$ 和 $\pm 2 \pi \hat{\mathrm{k}} / a$ 在它 们的中点：它们定义了一个立方体积;
• 体心立方晶格: 取垂直于向量的 12 个平面 $2 \pi(\pm \hat{j} \pm \mathrm{k}) / a, 2 \pi(\pm \hat{k} \pm \hat{\hat{i}}) / a$ ， 和 $2 \pi(\pm \hat{\mathrm{i}} \pm \hat{\mathrm{j}}) / a$ 在它们的中点: 它们定义了一个唟形十二面体体积;
• 面心立方晶格: 取垂直于向量的八个平面 $2 \pi(\pm \hat{i} \pm \hat{j} \pm \hat{k}) / a$ 在它们的中点，并 通过另一组六个平面进一步切割它们，将倒易晶格向量一分为二 $\pm 4 \pi \hat{\mathrm{i}} / a, \pm 4 \pi \hat{\mathrm{j}} / a ，$ 和 $\pm 4 \pi \hat{\mathrm{k}} / a$ : 生成的体积是截断的八面体。
可以识别出多条高对称线和点，如图2.16分别用红线和黑点表示。高对称点1BZ通常位于 区域、边豚和面的中心以及角点。它们在固态物理学中发挥着重要作用: 每当我们需要根 据波矢量可视化晶体物理特性时 ${ }^{11}$ ，传统的选择是沿着图 $2.16$ 中红色标记的路径，对应于 所佣的不可约部分的边緣 $1 \mathrm{BZ}$. 为了进一步参考，我们报告了用于 $\mathrm{fcC}$ 晶体的标准标记，以 指示一些高度对称的方向: 连接晶体的三个方向 $\Gamma$ 区域中心到 $X, K ，$ 和 $L$ 点表示为 $\Delta, \Sigma$ ，和 $\Lambda$ ，分别。

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

$$\Delta \mathcal{H f}=\mathcal{H}-\mathcal{H} 0,$$

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## MATLAB代写

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