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物理代写|固体物理代写Solid-state physics代考|Experimental determination of the band structure
Many experimental techniques used to measure the band structures in solids use magnetic fields to enable the Landau quantisation of electronic orbits. Other methodologies are instead optical: they measure either photon absorption or reflection phenomena occurring during the interaction between a solid specimen
and some electromagnetic probe. Since the detailed description of such magnetic or optical techniques can hardly be exploited by means of such an elementary theory of the solid state as presented in this Primer, we limit ourselves to outlining just the photoelectron spectroscopy (PES) technique, which is (at least conceptually) very simple, while directing the interested reader to other textbooks $[1,2,22,24]$ for a more thorough presentation.
PES is the solid state counterpart of the photoelectric effect $[7,8]$ in that an electron initially located on an occupied crystalline band state with energy $E_n$ is promoted, upon absorption of a photon with energy $\hbar \omega$, to an empty state with energy $E_{\text {empty }}$ above the vacuum level $E_{\text {vacuum }}$ of the investigated material ${ }^{16}$. The excited electron eventually escapes from the solid, moving as a free particle with kinetic energy $E_{\text {out }}$ simply given by the balance
$$
\begin{aligned}
E_{\text {out }} &=\underbrace{E_{\text {empty }}}-E_{\text {vacuum }} \
&=\left(E_n+\hbar \omega\right)-E_{\text {vacuum }} .
\end{aligned}
$$
While the photon energy $\hbar \omega$ is controlled by the experimental setup, a direct measurement of $E_{\text {out }}$ allows us to determine the energy $E_n$ of the crystalline band state (the vacuum level is known). The intensity of the PES signal is proportional to the number of electrons occupying the initial state or, equivalently, to the density of electronic states at energy $E_n$.
From this simplified explanation it is deduced that PES just provides the band diagram reported in figure $8.2$ left, but not the dispersion relations $E=E_n(\mathbf{k})$ which indeed require the knowledge of both the wavevectors and the corresponding energies. This limitation is overcome by a more advanced version of this experimental technique, known as angle-resolved photoelectron spectroscopy (ARPES). We observe that, in general, the photoemission process must conserve both energy, as reported in equation (8.49), and momentum. In particular, it is proved [1] that parallel to the crystal surface the electron momentum obeys the following conservation law
$$
\hbar \mathbf{k}n^{\text {(surf) }}=\hbar \mathbf{k}{\text {empty }}^{\text {(surf) }}+\hbar \mathbf{G}^{\text {(surf) }},
$$
where $\mathbf{G}^{\text {(surf) }}$ is a reciprocal lattice vector of the surface ${ }^{17}$, while $\hbar \mathbf{k}n^{\text {(surf) }}$ and $\hbar \mathbf{k}{\text {empty }}^{\text {(surf) }}$ are the surface projections of the initial and final electron momenta.
物理代写|固体物理代写Solid-state physics代考|Other methods to calculate the band structure
The accurate determination of the band structure of crystalline solids is an art extensively developed in the second half of the XXth century, in parallel with the development of increasingly powerful digital computers: advances in theoretical methods and numerical techniques have been tightly interlaced and mutually beneficial. A number of different methods have been set up, the tight-binding approach-here privileged for pedagogical reasons-being just one among many others. The mathematics of such methods represents a very technical issue of the solid state theory, which is fully exploited elsewhere $[2,12,13]$. Here we limit ourselves to outlining some general features.
The conceptual framework is that defined by the adiabatic, frozen core, nonmagnetic, and single-particle approximations. The Schrödinger problem to be solved is provided by equation (1.22), where the local potential $V_{\text {cfp }}(\mathbf{r})$ acting on the electron is typically determined by a self-consistent procedure. Also, the single-particle wavefunction must have the form of a Bloch wavefunction. Finally, core and valence wavefunctions have a remarkably different space dependence: they both display strong atomic-like oscillations near each ion, while in the interstitial regions core wavefunctions are vanishingly small and valence ones are instead slowly-varying plane-wave like. This ultimately dictates that core and valence wavefunctions are orthogonal.
A first class of band structure methods is based on the idea of representing the crystalline states as Bloch wavefunctions independent of the energy of the valence state of interest. This is the case of the tight-binding method where atomic orbitals are used to create a Bloch state; the same concept is also adopted by using orthogonalised plane waves (OPWs), where the orthogonality between core and valence states is enforced by constructing the valence Bloch state by means of plane waves suitably orthogonalised to core states.
A different choice is operated in the so called cellular methods, where just a single Wigner-Seitz cell is considered, where the single-electron potential is approximated within a sphere centred on each lattice site so as to describe an isolated ion (and, therefore, is spherically symmetric), while outside the sphere is taken to be zero. The radial Schrödinger equation for such a muffin tin potential is then solved and these solutions are used as a basis set for the crystalline wavefunctions. More specifically, a set of augmented plane waves (APWs) are generated, consisting in a combination of a spherical wave (describing the electron state within the Wigner-Seitz cell) and a plane wave (describing the electron state in the interstitial regions). Suitable boundary conditions are applied at the borders of the Wigner-Seitz cell.
物理代写|固体物理代写固态物理代考|带结构的实验测定
许多用来测量固体带结构的实验技术使用磁场来实现电子轨道的朗道量化。其他的方法则是光学的:它们测量固体样品之间相互作用时发生的光子吸收或反射现象
和一些电磁探头。由于这种磁性或光学技术的详细描述很难通过本入门中所介绍的这种基本的固体理论来加以利用，我们将自己局限于概述光电子能谱(PES)技术，它(至少在概念上)是非常简单的，同时引导有兴趣的读者查阅其他教科书$[1,2,22,24]$以获得更全面的介绍
PES是光电效应$[7,8]$的固态对应物，即最初位于能量为$E_n$的已占据晶体带态的电子，在吸收能量为$\hbar \omega$的光子后，被提升到能量为$E_{\text {empty }}$的空态，高于所研究材料${ }^{16}$的真空能级$E_{\text {vacuum }}$。被激发的电子最终逃离固体，以自由粒子的形式运动，其动能$E_{\text {out }}$简单地由平衡
$$
\begin{aligned}
E_{\text {out }} &=\underbrace{E_{\text {empty }}}-E_{\text {vacuum }} \
&=\left(E_n+\hbar \omega\right)-E_{\text {vacuum }} .
\end{aligned}
$$
给出，而光子能量$\hbar \omega$是由实验装置控制的，直接测量$E_{\text {out }}$可以使我们确定晶体带态的能量$E_n$(真空水平是已知的)。PES信号的强度与占据初始态的电子数成正比，或者等价地，与能量$E_n$时的电子态密度成正比 从这个简化的解释推断，PES只提供了图$8.2$左所示的带图，而没有提供色散关系$E=E_n(\mathbf{k})$，而色散关系确实需要对波矢和相应的能量都有了解。这种限制被这种实验技术的更先进版本所克服，被称为角度分辨光电子能谱(ARPES)。我们观察到，一般来说，光电发射过程必须同时保存能量(如式8.49所示)和动量。特别地，[1]证明了平行于晶体表面的电子动量服从以下守恒定律
$$
\hbar \mathbf{k}n^{\text {(surf) }}=\hbar \mathbf{k}{\text {empty }}^{\text {(surf) }}+\hbar \mathbf{G}^{\text {(surf) }},
$$
其中$\mathbf{G}^{\text {(surf) }}$是表面的倒一格向量${ }^{17}$，而$\hbar \mathbf{k}n^{\text {(surf) }}$和$\hbar \mathbf{k}{\text {empty }}^{\text {(surf) }}$是初始和最终电子动量的表面投影。
物理代写|固体物理代写固态物理代考|其他计算带结构的方法
.
晶体固体带结构的精确测定是在二十世纪下半叶与日益强大的数字计算机的发展同时得到广泛发展的一门艺术:理论方法和数值技术的进步紧密交错且相互有利。许多不同的方法已经建立起来，严格约束的方法——在这里由于教学原因而被优待——只是众多方法中的一种。这种方法的数学代表了固体理论的一个非常技术性的问题，这在其他地方被充分利用$[2,12,13]$。在这里，我们只概述一些一般的特性
概念框架是由绝热、冻结核、非磁性和单粒子近似定义的。要解决的Schrödinger问题由式(1.22)给出，其中作用于电子的局部电势$V_{\text {cfp }}(\mathbf{r})$通常由自洽程序确定。此外，单粒子波函数必须具有布洛赫波函数的形式。最后，核心波函数和价波函数具有明显不同的空间依赖性:它们都在每个离子附近表现出强烈的原子状振荡，而在空隙区，核心波函数几乎消失，而价波则是缓慢变化的平面波。这最终决定了核心波函数和价波函数是正交的。第一类带结构方法是基于用布洛赫波函数表示晶体态的思想，而不依赖于感兴趣价态的能量。这就是紧结合法的情况，原子轨道被用来创造布洛赫态;在使用正交平面波(OPWs)时也采用了同样的概念，其中核心和价态之间的正交性是通过适当正交于核心态的平面波来构建价布洛赫态来加强的
在所谓的胞法中，有一种不同的选择，其中只考虑一个单一的Wigner-Seitz胞，其中单电子势在以每个晶格点为中心的球内近似，以描述一个孤立的离子(因此，它是球对称的)，而在球外则被认为是零。然后求解松饼锡势的径向Schrödinger方程，并将这些解用作晶体波函数的基集。更具体地说，生成了一组增强平面波(APWs)，由球波(描述Wigner-Seitz单元内的电子状态)和平面波(描述间隙区中的电子状态)组成。在Wigner-Seitz单元的边界处应用合适的边界条件
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