# 物理代写|固体物理代写Solid-state physics代考|PHYSICS7544

## 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

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}

$$\hbar \mathbf{k}n^{\text {(surf) }}=\hbar \mathbf{k}{\text {empty }}^{\text {(surf) }}+\hbar \mathbf{G}^{\text {(surf) }},$$

## 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

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