#!/usr/bin/env python
# coding: utf-8

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import sys

sys.path.append("../../code")
from init_mooc_nb import *

init_notebook()


# # A quick review of band structures
# 
# For the material of this course we assume familiarity with basic linear algebra, quantum mechanics and solid state physics.
# In this chapter, we briefly review the concepts most relevant to this course. 
# If you think you know how you would proceed if you are given to compute the bandstructure of graphene then you can likely skip this chapter.
# 
# ## Quantum mechanics: electrons as waves
# Quantum mechanics begins with stating that particles such as electrons should really be treated
# as waves. These waves are described by the famous Schrodinger equation 
# $$i\hbar\partial_t \Psi = H\Psi,$$
# where at this point $\Psi$ is the "wave-function" and $H$ is the Hamiltonian. 
# The problem of analyzing this Schrodinger equation can be reduced to the eigenvalue problem in linear algebra, though in many cases the vector space might be infinite dimensional.
# In the following, we assume familiarity with basic finite dimensional linear (matrix) algebra.
# 
# ### Schrodinger equation besides electrons
# Our main focus is quantum-mechanical systems, however, as we will see, many ideas apply also in completely classical context of sound propagation and elasticity.
# To see this, let us  convert a familiar wave-equation for a string in to a Schrodinger-like form.
# You must have seen a wave-equation for a string that looks like 
# $$\partial_t^2 h-c^2\partial_x^2 h=0,$$
# where $h(x,t)$ is the vertical displacement of the string.
# This wave-equation is second order in time.
# Let's try to make it first order like the Schrodinger equation by defining $h_1(x,t)=c^{-1} \partial_t h(x,t)$ and $h_2(x,t)=\partial_x h(x,t)$. 
# After doing this we see that our wave-equation turns into a pair of equations that are linear order in time:
# $$\partial_t h_2 = c\partial_x h_1$$ 
# and 
# $$\partial_t h_1=-c\partial_x h_2.$$
# 
# We can turn this into the Schrodinger equation if we define:
# $$\Psi(x,t)=\left(\begin{array}{c}h_1(x,t)\\h_2(x,t)\end{array}\right)\quad H=c\left(\begin{array}{cc}0& i\\-i & 0\end{array}\right)(-i\partial_x).$$
# Now those of you who know basic quantum mechanics might say this is a very strange Schrodinger equation.
# But this indeed is the wave-function for helical Majorana particles that we encounter later on.
# 
# ### Applying the Schrodinger equation
# The wave-function $\Psi$ in the Schrodinger equation that describes electrons is typically a complex though the Hamiltonian is not a matrix (thankfully):
# $$H=-\frac{\hbar^2}{2m}\partial_x^2 + V(x),$$
# where $m$ is the mass of the electron and $V(x)$ is the background potential energy over which the electron is moving.
# 
# The main things that you should remember about wave equations for electrons are:  
# 
# * $\Psi(x,t)$ is complex, 
# * $H$ is a Hermitian  matrix or operator 
# * density of electrons are related to $|\Psi(x,t)|^2$.
# * If $N$ is the number of electrons, one must occupy $N$ orthogonal wave-functions.
# 
# The last point is more subtle and is called the **Pauli exclusion principle**. We elaborate on orthogonality later.
# 
# Since we are interested in static properties of electrons in materials for much of our course, it helps to make the simplifying ansatz: $\Psi=e^{-i E t/\hbar}\psi$.
# This ansatz simplifies the Schrodinger equation to a time-independent form:
# $$H\psi=E\psi,$$
# which is an eigenvalue problem in linear algebra.
# 
# We can often model electrons in materials within the  **tight-binding** approximation where electrons are assumed to occupy a discrete set of orbitals. 
# We then take $\psi_a$ to be the wave-function of the electron on orbital $a$. 
# The wave-functions $\psi_a$ can be combined into $\psi$, which is then a vector.
# In this case, the Hamiltonian $H$ becomes a matrix with components $H_{ab}$.
# These definitions transform the time-independent Schrodinger equation into a matrix eigenvalue problem from linear algebra.
# Once we know how to set-up the matrix $H_{ab}$ to model a particular material, we can extract the properties of the material from the wave-function components $\psi_a$ and energy (eigenvalue) $E$. 
# A few key properties of the Schrodinger equation $H\psi^{(n)}=E^{(n)}\psi^{(n)}$ are: 
# * if $H$ is an $N\times N$ matrix, the eigenvalue index $n$ goes from $n=1,\dots,N$. 
# *  $H$ is Hermitian i.e. $H_{ab}=H_{ba}^*$. 
# * Eigenstates are orthogonal i.e. $\psi^{(n)\dagger} \psi^{(m)}=0$ for $m\neq n$.
# 
# 
# Physicists have a convenient notation for doing linear algebra called the Dirac **bra-ket** notation. 
# In this notation, wave-functions such as $\psi$ are represented by **kets** i.e. $\psi\rightarrow |\psi\rangle$. 
# We construct the ket $|\psi\rangle$ from the components of the wave-function $\psi_a$ using the equation: $$|\psi\rangle=\sum_a \psi_a |a\rangle.$$
# Similarly, we turn the Hamiltonian $H$ in to  an **operator** using the equation :$$H=\sum_{ab}H_{ab}|a\rangle \langle b|,$$
# where $H_{ab}$ are the elements of the matrix $H$ from the last paragraph. 
# We call the object $\langle b|$ a **bra** and together with the ket it forms a bra-ket with the property $\langle b| a\rangle=\delta_{ab}$.
# The Schrodinger equation now looks like $$H|\psi\rangle = E|\psi\rangle,$$
# which can be checked to be the same equation as the linear algebra form. 
# 
# ### Example: Atomic triangle
# Let's now work out the simple example of electrons moving in a triangle of atoms, where each atom has one orbital. We label the orbitals as $|0\rangle,|1\rangle,|2\rangle$. 
# With this labeling, the **hopping** amplitude $t$ of electrons between orbitals has the Hamiltonian
# $$H=-t(|0\rangle \langle 1|+|1\rangle \langle 2|+|2\rangle \langle 0|)+h.c,$$
# where $h.c.$ stands for Hermitian conjugate, which means that you reverse the ordering of the labels and take a complex conjugate. 
# We can also write the Hamiltonian in matrix form 
# $$H_{ab}=-\left(\begin{array}{ccc}0&t&t^*\\t^*&0&t\\t&t^*&0\end{array}\right).$$
# Diagonalizing this matrix is a straightforward exercise that results in three eigenvectors $\psi^{(n)}_a$ (with $n=1,2,3$)
# corresponding to energy eigenvalues 
# $$E^{(n)}=-2 |t| \cos{\theta},|t|\cos{\theta}\pm |t|\sqrt{3}\sin{\theta}$$
# (where $t=|t|e^{i\theta}$).
# The corresponding eigenvectors 
# $$\psi^{(n)}_a=3^{-1/2}(1,1,1),3^{-1/2}(1,\omega,\omega^2),3^{-1/2}(1,\omega^2,\omega)$$
# where $\omega$ is the cube-root of unity (i.e. $\omega^3=1$).
# 
# ## Bloch's theorem for bulk electrons
# 
# Actually, we can even solve the problem of an electron in an N site ring (triangle being $N=3$).
# The trick to doing this is a neat theorem called Bloch's theorem.
# Bloch's theorem is the key to understanding electrons in a crystal. 
# The defining property of a crystal is that the atomic positions repeat in a periodic manner in space.
# We account for ALL the atoms in the crystal by first identifying a finite group of orbitals called the **unit-cell**.
# We choose the unit-cell so that we can construct the crystal by translating the unit cell by a discrete set of vectors called lattice vectors to $n$.
# We label the orbitals in the unit-cell by the index $l$, which takes a finite set of values.
# By combining the unit cell and the lattice vectors, we  construct positions $a=(l,n)$ 
# of all the orbitals in the crystal.
# For our example of an atomic ring of size $N$, the index $l$ wouldn't be needed since there is only one orbital per unit-cell and $n$ would take values $1$ to $N$.
# In a three-dimensional crystal, $n=(n_x,n_y,n_z)$ would be a vector of integers. 
# The Hamiltonian for a crystal has matrix elements that satisfy $H_{(l,n),(l',m)}=H_{(l,n-m),(l',0)}$ for all pairs of unit-cell $n$ and $m$. 
# 
# > Bloch's theorem states that the Schrodinger equation for such Hamiltonians in crystals can be solved by the ansatz: $$\psi_{(l,n)}=e^{i k n}u_l,$$
# where $u_l$ is the periodic part of the Bloch function which is identical in each unit-cell.
# 
# The parameter $k$ is called crystal momentum and is quite analogous to momentum (apart from a factor of $\hbar$)
# except that it is confined in the range $k\in [-\pi,\pi]$ which is referred to as the **Brillouin Zone**.
# You can now substitute this ansatz into the Schrodinger equation: $\sum_{l'm}H_{(l,n),(l',m)}u_{l'}e^{i k m}=E(k) e^{i k n}u_{l}(k)$. 
#  Thus the Bloch functions $u(k)$ and energies $E(k)$ are obtained from the eigenvalue equation (so-called Bloch equation) $$H(k)u(k)=E(k)u(k),$$
# where $$H(k)_{ll'}=\sum_{m}H_{(l,-m),(l',0)}e^{-i k m}.$$
# The Bloch equation written above is an eigenvalue problem at any momentum $k$. 
# The resulting eigenvalues $E^{(n)}(k)$ consitute the bandstructure of a material, where the eigenvalue label $n$ is also called a band index. 
# 
# ### Example: Su-Schrieffer-Heeger model
# 
# Let us now work through an example. 
# The Su-Schrieffer-Heeger (SSH) model is the simplest model for polyacetylene, which to a physicist can be thought of as a chain of atoms with one orbital per atom.
# However, the hopping strength alternates (corresponding to the alternating bond-length ) between $t_1$ and $t_2$.
# Ususally you could assume that since each orbital has one atom there is only one atom per unit cell.
# But this would mean all the atoms are identical. 
# On the other hand, in polyacetylene, half the atoms are on the right end of a short bond and half of them are on the left. 
# Thus there are two kinds of atoms - the former kind we label $R$ and the latter $L$. Thus there are two orbitals per unit cell that we label $|L,n\rangle$ and $|R,n\rangle$ with $n$ being the unit-cell label.
# 
# ![](figures/Trans-_CH_n.svg)
# 
# The Hamiltonian for the SSH model is 
# $$H=\sum_n \{t_1(|L,n\rangle\langle R,n|+|R,n\rangle\langle L,n|)+t_2(|L,n\rangle\langle R,n-1|+|R,n-1\rangle\langle L,n|)\}.$$
# This Hamiltonian is clearly periodic with shift of $n$ and the non-zero matrix elements of the Hamiltonian can be written as $H_{(L,0),(R,0)}=H_{(R,0),(L,0)}=t_1$ and $H_{(L,1),(R,0)}=H_{(R,-1),(L,0)}=t_2$. 
# The $2\times 2$ Bloch Hamiltonian is calculated to be: $$H(k)_{ll'=1,2}=\left(\begin{array}{cc}0& t_1+t_2 e^{i k}\\t_1+t_2 e^{-ik}&0\end{array}\right).$$
# 
# We can calculate the eigenvalues of this Hamiltonian by taking determinants and we find that the eigenvalues are 
# $$E^{(\pm)}(k)=\pm \sqrt{t_1^2+t_2^2+2 t_1 t_2\cos{k}}.$$ 
# Since $L$ and $R$ on a given unit-cell surrounded one of the shorter bonds (i.e. with larger hopping ) we expect $t_1>t_2$. As $k$ varies across $[-\pi,\pi]$, $E^{(+)}(k)$ goes from $t_1-t_2$ to $t_1+t_2$. Note that the other energy eigenvalue is just the negative $E^{(-)}(k)=-E^{(+)}(k)$. 
# > As $k$ varies no energy eigenvalue $E^{(\pm)}(k)$ ever enters the range $-|t_1-t_2|$ to $|t_1-t_2|$. This range is called an **band gap**, which is the first seminal prediction of Bloch theory that explains insulators.
# 
# 
# This notion of an insulator is rather important in our course.
# So let us dwell on this a bit further. Assuming we have a periodic ring with $2N$ atoms so that $n$ takes $N$ values, single valuedness of the wave-function $\psi_{(l,n)}$ requires that $e^{i k N}=1$. 
# This means that $k$ is allowed $N$ discrete values separated by $2\pi/N$ spanning the range $[-\pi,\pi]$. 
# Next to describe the lower-energy state of the electrons we can fill only the lower eigenvalue $E^{(-)}(k)$ with ane electron at each $k$ leaving the upper state empty. 
# This describes a state with $N$ electrons. Furthermore, we can see that to excite the system one would need to transfer an electron from a negative energy state to a positive energy state that would cost at least $2(t_1-t_2)$ in energy. 
#  Such a gapped state with a fixed number of electrons cannot respond to an applied voltage and as such must be an insulator. 
# 
# This insulator is rather easy to understand in the $t_2=0$ limit and corresponds to the double bonds in the polyacetylene chain being occupied by localized electrons. 
# 
# ## $k\cdot p$ perturbation theory
# 
# Let us now think about how we can use the smoothness of $H(k)$ to predict energies and wave-functions at finite $k$ from $H(k=0)$ and its derivatives.
# We start by expanding the Bloch Hamiltonian 
# $$H(k)\approx H(k=0)+k H^{'}(k=0)+(k^2/2)H^{''}(k=0)$$. 
# Using standard perturbation theory we can conclude that:
#   the velocity and mass of a non-degenerate band near $k\sim 0$ is written as 
#   $$v_n =\partial_k E^{(n)}(k)= u^{(n)\dagger} H^{'}(k=0) u^{(n)}$$
#   and
#   $$m_n^{-1}=\partial^2_k E^{(n)}(k)=u^{(n)\dagger} H^{''}(k=0) u^{(n)}+\sum_{m\neq n}\frac{|u^{(n)\dagger} H^{'}(k=0) u^{(m)}|^2}{E^{(n)}(k=0)-E^{(m)}(k=0)},$$
# where $E^{(n)}(k=0)$ and $u^{(n)}(k=0)$ are energy eigenvalues and eigenfunctions of $H(k=0)$. One of the immediate consequences of this is that the effective mass $m_n $ vanishes as the energy denominator $E^{(n)}(k=0)-E^{(m)}(k=0)$ (i.e. gap ) becomes small. This can be checked to be the case by expanding 
# $$E^{(-)}(k)\simeq -(t_1-t_2)-\frac{t_2^2}{(t_1-t_2)}k^2$$. 
# 
# ### Discretizing continuum models for materials
# The series expansion of $H(k)$ that we discussed in the previous paragraph is a continuum description of a material. 
# This is because the series expansion is valid for small $k$ that is much smaller than the Brillouin zone. 
# The continuum Hamiltonian is obtained by replacing $k$ in the series expasion by $\hbar^{-1}p$, where $p=-i\hbar\partial_x$ is the momentum operator. 
# 
# A continuum Hamiltonian is sometimes easier to work with analytically then the crystal lattice of orbitals.
# On the other hand, we need to discretize the continuum Hamiltonian to simulate it numerically. We can do this representing $k$ as a discrete derivative operator: $$k=-i\partial_x\approx -i(2\Lambda)^{-1}\sum_n (|n+1\rangle\langle n|-|n\rangle\langle n+1|).$$ 
# The label $n$ is discrete-analogous to the unit-cell label, where the unit cell has size $\Lambda$. 
# To check that this is a representation of the derivative, apply $i k=\partial_x$ to $|\psi\rangle$ as $i k|\psi\rangle\approx \sum_n \frac{\psi_{n+1}-\psi_{n-1}}{2\Lambda}|n\rangle$. 
# In addition, we need to represent the $N\times N$ matrix structure of $H(k=0)$. 
# This is done by introducing label $a=1,\dots N$ so that the Hamiltonian is defined on a space labeled by $|a,n\rangle$.
# Applying these steps to the the $k\cdot p$ Hamiltonian takes the discrete form:
# $$H(k)\approx \sum_{n,a,b} H(k=0)_{ab}|a,n\rangle \langle b,n| +i H^{'}(k=0)_{ab}(|a,n+1\rangle\langle b,n|-|a,n\rangle\langle b,n+1|),$$
# where we have dropped the $k^2$ term for compactness.
# For future reference, $k^2$ would discretize into $k^2=-\sum_n (|n\rangle \langle n+2|+|n+2\rangle\langle n|-2|n\rangle \langle n|)$.
# 
# But wait! Didn't we just go in a circle by starting in a lattice Hamiltonian and coming back to a discrete Hamiltonian? 
# Well, actually, the lattice in the newly discretized model has almost nothing to do with the microscopic lattice we started with. 
# More often than not, the lattice constant $\Lambda$ (i.e. effective size of the unit-cell) in the latter representation is orders of magnitude larger than the microscopic lattice constant.
# So the discrete model following from $k\cdot p$ is orders of magnitude more efficient to work with than tht microscopic model, which is why we most often work with these.
# Of course, there is always a danger of missing certain lattice level phenomena in such a coarse-grained model. 
# Practically, we often do not start with an atomistic lattice model, but rather with a continuum $k\cdot p$ model and then discretize it. 
# This is because, the latter models can often be constrained quite well by a combination symmetry arguments as well as experimental measurements. 
# For example the $k\cdot p$ model for the conduction band minimum state of a GaAs quantum well is 
# $$H(k)=\hbar^2 k^2/2m^*+\alpha_R (\sigma_x k_y-\sigma_y k_x),$$
# where $m^*$ is the electron effective mass, $\sigma_{x,y}$ are Pauli matrices and $\alpha_R$ is the Rashba spin-orbit coupling. This model is rather complicated to derive from the atomistic level (though it can be done). On the hand, it has also been checked experimentally through transport.
# 
# ## Summary
# The main goal of this section was to review the simplest models for how electrons in crystals can be described quantum mechanically.
# 
# Let us summarize this review of bandstructures: 
# * Quantum mechanics views electrons as waves described by the Schrodinger equation.
# * The Schrodinger equation written in a basis of orbitals becomes a matrix eigenvalue problem from linear algebra.
# * The Schrodinger equation for electrons in crystals can be solved using Bloch's theorem, where crystal momentum $k$ is a good quantum number.
# * The crystal momentum $k$ is periodic within the Brillouin zone.
# * We can treat the crystal momentum $k$ as a derivative when $k$ is small - called the $k\cdot p$ approximation.
# * We solve this Hamiltonian numerically by discretizing the derivative $k$.