PERIODIC POTENTIALS

AND THE EMERGENCE OF BAND STRUCTURES

A COLLECTION OF QUANTUM SYSTEMS

Free Particle

1D Box

Harmonic Oscillator

Hydrogen Atom

\hat{H} = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}

\hat{H} = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} +V(x)

V(x)= \begin{dcases} 0 \, -d/2 \leq x \leq d/2 \\ \infty \; otherwise \end{dcases}

V(x)= 0

\hat{H} = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} +V(x)

V(x)= \frac{1}{2} k x^2

\hat{H} = -\frac{\hbar^2}{2 \mu} \frac{1}{r}\frac{\partial^2}{\partial r^2}r + V(r)

V(r) = \frac{\hat{L}^2}{2 \mu r^2} - \frac{1}{4 \pi \varepsilon_0}\frac{e^2}{r}

E_k=\frac{\hbar^2k^2}{2m}

E_n=\frac{\pi^2 \hbar^2}{2m}n^2

E_n=\hbar \omega (n + \frac{1}{2})

E_n=-\frac{1}{4 \pi \varepsilon_0}\frac{e^2}{2 a_0}\frac{1}{n^2}

\Delta E=\frac{\pi^2 \hbar^2}{2m}(2n+1)

\Delta E=\hbar \omega

\Delta E=\frac{1}{4 \pi \varepsilon_0}\frac{e^2}{2 a_0} \frac{2n+1}{n^2(n+1)^2}

\Delta E \rightarrow 0

A COLLECTION OF QUANTUM SYSTEMS

Free Particle

1D Box

Harmonic Oscillator

Hydrogen Atom

\hat{H} = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}

\hat{H} = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} +V(x)

V(x)= \begin{dcases} 0 \, -d/2 \leq x \leq d/2 \\ \infty \; otherwise \end{dcases}

V(x)= 0

\hat{H} = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} +V(x)

V(x)= \frac{1}{2} k x^2

\hat{H} = -\frac{\hbar^2}{2 \mu} \frac{1}{r}\frac{\partial^2}{\partial r^2}r + V(r)

V(r) = \frac{\hat{L}^2}{2 \mu r^2} - \frac{1}{4 \pi \varepsilon_0}\frac{e^2}{r}

\langle [\hat{H},\hat{H}] \rangle = 0

E=Const.

Time Symmetry

\langle [\hat{H},\hat{H}] \rangle = 0

E=Const.

Time Symmetry

\langle [\hat{H},\hat{H}] \rangle = 0

E=Const.

Time Symmetry

\langle [\hat{H},\hat{H}] \rangle = 0

E=Const.

Time Symmetry

\langle [\hat{p},\hat{H}] \rangle = 0

p=Const.

Translational Symmetry

\langle [\hat{L}^2,\hat{H}] \rangle = 0; \,\langle [\hat{L}_z,\hat{H}] \rangle = 0

L^2,L_z=Const.

Rotational Symmetry

MATERIALS ARE (OFTEN) CRYSTALS

1D PERIODIC POTENTIALS

V(x)=V(x+na)

Symmetries

\hat{T}_a \psi(x) = \psi(x+a)

Discrete Translation Operator

\langle [\hat{H},\hat{H}] \rangle= 0

\hat{H} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x)

\(\hat{T}_a\) Eigenfunction and Conserved Quantities

\begin{cases} \hat{T}_a \phi(x) = e^{ika}\varphi(x) \\ \varphi(x) = e^{ikx}f(x) \\ f(x) = f(x+a) \end{cases}

\langle [\hat{T}_a,\hat{H}] \rangle = 0

\begin{cases} E_n=const. \\ e^{ika}; \, k=const. \end{cases}

BLOCH THEOREM

The solutions to the Schrödinger equation in a periodic potential can be expressed as plane waves modulated by periodic functions.

\begin{dcases} \hat{H} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x) \\[10pt] V(x) = V(x+na) \end{dcases}

\begin{dcases} \varphi_{nk}(x) = e^{ikx}f_{nk}(x) \\ f_{nk}(x) = f_{nk}(x+na) \end{dcases}

MOMENTUM CONSERVATION AND CRYSTAL MOMENTUM

Momentum is not conserved in a periodic crystal

[\hat{p},\hat{H}] \neq 0

p \neq \hbar k

\(k\) in \(\varphi_{nk}(x) = e^{ikx}f_{nk}(x)\) is not the traditional wavevector, and takes the name of crystal momentum

RECIPROCAL LATTICE

AND UNIQUE VALUES OF CRYSTAL MOMENTUM

\begin{dcases} \varphi_{nk}(x) = e^{ikx}f_{nk}(x) \\ f_{nk}(x) = f_{nk}(x+na) \end{dcases}

-\frac{a}{2}

\frac{a}{2}

g = \frac{2 \pi}{a}

-\frac{g}{2}

\frac{g}{2}

Bravais Lattice: \(\{\vec{R}\}, \, \vec{R}=ma, \, m \in \mathbb{Z}\)

Reciprocal Lattice: \(\{\vec{G}\}, \, \vec{G}=lg, \, l \in \mathbb{Z}\)

Unit cell

Brillouin Zone

\begin{dcases} \varphi_{nk}(x) \equiv \varphi_{nk'}(x) \\ k' = k + lg, \, l \in \mathbb{Z} \end{dcases}

RECIPROCAL LATTICE

AND UNIQUE VALUES OF CRYSTAL MOMENTUM

\begin{dcases} \varphi_{nk}(x) = e^{ikx}f_{nk}(x) \\ f_{nk}(x) = f_{nk}(x+na) \end{dcases}

\varphi_{nk_1}(x) = e^{ik_{1}x}f_{nk_{1}}(x)

\varphi_{nk_2}(x) = e^{ik_{2}x}f_{nk_{2}}(x)

k_2 = k_1 + g

FOURIER SERIES OF THE PERIODIC POTENTIAL

V(x) = \sum_n V_n e^{i k_n x} = \sum_{g_n} V_{g_n} e^{i g_n x}

\tilde{V}(k)

-g

-2g

Example: \(V(x)=V_0 \cos(gx)\)

FOURIER SERIES OF THE EIGENFUNCTION

\begin{dcases} \varphi_{nk}(x) = \sum_k c_k e^{i k x} \\[10pt] c_k = \tilde{\varphi}(k) \Delta k \end{dcases}

\tilde{\varphi}(k)

-g

-2g

\varphi_{nk}(x) = \int_{-\infty}^{\infty}\tilde{\varphi}(k) e^{i k x}dk

\varphi_{nk}(x) \simeq \sum_k \tilde{\varphi}(k) e^{i k x} \Delta k

PERIODIC SCHRODINGER EQUATION

\varphi_{nk}(x) = \sum_k c_k e^{i k x}

\left(-\frac{\hbar^2}{2 m} \frac{d^2}{d x^2}+V(x)\right) \varphi_{nk}(x)=E_n \varphi_{nk}(x)

V(x) = \sum_{g_n} \tilde{V}_{g_n} e^{i g_n x}

\left(-\frac{\hbar^2}{2 m} \frac{d^2}{d x^2}+\sum_{g_n} V_{g_n} e^{i g_n x}\right) \sum_k c_k e^{i k x}=E_n \sum_k c_k e^{i k x}

\frac{\hbar^2}{2 m} \sum_k c_k k^2 e^{i k x}+\sum_{g_n} \sum_k V_{g_n} c_k e^{i(g_n+k) x}=E_n \sum_k c_k e^{i k x}

PERIODIC SCHRODINGER EQUATION

\frac{\hbar^2}{2 m} \sum_k c_k k^2 e^{i k x}+\sum_{g_n} \sum_k V_{g_n} c_{k-g_n} e^{i k x}=E_n \sum_k c_k e^{i k x}

\sum_k\left(\frac{\hbar^2}{2 m} k^2 c_k+\sum_{g_n} V_{g_n} c_{k-g_n}\right) e^{i k x}=\sum_k E_n c_k e^{i k x}

\sum_k\left(\left(\frac{\hbar^2}{2 m} k^2-E_n\right) c_k+\sum_{g_n} V_{g_n} c_{k-g_n}\right) e^{i k x}=0

PERIODIC SCHRODINGER EQUATION

\left(\frac{\hbar^2}{2 m} k^2-E_n\right) c_k+\sum_{g_n} V_{g_n} c_{k-g_n}=0

\frac{\hbar^2}{2 m} k^2 c_k+\sum_{g_n} V_{g_n} c_{k-g_n}=E_n c_k, \; \forall k

Central Equation

AN EXAMPLE OF CENTRAL EQUATION

-g

V(x) = 2V_0 \cos(gx) = V_0e^{igx}+V_0e^{-igx}

-g

V_0

\begin{align*} & V_0 c_{k-3 g}+ \frac{\hbar^2}{2 m}(k-2 g)^2 c_{k-2 g}+V_0 c_{k-g}=E_n c_{k-2 g} \\ & V_0 c_{k-2 g}+ \frac{\hbar^2}{2 m}(k-g)^2 c_{k-g}+V_0 c_k=E_n c_{k-g} \\ & \textcolor{blue}{V_0 c_{k-g}+ \frac{\hbar^2}{2 m} k^2 c_k+V_0 c_{k+g}=E_n c_k} \\ & V_0 c_k+ \frac{\hbar^2}{2 m}(k+g)^2 c_{k+g}+V_0 c_{k+2 g}=E_n c_{k+g} \\ & V_0 c_{k+g}+\frac{\hbar^2}{2 m}(k+2 g)^2 c_{k+2 g}+V_0 c_{k+3 g}=E_n c_{k+2 g} \end{align*}

AN EXAMPLE OF CENTRAL EQUATION

\bgroup \def\arraystretch{2.0} \left[ \begin{array}{ccccc} \frac{\hbar^2}{2 m}(k-2 g)^2 & V_0 & 0 & 0 & 0 \\ V_0 & \frac{\hbar^2}{2 m}(k-g)^2 & V_0 & 0 & 0 \\ 0 & V_0 & \frac{\hbar^2}{2 m} k^2 & V_0 & 0 \\ 0 & 0 & V_0 & \frac{\hbar^2}{2 m}(k+g)^2 & V_0 \\ 0 & 0 & 0 & V_0 & \frac{\hbar^2}{2 m}(k+2 g)^2 \end{array}\right] \egroup \left(\begin{array}{c} c_{k-2 g} \\ c_{k-g} \\ c_k \\ c_{k+g} \\ c_{k+2 g} \end{array}\right)=E_n\left(\begin{array}{c} c_{k-2 g} \\ c_{k-g} \\ c_k \\ c_{k+g} \\ c_{k+2 g} \end{array}\right)

\varphi_{kn}(x) = \sum_{g_n}c_{k-g_n}e^{i(k-g_n) x} = e^{ikx}\sum_{g_n}c_{k-g_n}e^{-g_n x}

DRAWING THE BAND DIAGRAM

-\frac{g}{2}

\frac{g}{2}

E_g

ESTIMATING THE BAND GAP

V(x) = 2V_0 \cos(gx) = V_0e^{igx}+V_0e^{-igx}

\begin{gathered} V_0 c_{k-g}+\left(\frac{\hbar^2}{2 m} k^2-E_n\right) c_k+V_0 c_{k+g}=0 \\ V_0 c_k+\left(\frac{\hbar^2}{2 m}(k+g)^2-E_n\right) c_{k+g}+V_0 c_{k+2 g}=0 \end{gathered}

\left(\begin{array}{cc} \left(\frac{\hbar^2}{2 m} k^2-E_n\right) & V_0 \\ V_0 & \left(\frac{\hbar^2}{2 m}(k+g)^2-E_n\right) \end{array}\right)\left(\begin{array}{c} c_k \\ c_{k+g} \end{array}\right)=\left(\begin{array}{l} 0 \\ 0 \end{array}\right)

ESTIMATING THE BAND GAP

with \; k=-\frac{g}{2}

\left(\begin{array}{cc} \left(\frac{\hbar^2}{2 m}\left(-\frac{g}{2}\right)^2-E_n\right) & V_0 \\ V_0 & \left(\frac{\hbar^2}{2 m}\left(\frac{g}{2}\right)^2-E_n\right) \end{array}\right)\left(\begin{array}{l} c_{-g / 2} \\ c_{g / 2} \end{array}\right)=\left(\begin{array}{l} 0 \\ 0 \end{array}\right)

An homogeneous system has a non-trivial solution only if the determinant of the coefficient matrix is zero

\left|\begin{array}{cc} \frac{\hbar^2}{2 m}\left(\frac{g}{2}\right)^2-E_n & V_0 \\ V_0 & \frac{\hbar^2}{2 m}\left(-\frac{g}{2}\right)^2-E_n \end{array}\right|=0

E_{\pm}=\frac{\hbar^2}{2 m}\left(\frac{g}{2}\right)^2 \pm V_0, \\[10pt] E_{-}=E_{n}, \, E_{+}=E_{n+1}

E_g = E_{+}-E_{-} = 2V_0

NUMERICAL SOLUTION OF THE CENTRAL EQUATION

FAST FOURIER TRANSFORM

def square(x_arr, fill_f V_maximum):
	dim_x = np.shape(x_arr)[0]
	V = np.zeros(dim_x)
	V[0:int(dim_x*fill_f)] = V_maximum
	return V

dim = 256      # number sampling points in real space of for the periodic potential
x=np.linspace(-a/2,a/2,dim)  # x points of unit cell in real space
dx = x[1]-x[0]  # spatial sampling
V_max = 5.0 # potential amplitude
fill_fraction = 0.5 # fill fraction of the potential
V = square(x,fill_fraction,V_max)

NUMERICAL SOLUTION OF THE CENTRAL EQUATION

FAST FOURIER TRANSFORM

V_fft = np.fft.fft(V,norm='forward')  # compute full fft

V(x) = \sum_{g_n} V_{g_n} e^{i g_n x}

NUMERICAL SOLUTION OF THE CENTRAL EQUATION

FAST FOURIER TRANSFORM

V_fft = np.fft.fft(V,norm='forward')  # compute full fft
V_fft_shift = np.fft.fftshift(V_fft)  # shift fft

V(x) = \sum_{g_n} V_{g_n} e^{i g_n x}

NUMERICAL SOLUTION OF THE CENTRAL EQUATION

FAST FOURIER TRANSFORM

V_fft = np.fft.fft(V,norm='forward')  # compute full fft
V_fft_shift = np.fft.fftshift(V_fft)  # shift fft
k_fft = np.fft.fftshift(np.fft.fftfreq(len(V),dx*1e9))  # compute spatial fft frequencies

V(x) = \sum_{g_n} V_{g_n} e^{i g_n x}

NUMERICAL SOLUTION OF THE CENTRAL EQUATION

FAST FOURIER TRANSFORM

V_fft_r = np.fft.rfft(V,norm='forward')  # compute fft
k_fft_r = np.fft.rfftfreq(len(V),dx*1e9)  # compute fft spatial frequencies

V(x) = \sum_{g_n} V_{g_n} e^{i g_n x}

NUMERICAL SOLUTION OF THE CENTRAL EQUATION

FAST FOURIER TRANSFORM

We select the \(2 n_{max}+1\) orders, to correctly fill the central equation matrix

NUMERICAL SOLUTION OF THE CENTRAL EQUATION

FAST FOURIER TRANSFORM

We check the reconstructed periodic potential with the \(2 n_{max}+1\) orders.

NUMERICAL SOLUTION OF THE CENTRAL EQUATION

COFFICIENT MATRIX

\bgroup \def\arraystretch{2.0} \left[ \begin{array}{ccccc} \frac{\hbar^2}{2 m}(k-2 g)^2 & V_{g_{1}} & V_{g_{2}} & V_{g_{3}} & V_{g_{4}} \\ V_{g_{-1}} & \frac{\hbar^2}{2 m}(k-g)^2 & V_{g_{1}} & V_{g_{2}} & V_{g_{3}} \\ V_{g_{-2}} & V_{g_{-1}} & \frac{\hbar^2}{2 m} k^2 & V_{g_{1}} & V_{g_{2}} \\ V_{g_{-3}} & V_{g_{-2}} & V_{g_{-1}} & \frac{\hbar^2}{2 m}(k+g)^2 & V_{g_{1}} \\ V_{g_{-4}} & V_{g_{-3}} & V_{g_{-2}} & V_{g_{-1}} & \frac{\hbar^2}{2 m}(k+2 g)^2 \end{array}\right] \egroup

NUMERICAL SOLUTION OF THE CENTRAL EQUATION

COFFICIENT MATRIX

\bgroup \def\arraystretch{2.0} \left[ \begin{array}{ccccc} \frac{\hbar^2}{2 m}(k-2 g)^2 & V_{g_{1}} & V_{g_{2}} & V_{g_{3}} & V_{g_{4}} \\ V_{g_{1}}^* & \frac{\hbar^2}{2 m}(k-g)^2 & V_{g_{1}} & V_{g_{2}} & V_{g_{3}} \\ V_{g_{2}}^* & V_{g_{1}}^* & \frac{\hbar^2}{2 m} k^2 & V_{g_{1}} & V_{g_{2}} \\ V_{g_{3}}^* & V_{g_{2}}^* & V_{g_{1}}^* & \frac{\hbar^2}{2 m}(k+g)^2 & V_{g_{1}} \\ V_{g_{4}}^* & V_{g_{3}}^* & V_{g_{2}}^* & V_{g_{1}}^* & \frac{\hbar^2}{2 m}(k+2 g)^2 \end{array}\right] \egroup

NUMERICAL SOLUTION OF THE CENTRAL EQUATION

COFFICIENT MATRIX

\bgroup \def\arraystretch{2.0} \left[ \begin{array}{ccccc} 0 & V_{g_{1}} & V_{g_{2}} & V_{g_{3}} & V_{g_{4}} \\ V_{g_{1}}^* & 0 & V_{g_{1}} & V_{g_{2}} & V_{g_{3}} \\ V_{g_{2}}^* & V_{g_{1}}^* & 0 & V_{g_{1}} & V_{g_{2}} \\ V_{g_{3}}^* & V_{g_{2}}^* & V_{g_{1}}^* & 0 & V_{g_{1}} \\ V_{g_{4}}^* & V_{g_{3}}^* & V_{g_{2}}^* & V_{g_{1}}^* & 0 \end{array}\right] \egroup + \bgroup \def\arraystretch{2.0} \left[ \begin{array}{ccccc} \frac{\hbar^2}{2 m}(k-2 g)^2 & 0 & 0 & 0 & 0 \\ 0 & \frac{\hbar^2}{2 m}(k-g)^2 & 0 & 0 & 0 \\ 0 & 0 & \frac{\hbar^2}{2 m} k^2 & 0 & 0 \\ 0 & 0 & 0 & \frac{\hbar^2}{2 m}(k+g)^2 & 0 \\ 0 & 0 & 0 & 0 & \frac{\hbar^2}{2 m}(k+2 g)^2 \end{array}\right] \egroup

NUMERICAL SOLUTION OF THE CENTRAL EQUATION

BAND COMPUTATION

f_{bands}(k,n_{max},V) \Rightarrow \begin{dcases} \{E_1,E_2,...,E_{2 n_{max}+1} \} \\[10pt] \{\mathbf{c}_k^1,\mathbf{c}_k^2,...,\mathbf{c}_k^{2 n_{max}+1} \} \end{dcases}