12_band_structures_in_higher_dimensions.md

import numpy as np
import plotly.offline as py
import plotly.graph_objs as go

pi = np.pi

Tight binding and nearly free electrons

(based on chapters 15–16 of the book)

!!! summary "Learning goals"

After this lecture you will be able to:

- examine 1D and 2D band structures and argue if you expect the corresponding material to be an insulator/semiconductor or a conductor.
- describe how the light absorption spectrum of a material relates to its band structure.

Band structure

How are material properties related to the band structure?

For a material to be a conductor, there should be available electron states at the Fermi level. Otherwise all the states are occupied, and all the currents cancel out.

A band structure of a 1D material may look similar to this:

We see several energy bands that may be separated by a band gap or overlapping.

When the Fermi level lies in the band gap, the material is called a semiconductor (or dielectric or insulator). When the Fermi level is between different bands, it is a conductor (metal).

A simple requirement for insulators

In an insulator every single band is either completely filled or completely empty.

How many electrons may an insulator have per unit cell? To answer this we need to integrate the density of states. Integrating

$g(E)$

is hard, but integrating

$\rho(k)$

is easy.

For a single band

$$N = 2 \int_{BZ}dk_x dk_y dk_z [L\times W\times H] (2\pi)^{-3} = 2 LWH$$

So a single band has 2 electrons per unit cell (because of spin).

We come to the important rule:

Any material with an odd number of electrons per unit cell is a metal.

If the material has an even number of electrons per unit cell it may be a semiconductor, but only if the bands are not overlapping (see the figure above). For example: Si, Ge, Sn all have 4 valence electrons. Si (silicon, band gap 1.14 eV) and Ge (germanium, band gap 0.67 eV) are semiconductors, Sn (tin) is a metal. Interesting feature: the heaviest material is a metal, why?

Fermi surface using a nearly free electron model

Sequence of steps (same procedure as in 1D, but harder because of the need to imagine a 2D dispersion relation):

1. Compute
   $k_f$
   using the free electron model (remember this is our starting point).
2. Plot the free electron model Fermi surface and the Brillouin zones.
3. Apply the perturbation where the Fermi surface crosses the Brillouin zone (due to avoided level crossings).

The resulting band structure looks like this (in the extended Brillouin zone scheme):

def E(k_x, k_y):
    delta = np.array([-2*pi, 0, 2*pi])
    H = np.diag(
        ((k_x + delta)[:, np.newaxis]**2
        + (k_y + delta)[np.newaxis]**2).flatten()
    )
    return tuple(np.linalg.eigvalsh(H + 5)[:3])

E = np.vectorize(E, otypes=(float, float, float))

momenta = np.linspace(-2*pi, 2*pi, 100)
kx, ky = momenta[:, np.newaxis], momenta[np.newaxis, :]
bands = E(kx, ky)

# Extended Brillouin zone scheme
pad = .3
first_BZ = ((abs(kx) < pi + pad) & (abs(ky) < pi + pad))
second_BZ = (
    ((abs(kx) > pi - pad) | (abs(ky) > pi - pad))
    & ((abs(kx + ky) < 2*pi + pad) & (abs(kx - ky) < 2*pi + pad))
)
third_BZ = (
    (abs(kx + ky) > 2*pi - pad) | (abs(kx - ky) > 2*pi - pad)
)

bands[0][~first_BZ] = np.nan
bands[1][~second_BZ] = np.nan
#bands[2][~third_BZ] = np.nan

# Actually plotting

fig = go.Figure(
    data = [
        go.Surface(
            z=band / 5,
            # colorscale=color,
            opacity=opacity,
            showscale=False,
            hoverinfo='none',
            colorscale='Reds',
            x=momenta,
            y=momenta,
        )
        for band, opacity
        in zip(bands[:2],
               # ['#cf483d', '#3d88cf'],
               (1, 0.9))
    ],
    layout = go.Layout(
        title='Nearly free electrons in 2D',
        autosize=True,
        hovermode=False,
        margin=dict(
            t=50,
            l=20,
            r=20,
            b=50,
        ),
        scene=dict(
            yaxis={"title": "k_y"},
            xaxis={"title": "k_x"},
            zaxis={"title": "E"},
        )
    )
)
py.iplot(fig, show_link=False)

Observe that the top of the first band is above the bottom of the lowest band. Therefore if

$V$

is sufficiently weak, the material can be conducting even with 2 electrons per unit cell!

A larger

$V$

makes the Fermi surface more distorted and eventually makes the material insulating. Let's compare the almost parabolic dispersion of the nearly free electron model with a tight-binding model in 2D.

We now have a dispersion relation

$E = E_0 + 2t(\cos k_x a + \cos k_y a)$

, which looks like this:

momenta = np.linspace(-pi, pi, 100)
kx, ky = momenta[:, np.newaxis], momenta[np.newaxis, :]
energies = -np.cos(kx) - np.cos(ky)
fig = go.Figure(
    data = [
        go.Surface(
            z=energies,
            # colorscale='#3d88cf',
            opacity=1,
            showscale=False,
            hoverinfo='none',
            x=momenta,
            y=momenta,
        )
    ],
    layout = go.Layout(
        title='Tight-binding in 2D',
        autosize=True,
        hovermode=False,
        margin=dict(
            t=50,
            l=20,
            r=20,
            b=50,
        ),
        scene=dict(
            yaxis={"title": "k_y"},
            xaxis={"title": "k_x"},
            zaxis={"title": "E"},
        )
    )
)
py.iplot(fig, show_link=False)

Light adsorption

Photons of external light can be reflected, transmitted, or adsorbed by the material. Adsorption, in turn requires energy transfer from the photon to electrons. In a filled band there are no available states where energy could be transferred (that's why insulators may be transparent).

When transition between two bands becomes possible due to photons having high energy, the adsorption increases in a step-like fashion, see the sketch below for germanium.

Here

$E'_G\approx 0.9eV$

and

$E_G\approx 0.8 eV$

. The two steps visible steps are due to the special band structure of Ge:

The band structure has two band gaps: direct, the band gap at

$k=0$

,

$E'_G$

and indirect gap

$E_G$

at any

$k$

. In Ge

$E_G < E'_G$

, and therefore it is an indirect band gap semiconductor. Silicon also has an indirect band gap. Direct band gap materials are for example GaAs and InAs.

Photons carry very little momentum and a very high energy since

$E = c \hbar k$

and

$c$

is large. Therefore to excite electrons at

$E_G$

, despite a lower photon energy is sufficient, there is not enough momentum. Then an extra phonon is required. Phonons may have a very large momentum at room temperature, and a very low energy since atomic mass is much higher than electron mass.

A joint adsorbtion of a photon and a phonon collision may excite an electron across an indirect band gap, however this process is much less efficient, and therefore are much worse for optics applications (light emitting diodes, light sensors, etc).

Summary

• In periodic potential all electron states are Bloch waves
• Electron dispersion is organized into energy bands that may overlap, or may be separated by band gaps
• If the lattice potential is weak, the dispersion can be obtained by copying
  $p^2/2m$
  into different Brillouin zones, and opening gaps at every level crossing. Each gap is equal to the Fourier component of the lattice potential.
• If the number of electrons per unit cell is odd, the material must be conducting.
• Each band hosts
  $2N$
  eletrons, therefore a material with odd number of electrons is a metal; that with an even number of electrons may be an insulator.
• Light adsorption is a tool to measure the band gap, and it distinguishes direct from indirect band gaps.