13_semiconductors.md

from matplotlib import pyplot

import numpy as np
from scipy.optimize import curve_fit
from scipy.integrate import quad

from common import draw_classic_axes, configure_plotting

configure_plotting()

def sqrt_plus(x):
    return np.sqrt(x * (x >= 0))

# Band structure parameters.
E_V, E_C, E_F = -1.2, 1.8, .4
E_D, E_A = E_C - .7, E_V + .5
m_h, m_e = 1, .5

default_colors = pyplot.rcParams['axes.prop_cycle'].by_key()['color']
blue, red = default_colors[0], default_colors[3]

Semiconductor physics

(based on chapters 17–18 of the book)

!!! success "Expected prior knowledge"

Before the start of this lecture, you should be able to:

- Simplify integral expressions by Taylor expansion
- Compute the density of states of the free electron model
- Apply the concepts of group velocity and effective mass to solve problems

!!! summary "Learning goals"

After this lecture you will be able to:

- Describe the concept of holes, and apply this concept to describe the properties of semiconductors
- Compute the density of states of electrons in semiconductors
- Compute the density of charge carriers and chemical potential as a function of temperature

??? info "Lecture video"

<iframe width="100%" height="315" src="https://www.youtube-nocookie.com/embed/IxDp_JAtBQs" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>

Up until this point, we focused on calculating and understanding the band structures. However, the dispersion of a band is only part of the story. An empty band is not going to lead to any interesting physical properties no matter how sophisticated it is. Therefore, it is also important how bands are filled by the particles. By carefully controlling the distribution of particles in the bands, we are able to engineer material properties that we require. Without a doubt, the greatest example is semiconductors—the bedrock of modern electronics. In this lecture, we shall grasp the basics of semiconductors by learning how to treat bands at different levels of filling.

Review of band structure properties

Before proceeding further, let us remind ourselves of important band structure properties.

• Group velocity v=\hbar^{-1}\partial E(k)/\partial k. Descibes how quickly electrons move within the lattice.
• Effective mass m^* = \hbar^2\left(d^2 E(k)/dk^2\right)^{-1}. Tells us how hard it is to accelerate the particles and is related to the curvature of the band.
• Density of states g(E) = \sum_{\textrm{FS}} (dn/dk) \times (dk/dE). The amount of states per infinitesimal interval of energy at given energy. The quantity is vital in order to calculate any bulk property of the material such as conductivity, heat capacity, etc.

In order to check that everything makes sense, we apply the concepts to the free electron model:

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

The velocity is \hbar^{-1}\partial E(k)/\partial k = \hbar k / m \equiv p/m.
The effective mass is m^* = \hbar^2\left(d^2 E(k)/dk^2\right)^{-1} = m.

So in this simplest case the definitions match the usual expressions.

Filled vs empty bands

We distinguish three different band filling types: filled, empty and partially filled. Despite being two opposite extreme extreme cases, filled and empty bands are very similar. For example, both filled and empty bands carry no electric current:

\begin{align} j = 2e \frac{1}{2\pi} \int_{-\pi/a}^{\pi/a} v(k) dk = 2e \frac{1}{2\pi \hbar} \int_{-\pi/a}^{\pi/a} \frac{dE}{dk} \times dk = \ 2e \frac{1}{2\pi \hbar} [E(\pi/a) - E(-\pi/a)] = 0. \end{align}

On the other hand, a filled band has an equal number of electrons going forwards and backwards which thus cancel and lead to zero current. Similar results apply to many other physical quantities such as heat capacity and magnetization. Therefore, filled and empty bands do not affect most physical properties and can be disregarded. As a result, rather than to consider hundreds of bands that a material contains, we neglect most of them and just focus on the handful of partially filled bands around Fermi level.

From electrons to holes

Because completely filled or completely empty bands have simple properties, we may search for a convenient way to describe a band that only has a few electrons missing or extra. While keeping track of a few electrons has no tricks, even a few electrons missing from a band seem to require considering all the other electrons in a band.

A more efficient approach to describing a nearly filled band is motivated by the following analogy. Let us say we have 100 boxes: 99 are occupied and 1 is empty. To keep track which box is occupied/empty, we could write down the numbers of all 99 occupied boxes. If, on the other hand, we only keep track which single box is empty, we solve the problem with a lot less book-keeping.

The same approach applies to band filling. Instead of describing a lot of electrons that are present in an almost filled band, we focus on those that are absent. The absence of an electron is called a hole: a state of a completely filled band with one particle missing.

In this schematic we can either say that 8×2 electron states are occupied (the system has 8×2 electrons counting spin), or 10×2 hole states are occupied. Electron and hole pictures correspond to two different, but equivalent ways of describing the occupation of a band. Naturally, dealing with electrons is more convenient whenever a band is almost empty and with holes when a band is almost full.

Properties of holes

Let us compare the properties of an electron with energy E and a hole obtained by removing that electron. Since removing an electron reduces the total energy of the system, the hole's energy is opposite to that of an electron E_h = -E. The probability for an electron state to be occupied in equilibrium is given by f(E):

f(E) = \frac{1}{e^{(E-E_F)/kT} + 1}.

Since a hole is a missing electron, the probability for a hole state to be occupied is

f_h(E) = 1 - f(E) = 1 - \frac{1}{e^{(E-E_F)/kT} + 1} = \frac{1}{e^{(-E+E_F)/kT} + 1},

which is the Fermi distribution of particles with energy E_h= -E and E_{F,h} = -E_F.

The momentum p_h of a hole should give the correct total momentum of a partially filled band if one sums momenta of all holes. Therefore p_h = -\hbar k, where k is the wave vector of the electron.

Similarly, the total charge should be the same regardless of whether we count electrons or holes, so holes have a positive charge +e (electrons having -e).

On the other hand, hole's velocity is the same as that of an electron:

\frac{dE_h}{dp_h} = \frac{-dE}{-d\hbar k} = \frac{dE}{dp}.

Finally, we derive the hole effective mass from the equations of motion:

m_h \frac{d v}{d t} = +e (E + v\times B).

Comparing with