1

Basic ideas

What’s inside a solid? Huge numbers of electrons, nucleons, dirt (impurities). In a tiny piece of solid (~1mm×1mm×1mm), there are
◼ Number of nucleons: ~ 1020
◼ Number of electrons: ~ 10 × 1020
◼ Number of impurities: ~ 0.001 * 1020
Huge number of degrees of freedom, impossible to keep track of
◼ Each particle has three degrees of freedom (x, y, z) + additional internal degrees of freedom like spins. For classical particles, we can
simulate the motion of a few thousand particles, but 1020 is impossible.
◼ These particles are quantum particles. A quantum system is much more complicated than classical ones. We can simulate (exactly) about
20 particles at most. This number increase by about 1 every two years, according to Moore’s law.
How about utilizing statistical physics? (similar to what we did for quantum and classical gases)?
There is a problem: the high density.
◼ In statistical physics, we know very well how to handle nearly-independent particles (like particles in ideal gas)
◼ However, in solids, the particles are so dense, that they constantly bumps into each other, making it impossible to treat them as
independent particles.
What should we do:
◼ Starting from the quantum ground state
◼ Add a bit energy to the system (temperature, apply voltage etc), and then the system will be at an excited states
◼ Look at the excited states. Very luckily, in many solids, the excitations can be described by a bunch of particles (quasi-particles). These
quasi-particles have a much lower density and are nearly independent particles.
◼ We can do statistical physics for these quasi-particles: quasi-electrons/quasi-holes + phonons

1.1. The Born–Oppenheimer approximation

Also known as the adiabatic approximation (page 425)

1.1.1. A little bit of chemistry

Ne Element No. of electrons on each shell
1 H 1 s1
2 He 1 s2
3 Li 1 s2 2 s1
4 Be 1 s2 2 s2
5 B 1 s 2 2 s 2 2 p1
6 C 1 s 2 2 s 2 2 p2
… … …
10 Ne 1 s 2 2 s 2 2 p6
11 Na 1 s2 2 s2 2 p2 3 s1
12 Mg 1 s2 2 s2 2 p2 3 s2
... … ...
2 Phys520.nb

Ne Element No. of electrons on each shell

1 H 1 s1
2 He 1 s2
3 Li 1 s2 2 s1
4 Be 1 s2 2 s2
5 B 1 s 2 2 s 2 2 p1
6 C 1 s 2 2 s 2 2 p2
… … …
10 Ne 1 s 2 2 s 2 2 p6
11 Na 1 s2 2 s2 2 p2 3 s1
12 Mg 1 s2 2 s2 2 p2 3 s2
... … ...
Valence electrons:
Electrons in the low energy states (inner shells) are bonded tightly to the nucleon. (It costs about 1-10 eV to remove one of these electrons from
an atom. 1eV is 104 K. It is a very high energy cost in solid state physics).
Electrons in high energy states (outer shells) are loosely bonded to the nucleon, and thus are easy to remove. These electrons are called the
valence electrons and these energy states are called the valence shells.
Valence electrons are the electrons in a atom which can participate in the formation of chemical bonds. They are typically electrons in the
outermost (or second-outer-most) shell.
Valence: the number of valence electrons for an element. By definition, it should be an integer (but not always the case).
An element may have more than one valence (e.g. Fe 2 or 3 depending on the chemical compound). In a specific material, in most cases, the
valence is a uniquely determined integer. However, in some materials (known as mixed valence materials), an element show a non-integer
valence. This element has more than one possible integer valence (say Z1 and Z2 ). Instead of choosing a specific valence value, the element
stays in a mixed state, with some probability p having valence Z1 and the probability 1 - p to have valence Z2 . On average, it brings the valence
to p Z1 + (1 - p) Z2 , which is non-integer.

1.1.2. ions+valence electrons

An Atom =A nucleon+core electrons+valence electrons
Because core electrons are tightly bonded to the nucleon, we can treat the nucleon and inner electrons as one object, an ion. Thus
solid =ions+valence electrons

Electron density:
When we talk about electron density, mostly we are referring to valence electrons:
number of atoms × Z
electron density =
Vol

In metals, the electron density is often about 1022 - 1023  cm3

1.1.3. The adiabatic approximation

Ions are very heavy and thus slow. Electrons are much lighter and faster. As a result, their motions decouple, i.e.
◼ When we study the motion of electrons, we can treat ions as a stationary background
◼ When we study the motion of ions, we can treat electrons as a homogeneous background
Very similar to stones (ions) in water (electrons).

1.1.4. Motions of the ions

The ground state: form a lattice
Low energy excitations: elastic waves
High energy excitations: melting the solid, which will well not consider
QM tells us that waves are particles (e.g. lights or say E&M waves give us phonons). Elastic waves, in the quantum treatment, gives us
phonons. Phonon density in a solid is much lower than the density of atoms (low density) and phonons are almost free (non-interacting)
particles.
 Phys520.nb 3

Bottom line: motions of ions are described by a phonon gas, which is a nearly idea Bose gas.

1.1.5. Motions of the valence electrons

Ground state: the Fermi sea
Excitations: only fermions near the Fermi surface are involved for low energy excitations
Number of electrons near the Fermi surface << total number of valence electrons (about 0.01%-1% of total electrons).
Thus, electrons involved in excitations (quasi-electrons or quasi-holes) have a much lower effective density. Almost an idea Fermi gas.
Bottom line: motions of valence electrons are described by a Fermi gas

1.1.6. a solid = a phonon gas+a Fermi gas

Comment: the couplings between the phonon gas and the Fermi gas are weak, and often ignored in the leading order approximation. However,
it is not always negligible. Sometimes, these coupling is crucial, e.g. for conventional superconductors.

1.2. Approximations for the Fermi gas: (Chapter 3 in the text book)

1.2.1. Various approximations

For the motions of electrons, we can adopt (one of) the following approximations.
1. Ignore interactions between electrons, ignore ions, ignore quantum effects and treat electrons as classical particles: i.e. electrons form a
(classical) ideal gas
2. Ignore interactions between electrons, ignore ions, keep quantum effects and treat electrons quantum fermion particles: i.e. electrons form
an idea Fermi gas. In Ashcroft and Mermin, this is called the “free electron approximation”
3. Ignore interactions between electrons, keep the ions but assumes ions doesn’t move (the adiabatic approximation). Keep quantum effects:
In Ashcroft and Mermin, this is called the “independent electron approximation”, but in other literature, this is often called the “free
electron approximation”.
4. Keep interactions between electrons, keep the ions but assumes ions doesn’t move (the adiabatic approximation). Keep quantum effects.
This is the Landau’s Fermi liquid theory (will be discussed in 540. At the end of the day, the final conclusions of the Landau Fermi liquid
theory tells us that approximation #3 works very well).
Sometimes, we needs to take in account some additional effects. Typically, we introduce these additional effects as small perturbations and treat
them using the perturbation theory, e.g. scattering between electrons and phonons

1.2.2. Which approximation should be used?

Mathematically, #4 is the most accurate approximation and thus we should always use #4. However, in physics, the most accurate is not always
the best. Instead, it is often one of the worst option. This is because the most accurate is also the most complicated. In many cases, these extra
complications are unnecessary, and thus using such a complicated approach is a waste of time. In practice, we should always choose the
simplest approximation that can correctly capture the physics. This is very important in research!
1. Approximation #1 is the least accurate one, but also the simplest one. It gives good predictions for certain transport coefficients. For
transport coefficients, it works equally well as the approximation #2 (They produce almost the same numbers, but approximation #1 get the
right number for a wrong reason). This approximation gives terrible predictions on thermodynamics, e.g. the specific heat.
2. Approximation #2, in addition to transport coefficients, it works well for thermodynamical quantities, like specific heat. However, this
approximation predicts that almost any solid material is a metal, which is clearly wrong.
3. Approximation #3, it tells us which material is an insulator and which one is a metal. It is widely used in research.
4. Approximation #4, predictions from this approximation are qualitatively the same as approximation #3, but quantitatively much more
accurate (from simple materials). “Simple” here means, electron-electron interactions are not very strong. Most materials studied before
1960s belongs to this family.
What about “not-simple materials”? In late 1960s, especially after 1980s, lots of complex materials (strongly-correlated materials) are discov-
ered, where interactions between electrons are so strong that we can no longer treat the systems as nearly free gases. The most famous example
includes high temperature superconductors and fractional quantum Hall systems.