### Condensed Matter III

Electrons in metals: use the free electron model in which we treat the electrons as being free to move within the crystal (their Bohr radii overlap, so they are effectively delocalised. We ignore electron-electron interactions and assume electrons move in a periodic potential.

Free electrons can be considered as plane waves $\psi=Ae^{\pm ikx}$

Fermi level, μ(T) has occupation number ½. The Fermi level at T=0 is of course the Fermi energy.

Fermi energy and wavevector

$N=\int_0^{E_F}g(\epsilon)d\epsilon=\int_0^{k_F}g(k)dk$

kF is the Fermi wavevector, and $E_F=\frac{\bar{h}^2k_F^2}{2m}$

We can also define a Fermi velocity $mv_F=\bar{h}k_F$

In 1D $k_F=\frac{\pi N}{L}$

In 2D $k_F=(\frac{2\pi N}{A})^{\frac{1}{2}}$

In 3D $k_F=(\frac{3\pi^2 N}{V})^{\frac{1}{3}}$

k-space and the Fermi sphere: think of electrons filling a sphere in k-space, up to a radius kF – this is the Fermi sphere (the surface of which is known as, surprisingly enough, the Fermi surface). It is an idea that will be of use from now on.

Fermi surface and Brillouin Zone
In 3D, for a monovalent metal $k_F=\frac{(3\pi^2)^{\frac{1}{3}}}{a}$ and for a divalent metal, $k_F=\frac{(6\pi^2)^{\frac{1}{3}}}{a}$

If a Brillouin Zone is completely filled there are no empty states so conduction cannot occur. If there are both electrons and empty states, conduction can occur. We will go into this in more detail later as we need to consider electron-lattice interactions.

Electronic heat capacity: Apologies, but as I wasn’t happy with the derivation in the notes I’m using to construct this page I can’t complete this section until I find a better derivation.

Total heat capacity of a metal: $C=\alpha T+\beta T^3$

First term is due to electrons, second arises from Debye model for phonons. To find the constants α and β, we can plot C/T vs. T2 to get a graph with gradient β and y-intercept α.

Pauli paramagnetism

Start off with equal numbers of spin and spin ↑.

Apply a B-field, energy levels shift.

Electrons in the shaded region can now go into lower energy states- their spins flip. This imbalance causes an overall magnetisation.

Approximate the shaded area to be a box.

No. electrons for which spin flips $=\frac{1}{2}\mu_BBg(E_F)$

Change in magnetic moment per electron $= 2\mu_B$

Total magnetisation $M=\mu_B^2Bg(E_F)$

Total area under curves=N

Approximate this area to be a box $N=E_Fg(E_F)=kT_Fg(E_F)$

Gives $M=\frac{N\mu_B^2B}{kT_F}=\frac{\chi B}{\mu_0}$

$\chi=\frac{N\mu_0\mu_B^2}{kT_F}$ smaller than Curie law χ by a factor of $\frac{T}{T_F}$

The Hall effect for one carrier type

Force on electron $\bold{F}=q(\bold{E}+\bold{v}\times\bold{B})$

$F=qE_y-qv_xB_z$

When F=0, $|E_y|=|v_xB_z|$

Hall coefficient $R_H=\frac{E_y}{j_x}{B_z}=\frac{1}{nq}$ using $j_x=nqv_x$

$E_y=\frac{V_y}{d}$ measurable

$j_x=\frac{I}{A}$, I is input, A is known

Bz is applied.

Since all of the above quantities can be measured relatively easily, RH can be determined and hence carrier concentration n can be found.

Hall effect for multiple carrier types will be inserted later.

Experimental note for measuring the Hall effect:

when measuring Vd, there is no way the contacts can be exactly aligned; there will be small ‘across’ voltage and possibly a thermal voltage between the two contacts. To eliminate this offset, reverse the current after making the first set of measurements, retake the measurements and then take the average of the two.

Thermal conductivity in metals: in metals, heat is mainly carried by electrons. Using simple kinetic theory arguments

$\kappa=\frac{1}{3}C_V\lambda$

$C=\frac{\pi^2}{2}Nk\frac{T}{T_F}$, $\lambda=v_F\tau$, $=v_F$

vF is the Fermi velocity, which will be covered later on. For now, just take it as read that $v_F=\sqrt{\frac{2E_F}{m}}$

Substituting gives $\kappa=\frac{\pi^2}{6}NK^2\frac{T}{E_F}v_F^2\tau=\frac{\pi^2}{3}\frac{Nk^2T\tau}{m}$

($\tau \propto \frac{1}{N_{phonon}}$ when phonon scattering dominates. $N_{phonon}~\frac{E_{tot}}{kT}$

Low temperature: τ independent of T. Electrons mainly scatter off defects as there aren’t many phonons around, and those that are presents have long wavelengths and don’t see the electrons. $\kappa \propto T$

Intermediate temperature: low temp phonons obey the Debye model- $\tau \propto T^{-3}$, $\kappa \propto T^{-2}$

High temperature: ‘classical’ phonons- $\tau \propto T^{-1}$, κ constant.

Thermal conductivity in non-metals: electrons are not free, so now heat is mainly carried by phonons.

$\kappa=\frac{1}{3}C_V\lambda$ as before and $\lambda=\frac{1}{N\sigma}$

Low temperature: long wavelength phonons do not see impurities, so the mean free path is limited only by the size of the sample. $C \propto T^3$ from the Debye model, so overall $\kappa \propto T^3$

Intermediate temperature $N_{phonon}=\frac{1}{e^{\frac{\bar{h}\omega}{kT}}-1}\simeq e^{\frac{-\bar{h}\omega}{kT}}$

$\lambda \sim e^{\frac{\theta_D}{kT}}$ dominates T3 dependence of C.

$\kappa \propto T^3 e^{\frac{\theta_D}{kT}}$

High temperature: C tends to classical limit

$N_{phonon}\simeq \frac{1}{1+\frac{\bar{h}\omega}{kT}-1} \propto T$, so $\lambda \propto T^{-1}$

$\kappa \propto \frac{1}{T}$

Electrical conductivity, σ: where thermal conductivity is about energy transfer, electrical conductivity and resistivity are determined by momentum transfer. In order to discuss conductivity, however, we’re going to need to use the Fermi sphere.

Electrical resistance is caused by phonons moving the electrons to the other side of the Fermi sphere (i.e. the electron is slowed down by colliding with phonons).

Low temperature: electron-phonon collisions not present, electron-electron collisions only -> constant σ.

Intermediate temperature: phonons are present but have low momentum. An electron must collide with many phonons to be moved to the other side of the Fermi sphere. Resistivity ~T5 – there is a factor of T3 for phonon number from the Debye model (total energy ~T4 and each phonon has energy ~kT) and an additional factor of T2 to account for the ‘effectiveness’ of a collision (think of it like a cross-section).

Conductivity σ~T-5

High temperature: phonons have greater momentum- a single phonon can move an electron to the other side of the Fermi sphere.

Resistivity depends on phonon number $N_{phonon}\sim\frac{1}{e^{\frac{\bar{h}\omega}{kT}}-1} \sim \frac{1}{1-\frac{\bar{h}\omega}{kT}-1}\sim kT$

So σ~T-1

Expression for electrical conductivity: start from $J=\sigma E=\delta n v_d q$

We need to find vd, the drift velocity. Fortunately, we are spoiled for choice as to how to proceed.

1. Solve the equation of motion $\frac{dv}{dt}+\frac{v}{\tau}=\frac{eE}{m}$ where the v/τ term arises from collisions.
2. Say $\frac{dv}{dt}\sim\frac{v}{\tau}=\frac{eE}{m}$ (i.e. momentum=force x time)
3. $J=\delta n vq$

$\delta n=\frac{\delta n}{\delta k}\delta k=\frac{N}{k_F}\delta k$

Momentum $\bar{h}\delta k=eE\tau$

$\Rightarrow J=\frac{NvqeE\tau m}{\bar{h}k_Fm}$

But of course $mv=\bar{h}k$, so

$J=\frac{Ne^2\tau}{m}E$

Whichever way you choose to work it out, the important result to remember is that $\sigma=\frac{Ne^2\tau}{m}=Ne\mu$, where μ is the mobility, which we are just about to go ahead and define.

Mobility, μ is another useful quantity, defined as below.

$\mu=\frac{v_d}{E}=\frac{e\tau}{m}=\frac{\sigma}{Ne}$

This quantity shouldn’t be too difficult to calculate- we can get N from the Hall effect, and calculate σ from resistivity measurements (recall that $R=\frac{V}{I}=\frac{\rho L}{A}$ and $\rho=\frac{1}{\sigma}=\frac{VA}{IL}=\frac{E}{jx}$

Wiedemann-Franz Law: take the ratio of electron thermal and electrical conductivities.

Definte the Lorentz number $L=\frac{\kappa}{\sigma T}=\frac{\pi^2k^2}{3e^2}$ constant.

This works well at high and low temperatures but breaks down in the ‘Debye’ region where energy and charge scattering are different.

Effect of an electric field on the Fermi surface: Electrons are accelerated by an E-field, with $\delta k=\frac{-eE\tau}{\bar{h}}$ on average. The Fermi sphere is shifted by $\delta k << k_F$

Scattering occurs at EF; to relax momentum, k must be changed by ~kF, which requires phonons with large k. However, phonon energy is small, so only the fraction of electrons $\frac{kT}{E_F}$ can be scattered.

Drift velocity and Fermi velocity: an individual electron moves at the Fermi velocity vF, but in an E or B-field, the electrons as a whole move in a particular direction at the drift velocity, vd. To get an idea of what’s going on, think of a swarm of bees- if we just follow one bee, it might be moving quite quickly and in various directions within the swarm, but the swarm is a whole will be moving more slowly in one particular direction.

When to use what (don’t just blindly memorise this, try to think why we use a particular velocity in a given formula)- $\lambda=v_F \tau$; $\bold{F}=q(\bold{E}+\bold{v_d}\times\bold{B})$; $\frac{mv_d}{\tau}=eE$

Summary- successes and failures of the free electron model

Successes

• Temperature dependence of heat capacity
• Paramagnetic susceptibility (Pauli)
• Lorentz number
• Hall effect and heat capacity of simple metals

Failures

• Hall effect and heat capacity of many metals is not predicted correctly
• No explanation as to why RH can be positive
• No explanation as to why mean free paths can be so long
• Does not explain why some materials are metals whilst others are insulators or semiconductors