Thermal conductivity A material's ability to conduct heat. Temperature Electric current density (je = I/A) Heat current density Fourier's Law for heat conduction.  Thermal current density  = Energy per particle v = velocity n = N/V 2l

About half the particles are moving right, half to left. Thermal conductivity Heat current density Heat Current Density jtot through the plane: jtot = jright - jleft Limit as l gets small: Read this as the energy is a function of position, due to the temperature gradient (not multiplying by x+/- L) The slides I got this argument from said: “Heat energy per particle passing through the plane started an average of “l” away.” Seems like it should be l/2 to me. About half the particles are moving right, half to left. x

Thermal conductivity x v v v

Thermal conductivity (expanding to 3d) Heat current density x As you go to three dimensions dT/dx goes to a gradient and v… How does / depend on temperature?

Thermal conductivity 1/3 cvv2 1/3 cvmv2 ne2/m ne2 cvkBT ne2 = = Drude applied kinetic theory of gases ½ mv2 = 3/2 kBT Is there something we could do to simplify this? cvkBT The book jumps through claiming a value for cv = ne2

Classical Theory of Heat Capacity When the solid is heated, the atoms vibrate around their sites like harmonic oscillators. The average energy for a 1D oscillator is ½ kT. Therefore, the average energy per atom, regarded as a 3D oscillator, is 3/2 kBT, and the total energy density is 3/2 nkBT where n is the conduction electron density and kB is Boltzmann constant. Differentiation w.r.t temperature gives heat capacity 3/2 n kB Got to this slide (27), but was running out of time 3/2 n kB2T = = 3kB2T / 2e2 ne2

Thermal conductivity optimization To maximize thermal conductivity, there are several options: Provide as many conduction electrons as possible free electrons conduct heat more efficiently than phonons (=lattice vibrations). Make crystalline instead of amorphous irregular atomic positions in amorphous materials scatter phonons and diminish thermal conductivity Remove grain boundaries gb’s scatter electrons and phonons that carry heat Remove pores (air is a terrible conductor of heat) Or do the opposite if you want to minimize thermal conductivity

What happens?

The Seebeck Effect A temperature gradient generates an electric field E = QT, where Q is known as the thermopower = -cv / 3ne Heat it for a short time and then remove heat Versus Keep heating faster than can relax

Seebeck and the reverse (Peltier) Effects ~ millivolts/K for (Pb,Bi)Te The Seebeck effect is the conversion of temperature differences directly into electricity. Applications: Temperature measurement via thermocouples; thermoelectric power generators; thermoelectric refrigerators; recovering waste heat Lead telluride and or bismuth telluride are typical materials in thermoelectric devices that are used for heating and refrigeration. Demo: https://www.youtube.com/watch?v=bt5o_rn0FmU

Many open questions: Why does the Drude model work relatively well when many of its assumptions seem so wrong? In particular, the electrons don’t seem to be scattered by each other. Why? Why is the actual heat capacity of metals much smaller than predicted? From Wikipedia: "The simple classical Drude model provides a very good explanation of DC and AC conductivity in metals, the Hall effect, and thermal conductivity (due to electrons) in metals. The model also explains the Wiedemann-Franz law of 1853. "However, the Drude model greatly overestimates the electronic heat capacities of metals. In reality, metals and insulators have roughly the same heat capacity at room temperature.“ It also does not explain the positive charge carriers from the Hall effect. one thing we can understand right now is why Wiedemann Franz works more or less: we get rid of all the funny quantities, lambda, tau and so on. I don’t cite Wikipedia often, but it does a reasonable job summarizing here.

Objectives By the end of this section you should be able to: Apply Sch. Equation to a metal Apply periodic boundary conditions Start to understand k space Determine the density of states and Fermi energy Find the Fermi temperature, velocity, etc. K space will make a lot more sense after we talk about reciprocal lattices and diffraction, but we will start to be introduced to it here. Don’t worry if it is still confusing.

Improvement to the Drude Model Sommerfield recognized we needed to utilize Pauli’s exclusion principle Typically, this is the only difference Electrons cannot all be in the lowest energy state, since this would violate the Pauli Principle. In most applications of the Sommerfield model, this was the only difference. The book shows this graph with a kinetic energy/kBT axis instead of energy. Personally, I think energy makes more sense to most people.

Number of electrons per unit volume f(v) Fermi-Dirac Maxwell-Boltzmann = N/V To is some normalization temperature that we’ll come back to Normalization condition solves for constants Another common way to write is f(E)

Sommerfield still assumes the free electron approximation U(r) U(r) Neglect periodic potential & scattering Reasonable for “simple metals” (Alkali Li,Na,K,Cs,Rb) What does this remind you of?

The Quantum Analogy These conduction electrons can be considered as moving independently in a square well and the edges of well corresponds to the edges of the sample. (ignores periodic potential from atoms) A metal with a shape of cube with edge length of L Cube V=L3 Periodic boundary conditions with running wave in 1D L U Inside U=0, for 3 dimensions:

How do we go about solving this? L U Possible Boundary conditions 1. Common: Ψ(0)=0 and Ψ(L)=0 Standing waves. Wells aren’t really infinite 2. Periodic: Ψ(x,y,z)= Ψ(x+L,y,z) The transfer of charge and energy doesn’t just stand, it runs like running waves. We show that this eigenstate solves the solution, which also gives us these eigenvalues or energies Talk about what k is (units: radians/meter). Compare to plane wave sin[k(x-vt)]. Show E by taking derivative twice. What does V equal infinity mean? It’s never leaving (and has always been there, makes me think of universes). Even black holes radiate. What can we learn about the simple well? What would we like to know? How do we figure those out? What would happen if use negative k, try -1, same curve, just negative with eigenvalues Eigenstates Known as a running wave

To Compare, Let’s Remind Ourselves of the Standing Wave Solution L U Boundary conditions Ψ(0)=0 and Ψ(L)=0 Eigenstates with eigenvalues Write out what plugging in +L does to wavefunction on board and gives the condition above Talk about what k is (units: radians/meter). Compare to plane wave sin[k(x-vt)]. Show E by taking derivative twice. What does V equal infinity mean? It’s never leaving (and has always been there, makes me think of universes). Even black holes radiate. What can we learn about the simple well? What would we like to know? How do we figure those out? What would happen if use negative k, try -1, same curve, just negative where Or in 3D: Where nx, ny and nz are integers Similar idea for running waves: How to find A?

Wavefunctions: Ideal Quantum Well 1D standing waves What does the wavefunction mean? Is this a realistic well? Why not? What would a realistic well look like? In your group, write the wavefunction for the lowest three energies.

Semiconductor Quantum Well More about this diagram later today In

Optical Detection of Spin Polarization in Quantum Wells hot CoFe e GaAs/InGaAs/GaAs CoFe external magnetic field e Alumina tunnel barrier h n The electrons with spin aligned in the direction of the magnetic field are more easily able to travel across the tunnel barrier. Those that do get through, however, are reduced in energy due to the tunnel barrier and thus are less likely to get through the Schottky barrier. A laser excites the material, producing an equal number of up and down electron spins. The electrons recombine with holes in the quantum well. The polarization of the photons given off from this recombination provides the spin state of the electron just before recombination. This is a very simple spin-selective device. Electrons of one angular momentum are favored as they travel past the Schottky barrier due to the external magnetic field and spin filtering in the CoFe. They then fall into the quantum well and recombine with holes. Emission from the quantum well gives a good probe of spin. h InGaAs GaAs: n i i p

The wave vector k is very important! Eigenstates with eigenvalues To see why, note that  is an eigenstate of the momentum operator p Operate p on wavefunction to get eigenvalue for momentum P=hbark

k is the wave vector (Will explore more in Ch.5) Momentum space or k-space is the set of all wavevectors k, associated with particles - free and bound All points in a crystal that have an identical environment are described by one point in k space. This allows us to dramatically reduce the size of many atom systems.

The Density of Levels (Closely related to the density of states) As we’ll see next time, we will often need to know the number of allowed levels in k space in some k-space volume  If >>2/L, then the number of states is ~ / (2/L)3 (in 3d) Or V/ (2)3 Then the density of those levels is N/ or V/83

Will show an example later Summing over k space Since the volume of k space is V/83, summing any smooth function F(k) over k space can be approximated as: Will show an example later

Consider a spherical reason Let’s find the number of allowed k values inside a spherical shell of k-space of radius kF The number of allowed values of k Since there are two spin states for each k

The k-space sphere with radius kF is called the Fermi sphere. Warning: The Fermi level will be defined slightly differently for nonmetals. The Fermi Sphere The k-space sphere with radius kF is called the Fermi sphere. kz ky kx Fermi surface kF If we convert k-space to energy space, the resulting radius of the energy sphere surface would give us the cutoff between occupied and unoccupied energy levels. The surface of the Fermi sphere represent the boundary between occupied and unoccupied k states at absolute zero for the free electron gas.

Definition of the Work Function Additional energy above the Fermi level required to remove electrons from the solid fermi level =work function (3-4eV)

Fermi Energy in terms of the Bohr radius Bohr radius = mean radius of the orbit of an electron around the nucleus of a hydrogen atom at its ground state Recognize? Ground state energy of hydrogen atom

Electrons in 3D Infinite Potential Well Group: What is the ground state configuration of many electrons in the 3D infinite potential well? Consider the case of solid with 34 electrons. Determine the energy of each electron relative to . How many electrons are of each energy? Take the ground state to be when n1 = n2 = n3 = 1

Extra slides we may not have time to cover (just extra examples)

Summing the energy density over k space Since the volume of k space is V/83, summing any smooth function F(k) over k space can be approximated as: An extra factor of 2 because of spin dk=k2sin dk d d

Energy per Electron E/N in the ground state Combining: Fermi Temperature

Calculate the Fermi energy for copper. The density of copper is 8.96 gm/cm3, and its atomic weight is 63.5 gm/mole. (assume valence of 1) Calculate the Fermi energy for copper. Find the classical electron velocity from EF= ½ mvF2 and Fermi temperature.  

Pressure and Compressibility of an Electron Gas (Skip if time low, result most important, in book) Pressure is defined as E/ V for constant N, so the pressure on an electron gas is

Reminder: Effective Radius rs – another measure of electronic density = radius of a sphere whose volume is equal to the volume per electron (mean inter-electron spacing) Å – Bohr radius in metals rs ~ 1 – 3 Å (1 Å= 10-8 cm) rs/a0 ~ 2 – 6 Combine with above: Bohr radius = mean radius of the orbit of an electron around the nucleus of a hydrogen atom at its ground state Draw box with lots of circles in it on the board Å-1