Today’s Objectives Finish energy level example Derive Fermi Dirac distribution Define/apply density of states Review heat capacity with improved model Chapter 1: Applied Drude model to understand current density, the Hall coefficient, AC conductivity, heat capacity and thermal conductivity in simple metals. Required knowledge/determination of carrier density. Chapter 2: Derive and show how Fermi-Dirac statistics can be added to improve the Drude model.

34 Electrons in 3D Infinite Well  (n 1, n 1, n 1 )  The lowest energy for this system is 3E 0, which corresponds to n 1 = n 2 = n 3 = 1 Thus only 2 (two) electrons can have this energy: one with spin  and one with spin   Next energy level (6E 0 ), for which one of n’s is 2 [211] A total of 6 (six) electrons can have this energy  Next energy level (9E 0 ) [221] What are the combinations of n’s for this energy level?  Next energy level (11E 0 ) [311] Accommodates 6 electrons  Next energy level (12E 0 ) is two-fold degenerate [222]

34 Electrons in 3D Infinite Well  So far we have placed 22 electrons, so we need to add another 12 electrons  What is the next energy level?  The next energy level is 14E 0 What are n 1, n 2, and n 3 ? What is the degeneracy? This energy can be had by 12 electrons  We have placed all 34 electrons!! 1, 2, 3

34 Electrons in 3D Infinite Well  In this configuration, What is the probability at T =0 that a level with energy 14E 0 or less will be occupied? It is 1! What is the probability that a level with energy above 14E 0 will be occupied at T=0? It is 0!

Fermi Energy In a metal:  The highest filled energy is called the Fermi Energy  It is often denoted as E F  In our case: E F = 14E 0  An electron with E = 14E 0 is said to be at the Fermi level

Other Fermi Terms  Only at temperatures above E F =k B T F will the free electron gas behave like a classical gas.  Fermi momentum  And Fermi velocity  These are the momentum and the velocity values of the electrons at the states on the Fermi surface of the Fermi sphere.  The Fermi Sphere plays important role on the behavior of metals.

Example: monovalent potassium metal Only at temperatures above E F =k B T F will the free electron gas behave like a classical gas.

T=0 Thermal Properties At T=0, all levels up to E F are filled. What happens when T is greater than zero?

Probability with Temperature Start Classical: At finite temperature T the probability that state N is occupied depends on the Boltzmann distribution: To calculate properties, we have to take the average over all states Partition function Z Helmholtz free energy F U=internal energy, S=entropy

Combining Due to exclusion, fi is probably of an electron being in a particular one electron level i Or one minus all E>E F states 

What if we add one more electron The energy difference between this 34 electron and a 35 electron system is the energy in state (3,2,2) Or more generically…

Some algebra manipulation Chemical potential

Solving for f i The probably should not change that much from one electron

Temperature affect on probability At the chemical potential, there is a 50% chance of finding an electron As T0, E F

How does this help us understand the problem with Drude’s heat capacity?  When a metal is heated, electrons are transferred from below E F to above E F. The rest of the electrons deep inside the Fermi level are not effected. Will come back to if time. Otherwise, see extra slides

Reminder: The Density of Levels (Closely related to the density of states) As discussed last 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 Or V/ (2) 3 Then the density of those levels is N/ or V/8 3

Reminder: 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: Since electrons, extra factor of 2 because of spin dk=k 2 sin dk d d n= Carrier density

Calculating the carrier density n= dk=k 2 sin dk d d g(E) is the density of levels per unit volume Let’s find g(E)! (Just given in book)

Density of Levels The number of running waves N: per dimension k Space N=g(k) Result same for standing waves! L/2 4k 2 dk Times 2 for spin!

Group: dnkdnk n= Find d n k for n=1 and 2 dimensions

Finding the 3 dimensional density of states D(E) g(k)g(E) g(k) g(E)

Free electrons in 3D Group: Find g(E) = g(k) dk/dE More than one way to approach g(E)

Density of levels

The free electron gas at T < Fermi temp Big difference at high temps k

T>0 T=0 n(E,T) E g(E) EFEF n(E,T)  number of free electrons per unit energy range is just the area under n(E,T) graph. The shaded area shows the change in distribution between absolute zero and a finite temperature. Fermi-Dirac distribution function is a symmetric function, meaning: At low temperatures, the same number of levels below E F is emptied and same number of levels above E F are filled by electrons.

A knowledge of the thermal energy is fundamental to obtaining an understanding many of the basic properties of solids. For example, the heat energy necessary to raise the temperature of the material, called Specific Heat or Heat Capacity Application: Determining Metallic Properties

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 k B T, and the total energy is 3/2 nk B T where n is the conduction electron density and k B is Boltzmann constant. Differentiation w.r.t temperature gives a constant for the heat capacity: 3/2 n k B Want to Contrast with Drude Model of Heat Capacity

Heat capacity of the free electron gas  From the diagram of n(E,T) the change in the distribution of electrons can be resembled into triangles of height ½ g(E F ) and a base of 2k B T so the area gives that ½ g(E F )k B T electrons increased their energy by k B T. ~ The difference in thermal energy from the value at T=0°K For an exact calculation:

 Differentiating with respect to T gives the heat capacity at constant volume: Heat capacity of Free electron gas While this works great for metals, we will find that it’s very different for other materials. ~

Comparison with Data Classical Model suggests one constant value and same value for all materials T F =65,000 K To improve upon this model, we’ll have to consider phonons (later)

Measuring specific heat on a budget Step 1: Set-up the calorimeter Step 2: Boil the water containing metal, Pour Step 3: Stir while measuring temperature Energy conserved: q metal = q water + q coffeecup q cup can be ignored c metal m metal  T system = c water m water  T water

constant volume or 'bomb' calorimeter Modern calorimetry works on the same principles, just looks more fancy.

Why Specific Heat? Profile of Frances Hellman Physics Professor at University of California, Berkeley Previous chair of the physics department “My research group is concerned with the properties of novel magnetic and superconducting materials especially in thin film form. We use specific heat, magnetic susceptibility, electrical resistivity, and other measurements as a function of temperature in order to test and develop models for materials which challenge our understanding of metallic behavior. Current research includes: effects of spin on transport and tunneling, including studies of amorphous magnetic semiconductors and spin injection from ferromagnets into Si; finite size effects on magnetic and thermodynamic properties…”

Free Electron Models Classical Model:  Metal is an array of positive ions with electrons that are free to roam through the ionic array Electrons are treated as an ideal neutral gas, and their total energy depends on the temperature and applied field In the absence of an electrical field, electrons move with randomly distributed thermal velocities Quantum Mechanical Model:  Electrons are in a potential well with infinite barriers: They do not leave metal, but free to roam inside Electron energy levels are quantized and well defined, so average energy of electron is not equal to (3/2)k B T Electrons occupy energy levels according to Pauli’s exclusion principle

Homework 2.3: When the Fermi-Dirac distribution is close to the MB, what does that mean for r s ? Maxwell-BoltzmannFermi-Dirac r s – another measure of electronic density = radius of a sphere whose volume is equal to the volume per electron (mean inter-electron spacing)

(If time) Strength of Metals (p. 39) The book nicely derives (using P=-dE/dV) that the pressure need to change the metal volume P =(V/V) 2/3 n E F As n is quite dense, need pressures over 10,000 atmospheres for a volume change of 1%! This Fermi pressure is what makes solids stable.