Last Time Free electron model Density of states in 3D Fermi Surface Fermi-Dirac Distribution Function Debye Approximation.

Today Measuring the occupied density of states Effective Mass Electrical Conductivity Thermal Conductivity Wiedemann-Franz Ratio Heat Capacity ElectronsPhonons

Fermi-Dirac Distribution Function Becomes a step function at T=0. Low E: f ~ 1. High E: f ~ 0. Go play with the Excel file “fermi.xls” at:  = chemical potential = “Fermi Level”  (T=0)=  F Fermi energy Right at the Fermi level: f = 1/2.

Number of electrons per energy range Fermi function Density of states Implicit equation for  N is conserved Shaded areas are equal  room temp Density of Occupied States

Heat Capacity Width of shaded region ~ kT Room temp T ~ 300K, T F ~ 10 4 K  Small width  Few electrons thermally excited How many electrons are excited thermally? Shaded area  triangle. Area = (base)(height)/2 Number of excited electrons:  (g(  F )/2)(kT)/2  g(  F )(kT)/4 Excitation energy  kT (thermal) Total thermal energy in electrons: C ~ T Heat Capacity in a Metal

How you would do the real calculation: Implicit equation for   fully determines n( , T) Then In a metallic solid, C ~ T is one of the signatures of the metallic state ElectronsPhonons Correct in simple metals. Heat Capacity

Measuring n( , T) X-ray Emission (1)Bombard sample with high energy electrons to remove some core electrons (2)Electron from condition band falls to fill “hole”, emitting a photon of the energy difference (3)Measure the photons -- i.e. the X-ray emission spectrum n( , T) is the actual number of electrons at  and T

Measuring n( , T) X-ray Emission n( , T) is the actual number of electrons at  and T Emission spectrum (how many X-rays come out as a function of energy) will look like this. Fine print: The actual spectrum is rounded by temperature, and subject to transition probabilities. Void in New Hampshire.

EFFECTIVE MASS Real metals: electrons still behave like free particles, but with “renormalized” effective mass m * In potassium (a metal), assuming m * =1.25m gets the correct (measured) electronic heat capacity Physical intuition: m * > m, due to “cloud” of phonons and other excited electrons. Fermi Surface At T>0, the periodic crystal and electron-electron interactions and electron-phonon interactions renormalize the elementary excitation to an “electron-like quasiparticle” of mass m *

Electrical Conductivity Collisions cause drag Electric Field Accelerates charge   mean time between collisions Steady state solution:  =mobility Electric current density (charge per second per area) Units: n=N/V ~ L -3 v ~ L/S current per area average velocity

Electrical Conductivity Electric current density (charge per second per area) current per area Electrical Conductivity OHM’s LAW (V = I R ) n = N/V m e = mass of electron e = charge on electron  = mean time between collisions

What Causes the Drag?

Bam! Random Collisions On average, I go about  seconds between collisions with phonons and impurities electron phonon

Scattering It turns out that static ions do not cause collisions! What causes the drag? (Otherwise metals would have infinite conductivity) Electrons colliding with phonons (T > 0) Electrons colliding with impurities  imp is independent of T

Mathiesen’s Rule how often electrons scatter total how often electrons scatter from phonons how often electrons scatter from impurities Independent scattering processes means the RATES can be added. 5 phonons per sec. + 7 impurities per sec. = 12 scattering events per second

Mathiesen’s Rule Resistivity If the rates add, then resistivities also add: Resistivities Add (Mathiesen’s Rule)

Thermal conductivity Heat current density  = Energy per particle v = velocity n = N/V Electric current density Heat current density

Thermal conductivity Heat current density x Heat Current Density j tot through the plane: j tot = j right - j left j right j left Heat energy per particle passing through the plane started an average of “l” away. About half the particles are moving right, and about half to the left. x

Thermal conductivity Heat current density x Limit as l goes small:

Thermal conductivity Heat current density x

Thermal conductivity Heat current density x How does it depend on temperature?

Thermal conductivity 

Wiedemann-Franz Ratio Fundamental Constants ! Cu: = 2.23  W  /  2 (Good at low Temp) Major Assumption:  thermal =  electronic very hi T & very low T (not at intermediate T)

Homework Problem 3 “r s ” Radius of sphere denoting volume per conduction electron n=N/V=density of conduction electrons In 3D Defines r s

Solid State Simulations Go download these and play with them! For this week, try the simulation “Drude”

Today Measuring the occupied density of states Effective Mass Electrical Conductivity Thermal Conductivity Wiedemann-Franz Ratio Heat Capacity ElectronsPhonons