Last Time Debye Approximation. Free electron model Fermi Surface h2 k F EF = 2m 2 Fermi-Dirac Distribution Function Density of states in 3D g~ 1/ 3 3 N kF = V
2 Today Heat Capacity C = AT + BT 3 Electrons Phonons Measuring the occupied density of states Effective Mass Electrical Conductivity Thermal Conductivity Wiedemann-Franz Ratio Fermi-Dirac Distribution Function f ( , T )= 1 e ( ) + 1 Becomes a step function at T=0. Low E: f ~ 1.
High E: f ~ 0. = chemical potential = Fermi Level (T=0)=T=0)=F Fermi energy Right at the Fermi level: f = 1/2. Go play with the Excel file fermi.xls at: http://ece-www.colorado.edu/~bart/book/distrib.htm#fermi Density of Occupied States Density of states Fermi function n ( , T ) = g( ) f( , T ) Number of electrons per energy range N = n ( , T )d = g( ) f( , T )d = N 0
Implicit equation for 0 N is conserved Shaded areas are equal (T > 0 ) < F (T ) = F 1 o (T 2 ) 0.01% @ room temp Heat Capacity Width of shaded region ~ kT Room temp T ~ 300K, TF ~ 104 K Small width Few electrons thermally excited How many electrons are excited thermally? Shaded area triangle. Area = (T=0)=base)(T=0)=height)/2 Number of excited electrons: (T=0)=g(T=0)=F)/2)(T=0)=kT)/2 g(T=0)=F)(T=0)=kT)/4 Excitation energy kT (T=0)=thermal) 1
4 1 4 2 Total thermal energy in electrons: E g( F ) kT (kT ) = g( F )(kT ) C= dE 1 3N 2 g( F ) k2T = kT dT 2 4 F Heat Capacity in a Metal C ~ T Heat Capacity
How you would do the real calculation: N = n ( , T ) d Implicit equation for fully determines n(T=0)=, T) 0 E = n ( , T ) d = f( , T ) g( ) d 0 Then 0 2 dE =C = g( F ) kB2T
dT 2 In a metallic solid, C = AT + BT 3 Electrons Phonons C ~ T is one of the signatures of the metallic state Correct in simple metals. Measuring n(T=0)=, T) N = n ( , T ) d = g( ) f( , T )d = N 0 0
n(T=0)=, T) is the actual number of electrons at and T X-ray Emission (T=0)=1) Bombard sample with high energy electrons to remove some core electrons (T=0)=2) Electron from condition band falls to fill hole, emitting a photon of the energy difference (T=0)=3) Measure the photons -- i.e. the X-ray emission spectrum Measuring n(T=0)=, T) N = n ( , T )d = g( ) f( , T )d = N 0 0 n(T=0)=, T) is the actual number of electrons at and T X-ray Emission
Emission spectrum (T=0)=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* h2 k2 E= * 2 In potassium (T=0)=a metal), assuming m* =1.25m gets the correct (T=0)=measured) electronic heat capacity Physical intuition: m* > m, due to cloud of phonons and other excited electrons. 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* Fermi Surface
Electrical Conductivity *v v v dv m v F =m * =eE dt Collisions cause drag Electric Field Accelerates charge mean time between collisions Steady state solution: &=0 v v
v v = * E average velocity e e = * =mobility m Electric current density (T=0)=charge per second per area) v ne 2 v j =nev = * E m Units: n=N/V ~ L-3 current per area v ~ L/S Electrical Conductivity Electric current density (T=0)=charge per second per area) v ne 2 v j =nev = * E m
v j E current per area OHMs LAW ne 2 = * m (T=0)=V = I R ) Electrical Conductivity n = N/V e = charge on electron me = mass of electron = mean time between collisions
What Causes the Drag? On average, I go about seconds between collisions electron Bam! phonon Random Collisions with phonons and impurities 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)=T > 0)
Electrons colliding with impurities ph (T =0) imp is independent of T Mathiesens Rule how often electrons scatter from impurities 1 tot how often electrons scatter total =
1 ph (T ) + 1 imp how often electrons scatter from phonons Independent scattering processes means the RATES can be added. 5 phonons per sec. + 7 impurities per sec. = 12 scattering events per second 1 Mathiesens Rule me 1 = 2
ne ne 2 = me tot = 1 ph (T ) + Resistivity If the rates add, then resistivities also add: ph
me 1 = 2 ne ph tot = ph + imp imp = me = 2 ne 1 1 + ph imp me 1 ne 2 imp
Resistivities Add (Mathiesens Rule) 1 imp Thermal conductivity Electric current density Heat current density v E jt = secarea Heat current density = Energy per particle
jt =vn v = velocity n = N/V Thermal conductivity Heat current density jleft jright x Heat Current Density jtot through the plane: jtot = jright - jleft 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 v T T x v 1 2 jt = v cv T 3 v
2 = vx 2 + vy 2 + vz 2 =3 v x 1 2 k = v c v 3 How does it depend on temperature? 2
Thermal conductivity 1 2 k = v c v 3 1 2 E F = me v F 2 T 2 cv = nkB 2 TF T 1 2EF 2 k= nkB
3 me 2 TF 2 nkB 2T = = 3me 2 k B 2T = 3 n m e Wiedemann-Franz Ratio 2 2 k B T n = 3 me 2 n
=e me 2 k kB 8 = =2.45 10 T 3 k2 Fundamental Constants ! Cu: = 2.23 10-8 W/k2 (T=0)=Good at low Temp) Major Assumption: thermal = electronic Good @ very hi T & very low T (T=0)=not at intermediate T) 2
Homework Problem 3 rs Radius of sphere denoting volume per conduction electron Defines rs V 1 4 3 = N n 3 n=N/V=density of conduction electrons 1/ 3 In 3D 3 rs = 4n Solid State Simulations http://www.physics.cornell.edu/sss/ Go download these and play with them!
For this week, try the simulation Drude Today Heat Capacity C = AT + BT 3 Electrons Phonons Measuring the occupied density of states Effective Mass Electrical Conductivity Thermal Conductivity Wiedemann-Franz Ratio
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