Chapter 6: The Free Electron Gas

Question 1

Specific Heat in Metals: Experimental Results

Answer 1

Typically plot C/T vs T²

Question 2

Fermi Energy

(3 points)

Answer 2

• From Pauli Principle, each energy level can only contain two electrons. The energy level of the highest filled state is the Fermi level
  
  • Corresponds to a sphere in k-space
  • Radius k_F = (3π​²n)^1/3

Question 3

Chemical Potential

(5 points)

Answer 3

• Unlike phonons, number of electrons N is constant
  
  • Leads to chemical potential included in statistics
• Temperature dependence very small
  
  • µ(300K) ≈ E_F
• When two objects brought into contact µ_a = µ_b

Question 4

Free Electron Gas: Ground State

(Take-Away: 2 points)

Answer 4

• Energy E = (h-bar)²k²/(2m)
• Well-defined momentum p = (h-bar)k

Question 5

Drude vs Sommerfeld

(4 points)

Answer 5

• Drude assumed free mean path l = v_thτ
  
  • ​Scattering at ions → WRONG!
• Sommerfeld assumed free mean path l = v_Fτ
  
  • ​Scattering at imprefections

Question 6

Free Electron Gas: Boundary Conditions

(Take-Away: 2 points)

Answer 6

• k_d = 2πn_i/L_d for d = x, y, z and i = 0, ±1, ±2, …
• For every k, there exists two electrons (because of spin) with energy

Question 7

Hall Effect

(Setup: 4 points)

Answer 7

• Smaple has E-field in +x-direction and B-field in the +z-direction
• Current in +x-direction leads to E-field in +y-direction → Hall Field
• Lorentz force in -y-direction until compensated by E-field
• Leads to charge separation

Question 8

Temperature Dependence of Electrical Conductivity

(2 points)

Answer 8

• σ = 1/ρ so helpful to understand dependence on resistivity
  
  • Due to scattering of at phonons and defects

Question 9

Specific Heat - Electronic Contribution: Sommerfeld Approximation

(Assumptions: 2 points)

Answer 9

• Actual internal energy U not solvable analytically
• F(E) only deviated from F(E(T=0)) in region ±k_BT around E ≈ µ

Question 10

Temperature Dependence of Resistivity

(Graph)

Answer 10

Question 11

Free Electron Gas

(Assumptions: 3 points)

Answer 11

• Electrons in metal are delocalized
• No electron-ion interactions
• No electron-electron interactions

Question 12

Hall Effect

(Take-Away)

Answer 12

• Can obtain sign and charge of carriers feom Measure Hall coefficient R = -1/(ne)

Question 13

Free Electron Gas: Ground State

(Assumptions: 4 points)

Answer 13

• N free electrons
• Volume V = L³
• Temperature T = 0
• Becuase no electron-electron interactions, can solve problem of single electron in a box with volume V

Question 14

Sommerfeld Model

(Take-Away: 4 points)

Answer 14

• Again σ = neµ
  
  • ​But, µ = eτ /m with l = v_Fτ
  • Results in higher conductivity
• Fermi velocity used, because only electrons near Fermi energy can participate

Question 15

2D Electron Gas

(Assumptions: 2 points)

Answer 15

• Potential walls at z = ±L/2
• k_z restricted to πn/L for n = 1, 2, 3, …

Question 16

Temperature Dependence of Resistivity: Phonon Scattering

(6 points)

Answer 16

• High temperature T >> Θ_D
  
  • ​ρ ∝ <n> ∝ T</n>
• Low temperature T << Θ_D
  
  • ​ρ ∝ <n> ∝ T³</n>
  • Experiment shows ρ ∝ T^{<span>5</span>}
  • Additional T² comes from scattering angle being small

Question 17

Low-Dimension Electronic Systems

Answer 17

Obtained by placing potential walls in one or more directions

Question 18

Temperature Dependence of Thermal Conductivity in Metals

Answer 18

• From Wiedemann-Franz Law: K(T) = LσT ∝ T /ρ

Question 19

Fermi Gas: Pressure

Answer 19

Question 20

Electron Density of States in Momentum Space

(2 points)

Answer 20

• One state per unit volume
• Two electrons per state

Question 21

Drude Model

(Take-Away)

Answer 21

• σ = neµ

Question 22

Fermi-Dirac Distribution

(3 points)

Answer 22

• Gives the probability that a state with energy E is occupied for a given temperature
  
  • µ ≡ chemical potential
  • Because k_BT << E_F , only small number of electrons redistributed

Question 23

Fermi Gas: Total Energy per Electron

Answer 23

• Even at T = 0, still very high

Question 24

Drude Model

(Assumptions: 6 points)

Answer 24

• Electrons in metal behave clasically like a gas of particles
  
  • Move with thermal velocity v_th
  • Collide with atomic cores
• Acceleration comes from electric field
• Deceleration comes from collision
  
  • Relaxes velocity within τ

Question 25

Wiedemann-Franz Law | (2 points)

Answer 25

* Gives relationship between thermal conductivity *K* and electrical conductivity *σ* * *K/(σT) = L* → constant: Lorentz number

Question 26

Other Fermi Quantities

Answer 26

* Fermi temperature *k_BT_F = E_F* * Fermi wavelength *k_F λ_F = 2π* * Fermi velocity: *mv_F = p_F = (h-bar)k_F*

Question 27

Specific Heat - Electronic Contribution: Classical Treatment | (Take-Away)

Answer 27

100x larger than experiment, because of Pauli Principle

Question 28

Specific Heat - Electronic Contribution: Classical Treatment | (Assumptions: 2 points)

Answer 28

* Each electrons has 3*/2\*k_BT* energy * Factor of 2 from spin

Question 29

Electron Density of States in Energy Space | (Graphs)

Answer 29

Question 30

Sommerfeld Model | (Assumptions: 5 points)

Answer 30

* Electrons are gas of free *fermions* * Obey Schrodinger's equation * Obey Pauli Principle * Entire Fermi sphere responds to external ***E*** * When external ***E*** turned off, relaxation occurs ∝ *e^−t/τ* due to scattering

Question 31

Electron Density of States in Energy Space | (Equations)

Answer 31

Question 32

Free Electron Gas

Answer 32

An ideal gas of free, non-interacting electrons

Question 33

Free Electron Gas: Boundary Conditions | (Boundary Condition)

Answer 33

* *ψ_k(x,y,z) = ψ_k(x + L_x,y,z) = ψ_k(x,y + L_y,z) = ψ_k(x,y,z + L_z)*

Question 34

Thermal Conductivity in Metals

Answer 34

* Can be calculated according to kinetic gas theory like phonons * *K_ph \<\< K_el* for metals

Question 35

2D Electron Gas | (Take-Away)

Answer 35

* There exists confinement energy ∆E ∝ L⁻² * Creates 2D parabolic sub-bands in xy-plane

Question 36

Fermi Gas at Finite Temperature | (2 points)

Answer 36

* For *T \> 0,* smearing out of Fermi edge * Distributed according to Fermi-Dirac statistics

Question 37

Fermi Gas: Compressibility

Answer 37

Question 38

Temperature Dependence of Electric Conductivity | (Graph)

Answer 38

Question 39

Specific Heat - Electronic Contribution: Sommerfeld Approximation | (Take-Away: 3 points)

Answer 39

* Only electrons near *E_F* can redistribute and contribute to specific heat * Number of electrons *N_th ≈ D(E_F)k_BT* * Justification for *C_V ∝ T*

Question 40

Drude Model | (Overview: 4 points)

Answer 40

* Historically oldest model * Correctly predicts * ***J**_q ∝ **E*** * Weidemann-Franz Relation

Question 41

Temperature Dependence of Resistivity: Defects Scattering | (3 points)

Answer 41

* Number of defects is constant * ρ₀ ≡ residual resistance is constant (independent of T) * At low *T,* phonon scattering vanishes, and only residual remains

Question 42

Electrical Conductivity

Answer 42

Question 43

Electron Motion in Magnetic Field

Answer 43

* Electrons now feel Lorentz force