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Phys 460 Lecture 12
(
pdf version - 6 slides/page
)
Wednesday, October 4, 2006
Lecturer: Richard Martin
No new Homework today
Reading:
Kittel, Chapt. 6
Start of Part II of Course - Electronic Properties of Solids
The Electron Gas
Outline
From previous lectures:
Typical Crystal Structures
Diffraction, Fourier Analysis, and the Reciprocal Lattice
Crystal binding, elastic waves
Vibration waves in crystals: dispersion curves, quantization
Thermal properties of crystals due to vibrations
Role(s) of electrons in solids
Electrons hold the solid together!
Recall the binding mechanisms from Ch. 3
Electrons also determine electrical conductivity, magnetism, optical properties, ...
History
Failure of classical mechanics (Drude-Lorentz model)
Start of quantum mechanics
Electrons in atoms and solids played a key role in establishing quantum mechanics
Electrons obey Pauli exclusion principle and Fermi statistics
Leads to Fermi distribution f(E) = 1/[exp(E-mu)/kT) +1] where mu = chemical potential for electrons = Fermi energy at T=0
Simplest model - non-interacting electron gas
Electrons in a box
E = p
2
/2m
In this case we can solve the Schrodinger Eq.
Because waves must fit into the box they are quantized with E = (hbar
2
/2m) k
2
, where k = m pi/L
Extends simply to three dimensions
Easiest to use traveling waves that obey periodic boundary conditions (just as for phonons) psi ~ exp(i k dot r), where k = plus/minus m (2pi/L)
Key point: Each state corresponds to a volume in k space of (2pi/L)
3
= (2pi)
3
/V - exactly the same as for phonons!
Fermi energy and momentum
Because electrons obey exclusion principle, the lowest energy state is for all state filled up to the Fermi energy (or Fermi momentum)
All other states empty (infinite number of states extending to infinite energy)
This is the "ground state" - the state of the system at T=0
Density of states, Internal energy, Heat capacity
Density of states, D(E) = # states for a given spin per unit energy per unit volume
= (1/2pi
2
) E
1/2
(2m / h
2
)
3/2
Key point: D(E) ~ square root of E in 3 dimensions
U = Integral over E of D(E) * f(E)
C = dU/dT
By simple arguments one can see that C ~ k
B
(T/T
F
)
Direct mathematical integration gives coefficient, C = (pi
2
/3) D(E
F
)k
B
2
T
For the gas, this is (pi
2
/2) k
B
(T/T
F
) per electron
Typical values
Define the Fermi temperature T
F
= E
F
/k
B
Key point: T << T
F
at ordinary T
Comparison of heat capacity of phonons and electrons in a metal
Electrons dominate at low T (C ~ T for electrons compared to T
3
for phonons)
Phonons dominate at high T because of the reduction factor (T/T
F
) for electrons due to the exclusion principle
Email clarification questions and corrections to
rmartin@uiuc.edu
Email questions on solving problems to
xin2@.uiuc.edu