MTLE-6120: Advanced Electronic Properties of Materials Electron - - PowerPoint PPT Presentation
1 MTLE-6120: Advanced Electronic Properties of Materials Electron transport: phonons and electron-phonon scattering Contents: Phonon density of states and heat capacity Thermal conductivity: electronic and lattice Electron-phonon
◮ Phonon density of states and heat capacity ◮ Thermal conductivity: electronic and lattice ◮ Electron-phonon momentum relaxation time
◮ Kasap: 4.10
◮ 1D chain of atoms of mass M connected by springs K ◮ Frequencies ω = 2
M
2
M , the sound velocity
2(K/M)1/2
+π/a Frequency ω Wavevector k
◮ Phonons have a band structure, just like electrons ◮ Key difference: linear near k = 0(Γ) rather than quadratic ◮ In 3D isotropic mateirals: two sound speeds: longitudinal (vL) and
◮ Two tranverse modes degenerate near k = 0 only (three polarizations) ◮ Compare energy scale against electrons: factor of 100 smaller
◮ Number of states per unit volume in
1 (2π)d ◮ Phonon dispersion relations ω = ωn(
◮ Therefore density of states g(ω) = n
k (2π)d δ(ω − ωn(
◮ Need a simple model to evaluate analytically (analogous to free electrons)
◮ How many total phonon states in one band per unit cell? One! ◮ Debye model: keep linear dispersion with correct total number of states ◮ If ω = v|
◮ For linear phonon dispersion in 3D
◮ If this is true for all ω, then total number of modes is
◮ Debye model: cutoff model at ωD to keep correct number of states ◮ Number of states per unit volume (one band) = nion (# ions / volume) ◮ Therefore impose condition:
D
D
◮ Corresponding cutoff in wave vector, kD = ωD/v = (6π2nion)1/3 ◮ Note similarity to kF = (3π2n)1/3 for electrons
◮ Using two Debye models for velocities vL and vT ◮ Note energy scale ED = kBTD = ωD
◮ At temperature T each phononic state of energy E has average occupation
E kBT − 1 ◮ For massless bosons like photons and phonons, µ = 0 because number not
◮ Density of phonon modes g(ω) = 4πω2 (2πv)3 Θ(ωD − ω) (Debye model) ◮ Phonon modes occupied by Bose function nph(ω) at temperature T ◮ Internal energy per unit volume:
ω kBT − 1
ω kBT − 1
◮ At high temperatures ω ≪ kBT ⇒ exp ω kBT − 1 ≈ ω kBT ◮ Correspondingly, internal energy is approximately
ω kBT
D
D
◮ Accounting for the three polarizations, U = 3nionkBT and CV = 3nionkB ◮ This is (of course) the equipartition result: high T → classical limit
◮ At low temperatures,
ω kBT − 1
ωD kBT
BT 4
BT 4
BT 4
◮ Accounting for one longitudinal and two transverse velocities:
B
L
T 1
T 2
◮ And corresponding heat capacity
B
L
T 1
T 2
◮ Energy current due to one electron E
◮ At energy E, energy flux g(E)f(E)(Ev(E)) in random directions,
◮ Non-uniform temperature ⇒ average energy flow in one direction ◮ Assume constant temperature gradient T(x) = T0 + x dT dx ◮ What is the net flux of energy across x = 0?
2πd cos θ 4π
λ
−1 2πd cos θ 4π
λ
d cos θ 2
λ
−1 d cos θ 2
λ
d cos θ 2
λ
d cos θ 2
λ
d cos θ 2 −∞ dxx ex/(λ cos θ) λ
d cos θ 2
λ
d cos θ 2
d cos θ 2
F τ
F τ
F τ
BT
F g(EF )τ
BT
F g(EF )τ
3
B
B
◮ Phonons cannot transport charge, but they can transport heat. ◮ Our derivation for electrons didn’t assume electrons till this point:
◮ Lattice contribution: phonon DOS and Bose occupations instead
◮ Which is larger? ◮ For semiconductors / insulators: few electrons (and holes) ⇒ κL dominates ◮ For metals:
◮ n ≈ nion ◮ vF ∼ 106 m/s ≫ vL, vT ∼ 103 − 104 m/s (electrons win) ◮ τph ∼ ps, while τe ∼ 10 fs (phonons win)
◮ Net result: κe dominates at room temperature, κL important at high T ◮ Small κL ⇒ Lorenz number (based only on κe) close to ideal
◮ All transport coefficients depend on scattering time τ ◮ For electrons: τ determined by electron-phonon scattering ◮ Crude argument during Drude discussion:
◮ Electrons only scatter against displaced ions (now we know why: band
◮ Cross-section ∝ mean-squared displacement ∝ kBT (equipartition) ◮ Therefore τ −1 ∝ T, and ρ =
m ne2τ ∝ T
◮ Why is this not true at low T? ◮ Equipartion no longer valid for T ≪ TD
100 200 300
T emperature [K]
1 2 3 4
Resistivity [10-8 m]
P u r e c
p e r C
d w
k e d + 1 % N i + 2 % N i
◮ Process: electron in one state absorbs (or emits) a phonon to go into
◮ Phonon absorption rate of electron in state i (Fermi’s Golden rule):
◮ Ingredient 1: sum over states j ◮ Count up electronic states using
◮ State must be unoccupied (Fermi statistics): (1 − f(Ej))
◮ Also need to sum over phonons available to absorb:
◮ Net result:
◮ So far assumed that final states counted as:
◮ But many pairs of final states will not satisfy
◮ Instead of all pairs, only count those with correct momentum ◮ RHS on sphere of radius kj ≈ ki (phonon energies negligible) ◮ LHS on sphere of radius kph centered on
◮ Total pairs of
j) · (4πk2 ph) ◮ Momentum conserving set: 2πkph ◮ Fraction conserving momentum:
j) · (4πk2 ph) =
jkph
i ω ◮ Net final state counting:
i ω
◮ Does electron-phonon scattering randomize momentum?
◮ Change in angle θ between
i + k2 j − k2 ph
ph
i ◮ Extent of randomization proportional to:
i v2 ph ◮ To calculate momentum-relaxtion time instead of
i ω ·
i v2 ph
◮ Matrix element:
j (
◮ For estimate, make similar to argument made in classical case ◮ Instead of equipartition, apply to single phonon mode ◮ Mean-squared atom displacement ∝ ω ◮ ∆V (
◮ Therefore assume
◮ Collecting ingredients together:
i ω ·
i v2 ph
◮ Neglect phonon energies ω ≪ Ej, Ei (two orders smaller)
i vph
i vph
ω kBT − 1
◮ For high temperatures T ≫ TD, exp ω kBT − 1 ≈ ω kBT :
ω kBT
◮ For low temperature T ≪ TD,
ω kBT − 1
◮ Temperature dependence:
F
F
◮ Reduction from classical result because few phonon quanta
◮ Transition metals ⇒ low d-band velocities at EF ⇒ high resistivity
100 200 300
T emperature [K]
1 2 3 4
Resistivity [10-8 m]
P u r e c
p e r C
d w
k e d + 1 % N i + 2 % N i