Thermoelectricity: From Atoms to Systems
Week 2: Thermoelectric Transport Parameters Lecture 2.5: Lattice Thermal Conductivity
By Mark Lundstrom Professor of Electrical and Computer Engineering Purdue University
Lundstrom nanoHUB-U Fall 2013 1 review: coupled charge and heat currents electrical current:
heat current (electronic):
heat current (lattice):
Lundstrom nanoHUB-U Fall 2013 2 heat flux and thermal conductivity
1) Electrons can carry heat, and we have seen how to evaluate the electronic thermal conductivity.
2) In metals, electrons carry most of the heat.
3) But in semiconductors and insulators, lattice vibrations (phonons) transport most of the heat.
Lundstrom nanoHUB-U Fall 2013 3 Lecture 5 topics
1) Phonons
2) Landauer approach to phonon transport
3) Thermal conductivity vs. temperature
1) Electron vs. phonon transport
Lundstrom nanoHUB-U Fall 2013 4 electron dispersion
Electrons in a solid behave as both particles E (quasi-particles) and as waves.
Electron waves are described by a “dispersion:”
Because the crystal is periodic, the dispersion is periodic in k (Brillouin zone). k Particles described by a “wavepacket.” −π a π a The “group velocity” of a wavepacket is determined by the dispersion:
Lundstrom nanoHUB-U Fall 2013 5 mass and spring
M iωt x(t)− x0 = Ae
Lundstrom nanoHUB-U Fall 2013 6 phonon dispersion
Lattice vibrations behave both as particles ω (quasi-particles) and as waves. Einstein
Lattice vibrations are described by a “dispersion:” Debye Because the crystal is periodic, the dispersion is periodic in k (Brillouin zone). BW Particles described by a “wavepacket.”
The “group velocity” of a wavepacket is −π a π a determined by the dispersion:
Lundstrom nanoHUB-U Fall 2013 7 real dispersions
3 TO (2) 0.06 2 cb LO (1) 1 0.04 LA (1) (eV) (eV) l h e 0 p E E -1 0.02 hh TA (2) lh -2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 k (x π/a ) q (x π/a ) x 0 x 0 note the different energy scales! electrons in Si (along [100]) phonons in Si (along [100])
8 Lundstrom nanoHUB-U Fall 2013 8 general model for lattice thermal conduction
(electrons)
I I 0 channel V
9 Lundstrom nanoHUB-U Fall 2013 9 near-equilibrium heat flux
watts = (watts/K) x K thermal conductance
10 Lundstrom nanoHUB-U Fall 2013 10 window functions
Recall the electrical conductance:
“window function”:
Wel (E)= (− ∂f0 ∂E)
11 Lundstrom nanoHUB-U Fall 2013 11 heat conduction
1) Fourier’s Law of heat conduction:
2) Thermal conductance:
3) Quantum of heat conduction:
4) Window function for phonons:
12 Lundstrom nanoHUB-U Fall 2013 12 electrical conduction
1) Electrical current: I = G∆V
2) Electrical conductance:
2 3) Quantum of electrical conduction: 2q h
4) Window function for electrons: Wel (E)= (− ∂f0 ∂E)
13 Lundstrom nanoHUB-U Fall 2013 13 window functions: electrons vs. phonons
Electrons Phonons
60 60 50K )
) 50K -1 -1 40 40 (eV h (eV l p e W W 20 300K 20 300K
0 0 -0.1 0 0.1 0 0.1 0.2 E -E (eV) E (eV) el F ph
Wel (E)= (− ∂f0 ∂E)
14 Lundstrom nanoHUB-U Fall 2013 14 diffusive heat transport (3D)
(Watts)
(Watts/K)
(diffusive phonon transport)
(Watts)
(Watts/m-K)
15 Lundstrom nanoHUB-U Fall 2013 15 diffusive heat transport (3D)
(Watts / m2)
(Watts/m-K)
(Amperes / m2)
(1/Ohm-m)
16 Lundstrom nanoHUB-U Fall 2013 16 diffusive heat transport (3D)
(Watts / m2)
(Watts/m-K)
(Amperes / m2)
(1/Ohm-m)
17 Lundstrom nanoHUB-U Fall 2013 17 connection to traditional view
(1/Ohm-m)
(Watts/m-K)
18 Lundstrom nanoHUB-U Fall 2013 18 effective mass model for electrons
E As long as the BW >> kBT, the effective mass model generally works ok.
Typically, only states near the band edge matter, and these k regions can be described by −π a π a an effective mass approximation.
Lundstrom nanoHUB-U Fall 2013 19 Debye model for acoustic phonons
Linear dispersion model ω
Debye
BW
−π a π a If acoustic phonons near q =0 mostly contribute to heat transport, Debye model works work well.
Lundstrom nanoHUB-U Fall 2013 20 limitation of Debye model
2 20 silicon Debye (Si)
) M ph full band (Si) -2 1.5 15
) m 50 K
- 1 Wph 19 300 K
1 10 (eV
ph (x10 h W p 0.5 5 Window function spans the
M entire BZ at room temp.
0 0 0 0.05 0.1 Debye model works well at low E (eV) ph temperatures.
Lundstrom nanoHUB-U Fall 2013 21 effective mass model for electrons
Parabolic dispersion assumption for electrons works well at room temperature.
electrons in Si 0.4 20 Mel, EMA Mel, full band ) Wel, 50 K 15
-2 0.3 W , 300 K el ) m - 1 19
10 (eV
0.2 el W (x10 l e 0.1 5 M
0 0 0 0.1 0.2 E - E , E - E (eV) el C el F
Lundstrom nanoHUB-U Fall 2013 22 lattice thermal conductivity
23 Lundstrom nanoHUB-U Fall 2013 23 scattering
Electrons scatter from: Phonons scatter from:
1) defects 1) defects -e.g. charged impurities, neutral -e.g. impurities, dislocations, impurities, dislocations, etc. isotopes, etc.
2) phonons 2) other phonons
3) surfaces and boundaries 3) surfaces and boundaries
4) other electrons 4) electrons (“phonon drag”)
24 Lundstrom nanoHUB-U Fall 2013 24 phonon-phonon scattering
i) momentum conservation:
ii) energy conservation:
little effect on thermal conductivity!
25 Lundstrom nanoHUB-U Fall 2013 25 N and U processes
High q implies short wavelength. Unphysical because wavelength would be less than lattice spacing.
Normal (N) process Umklapp (U) process (momentum not (momentum conserved) conserved). Lowers κL. Little effect on κL.
26 Lundstrom nanoHUB-U Fall 2013 26 scattering summary
1) point defects and impurities: “Raleigh scattering”
2) boundaries and surfaces:
3) Umklapp scattering:
27 Lundstrom nanoHUB-U Fall 2013 27 temperature-dependent thermal conductivity
4 10 cal. expt. )
-1 3
K 10 defects
-1 phonon scattering by U-processes
(Wm 2 10 ph κ
Si 1
10 0 1 2 3 10 10 10 10 T (K) population of modes and boundary scattering
Lundstrom nanoHUB-U Fall 2013 28 electron vs. phonon conductivities
The expressions look similar:
In practice, the mfps often have similar values. The difference is
For electrons, the location EF can vary
But even when EF = EC,
Lundstrom nanoHUB-U Fall 2013 29 electron vs. phonon conductivities
Example 1: Bi2Te 3:
Phonon MFP engineering Large mass, low sound velocity small BW
Lundstrom nanoHUB-U Fall 2013 30 n-type Bi2Te 3 vs. Si
Example 1: Bi2Te 3:
Example 2: Si:
Lundstrom nanoHUB-U Fall 2013 31 Phonon MFP engineering
Jiaqing He, Mercouri G. Kanatzidis, and Vinayak P. Dravid, “High performance bulk thermoelectrics via a panoscopic approach,” Materials Today, 16, 166-176, 2013.
Lundstrom nanoHUB-U Fall 2013 32 summary
1) Model for electrical conduction can readily be extended to phonons. The mathematics are very similar. Lattice heat conduction is quantized just as electronic conductivity is.
2) The different BW’s of the electron and phonon dispersions have important consequences. For electrons, a simple dispersion (effective mass) is often adequate, but for phonons, the simple dispersion (Debye model) is not very good.
3) There is no Fermi level for phonons, so the lattice thermal conductivity cannot be varied across several orders of magnitude like the electrical conductivity, but MFPs can be engineered.
Lundstrom nanoHUB-U Fall 2013 33