First created: Nov. 26, 2003     Updated: January 28, 2008
Heat Conduction
Heat conduction in a nanoparticle
Electronic Specific Heat

The electronic specific heat is given, in the free electron approximation by:
Cel [J m-3 K-1] = (1/2) π2 n kB T / TF = (1/2) π2 n kB kB T / εF
Cel [J kg-1 K-1] = (1/2) π2 (n/ρ) kB T / TF = (1/2) π2 (n/ρ) kB kB T / εF
Cel [J mol-1 K-1] = (1/2) π2 NA kB T / TF = (1/2) π2 NA B kB T / εF
Cel = γ T
where
n = electron density of metal [ m-3 ]
ρ = mass density of metal [ kg/m3 ]
NA = Avagadro's number = 6.0221x1023
kB = Boltzmann's constant = 1.3807x10-23 J/K
T = temperature in Kelvin
TF = Fermi temperature of metal
εF = Fermi energy of metal
Source: Kittel, page 252 (eq. 38).
Table 1: Properties of Metals
 free
electron
number
density		 free
electron
Fermi
velocity		 Fermi
energy		Fermi
temper-
ature		electronic
specific
heat
constant		electronic
specific
heat
constant		electronic
heat
capacity
at 293K		total
heat
capacity
at 293K		total heat
capacity
at 273K		Drude
colli-
sion
time		total
thermal
conduc-
tivity		electron
phonon
coupling		Bulk
electron
phonon
relax.
time
 n vF εFTFγγγ TCvCvτ DRκge-phτe-ph
 (m-3)(m/s) (eV)(K)J mol -1 K -2J m-3 K-2J/kg-KJ/kg-KJ/m3·K(fs)W/m-KW m-3 K-1(fs)
Ag 5.85x1028 1.39x106 5.48 6.36×104 6.46×10-462.8
by me		1.75
by me		2362.45×106 404182.16×1016
by me		850
Note 7
Au 5.90×10281.39x106 5.516.39×1047.29×10-471
Note 6		1.08
by me		1292.49×106303152.95×1016
Note 5
2.3×1016
Note 9
3.3×1016
Note 12		 650
Note 2
1100
Note 7
600
Note 12
Ni 9.14×1028     7.02×10-31065
Note 6		35.1
by me		4454.1×106 913.6×1017
Note 6		867
by me
Pt 6.62×1028      6.8×10-3745
by me		10.2
by me		1332.85×106 71.41.3×1018
Note 10
2.5×1017
Note 11		175
Note 10
Mo 6.42 ×1028      2.0×10-3213
by me		6.1
by me		2512.56×106 1231.3×1017
Note 6		480
by me
Cu 8.45×1028      6.95×10-498
Note 9		? ?
by me		384??×10627401
Note 9		1.0×1017
Note 9		????
by me
Cr 8.33×1028      1.40×10-3194
Note 9		? ?
by me		453??×106 94
Note 9		4.2×1017
Note 9		????
by me
Ru 7.36×1028      3.3×10-3400
Note 9		? ?
by me		???×106 117
Note 9		1.85×1018
Note 9		????
by me
  atoms/m3
Note 8		       Note 1   Note 3 Note 4Note 3  
Note 1: Experimental values from Table 2 page 254 of Kittel 4th ed. 1971
Note 2: From J. H. Hodak et al. J. Phys. Chem. B 104 9954-9965 2000
Note 3: Page 632 of A. Bejan "Heat Transfer" Wiley 1993    (at 293 K)
Note 4: Table 1.3 on page 10 of Ashcroft & Mermin, Holt, Rinehart and Winston 1976    (at 273 K)
Note 5: (bulk value) R. H. M Groeneveld, R. Sprik and A. Lagendijk, Phys. Rev. Lett. 64 1990 784
            and R. H. M. Groeneveld, R. Sprik and A. Lagendijk Phys. Rev. B 51 11433-11445 1995 (Au and Ag)
Note 6: S.-S. Wellershoff et al. "The role of electron-phonon coupling in femtosecond..."
            www.svante-home.de/pdf/whg99.pdf ::
Note 7: PRL 90 177401 2003
Note 8: (atomic concentrations) Table 5 on page 39 of Kittel, 1971
Note 9: M. Bonn et al. PRB 61 1101 2000
Note 10: J. H. Hodak et al. JCP 114 2760 2001  (Pt ge-ph and τe-ph from Kaganov model estimate)
Note 11: J. Hohlfeld, S.-S. Wellershoff, J. Gudde, U. Conrad, V. Jahnke and E. Matthias,
               Chem. Phys. 251, 237 (2000)   [experimental ge-ph for Pt, as cited in Hodak JCP 114 2001]
Note 12: J. H. Hodak, A. Henglein and G. V. Hartland, Pure Appl. Chem. 72 pp.189-197 (2000) 8 nm diam Au

   For a metal at low temperature:
C = γ T + A T3
where
γ = electronic specific heat constant

   Thermal conductivity is given approximately by:
κ = (1/3) vrms2 τDR cv = (1/3) vrms cv l
where
κ = thermal conductivity [ W / m - K ]
vrms = rms carrier speed
τDR = mean time between collisions
cv = heat capacity at constant volume [ J m-3 K-1 ]
l = mean free path

   To obtain the bulk electron-phonon relaxation time:
τe-ph = γ Tambient / ge-ph

   The Kaganov model for the electron-phonon coupling is:
ge-ph = π2

 6 		me s2 n0

  τ(T) T
This is equation (4) in JCP114_2760_2001. s = average speed of sound in the metal. me is the electron mass. τ(T) is the intrinsic relaxation time. n0 is the free electron density. τ(T) can be estimated from the electrical conductivity σ. Using parameter values for Au, this gives ge-ph = 3×1016 W m-3 K-1, agreeing well with experimental values. The original reference is: M. I. Kaganov, I. M. Lifshiftz and L. V. Tanatarov, Sov. Phys. JETP 4, 173 (1957).

   Another formula for ge-ph is:
ge-ph = kB m2 qD4 Ξ2

16 ρ π2 hbar3
where Ξ is the effective e-ph deformation potential. (Ξ = 2.9 eV for Ag, 3.7 eV for Au) For details see N. Del Fatti et al. PRB61_16956_2000 equation (17).

   e-e scattering times in noble metals are on the order of 10 to 500 fs, depending on the electron energy [JCP114 2760 2001]. Thus, the equilibration of the electrons may be incomplete while the decrease of electron energy through electron-phonon coupling is already significant. So the concept of "electron temperature" may not be fully applicable. The electrons may not have a chance to reach their own equilibrium before they cool down to the lattice temperature.

References

C. Kittel, "Introduction to Solid State Physics" 4th edition, John Wiley, 1971

Ashcroft and Mermin, "Solid State Physics" Holt, Rinehart and Winston, 1976
Daniel Murray
Associate Professor
Math, Stats & Physics Unit
University of British Columbia - Okanagan
Kelowna, BC, Canada
daniel "dot" murray "at" ubc "dot" ca

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