The Wiedemann-Franz Law, Nanowires, and Quantum Mechanics

Oct. 30, 2012 - PRLog -- Background

In 1853, Wiedemann and Franz (W-F) proposed the ratio of thermal (sigT) to electrical (sigE) conductivity of metals to be proportional to the absolute temperature T, the proportionality constant given by the Lorenz number L,  

sigT / sigE = L * T = Pi²(k/e)²/3 = 2.45 x10^-8 W-ohm/K²

where, k is Boltzmann’s constant and e is the electronic charge. As shown in the Table at 300 K, the Lorenz number L is about the same [1] for most bulk metals.
Metal                         Lorenz number L x10^-8
                                        (W- ohm/K²)

Cu                                          2.20
Ag                                           2.31
Au                                           2.32
Mg                                          2.14
Fe                                           2.61
Al                                            2.14
Pb                                           2.64
Pt                                            2.51

Although the W-F law is verified for the bulk, theoretical calculations [2] based on the Boltzmann transport equation that include electron surface scattering have for some time claimed the validity of the W-F law even for very thin film thicknesses. Generally, the electrons are thought the dominant carriers of charge and heat in metallic samples, and therefore, a decrease of electron mobility due to scattering should reduce thermal conductivity and electrical conductivity by the same factor, and consequently their ratio should be equal to the bulk Lorenz number.

Recently, the Lorenz number for Pt was found [3] reduced 30% from bulk in experiments on 100 nm diameter nanowires (NWs). The electrical conductivity sigE was measured directly and found reduced by a factor of 2.46 from bulk. However, a direct measurement of the thermal conductivity sigT based on measurements of temperatures could not be made because of the nanoscale size of the NW. Therefore, the temperatures along the NW were inferred from solutions of the Fourier equation based on balancing the Joule heat by conductive heat flow along the NW and the Stefan-Boltzmann (S-B) thermal radiation loss to the surroundings.

The Fourier solutions for the NW showed S-B radiation losses to be negligible leaving Joule heat to be conserved solely by thermal conduction.  Based on the calculated temperature differences over the NW length, the thermal conductivity sigT of Pt was found reduced by about a factor of 3 from bulk.  Both reduced sigE and sigT were explained [3] by grain boundary scattering of electrons that taken together gave a 70 % reduction in the bulk Lorenz number.

Problem

The Fourier equation by assuming the NW has heat capacity is invalid at the nanoscale as QM requires the heat capacity to vanish. QM stands for quantum mechanics. Explanations of reduced thermal conductivity sigT by electron scattering at grain boundaries is simply not correct.

QED Solution

The QM requirement that the heat capacity of the atom vanishes in NWs invalidates the temperatures derived by the Fourier equation. Indeed, Planck’s QM given by the Einstein-Hopf expression for the harmonic oscillator shows the thermal kT energy of the atom to vanish in  nanostructures having TIR confinement wavelengths < 1 micron.  TIR stands for total internal reflection.  Recall classical phsics by statistical mechanics (SM) on which the Fourier equation is based allows the atom to erroneously have heat capacity at the nanoscale as shown in the thumbnail. Lacking heat capacity by QM, the Joule heat cannot be conserved by increasing the NW temperature. This means temperature changes implied in the Fourier equation do not occur, i.e., there is no conductive heat flow in NW’s.

In contrast, the theory of QED radiation asserts the Joule heat is conserved at the TIR frequency of the NW by the creation of charge and QED radiation inside the NW. QED stands for quantum electrodynamics. Unlike the S-B radiation, the QED radiation is significant being almost entirely equal to the Joule heat. See http://www.nanoqed.org, 2008-2012.  

Conclusions

1.  By QM, NWs have vanishing heat capacity that precludes the conservation of Joule heat by an increase in temperature. Instead, conservation proceeds by the creation of QED radiation inside the NW at its TIR resonance. The QED radiation produces charge by the photoelectric effect or is emitted to the surroundings as non-thermal EM radiation. However, the QED radiation losses were not included in the heat balance [3] of the Fourier equation.  

2.  The QED radiation created at the speed of light is far faster than phonons can respond at acoustic velocities, thereby effectively negating conductive heat flow Q along the NW. What this means is the Joule heat is almost all conserved by the creation of QED radiation. However, if QED radiation is included in the heat balance, solutions of the Fourier’s equations will show the conductive heat Q = 0.
     
3. In the NW experiments, the thermal conductivity sigT should remain at bulk, and therefore there is no temperature difference across the length of the NW.  The electrical conductivity sigE of the NW is indeed reduced by electron scattering at grain boundaries, but otherwise is not affected by the QM restriction on the heat capacity of the atom.  The NW having bulk sigT and reduced sigE therefore gives a Lorenz number L for the NW > bulk.

4.  Thermal conductivity of NWs cannot be inferred from solutions of Fourier’s equation if QED radiation is not included in the heat balance. Verification in NWs requires measurements of charge and the emission of QED radiation, the latter difficult because the TIR resonance is beyond UV frequencies, although charge is readily observed at the nanoscale.

5.  The W-F law is applicable to NW’s provided the thermal conductivity sigT for the bulk is assumed. The Lorenz number therefore depends entirely on the NW electrical conductivity sigE.

6.  In support of the W-F law, explanations of reduced thermal conductivity by electron scattering at grain boundaries may be safely dismissed by QM.    

References

[1] C. Kittel, Introduction to Solid State Physics, 5th Ed., Wiley, New York, p. 178, 1976.
[2] C. Tellier, et al., “Effects of electron scattering on thermal conductivity of thin films,” J. Mater. Sci., 16, 2287, 1981.
[3] F. Volklein, et al., “The experimental investigation of thermal conductivity and the Wiedermann–Franz law for single metallic nanowires,” Nanotechnology, 20, 325706, 2009.