Electron thermal conductivity

One can calculate the contribution to the thermal conductivity from the electrons classically and almost obtain the correct result. Classically, the heat capacity of an electron gas with n electrons per unit volume, C = 3nk/2. The mean free path is given by vt, where v is the average thermal velocity and t is the time between collisions. Therefore 

Now if the correct value for the electronic heat capacity given by Equation 17.22 is put into Equation 17.28 along with v = Vf. and A=z pT, where v = 2sf lm, the thermal conductivity of the electrons becomes 

Note that the smaller heat capacity of the electron gas is compensated for by the higher Fermi velocity and an expression close to the classical result is obtained but for totally different reasons. 

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The two contributions give rise to the electron thermal conductivity ks (eq. (3.32)). 

Fig. 24. Magnetic field dependence of the electronic thermal conductivity at T - 0, normalized to its value at Hc2- Circles are for LuNi2B2C, squares for UPt3 and diamonds for Nb. Note the qualitative difference between the activated thermal conductivity of the s-wave superconductor Nb and the roughly linear growth seen in UPt3, a superconductor with a line of nodes (Boaknin et al. 2001).

Usually, the electronic thermal conductance re can be calculated from the Wiedemann - Franz law, re TG/e2. However, as shown in Ref. [8, 9] for the ballistic limit f > d, this law gives a wrong result for Andreev wires if one uses an expression for G obtained for a wire surrounded by an insulator. Andreev processes strongly suppress the single electron transport for all quasiparticle trajectories except for those which have momenta almost parallel to the wire thus avoiding Andreev reflection at the walls. The resulting expression for the thermal conductance... 

In this equation, 5 is the Seebeck coefficient, p is the resistivity (p =1/ a, where a is the electrical conductivity), and iris the thermal conductivity. Metals are typically poor themioelectrics because of a low Seebeck coefficient and a large contribution to the thermal conductivity by the conduction electrons. In contrast, insulators have a large Seebeck coefficient and a small electronic thermal conductivity, but the carrier density is low, leading to a high resistivity. Mahan, et al. have shown that a carrier density intermediate between that of a metal and that of an insulator is optimum (N-lO cm"3) (1). Typically, doped semiconductors make the best thermoelectrics. 

In ceramics with mobile electrons or holes there is a third mechanism that can contribute to thermal conductivity. The electronic thermal conductivity is... 

By analyzing the temperature dependence of the electrical properties, using our results (Fig. 5) and published data [9,10], another characteristic feature of the structure of CrSi2 crystals becomes apparent, which makes the energy spectrum of valence electrons in this compound more precise. A calculation of the lattice thermal conductivity of single crystals, taken as the difference between the total and electronic thermal conductivities (Fig. 5), as a function of temperature, shows that it decreases continuously up to the maximum measurement temperature. This Indicates the absence of an additional heat transfer component due to ambipolar diffusion of carriers [18] in the intrinsic conduction range. 

While thermal energy can be transported through a solid via a variety of different mechanisms [24], the two most important for TE applications are diffusive transport of energy by the mobile charge carriers (electronic thermal conductivity, iCg) and phonons (lattice thermal conductivity, Kl). Since it is a relatively good approximation to treat and Kl as independent for many solids, significant emphasis has been placed on the development of TE materials with low icl values [6]. Amorphous solids are typically characterized by low carrier mobilities and thus low a values, however they also exhibit some of the lowest known thermal conductivities (excluding porous materials) and serve as useful benchmark materials for TE materials research. 

At low altitudes, below about 250 km, the electron temperature can be obtained simply by equating local heating and cooling rates. However, above this altitude, electron thermal conduction is important, and the equation governing the electron temperature becomes more complicated (see Schunk and Nagy, 1978, 2000). 

The electron thermal conductivity of metals and semiconductors is determined by the Wiedemann-Franz law... 

The most important properties of superconducting electrons are that they do not transport energy and do not interact with phonons. At T < 7 the number of heat transporting electrons decreases by an exponent determined by the energy gap d(T) at the Fermi surface and the electron thermal conductivity... 

Smirnov, I.A., and V.I. Tamarchenko, 1977, Electronic Thermal Conductivity in Metals and Semiconductors (Nauka, Leningrad). In Russian. 

The thermal conductivity or k (i.e., the time rate of transfer of heat by conduction) of interstitial carbides is different from that of most other refi actory materials as k increases with increasing temperature as shown in Fig. 4.2.l l Typically, the mechanism of thermal conductivity involves two components electron thermal conductivity and phonon (lattice) conductivity kp. As shown in Fig. 4.3 (in this case for titanium carbide), k increases markedly with temperature. This behavior is believed to be the... 

Recent experimental results on a compressional Z-pinch and on a laser-initiated gas-embedded Z-pinch show considerable enhancement of MHD stability over conventional theory. It is thought that this could be due to finite ion Larmor radius effects. Several theoretical models of energy and pressure balance of a linear Z-pinch with end-losses have been made electron thermal conduction with (jot = 0, with ojT = oo, and singular thermal ion transit time loss. 

On the other hand, we notice that the type of derivation given in Ref. 19 for the electron thermal conductivity does not apply when ohmic heating is negligible as in the case of the experiments we consider. A new analytical expression that can be adopted for this case has been proposed in Ref. 20 on the basis of very limited experimental information that indicates a deterioration of the... 

The major concern with these observations is the low electron temperatures. In an effort to increase the temperature, a snipper coil was added to the entrance region to speed up the field reconnection. Faster reconnection would reduce the time for parallel electron thermal conduction back to the source and would also reduce the possibility that metallic impurities evolving late... 

K number of nearest neighbor ions electronic thermal conductivity... 

A., electronic thermal conductivity of < H Hall conductivity tensor... 

Aj and A, denote the total electronic thermal conductivity and the thermal conductivity due to one of the scattering mechanisms (/), respectively. The subscript e in denotes the electron-diffusion thermopower (see below). Assuming the validity of the Wiedemann-Franz law ... 

The matrix elements Q11 determine the temperature dependence of the electronic thermal conductivity. Since the spectral function aiF is proportional to Qu is proportional to for T 0 and Qu is proportional to T for For the... 

ITj spd contributions to the total electronic thermal resistivity. As for the electrical resistivity, the residual thermal resistivity, W, dominates at low temperatures. Wq and Po related via the Wiedemann-Franz law (eq. (36)). The temperature dependence of was first calculated in the scope of the Debye approximation by Wilson (see, e.g. Ziman 1960). The dashed line in fig. 2 gives a schematic graph of the temperature variation according to Wilson s formula. For a non-magnetic compound at low temperatures the electronic thermal conductivity can be approximated as... 

The subscript e indicates the electronic thermal conductivity or resistivity. 

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