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Charge current versus entropy current: A new way to get the Wiedemann-Franz ratio

The electron fluid in a metal is an excellent conductor of electrical charge.  It is also an excellent conductor of heat.  In an insulator, such as diamond, phonons (quantized lattice vibrations) carry all the heat current.  By contrast, in the familiar metals gold, copper, aluminum, lead, the electron fluid conducts nearly the entire heat current (the phonon current still exists but constitutes less than 1 percent of the observed current).

Although the electrical and thermal currents carry distinct physical quantities (charge and entropy), it is very fruitful to compare their magnitudes.  The earliest comparison of the electronic thermal conductivity ke and electrical conductivity s was made by Wiedemann and Franz in 1853.  They found that, in the majority of metals, the  ratio ke/Ts, the Wiedemann-Franz ratio (expressed in universal units), falls in the range 3.1-3.3 near room temperature.  Following early applications of quantum mechanics to the electron gas problem, Mott and Sommerfeld showed that this number (also called the Lorenz number L) is p2/3 = 3.29.

Forward versus back-scattering
When an electrical current exists in a metal, the electron population has a net drift velocity anti-parallel to the E-field (or parallel if the FS is hole-like).  In familiar metals, the (out-of-equilibrium) electrons are brought back to equilibrium by scattering from phonons (in novel metals, such as the cuprates, scattering from other electrons, ee scattering, are more important).  The situation for a heat current is quite different; the FS is left undisplaced.  By measuring the Wiedemann-Franz ratio, we may learn about the different scattering processes that affect the two types of currents.  It is useful to represent the difference between a charge current and an entropy current in momentum k-space.

In the absence of currents, the Fermi Surface is undisplaced (light blue sphere in left figure).  When an E-field is applied, the Fermi Surface (FS) is displaced from its equilbrium position (dark blue sphere).  For a given E field, the displacement is determined by the rate at which the 'fast' electrons moving to the right are scattered to the left, i.e. across the FS diameter (the displacement is proportional to the electrical current).  These 'back-scattering' events require phonons with large momenta q.  By contrast, scattering by phonons with small q (forward scattering events) are ineffective in relaxing the charge current.  Hence, at low temperatures when high-energy (large-q) phonons are scarce, the displacement becomes very large, and we have a high electrical current.  The conductivity s is largely limited by back-scattering from phonons.

In contrast, when a thermal gradient generates a heat current, the FS is not displaced (red sphere in the right figure).  Assuming the left end of the sample is warmer than the right, right-moving electrons are at a temperature warmer than average, while left-moving ones are cooler.  As the Fermi-Dirac distribution is a step-wise function at absolute zero, but becomes rounded at high temperatures, we depict the difference between right and left-moving electrons by blurring the right surface of the sphere.  Since the FS is undisplaced, there is nearly zero electrical current (see below).  Instead, the asymmetry in entropy content between the right and left moving electrons implies a large entropy current to the right.  This is measured as ke.  Now, we find that this heat current is equally sensitive to relaxation by back-scattering and forward-scattering events.  The former reverses the velocity of the hot electrons, while the latter 'cools' the hot electrons without changing their momentum.  Unlike s, the thermal conductivity ke is sensitive to both types of scattering.

[The displaced FS on the left actually generates a very weak heat current called the Peltier heat current, which should be carefully distinguished from the large heat current produced in the right figure.  Likewise, the undisplaced but fuzzed FS on the right generates a weak Peltier electrical current, which is irrelevant here, but all-important in thermopower experiments.]

As we cool a metal below 100 K, large-q phonons become increasingly rare.  Without these efficient scatterers, s increaes rapidly, as discussed above.  Likewise ke also increases.  However, small-q phonons, which are plentiful, are effective in scattering the heat current.  Therefore, the increase in ke is relatively weaker than that in s.  The implies that the Lorenz number ke/Ts should decrease significantly from p2/3 with decreasing T.  This is observed in all familiar metals.

The Lorenz number has been very difficult to pin down in the cuprate superconductors because, in their normal state (above Tc), ke cannot be measured directly.  The electron density in cuprates is so low that the heat current is carried nearly entirely by phonons; we have a situation opposite to that of the familiar metals.  To get around this obstacle, we need a method that selectively picks out the electronic heat current while ignoring that of the phonons.  The thermal Hall effect is tailor-made for this task.

In an applied temperature gradient along x, electrons flow to the cooler region, generating an entropy current parallel to x.  When a magnetic field H is applied parallel to z, the Lorentz force deflects the path of the electrons towards -y.  The deflected electrons produce a small temperature gradient along y, i.e. transverse to the applied gradient direction. The deflected heat current is expressed as the thermal Hall conductivity kxy.  Just as in the familiar electrical Hall effect, the transverse gradient reverses in sign with H.  By contrast, phonons do not produce a thermal Hall effect since they are not affected by the field.  The physics of the thermal Hall conductivity kxy produced by asymmetric scattering of quasiparticles by vortices below Tc is quite similar.  Instead of the diagonal conductivities, ke and s, we may compare the thermal Hall conductivity kxy with the electrical Hall conductivity sxy to form the Hall-Lorenz number, viz.


To test this idea, we measured kxy in elemental copper, and found good agreement between Lxy and the standard L, both in their respective magnitudes and T dependences.


Figure 1   Traces of the thermal Hall conductivity kxy versus H in untwinned YBa2Cu3O7 at temperatures above the critical temperature Tc (= 93 K).  The H-linear variation implies that the mean-free-path l is very short (less than 100 Angstrom).  Comparing the values of kxy with the electrical Hall conductivity sxy (measured in the same sample), we may calculate the Hall-Lorenz number Lxy.  [See Zhang et al. Phys. Rev. Lett. 84, 2219 (2000)].

To compute the Hall-Lorenz number, we need to measure the thermal Hall conductivity kxy at elevated temperatures to high resolution.  Traces of kxy versus applied field are shown in Fig. 1 for temperatures between 95 and 320 K (measurements by Yuexing Zhang).  The short mean-free-path of the charge carriers in the normal state implies a very small electronic heat conductivity ke (about 10 times smaller than that of the phonons).  Since the Hall response is a further factor of 50 smaller, we have to resolve a very small temperature gradient in the transverse direction.  For example, in the trace at 320 K, the Hall signal at 14 Tesla corresponds to dT of 1.5 mK across the width of the sample (500 microns).  We have managed to obtain the required resolution only recently.  The T dependence of the slope kxy/B is plotted in Fig. 2.







Figure 2   The temperature dependence of the slope kxy/B extracted from Fig. 1 (open circles) and the Hall-Lorenz number Lxykxy/Tsxy in an optimally doped YBaCuO7 crystal (red solid circles), and in an underdoped YBaCuO6.6 crystal (blue).   The Hall conductivity kxy/B varies as 1/T1.2 while Lxy in both samples is close to T-linear (it should be nearly constant at the value p2/3 above 200 K).  The suppressed value of Lxy directly reflects the strange-metal behavior of the cuprates in their normal state.  [Zhang et al. Phys. Rev. Lett. 84, 2219 (2000), and unpublished].

With the measured kxy, we may calculate the Hall-Lorenz number.  We find that Lxy is anomalously suppressed from the number p2/3.  Moreover, it displays an unexpected T-linear dependence over a wide range of T (red circles)   As a follow up, we have recently completed the same measurements in an underdoped YBCO crystal with a reduced Tc = 60 K.  We find that its values for Lxy (blue circles) also fall on the same line as the optimally-doped crystal.  This strongly suggests that the T-linear dependence of Lxy reflects an intrinsic transport property of charge carriers in the copper-oxide layer.  The unusually small value of the Lorenz number and its strong T dependence reflects the strange-metal properties of the normal state of the cuprates.


Y. Zhang, N.P. Ong, Z.A. Xu, K. Krishana, R. Gagnon, and L. Taillefer, “Determination of the Wiedemann-Franz ratio from the thermal Hall conductivity: Application to Cu and YBa2Cu3O6.95.”, Phys. Rev. Lett. 84, 2219 (2000).