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
*k*_{e} and electrical conductivity s was made by Wiedemann and Franz in 1853. They found
that, in the majority of metals, the ratio *k*_{e}/*T*s, 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 p^{2}/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 *k*_{e}. 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 *k*_{e} 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 *k*_{e}
also increases. However, small-*q* phonons, which are plentiful, are
effective in scattering the heat current. Therefore, the increase in *k*_{e}
is relatively weaker than that in *s*.
The implies that the Lorenz number *k*_{e}/*T*s
should decrease significantly from p^{2}/3
with decreasing *T*. This is observed in all familiar metals.

**Cuprates**

The Lorenz number has been very difficult to pin down in the cuprate
superconductors because, in their normal state (above *T*_{c}), *k*_{e}
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 *k*_{xy}.
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 *k*_{xy}
produced by asymmetric scattering of quasiparticles by vortices below *T*_{c}
is quite similar. Instead of the diagonal conductivities, *k*_{e}
and s, we may compare the thermal Hall
conductivity *k*_{xy} with
the electrical Hall conductivity s_{xy} to form the Hall-Lorenz number, viz.

*L*_{xy}
= *k*_{xy}/*T*s_{xy}.

To test this idea, we measured *k*_{xy
}in elemental copper, and found good agreement between *L*_{xy}
and the standard *L*, both in their respective magnitudes and *T*
dependences.

**Results**

**Figure 1 **
Traces of the thermal Hall conductivity *k*_{xy} versus *H* in untwinned YBa_{2}Cu_{3}O_{7}
at temperatures above the critical temperature *T*_{c} (= 93
K). The *H*-linear variation implies that the mean-free-path *l*
is very short (less than 100 Angstrom). Comparing the values of *k*_{xy} with
the electrical Hall conductivity s_{xy} (measured in the same sample), we may calculate the
Hall-Lorenz number *L*_{xy}. [See Zhang et al. Phys. Rev.
Lett. **84**, 2219 (2000)].

To compute the Hall-Lorenz number, we need to measure the thermal Hall
conductivity *k*_{xy} at
elevated temperatures to high resolution. Traces of *k*_{xy} 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 *k*_{e} (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 d*T* 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 *k*_{xy}/*B*
is plotted in Fig. 2.

**Figure 2**
The temperature dependence of the slope *k*_{xy}/*B* extracted from Fig. 1 (open circles) and the
Hall-Lorenz number *L*_{xy} = *k*_{xy}/*T*s_{xy} in an
optimally doped YBaCuO7 crystal (red solid circles), and in an underdoped
YBaCuO6.6 crystal (blue). The Hall conductivity *k*_{xy}/*B*
varies as 1/*T*^{1.2} while *L*_{xy} in both samples
is close to *T*-linear (it should be nearly constant at the value p^{2}/3 above
200 K). The suppressed value of *L*_{xy} 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 *k*_{xy},
we may calculate the Hall-Lorenz number. We find that *L*_{xy}
is anomalously suppressed from the number p^{2}/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 *T*_{c}
= 60 K. We find that its values for *L*_{xy} (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 YBa_{2}Cu_{3}O_{6.95}.”, Phys. Rev. Lett. **84**,
2219 (2000).