How to detect quasiparticle entropy current with lots of phonons around

In superconductors, electrons form Cooper pairs, which are collectively
called the condensate. The condensate displays all the spectacular
electronic properties that one associates with superconductivity, such as flux
expulsion (Meissner effect). At finite temperature, however, a small
fraction of the pairs is 'broken' to form quasiparticles (entropy must be
finite!). The quasiparticle is a quantum superposition of a pure-particle
state and a pure-hole state. For many purposes, however, we may simply
regard it as an 'electron' drifting inside the condensate. There is
enormous interest in the properties of quasiparticles because they are the
fundamental, low-energy excitations of the condensate. Moreover, in the
cuprates, the quasiparticles have a linear *E*(**k**) vs. **k**
dispersion. This Dirac-like dispersion adds to their interest.
However, they are difficult to 'see' experimentally. A powerful approach
is to investigate the heat (or entropy) current of the quasiparticles.
The condensate is indifferent to a thermal gradient (since it carries zero
entropy), whereas quasiparticles drift down the gradient to produce an electronic
heat current. A drawback to overcome is that the thermal gradient also
produces a phonon current (lattice vibrations), which may be much larger than
the electronic heat current.

So, here's the problem. How do you see the quasiparticle entropy
current against the large background of the phonons? Let's consider
piercing the condensate with an array of vortices (quantized flux lines) by
applying a magnetic field. Each vortex core is surrounded by a tight
supercurrent loop whose sense of circulation is fixed by the applied field
direction. The vortex strongly scatters quasiparticles (figure).
Because of the superfluid circulation around the vortex, the quasiparticles
suffer a left-right scattering asymmetry. More of them end up, say, to
the left of the core compared with the right (viewed from the incident
direction). This unbalance leads to a transverse temperature gradient
that translates into a large thermal Hall conductivity k_{xy}. Phonons, which are electrically neutral,
do not suffer this asymmetric scattering. Thus, by monitoring k_{xy}, we measure a 'Hall' heat current
entirely from the quasiparticles. With this technique (and some simple
assumptions) we can objectively back out the quasiparticle lifetime and
density, and learn a lot about their behavior in an intense field.

**Figure
1** Traces of the thermal Hall conductivity k_{xy} versus field *H* at
selected temperatures above 35 K in a BZO-grown crystal of optimally-doped
superconductor YBa_{2}Cu_{3}O_{7} with *T*_{c}
= 92 K. At each temperature, k_{xy}
rises steeply in weak fields, reaches a broad maximum and then decreases slowly.
The rate of the initial increase is proportional to the square of the
zero-field mean-free-path *l*. The inset compares the thermal conductivity
k_{a} in this crystal (solid
circles) with that in a non-BZO grown crystal (open circles). Both are
detwinned. These curves were taken by Yuexing Zhang at

The traces in Figure 1 represent the thermal Hall conductivity k_{xy} versus field *H* at
various temperatures. k_{xy}
is the qp heat current deflected to the left by the vortices (it changes sign
when **H** is reversed in direction). At high temperatures (85 K) it
is simply proportional to *B*, the vortex density. At lower
temperatures, however, k_{xy}
goes through a broad maximum. A striking feature here is the very rapid
increase in the initial slope dk_{xy}/d*B*,
which grows a *thousand*-fold between 90 K and 30 K. This reflects a
very steep increase in the zero-field mean-free-path of the quasiparticles.

Figure 2The zero-field mean-free-pathlextracted from the initial slope of k_{xy}displayed in Fig. 1. Values of the weak-field Hall angle q/B(proportional tol) is shown on the right scale. The open and closed circles show two alternate ways of calculatingl. The expanded scale shows the remarkably steep increase inljust belowT_{c}. From Ref. 2 below.

As shown in Fig. 2, *l *is not strongly *T* dependent above *T*_{c},
but accelerates rapidly in the superconducting state. Between *T*_{c}
and 20 K,* l* increases 200 fold from about 80 Angstroms to nearly 1
micron. The steep increase in quasiparticle lifetime below *T*_{c}
is one of the characteristics of the cuprates, and is not at all understood.

**References**

1. K. Krishana, J. M. Harris, and N. P. Ong, “Quasi-particle
mean-free-path in YBa_{2}Cu_{3}O_{7} measured by the
thermal Hall conductivity.”, Phys. Rev. Lett. **75**, 3529 (1995).

2. Y. Zhang, N.P. Ong, P. W. Anderson, D. A. Bonn, R. X. Liang, and W. N.
Hardy, “Giant enhancement of the thermal Hall conductivity in high-purity
YBa_{2}Cu_{3}O_{7}.”, Phys. Rev. Lett. **86**,
890 (2001).