Page 421 - Electrical Properties of Materials

Exercises 403

superconductors. Just as with cuprates, the parent compound LaOFeAs was
not superconducting, but upon replacing some of the oxygen by fluorine, it be-
came superconducting. Its phase diagram is similar to that shown in Fig. 14.22.
At low doping density it is an insulating antiferromagnet, but as the density in-
creases further it turns into a superconductor. The story is also similar. The
quest started with the somewhat different LaOFeP, which became supercon-
ducting at T c = 5 K. Replacing phosphorous by arsenic raised the critical
temperature to T c = 26 K, which then rose to 43 K when lanthanum was
replaced by samarium, and to 55 K for the same compound under pressure.
However, the electrical properties of cuprates and pnictides are different at
room temperature. The latter compounds conduct electricity; the former do
not.
Our final conclusion? Anything is possible. Experimenters and theoreticians
will all be busy in the next few decades.

Exercises
14.1. It follows from eqn (1.15) that in the absence of an in Fig. 14.13(b)] is given as
electric field the current density declines as
E
C √ ,
2
J = J 0 exp(–t/τ), E – 2
where C is a constant and E is the energy measured from the
where τ is the relaxation time related to the conductivity by
Fermi level (middle of the gap). Show that at T = 0 the tun-
eqn (1.10).
nelling current is zero when U < 2 /e, and the tunnelling
In an experiment the current flowing in a superconducting current is proportional to
ring shows no decay after a year. If the accuracy of the meas-
eU–
urement is 0.01%, calculate a lower limit for the relaxation eU – E E dE
28
–3
2
2 1/2
2
time and conductivity (assume 10 electrons m ). How many [(eU – E) – ] 1/2 [E – ]
times larger is this conductivity than that of copper?
for U > 2 /e.
14.2. What is the maximum supercurrent that can be passed 14.6. A lead-insulator-tin superconducting tunnel junction
through a 2 mm-diameter lead wire at 5 K (use data from has a current–voltage characteristic at 1 K similar to that
Table 14.1). shown in Fig. 14.17, with the current maximum at U =
0.52 mV and the point of sudden upsurge at U =
14.3. In the first phenomenological equations of supercon- 1.65 mV.
ductivity, proposed by F. and H. London in 1935, the current
density was assumed to be proportional to the vector potential (i) Find the energy gaps in lead and tin at zero temperature.
and div A = 0 was chosen. Show that these assumptions lead (ii) At what temperature will the current maximum disap-
to a differential equation in A of the form of eqn (14.63). pear?
14.4. The parameter λ, defined in eqn (14.60), may be re- 14.7. If a microwave cavity made of tin is cooled to 1 K, can
~
garded as the penetration depth for κ = 0. A typical value you expect the losses to be substantially less than at 4 K?
for the measured penetration depth is 60 nm. To what value of At what frequency would you expect superconductive ef-
2
ψ does it correspond?
0 fects to completely disappear in tin held at 1 K?
14.5. The energy diagram for a tunnel junction between 14.8. What is the frequency of the electromagnetic waves ra-
two identical superconductors is shown in Figs. 14.13 diated by a Josephson junction having a voltage of 650 μV
and 14.14. The superconducting density of states [sketched across its terminals?