The electrical engineering handbook CH053

© 2000 by CRC Press LLC

are actually two sets of interpenetrating electron fluids: the uncorrelated electrons providing ohmic conduction
and the correlated ones creating supercurrents. This two-fluid model is a useful way to build temperature effects
into the London relations.
Under the two-fluid model, the electrical current density,

J

, is carried by both the uncorrelated (normal)
electrons and the superelectrons:

J

=

J

n

+

J

s

where

J

n

is the normal current density. The two channels are

modeled in a circuit as shown in Fig.



53.2 by a parallel combination of a resistor (representing the ohmic
channel) and an inductor (representing the superconducting channel). To a good approximation, the respective
temperature dependences of the conductor and inductor are

(53.10)

and

(53.11)

where

␴

o

is the dc conductance of the normal channel. (Strictly speaking, the normal channel should also
contain an inductance representing the inertia of the normal electrons, but typically such an inductor contrib-
utes negligibly to the overall electrical response.) Since the temperature-dependent penetration depth is defined
as

␭

(

T

) =

,

the effective conductance of a superconductor in the sinusoidal steady state is

(53.12)

where the explicit temperature dependence notation has been suppressed.
It should be noted that the temperature dependencies given in Equations (53.10) and (53.11) are not precisely
correct for the high-

T

c

materials. It has been suggested that this is because the angular momentum of the
electrons forming a Cooper pair in high-

T

c

materials is different from that in low-

T

c
ones. Nevertheless, the
two-fluid picture of transport and its associated constitutive law, Eq. (53.12), are still valid for high-T
c
super-
conductors.
Most of the important physics associated with the classical model is embedded in Eq. (53.12). As is clear
from the lumped element model, the relative importance of the normal and superconducting channels is a
FIGURE 53.2A lumped element model of a superconductor.
˜
ss
ooc
c
c
TT
T
T
TT
(
)
=
(
)
æ
è
ç
ö
ø
÷
£
4
for
LLT
TT
TT
c
c
(
)
=
(
)
-
(
)
æ
è
ç
ç
ö
ø
÷
÷
£0
1
1
4
for
LT()m
o
¤
ss
wl
=+
m
˜
o
o
j
1
2
© 2000 by CRC Press LLC
function not only of temperature but also of frequency. The familiar L/R time constant, here equal to⌳␴
~
o
,
delineates the frequency regimes where most of the total current is carried by J
n
(if ␻⌳␴
~
o
>> 1)or J
s
(if
␻⌳␴
~
o
<< 1).This same result can also be obtained by comparing the skin depth associated with the normal
channel, ␦ = , to the penetration depth to see which channel provides more field screening. In
addition, it is straightforward to use Eq. (53.12) to rederive Poynting’s theorem for systems that involve
superconducting materials:
(53.13)
Using this expression, it is possible to apply the usual electromagnetic analysis to find the inductance (L
o
),
capacitance (C
o
), and resistance (R
o
) per unit length along a parallel plate transmission line. The results of such
analysis for typical cases are summarized in Table 53.1.
53.3 Superconducting Electronics
The macroscopic quantum nature of superconductivity can be usefully exploited to create a new type of
electronic device. Because all the superelectrons exhibit correlated motion, the usual wave–particle duality
normally associated with a single quantum particle can now be applied to the entire ensemble of superelectrons.
Thus, there is a spatiotemporal phase associated with the ensemble that characterizes the supercurrent flowing
in the material.
If the overall electron correlation is broken, this phase is lost and the material is no longer a superconductor.
There is a broad class of structures, however, known as weak links, where the correlation is merely perturbed
locally in space rather than outright destroyed. Coloquially, we say that the phase “slips” across the weak link
to acknowledge the perturbation.
The unusual properties of this phase slippage were first investigated by Brian Josephson and constitute the
central principles behind superconducting electronics. Josephson found that the phase slippage could be defined
as the difference between the macroscopic phases on either side of the weak link. This phase difference, denoted
as ␾, determined the supercurrent, i
s
, through and voltage, v, across the weak link according to the Josephson
equations,
(53.14)
and
(53.15)
where I
c
is the critical (maximum) current of the junction and ⌽
o
is the quantum unit of flux. (The flux
quantum has a precise definition in terms of Planck’s constant, h, and the electron charge, e: ⌽
o
ϵ h/(2e) »
2.068 ´ 10
–15
Wb). As in the previous section, the correlated motion of the electrons, here represented by the
superelectron phase, manifests itself through an inductance. This is straightforwardly demonstrated by taking
the time derivative of Eq. (53.14) and combining this expression with Eq. (53.15). Although the resulting
inductance is nonlinear (it depends on cos ␾), its relative scale is determined by
(53.16)
2 wm
o
s
˜
o
()¤
-Ñ×´
(
)
=+m+
(
)
æ
è
ç
ö
ø
÷
+
(
)
òò
ò
EH E H J
J
22
dv
d
dt
Tdv
T
dv
V
os
V
o
n
V
1
2
1
2
1
2
1
2
2
e
s
L

˜
iI
sc
= sinf
v
d
dt
o
=
F
2p
f
L
I
j
o
c
=
F
2p

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