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Fundamental Properties of Superconductors

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1
Fundamental Properties of Superconductors
The vanishing of the electrical resistance, the observation of ideal diamagnetism, orthe appearance of quantized magnetic flux lines represent characteristic propertiesof superconductors that we will discuss in detail in this chapter. We will see that allof these properties can be understood, if we associate the superconducting statewith a macroscopic coherent matter wave. In this chapter we will also learn aboutexperiments convincingly demonstrating this wave property. First we turn to thefeature providing the name “superconductivity”.
1.1
The Vanishing of the Electrical Resistance
The initial observation of the superconductivity of mercury raised a fundamentalquestion about the magnitude of the decrease in resistance on entering the super-conducting state. Is it correct to talk about the
vanishing
of the electrical resistance?During the first investigations of superconductivity, a standard method formeasuring electrical resistance was used. The electrical voltage across a samplecarrying an electric current was measured. Here one could only determine that theresistance dropped by more than a factor of a thousand when the superconductingstate was entered. One could only talk about the vanishing of the resistance in thatthe resistance fell below the sensitivity limit of the equipment and, hence, could nolonger be detected. Here we must realize that in principle it is impossible to proveexperimentally that the resistance has exactly zero value. Instead, experimentally, wecan only find an upper limit of the resistance of a superconductor.Of course, to understand such a phenomenon it is highly important to test withthe most sensitive methods, to see if a finite residual resistance can also be found inthe superconducting state. So we are dealing with the problem of measuringextremely small values of the resistance. Already in 1914 Kamerlingh-Onnes usedby far the best technique for this purpose. He detected the decay of an electriccurrent flowing in a closed superconducting ring. If an electrical resistance exists,the stored energy of such a current is transformed gradually into Joule heat. Hence,we need only monitor such a current. If it decays as a function of time, we can becertain that a resistance still exists. If such a decay is observed, one can deduce an
Superconductivity: Fundamentals and Applications,
2nd Edition. W. Buckel, R. KleinerCopyright ©2004 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 3-527-40349-3
11
t L R
0
e
I
)
t
(
I
–
=
upper limit of the resistance from the temporal change and from the geometry of the superconducting circuit.This method is more sensitive by many orders of magnitude than the usualcurrent-voltage measurement. It is shown schematically in Fig.1.1. A ring madefrom a superconducting material, say, from lead, is held in the normal state abovethe transition temperature
T
c
. A magnetic rod serves for applying a magnetic fieldpenetrating the ring opening. Now we cool the ring below the transition tem-perature
T
c
at which it becomes superconducting. The magnetic field
1)
penetratingthe opening practically remains unchanged. Subsequently we remove the magnet.This induces an electric current in the superconducting ring, since each change of the magnetic flux
F
through the ring causes an electrical voltage along the ring. Thisinduced voltage then generates the current.If the resistance had exactly zero value, this current would flow without anychange as a “permanent current” as long as the lead ring remained super-conducting. However, if there exists a finite resistance
R
, the current would decreasewith time, following an exponential decay law. We have(1-1)Here
I
0
denotes the current at some time that we take as time zero;
I
(
t
) is the currentat time
t
;
R
is the resistance; and
L
is the self-induction coefficient, depending onlyupon the geometry of the ring.
2)
1
Throughout we will use the quantity
B
to describe the magnetic field and, for simplicity, refer to it as“magnetic field” instead of “magnetic flux density”. Since the magnetic fields of interest (also thosewithin the superconductor) are generated by macroscopic currents only, we do not have to distinguishbetween the magnetic field
H
and the magnetic flux density
B
, except for a few cases.
2
The self-induction coefficient
L
can be defined as the proportionality factor between the inductionvoltage along a conductor and the temporal change of the current passing through the conductor:
U
ind
= –L
d
I
d
t
. The energy stored within a ring carrying a permanent current is given by
1
⁄
2
LI
2
. Thetemporal change of this energy is exactly equal to the Joule heating power
RI
2
dissipated within theresistance. Hence, we have –
dd
t
(
12
LI
2
) =
R I
2
. One obtains the differential equation –
d
I
d
t
=
RL
I
, thesolution of which is (1.1).
Fig. 1.1
The generation of a permanent current in a superconducting ring.
1Fundamental Properties of Superconductors
12
For an estimate we assume that we are dealing with a ring of 5cm diameter madefrom a wire with a thickness of 1mm. The self-induction coefficient
L
of such a ringis about 1.3
V
10
–7
H. If the permanent current in such a ring decreases by less than1% within an hour, we can conclude that the resistance must be smaller than 4
V
10
–13
V
.
3)
This means that in the superconducting state the resistance has changedby more than eight orders of magnitude.During such experiments the magnitude of the permanent current must bemonitored. Initially[1] this was simply accomplished by means of a magneticneedle, its deflection in the magnetic field of the permanent current being observed.A more sensitive setup was used by Kamerlingh-Onnes and somewhat later byTuyn[2]. It is shown schematically in Fig.1.2. In both superconducting rings 1 and2 a permanent current is generated by an induction process. Because of this currentboth rings are kept in a parallel position. If one of the rings (here the inner one) issuspended from a torsion thread and is slightly turned away from the parallelposition, the torsion thread experiences a force srcinating from the permanentcurrent. As a result an equilibrium position is established in which the angularmoments of the permanent current and of the torsion thread balance each other.This equilibrium position can be observed very sensitively using a light beam. Anydecay of the permanent current within the rings would be indicated by the lightbeam as a change in its equilibrium position. During all such experiments, nochange of the permanent current has ever been observed.A nice demonstration of superconducting permanent currents is shown inFig.1.3. A small permanent magnet that is lowered towards a superconducting leadbowl generates induction currents according to Lenz’s rule, leading to a repulsiveforce acting on the magnet. The induction currents support the magnet at anequilibrium height. This arrangement is referred to as a “levitated magnet”. Themagnet is supported as long as the permanent currents are flowing within the leadbowl, i.e. as long as the lead remains superconducting. For high-temperaturesuperconductors such as YBa
2
Cu
3
O
7
this demonstration can easily be performedusing liquid nitrogen in regular air. Furthermore, it can also serve for levitatingfreely real heavyweights such as the Sumo wrestler shown in Fig.1.4.The most sensitive arrangements for determining an upper limit of the resistancein the superconducting state are based on geometries having an extremely smallself-induction coefficient
L
, in addition to an increase in the observation time. Inthis way the upper limit can be lowered further. A further increase of the sensitivityis accomplished by the modern superconducting magnetic field sensors (see Sect.7.6.4). Today we know that the jump in resistance during entry into the super-conducting state amounts to at least 14 orders of magnitude[3]. Hence, in thesuperconducting state a metal can have a specific electrical resistance that is at mostabout 17 orders of magnitude smaller than the specific resistance of copper, one of
3
For a circular ring of radius
r
made from a wire of thickness 2
d
also with circular cross-section(
r
>>
d
), we have
L =
m
0
r [ln(8
r/d
)–1.75] with
m
0
= 4
p
V
10
–7
V s/Am. It follows that
R
≤
–ln0.99
V
1.3
V
10
–7
3.6
V
10
3
VsAm
?
3.6
V
10
–13
V
.
1.1The Vanishing of the Electrical Resistance
13
our best metallic conductors, at 300 K. Since hardly anyone has a clear idea about“17 orders of magnitude”, we also present another comparison: the difference inresistance of a metal between the superconducting and normal states is at least aslarge as that between copper and a standard electrical insulator.Following this discussion it appears justified at first to assume that in thesuperconducting state the electrical resistance actually vanishes. However, we mustpoint out that this statement is valid only under specific conditions. So theresistance can become finite if magnetic flux lines exist within the superconductor.Furthermore, alternating currents experience a resistance that is different fromzero. We return to this subject in more detail in subsequent chapters.
Fig. 1.2
Arrangement for the observa-tion of a permanent current (after[2]).Ring 1 is attached to the cryostat.
1Fundamental Properties of Superconductors
14
This totally unexpected behavior of the electric current, flowing without resistancethrough a metal and at the time contradicting all well-supported concepts, becomeseven more surprising if we look more closely at charge transport through a metal. Inthis way we can also appreciate more strongly the problem confronting us in termsof an understanding of superconductivity.We know that electric charge transport in metals takes place through theelectrons. The concept that, in a metal, a definite number of electrons per atom (forinstance, in the alkalis, one electron, the valence electron) exist freely, rather like agas, was developed at an early time (by Paul Drude in 1900, and Hendrik AntonLorentz in 1905). These “free” electrons also mediate the binding of the atoms inmetallic crystals. In an applied electric field the free electrons are accelerated. After
Fig. 1.3
The “levitated magnet” for demonstrating the permanent currents that are generated in superconducting lead by induction during the loweringof the magnet. Left: starting position. Right: equilibrium position.
Fig. 1.4
Application of free levitationby means of the permanent currentsin a superconductor. The Sumowrestler (including the plate at thebottom) weighs 202 kg. The super-conductor is YBa
2
Cu
3
O
7
. (Photo-graph kindly supplied by the Inter-national Superconductivity ResearchCenter (ISTEC) and Nihon-SUMOKyokai, Japan, 1997).
1.1The Vanishing of the Electrical Resistance
15

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