Dresner, Stability of superconductors.2002
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M = 2a[H – Hc2(0)(1– T2/T2co )]/[ Hc2(0)]2
Inserting Eq. (2.6.4) into Eq. (2.6.1) we find
so that finally
Cs(T,H) = Cn(T) + 3γ T3/T 2co – γ T+ γ TH/Hc2(0) = (β + 3γ /Tco2 )T 3 + γ TH/Hc2(0)
CHAPTER 2
(2.6.4)
(2.6.5)
(2.6.6)
(2.6.7)
2.7. MATRIX RESISTIVITY; BLOCH–GRÜNEISEN FORMULA
Electrical resistance is caused by the scattering of electrons as they move through a wire under the action of a potential difference. The electrons can be scattered by solid-state defects, by impurities, and by the phonons themselves. The phonon contribution to the resistivity rr is given by a formula similar in appearance to the Debye formula and called the Bloch–Grüneisen formula (Reed and Clark, 1983):
Figure 2.6. A log-log plot of the quantify y, which is proportional to the phonon resistivity, versus the dimensionless temperature T/θ according to the theory of Bloch and Grüneisen (Redrawn from an original appearing in Dresner (1993, “Stability”) with permission of Butterworth-Heinemann, Oxford, England.)
Material Properties |
31 |
Table 2.3. The Ice-Point (273 K)
Resistivities of Several Metals
Material Resistivity(µΩ -cm)
cu |
1.55 |
Ag |
1.48 |
A1 |
2.43 |
|
|
where K is a constant yet to be determined. Again the quantity in the brackets has been tabulated by Abramowitz and Stegun (1965).
The quantity y plotted against x = 1/z = T/θ as calculated using Eq. (2.7.1) is shown in Fig. 2.6. Again there are two asymptotes, namely,
T << θ: |
y = 124.4x5 |
(2.7.2) |
|
1 |
|
T >>θ: |
y2 = x/4 |
(2.7.3) |
The approximation
(2.7.4)
fits the exact curve quite well when n = 0.65. The constant K in Eq. (2.7.1) is usually fitted to the room-temperature (300 K) or the ice-point (273 K) resistivity. Typical values of the latter for several matrix materials are shown in Table 2.3.
In addition to scattering by phonons, the conduction electrons are scattered as they migrate by impurities and solid-state defects. Whereas the phonon contribution to the resistivity ρρ falls strongly with decreasing temperature, the contribution of the impurities and defects, called the residual resistivity and denoted by rr , is independent of temperature. According to Matthiesen’s rule, the total resistivity is the sum of the residual and phonon contributions; above about 30 K the phonon resistivity dominates. It is worth noting here that annealing (such as might occur in a heat treatment) decreases the residual resistivity whereas cold-working (such as might occur in drawing or extrusion) increases it.
2.8. MAGNETORESlSTIVITY
When a metal carries current in a magnetic field, its electrical resistivity is larger than when there is no field. This is because the Lorentz force on the moving
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CHAPTER 2 |
conduction electrons changes their trajectories. The additional resistivity is called the magnetoresistivity and is denoted by ρm. The magnetoresistivity depends on the direction of the magnetic field relative to the current encountering the resistance. In the high-field region of solenoids, the magnetic field is perpendicular to the current in the conductor (transverse magnetic field). In transverse magnetic fields, the magnetoresistivity is related to the applied magnetic field B by Köhler’s rule, namely,
ρm/ρ = F(BρRT /ρ) |
(2.8.1) |
where ρRT is the room-temperature resistivity and F is an empirically determined function (which of course must vanish when its argument vanishes). For many metals, among them copper and silver, the function F is nearly a straight line (see Figs. 2.7 and 2.8). When F is linear, the transverse magnetoresistivity is directly proportional to B. The proportionality constants for copper and silver are, respectively, 4.5 x 10¯11 ohm-m/T and 3.0 x 10¯11 ohm-m/T.
The transverse magnetoresistivity for A1 is more complicated and does not follow Köhler’s rule; instead it rises at low fields, but saturates quickly at relatively modest fields. According to Reed and Clark (1983), the longitudinal magnetoresistance is lower than the transverse, usually by a factor of two or more. Furthermore, it saturates at high enough fields.
Figure 2.7. A Köhler plot of the transverse magneto-resistance of copper (Reed and Clark, 1983). (Redrawn from an original provided courtesy of the American Society for Metals.)
Material Properties |
33 |
Figure 2.8. A Köhler plot of the transverse magneto-resistance of silver (Iwasaet al., 1993). (Redrawn from an original appearing in Iwasa et al. (1993) with permission of Butterworth-Heineman, Oxford, England.)
At low temperatures, the transverse magnetoresistivity may be comparable with or even exceed the residual resistivity and so must be accounted for in studies of superconductor stability.
2.9. THE WIEDEMANN–FRANZ LAW
Because both heat and electric charge are transported in metals by the electrons, there is a relation between the thermal and electrical conductivities. This relation, called the Wiedemann–Franz law, is most often written
kρ =LoT |
(2.9.1) |
where k is the thermal conductivity, r the electrical resistivity, Lo the Lorenz constant, equal to 2.45 x 10-8 V2K-2, and T is the temperature. The Wiedemann– Franz law is an approximation based on a simplified picture of the electronic structure of metals, and its accuracy depends both on the temperature T and the purity of the metal. For ordinary commercial metals, which are relatively impure,
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CHAPTER 2 |
the law is reasonably accurate both at low temperatures < O.1θ and at high temperatures > q. Between these two extremes, the effective value of the Lorenz constant Lo falls somewhat, the fall being greater the greater the punty of the material.
There is some question about whether the Wiedemann–Franz law holds in a magnetic field. We know that the resistivity is augmented by the field (magnetoresistance). Does the thermal conductivity fall in a magnetic field? According to Fevrier and Morize (1973), in copper it does and in such a way that the effective Lorenz constant Lo increases with increasing transverse field in their two samples by 4% and 13% up to 5 T (roughly 1–2% per tesla). Arenz, Clark, and Lawless (1982), also studying copper in fields up to 12 T, find an increase in the effective value of Lo with increasing transverse magnetic field, in their case by about 7% per tesla.
In spite of these drawbacks, the Wiedemann–Franz law is widely used in studies of superconductor stability because of its simple form.
3
Flux jumping
3.1. THE CRITICAL-STATE MODEL
As noted earlier, as the temperature or the magnetic field crosses the upper boundary in Fig. 1.2 there is a change of phase accompanied by a sudden change in some physical properties, e.g., resistivity and, as we saw in the last chapter, specific heat. But as the current density exceeds the critical current density and the fluxoid lattice is torn loose by the Lorentz force from its pinned attachment to the defects in the solid, no true phase change occurs,1 although the resistivity of type-II superconductors undergoes a sharp (but continuous) change (see Fig. 3.1).
The sharpness of this change allows us to describe the behavior of type-II superconductors in terms of the so-called critical-state model (Bean, 1962; Anderson and Kim, 1964), in which the superconductor either carries a current density Jc, the critical current density, or no current at all. To see how such a situation comes about, let us consider the parallel circuit shown in Fig. 3.2, in which one conductor is copper and the other is a superconductor, for example, NbTi. Such a circuit in fact represents a multifilamentary superconductor in which many fine filaments of NbTi run longitudinally through a matrix of copper.
Let the critical current in the superconductor (at the ambient field and temperature) be Ic and let I be the total current being driven through both branches. (The usual definition of Ic (or Jc) is the current (or current density) at which the superconductor resistivity reaches a specified value, typically 10-13 ohm-m.) If I < Ic, all the current flows in the superconductor. When I > Ic, the current divides according to the resistances of the two branches. But what is the resistance of the superconducting branch? Because of the steepness of the resistivity curve in Fig. 3.1, no matter what value the resistivity has (as long as it is not zero), the current in the superconductor is very close to Ic . Therefore, the current I – Ic must flow through the copper. The voltage drop across the copper is then V = Rcu(I – Ic). Since the copper and the superconductor are electrically in parallel, the superconductor
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CHAPTER 3 |
Figure 3.1. A sketch of the sharp but continuous increase with current of the resistivity of type-II superconductors.
also experiences the voltage drop V, and its resistance consequently is V/Ic. If we raise the current I, the voltage V = Rcu(I – Ic) goes up and so does the resistance of the superconductor. This requires an increase in the current in the superconductor, but because of the steepness of the curve in Fig. 3.1, the required increase in the superconductor current is very small. For practical purposes, then, the superconductor can be considered always to be carrying the current Ic when it is resistive.
The italicized statements above give the essence of the critical-state model, which brings with it enormous simplifications in computation, and which, it can be fairly said, underlies the entire field of applied superconductivity.
3.2. THE THREE-PART CURVE OF JOULE POWER
The parallel circuit in Fig. 3.2 generates resistive (Joule) power when I > Ic and some of the current is carried in the copper. Since the voltage V spans both
Figure 3.2. A parallel circuit in which one conductor is copper and the other is a superconductor.
Flux Jumping |
37 |
Figure 3.3. (a) A plot of critical current Ic versus temperature T showing a lineardeclineof Ic with T.
(b) The three-part powergeneration curve that is the basis of most studies of superconductor stability.
branches of the parallel circuit, the entire current falls through that voltage drop. Then the Joule power Q being produced is IV = IRcu(I – Ic).
Now empirical observations of many superconductors show that the critical current (or current density) falls linearly with increasing superconductor temperature T. Shown in Fig. 3.3a is a plot of Ic versus T showing a linear decline of Ic with T that reaches zero when T = Tc (where the superconductor becomes normal). Shown also is the current level I and the temperature Tcs at which Ic = I. This temperature is the largest at which the superconductor can still carry all of the impressed current I. At higher temperatures, some of the current I spills over into the copper. This is called current sharing and Tcs is called the current-sharing threshold temperature. Clearly, for T < Tcs , Q = 0 (see Fig. 3.3b).
When T > Tc , Ic = 0, and Q has the constant value I2Rcu, all of the current being in the copper (since the normal-state resistivity of NbTi is much greater than that of copper). Between Tcs and Tc, Q = IRcu(I – Ic) is taken to vary linearly with T because Ic itself varies linearly with T. The resulting three-part curve of Fig. 3.3b, made up of three straight-line segments, plays a fundamental role in the science of applied superconductivity and is the basis of most studies of superconductor stability.
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CHAPTER3 |
3.3.CHARGING OFA SUPERCONDUCTOR: CRITICAL-STATE MODEL
When any conductor is exposed to a changing magnetic field, currents are induced in it: this is Faraday’s law of induction. These currents are called eddy currents or shielding currents. The latter term is a consequence of Lenz’s law that tells us that the induced currents flow in such a direction as to oppose the change in the external magnetic field. If the conductor is an ordinary conductor, e.g., copper, the induced shielding currents encounter resistance. If the external field change stops, the shielding currents decay as a consequence of the resistance.
If the conductor is a type-II superconductor in which no pinning is present, the shielding currents interact via the Lorentz force with the fluxoids, causing them to move. The motion of the fluxoids creates an electric field that opposes the flow of the shielding currents, so that when the external field change stops, the shielding currents decay. (The opposing electric field induced by the motion of the fluxoids is often represented as though the shielding currents were flowing through a resistance, which is named the flux-flow resistance.) When the shielding currents have decayed and the magnetic field has become uniform throughout the sample (in the form of a uniform fluxoid lattice), a state of thermodynamic equilibrium is reached.
When strong pinning is present, it restrains the motion of the fluxoid lattice, prevents the shielding currents from decaying, and greatly delays the attainment of thermodynamic equilibrium. In this case, the pattern of shielding currents is again determined by the critical-state model. If the shielding current density locally exceeds the critical current density, pinning fails and flux-flow resistance appears. But as soon as the current density decays to the critical value, pinning again becomes effective and the flux-flow resistance disappears. Furthermore, at any place in the sample where the magnetic field is changing and an electric field exists, the shielding current rises without hindrance (owing to the lack of resistance in the superconductor) until it reaches the critical value. So the current density in the sample always has the critical value, though at different points it may flow in different directions. If there are places where the magnetic field is completely shielded, the current density there is zero. But the only finite value the current density can have is the critical value.
The application of the critical-state model to the interior of a superconductor can be illustrated by the following example. Let us consider a slab of superconductor of thickness 2d that is being charged with current (Fig. 3.4). Let I be the z-directed current per unit length of slab extending in the x-direction. If we apply the right-hand rule to the current, we see that the magnetic field created as the current increases is in the x-direction on the left-hand side of the slab and in the
–x-direction on the right-hand side of the slab. If we apply the left-hand rule to the rising magnetic flux through the rectangle ABCD, we see that the induced voltage
Flux Jumping |
39 |
Figure 3.4. A sketch illustrating the development of layers of critical current density in a slab of superconductor being charged with current.
around the rectangle opposes the flow of the current I in the center of the slab and promotes it near the edges of the slab. These relations, which apply to any conductor, are the origin of the skin effect in ordinary conductors. In a superconducting slab, they force the current density to adopt the critical value Jc in two layers of thickness a = 1/2Jc adjacent to each edge of the slab.
According to the z-component of Maxwell’s equation
(3.3.1)
and according to the x-component of Maxwell’s equation
(3.3.2)
The values of J, B, ∂B/∂t, and E are as shown in Fig. 3.5. The source of the electric field E is the external power supply, which must maintain it as long as the current I is increasing. The penetration depth, a, increases as I increases until at full penetration (a = d), the current I reaches the critical value 2Jcd.
3.4.CHARGING OF A SUPERCONDUCTOR: POWER-LAW RESlSTlVlTY
The foregoing example has been chosen because more elaborate computations are possible for this example that enable us to understand in greater detail the nature of the critical-state model. The critical-state model as used above is based on the