Dresner, Stability of superconductors.2002

The value n = 0.85 gives an excellent fit to the exact curve and is a useful approximation for numerical calculation. Eq. (2.1.2), known as the Debye T3-law, is often written Cv1 = βT3, where b = (12π4/5)rR/θ3 = 1944r/θ3 J mol¯1 K¯4.

The only physical property of the solid that enters the Debye theory is the Debye temperature q. Table 2.1 shows its value for several materials of interest in applied superconductivity.

Debye’s theory accounts for the contribution to the specific heat of the vibrations of the atomic lattice of which the solid is composed. (These vibrations are known conventionally as phonons, and Debye’s theory is said to give the phonon contribution to the specific heat.) If the solid is a metal, as are all the entries in Table 2.1, there is an additional contribution to the specific heat from the electrons. This additive contribution, which plays a significant role only at low temperatures, has

where g is the so-called Sommerfeld constant. Its value is given in Table 2.2 for the same materials that appear in Table 2.1.

Table 2.1. The Debye Temperature qfor

2.2. SPECIFIC HEATS OF TYPE-II SUPERCONDUCTORS

The specific heat of a type-II superconductor in the normal state is also described by the Debye-Sommerfeld theory given in the previous section. In the superconducting state, it is given approximately by

where Cs(T,H) is the specific heat in the superconducting state at temperature T and applied magnetic field H, Cn(T) is the specific heat in the normal state at temperature T, Tco is the critical temperature at zero applied field, and Hc2(0) is the (upper) critical field at zero temperature. The second line applies at low temperatures, when

the Debye T 3-law, Cphonon = βT 3, is a satisfactory approximation. At the critical temperature Tc (at applied field H), where the phase change occurs, the specific heat

undergoes a jump

Such jumps in the specific heat (so-called l-anomalies) are characteristic of second-order phase changes in which there is no latent heat.

Figure 2.2 shows a plot of measured values of C/T versus T 2 for the low-tem- perature superconductor Nb 44.6-wt-% Ti reported by Elrod, Miller, and Dresner (1982). The lines are best fits to Eq. (2.2.2) using the values γ = 0.145 J kg¯1 K¯2 , β = 2.3 x 10¯3 J kg¯1 K¯4 , and µ oHc2(0) = 14 T. The fit is excellent and reinforces our belief that Eq. (2.2.2) is a description of superconductor specific heats at low temperatures adequate for use in the theory of superconductor stability. It is perhaps worth noting here that our goal in this chapter is to find, where possible, simple, widely applicable formulas that allow approximate computation of the physical properties we use to study the stability of superconductors.

The Debye theory is based on an approximate phonon spectrum in which the density of phonon states varies as the square of the phonon frequency. At low

Figure 2.2. Measured values of C/T versus T2 for Nb 44.6-wt-% Ti (Elrod et al., 1982). (Redrawn from an original appearing in Elrod et al. (1982) with permission of Plenum Publishing Corp., New York.)

temperatures, only low-frequency phonons are excited, and at this end of the phonon spectrum, the quadratic approximation is accurate. At higher temperature, such as might typify the new high-temperature superconductors, the fit to Eq. (2.2.1) might not be as good. Indeed, deficiencies in the Debye theory due to its simplified phonon spectrum have been known for some time—long ago, Born and von Karman (Reed and Clark, 1983) sought to improve the Debye theory by invoking a more realistic phonon spectrum, but only at the cost of increased computational complexity. Occasionally, the difficulties with the Debye theory have been circumvented in an ad hoc way by letting the Debye temperature q vary with T, but such a “cure” is only applicable when the specific heat is already known experimentally over a wide range of temperatures.

This is exactly the situation with recent measurements of the specific heat of YBa2Cu3O7 recently published by Pavese and Malishev (1994): see Fig. 2.3. Shown in Fig. 2.4 are plots of the specific heat calculated using Eqs. (2.1.1) and (2.2.1) for five values of the Debye temperature together with seven points taken from Fig. 2.3. At low temperatures q ~ 375 K, whereas at high temperatures q ~ 525 K. Choosing the smaller value leads to a maximum error in the neighborhood of 100 K of about 30%.

We shall not abandon Eqs. (2.1.1) and (2.2.1) for high-temperature superconductors, but rather accept the facts of the last paragraph as a caution on the limitation of their accuracy.

The next four optional sections present a discussion of the thermodynamics of superconductors that culminates in the derivation of Eq. (2.2.1).

2.3. FIRST LAW OF THERMODYNAMICS FOR A MAGNETIZABLE BODY

Presented below is a phenomenological theory of the specific heat of the superconducting state. In this theory, the specific heat is deduced by thermodynamic arguments from the measured value of another, more accessible quantity, the magnetization M of the superconductor in the presence of an applied field

H.

The state of a magnetizable body is described by Maxwell’s equations

where E is the electric field, B is the magnetic induction, H is the applied magnetic field, created by J, the current density produced by the external power supply, and

from the right-hand side of Eq. (2.3.2).

In MKS units, M has the dimensions of A/m = Am2/m3 or the dimensions of dipole moment per unit volume. In a conceptual sense, M can be thought of as an increase in the total field inside the body brought about when the applied field orients the body’s atomic dipoles.

To write the first law of thermodynamics for a magnetizable body, we need to know the increment of work the body does when its magnetic state changes. Change in the magnetic state, i.e., change in H and M, causes change in the induction B, which in turn creates the electric field E. This field applies a force Ee to the charge carriers produced by the external power supply that create the external field H. This force (the back emf) is opposed by an equal and opposite force –Ee produced by the external power supply. If the velocity of these charge carriers is v, –eE·v is the rate at which the external power supply does work on them. If their density is n, –neE·v = –E·J is the rate at which the external power supply does work on the charge carriers in a unit volume. The quantity d(vol) is thus the rate at which the external power supply does work when the state (H,M) is changing.

To relate the quantity E·J to the state variables H and M, we scalar multiply

integrate Eq. (2.3.4) over a large volume V whose surface S surrounds the body and the external power supply, we find

(2.3.5)

As the volume V grows large, the magnitude of the product E x H falls faster than the surface S increases, and the surface integral vanishes in the limit. Thus

(2.3.6)

(2.3.7)

The passage from Eq. (2.3.6) to Eq. (2.3.7) has been achieved by substituting the right-hand side of Eq. (2.3.3) for B. The second volume integral only extends over the volume of the body, for only there is M different from zero.

The left-hand side of Eq. (2.3.6) is the rate at which the external power source is supplying power. The first term on the right-hand side of Eq. (2.3.7) is the rate of increase of the energy of the magnetic field and the second term is the rate of increase of the energy of the body, i.e., the rate as which the external power supply does work on the body. For thermodynamic purposes, we need the rate at which work is done by the body during a change of state; it is

(2.3.8)

In the simplest experiment to measureM, the external field H is applied parallel to the axis of a long wire so that both H and M are parallel and uniform over the body. Then the rate of work done by the body is

where V is the volume of the body. Thus, finally, the work done by the body in a change of state is

(2.3.10)

2.4. GlBBS FREE ENERGY OF A MAGNETIZABLE BODY

The two laws of thermodynamics may be written together for the magnetizable body as

(2.4.1)

where T is the temperature, s the entropy of the body per unit mass, u its internal energy per unit mass, and δ its density. The most convenient independent variables to use in the thermodynamic discussion are the applied field H and the temperature T since they are under our control. Accordingly, we introduce the function

(2.4.2)

which is the Gibbs free energy per unit mass of the body. Then

(2.4.3)

so that

(2.4.4)

Finally, the specific heat at constant applied field CH is given by

(2.4.5)

According to Eq. (2.4.4),

(2.4.6)

where the subscript s on g indicates the superconducting state. When the pair of values (T,H) lies on the phase boundary between the mixed and the normal states, the Gibbs free energy of the normal and superconducting states are equal (remember, the transition is second-order). Thus

(2.4.7)

where Hc2 is the upper critical field at temperature T (see Fig. 1.2). If we subtract Eq. (2.4.7) from Eq. (2.4.6), we find