Abstract
In the following the main steps in the study of the superconductive state along the twentieth century are schematically collected.
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Notes
- 1.
The use of Hg, a liquid metal at room temperature, was related to the possibility to perform a series of distillation processes at relatively low temperature, since at that time a role of the impurities in levelling to a finite value the resistivity in metals at very low temperatures was suspected. Kamerlingh-Onnes first claimed that the drop to zero of the resistance supported his point of view, that without the impurities the resistance had to tend to zero on cooling towards zero temperature. Then he corrected himself, observing that in reality a new state, that he called superconductivity, was induced on crossing the critical temperature of about 4.2 K.
- 2.
The increase of the superconducting temperature and of the critical fields and critical currents in the alloys is related to the reduction of the coherence length accompanying the decrease of the mean free path of the electrons . The alloying process transforms the metallic superconductors from first- type to second type (Sects. 18.7 and 18.9).
- 3.
The resistance does actually drop to zero, provided that competing processes like flux flow of the vortices are properly avoided. From precise measures by means of NMR spectra (see Sect. 6.3) of the magnetic field due to superconducting current launched in a solenoid, it can be proved that no appreciable variation would occur at least for a period of about 10\(^5\) years. However the resistance is not zero for alternate current and in the presence of external magnetic field.
- 4.
The maximum current in a wire can be related to the critical field. Approximately one has \(I_{{ max}}\simeq 5 RH_c\), in Amp., the radius R of the wire being in cm and \(H_c\) in Oe.
- 5.
This equation has to be taken with a certain care since in its general form it is valid only in stationary condition, while a variation of the magnetic field launches a transitory current. On the other hand the currents launched in the transient circulate in a narrow outer sheet of the material and the electric field in the bulk has to be considered zero.
- 6.
Thus in the condition of total conductivity only, one would not have a single thermodynamic state and the equilibrium thermodynamics could not be applied. This is another relevant difference between the total conductor and the superconductor. In the superconductor the magnetic field is anyway repelled, irrespective of the order of the operations, cooling in the magnetic field or applying the field below \(T_c\) (the field to be considered is smaller than the critical field).
- 7.
Such a strong value of the diamagnetic susceptibility makes possible the phenomenon of the diamagnetic levitation of large masses. It can be remarked since now that for strongly type II superconductors, as the HTcSC cuprates, a similar impressive phenomenon is the diamagnetic suspension : the superconductor follows a magnet lifted up, thus appearing magically suspended.
- 8.
One could observe that an electron, after having caused a local shrinking of the positive lattice, leaves the place at a speed of the order of the Fermi velocity while the distortion evaporates in much longer times, the process involving the sound velocity.
- 9.
The relationship between transition temperature \(T_c\) and the Debye temperature to a good approximation is provided by the semi-empirical formula due to McMillan
$$ T_c= \frac{\varTheta _D}{1.45} exp\biggl (\frac{-(\lambda ^*+1)}{[\lambda ^*- \mu ^*(1+ 0.62\lambda ^*)]}\biggr ), $$where the electron-phonon coupling parameter \(\lambda ^*\) typically varies in the range 0–2 while the repulsive Coulombic parameter \(\mu ^*\) is between 0.1 and 0.2. \(\lambda ^*\) and \(\mu ^*\) are usually determined by means of electron tunneling experiments.
- 10.
At Sect. 18.9, in describing the Ginzburg-Landau theory in the general framework of phase transitions, it will be shown that the “correlation range” can be considered the equivalent of the coherence length introduced at Sect. 15.2. It is remarked that in most cases the distance over which the correlation is effective is limited to an almost temperature-independent mean free path of the electrons . The Cooper pair coherence length is smaller than the Ginzburg-Landau coherence length that we shall introduce at Sect. 18.9. Only when the mean free path of the electrons can be assumed infinite the two coherence lengths (Cooper and Ginzburg-Landau) coincide.
- 11.
One could argue that this form of the wavefunction, being based on the condition of very large correlation range, could break down in superconductors where the range of the interaction in the pair (namely the correlation length) is a few lattice steps, as for instance in the high temperature superconductors (Sect. 18.8). In reality, the Ginzburg-Landau theory , valid both in the BCS superconductors at very large coherence length as well as in high \(T_c\) superconductors, is based on the inspiring guess that the density of superconducting carriers (the order parameter in the scenario of the phase transition, see Sect. 15.2) can be written as the modulus square of some effective wave function.
- 12.
See note for Eq. (18.1). The currents launched in the transient circulate in a narrow sheet close to the surface of the ring, while the electric field is zero along the internal line.
- 13.
Such an approach as been devised by Feynman and it is often reported as the two-states Feynman model.
- 14.
\(n_A\) and \(n_B\) are constant while a non-zero charge flows because the source acts in order to feed the junction of further pairs. To take into consideration also the currents provided by the source would not modify the current related to the tunnelling of the insulating layer, which is the one we are evaluating here (see the book by Feynmann).
- 15.
The magnets with conventional (BCS) type II superconductors, as NbTi or Nb\(_3\)Sn, can reach up to 10 T and are the ones generally used in NMR imaging (see Sect. 6.3). The magnetic levitation force can be of the order of 10\(^4\) N/m\(^2\). The quality factor of superconducting microwave cavities can be increased up to 10\(^{10}\), as in the ones currently used in particle accelerators. For extensive description of the applications, most for BCS superconductors, see the book by Buckel .
- 16.
This is true for the family of the cuprate superconductors. In \({MgB}_2\) the relatively high \(T_c\) (about 39 K) is still related to conventional BCS mechanism, in the presence of particular electronic band structure and phonon bath. In the iron-based superconductors discovered in 2008 (the Fe-oxypnictides) the magnetic ion is Fe and the pairing mechanism possibly involves magnetic excitations.
- 17.
With reference to Chap. 12, the Mott insulator is somewhat opposite to the band insulator, its existence being due to the strong repulsive correlation, an effect that can hardly be described in the band-type theories.
- 18.
In anisotropic superconductors, in the reference frame aligned with the principal axes, there are three coherence lengths \(\xi ^{(i)}\) and three penetration lengths \(\lambda _L^{(i)}\). These six lengths, not independent from each other, are commonly introduced in order to extend the isotropic Ginzburg-Landau theory by referring to \(\lambda _L= [\lambda _L^{(1)}\lambda _L^{(2)}\lambda _L^{(3)}]^{1/3}\) and \(\xi = [\xi ^{(1)}\xi ^{(2)}\xi ^{(3)}]^{1/3}\) (see Sect. 18.9).
- 19.
Below \(T_{{ irr}}\) one observes a difference between the diamagnetic susceptibility \(\chi _{{ dia}}\) (Meissner effect, see Sect. 18.3) measured after cooling in the presence of the magnetic field (FC condition) and the one measured at the same temperature after cooling in zero field (ZFC condition). In a certain temperature range below \(T_{{ irr}}\) the ZFC diamagnetism is time-dependent, since the vortices are forced to penetrate inside the bulk, driving the relaxation process leading from the ZFC to the FC condition.
- 20.
The phase and the magnetic vector potential \(\mathbf {A}\) depend on the choice of the gauge but all the physical variables, including the magnetic field \(\mathbf {B}\), are gauge-invariant (see Problem 18.14).
- 21.
Equation (18.38) should be considered a masterpiece of physical intuition. It has a possible justification in the light of the analogy with the Schrodinger equation by assigning to \(\psi \) the character of wavefunction. The BCS theory has clarified many aspects related to that equation. Gor’kov has shown that the GL theory can be derived from the BCS theory when the latter is generalized to include spatially varying situations, near the transition. \(\psi \) corresponds to the wavefunction of the centre of mass of the Cooper pair (see Sect. 18.2.3).
- 22.
\(\partial F/\partial \psi (\mathbf {r})^*= 0\) involves the functional derivative, the free energy being a function of infinitely many variables. By deriving with respect to \(\psi (\mathbf {r})\) the complex conjugate of Eq. (18.41) would be obtained.
- 23.
These conditions are also rather well verified in nanoparticles , namely particles of size much smaller than the coherence length (see Sect. 18.12), this case being often called zero-dimensional condition.
- 24.
Over the whole temperature range below \(T_c\) more appropriate temperature dependencies are \(H_c(T)\propto [1- (T/T_c)^2]\) and \(|\psi |^2= n_c\propto [1- (T/T_c)^4]\).
- 25.
Return to the note n.10 of the present Chapter.
- 26.
\(\kappa \) is the only parameter that really plays the crucial role in the GL theory.
- 27.
For details and for considerations on the effects of the magnetic field in other experimental conditions, see the book by Poole, Farach and Creswick. One could remark that an entire volume would not be sufficient in order to illustrate all the applications of the Ginzburg-Landau theory in a variety of circumstances, as remarked by the authors.
- 28.
It is reminded that for material strongly anisotropic in their superconducting properties, as the HTcSC, significant modifications to the expressions derived above have to be taken into account. For further details see the book by Poole, Farach and Creswick .
- 29.
Pinning has technological importance in order to lock the vortices and avoid dissipation. For a current \(\mathbf {j}\) flowing perpendicular to the field \(\mathbf {H}\) the Lorentz force pushes the vortices along the \(\mathbf {j} \times \mathbf {H}\) direction. The flux variation implies electric field parallel to \(\mathbf {j}\) and then an effective electrical resistance. Also the Magnus sideway acting force can be involved in the vortex motions.
- 30.
Here we write \(\alpha \) and \(\beta \) instead of a and b to emphasize that in this section we deal with the fluctuations around the equilibrium values.
- 31.
The sum over \(\mathbf {k}\) should be limited to a cut-off value of the order of the inverse coherence length \(\xi _0^{-1}\), somewhat in analogy with the cut-off at the boundary of the Brillouin zone or at the Debye wave vector (see also footnote 33).
- 32.
The Ginzburg-Landau (GL) correlation time is often written \(\tau _{{ GL}}=(\gamma \hbar /\alpha )\) by adding \(\hbar \) to the coefficient in Eq. (18.74) in order to get correspondence with the expression \(\tau _{{ GL}}= \hbar /8k_B(T-T_c)\) derived in the microscopic theories. It can be observe that in this way, without the non-linear term, Eq. (18.73) takes a form consistent with the time-dependent Schrodinger equation.
- 33.
A variety of data and extensive theory can be found in the book by Larkin and Varlamov . A comprehensive description of fluctuation effects, at a level comparable to the present text, is given at Chap. 8 of the book by Tinkham .
- 34.
To give a quantitative estimate of the temperature range where this assumption is appropriate, it is mentioned that the Ginzburg-Levanyuk parameter above which the first-order fluctuation correction is substantially valid, in zero-dimension is given by
$$ G_i(0)\simeq 13.3 [T_c(0)/T_F][\xi _0/d]^{3/2} $$with \(T_F\) Fermi temperature. (For details see Chap. 2 of the book by Larkin and Varlamov .)
- 35.
This is the expression coming as functional integral \(\int d^2\psi (\mathbf {r}) exp(-f[\psi (\mathbf {r})]/k_BT)\) over the “field” \(\psi (\mathbf {r})\). Thus \(Z_0= \int _0^{\infty } d|\psi _0|^2exp(-\alpha |\psi _0|^2/k_BT)= \pi k_BT/\alpha (T)\).
Specific References and Further Reading
G. Grosso and G. Pastori Parravicini, Solid State Physics, 2\(^{\rm nd}\) Edition, (Academic Press, 2013).
J.F. Annett, Superconductivity, Superfluids and Condensates, (Oxford University Press, Oxford, 2004).
W. Buckel, Superconductivity- Fundamental and Applications, (VCH Weinheim, 1991).
D.L. Goodstein, States of Matter, (Dover Publications Inc., 1985).
H. Ibach and H. Lüth, Solid State Physics: an Introduction to Theory and Experiments, (Springer Verlag, 1990).
C.P. Poole, H.A. Farach, R.J. Creswick, Superconductivity, (Academic Press, San Diego, 1995).
R.P. Feynman, R.B. Leighton and M. Sands, The Feynman Lectures on Physics Vol. III, (Addison Wesley, Palo Alto, 1965).
M. Tinkham, Introduction to Superconductivity, (Dover Publications Inc., New York, 1996).
A. Larkin and A.A. Varlamov, Theory of Fluctuations in Superconductors, (Oxford Science Publications, Clarendon Press, Oxford, 2005).
N.W. Ashcroft and N.D. Mermin, Solid State Physics, (Holt, Rinehart and Winston, 1976).
A. Barone and G. Paternó, Physics and Applications of the Josephson Effect, (John Wiley, New York, 1982).
J.G. Bednorz and K.A. Müller, (Eds.), Early and Recent Aspects of Superconductivity, (Springer-Verlag, Berlin, 1990).
G. Burns, High Temperature Superconductivity - An Introduction, (Academic Press Inc., 1992).
M.Cyrot and D.Pavuna, Introduction to Superconductivity and High-\(T_c\) Materials, (World Scientific, Singapore, 1992).
P.G. de Gennes, Superconductivity of Metals and Alloys, (Addison-Wesley, 1989).
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Rigamonti, A., Carretta, P. (2015). Superconductors, the Superconductive Phase Transition and Fluctuations. In: Structure of Matter. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-17897-4_18
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