Type-II Superconductivity∗ Dr Demetris Charalambous Meteorological Service, Ministry of Agriculture, Natural Resources and Environment, 1418 Lefkosia, CYPRUS and Department of Physics, University of Lancaster Lancaster, LA1 4YB, UNITED KINGDOM Email: [email protected] Last updated: March 2007 §1. General introduction A type-II superconductor allows the penetration of a large-enough magnetic field into its bulk. Below Hc1 , also known as the lower critical field, a complete MeissnerOchsenfeld effect is observed: an applied magnetic field is completely expelled from the bulk of a specimen and only penetrates within a distance λL from the surface. λL is known as the London penetration depth. At the upper critical field H c2 the material returns to the normal state through a second-order phase transition. The thermodynamic critical field Hc of a type-II superconductor provides a measure of the free energy difference between the normal state (Fn ) and the superconducting state (Fs0 ) in the absence of a magnetic field, that is (1)

Hc2 = Fn − Fs0 . 8π

The quantity Hc satisfies Hc1 < Hc < Hc2 . For intermediate applied fields Hc1 < H < Hc2 , the magnetic field penetrates a sample in the form of small filaments, each carrying a quantum of magnetic flux Φ0 . This is because the formation of an interface between a normal and superconducting region becomes thermodynamically favourable at a field less than Hc when λL & ξ, where ξ is the superconducting coherence length: the destruction of superconductivity in a region ∼ ξ is associated with an increase in the energy of the system ∼ (Hc2 /8π)πξ 2`, the energy required to break up Cooper pairs. At the same time, the magnetic field penetrates a region ∼ λL . This is associated with an energy loss ∼ (H 2 /8π)πλ2L ` since work must be done to expel the field lines from the bulk of the material. Hence, the formation of normal-superconducting interfaces becomes thermodynamically favourable when λL & ξ. In the above expressions ` is the length√of the (cylindrical) region in question. An exact calculation gives κ = λL /ξ > 1/ 2, where κ is known as the Ginzburg-Landau parameter . These flux filaments, also known as vortices or vortex lines, repel each other and at equilibrium form a lattice, the symmetry and orientation of which depends on the direction of the applied magnetic field relative to the underlying crystal lattice. The vortex lattice spacing is determined from the flux quantisation condition. For triangular and square lattices, it is easy to show that the corresponding lattice parameters are ∗

See Charalambous, D., On the dynamics of vortex lines in type-II superconductors, Ph.D. Thesis, University of Birmingham, Birmingham, 2004. 1

given by (2)

a4 =

s

2 Φ √ 0 3 B

a =

r

Φ0 . B

§2. Landau theory of phase transitions In 1937 Lev Landau developed a phenomenological theory of continuous phase transitions, also known as transitions of the second-order (Landau, 1937). These phase transitions involve a change in the symmetry of the system under consideration. Although the change in the state of a system changes continuously, the change in symmetry is discontinuous: a system either possesses a symmetry property or it does not. Since the state of the system changes continuously, thermodynamic functions of state also change continuously. Their derivatives are, however, discontinuous. At the transition point, the symmetry group of the system must contain all the symmetry elements of both phases. The symmetry of one of the phases is thus higher than that of the other, i.e. the symmetry group of the less symmetrical phase is a subgroup of that of the more symmetrical phase. Many second-order phase transitions, such as the one from the normal to the superconducting state, involve a transition from a ‘high-temperature’ phase (more disordered, higher symmetry) to a low-temperature phase (ordered, lower symmetry). The central idea of the Landau theory is the representation of the thermodynamic potential Φ of the system as a function of thermodynamic variables (e.g. pressure P and temperature T ) and an order parameter η. The expression η = 0 holds in the more symmetrical phase and η takes non-zero values in the less symmetrical phase. However, η is not an independent variable, like P and T . It is determined from the condition of thermal equilibrium at a given pressure and temperature. Since the transition is continuous, η can take arbitrarily small values near the transition point. Hence, Φ can be expanded in powers of η (Landau & Lifshitz, 1980): (3)

Φ = Φ0 + αη + Aη 2 + Cη 3 + Bη 4 + . . . .

The parameters α, A, C and B are, in general, functions of P and T . The equilibrium value of η is determined by minimising Φ, i.e. from the condition ∂Φ/∂η = 0. Then α = 0 otherwise the equilibrium condition does not contain the solution η = 0, as required. In other words, when α 6= 0 the symmetry of the high-temperature phase is lowered. In the symmetrical phase, η = 0 must correspond to a minimum value of Φ, i.e. (∂Φ/∂η)η=0 = 0 and (∂ 2 Φ/∂η 2 )η=0 > 0. These conditions suggest that A ≥ 0 in the high temperature phase, i.e. for T ≥ Tc , where Tc is the critical temperature. The same conditions, when applied for T < Tc give A < 0. Since A(T > Tc ) > 0 and A(T < Tc ) < 0, then A(Tc ) = 0, with Tc = Tc (P ). The transition point itself must correspond to a stable minimum of Φ. Hence, C(Tc ) = 0. Φ must be bounded from below, i.e. | inf η {Φ}| < ∞, hence B > 0. As a result, the transition point is determined by the vanishing of A and C, subject to B being positive. If C is not identically zero, the A = C = 0 at Tc corresponds to an isolated point (Pc , Tc ). If C ≡ 0, then one obtains a line of transition points, i.e. Tc = Tc (P ). It will be assumed that C ≡ 0 unless otherwise stated. The above expression for Φ reduces to (4)

Φ = Φ0 (T, P ) + A(T, P )η 2 + B(T, P )η 4 . 2

Assuming that A(T, P ) is not singular at Tc , it can be expanded about T = Tc to give A = a(P )(T − Tc ), where a(P ) = ∂A/∂T (P ). For equilibrium, one then obtains (5)

η = η¯ =

r

a(P ) (Tc − T )1/2 , 2B

i.e. η is non-analytic at T = Tc . It is worth noting that although η is non-analytic at the transition point, Φ was nevertheless expanded in series of η about T = Tc . This places another important limitation on the Landau theory, in addition to the requirement that Tc − T  Tc : the Landau theory is not valid within a narrow region about Tc , known as the fluctuationdominated regime. Qualitatively, as the transition point is approached, minη {Φ} becomes flatter, and as a result, the ‘restoring force’ that keeps the body in equilibrium becomes very small. Therefore, both the amplitude of fluctuations and the relaxation time of the system become unbounded. It is easy to show that the mean squared amplitude of the fluctuations in η is given by (Landau & Lifshitz, 1980) h(η − η¯)2 i =

(6)

kB T . 8a(P )(T − Tc )

The above expression diverges as T → Tc . §3. Ginzburg-Landau theory of superconductivity The Ginzburg-Landau (gl) theory of superconductivity was developed in 1950 jointly by Vitali Ginzburg and Lev Landau to account for the properties of superconductors near the transition temperature (Ginzburg & Landau, 1950). The gl theory is phenomenological and based on the Landau theory of phase transitions. gl realised that the condensate wavefunction Ξ (r, t) provides the natural choice for the order parameter η. Ξ can be regarded as an effective wavefunction of the superconducting electrons. It is then possible to define another (complex) quantity ψ which has the same phase as Ξ but is normalised such that |ψ|2 = ns /2, where ns is the density of superconducting electrons, i.e. the density of Cooper pairs of mass 2m, where m is the electron mass. The symmetry group of the normal state ‘contains’ the unitary group U(1). The symmetry corresponding to the latter is spontaneously broken in the superconducting state. Hence, the contribution of the superconducting condensation energy to the free energy of the system must remain invariant under a global gauge transformation. As a result, ψ must appear in combinations of ψ ∗ ψ. Therefore, (7)

F = Fn +

Z

Ω⊂R3

n o 1 h ¯2 d3 r α|ψ|2 + β|ψ|4 + |∇ψ|2 , 2 4m

where integration is carried out over the volume of the superconductor, Ω. α and β are phenomenological parameters, corresponding to A and B of eq. (4)—they can be determined from the microscopic BCS-Gor’kov theory. Fn is the free energy of the system in the normal state. The gradient term provides an energy cost for rapid variations of ψ. Note that odd powers of ψ, such as |ψ|3 are not present since the superconducting transition occurs over a continuous line in the magnetic field-temperature phase diagram. 3

Also, such terms result in functions of the form (ψ ∗ ψ)3/2 which are multivalued and hence non-analytic at ψ = 0. Ignoring spatial variations of ψ, the equilibrium value is given by |ψ|2 = −

(8)

α0 α = (Tc − T ), β β

where α = α0 (T − Tc ) and α0 is a constant, in agreement with eq. (5). In the presence of a magnetic field, the free energy must be gauge invariant. Since the vector potential is defined with respect to some arbitrary function χ(r), F should not change when A → A + ∇χ(r). Eq. (7) is then modified to give (9)

F = Fn0 +

Z

Ω⊂R3

d3 r

2ie  2 o 1 h ¯ 2  + α|ψ|2 + β|ψ|4 + A ψ . ∇− 8π 2 4m h ¯c

n B2

Fn0 is the free energy in the normal state, in the absence of a magnetic field and B 2 /8π is the field energy density, with B being the magnetic induction. The free energy expression has been corrected to account for the fact that current carriers are Cooper pairs. It should be pointed out that eq. (9) ignores the normal state susceptibility: for superconductors with very high upper critical fields, the Clogston field limit (also known as the paramagnetic limit) may be reached for large applied fields, leading to destruction of superconductivity by Cooper pair breaking (Rose-Innes & Rhoderick, 1978). By writing ψ = |ψ|ei arg{ψ} = φ1 + iφ2 , it can be seen that the free energy is determined by both ψ and ψ ∗ . Hence, the equilibrium value of the order parameter is determined by the variational expressions (10)

δψ F = δψ ∗ F = δA F = 0.

Straightforward algebra leads to the celebrated gl equations: 1  2e 2 −i¯ h∇ − A ψ + αψ + β|ψ|2 ψ = 0, 4m c

(11)

(12)

(13)

j=

ie¯ h ∗ 2e c ∇∧∇∧A= (ψ ∇ψ − ψ∇ψ ∗ ) − |ψ|2 A, 4π 2m mc   2e ˆ · −i¯ n h∇ψ − Aψ = 0. c

ˆ is a unit vector normal to ∂Ω, the boundary of Ω. Eq. (13) implies that n ˆ · j = 0, n i.e. the component of the supercurrent density perpendicular to the surface of the body should vanish, provided the superconductor is surrounded by vacuum. In the case of a normal metal-superconductor junction, the gl boundary condition is modified to give (14)

  1 2e ˆ · −i¯ n h∇ψ − Aψ = i ψ. c b 4

b is known as the BCS-de Gennes extrapolation length and can be determined from microscopic theory (Lifshitz & Pitaevskii, 1980). It is easy to show that in the limit of uniform ψ, the gl equations reduce to the London equation B + λ2L ∇ ∧ ∇ ∧ B = 0,

(15) where (16)

λL =



mc2 β 8πe2 |α|

1/2

is the London penetration depth. In the absence of a magnetic field and for very small values of ψ such that |ψ|2 → 0, eq. (11) takes the form ∇2 ψ =

(17)

4mα ψ. h ¯2

Therefore, in addition to λL , the gl equations contain another characteristic length given by h ¯ ξ(T ) = p . 2 m|α|

(18)

It can also be shown that ξ(T ) is the correlation radius of the thermal fluctuations of the order parameter ψ. Note that for the gl theory to be valid, both λL and ξ must be much larger than the size ξ0 of a Cooper pair, i.e. λL and ξ should vary slowly in space. At Tc , ξ(T ) diverges as expected. The gl parameter κ is given by (19)

κ = λL /ξ =

√ |e| mcβ 1/2 = 2 2 Hc (T )λ2L (T ), 1/2 h ¯c (2π) |e|¯ h

and is constant within the gl theory. Ginzburg derived a criterion which determines the size of the fluctuation region near Tc (Kopnin, 2002): the gl equations can be applied provided that (20)

Tc − T 

β 2 Tc2 . α(¯ h2 /m)3

With the aid of BCS theory, this gives  k T 4 Tc − T B c ,  Tc µ

(21)

where µ is the chemical potential. For conventional superconductors, kB Tc /µ ∼ 10−3 − −10−4 . Hence, the above condition holds almost up to the critical point, in other words the fluctuation-dominated regime practically disappears. The gl equations can be used to calculate the critical fields Hc1 and Hc2 . The result is (Lifshitz & Pitaevskii, 1980) (22)

Hc1 ≈

Φ0 (ln κ + 0.08) for κ  1, 4πλ2L 5

Hc2 =

Φ0 . 2πξ 2

In the case of a semi-infinite or finite superconductor, the presence of boundaries gives rise to a third critical field, Hc3 ≈ 1.695Hc2 , when the field is applied parallel to the surface. In other words, superconductivity persists in a surface layer up to a field Hc3 . The width of this surface sheath is of the order of ξ(T ). §4. The Vortex Lattice from Ginzburg-Landau theory

In the original paper by Ginzburg √ and Landau (Ginzburg & Landau, 1950), solutions to their equations for κ > 1/ 2 were not considered interesting. In the authors’ own words: “Since from the experimental data κ  1, the solution of the equations possible for another limiting case when κ → ∞ does not offer any intrinsic interest, we shall not discuss it.” √ It turned out that solutions for κ > 1/ 2 were very interesting indeed. In 1953, Alexei Abrikosov obtained a solution to the gl equations in the vicinity of Hc2 (Abrikosov, 1957). The solution was periodic and contained points where the complex order parameter vanished and its phase changed by 2π around a contour surrounding such points. Owing to its importance in this thesis, the Abrikosov solution is briefly outlined below. For fields very close to the upper critical field Hc2 , the order parameter ψ is very small. Hence, the non-linear term |ψ|2 ψ may be neglected from the gl equations which become linear and equivalent to Schr¨ odinger’s equation for a free particle in a magnetic field. The eigenfunctions are well-known and may be expressed in the form (Ketterson & Song, 1999)    (x − x0 )2 x − x0  (0) (23) ψn,ky ,kz (x, y, z) = exp i(ky y + kz z) − H , n 2a2H aH

and the corresponding eigenvalues are given by  1 (24) n = n + h ¯ ωc , 2

where Hn is a Hermite polynomial of the nth order with the integral representation (Lebedev, 1972) (25)

2n (−i)n ex √ Hn (x) = π

2

Z∞

dt tn e−t

−∞

2

+2itx

,

n ∈ Z, n > 0.

The quantities ky and kz are constants, x0 = a2H ky , a2H = Φ0 /(2πH) is a (magnetic length)2 , H is the applied magnetic field and ωc = eH/mc. Since this is a solution of a linear partial differential equation, any linear combination of (23) also satisfies the linearised gl equations. Note that at Hc2 , n = 0. Just below Hc2 , Abrikosov assumed that the solutions to the original non-linear equations would differ only slightly from the above and wrote ψ = ψ (0) + ψ (1) and A = A(0) + A(1) to first-order. Quantities with the superscript (0) refer to H = Hc2 whereas those with superscript (1) correspond to first-order corrections. Using these ‘trial’ solutions, the non-linear gl equations can be solved self-consistently. Ignoring the z-dependence, the general solution takes the form   X (x − x` )2 , (26) ψ(x, y) = C` exp i`ky y − 2ξ 2 `∈Z

6

where x` = `ξ 2 ky . Eq. (26) is simplified considerably if it is assumed that C` = C`+N , where N ∈ Z, N > 0. The minimum energy solution corresponds to N = 2 with C1 = iC0 and represents a triangular vortex lattice. N = 1 corresponds to a square vortex lattice. The free energy is given by the expression (27)

  V (B − Hc2 )2 2 F = Fn + B − . 8π 1 + βA (2κ2 − 1)

Here, B denotes the average field, V is the volume of the superconductor and βA is a parameter introduced by Abrikosov which describes the structure of the vortex lattice. It is given by (28)

h|ψ|4 i βA ≡ , h|ψ|2 i2

4 where h· · ·i represents a volume average. For a triangular lattice, βA ≈ 1.16 whereas  βA ≈ 1.18.

References [1] Landau, L.D., On the Theory of Phase Transitions, Fiz. Z. Sowejetunion 11 (1937), p. 26. [2] Landau, L.D. & E.M. Lifshitz, Statistical Physics, Part 1, 3rd ed., Pergamon Press, Oxford, 1980. [3] Ginzburg, V.L. & L.D. Landau, On the Theory of Superconductivity, Zh. Eksp. Teor. Fiz. 20(1950), p. 1064. [4] Rose-Innes, A.C. & E.H. Rhoderick, Introduction to Superconductivity, 2nd ed., Pergamon Press, Oxford, 1978. [5] Lifshitz, E.M. & L.P. Pitaevskii, Statistical Physics, Part 2, Pergamon Press, Oxford, 1980. [6] Kopnin, N.B., Introduction to Ginzburg-Landau and Gross-Pitaevskii Theories for Superconductors and Superfluids, J. Low Temp. Phys. 129(2002), pp. 219– 262. [7] Abrikosov, A.A., On the magnetic properties of superconductors of the second kind , Zh. Eksp. Teor. Fiz. 32(1957), pp. 1442–1452 [Sov. Phys. JETP 5(1957), pp. 1174–1182] [8] Ketterson, J.B. & S.N. Song, Superconductivity, Cambridge University Press, Cambridge, 1999. [9] Lebedev, N.N., Special functions and their applications, Dover Publications Inc., New York, 1972.

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