Ginzburg-Landau length scales

Transcription

1 597 Lecture 6. Ginzburg-Landau length scales This lecture begins to apply the G-L free energy when the fields are varying in space, but static in time hence a mechanical equilibrium). Thus, we will be minimizing the G-L functional. Among other things, the solutions may include supercurrents that are constant with time. There first outcome of these calculations, and a key idea in all further applications of G-L, are the two length scales one can construct from the G-L parameters: the penetration depth associated with the vector potential field Ar), and the coherence length, associated with the order parameter field Ψr). The physical meaning of either one is the healing lengths of its respective field the minimal spatial scale over which it can vary. In a neutral superfluid, of course, ξ is the only length scale. Conversely, for superconductors, there is a London approximation in which Ψ 0 is fixed and A and θ are the only fields, in which case λ is the only length scale. We ll also derive the London equations relating currents to the vector potential. As in all other problems where we are after a spatially can approace the spatial dependences in two ways. A simple-minded variational approach, using the special case where only one of the fields is varying, captures the essence. The accurate approach uses the calculus of variations to write the differential equations that must be satisfied by the spatially varying and coupled) Ψr) and Ar) in any global configuration that minimizes the total free energy. As there are just a few basic GL parameters recall Table 6.1.1) but many physically meaningful ways to combine them, it follows there are many not obviously) equivalent ways to rewrite any given formula. For many problems, the key step is to find the best combination. Most frequently this is a length scale. One reason is that the given geometry often has finite dimensions in some directions.) Ratio of lengths Since there are two length scales, we immediately know that G-L models are not all equivalent by some rescaling. The one nontrivial parameter is the Ginzburg-Landau ratio κ = λ/ξ 6..1) As we will see Lec. 6.3 and Lec. 6.6 ), Type I and Type II superconductors correspond respectively to κ less than or greater than 1/. Copyright c 011 Christopher L. Henley

2 598 LECTURE 6.. GINZBURG-LANDAU LENGTH SCALES and Shorthand for the gauge-invariant derivatives For future use, it is convenient to define the notations A θ θ e ) c Ar). 6..) 6. A Coherence length A Ψ ie ) c Ar) Ψr) 6..3) In any charged or neutral superfluid described by G-L theory, the coherence length ξ is defined by α = m ξ 6..4) The coherence length parametrizes, in general, the range over which superconducting order is reduced affected by a local perturbation. Clearly, a stronger superconductor has a smaller ξ, too, since the order parameter rebounds more rapidly in space to its preferred magnitude Ψ 0. From a statistical physics perspective, it turns out, ξ is simply the correlation length in the order-parameter correlation defined in Lec That is, if Ψr) thermally fluctuates, how large a region is carried along with it, typically?) In the microscopic theory of a paired BCS-type superconductor, it turns out that ξt = 0) is the typical Cooper pair radius; at higher temperatures it is larger. Variational estimate To see the physical meaning of ξ, say there is an interface at x = 0 with the boundary condition Ψx = 0) = 0. Imagine a neurral superfluid or a charged one without magnetic fields and currents.) How does Ψz) approach its bulk value Ψ 0? In the absence of currents, θ = constant so we take Ψr) real without loss of generality, The total free energy is F tot d 3 rf GL r), where The free energy density Lec. 6.1 ) reduces to F GL = α ξ Ψr) + F L Ψ) 6..5) Now make a crude variational assumption 1 the profile is linear shaped over a width l which is the variational parameter) { Ψ 0 x/l for 0 < x < l; Ψx) = 6..6) Ψ 0 for x > l. The first term in 6..6) favors l to keep the gradients small, whereas the second term favors l 0 so that F L takes its minimum value in almost the entire volume. Substituting 6..5) into 6..6), and subtracting off the bulk free energy density F cond, we get a free energy per surface area A of α ξ F tot /A = l Ψ αψ ) 10 βψ4 0 ) + F cond l 6..7) 1 One similarly estimates the width of any wall or interface, e.g. a discommensuration wall in the Frenkel-Kontorova model, Lec. 3.3, or a domain wall in a ferromagnet, Lec. 5.3 Z?).

3 6. B. LONDON LIMIT AND PHASE ANGLE FIELD 599 But αψ 0 F cond and βψ 4 0 F cond so it simplifies to F tot ξ = F cond + 8 ) A l 15 l 6..8) Taking the minimum with respect to l gives l = 15/)ξ. Substituting back gives an interface energy per unit area F tot A ξ F cond 6..9) That makes sense: we lose the condensation energy in the volume where Ψ deviates much from Ψ 0, which happens in a layer of width ξ. MOVE Thus ξ represents the healing length over which Ψ returns to the bulk value from some disturbed value. Nonlinear Schrödinger equation The second approach for the interface profile and energy is to solve it exactly. Set the variation of the total LG free energy to zero, to obtain the appropriate Euler-Lagrange) equation from 6..5). The gradient-squared term, as always, must be integrated by parts). For the neutral case, you get δf tot δψr) Ψ + α + β Ψr) ) Ψr) = ) m Eq ) is called the nonlinear Schrödinger equation because it looks exactly a Schrödinger equation with a self-consistent potential v eff r) δf L /δψ = α+β Ψr). For a charged superfluid, replace A ). Eq ) is also the special case of the time-dependent GL equation see Sec. 6.1 B ) when dψr)/dt 0. Asymptotic exponential decay At the minimum of F L, Ψ = Ψ 0 = α /β, where α = α. We can Taylor expand around Ψ 0 to obtain α + βψ )Ψ α δψ where δψ Ψ Ψ 0. When we substitute this back into 6..10), and divide through by the prefactor of the gradient term, we obtain + ξ )δψr) = ) Thus, if x is the distance into the normal domain from the boundary of a superconducting region, asympotically the distortion decays as δψx) e x/ξ 6..1) See Fig. 6..1a).) 6. B London limit and phase angle field The London limit is when the order parameter magnitude Ψx) takes its bulk value nearly everywhere, so that Ψr) Ψ 0 e iθr) 6..13) This is true, for example, away from boundaries or vortices in a sample, that would force Ψr) to zero somewhere; it is also a very good approximation for extreme Type

4 600 LECTURE 6.. GINZBURG-LANDAU LENGTH SCALES a). b). Ψ x) ~e x/ξ B x) z J yx) Ψ 0 B z = H c ~e x/ λ x Figure 6..1: Coherence length ξ and penetration depth λ as scales for decay to the bulk values. a). With a boundary condition Ψ0) = 0, the order parameter rises to its bulk magnitude with a healing length equal to ξ/. b). Meissner effect: in the London approximation n s equal to bulk value), the magnetic field Bx) and screening currents J sx) have a decay length equal to λ. II superconductors such as high-t c cuprates, in which the healing length ξ is so short that Ψ rebounds to its bulk value only a short distance away from any such defect or wall. The only remaining freedom of the order parameter is the phase field θr). The London limit was first considered 1930s) before the order parameter was imagined. Indeed, all results in the London limit may be massaged so as to replace all reference to Ψ 0 and θr) by the superfluid density n s and velocity v s r), respectively; these look deceptively) similar to the variables of elementary Drude transport theory. The first substitution is obvious, Ψ 0 n s ) For the second one, recall the formula for supercurrent, 6.1.1). By simply substituting Ψr) = n s r)e iθr), one obtains J s = n s e v s n s e m A θ 6..15) The gradient part of the free energy density becomes F grad = m n s A θ ; London) 6..16) with 6..15), this takes on exactly the form of a kinetic energy of the bosons, The free energy density is Currents from G-L: London equations F grad = 1 n sm v s r) ) F GL = F cond + F grad + 1 8π Br) 6..18) We are looking for static minima with J s 0) by setting the variational derivatives of d 3 rf GL with respect to A equal to zero. Euler-Lagrange equations. ) The F grad term is done using the chain rule since A depends directly on A, thus v s µ r)/ A ν r) = e /m cδ µν. The second term in F GL requires an integration by

5 6. C. PENETRATION DEPTH 601 parts, which gives an extra minus sign since the the order of a cross-product gets reversed in the process. We get δf tot δar) = n se v s 1 A) = ) c 8π The first term in 6..19) is J s /c. Two of Maxwell s equations say that A B and B = 4π c J. Substituting all this, 6..19) simply says J = J s static equilibrium) 6..0) Eq. 6..0) is the Second) London equation. It says we can identify the supercurrent with the total current. Eq. 6..0) expresses a simple-minded but useful picture: there is a superfluid with a superfluid density given by 6..0) a number density); the superconducting electrical response will turn out to be proportional to n s. More generally we could write J = J s + J n, where J n represents a normal fluid that may flow independently of the superfluid. In reality it represents the gas of thermally excited elementary excitations in the superfluid at T > 0: see Lec. 7.5.) However the content of 6..0) is that J n can be nonzero only at AC, not at DC. Current conservation We did variational minimizations of the free energy with respect to the vector potential field and the magnitude part of the order parameter field; what remains is the phase angle field. Starting from 6..16), this gives: δf tot δθr) = δ δθr) [ d 3 rf θ = θ e ] A c Ψ = J s = ) using 6..15). This just expresses supercurrent conservation. This is not exactly a physical result; rather, it reveals that F grad was set up so as to guarantee that J s would be the current and satisfy 6..1).) 6. C Penetration depth Now I turn to the other length scale, still staying in the London limit. NOTE sorry, this is first use of London Limit. Taking the curl ) of 6..15) and 6..) the curl of θ gives zero we get J s = n se m c B 6..) Notice how the factors have canceled! This is a different relation between current and magnetic field than Ampère s law B = 4π c J 6..3) Next apply another curl to both sides of 6..) and using Ampère s law with J = J s, we get J s n s e ) 4π m c J s 6..4)

6 60 LECTURE 6.. GINZBURG-LANDAU LENGTH SCALES By a vector calculus identity )J s r) = J s ) J s and using the conservation of supercurrent 6..1) we get where λ is the penetration depth Note that λt ) depends on T since n s T ) does.) Meissner effect λ )J s = ) λ = m c 4πn s e 6..6) By taking yet another curl of 6..5) and using Faraday s law [?]) we also get λ )B = ) within the superconductor. Now consider the behavior of magnetic field near the surface of a superconducting sample, parametrized as a function of x with a constant magnetic field B 0 at x < 0 and superconductor at x > 0. The magnetic field vanishes deep inside the superconductor, and magnetic field lines cannot terminate, so B 0 must be oriented tangent to the surface; thus Br) = Bx)ẑ without loss of generality. Correspondingly Ar) = Ax)ˆx and Bx) dax)/dx. The decay of the magnetic field and the screening supercurrent are illustrated in Fig Then 6..7) tells us the fields don t decay instantaneously at the surface. How could they? the screening currents required would be strictly on the surface corresponding to infinite current density). Instead, 6..5) and 6..7 imply J s x) e x/λ 6..8) Bx) e x/λ 6..9) where x is the distance from the boundary. It turns out λ is less than a micron, so 6..8) and 6..9) explain the Meissner effect. All magnetic fields decay to zero within a penetration depth λ. The flux-excluded state is associated with permanent and, necessarily, dissipationless) surface currents that screen the fields. As a corollary, all currents also are confined to this thin layer near the boundary of the superconductor. If there were any currents deep in the bulk, they d make a field, which is a contradiction.) Clearly, a smaller value of λ indicates a stronger superconductor it has a larger superconducting density and is screens out fields more rapidly as a function of distance). A variational estimate can also be made for λ see Ex. 6..1) as was done for ξ in Sec.??, above).