Critical fields and intermediate state

Transcription

1 621 Lecture 6.3 Critical fields and intermediate state Magnetic fields exert pressure on a superconductor: expelling the flux from the sample makes it denser in the space outside, which costs magnetic field energy. At some point all, or part, of the superconducting sample will go to the normal phase. This lecture is intended to show some of the ways this may happen. I m sorry that I only have time to write an outline. The key points here are (i) the critical field, H c, defining the bulk coexistence of normal (N) and superconducting (S) phases; (ii) the N-S domain wall profile and surface energy σ W, estimated below; and (iii) the part N and part S configuration which is often the equilibrium state of a sample with a generic shape and orientation to the field. The intermediate state is a particular case of (iii) with a periodically repeating pattern of alternating N and S domains. Coexistence is possible, of course, only at a first-order transition. So it happens only with Type I superconductors along the line H = H c (T ). N-S domain walls Now we consider what happens if you have a homogeneous superconducting sample, but part of it is in the normal state with the magnetic field exactly at the critical value, so that the bulk free energy on both sides is the same. Of course, no kind of wall could ever be stable if the free energies were different on the two sides. We can have a wall in equilibrium only if the N and S phases are coexisting. That is, when one includes the magnetic pressure term due to the uniform field Happ present on the N side of the wall they have the same bulk free energies. This only happens when Happ = H c. The entire behavior of the wall depends on the dimensionless ratio κ λ/ξ defined in Lec It turns out that when κ > 1/ 2 then σ W < 0. The case σ W < 0 is called type II. It means that the system wants to make as much wall as possible! If a region with field adjoins the superconducting part, the flux will dissolve into the superconductor and spread itself as uniformly as possible. This is a thermodynamic phase and (rather misleadingly) it s called the mixed state ; it will be studied in detail in Lec. 6.6 Our concern in this lecture is with the type I case, σ W > 0. In many situations, the minimum energy is for the field to penetrate partially, but this competes with the wall energy cost. Thus the system is a coexisting mixture of N and S regions. This is called Copyright c 2011 Christopher L. Henley

2 622 LECTURE 6.3. CRITICAL FIELDS AND INTERMEDIATE STATE the intermediate state. (a) normal normal B 0 super conducting super conducting X X+ X B B Figure 6.3.1: Magnetic field pressure and critical field. The scale in these figures is macroscopic so the wall thickness is invisible. Here and later in this Lec., lines of flux do not Arrowed lines represent the (continuous) magnetic field strength. (a) Wall (heavy line) between coexisting normal (with field B 0) and superconducting domains. To estimate the field pressure, we assume the former domain has a (large) width X and imagine a virtual displacement by X. (b) Impossible configuration in which a cube-shaped sample, placed in a magnetic field less than H c, would be superconducting throughout its volume; the field strength diverges next to the corners. (c) Correct configuration with partial field penetration; everywhere along the domain wall, the field is tangent and its strength is exactly H c. Critical field The thermodynamic critical field is defined by (6.3.4) (below). In a thin needle or slab shaped sample, aligned along the field, that s exactly the applied field at which the entire sample will go normal. But things are more complicated for general shapes. A cube, for example, would go normal around its edges while its bulk would stay superconducting. The thing to remember is that the boundary condition is always that the field lines immediately adjacent to the N-S interface are just like the needle/slab cases. That is, the field runs tangential to the interface, and its magnitude is exactly equal to H c. This boundary condition, in principle, suffices to solve the any field configuration problem, as long as it s a type I superconductor. 6.3 A Critical field Free energy in presence of applied field I want to argue that the effective magnetic energy density (assigned to the N region) is B 2 /8π. This looks like it has the wrong sign! By placing our sample in a uniform field we constrain a certain fixed flux to pass through the volume surrounding the S domain. Then expanding the Normal region will decrease the total magnetic energy, by an amount B 2 /8π per unit volume. Here is the general argument for what we actually want to minimize, for a superconductor placed in a uniform field H app created by sources far away. That implies we have a somewhat funny boundary condition on B(x). The bottom line is that the

3 6.3 B. NORMAL-SUPERCONDUCTOR DOMAIN WALL ENERGY 623 effective magnetic energy is U mag = d 4 r 1 8π [B(r) H app] 2 (6.3.1) Physically this says that the lowest energy thing to do is to have a uniform field; any deviations from it cost an energy. This will be used extensively in Lec As a simple case, imagine a geometry where all fields depend only on x and x > 0 is superconducting and excludes all field. So say B 0 outside the sample is confined to X < x < 0, where the boundary X is very far away. Since + B(x)dx is fixed, the field in the region ( X, 0) is reduced B 0 = B 0 ( B(x)dx)/X. The corresponding 0 total energy is changed by 1 8π (B 2 0 B0 ) 2 XA = 1 4π + In other words, we modify the magnetic energy density 0 dxb 0 B(x) (6.3.2) U mag 1 4π B 0 B(r) (6.3.3) and now we only integrate it over the superconducting volume (x, + ). 1 The critical field Consider a normal region coexisting with a superconducting one. For the N and S phases to have the same free energy, as they are when H = H c. Then (as already asserted in Lec. 6.1 ) H 2 c /8π = F cond (6.3.4) and F cond is the condensation energy, Typical values are H c = 0.1 Tesla (or less) for the strongest ordinary superconductors (Nb, Pb, Hg); for high-t c s H c is up to 1 Tesla. In the type II case (including the high- T c s), (6.3.4) defines a thermodynamic critical field. This it is not a critical field in the real material since flux starts penetrating at H c1 and superconductivity persists to H c2, where H c1 < H c < H c2. Rather, H c is just a convenient way to convert the condensation energy into magnetic field units. 6.3 B Normal-superconductor domain wall energy The domain wall (free) energy is defined as the extra free energy cost of imposing a domain wall, over that of the state without a domain wall. (As long as the two bulk phases have the same free energy, the system s total free energy will be unaffected by translations of the wall.) The aim of this section is to compute the wall free energy σ W, such that the difference in free energy between these states is F = σ W A W (6.3.5) 1 This derivation is, of course, equivalent to much more general formulas about the interaction of an applied field with the magnetization in a medium. I have deliberately avoided the use of macroscopic electromagnetism, in order to remind the reader that these formulas can be derived directly from the Ginzburg-Landau free energy.

4 624 LECTURE 6.3. CRITICAL FIELDS AND INTERMEDIATE STATE where A W is the total wall area. Note that σ W is not even defined, except exactly at the coexistence of S and N phases i.e. at H = H c. In Lec. 6.2, we imagined the deviations of the order parameter from its bulk value Ψ 0 and of the magnetic field from its bulk value 0, around the surface of a superconductor. We also found the associated surface energy costs were O(ξ F cond ) and O(λ F cond ). As we shall calculate below, the N-S wall free energy (per unit area) looks like a sum of those two results: σ W F cond [O(ξ) O(λ)] (6.3.6) The analytic theory can be set up, similar to calculations we did for the Frenkel- Kontorova model, but more complicated since we now have two spatially varying fields and two length scales. (a). (b). Type I: κ λ << 1 ξ 2 Type II: κ λ ξ >> 1 2 H c B(x) ξ Ψ(x) Ψ 0 λ x H B(x) Ψ(x) c Ψ0 ξ λ x Figure 6.3.2: Profile of order parameter field Ψ(x) and magnetic field strength B(x) across a domain wall; the magnetic field is B(x)ẑ. (a). Type II case, κ = λ/ξ 1/ 2. (b). Type I case, κ 1/ 2; in this case the configuration shown is physically incorrect, since a spatially modulated S state (the Abrikosov vortex lattice) has a lower free energy. Set-up We imagine a geometry where the field runs in the z direction and the wall is normal to the x direction. Thus the fields, currents, and order parameter are only functions of x, and furthermore by symmetry the field can only point in the z direction. That means the currents (and, in London gauge, the vector potential) must point in the y direction: B B(x)ẑ = da(x) dx ẑ (6.3.7) A A(x)ŷ (6.3.8) Ψ(r) Ψ(x) (6.3.9) where Ψ(x) is real (in this geometry, and in the gauge adopted.) Take the magnetic free energy density derived in Sec. 6.3 A specializing to the current set-up with Happ = H c. Then F mag = 1 8π (B(x) H c) 2. Thus the total free energy density is [ F (x) = 1 ( d ) 2 8π (B z(x) H c ) 2 + Ψ(x)] dx A(x) + F L (Ψ(x)) (6.3.10) Deep in the bulk on the N side, the three terms of (6.3.10) gives F (x ) = since (respectively) B z ( ) = H c ; Ψ( ) = 0; Ψ( ) = 0 (6.3.11)

5 6.3 B. NORMAL-SUPERCONDUCTOR DOMAIN WALL ENERGY 625 On the S side it gives F (x + ) = F cond + 0 F cond since (respectively) Remarks: shape of the wall B z = 0; J s (x + ) = 0; Ψ(x + ) = Ψ 0. (6.3.12) The wall is closely analogous to the Frenkel-Kontorova domain wall done in Lec. 3.3 [Undone 2003] in that case the two domains differed simply by shifting the overlayer by one lattice constant of the substrate. It is also analogous to the Bloch wall between two domains of a ferromagnet that has easy axes (due to anisotropy), in Lec. 5.3 Z(?)(omitted). Also in Ashcroft and Mermin, p In both of those earlier examples, there was a symmetry relating the phases on the two sides. But here, the two phases are different; the gas-liquid interface would be a closer analogy. Furthermore, in both of those earlier examples, there was just one order parameter field changing across the wall (the displacement field in the Frenkel-Kontorova case, and the magnetization vector in the ferromagnet case.) It turned out that you could rescale the length units and energy units so as to get a single universal shape of wall. Its thickness was given by a single length scale l, determined by balancing the gradient energy and the locking or anisotropy energy favoring the discrete values of the order parameter field which occur in the domains. But here, we have two length scales the penetration depth λ and the coherence length ξ. Consequently the shape of the wall (indeed, even the sign of the wall energy!) depends on the ratio κ λ/ξ. In the Frenkel-Kontorova model (Lec. 3.3 ), the displacement field of a domain wall has a tail decaying exponentially with a scale l W ; we ve already seen (Lec. 6.2 ) that the analogous length scales in superconductivity are λ and ξ. In the F-K model that same l W also governed the width of a wall; it s a reasonable and correct guess that this carries over to superconductivity. Within the wall, the regions where B(x) and Ψ(x) are changing have respective widths λ and ξ. Results I omit the algebra of the exact solution. Along the lines of Lec. 3.3 and the associated exercises for the Frenkel-Kontorova domain wall, you can find the Euler-Lagrange variational equations and then find a constant of motion (with x playing the role of a time variable). 2 Here, I ll content myself with making estimates, assertions, and pictures (see Fig ). Now, let s estimate the three terms in (6.3.10). They change from the values in (6.3.11) to those in (6.3.12) at different places, because B and Ψ change at different places. The first and thrid terms give F mag F cond for x > λ; F grad F cond for x > ξ (6.3.13) The second term in (6.3.10) is zero, except where the others are changing; in that case, as we know to expect in balancing arguments, it turns out to be proportional to the other two terms in (6.3.13). So, doing the integral of (6.3.13) gives σ W F cond [O(1)ξ O(1)λ] (6.3.14) 2 This is worked through, for example, in de Gennes book, or in Geballe and White Long Range Order in Solids.

6 626 LECTURE 6.3. CRITICAL FIELDS AND INTERMEDIATE STATE In other words, for λ sufficiently large, the domain wall has negative energy! More precisely, 1.89ξ F cond for κ 1/ 2 (Type I limit); σ W = 0 for κ = 1/ 2; 1.10λ F cond for κ 1/ 2 (Type II limit). (6.3.15) 6.3 C Intermediate state Consider a flat thin plate of diameter L oriented normal to the applied field H (much less than H c ). As we make the plate bigger, and make the field bigger, the field exclusion cost grows as H 2 L 3 while the cost of making normal regions grows only as L 2. What happens is that the system develops alternating stripes of N and S phases. (See Fig ) The N parts, of course, must have field exactly H c ; from the total flux we can figure out the area fraction of N domains. There is a tradeoff of wall energy and of field energy which determines the peroiodicity of this stripe array. It is closely analogous to the magnetic domain theory the field energy is mathematically equivalent to the dipole energy of interaction between the magnetizations of the N and S stripes. Dipole energy is minimized when the flux is uniform at coarse grained scales, i.e. a short wavelength of the stripe pattern, but wall energy prefers a long wavelength. The intermediate state is discussed by Tinkham, Section 3-4. It is also possible for a current distribution to favor the intermediate state. For certain geometries and current values, there is no self-consistent solution if you assume a uniformly superconducting or a uniformly normal sample. See Tinkham, section 3-5. B L S L N d S N S N S N S Z X Figure 6.3.3: A plate showing intermediate state. The scales X, Z, and the width of the normal (N) and superconducting (S) domains l N and l S, are the set-up for an variational estimation. 6.3 X Intermediate state geometry as visualized in magneto-optical imaging Since the mid-20th century, it has been possible to image the intermediate state, using the magneto-optic Kerr effect (the polarization of reflected light is rotated by the metal s magnetization within the skin depth). In practice, the domain walls have a complicated statistical geometry, so that hysteresis is possible even in the absence of

7 6.3 X. INTERMEDIATE STATE GEOMETRY AS VISUALIZED IN MAGNETO-OPTICAL IMAGING627 non-superconducting inclusions (that would pin the normal-state domains). Specifically, when magnetic field is entering the sample as in Fig (a), the domains organize into tubes of normal phase, whose cross-section forms a froth, much as soap films do; only when field is leaving the sample, as in Fig (b) and (c) do the domains form layers (laminae) as imagined in theoretical models such as Fig Figure 6.3.4: Intermediate state in a thick sample. These are taken using magneto-optical imaging. Sample is Pb with diameter 5mm and thickness 1mm; superconducting regions are dark. (a). Increasing field: domain pattern is a froth of tubes (seen end-on) (b). Decreasing field: domains are laminae, as presented in the theory (c). Same as (b), but smaller field, leading to a shorter length scale. This maze pattern is similar to those emerging in magnetic films or in many other situations where long-range repulsions frustrate phase separation. [From R. Prozorov, Phys. Rev. Lett. 98, (2007).