For clean sample, in the mean field approach, for a given effective anisotropy η, the system will prefer a rhombic lattice with apex angle θ. The numerical results of transition condition (4.22) are presented in Figs.4.2. We found that for a given ηm, the magnetic field enhance the anisotropy strength η,the apex angle θ deviates from 60◦ and decrease to 45◦ continuously and situate till the magnetic field destroy the superconductivity.
The system undergo a second order phase transition.
The free energy with disorder correction 4.33 is plotted in Fig. 4.3. The system undergo a 1st order phase transition, square lattice is not stable while dimensionless temperature t increasing. When effective anisotropy is stronger, it suppresses the barrier between two angle, the phase transition became continuous. The phenomenon happens
0.010 0.015 0.020 0.025 0.030 45
50 55 60
Apex Angle (θ)
η θ
Figure 4.2: Evalution of rhombic lattice in magnetic field. In mean field approach, the square to rhomb phase transition occure at η = eηb = 0.02932. After the critical field, the vortex exhibits perfect square vortex lattice till the upper critical field.
at the vicinity of the S-N phase transition line, one notice that the square lattice is not stable while temperature increase. The SPT transition happen at ∂θ2F (θ) |45◦ = 0, where F are free energy of clean sample plus disorder correction , F = Fclean + Fdis. Thus, it is convenient to collect the all the coefficients to p (t, b) = nγ4πb√
1 − t − b , such that (∂θ2Fclean′ + p (b, t) ∂θ2Fdis′ )45◦ = 0 . The notation Fclean′ = B10 and Fclean′ is the integration in F . One found that ∂θ2Fdis′ has different sign as ∂θ2Fclean′ . The structural phase transition line with various disorder n are plotted in Fig. 4.4. Unlike the case in a clean sample where SPT line is t independent, at low temperature, disorder enhance the anisotropic effect. Near the S-N transition line where the microscopic fluctuation coupled with disorder smear out the effect of anisotropy, the system restores the rhombic lattice from square lattice. Indeed, when near the S-N phase transition, the higher order correction in free energy functional, the D4 symmetric correction, is less important then its isotropic part. However, due to the applicability of the perturbation approach, the
theoretical line failed when in strong disorder.
Figure 4.3: Energy difference between θ = 44 and other angle. The material paramter for anisotropy eη = 0.5, disorder stength n = 0.8 . Mangetic field is fix at b = .5, system undergo 1st phase tranistion while decreasing temperature: the preference angle jump from rhombic (θ˜52◦) to perfect square (θ = 45◦) .
Now we turn to the thermal fluctuations influences on structural phase transition in clean system. In fig. 4.6 we show both perturbation (one loop ) result and Gaussian variational approach. The laster approach is minimizing the free energy f (θ, n = 0, eη) with respect to variational functional which we use the mode expansion to simplified the functional equation to algebra equations. It is interesting that at low temperature two method shows the same tendency, however, while temperature increasing, two SPT line depart from each other and result in totally different curvature. In both methods, they doesn’t cross the melting line before the acoustic spectrum become negative. In spite of the different curvature, both of them show that for strong Gi the influence area increases. It is know that at the vicinity of critical the perturbation method is not valid. Moreover, for strong anisotropy case, the system undergo SPT at low field and
Figure 4.4: The phase diagram of structural phase transition. The voretx lattice struc-ture of the system with b, t at the upper plane of the diagram are square lattice, while at the lower plan are rhombic lattice. The material anisotropy eη = 0.5. System with stonger disorder depart further from the SPT line for clean sample (n = 0).
the thermal fluctuation influence is suppressed.
When disorder influence are considered via ensemble average over all possible state, we expected that the variational method will be able to give better result then pertur-bation.
4.8 Discussion
We analyze the rhombic to square vortex lattice phase transition in anisotropic super-conductors using a variant of Ginzburg-Landau theory. The mean-field phase diagram is determined to first order in the anisotropy parameter, and shows a reorientation tran-sition of the square vortex lattice with respect to the crystal lattices. introduce both
Figure 4.5: Comparison with experiment Dewhust’s experimend. The fitting parameter n = 3.7 amd eη = .5.
thermal fluctuation effect and disorder effect by both perturbation and variational ap-proach to show that thermal fluctuations and disorder produce a reentrant rhombic to square lattice transition line at the vicinity of S-N phase transition line, similar to recent studies which used a nonlocal London model. Moreover, we show that for material with small Gi the reentrant rhombic is due to the quench disorder.
An in-plane anisotropic superconductor has a potential to have vortex lattice config-uration other then hexagonal lattice. The coupling strength between vortex lattice and the underlying material properties depend no only on the material anisotropy coefficient ηm but also strongly depends on the external magnetic field. For a given anisotropy ma-terial with 4-fold symmetry, the field induced structural distortion will eventually reach a square lattice at high field (H > H2). It should be noted that at low temperature,
0.00 0.050.10 0.15 0.20 0.25 0.30 0.35 0.40 0.450.50
(thin line for Gi=10 -5
)
melting line
P
G
Figure 4.6: Structual Phase Transition with thermal fluctuations influece at eη = .1.
Result of both one loop perturbation result (P ) and variational gaussian (G) are shown.
The insert shows material with different Gi.
the structural phase transition is hardly depend on temperature. In the presence of dis-order which is introduced as a random pinning potential ,ahW (x) |ψ (x)|2, the system has irregularity of minority vortices. We average all possible configuration of disorder, namely we take into account all possible system with different disorder distribution in space, W (x) , and make ensemble average. The result shows the temperature depen-dency appear on the SPT line. The physical interpretation is that the disorder coupled with microscopic thermal fluctuations in the quadratic term smears out the field induced anisotropy. Hence, the vortices rearranged themselves according to the effective coupling with underlying crystal which now has temperature dependency. This model works suc-cessfully explain the SPT characteristic line for low temperature superconductors (with small Gi ).
0.0 0.2 0.4 0.58
0.60 0.62
Gi=10 -6
, m
=0.04
n d
0
10 -5
10 -4
10 -3 b=H/H c2
t=T/T c
Figure 4.7:
Considering the thermal motion of vortices, at low temperature, the system appears to follow the symmetry of Hamiltonian, therefore, at the vicinity of SPT line, rhombic lattice could become square lattice due to the thermal fluctuation. And indeed, with larger Gi which characterize the thermal fluctuations strength of a material, the influence area on the phase diagram is larger. However, at high temperature when it approach the melting line, the dramatic vibration of vortex smear the anisotropic effect from the symmetry of underlying structure, the repulsion force became less anisotropic and restore the rhombic lattice structure and the apex angle increase as temperature increase. The anomaly in the dynamic magnetic susceptibility and paramagnetic [20] on borocarbides crystals is associated with a structural transition in the vortex lattice and satisfied with this picture. Theoretical results proposed by Dorsey et al [24], Kogan et al [20] proposed that the appear of rhombus at the vicinity of Hc2 line is due to the thermal fluctuation smear out the anisotropic effect of underlying vortex. For borocarbides the Gi˜10−5, it
is questionable if the thermal fluctuations plays the major role.
Chapter 5 Summary
Mesoscopic vortex motions influence the thermodynamic properties of superconducting state dramatically. Reduced symmetries of the underlying material lead non-trivial and complicate phenomena. Thermodynamic phase diagram of vortex matter includes: liquid state, solid state, glass state and other subregion. The complexity mainly depends on the intrinsic material properties ( Gi, Hc2, Tc and κ) and the density of disorder of a system. In a GL model, various approaches are proposed to understand the physics. In a liquid state, one can start from a solvable “noninteracting” field theory at very high temperatures and develop a theory of liquid by resummation of Feynman diagram and other resummation technique [90] [118] [63]. In solid state, one can study the “harmonic”
solid at low temperatures and trace its destruction by fluctuations [48][113][112]. These two approaches are consequently the one phase theories. Direct numerical simulation can provides a valuable information on both the liquid and the solid side of the transition line.
In this work, by considering the thermal fluctuations and quenched disorder, we show non-trivial consequence of anisotropic effects on vortex matters. The anisotropic effect of layered superconductor ruins the high field scaling behavior in liquid phase because the effective dimensionality of the layered superconductor have crossover between 2D to 3D which can be influenced by the external field. For strongly 2D system, the thermal fluctuations excitation can go beyond lowest Landau level. We show that the higher
landau level contribution around Tc is important, it is responsible for the failure of the LLL scaling.
The microscopic theory for a system with 4-fold in-plane anisotropy is not yet estab-lished, and nor does the direct link from non-Local Landon model to 4-fold symmetric Ginzburg-Lanau model. From the phenomenological GL we adopted here, at certain point in parameter space, a structural phase transition between rhombus and square can occur. The field dependent coupling between vortex lattice and atomic crystalline lattice will be enhanced by increasing magnetic field. At high field it will saturate with square lattice before Hc2(T ). In our theory we assume the coupling ηm between vortex lattice and atomic lattice is temperature independent, for a clean low-Tc superconductor which mean field approach is valid, we show that the temperature dependence of SPT is not expected. The temperature dependency of SPT for small Gi material is not a result of mesocopic thermal fluctuations, but due to the microscopic fluctuations from quench disorder.
Because of the ability to tune the interaction forces between individual vortex can be easily done by changing applied magnetic field and temperature of the system, I believe that the physics of structure phase transition can be study in vortex lattice while lack of the ability to control the interaction between atoms in materials. Such a system is very interesting, and due to possible realization of the system it may serve as an excellent experimental tool to examine well-developed theories with experiment.
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