國
立
交
通
大
學
電子物理學系
博
士
論
文
異向性超導體之渦旋物質的熱力學性質
-Ginzburg - Landau 理論
Ginzburg-Landau theory—
The thermodynamic properties of the anisotropic superconductors
研 究 生：林佩真
指導教授：儒森斯坦
教授
異向性超導體之渦旋物質的熱力學性質
-Ginzburg - Landau 理論
Ginzburg-Landau theory—
The thermodynamic properties of the anisotropic superconductors
研 究 生：林佩真
Student：Pei-Jen Lin
指導教授：儒森斯坦
Advisor：Baruch Rosenstein
國 立 交 通 大 學
電子物理 學 系
博 士 論 文
A ThesisSubmitted to Department of Electrophysics College of Science
National Chiao Tung University in partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
in Electrophysics
June 2008
Hsinchu, Taiwan, Republic of China
異向性超導體之渦旋物質的熱力學性質
---Ginzburg - Landau 理論
學生：林佩真
指導教授
：儒森斯坦
國立交通大學電子物理學系博士班
中文摘要
本論文主旨要探討在熱微擾下二類超導體在大磁場下的物理性質，以增廣超導應用上的知識。主 要研究的材料是針對高溫超導和新的非傳統超導，此類材料在空間上的異向性很強，強烈的影響 到費米面的不對偁性，使得材料在超導態的物理性質與傳統超導體有很大的不同。在磁場下超導 所形成的奈米級的『渦旋物質』，它們可以大小彼此間的影響力是可以由調變溫度和磁場所控 制，此外它們的活動深深影響在臨界區的物理現象。超導態的相變屬於二類相變的範疇, 因此我 們採用 Ginzburg – Landau 理論作為基本模型，根據不同的系統加以變化，並以統計力學的技巧考 慮高溫效應和雜質的影響在一項性強的材料下的物理性質。 材料在空間異相性可以分為ｘｙ平面上的異相性和ｚ軸上層狀結構。在ｘｙ平面上的異相性我們 探討了四方長柱形底材的渦旋物質之結構相變。根據我們的計算發現可以藉由調變磁場強度控制 渦旋物質結構使其產生由菱形到矩形的相變。在此研究中我們分別考慮了純系統和無序系統下對 此結構相變的影響，結果顯示雜質影響下原本為第二類相變結構相變成為第一類相變，同時雜質 也影響了結構相變曲線的溫度相依性等其他現象。 在研究ｚ軸上層狀結構我們探討此異性相對超導反磁性的影響。所採用的模型有Laerence-Doniach 模型和準二維的 Ginzburg – Landau 模型。一般勻稱的材料在 Hc2(T)附近的強磁場區裡
物理量在不同磁場下的溫度特性曲線會有最低 Landau 能級的 scaling 行為,我們發現底材層狀結 構和在準二維系統強大的熱擾動皆會破壞這個 scaling 行為,尤其在臨界曲線附近一般常用的最 低 Landau 能級的近似法無法使用描述具有強熱效應的準二維系統。我們將理論的結果與很多實 驗比照得到很好的應證。
Ginzburg-Landau theory—The thermodynamic properties of the anisotropic
superconductors
Student：Lin, Pei-Jen
Advisors：
Prof. Rosenstein, BDepartment of Electrophysics
National Chiao Tung University
ABSTRACT
Vortex matter plays an important role in superconductivity state especially for high-Tc cuprate
superconductors with layered structure. For an homogeneous system, the isotropic repelled vortices
form the Abrikosov lattice. It was first proposed by Abrikosov for temperatures close to Tc, and then it
was extended to all temperatures [56]. These theories ignore the fluctuations of the order parameter, called mean-filed theory. It is a very good approximation for the conventional superconductors. However, under fluctuations influence at high temperature, the motions of vortices are responsible for the thermodynamic properties and transport properties of systems especially for those unconventional superconductors due to it's strong anisotropic magnetic properties.
The most efficient way to study the mesoscopic phenomena is the effective Ginzburg-Landau (GL) functional. Unfortunately, because of the nonlinear term, even having the effective functional one can hardly calculate the free energy exactly. It is a typical problem for the critical phenomena. Lowest Landau level approximation (LLL) is a common way to simplify the question and its practical region is
valid all the way down to H = Hc2(T)/13 [89]. The LLL degeneracy results in a high field scaling was
observed in many experiments. For various physical quantities (static quantities such as magnetization curves, specific heat etc. and dynamic quantities such as electrical conductivity etc.) as function of temperature will collapse to a scaling function for various fields. However, recent experiments show the failure of high field scaling behavior and the GL model is under examination. To understand them, in this thesis, we consider two cases in liquid phase: For strongly quasi 2D system, higher landau levels contribution is taken into account. For the layered superconductors, the coupling between layers which changes the dimensionality of the system is considered. Our results show a good agreement with several experiments.
In the second part of this dissertation, the structural transition of vortex solid state is discussed. For a 4-fold symmetric system, it transpires that the coupling between magnetic flux and underlying crystal lattice influences the vortex lattice configuration in a complicate way. The distortion of vortex lattice from hexagonal lattice to square lattice depends on an external magnetic field which enhances the effect of anisotropy and temperature. At a sufficient high field, the configuration is a perfect square lattice.
Temperature dependency of the structural phase transition is under debate. Experiments shows that even for small Gi materials, the structural phase transition has a strong temperature dependency which is inconsistent with theoretical prediction base on the thermal fluctuations influences. It is found that quenched disorder is responsible for the departure from the mean field result in clean sample. Thermal fluctuations merely smear the anisotropy effect at the vicinity of its melting line.
Acknowledgements
I wish to express my deepest gratitude to Professor B. Rosenstein for his guidance and support throughout my research. I appreciate the scientific collaboration with him.
My special thanks to Professor D. Li for his help on some parts of my work.
I would like also to thank Dr. G. Bel for his fruitful, guidance and friendship through-out my research.
Contents
Approval ii Abstract iii Acknowledgements vi Contents vii 1 Introduction 1 1.1 Superconductivity . . . . . . 5 1.2 Spontaneous Symmetry Breaking and Phase Transitions . . . 10
1.3 Thermal fluctuations in anisotropic/ disordered superconductors . . . 12
2 Mesoscopic description of superconductor in magnetic field 17
2.1 Landau Expansion of the Free Energy in homogeneous Superconductors . . . 18
2.2 Landau Expansion of the Free Energy in Anisotropic Superconductors . . . 20
2.3 Applicability of Lowest Landau level approximation within the mean field
approach . . . 24 2.4 Optimized perturbation approach and the Bore-Pad´e approximation. . . .. . . 26 2.5 Disorder . . . 30
3 Magnetization in Layered Superconductors 33
3.1 Introduction. . . 33
3.2 Basic equations and assumptions. . . 35
3.3 Dimensionality effects of layered superconductors. . . 40
3.4 Sum of contributions of all Landau levels in quasi-2D systems. The 1 OPA method. . . 42
3.5 All Landau level contributions in quasi-2D systems. An effective LLL model . . . 45
3.6 Results and Comparisons with Experiments . . . 48
3.7 Discussion . . . 52
4 Structural Phase transition in Fourfold Symmetric Superconductors 54 4.1 Introduction . . . 54
4.2 Basic equations and symmetries . . . . 56
4.3 Structure of the vortex lattice in the clean system and its phase transition . . . 57
4.4 Effects of weak disorder on structural phase transition . . . . 60
4.5 Spectrum of rhombic lattice in a clean system . . . 63
4.6 Hartree-Fock-like approximation and Replica trick in disordered system . . . 66
4.7Results . . . 71 4.8 Discussion . . . 74 5 Summary 79 Bibliography 81 viii
Chapter 1
Introduction
A two steps transition from the superconducting to the normal state with two critical fields was first observed in the magnetization of superconducting alloys by De Haas and Casimir-Jonker back in 1935 [2]. It was originally “explained away” by inhomogeneity of their samples. Correct explanation proposed on 1957 when Abrikosov predicted theoret-ically the existence of type II superconductors that magnetic field can penetrate into a superconducting material [4]. Each “individual” magnetic flux is quantized by surround-ing supercurrent thus to form a vortex. Abrikosov also suggested a periodic magnetic field distribution, transverse to the applied field, occurs near the critical region of super-conducting -normal transition. Due to this structure, the mixed state is sometimes called “the vortex lattice phase”. The existence of the vortex lattice was not widely accepted at that time. In 1964, at the suggestion of De Gennes & Matricon, Cribier, Jacrot, Rao and Farnoux performed the first experiment in which the triangular vortex configuration in a single crystal was observed by neutron scattering [11]. At different magnetic field H and temperature T , multitude of vortex lattice configurations were observed in N b which has a simple cubic underlying atomic crystalline structure. All the vortex lattices were locked in a certain crystal direction [13]. Theoretically, the energy difference between various lattices is very small. For example, it amounts to just 2% between the square and the tri-angular lattices. In real substances, the crystalline symmetry can make the square lattice more favorable. In 1966, Neumann and Tewordt proposed to incorporate nonlocal effects
by including higher order derivatives terms in the general GL model which reflect the material anisotropies. Since in original GL or London model, to incorporate anisotropy terms is via local effective mass tensor, m. And N b has cubic crystalline atomic struc-ture whose m is isotropic, thus the nonlocal effect is important [12]. In the new class of heavy fermion and cuprate superconductors, one would expect even richer behaviors of vortex configuration as these materials exhibit highly anisotropic electronic structures and order parameters with unconventional symmetries involving nodes in superconduct-ing gap. Microscopically, the Fermi surface anisotropy, anisotropy of the effective mass, and asymmetry in the superconducting gap remove the degeneracy of vortex lattice ori-entation with respect to atomic lattice and also other lattice configurations will appear. Renewed interest in the vortex configurations is due to experimental observations of the rhombic-to-square lattice transition in high-κ materials in s-wave superconductors, such as Boracabides (RE = Er, Y, Lu) N i2B2C [14], V3Si [15] and N b [13], and in d-wave
high Tc superconductors, such as La2−xSrxCuO4 [16] and Y Ba2Cu3O7[18] , and in
re-cently discovered d-wave heavy fermion superconductor such as CeCoIn5[19]. Roughly
speaking, for a system with a four-fold symmetry, magnetic field will enhance the cou-pling with underlying crystal, therefore, square vortex lattice appears at higher magnetic fields, compare with the magnetic fields where triangular lattices were observed. The current puzzling issue is whether the temperature dependence of the structural phase transition (SPT) line is a consequence of thermal fluctuations of vortex lattice or not. This contrasts with the situation in low-Tc superconductors, such as Boracabides etc.
in which the thermal fluctuation influence is negligible. In the present thesis, I studied how both the thermal fluctuation and the quenched disorder influence the structural phase transitions in the vortex lattice based on the extension of the GL model for 4-fold symmetric system[22][26]. Unlike the existing theoretical interpretations[23][24] [20], we proposed that quenched disorder dominates the temperature dependence of SPT tran-sition. Near the mean field SPT line, while having a square lattice structure at low temperatures, rhombic lattice is restored at a higher temperature region in the vicinity of Hc2(T )[21][25]. In addition to the in-plane anisotropy which results in the
struc-tural phase transition in vortex solid state, the anisotropy in c-direction dramatically enhances the thermal fluctuations in the presence of magnetic field at high temperatures. It is typical for high-Tc layered superconductors that thermal fluctuations change the
morphology of vortex matter. The intrinsic properties of the high-Tc superconductor
which enhanced fluctuations of order parameter are (a) the high-κ , the ratio between coherence length and magnetic penetration depth, and (b) the large effective distance (or weak coupling) between superconducting layers and (c) high critical temperature Tc
and (d) high critical field Hc2. A widely accepted way to characterize the thermal
fluctu-ations influence of a material is using a dimensionless quantity Gi (see section 1.3). Due to the relatively large Gi of high-Tc layered superconductor (compare to conventional
superconductor), an additional critical scaling behavior which is very different from the ordinary universality of critical phenomena arise from thermal fluctuations. In 1991, the magnetization and resistivity in Y Ba2Cu3O7 by Welp et al [28] reported scaling
behaviors in the variable of (T − Tc(H)) / (T H)2/3 around the critical temperature in
the vicinity of the Hc2(T ) . The temperature dependent physical quantities of various
magnetic fields collapse into a single scaling function (see Fig. 3.4). The scaling func-tion is universal for various anisotropy 3D materials. Li and Suenaga noticed that the magnetization of highly anisotropic Bi2Sr2Ca2Cu3O10 crystals can be described by the
2D version of the scaling function in the variable (T − Tc(H)) / (T H)1/2 [29]. Due to the
scaling behavior, there exist a crossing point (T∗, H∗) where T dependent physical
quan-tity curves, O (T, H) , of various H (or T ) interact at the point for a given dimension. The fixed point characterize the dimensionality of the system. The scaling called Lowest Landau level (LLL) scaling is due to the situation that the fluctuations near Hc2(T )
can be represented in terms of the GL field theory on a degenerate manifold spanned by the LLL for Cooper pairs. The LLL scaling is formally valid in a wide range of the H − T parameters space (see section 2.3 for details). Therefore, LLL approximation is a general adopted approximation to simplified the nonlinear GL theory[48, 89, 83]. However, when the interaction term in the GL theory is larger then the cyclotron gap of Cooper pairs and the fluctuations from excited Landau levels become significant. For
layered materials such as LaSCO [32], HgBCCO[31] and underdoped Y BCO[86], while the coherence length ξc(T ) is comparable to interlayer spacing d at certain temperature
near Hc2(T ), the change of dimensionality of the system results in the breaking of the
general LLL scaling. Namely, the crossing point of the magnetization curves can ”moves” from its 2D to 3D position while approaching Hc2(T ) from the superconducting phase.
In 2000, underdoped LaSCO experiment by Huh shows the motion of crossing points in the opposite directions to that predicted theoretically and eventually it exceed Tc
[32]. We mathematically define the intersection point and study its motion base on a layered model, the LD model. It is shown that the intersection point always occurs below Tc [36]. And the theory is in agreement with other recent experiments on layered
superconductors on HgBCCO[31] and LaSCO [32]. Recently experiments on strongly anisotropic quasi-2D material BSCCO done by Ong et al. on 2005 provided a “new” behavior of High-Tc superconductors[30], in which the data has been found to be in
dis-agreement with the theory based on the thermal fluctuations scenario. In this study, we proposed that the anomalous behavior is due to the thermal fluctuations from excited Landau levels in which we take into account the contribution from all Landau levels and use resummation technique to obtain the theoretical curves. In the following, the basic idea and the current understanding of the vortex matter in type II superconductor are discussed in this chapter. The second chapter will discuss the various modifications of homogeneous GL model for particular materials. In section 2.3, the valid region of LLL approximation is discussed. In section 2.4, an efficient approach, the optimization variational approach, and a resummation technique, a Borel-Pade approximation, are introduced and the convergency of the series is discussed. In section 2.5 the disorder model which will be used in SPT is discussed. The fluctuations influence of different di-mensionality high-Tc superconductors are presented in Chapter 3. The structural phase
transition will be discussed in Chapter 4. As a result I can state that the “mystery” part at the critical regime of high-Tc superconductivity can be understood as the thermal
1.1
Superconductivity
Superconductivity was first discovered in 1911 by H. Kamerlingh Onnes in Leiden [1]. A few years after he had first liquefied helium and reach temperatures of a few degrees of Kelvin, he observed that the electrical resistance of various metals such as mercury, lead, and tin disappeared completely in a small temperature range at a critical temperature, Tc, which is characteristic of the material. This perfect conductivity is the first traditional
hallmark of superconductivity.
The next hallmark to be discovered was perfect diamagnetism, found in 1933 by Meissner and Ochsenfeld [7]. They found that not only does magnetic field not enter a superconducting sample, as might be explained by perfect conductivity, but also that a field in an originally normal sample is expelled as the sample is cooled through Tc.
The existence of such a reversible Meissner effect implies that superconductivity will be destroyed by a magnetic field above certain critical field Hc. The new thermodynamic
state is call superconducting state. This critical field, named as thermodynamic critical field, is determined by the difference in free energies of the superconducting state and the normal state.
The disappearance of the electrical resistance below Tc, has numerous important
ap-plications. Since it allows the existence of non-decaying electric currents, supercurrent, stay permanently inside the material. It makes possible the production of many im-portant devices such as extremely powerful electromagnets, energy reservoirs, and much more. However, the supercurrent density allowed to flow through a sample is limited by a critical value Jc, which is an upper limit of the current consumption of these devices
and, therefore, on their maximum output power.
Various theories were suggested in order to describe and explain superconductivity. In 1935 F. and H. London [5][6] proposed a phenomenological theory based on clas-sical Maxwell electromagnetism, which was able to describe the basic electromagnetic properties of a homogeneous superconductor. The London theory and its future gen-eralizations [10] introduced two important scales: the concepts of correlation length, ξ, and penetration depth, λ. Those two parameter characterize many physical properties
of a system.
Figure 1.1: The H-T phase diagram of Type I and Type II superconductors
Even though the two quantities are both material and temperature-dependent, their quotient is effectively temperature-independent and can be considered as a material characteristic. The new dimensionless quantity is called the Ginzburg-Landau parameter and is denoted by κ = λ/ξ. As was shown by Abrikosov [4], two types of superconductors exist, differing by the value of κ and behaving differently in the presence of a magnetic field. Superconductors with κ < 1/√2 are called Type-I, and those with κ > 1/√2 are called Type-II. The values 1/√2 is an exact solution where the interface energy( between superconductivity and normal state) vanished.
TypeI superconductors can exist in one of two thermodynamically stable states -either in the normal, or in the superconducting state. The superconducting state is energetically favorable at T < Tc and H < Hc. Hc and Tc are mutually dependent,
see Fig. 1.1. Applying an external magnetic field to the system turns on the surface supercurrents, which screen the field from the interior of the superconductor. It does not allow external magnetic field to penetrate deeper than λ. This phenomenon is called the Meissner effect [7], and the whole state is sometimes called the Meissner state. In this state the material has perfect diamagnetism The magnetization defined as 4πM = B (r) − H( where B = 0 in Meissner state) is negative and proportional to up to Hc. In idea sample, it has a reversible hysteresis curve.
When the external magnetic field approaches Hc and the screening surface
supercur-rents approach Jc (Hc and Jc are therefore mutually dependent), the superconducting
state is no longer energetically favorable, and a second order phase transition into the normal state takes place. The opposite process is possible as well. The quantity H2
c/8π
is the condensation energy density of the system. It should be emphasized that this sce-nario is exact only for the case of an infinite cylinder, while arbitrary geometry Type-I superconductors transform into an intermediate state [50, 51], consisting of large super-conducting and normal domains separated by domain walls.
Unlike Type-I, Type-II superconductors have an extra thermodynamically stable state - the mixed state [4], in which the external magnetic field partially penetrates the bulk of the superconductor, locally destroying superconductivity. In this case two critical magnetic fields exist, Hc1and Hc2(see Fig. 1.1). Hc1is the lower critical magnetic
field, at which the magnetic field starts penetrating into the bulk of the superconductor and superconductivity begins to decline, and Hc2 is the upper critical field, at which the
magnetic field fills the whole sample, i.e. superconductivity is destroyed while the normal metallic state is recovered . The Hc1 is mainly determined by the London penetration
depth λ, which is the length scale determining the electromagnetic response of the superconductor. From the London equation set, one got Hc1 = (Φ0/4πλ2) log (κ) . The
upper critical field Hc2 is determined by the coherence length ξ of superconductor, which
determines the spatial response of the macroscopic field Ψ. The relation between Hc2
and ξ are given by Hc2= Φ0/2πξ2, where Φ0 is a fluxon. The transition to normal state
is of second order.
The differences in the behavior of Type-I and Type-II superconductors can be ex-plained if one examines the transitional energy between the normal and the supercon-ducting domains, which is positive in Type-I and negative in Type-II superconductors. In this study, we have interest on physics of the mixed state.
In the mixed state, the penetration of the magnetic flux into the superconductor takes place in the form of long thin flux lines, called Abrikosov vortices or fluxons (see Fig.1.2). At the center of each vortex a normal core exists, bearing the magnetic flux
Figure 1.2: The order parameter and the magnetic field profiles of a single Abrikosov vortex
created by supercurrents moving around the core. The characteristic radius of the core, i.e., the radius at which the order parameter decay from its maximal value to zero is ξ, while the magnetic field and the supercurrents, which surround the core, spread as far as λ from it. The amount of magnetic flux Φ carried by each vortex is quantized and equal to an integer number of unit quanta Φ0 = hc/2e = 2.07 · 10−7(G · cm2) [61]. For
a single vortex, with magnetic field, H, apply to z direction, the induced magnetic field
B (r) = Φ0 2πλ2K0 |r| λ
where K0 is Hankel function has the following properties:
K0 r λ ≈ log (κ) , r ≤ ξ − log (r/λ) , ξ ≪ r ≪ λ π 2 λ r 1/2 e−r/λ , r ≫ λ
The core cutoff is introduced to prevent unphysical divergence of magnetic field.
Vortex interaction consists of two parts, electromagnetic interaction due to the Lorentz force acting between the current loops, and interaction due to the gradient of the order parameter in the vicinity of the core. The electromagnetic interaction between fluxons and antifluxons (fluxons with an opposite direction of supercurrent and magnetic field) is attractive, while the electromagnetic interaction between fluxons of the same sign is repulsive. The cores interaction is always attractive but is usually neglected due to
the fact that it decays over short distance compare to the electromagnetic interaction, in extremely Type II superconductors (κ >> 1). Let’s look at a simplified model in London limit where the validity is r ≫ ξ. Consider two parallel straight vortices , the London equation is linear in magnetic field within range of its validity. The interaction line energy density (Gibbs energy) between two straight vortices line which is defined as ∆g = g (x1, x2) − g (x1) − g (x2) where xi is the position of the core i is
∆g = Φ 2 0 8π2λ2K0 r λ . The interaction force per unit length is
ff orce= − dg dr = Φ2 0 8π2λ2 r−1 , ξ ≪ r ≪ λ 1 2 π 2λr 1/2 e−r/λ , r ≫ λ .
One can see that the vortex repulsion is isotropic. In dense region, the vortex should form the hexagonal Abrikosov lattice, which provides the maximal vortex spacing for a given flux density B/Φ0. For interaction of curving vortices line please find the detail in
[99].
Figure 1.3: Profiles of correlation length ξ and penetration depth λ of Type-I (κ ≪ 1) and Type-II (κ ≫ 1) superconductors on a normal-superconducting interface the magnetic field profiles of a single Abrikosov vortex
1.2
Spontaneous Symmetry Breaking and Phase
Tran-sitions
Phase transitions [120] are usually described with an order parameter function, retain a finite value at the ordered (non-symmetric) phase and vanish in the disordered (sym-metric) phase. The order parameter depends on the system it describes, and bears the symmetries of the ordered state. As an example, one can think of a conventional su-perconductor, whose order parameter is a complex scalar having the U (1) symmetry, which matches the local gauge invariance of superconductors (for detailed discussion of the order parameter properties see discussion below).
There are two kinds of phase transitions, named the first order and the second order. The difference between them is that during a first order phase transition the system changes its state immediately rather than gradually, i.e., the order parameter is discon-tinuous at the transition point. The liquid-gas transition is a typical example of such a process where the density serves as an order parameter, and at the critical pressure there is a sudden increase in the density even if the pressure changes slightly. This, however, is not the case for the second order phase transition where various physical quantities either vanish or diverge at the transition point continuously. The order parameter de-clines smoothly as the system nears the transition point, while completely disappearing at the point itself. In the critical region, the scale of correlation is unbounded, namely, large scale correlation such as universality is observed. In the language of field theory, one is approaching a zero mass theory. At such point the first derivative of the free energy-like entropy, volume, magnetization, etc.-behave continuously. Note that is pos-sible there is no change in symmetry at transition point. For example the critical point of the gas-liquid transition involves no symmetry change at all.
Second order phase transitions can be modeled with a potential, proposed by Landau [3]:
V = α |Ψ|2+β 2 |Ψ|
4
(1.1) where Ψ is the order parameter function anda and b′ are phenomenological parameters.
Figure 1.4: The effective potential V (ϕ) describing second order phase transitions
Minimizing the potential one can find the values of |Ψ| corresponding to the vacuum state (see Fig.1.4). Given b′ > 0 (a necessary condition for the existence of a minimum),
this potential has two modes differing by the sign of a. If a > 0 there is only one minimum at |Ψ| = 0, and this mode corresponds to the disordered phase. However, if a < 0 , the minimum is at |Ψ| = p|a| /b′, while |Ψ| = 0 turns into a maximum, and
this mode corresponds to the ordered phase. α is therefore a crucial parameter that triggers the phase transition. The condensation energy is (αTc)2/2b′. It is temperature
dependent and has the following form at T ∼ Tc:
a(T ) ≃ αTc T Tc − 1 (1.2) As the temperature declines and passes through Tc the vacuum state of the system
changes, although it is possible that the new vacuum has a degeneracy. The type of this degeneracy strongly depends on the properties of the order parameter in the ordered phase. In a ferromagnet, for example, the order parameter is a real vector, namely the vector of magnetization, but its energy, according to Eq. (1.1), is not affected by the direction of magnetization, only by its magnitude. In 1D ferromagnets this situation corresponds to a 2-fold degeneracy of the ordered state - the magnetization vector points either up or down. The order parameter of a conventional superconductor is a complex scalar. Looking at Eq. (1.1) one finds that the superconducting vacuum state is infinitely degenerate, since the phase of order parameter does not appear in the potential and
therefore does not affect the energy.
Abrikosov vortices is a kind of topological defect. The systems with a complex order parameter are strings - 1D objects that are encircled by areas of different order parameter phase in such a way that its total change around the string at every point of its length is equal to 2πn, where n is an integer. The latter condition is required in order to ensure that the order parameter is a single-valued function.
1.3
Thermal fluctuations in anisotropic/ disordered
superconductors
The main phenomenon determining the physics of the vortex lattice is thermal motion of vortices about their equilibrium position. At high temperature thermal fluctuation increase the amplitude of the vortices vibration. I first exemplify the qualitatively the phenomenon using the London approximation limit, which is different from the so called lowest Landau level approximation limit mainly employed in the following sections. An isolated flux line acting as a stretched string can undergo both longitudinal or transverse vibrations. At the region of parameter space in which vortices are densely packed the vibrations are more localized along the length of the vortex core with numerous nearby vortices participating in collective. The vortex motion strongly depends on the material characteristics, impurities, external magnetic field and temperature.
A naive idea to estimate the influence of thermal fluctuation is via a characteristic length LT of segment of the vortex associated the quantized flux energy to the thermal
energy. The energy UM of magnetic field in a region of volume V is given by
UM = Φ2 0 8π V A2,
where B = Φ0/A and A is the area of the unit cell of the vortex lattice. If this is equated
to the thermal energy kBT for a quantum flux, and if we write A2/V = 2πLT, we obtain
the characteristic length
LT = Φ2 0 16π2k BT = 1.79 T (Kelvin)cm.
This is much larger than other characteristic lengths, such as ξ and λ, except in the case of temperature extremely close to Tc both ξ and λ diverged as (Tc− T )−1. Therefore,
fluctuation can be expect to be small when at low temperature. In high temperature cuprates several factors combine to enhance the effects of thermal fluctuations: (1) higher transition temperature, (2) shorter coherence length ξ, (3) large magnetic penetration length λ (4) quasi−2D− dimensionality, and (5) high anisotropy.
A proper fundamental material parameter describing the strength of thermal fluctu-ations is the Ginzburg number Gi = (Tc/Fcon)2/2 [62], which is the ratio of the minimal
(T = 0) condensation energy Fcon = (Hc2/8π) ξcξ2 within a coherence volume (ξcξ2) and
the critical temperature Tc. It is a dimensionless quantity. For typical superconducting
metals Gi is very small (of order 10−6). It becomes significant for relatively isotropic
high Tc cuprates Y BCO (10−3) and even quite large for strongly anisotropic cuprates
BSCCO (up to Gi = .1 − .5).
Diroder of a sample originally comes from point defect, dislocation, oxygen vacncy, grain boundary...etc.. It can change the local properties of the sample, sush as critical temperature, the effective mass and coupling between vortices. A vortex will experience a short range pinning force that hold the core of a vortex at a defect. The pinning energies have been reported in the range of hunreds of meV . When sufficient pinning center are present, the spatial structure of vortices will reflect the distribution of pinning center and the long-range order of vortex lattice is disturbed. It will result in glassy state. Disorder dramatically influence the vortex motion. Fig 1.5 shows the glass transi-tion in the driven case. In the flux-flow regime, the driven lattice hits the defects and pushes the vortex from the pinning centre. The onset motion of vortex is determined by competition between driving forces, usually Lortz forces, and pinning force. In the steady-state motion, the viscous force which result in dissipation are presense. In the presense of thermal excitations, vortices can undergo thermal hopping between pinning centers.
Theoretical study of those phenomena predict a complicate phase diagram of vortex matter shown in Fig.1.6, which is more complicate then the mean field diagram for clean
Figure 1.5: Diagram of Vortex Motion
idea sample. There are five main distinct phases: unpinned solid (solid), pinned solid (Bragg glass), pinned liquid (vortex glass or amorphous solid), weakly pinned solid with marginal glassy dynamics, and unpinned liquid (or simply liquid). With anisotropic effect on a − b plane, the phase diagram will have additional complexity, the vortex solid will encounter structural phase transition.
Melting of the vortex lattice happened when thermal agitation induces a wander-ing of vortex filaments and leads to an entangled flux liquid phase. Pairs of flux lines passing close to each other can be cut, exchanged, or reattached. Qualitatively, ac-cording to Lindermann criterion, melting occurs when the root mean-square fluctuation amplitude urms exceed the quantity ≈ 10−1s, where s is the average vortex separation
distance. Evidence of the first-order vortex-lattice melting transition is confirmed by experiments in single crystals of Y Ba2Cu3Oy[44] [33][28][39], Bi2Sr2CaCu2Oy [94][34],
and La2−xSrxCuOy[40][41][42] and other high−Tc materials. The transition type be
determined by magnetization jumps [94], spikes in specific heat [33], SQUID[34] and etc..
Figure 1.6: Phase diagram of vortex matter: The complixity of the phase diagram is due to the influence of thermal fluctuations, disorder and anisotropy.
translation and rotation symmetry. The intensity of the first Bragg peak can be used as an order parameter. The first order ODO line has two parts: the melting segment and the “second magnetization peak” segment, separates the homogeneous and the crystalline phases. The broken symmetry is not directly related to pinning, but the location of the line is sensitive to the strength of disorder. Thermal fluctuations, on one hand side, make quenched disorder less effective in destroying crystalline structure, thereby favoring solid, but, on the other hand, they themselves destabilize vortex lattice. The low temperature segment of the ODO line, the second peak line, is disorder dominated and weakly depends on temperature.
The memory effect is one of characteristics in glass state. The history-dependent property was observed by M¨uller at the beginning of the high-Tc era. The irreversibility
line is not very sensitive to the type and distribution of defects, although these defects have a pronounced effect on the critical current density. Moreover, there are two dis-tinguishable liquid phases (refers as liquid 1 and liquid 2) which are classified by its
dynamic properties. This feature receives a natural explanation within the developed description. The line separating two regimes coincides with the melting line of a clean sample. The theoretical argument can be found in [47].
Chapter 2
Mesoscopic description of
superconductor in magnetic field
A phenomenological description, typically valid near the transition temperature, was originally proposed by the Ginzburg and Landau [3] to describe any second order transi-tion. The remarkable success of GL theory is due to the long-range character of critical fluctuations leading to universality of the critical properties at the vicinity of transi-tion temperature. The effective theory can be derived from a microscopic one which determines the limited number of coefficients of the GL. The GL model is by far more efficient to study mesoscopic phenomena rather then a microscopic theory. The origi-nally proposed GL model described a material which is homogeneous in all direction. Microscopically it followed from assumptions of the spherical Fermi surface and isotropic pairing potential. Most of the newly discovered type II superconductors addressed in this work (like high Tc cuprates, borocarbides etc.) are far from being isotropic. The
symmetry of the underlying structure influences profoundly the mesoscopic behaviors. Here we introduce various commonly used modifications of the GL model describing the anisotropy.
Within the GL approach the free energy is expanded in order parameter up to a quartic term in potential, namely becomes a φ4 type theory. As a nonlinear field theory
The most commonly used is the lowest Landau level approximation. We will show that the bifurcation expansion on distance between the S-N phase transition line works well down to H = Hc2(T )/13 which is used very far from transition line [89] . In the
last section, we will demonstrate an resummation technique, the Borel-Pad´e series, to extract more physics from a expansion coefficients of a function of coupling. The factorial growth of the power series is taken into account by means of Borel transformations. As quotients of polynomials, the Pad´e approximate can describe functions with poles in lowest order. And a sequence of Pad´e approximates, which converges outside the region of power series, can be used to define an analytical continuation of the function outside the converge region of a power series.
2.1
Landau Expansion of the Free Energy in
homo-geneous Superconductors
The GL theory has both quantum and phase-transitional properties. It describes su-perconductivity as a gauge-invariant quantum phenomenon, and that its nucleation in a normal domain can be treated as a second order phase transition from the disordered (normal) to the ordered (superconducting) state. The basic feature of the theory is that the change in the Gibbs free energy density functional between normal and supercon-ducting states can be expanded in powers of the order parameter . Since the electrons forming the Cooper pairs are charged particles, the electromagnetic vector potential A must also be introduced. In the case of conventional superconductivity, this expansion has the form
Fcond = Fs− Fn= ~2 2m∗ |Dψ| 2 − a (T ) |ψ|2+ 1 2b ′|ψ|4 + 1 8π (∇ × A) 2 (2.1)
where Fn is the free energy of the normal state and D = ∇ − ie
∗
~cA is the covariant
derivative, Bj ≡ (∇ ×−→A )j is the magnetic induction. The magnetic energy term is also
present. m∗ and e∗are the effective mass and charge of the Cooper pair, respectively; c is
All those coefficient can be obtained from a microscopic theory. The stability of the superconducting state requires that b′ is always positive, while a (T ) is temperature
dependent and can be either positive (superconducting state) or negative (normal state), as shown in the Introduction.
Varying the free energy functional with respect to ψ∗ and A and equating the
vari-ation to zero the two steady state Ginzburg-Landau equvari-ations are achieved: a (T ) ψ + b′|ψ|2ψ + ~ 2 2m∗ −i∇ − ~e∗cA 2 ψ = 0 (2.2) c 4π∇ × ∇ × A ≡ Js = − ie∗~ 2m∗ (Ψ∗∇Ψ − Ψ∇Ψ∗) − e∗2 2m∗c|Ψ| 2 A (2.3)
where Jsis the supercurrent density or simplily Js = e
∗~ m∗ |ψ| 2 ∇φ − e~∗cA . It is clear that in the absence of fields (Meissner state) these equations have a homogeneous solution ψ = p|α| /β exp (iφ), where φ is constant. It is no longer the case in the mixed state. While at the center of the vortex core the order parameter ψ vanishes, at distances sufficiently far from it the absolute value of the order parameter is again constant |ψ| = p
|α| /β, but not the phase.
Requiring the supercurrents to encircle the core (i.e., ∇ · Js = 0) and choosing a
gauge in which ∇ · A = 0, one obtains the following equation for the order parameter phase: ∇2φ = 0. Making another requirement of azimuthal symmetry, this equation has
the following solution: φ = nϕ + ϕ0, where ϕ0 is a constant, ϕ is the azimuthal angle
and n is an integer, since φ (ϕ) = φ (ϕ + 2π) is a periodic function. From this result the quantization of magnetic flux in a superconductor is deduced:
Φ = I CA · dl + 4πλ2 c I C Js· dl = nΦ0 (2.4)
This phenomenon was first observed by Little and Parks [61] and confirmed by numerous experiments [75].
The quantization of magnetic flux in superconductors imposes an important con-straint - the topological charge conservation. The topological charge of the Abrikosov vortex is defined as:
n = Φ Φ0 = 1 2π I C|ψ| 2 ∇φ · dl (2.5)
where the integration is performed over an arbitrary path that encircles the vortex and lies infinitely far from its core. The topological charge conservation means that vortices can appear and disappear only by crossing the domain boundaries or, alternatively by vortex-antivortex pair production or annihilation.
The GL free energy has the property of local gauge invariance, which leaves the GL equations invariant under the following simultaneous gauge transformations:
ψ → ψ exp(iχ), A → A +~2ec∇χ, µ → µ − 2e~ ∂t∂ χ . (2.6)
One can see that these transformations do not affect the physical quantities, such as the Cooper pair density |ψ|2, the magnetic induction B = ∇ × A or the supercurrent density Js. In order to eliminate the additional degree of freedom, appearing because of
the local gauge invariance, Eqs.(2.2-2.3) must be accompanied by a gauge condition. In the absence of applied electric field a gauge that eliminates the scalar potential is used, demanding µ = 0 as well as ∇ · A = 0.
2.2
Landau Expansion of the Free Energy in
Anisotropic Superconductors
For isotropic superconductors, the GL model discussed in previous section is sufficient. However, in real materials, even in conventional superconductor such as N b [13], ex-perimental evidence had shown complicate behaviors of vortex matter. The symmetries of superconductors are strongly related to the crystallographic symmetry group of the material, producing an anisotropy of the Fermi surface, and to the effective attraction mechanism of Cooper pairing [69, 70]. Both of these are omitted from the Gor’kov equations and so does the GL model as Eq.( 2.1).
All High−Tcsuperconductivity arises in a family of layered copper oxides that all
fea-ture weakly coupled square-planar sheets of CuO2 Fig. 2.1 which are separated by other
materials. According to the current understanding, it seems that most of the proper-ties are determined by electrons moving within the copper-oxide planes. The remaining
components play structural roles and provide screening and doping environments. It may be considered as an experimental fact that the copper-oxide plane that determines the Fermi surface and low-energy electronic properties.
Figure 2.1: Crystal structure of La2−xSrxCuO4+δ and the Lawrence - Doniach model.
The basic building block in Fig.2.1 is a perovskite structure consisting of the CuO6
octahedron seen in the centre of the unit cell, surrounded by eight La ions. The CuO6octahedra are elongated, giving rise to a layered structure with quasi-two-dimensional
CuO2sheets. In this particular cuprate, a rocksalt-structured layer is stacked in between
the perovskite cells, but there are many variations that give rise to different families of high-temperature superconductors. In La2−xSrxCuO4+δ, Sr substitution for La, or the
addition of interstitial oxygen, changes the carrier concentration on the CuO2 planes,
drastically altering the electronic properties of this compound.
To reflect the fact of layered structure, the simplest model is to assume infinitesimally thin superconducting layered which are coupled via order parameter tunneling , the Josephson coupling, through insulating layers of thickness d. The modified GL model is called Lawrence-Doniach (LD) free energy functional, FLD [119]:
FLD = X n d Z d2r " ~2 2m∗ ab ∇ − i~e∗cA⊥ ψn 2 − a (T ) |ψn|2 +b ′ 2 |ψn| 4 + ~ 2 2m∗ cd2 ψn+1exp −i~e∗ cAz − ψn 2 + (B − H) 2 8π # . (2.7)
The order parameter ψn has discrete dependence on the index n in z direction and a
continuous dependence on x, y directions; the total vector potential A⊥+ Azk is definedb
at all points, where Az = (1/d)
RR(n+1)d
ns Azdz. The structure of this equation is similar
to GL energy with respect to the in-plane components while the interplane coupling is seen on expanding the exponent to be the operator ∂z− i (e∗/~c) Az to the case of finite
differences. The LD model is a practical tool for layered materials such as LSCO. Suppose the temperature dependent coherence length(ξc) is larger then the layered
distance (d), system can be regarded as a continuous 3D system but has anisotropy in c direction. A commonly used phenomenological GL model is introducing an effective-mass m∗. The asymmetry factor is defined by the ratio of effective mass
γ = q
m∗
c/m∗ab. (2.8)
For conventional superconductor, P b (γ = 1), N bSe2(γ = 11) and unconventional
super-conductor, Y Ba3Cu3O6.92 (γ = 5) , Y Ba3Cu3O6.7 (γ = 13), La2−xSrxCuO4+δ(γ = 15)
and Bi2Sr2CaCu2O8+δ(γ = 70) . With the anisotropy in c direction, the coherence
length ξc(T ) is typically much smaller
ξc2(T ) = ~
2
2mcαTc(1 − T/Tc)
= ξab2 (T ) /γ2 (2.9)
while the corresponding penetration depth λc(T ) is larger:
λ2c(T ) = m∗cc
2β
4πe∗2αTc(1 − T/Tc) = λ 2
ab(T ) γ2. (2.10)
In slightly anisotropic layered SC such as Y BCO-type, d ≪ ξc,system can be described
by the anisotropic 3D GL Model [119]. Its free energy functional has the form: F3D = Z d3x ~ 2 2mab |Dψ| 2 + ~ 2 2mc|∂ zψ|2+ a (T ) |ψ|2+ b′ 2|ψ| 4+(B − H) 2 8π . (2.11)
In extreme layered high-Tc materials such as Bi2Sr2CaCu2O8+δ-type, d ≫ ξc, system
can be regarded as qusi-2D system (since the tunneling between nearby CuO2-layered
are negligible) [119]. A commonly used 2D GL model has the form: F2D = Z d2x ~ 2 2mab |Dψ| 2 + a (T ) |ψ|2+b ′ 2|ψ| 4+(B − H) 2 8π . (2.12)
Now we will consider the system which breaks the in plane O(2) rotational symmetry. It is an experimental fact that the for high Tc superconductors such as Y BCO has a
mixture of d-wave and s-wave components [38], newly discovered Sr2RuO4 [37] has
p-wave pairing and etc.. Experimentally evidences show those influences can result in different configurations of vortex matter. In Y BCO both square and rhombic phases exists [18] . In overdoped LaSCO, at low temperatures, the square and rhombic lattices were observed in SANS experiments [16]. Asymmetry is not always related to the non s-wave nature of pairing, it can strongly relate to the structure of the Fermi surface, which is a consequence of the crystallographic symmetry group of the material. In borocarbides (RE = Lu, Y, Er) N i2B2C[14], N b [13], and V3(Si) [15], square vortex lattice is observed
using techniques such as decoration, STM, SANS or µSR etc.
Taking in to account the coupling with underlying material from microscopic theories is a formidable task. The current exist full microscopic models contain a large number of unknown parameters and are rather cumbersome to work with numerically [106][27]. To study it in phenomenological approach, several ideas are proposed [20][22][26].
To include the in-plan anisotropy to GL model one need to take into account higher order term. Since there is no quadratic in covariant derivative terms that break O(2). In this study, we consider the material has fourfold (D4 ) symmetric. One can use four
derivative terms to modify the model. There are three such terms
D2
x+ D2y
Ψ 2, D2zΨ∗ D2x+ D2yΨ, D2x− Dy2Ψ 2, (2.13) but only the last term which breaks the O(2) is irrelevant. This term leads to anisotropic shape of a vortex and an angle dependent vortex – vortex interaction thus the emergence of lattices other than hexagonal, a symmetric rhombic lattices. One therefore can add
the following gradient term for a in-plane 4-fold symmetric materail: Fab−aniso= ηm D2 x− D2y Ψ 2 (2.14)
with dimensionless constant ηm characterizing the strength of the in-plan asymmetry.
Since the ultimate microscopic theory is not known as yet, the coefficient ηmis considered
as phenomenologically fixed parameters.
2.3
Applicability of Lowest Landau level
approxima-tion within the mean field approach
The most common additional assumption in most of theoretical studyies is the lowest Landau level (LLL) approximation, namely, only the lowest Landau level (LLL) signif-icantly contributes to physical quantities of interest [109, 107, 108, 48, 111, 112]. To understand the valid region of the LLL approximation, we will discuss the higher Lan-dau level correction to the free energy of superconducting state. Due to symmetries of the problem leading to numerous cancellations the range of validity of the LLL approx-imation in the mean field approach is much wider then a naive range but extends all the way down to H = Hc2(T )/13. We will show that the contribution of higher Landau
levels is significantly smaller compared to LLL.
Now we discuss the solution of the Ginzburg-Landau equations obtained from Eq. 2.12 a perturbation method expend in ah, indicate as the distance near the mixed state
- normal phase transition line. This has been done before in Ref. [114] to the second order and higher order by Li and Rosesntein [89]. Our starting point is the rescaled free energy functional density Eq.(3.10). The expansion parameter is ahwhich is defined
in Eq.(3.9). Rewriting the quadratic part in terms of operator (“Hamiltonian” ) H ≡ −12(D2+ b) whose spectrum starts from zero, the equation of motion is therefore
Hψ − ahψ + ψ|ψ|2 = 0 (2.15)
Φ = (ah)1/2[Φ0+ ahΦ1+ ...] (2.16)
It is convenient to represent Φ0, Φ1, ... in the basis of eigenfunctions of H, Hϕn = nbϕn,
normalized to unit ” Cooper pairs density” < |ϕN|2 >r≡
R
celld
2x|ϕ
N|2 b2π = 1, the
inte-gration go over a unit cell of the vortex lattice. Assuming hexagonal lattice symmetry, one explicitly has
ϕN(x, y) = s √ 2σ 2NN ! +∞ X l=−∞
expiπρ(l2− l)exphi√2πσlyi
HN x √ b − √ 2πσbl × exp " −12 x √ b − √ 2πσbl 2# . (2.17)
with σ = √3/2 and ρ = 1/2. Function HN(x) is Hermite polynomials. Insert the Eq.(
2.16) to Eq.(2.15 ). To the order a1/2h , one get
HΦ0 = 0 (2.18)
and Φ0 is proportional to the Abrikosov vortex lattice solution ϕ0, namely Φ0 = g00ϕ0.
The general form expended in all Landau level for Φi is
Φi = g0iϕ0+ ∞
X
N =1
gNi ϕN. (2.19)
Inserting into Eq.(2.15) to order a3/2h , one obtains
HΦ1 = g0ϕ0− g0|g0|2ϕ0|ϕ0|2. (2.20)
Taking the inner product with ϕ0 one finds that
g00 =
1 √
βA
, (2.21)
where the Abrikosov’s constant is the following average over primitive cell: βA ≡<
|ϕ0|4 >r≈ 1.159. Inner product with ϕN determines gN0:
gN0 = − βN
where βN ≡< |ϕ0|2ϕNϕ∗0 >r . To fix the g01coefficient in Φ1, we need in addition also the
order a5/2h equation:
HΦ2 = Φ1− (g0)2(2Φ1|ϕ0|2+ Φ∗1ϕ20) (2.23)
Inner product with ϕ0 gives:
g10 = 3 2 ∞ X N =1 βN2 N bβA5/2. (2.24)
The mean field expression for the free energy to order a3
h can be obtained by inserting
the next correction Φ1 in Eq.(2.16 ) into Eq.(2.15) one obtains the free energy density:
Fmf T = 1 ω − a2 h 2β − a3 h β3b ∞ X N =1 βN2 N ! = 1 ω −.43a2h− .0078 a3 h b . (2.25)
Due to hexagonal symmetry of the vortex lattice [114], βN 6= 0 only when N = 6j, where
j ∈ N. For N = 6j it decreases very fast with j: β6 = −.2787, β12 = .0249. Because of
this the coefficient of the next to leading order is very small (additional factor of 6 in the denominator). We might preliminarily conclude therefore that the perturbation theory in ah works much better that might be naively anticipated H = Hc2(T )/13 and can be
used very far from transition line.
2.4
Optimized perturbation approach and the
Bore-Pad´
e approximation
To solve the nonlinear problem Eq.(3.13) by field theory, we use optimized perturbation approach (OPA) to obtain an converged series and then apply resummation technique to extract more physics from the series. Resummation process we adopted here is a Bore-Pad´e approximation [118].
OPA is first developed in field theory, one introduce an auxiliary parameter ε as
a variational parameter, then the free energy f can be interpolated as f = ε|ψ|2 +
α (f − ε|ψ|2). The artificial parameter α here is to generate a perturbation expansion by
optimize the free energy with respect to ε. To calculate the free energy fef f = −ω0log Z,
one starts from expanding the logarithm of the statistical sum Z in α,
Z = ZZ Dψ exp (−K) exp (−αV ) (2.26) = ZZ Dψ ∞ X i=0 (−αV )i i! exp (−K) .
For convenience in writing we defined ef as fef f = ωV0f . In OPA, we have ee f ≡ efn+
O(αn+2) , the nth order of OPA, ef
n, is e fn = − log ZZ Dψ exp (−K) − n+1 X i=1 (−α)i i! Vi K. (2.27)
where h...iK denotes the sum of all the connected Feynman diagrams. The first two
order of ef as a function of ε are e f0 = 2 log ε 4π2 + aε ε + 1 ε2 , (2.28) e f1 = ef0− 1 ε4 18 + 8ε + a 2 εε2 , (2.29)
where aε = aT − ε. The solution of nth order of OP denotes as εn is obtained from the
minimization equations, (∂ε− ∂aε) efn = 0, (2.30) such that fn = efn(εn) = min h e fn(ε) i .
The fef f series are calculated by several groups. The original paper by Thouless and
Ruggeri [110] reached 6thorder; and expended to 13th order by Hu and MacDonald [115].
The free energy has the form fef f = 2 log ε1 4π2 + 2fn(x) + O x n+1, with (2.31) fn(x) = n X i=1 hixi and x = 1 ε2 (2.32)
where ε = aT +
p a2
T − 4zn
/2. The coefficients hk can be found in [116] and zn can
be found in [85].The consecutive approximates are plotted on Fig.2.2. Dashed dotted lines represent a optimized perturbation theory (denoted by the order numbers). One can see clearly that the series are convergent with radius of convergence at aT = −5.
Figure 2.2: The free energy calculated using two different approximation schemes. Dashed dotted lines represent a optimized perturbation theory (denoted by the order numbers).
It is known that in the theory of critical phenomena one can extract more physical results from a divergent series by resummation processes. Bore-Pad ´e approximation is a good option. By applying Bore-Pad´e approximation to perturbation expansion at “weak coupling” [118], an accurate description in the critical region is obtained, since there exists a renormalization group flow from the weak coupling fixed point towards the strongly couple one [117]. We use the following strategy: The factorial growth of h (x) is taken into account by means of Borel transformations, then Pad ´e approximation is used to approach the less divergent series. Observing that in our case the hk of the
asymptotic series growth factorial for large k, we divide each term in the expansion by a factor k!, a less divergent series is obtained. It is so called Borel sum:
B (x) ≡
∞
X
k
i hn zn−1 1 −2 −4 2 −1 −6 3 389 −12.239721181139888 4 −39 − 2930 −7.508888400035477 5 471.39659451659446 −7.349933383279474 6 −6471.5625749551446 −14.152646217045422 7 101279.32784597063 −9.961364397930787 8 −1779798.7875947522 −9.174960576928443 9 34709019.614363678 −15.232548389083844 10 −744093435.66822231 −11.629924499110746 11 17399454123.559521 −10.8399817525306 12 −440863989257.28510 −15.9366927661989 13 12035432945204.531 −12.753308785106007
Figure 2.3: Coefficients hi and zi in 2D.
The Pad´e approximation for a series expansion B (x) =Pki=0Bixi+ O xk+1
up to the order k is given by [k, k − 1] ≡ k X i=0 aixi ,k−1 X i=1 bixi (2.34)
where ai and bi are chosen such that the series expansion of [k, k − 1] up to the order k
equals to the original series, B (x) = [k, k − 1] + O xk+1, or k
X
i=0
Bixi = [k, k − 1] . (2.35)
By doing the Borel transformation, the Borel sum , B (x), can be integrated to restore f (x) ,
h (x) =
ZZ ∞
0
dte−tB (xt) . (2.36)
The Bore-Pad´e series for different orders are shown in Fig. 2.2 with solid line denoted by ” BP” plus the order number represent the [k, k − 1] Borel - Pade series for k = 3, 4, 5.
The k = 4 and k = 5 practically coincide on the scale of the plot. For k = 4 and k = 5, the liquid energy converges to required precision (0.1%). The liquid energy extract by BP approximation agrees with the optimized Gaussian expansion results [90] till its radius of convergence at aT = −5.
2.5
Disorder
To describe the disorder (pinning) potential one adds a random component αTc(1 − t) W (r) |ψ(r)|2
or
αTc(1 − t − b) W (r) |ψ(r)|2
to the GL functional. The first case is indicate the random pinning potential change the local critical temperature; the later includes the a random effective mass due to defects (in Lowest Landau level approximation). We assume that W (r) has a Gaussian random distribution with variance,
W (r)W (r′) = nδ(3)
(r − r′), (2.37)
n is related to the average number of pinning center per volume.The ensemble average of a quantity O on all possible disorders is read as
hOi = R W Oexp h −R drW (r)2n2i R W exp h −R drW (r)2n2i . (2.38)
Disorder can pin vortices therefore it affects both dynamic and statistical behaviors of the system. Thermal fluctuation of the individual vortex lines lead to a dynamic sampling and hence average the disorder potential over the spatial extent of the thermal displacement hu2i1/2
th . Thermal fluctuation effect reduce the effect of quenched disorder
as thermal depinning which is a continuous crossover from a pinned to a an unpinned situation.
The number of pinning centers over the area of the sample L2is N = n ·L2. Potential
of N small pinning centers located at xi can be represented by a sum of delta functions:
V (−→x ) = V0 N
X
i=1
δ(−→x − −→xi) (2.39)
The constant V0 is proportional to the gap between normal and superconducting energy.
The average of the probability over disorder potential assumed random is:
V (−→x ) = 1 L2 NZ x1...xN V0 N X i=1 δ(−→x − xi) = 1 L2NV0× L 2(N −1)× N = 1 L2V0N = V0n (2.40)
We see that V (−→x ) ∝ n is independent of −→x . The correlator can be computed in similar way: V (x)V (0) = (V0)2 1 L2 NZ x1...xN X xi X x′ i δ(−→x − xi)δ(−xi′) = (V0)2 1 L2 NZ x1...xN X xi6=xi′ δ(−→x − xi)δ(−xi′) + X xi=xi′ δ(−→x − xi)δ(−xi′) = (V0)2 1 L2 N [L2(N −2)× (N2− N) + L2(N −1)× N × δ(−→x )] = (V0)2 N2− N L4 + N × δ(−→x ) L2 (2.41) Since N >> 1, we obtain V (x)V (0) ≈ (V0)2[n2+ nδ(−→x )] (2.42)
It is convenient to define a shifted random disorder field:
Its correlator is proportional to a delta function (white noise): W (x)W (0) = (V (x)− V (x))(V (0) − V (0))
= V (x)V (0)− V (x)V (0) − V (x)V (0) + V (x)V (0) = V (x)V (0)− V (x)V (0) = (V0)2[n2+ nδ(−→x )] − (V0n)2
Chapter 3
Magnetization in Layered
Superconductors
3.1
Introduction
Fluctuation diamagnetism is one of the characteristic feature of a strongly fluctuating
superconductor. For high-Tc superconductors, due to the enhanced thermal
fluctua-tions, diamagnetism is observed even above the transition temperature, Tc(H) [59, 62]
.At high field region ∼ Hc2(T ), Thouless and Ruggeri [109, 110] demonstrated that
thermodynamic and transport physical quantities have a good high-field scaling behav-ior as function of the dimensionless scaling temperature aT ∼ (T − Tmf (H)) / (T H)r,
for quasi−2D system r = 1/2, while for 3D system r = 2/3. This is a consequence of the dominant role of the Landau quantization: main contribution is coming from the lowest Landau level. It was studied and elaborated later by Teˇsanovic et al and others [48, 83]. For example, the magnetization experiments provide an evidence of the LLL scaling in of both 3D and 2D systems. Data shows that at different magnetic fields and temperatures magnetization curves collapse on a universal curve M/ (HT )r ∝ fsc(aT)
(see, for example, Fig.3.4) [29, 31, 44, 79].
Another striking feature is the magnetization curves M (T ) is that they intersect at the same point (T∗, H∗) for a wide range of magnetic fields, see Fig.3.6. It was observed
both in extremely anisotropic quasi two dimensional (quasi-2D) layered materials such
as BSCCO [29] and the T l based high Tc superconductors [79] and in more isotropic
quasi 3D ones, such as the optimally doped Y Ba2Cu3O7−δ [44] and Y Ba2Cu4O8 [43].
However recently it was found that magnetization curves of several other classes of lay-ered high Tc superconductors, including HgBa2Ca2Cu3O8+δ [31], strongly underdoped
Y Ba2Cu3O7−δ [86] and La1.92Sr0.08CuO4 [32], the intersection point is no longer the
same for all the magnetic fields. It rather moves a bit from its “3D” position at low fields to its “2D” position at high fields.
Theoretically, the phenomenon of the intersection points in 2D [48] and 3D materials [113] was first described in the framework of Ginzburg - Landau theory based on the fluctuations dominance scenario. Later, using the systematic expansion, it was shown that although the magnetization curves do not intersect at the same point at all the fields, it can move a negligible distance on the phase diagram in both 2D and 3D [83] . For layered Lawrence-Doniach (LD) model [86] , the theory shows a observaible migration
of the crossing points moving from 2D to 3D position while approach Hc2(T ) from
supercoducting state.
Moreover recent measurement on the strongly underdoped La1.92Sr0.08CuO4 [32]
shows the migration of crossing points move in an opposite direction, namely at low fields the intersection point is below Tc; while increasing magnetic field, the intersection
point moves in the opposite direction and eventually it exceeds Tc.In Ref. [32] the
the-oretical formulas of Ref. [86] in limits of quasi 2D and quasi 3D were used to quantify the data on LaSCO. While it was possible to fit the data in the optimally doped case, it was impossible to fit the data in the far underdoped cases.
In this study we will use LD for layered superconductor to investigate the movement of the intersection points. By following Hartree-Fock approximation in LLL approxi-amtion in Ref. [86], we mathematically defined the field dependent curve of “intersection point” for different coupling constant. Our result shows that with increasing magnetic field, the migration of the intersection points move from its 3D position to 2D position. More, the intersection doesn’t exceed Tc is proved. Therefore, the results of strongly
underdoped LaSCO are very puzzling and irreconcilable with the general LD theory. Another topic we will study here is inspired by the experiment done by Ong et
al [30]. In which they claimed that for strongly layered superconductor BSCCO[30]
the diamagnetism at the vicinity of the critical temperature in type II superconductors significantly differ from the behavior predicted by fluctuation diamagnetism theories based on Ginzburg-Landau model [80, 83, 48, 108, 84].
Theoretically, in the vortex liquid of 3D system,such as Y BCO, the HLL contribu-tion has been studied by Lawrie in the framework of the Gaussian (Hartree-fock like) approximation [49] . The result shows that the region of validity of LLL is very limited. Recently, the leading (Gaussian) contribution of HLL was combined with more refined treatment of the LLL modes in 3D system by Li and Rosenstein [87]. In the vortex liquid
of 2D system, the phenomenon above Tc was studied by Prange [88]and later by [80].
The later study show the physical quantity is strongly dependence on the UV cut-off which is puzzling.
In this study, we extend previous theoretical result [80] to T < Tc region by the
approach developed in Ref. [87]. Instead of using the traditional lowest Landau level approximation we integrated out higher landau modes to obtain an effective lowest Lan-dau level free energy functional and map it to existed lowest LanLan-dau result. Comparison with experimental results shows a good agreement with our theory for an optimally doped sample while underdoped sample differ from the theoretical results at low tem-perature.
3.2
Basic equations and assumptions
The Lawrence−Doniach model for layered superconducts
(LSCO-like material)
The material parameters of layered material LSCO are ξab ≈ 34 ˚A, γ ≈ 15 and the
interlayer spacing, d ≈ 15 ˚A. The measurement temperature |1 − t| < 0.1, the ξc(T )
system is the Lawrence-Doniach (LD) free energy functional, Eq.(2.7 ). We choice Tc
as a unit of temperature, T = tTc, and 2πξΦ02 as a unit of magnetic field, B = b
Φ0
2πξ2, the
coherence length ξab =
p
~2/ (2m
abαTc) as a unit of length in ab plane. The gauge is
choice to be the Landau gauge, A = (0, Hx, 0). In c direction is in ξc= ξab/γ. Namely,
x → ξabx, y → ξaby and d → ξcd . The order parameter is rescaled as ψn2 → 2αTb′cψ 2 n. The
Boltzmann factor of LD can be formulated as: fLD = FLD T = 1 ωt X n Z d2x1 2|Dψn| 2+ 1 2d2|ψn− ψn+1| 2 −1 − t2 |ψn|2+ 1 2|ψn| 4 (3.1)
The dimensionless coefficient ω,
ω =p2Gi2Dπ (3.2)
where the Ginzburg number is defined by Gi2D ≡ 1 2 32π2e2κ2ξT cγ c2h2d 2 . (3.3)
The nonlinear term |ψn|4 becomes very important in the temperature range |1 − t| ∼ Gi.
We will make the following assumption: (1) lowest Landau level approximation: the essential contribution is from lowest landau level as we mentioned in the previous section. (2) the magnetic fluctuation is negligible due the superposition of the magnetic field from individual vortex. It is convenient to expand the order parameter in terms of the Landau levels eigenfunctions basis, φN
kz,ky(r)[91] ψN n(x) = X N,k,q φN k,q(x)ψk,qN (3.4) where φNk,q(x) = p 1 LzLy 1 π1/4√2NN !exp " iqy + ikdn −12 x√b − √q b 2# (3.5) ×HN x√b − √q b exp " −12 x√b − √q b 2# .
The N stands for the Nth Landau level, q is the momentum in ab plane and k is the
one in c direction. For magnetic field is close the Hc2, the lowest Landau level modes
