Ginzburg-Landau 理論對於 Type-II 超導體的傳輸性質與漲落之研究

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國立交通大學

電子物理學系

博士論文

Ginzburg-Landau 理論對於 Type-II 超導體的傳輸性質與漲落之研究

Ginzburg-Landau theory of transport properties and fluctuations in

Type-II superconductors

博究生：卜德廷

指導教授：儒森斯坦教授

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Ginzburg-Landau 理論對於 type-II 超導體的傳輸性質與漲落之研究

Ginzburg-Landau theory of transport properties and fluctuations in

type-II superconductors

博究生：卜德廷 Student：Bui Duc Tinh

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 Thesis

Submitted to Department of Electrophysics College of Science

National Chiao Tung University in partial Fulfilment of the Requirements

for the Degree of Doctor of Philosophy

in Electrophysics November 2010

Hsinchu, Taiwan, Republic of China

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Acknowledgements

I wish to express my deepest gratitude to Professor Baruch Rosenstein for his guidance and support throughout my research. I appreciate the scientific collaboration with him.

My special thanks to Professor Dingping Li for his help on some parts of my work.

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國立交通大學電子物理系摘要高溫超導的發現有趣地修改了超導體熱擾動性質,尤其是熱力學性質和傳輸性質 (尤其是熱流傳輸)。我們在這工作理考慮在磁場下的超導體的擾動效應。在高溫擾動的影響下,特別是對一些非傳統超導體中很強的各向非均勻性的磁特性,系統的熱力學性質和傳輸特性會影響渦流的運動。高溫超導的 Ginzburg-Landau (GL)現象描述顯著成功地描述各種熱力學和傳輸特性。在平均場近似下,擾動可以忽略的情況下它變得相當容易。然而當擾動不可被忽略時,即使是等效描述也會變的相當複雜。當增加需要的假設時,有些進展已經被達成。額外的假設通常用於解析的計算,只有最低的 Landau 能階顯著的貢獻到我們關注的物理量上。對最近實驗研究的大範疇外來的參數(磁場,溫度), 這種近似是無效的,因此必須推廣理論到包含所有的 Landau 能階。過去只有電性傳輸的理論被發展,然而最近實驗已經能夠觀察到磁熱效應和熱流傳輸現象,像是 Nernst 和熱電功率,我們需要延伸擾動理論去包含這些現象。當只考慮高斯擾動時,熱電阻率和電阻率在平均場的轉換溫度下是被預測會發散, 這個理論與實驗結果矛盾。其中一個重要的結論是要符合實驗結果必須要考慮擾動間的交互作用。之前的工作是 S. Ullah 和 A. T. Dorsey 在 GL 方程式中,在最低的 Landau 能階中應用 Hartree-Fock 近似去處理四次方項計算熱流傳輸問題。在這個論文中,我的工作使用更有系統的高斯近似法去推廣包含所有 Landau 能階,並且計算有趣的物理量,像是橫向熱電阻率和 Nernst 訊號去描述 Nernst 效應, 也算了在線性響應下的第二類超導體的渦流範疇中之交流電阻率。我們使用包含雜訊的時變 GL(TDGL)方程式。我們的理論數值結果可以吻合數個高溫超導中的實驗數據。我也使用線性響應研究在外加磁場下層狀第二類超導體中的傳輸特性。使用 TDGL 方程式以及熱雜訊可以得到電阻率和霍爾電阻率。我們的理論結果定量上吻合在強電場下高溫導體的實驗數據。

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FLUCTUATIONS IN TYPE-II SUPERCONDUCTORS

Student: Bui Duc Tinh Advisor: Baruch Rosenstein

Department of Electrophysics National Chiao-Tung University

Abstract

The discovery of high-temperature superconductors (HTSC) has revived interest in ther-mal fluctuation effects in superconductors, both in thermodynamic properties and in transport properties, with emphasis on the heat transport. In this work we shall be con-cerned with the effect of fluctuations on the transport properties of a superconductor in a magnetic field. Under the influence of fluctuations 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 Ginzburg-Landau (GL) phenomenological description of high superconductors has been remarkably successful in describing various thermodynamic and transport properties. In the mean field approximation, when the fluctuations are neglected, it is relatively sim-ple. However when fluctuations are not negligible, even this effective description becomes very complicated. Some progress can be achieved when certain additional assumptions are added. An additional assumption, often made in analytical calculations, is that only the lowest Landau level significantly contributes to physical quantities of interest. However, in a large domain of external parameters (magnetic field, temperature) currently under experimental investigation, this approximation is not valid and now there is a need to generalize the theory to include all Landau levels.

In the past only electric transport has been thoroughly developed theoretically. How-ever recent experimental advance in observing various magnetocaloric coefficient and heat

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are considered, then the thermoelectric conductivity and the electrical conductivity were predicted to diverge at the mean-field transition temperature, in conflict with the experi-mental results. One of important conclusions is that interactions between the fluctuations must be considered in order to obtain even qualitative agreement with the experimental results. An early work on this subject was application of the Hartree-Fock approxima-tion to treat the quartic term within the lowest Landau level approximaapproxima-tion in the GL Hamiltonian for heat transport current by S. Ullah and A. T. Dorsey.

In this thesis I have extended the work to include all Landau level, use more system-atic Gaussian approximation and calculate physical quantities of current interest like the transverse thermoelectric conductivity and the Nernst signal , describing the Nernst effect, as well as ac conductivity in linear response in Type-II superconductor in the vortex-liquid regime. The time-dependent Ginzburg-Landau (TDGL) equation with thermal noise is used. Our results show a good agreement with several experiment and numerical simula-tion on HTSC.

I also studied the transport properties in a layered Type-II superconductor under magnetic field beyond the linear response. By using TDGL equation with thermal noise is to obtain electrical conductivity and Hall conductivity. Our results are in good qualitative and even qualitative agreement with experimental data on HTSC in strong electric fields.

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2.6 Points are σxx(ν − νn) for different samples of LaSCO, with x = 0.12 (un-derdoped, T_c=29 K), x = 0.17 (near optimal doping, T_c=36 K), and x = 0.2 (overdoped, T_c=27 K). The solid line is the theoretical value of α_xy/B, using ξ=30 ˚A and an anisotropy of γ = 20. The dashed line is obtained using a Hartree approximation. . . 19

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2.7 The thermodynamic phase diagram of BSCCO accommodates four distinct phases, separated by a first order melting line Hm(T ) (open circles), which is intersected by the second-order glass line H_g(T ) (solid dots). The inset plots an equivalent phase diagram, calculated based on Ref. [29], consisting of a second-order replica symmetry breaking lines Hg(T ) both above (dotted line) and below (dashed line) the first-order transition Hm(T ) (solid line). . . . 20

2.8 Points are αxy for different temperatures of LaSCO in Ref. [21], with x=0.2 (overdoped, Tc=28 K). The dashed line is the simulation value of αxy in Ref. [14]. The solid line is the theoretical value of αxy with fitting parameters (see text). . . 39 2.9 Comparison of the experimental melting line for overdoped LaSCO in Ref. [21]

with our fitting. . . 40 2.10 Points are eN for different temperatures of YBCO in Ref. [21], with y=6.5

(underdoped, Tc=50 K). The solid line is the theoretical value of eN with fitting parameters (see text). . . 41 2.11 Comparison of the experimental melting line for underdoped YBCO in Ref. [21]

with our fitting. . . 42 2.12 Points are e_N for different temperatures of YBCO in Ref. [21], with y=6.99

(overdoped, Tc=93 K). The solid line is the theoretical value of eN with fitting parameters (see text) . . . 43 2.13 Comparison of the experimental melting line for overdoped YBCO in Ref. [21]

with our fitting. . . 43 3.1 Points are resistivity for different electric fields of an optimally doped YBCO in

Ref. [49]. The solid line is the theoretical value of resistivity calculated from Eq. (3.52) with fitting parameters (see text). . . 62

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3.2 The current-voltage curves calculated from Eq. (3.50) by using the param-eters (see text) for different magnetic fields b = B/Hc2 at temperature

t = 0.75. . . 63

3.3 The current-voltage curves calculated from Eq. (3.50) by using the pa-rameters (see text) for different temperatures t = T /T c at magnetic flied

b = 0.5. . . 63

3.4 Points are resistivity for different magnetic fields of Bi2212 in Ref. [48]. The solid line is the theoretical value of resistivity calculated from Eq. (3.52) in linear case with fitting parameters (see text). . . 64 4.1 Experimental arrangement for Hall effect measurements. . . 67 4.2 Points are Hall conductivity for different electric fields of an optimally doped

YBCO in Ref. [49]. The solid line is the theoretical value of resistivity calculated from Eq. (4.26) with fitting parameters (see text). . . 77 5.1 Points are resistivity for different temperatures of an overdoped YBCO in Ref.

[114]. The solid line is the theoretical value of resistivity calculated from Eq. (5.38) with fitting parameters (see text). . . 89 5.2 Points are resistivity for different temperatures of an overdoped Bi2212 in Ref.

[115]. The solid line is the theoretical value of resistivity calculated from Eq. (5.38) with fitting parameters (see text). . . 90

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Chapter 1 Introduction

1.1 Superconductivity

Superconductivity was first discovered in 1911 by Heike Kamerlingh Onnes while he was studying the resistance of mercury. At the temperature of 4.2 K, he observed that the re-sistance suddenly disappeared and became unmeasurable in a small temperature regime. For some decades later there was no theoretical understanding of the superconducting mechanism except the classical interpretation of London’s equations of the Meissner ef-fect [1], which was able to describe the basic electromagnetic properties of a homogeneous superconductor. The London theory and its future generalizations introduced two im-portant scales: the concepts of correlation length ξ and penetration depth λ. Those two parameter characterize many physical properties of a system. Only in 1950, the first phe-nomenological theory of superconductivity was proposed by GL [2]. This theory, which is called GL theory of superconductivity had great success in explaining the macroscopic properties of superconductors. In particular, Abrikosov [3] showed that the GL theory predicts the division of superconductors into the two categories now referred to as Type-I and Type-II. Seven years later, the complete microscopic theory of superconductivity was finally proposed by Bardeen-Cooper-Schrieffer (BCS) [4]. The BCS theory explained the superconducting current as a superfluid density of Cooper pairs, i.e., pairs of electrons

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interacting through the exchange of phonons. This theory is successfully applied to most superconducting elements which are now called conventional superconductors.

As was shown by Abrikosov [3], two types of superconductors exist, differing by the value of κ = λ/ξ called the GL parameter 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. Type-I superconductors

Figure 1.1: The H-T phase diagram of Type-I and Type-II 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. Hcand Tcare mutually dependent, see Fig. 1.1. Applying an external

mag-netic 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, 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

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negUnlike TypeI, TypeII superconductors have an extra thermodynamically stable state

-Figure 1.2: The order parameter and the magnetic field profiles of a single Abrikosov vortex.

the mixed state [3], in which the external magnetic field partially penetrates the bulk of the superconductor, locally destroying superconductivity. In this case two critical mag-netic fields exist, Hc1 and Hc2 (see Fig. 1.1). Hc1 is the lower critical magnetic field, at

which the magnetic field starts penetrating into the bulk of the superconductor and super-conductivity 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. The relation between Hc2 and ξ are given by

Hc2= Φ0/4πξ2, 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.

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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 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 supercur-rents, 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 (Gcm2) magnetic flux

1.2 Fluctuation phenomena in superconductor

The fluctuation phenomena in clean bulk superconductors become important only in a very narrow ( 10−12 _{K) region in the vicinity of the transition temperature [5]. Aslamazov}

and Larkin [6] demonstrated that the fluctuation region in dirty superconducting films is determined by the resistance per film unit square and could be much wider than in bulk samples. Even more importantly they demonstrated the presence of fluctuation effects beyond the critical region, and not only in thermodynamic but in kinetic characteristics of superconductors too. They have discovered the phenomenon which is called paraconduc-tivity today: the decrease of the resistance of superconductor in the normal phase, still

at T > Tc. Simultaneously this phenomenon was experimentally observed by Glover [7]

and his results were found in perfect agreement with the Aslamazov-Larkin (AL) theory. Since this time the variety of fluctuation effects have been discovered. Their manifestation have been investigated also today, especially in new superconducting systems.

The characteristic feature of superconducting fluctuations is their strong dependence on temperature and magnetic fields in the vicinity of phase transition. This allows us to definitely separate the fluctuation effects from other contributions and to use them as the source of information about the microscopic parameters of a material. Accounting

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for fluctuation effects is necessary in the design of superconducting devices. Many ideas of the theory of superconducting fluctuations have been used in other fields of condensed matter theory, e.g. in developing of the theory of quantum fluctuations.

In the fluctuation theory, as in modern statistical physics on the whole, two meth-ods have been mainly used: they are the diagrammatic technique and the method of functional (continual) integration over the order parameter. Each of them has its own advantages and disadvantages [8]. The years of the fluctuation boom coincided with the greatest development of the diagrammatic methods of many body theory in condensed matter physics. These methods turned out to be extremely powerful: any physical prob-lem, after its clear formulation and the writing down of the Hamiltonian, can be reduced to the summation of some classes of diagrams. The diagrammatic technique allows us in a unique way to describe the quantum and classical fluctuations, the thermodynamical, and transport effects. The diagrammatic technique is especially suited to problems containing a small parameter: in this case it is possible to restrict their summation to the ladder ap-proximation only. In the theory of superconducting fluctuations one such small parameter

exists: as we will show below, it is the so-called Ginzburg-Levanyuk number Gi(D) which

is expressed as some power of the small parameters Tc/EF. In the vicinity of transition,

superconducting fluctuations in influence different physical properties of metal and lead to the appearance of small corrections to corresponding physical characteristics in a wide

range of temperatures. Due to the above mentioned smallness of Gi(D) these corrections

can be evaluated quantitatively in the wide enough temperature region. On the other hand, their specific dependence on the nearness to the critical temperature T − Tc allows

us to separate them in experiments from other effects.

In the description of the effect of fluctuations on thermodynamic properties of the system the method of functional integration turned out to be simpler. The ladder ap-proximation in the diagrammatic approach is equivalent to the Gaussian apap-proximation in functional integration. The method of functional integration turns out to be more effective in the case of strong fluctuations, for instance, in the immediate vicinity of the

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phase transition. The final equations of the renormalization group carried out by means of functional integrations turn out to be equivalent to the result of the summation of the parquet diagrams series. Nevertheless, the former derivation is much simpler.

1.3 Phenomenology of fluctuation thermodynamics

and transport

The problem of fluctuation smearing of the superconducting transition was not even considered during the first half of the century after the discovery of superconductivity. In

bulk samples of traditional superconductors the critical temperature Tc sharply divides

the superconducting and the normal phases. It is worth mentioning that such behavior of the physical characteristics of superconductors is in perfect agreement with both the GL phenomenological theory (1950) [2] and the BCS microscopic theory of superconductivity (1957) [4].

The characteristics of high temperature and organic superconductors, low-dimensional and amorphous superconducting systems studied today strongly differ from those of the traditional superconductors discussed in textbooks. The transitions turn out to be much more smeared out. The appearance of superconducting fluctuations above the critical temperature leads to precursor effects of the superconducting phase occurring while the

system is still in the normal phase, sometimes far from Tc. The conductivity, the heat

capacity, the diamagnetic susceptibility, the sound attenuation, etc. may increase consid-erably in the vicinity of the transition temperature.

The first numerical estimation of the fluctuation contribution to the heat capacity of

a superconductor in the vicinity of Tc was done by Ginzburg in 1960 [5]. In that paper

he showed that superconducting fluctuations increase the heat capacity even above Tc.

In this way fluctuations change the temperature dependence of the specific heat in the vicinity of the critical temperature where, according to the phenomenological Landau

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atures where the fluctuation correction to the heat capacity of a bulk, clean, conventional superconductor is relevant was estimated by Ginzburg to be

Gi ∼ 10−12÷ 10−14, (1.1)

The correction occurs in a temperature range δT many orders of magnitude smaller than that accessible in real experiments.

In the 1950s and 1960s the formulation of the microscopic theory of superconductivity, the theory of Type-II superconductors, and the search for HTSC attracted the attention of researchers to dirty systems, superconducting films and filaments. In 1968, in papers by Aslamazov and Larkin [6], and Maki [9], and a little later in a paper by Thompson [10], the fundament of the microscopic theory of fluctuations in the normal phase of a super-conductor in the vicinity of the critical temperature were formulated. This microscopic approach confirmed Ginzburg’s evaluation [5] for the width of the fluctuation region in a bulk clean superconductor. Moreover, it was found that the fluctuation effects increase drastically in thin dirty superconducting films and whiskers. In the cited papers it was demonstrated that fluctuations affect not only the thermodynamical properties of a super-conductor but its dynamics too. Simultaneously the fluctuation smearing of the resistive transition in bismuth amorphous films was found experimentally by Glover [7], and it was perfectly fitted by the microscopic theory.

In the BCS theory [4] only the Cooper pairs forming a Bose-condensate are considered. Fluctuation theory deals with the Cooper pairs out of the condensate. In some phenomena these fluctuation Cooper pairs behave similarly to quasiparticles but with one important difference. While for the well defined quasiparticle the energy has to be much larger than its inverse life time, for the fluctuation Cooper pairs the “binding energy” E0 turns out

to be of the same order. The Cooper pair life time τGL is determined by its decay into

two free electrons. Evidently, at the transition temperature the Cooper pairs start to

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τGL∼ ~/kB(T − Tc). The microscopic theory confirms this hypothesis and gives the exact coefficient: τGL= π~ 8kB(T − Tc) . (1.2)

Another important difference of the fluctuation Cooper pairs from quasiparticles lies in their large size ξ(T ). This size is determined by the distance by which the electrons forming the fluctuation Cooper pair move apart during the pair lifetime τGL. In the case

of an impure superconductor the electron motion is diffusive with the diffusion coefficient

Ddif f ∼ vF2τscatt (τscatt is the electron scattering time), and ξdir(T ) =

p

Ddif fτGL ∼

vF√τscattτGL. In the case of a clean superconductor, where kBT τscatt À ~ , impurity

scattering no longer affects the electron correlations. In this case the time of electron

ballistic motion turns out to be less than the electron-impurity scattering time τscatt

and is determined by the uncertainty principle: τbal ∼ ~/kBT . Then this time has to

be used in this case for the determination of the effective size instead of τscatt: ξcl(T ) ∼

vF

p

~τGL/kBT . In both cases the coherence length grows with the approach to the critical

temperature as ²−1/2_{, where} ² = ln T Tc ≈ T − Tc Tc , (1.3)

is the reduced temperature. The coherence length can be written in the unique way (ξ = ξcl,dir):

ξ(T ) = √ξ

². (1.4)

Finally it is necessary to recognize that fluctuation Cooper pairs can really be treated as classical objects, but that these objects instead of Boltzmann particles appear as clas-sical fields in the sense of Rayleigh–Jeans. That is why the more appropriate tool to study fluctuation phenomena is not the Boltzmann transport equation but the GL equation for

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classical fields. Nevertheless, at the qualitative level the treatment of fluctuation Cooper pairs as particles. it was demonstrated [8], in the framework of both the phenomenological GL theory and the microscopic BCS theory, that in the vicinity of the transition.

The complete description of its thermodynamic properties can be done through the calculation of the partition function [8]:

Z = tr ( exp Ã −Hb T !) . (1.5)

In the vicinity of the superconducting transition, side by side with the fermionic electron excitations, fluctuation Cooper pairs of a bosonic nature appear in the system. As already mentioned, they can be described by means of classical bosonic complex fields Ψ(r) which can be treated as “Cooper pair wave functions”. Therefore the calculation of the trace in (1.5) can be separated into a summation over the “fast” electron degrees of freedom and a further functional integration carried out over all possible configurations of the “steady flow” Cooper pairs wave functions:

Z = Z DΨDΨ∗_{(r) exp} µ −F [Ψ(r)] T ¶ , (1.6) where F [Ψ(r)] is GL functional.

The appearance of fluctuating Cooper pairs above Tc leads to the opening of a “new

channel” for charge transfer. In the Introduction the fluctuation Cooper pairs were treated

as carriers with charge 2e while their lifetime τGL was chosen to play the role of the

scattering time in the Drude formula. The generalization of the phenomenological GL functional approach to transport phenomena was presented in [8]. Dealing with the fluctuation order parameter, it is possible to describe correctly the paraconductivity type fluctuation contributions to the normal resistance and magnetoconductivity, Nernst effect, Hall effect, thermoelectric power and thermal conductivity at the edge of the transition [8].

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Theory of Nernst effect in high-T

_c

superconductor

2.1 Introduction

The electric field is induced in a metal under magnetic field by the temperature gradient

∇T perpendicular to the magnetic field H, phenomenon known as Nernst effect [8]

(di-rection of the electric field being perpendicular to both ∇T and H). Recently the Nernst

effect in high-Tc superconductors and low-temperature superconductor attracted

atten-tion both theoretically [8, 11–15] and experimentally [16–24]. In these materials effect of thermal fluctuations is very strong leading to depinning of Abrikosov vortices created by the magnetic field in Type-II superconductor below second critical field Hc2(T ). In the

mixed state the Nernst effect is large due to vortex motion, while in the normal state and in the vortex lattice or glass states it is typically smaller. The Nernst effect therefore is a probe of thermal fluctuations phenomena in the vortex matter, but in principle could shed some light on the underlying microscopic mechanism of superconductivity in cuprates. In the vortex-liquid state, a gradient −∇T drives the vortices to the cooler end of the sample because a normal vortex core has a finite amount of entropy relative to the zero-entropy condensate Fig. 2.1 [25, 26]. Because of the 2π phase singularity at each vortex core,

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Figure 2.1: The vortex-Nernst effect in a Type-II superconductor. Concentric circles represent vortices.

vortex motion induces phase slippage [27]. By the Josephson equation 2eVJ = ~ ˙θ, the

time derivative of the phase ˙θ produces an electrochemical potential difference VJ. We

have ˙θ = 2π ˙Nυ, where ˙Nυ is the number of vortices crossing a line kˆy. per second. The

Josephson voltage VJ may be expressed as a transverse electric field E = B × v which is

detected as the Nernst signal.

Figure 2.2 shows the setup in the Nernst experiment [23]. One end of the crystal is glued with silver epoxy onto a sapphire substrate, which is heat-sunk to a copper cold finger. A thin-film heater, silver-epoxied to the top edge of the crystal, generates the heat current flowing in the ab plane of the crystal. The temperature difference ∆T is measured by a pair of fine-gauge Chromel-Alumel thermocouples. A pair of Ohmic contacts are prepared on the edge of the sample by annealing different kinds of conductive materials. After the bath temperature is stabilized, the gradient is turned on. The Nernst voltage is preamplified and measured by a nanovoltmeter as the magnetic field is slowly ramped up. To remove stray longitudinal signals due to misalignment of the contacts, the magnetic field is swept in both directions. Only the field-asymmetric part of the raw data is taken as the Nernst signal.

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Figure 2.2: Crystal mounting geometry in the Nernst experiment.

Measurements of eN in fields H up to 45 T [23] reveal that the vortex Nernst

sig-nal eN has a characteristic “tilted-hill” profile, which is qualitatively distinct from that

of quasiparticles. The hill profile, which is observed above and below Tc, underscores

the continuity between the vortex-liquid state below Tc and the Nernst region above Tc.

Recently, the study of the Nernst effect in NbSe2 reveals a large quasiparticle

contribu-tion with a magnitude comparable and a sign opposite to the vortex signal [24]. A large negative Nernst coefficient, persisting at temperatures well above Tc=7.2 K, was found

in this metal. However, we will concentrate on the vortex-Nernst effect in Type-II su-perconductor of the overdoped La2−xSrxCuO4 (LaSCO) [21], underdoped and overdoped

YBa2Cu3Oy (YBCO) [21], where eN is intrinsically strongly nonlinear in H and generally

much larger than in nonmagnetic normal metal.

In this study we revisit the calculation in TDGL originally performed in Ref. [11] to obtain explicit expressions for the transverse thermoelectric conductivity αxy and the

Nernst signal eN in both a two dimensional 2D) and a three dimensional (3D) model.

Typically only the lowest Landau level (LLL) contribution was investigated [28]. We extend it to higher Landau levels necessary for exploring the experimentally accessible

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made, disorder and crystallization. In this theory the strength of the thermal fluctuations is described by just one dimensionless adjustable parameter η (closely related to the Ginzburg number Gi). This parameter determines simultaneously the location of the melting line measured on the same samples in recent experiments on Nernst effect. The expression of Ref. [29] for the melting line is in good agreement with many experiments in very wide range of materials (as was established recently in [30]) and MC simulation. Then fitting of the transverse thermoelectric conductivity and related quantities practically has no free parameters (of course there is a certain freedom in determining mean field parameters like Hc2 and Tc, but the range is limited by experimental values). The value

fitted from the Nernst effect turns out to be consistent with that derived from the melting line calculated in [29]. We will present the fitting of the melting line for the overdoped LaSCO [21], underdoped and overdoped YBCO [21].

2.2 Heat transport and Nernst effect

When a temperature gradient exists in a metal, the motion of the conduction electrons provides the of heat (in the form of kinetic energy) from hotter to cooler regions. In good conductors such as cooper and silver this transport involves the same phonon collision processes that are responsible for the transport of electric charge. Hence these metals tend to have the same thermal and electrical relaxation times at room temperature. An additional complication in the heat transport case is that the carriers of heat can be either charge carriers like electrons or electrically neutral phonons, whereas electrical current arises only from charge carrier transport. The transformation to the superconducting sate changes the nature of the carriers of the electric current, so it is to be expected that the transport of heat will be strongly affected.

The thermal current density J is the thermal energy per unit time crossing a unit area aligned perpendicular to the direction of heat flow. It is a vector representing the

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transport of entropy density Sφ at the velocity v,

J = T Sφv, (2.1)

from the hotter to the cooler regions of the material [31]. It is proportional to the gradient of the temperature ∇T through Fourier’s law,

J = −K∇T, (2.2)

where K is the coefficient of thermal conductivity.

In the normal state, electrical conductors are good conductors of heat in accordance with the law of Wiedermann and Franz. In the superconducting state, in contrast, the heat conductivity can be much lower because, as Uher [32] points out, Cooper pairs carry no entropy and do not scatter phonon.

In normal state, the principal carriers of thermal energy through metals in the normal state are conduction electrons and phonons. Heat conduction via each of these two chan-nels acts independently, so that the two chanchan-nels constitute parallel paths for the passage of heat. A simple model for the conduction of heat between two points A and B in the

sample is to represent the two channels by parallel resistors with conductivities Ke and

Kph for the electronic and phonon paths, respectively. The conductivities add directly, as

in the electrical analogue of parallel resistors, to give the total thermal conductivity K,

K = Ke+ Kph, (2.3)

In superconducting state, thermal conductivity involves the transport of entropy Sφ; super

electrons, however, do not carry entropy nor do they scatter phonons. Thus the thermal conductivity can be expected to decrease toward zero.

The heat current is sophisticated and has been the subject of a 30 years discussion. This is due to the fact that the notion of heat itself is not well-defined in the Hamiltonian

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formalism, so in order to be consistent A. Larkin and A. Varlamov defined the heat current, the heat transport current and the magnetization current from the basic principles of thermodynamics [8].

The Nernst effect is well-known in semiconductors and turns out to be small in good metals. The problem of the fluctuation contribution to the Nernst (and also in the re-lated Ettinghausen) effect attracted special attention after the experiments [16–18] which demonstrated the appearance of a fluctuation tail above the critical temperature in the Nernst signal of the high temperature superconductor. More recently, the Nernst effect again returned to the center of attention with the appearance of measurements showing a sizeable signal well above Tc, in particular in the underdoped regime [19, 20, 23, 24].

2.3 Nernst effect in superconductor

The appearance of a fluctuation tail above the critical temperature in the Nernst signal

was observed in strongly Type-II superconductors, both low-Tc like NbSe2 and NbSi films

[24] and several different high-temperature materials [17–20, 23]. The related Etting-shausen effect was detected as well [16]. In particular, the Nernst effect was observed well above Tc2(H) and even above Tc in Bi2Sr2CaCu2O8+δ (Bi2212) [23] , strongly

un-derdoped YBCO [20, 21, 23] and LaSCO [19, 20, 22, 23]. With the overdoped regime

(LaSCO with x = 0.20 and Tc=28 K) in Fig. 2.3, the signal rises steeply at each

temper-ature T , attaining a prominent maximum before decreasing. The total data set defines experimentally the region in H and T where vorticity is strongly present. At high fields, all the curves below 14 K are observed to follow a common curve towards zero (dashed line). Hence all the low-T curves vanish at the intercept of the common curve with the

field axis (45-50 T), which corresponds to Hc2(0). Going to higher T , we immediately

encounter an anomaly we immediately encounter an anomaly. Conventionally, the Hc2

line goes linearly to zero at Tc. Hence, ey ought to be finite in a field interval that → 0

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Figure 2.3: The field dependence of ey at indicated T in samples LaSCO.

large and nearly unchanged up to intense fields for close to Tc. The anomalous features

of the Nernst signal become more pronounced when we go to the underdoped regime. The results in underdoped YBCO (with y = 6.50 and Tc=50 K) are showed in Fig. 2.4.

As H increases above melting line, ey rises rapidly, but attains a very broad maximum

that extends undiminished to 30 T. These layered materials are highly anisotropic and

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effect of thermal fluctuations is enhanced. However in less anisotropic materials like the

hole-doped cuprate Nd2−xCexCuO4 (NCCO) [23] and weakly anisotropic and overdoped

or fully oxidized YBCO6.99[23] the effect persists. Fluctuations in these materials cannot

be described by a 2D model and generalization to anisotropic 3D model is required. The quasiparticle contribution to the Nernst signal attains a magnitude comparable to the vortex signal in the superconducting state. More recently, in experiment on amorphous thin films of the conventional low temperature superconductor Nb0.15Si0.85[24], a Nernst

signal generated by short-lived Cooper pairs in the normal state in Fig. 2.5. In these

Figure 2.5: (Color) Nernst signal (N ) as a function of magnetic field for temperatures ranging from 0.180 K to 0.360 K (upper left panel) and from 0.56 to 4.3 K (upper right panel) measured on thin films of Nb0.15Si0.85 (with Tc=380 mK and thicknesses 35 nm).

amorphous films, the contribution of free electrons to the Nernst signal is negligible. In-deed, the Nernst coefficient of a metal scales with electron mobility. The extremely short mean free path of electrons in amorphous Nb0.15Si0.85 damps the normal-state Nernst

ef-fect and allows a direct comparison of the data with theory. In the zero-field limit and close to Tc, the magnitude of the Nernst coefficient was found to be in quantitative

agree-ment with a theoretical prediction [12] by Ussishkin et al, invoking the superconducting correlation length as its single parameter. At high temperature and finite magnetic field,

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the data were found to deviate from the theoretical expression. In electron-doped cuprate NCCO the quasiparticle contribution to the Nernst signal is large [23]. The quasiparticle contribution actually dominates the Nernst signal far below Tc. Nevertheless, the vortex

signal retains its characteristic tilted-hill profile which is easily distinguished from the monotonic quasiparticle contribution.

The observation of the Nernst effect above Tc along with other strong fluctuation

effects was interpreted as a support for the preformed pairs scenario for the mechanism of the transition to the superconducting state. At the same time thermal fluctuations in

high-Tc materials lead to many other remarkable phenomena, most notably vortex lattice

melting and thermal depinning well studied both experimentally and theoretically over the last two decades, so that the theory of the Nernst effect should be consistent with the theory of these phenomena. Most importantly, the material parameters determining the fluctuation strengths can be determined from these better studied effects since in many recent experiments at least the melting line was measured on the same samples.

Theory of the electronic and the heat transport (including the Nernst effect) based on the phenomenological TDGL equations with thermal noise describing strongly fluctuating superconductors was developed long time ago [8, 11]. More recently within the same framework I. Ussishkin et al. [12] calculated perturbatively the low-field Nernst effect for

T > Tc due to contribution of Gaussian fluctuations and obtained results in agreement

with a microscopic Aslamazov-Larkin [8] calculation. They obtained the result for αSC

xy ,

which diverges as the conductivity, and in reasonable agreement with experimental data on LaSCO in Fig. 2.6. αSC xy ∝ σSCxy ∝ 1 (T − Tc)(d−4)/2 . (2.4)

If only Gaussian fluctuations are considered then, αxy, diverges at the mean-field

transi-tion, in conflict with the experimental results. One of important conclusions that inter-actions between the fluctuations must be considered in order to obtain even qualitative

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Figure 2.6: Points are σxx(ν − νn) for different samples of LaSCO, with x = 0.12 (underdoped,

Tc=29 K), x = 0.17 (near optimal doping, Tc=36 K), and x = 0.2 (overdoped, Tc=27 K). The solid line is the theoretical value of αxy/B, using ξ=30 ˚A and an anisotropy of γ = 20. The dashed line is obtained using a Hartree approximation.

agreement with the experimental results. S. Ullah and A. T. Dorsey [11] applied the Hartree approximation to treat the quartic term in the GL Hamiltonian within LLL. In the limit of high magnetic fields, they found a smooth crosser from a regime dom-inated by two-dimensional Gaussian fluctuation for T > Tc2(H), to mean-field results

for T < Tc2(H), with no intervening divergence, in agreement with the experimental

re-sults. The absence of such a divergence is due to the one-dimensional character of the fluctuations-fluctuations transverse to the applied magnetic fields.

S. Mukerjee et al. [14] numerically simulated the two dimensional TDGL equation with Langevin thermal noise for T < Tcand obtained results in reasonable agreement with

experimental data on LaSCO [21] at lower temperature, but the transverse thermoelectric conductivity became independent of magnetic field at higher temperatures in contrast to experiment. The simulation of this system, even in 2D, is difficult and it was one of our goals to supplement it with a reliable analytical expression in the region of the vortex liquid, namely in the region above the melting line (see Fig. 2.7) at which the vortex matter becomes homogeneous on a scale of several lattice spacings and the crystalline symmetry is lost. In this phase the pinning is ineffective and, unlike in the vortex glass

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phase, vortices actively promote the Nernst effect. Recent understanding of the vortex matter phase diagram is summarized in Fig. 2.7. There are four phases separated by two transition lines [33]: the first order melting line (sometimes called the order-disorder line at lower temperatures, Hm(T ) line in Fig. 2.7) and the irreversibility (or glass) continuous

transition. The melting line separates crystalline phases from a homogeneous phases,

Figure 2.7: The thermodynamic phase diagram of BSCCO accommodates four distinct phases,

separated by a first order melting line Hm(T ) (open circles), which is intersected by the second-order glass line Hg(T ) (solid dots). The inset plots an equivalent phase diagram, calculated based on Ref. [29], consisting of a second-order replica symmetry breaking lines Hg(T ) both above (dotted line) and below (dashed line) the first-order transition Hm(T ) (solid line).

while the glass line (Hg(T ) line in Fig. 2.7) separates pinned phases from the unpinned

ones. The mean field Hc2(T ) line in strongly fluctuating superconductors becomes a

crossover. Both pinning and crystalline order lead to a strong reduction of the Nernst signal and therefore these phases will not be considered here. We concentrate on the vortex liquid phase (see Fig. 2.7) and discuss the melting line and disorder only as limits of applicability of the theory and for determining the material parameters. The quantitative GL theory of the vortex liquid have been developed recently and it was established that the Hartree-Fock approach for the thermodynamic is close to the convergent Borel-Pade one in the wide region of the vortex liquid phase [29].

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2.4 The Ginzburg-Landau Model in 2D

2.4.1 Free energy

To describe fluctuation of order parameter in thin films or layered superconductors one can start with the GL free energy:

F = s0 Z dr ~2 2m∗|DΨ| 2_{+ a|Ψ|}2₊ b0 2|Ψ| 4_, _(2.5)

where A = (−By, 0) describes a constant and practically homogeneous magnetic field (we generally neglect small fluctuations of the magnetic field due to magnetization which are of order 1/κ2 _{<< 1 in the region of interest) in Landau gauge and the covariant derivative}

is defined by D ≡ ∇+i(2π/Φ0)A, with Φ0 = hc/e∗ being the flux quantum, e∗ = 2|e|. For

simplicity we assume linear dependence a(T ) = αTmf

c (tmf − 1), tmf = T /Tcmf, although

the temperature dependence can be easily modified to better describe the experimental

coherence length. The “mean field” critical temperature Tmf

c depends on the ultraviolet

(UV) cutoff, Λ, specified later. It is higher than measured critical temperature Tc due

to strong thermal fluctuations on the mesoscopic scale. The order parameter effective “thickness” of a layer, s0_{, is assumed to be small enough, so that order parameter does not}

vary considerably inside the layer (namely does not exceed the coherence length ξz(T )

along the field direction) and layers are nearly independent. We apply this model to describe experiments not just in BiSCCO and other highly anisotropic materials, but also in overdoped LaSCO [21] and strongly underdoped YBCO [21]. For more isotropic optimally doped or fully doped YBCO [21] an anisotropic 3D GL model (neglecting the layered structure) would be more appropriate. For materials between the two extremes, a more complicated model like the Lawrence-Doniach one should be used.

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2.4.2 Relaxation dynamics and thermal fluctuations

Since we are interested in transport phenomena, it is necessary to introduce some kind of dynamics for the order parameter. The simplest is a gauge-invariant version of the “Type A” relaxational dynamics [8, 34]. In the presence of thermal fluctuations, which on the mesoscopic scale are represented by a complex white noise [8, 35], it reads:

~2_γ0

2m∗DtΨ = −

δF

δΨ∗ + ζ, (2.6)

called in the present context TDGL equation. Explicitly the TDGL equation for the superconducting order parameter is

~2_γ0

2m∗DtΨ =

~2

2m∗D

2_{Ψ − aΨ − b}0_|Ψ|2_{Ψ + ζ,} _(2.7)

where Dt ≡ ∂/∂t − i(e∗/~)Φ is the covariant time derivative with φ (r) = −Ey being

the scalar potential describing electric field. To incorporate the thermal fluctuations via Langevin method, the noise term ζ (r, t), having Gaussian correlations

hζ∗(r, t)ζ(r0, t0)i = ~

2_γ0

m∗_s0T δ(r − r

0_{)δ(t − t}0_), _(2.8)

is introduced. Here δ(r − r0_{) is the two dimensional δ function of the in-plane coordinates,}

and the inverse diffusion constant γ0_{/2, controlling the time scale of dynamical processes}

via dissipation, is real, although a small imaginary (Hall) part is also generally present [36].

Throughout most of the thesis we will use the coherence length, ξ = (~2_/2m∗_αTmf c )1/2,

the zero-temperature correlation length as a unit of length, and Hc2(0) = Φ0/2πξ2 being

the zero-temperature critical field (extrapolated by the linear formula from Tc, actual

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rescaled as Ψ2 _{→ (2αT}mf

c /b0)ψ2. The dimensionless Boltzmann factor in these units is:

F T = 1 η_2Dmftmf Z dr · 1 2|Dψ| 2_{− (a} h+ b 2) |ψ| 2₊1 2|ψ| 4 ¸ , (2.9)

where the covariant derivatives in dimensionless units in Landau gauge are Dx = _∂x∂ − iby,

Dy = _∂y∂ with b = B/Hc2(0), and the constant is defined as ah = (1 − tmf − b)/2. The

dimensionless fluctuations’ strength coefficient is

η_2Dmf = q

2Gimf_2Dπ, (2.10)

where the Ginzburg number is defined by

Gimf_2D = 1 2 µ 8e2_κ2_ξ2_Tmf c c2_~2_s0 ¶₂ . (2.11)

In analogy to the coherence length and the penetration depth, one can define a characteristic time scale. In the superconducting phase a typical “relaxation” time is

tGL = γ0ξ2/2. It is convenient to use the following unit of the electric field and the

dimensionless field: EGL = Hc2ξ/ctGL, E = Ey/EGL. The TDGL Eq. (2.7) written in

dimensionless units reads

∂tψ − 1 2D 2_{ψ − (a} h+ b 2)ψ + |ψ| 2_{ψ − iEyψ = ζ,} _(2.12)

In terms of dimensionless quantities the Gaussian correlations read:

hζ∗(r, t)ζ(r0_{, t}0_{)i = 2η}mf

2Dtmfδ(r − r0)δ(t − t0), (2.13)

where the thermal noise was rescaled as ζ = ζ(2αTmf

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2.4.3 The heat and the electric total and transport currents

The total heat current density is written by [8, 11, 37]:

Jh _{= −} ~2 2m∗ ¿µ ∂ ∂t− i e∗ ~φ ¶ Ψ∗ µ ∇ − i2π Φ0 A ¶ Ψ À + c.c., (2.14)

while the total electric current is

Je_{= i}e∗~ 2m∗ ¿ Ψ∗ µ ∇ − i2π Φ0 A ¶ Ψ À + c.c. (2.15)

In terms of dimensionless quantities the currents read:

Jh = J_GLh jh, jh = − ¿µ ∂ ∂t − iEy ¶ ψ∗(∇ − iA) ψ À + c.c, (2.16) and Je= J_GLe je, je = i 2hψ ∗_{Dψi + c.c.} _(2.17) with Jh

GL = c~Hc2/(4πξ3γe∗κ2) and JGLe = cHc2/(2πξκ2) being the unit of the heat and

electric current density, respectively. Consistently the conductivity will be given in units of σGL= JGL/EGL= c2γ0/(4πκ2). This unit is close to the normal state conductivity σn

in dirty limit superconductors [38]. In general there is a factor k of order one relating the two: σn= kσGL.

An important aspect of the calculation of the electrothermal conductivity, discussed in detail [39], is the need to account for bulk magnetization currents. In the presence of a magnetic field, the system has magnetization current in equilibrium. The total heat current defined by Eq. (2.14) is thus a sum of transport and magnetization parts,

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The magnetization current is current that circulates in the sample and does not contribute to the net current which measured in a transport experiment. On the other hand, it does contribute to the total microscopic current, and it is thus necessary to subtract it from the total current to obtain the transport current response. In the presence of an applied electric field, it was shown in Ref. [39] that the magnetization current is given by

Jh

mag = cM × E, (2.19)

where M is the equilibrium magnetization.

Generally, to define the transport coefficients, the electric and heat transport current densities are related to the applied (sufficiently weak) electric field and the temperature gradient by

J_tr(e)i = σij_Ej_{− α}ij_∇j_T, _(2.20)

J_tr(h)i= eαijEj − κij∇jT, (2.21)

where σ, α, eα, and κ are the electrical, the thermoelectric, the electrothermal, and the

thermal conductivity components of the conductivity tensor (i, j = x, y). The Onsager relation implies eα = T α. The Nernst coefficient νN, under the condition J(e)tr = 0 is

expressed in terms of the above coefficients as

νN = Ey (−∇T )xB = 1 B αxyσxx− αxxσxy σ2 xx+ σxy2 . (2.22)

If the system shows no significant Hall effect (only such systems will be considered), then

σxy = 0 and the expression simplifies:

νN =

αxy

Bσxx

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The Nernst signal is defined

eN =

Ey

(−∇T )x

= BνN. (2.24)

For comparison with experiment, the fluctuation contribution, σxx and eN, should be

added to the normal sate contribution, σn _{and e}n

N. However, the normal state the Nernst

signal en

N is very small in these materials [12, 23] and will be largely ignored in what

follows.

It then follows that the electrothermal conductivity is given by

e αxy ≡ Jh (tr)x Ey = Jxh Ey + cMz. (2.25)

The both terms contribute as will be shown in the following Sections.

2.5 The transverse thermoelectric conductivity in the

vortex liquid phase

2.5.1 Melting of the vortex solid, vortex glass and the range of

validity of the gaussian approximation

At low temperatures vortex matter organizes itself into a (usually, but not always) hexag-onal vortex lattice. When disorder can be effectively neglected (either in very clean materials or when thermal depinning occurs), one can consider transport of the vortex lattice as a whole. Expressions for the electric and the thermal conductivities near Hc2(T )

neglecting thermal fluctuations were obtained in [11], and according to results the Nernst effect is generally very small compared to one in the vortex liquid. This can be qualita-tively understood as a result of rigidity of the lattice. Below the melting line the situation in this respect does not change much. Moreover due to unavoidable presence of disorder,

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in high-Tc superconductors thermal fluctuations are strong enough (especially for high

anisotropy and high magnetic fields) to destroy the expectation value of the condensate

hψi = 0. We always assume that thermal fluctuations melted away and in addition

tem-perature is high enough to thermally depin the vortex liquid (avoiding the “vortex glass”). As a consequence impurities in the vortex liquid are neutralized. To determine the range of validity of the above assumptions one has to estimate the location of the melting and the irreversibility lines. Within the LLL approximation (which is valid near melting in wide range of parameters [29]) the line separating the crystalline and the homogeneous phases is given in 2D by

a2D_T ≡ − (2Gi2D)−1/4

¡

bt¢−1/2¡1 − t − b¢= −13.6, (2.26)

where t = T /Tc and aT is the dimensionless “LLL scaled” temperature with

Gi2D ≡ 1 2 µ 8e2_κ2_ξ2_T c c2_~2_s ¶2 , (2.27)

being a 2D analog of the Ginzburg parameter characterizing the strength of thermal fluctuations on the mesoscopic scale. Eq. (2.26) determines the melting line in Fig. 2.7 and in turn the melting line fixes the Gi in all the fits to experimental data below. This expression was obtained from the comparison of the calculated free energies of the vortex lattice (expansion to two loop order) and of the vortex liquid within the Borel-Pade approach. The corresponding value and definition for 3D are

a3D

T = −21/3(Gi)−1/3(bt)−2/3(1 − t − b) = −9.5, (2.28)

where the Ginzburg number in 3D is defined as

Gi ≡ 1 2 µ 8e2_κ2_ξT cγ c2_~2 ¶₂ , (2.29)

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and γ ≡ pmc/m∗ is an anisotropy parameter. Note that here we use the standard

definition of the Ginzburg number different from that in Ref. [29].

In the presence of disorder, vortex matter can be pinned. It leads to several phe-nomena. On the one hand side the vortex lattice is destroyed effectively at large fields, but on the other hand side vortices are pinned and cannot take advantage of thermal fluctuations. The irreversibility or the vortex glass line determining the region in which thermal fluctuations overpower the quench by disorder is given in 2D by [33]

ag_T ≡ 42r − 1√ 2r , (2.30) where r = Gi −1/2 2D 4t (1 − t) 2_n, _(2.31)

and dimensionless parameter n characterizes the disorder strength (similar formulas exist in 3D) [29]. This determines the dotted line Hg(T ) in Fig. 2.7.

2.5.2 Magnetization in the vortex liquid within the Gaussian

approximation

In order to calculate magnetization, it is simpler to use the statistical mechanics rather than the time dependent approach.

f = F T = 1 η_2Dmftmf Z dr · 1 2|D| 2_{ψ − (a} h+ b 2) |ψ| 2₊1 2|ψ| 4 ¸ , (2.32)

In the framework of this approximation, free energy, Eq. (2.32), is divided into an opti-mized quadratic part K, and a “small” part V . Then K is chosen in such a way that the energy of a Gaussian state is minimal [29]. In liquid phase with an arbitrary homogeneous

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U(1) symmetric state, one variational parameter ε is sufficient K = 1 ηmf_2Dtmf Z dr · ψ∗ µ −1 2D 2₋ b 2+ ε ¶ ψ ¸ . (2.33)

The small perturbation is therefore

V = 1 ηmf_2Dtmf Z dr · (−ah− ε) |ψ|2+ 1 2|ψ| 4 ¸ . (2.34)

The eigenvalue of nth _{level is −}1

2D

2_{ϕ = (n +}1

2)ϕ. For simplified in writing, we introduce

ggauss(ε) = gtrlog(ε) + hV (ε)iK which is relative to the free energy density as fef f =

−η_2Dmftmfggauss, where gtrlog ≡ − log ·Z DψDψ∗exp(−K) ¸ = b 2π ∞ X n=0 log(nb + ε), (2.35) hV i_K = −(ah + ε) b 2π ∞ X n=0 1 nb + ε+ η mf 2Dtmf Ã b 2π ∞ X n=0 1 nb + ε !₂ , (2.36)

( Magnetic field independent term appear in the free energy density is dropped because it is irrelevant to our study on magnetization.) Both terms has ultraviolet divergency,

namely at large n the sums diverge. An UV cutoff Nf + 1 = Λ_b are introduced for

regularization. To extract the divergent part, one can interpolate the gtrlog to two terms:

gtrlog = b 2π ( _N f X n=1 " log (nb + ε) − Z _n+1/2 n−1/2 log (xb + ε) dx # + log ε + Nf X n=1 Z _n+1/2 n−1/2 log (xb + ε) dx ) . (2.37)

The last term is divergent and for large n, it can be approximated by log(1 + x) ∼ x:

b Nf X n=1 Z _n+1/2 n−1/2 log (xb + ε) dx

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Therefore one can divided gtrlog to an infinite part with Λ and a finite part, u: gtrlog = 1 2 · Λ(log Λ − 1) + (ε − b 2) log Λ ¸ + u(ε, b). (2.39)

The finite part u can be simplified as

u(ε, b) = b

2πfs(ε/b) +

b

2π(1/2 − ε/b) log b, (2.40)

where the function fs is defined as

fs(x) = log x−(x+1/2)(log(x+1/2)−1)+ ∞ X n=1 " log(n + x) − Z _n+1/2 n−1/2 log(y + x)dy # , (2.41)

which is basically − ln Γ(x) plus a constant.

Turning to the interactions part, we perform the “bubble” integral which diverges logarithmically: b 2π ∞ X n=0 1 nb + ε = 1 2πlog Λ + u 0_, _(2.42) where u0 _≡ ∂ ∂εu(ε, b) = 1

2π[ fs0(ε/b) − log b], and the derivative of fs is a polygamma

function, ψp, i.e. f_s0 = ∞ X n=1 " 1 n + x − Z _n+1/2 n−1/2 1 (y + x)dy # + · 1 x − log (x + 1/2) ¸ = −ψp(x). (2.43)

The total free energy in Gaussian variational approximation for all Landau levels is obtained, ggauss(ε) = 1 2πΛ(log Λ − 1) − η2Dt µ 1 2π log Λ ¶₂ − (ar_h+ ε) µ 1 2πlog Λ ¶ −(ar

h+ ε)u0+ u(ε, b) + η2Dt(u0)2. (2.44)

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ah−η2D_π tlog Λ = (1 − t − b)/2. The first three terms are divergent, however, they will not

contribute to physical quantities such as magnetization, specific hear ...etc. Minimizing the energy, we get the gap equation

ε = −ar_h + 2η2Dtu0(ε, b). (2.45)

Substitute the solution εs to ggauss one get the minimized free energy density fef f =

−η2Dtg: g = 1 2πΛ(log Λ − 1) − η2Dt µ 1 2πlog Λ ¶₂ − (ar_h + ε) µ 1 2πlog Λ ¶ +u(ε, b)|εs− η2Dt(u 0₎2_| εs. (2.46)

Magnetization 2D can be obtained by taking the first derivative of Gibbs energy with respect to magnetic field b.

M2D = − Hc2 4πκ2η2Dt∂bg = − T 2Hc2ξ2s0 (∂bu − 2η2Dtu0∂bu0). (2.47)

Similar calculation, magnetization 3D takes a form

M3D = − Hc2 4πκ2ηt∂bg = − T 2Hc2ξ2ξz (∂bu3D− 2ηtu03D∂bu03D). (2.48)

The function u(ε, b) can be written in the following form

u3D(ε, b) = 1 √ 2πb 3/2_v³ε b ´ , (2.49) where v (x) = ∞ X n=0 · √ n + x − 2 3(x + n + 1 2) 3 2 +2 3(x + n − 1 2) 3 2 ¸ − 2 3(x − 1 2) 3 2. (2.50)

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2.5.3 Vortex liquid within the Gaussian approximation

Due to thermal fluctuations the expectation value of the order parameter in vortex liquid is zero hψ(r, t)i = 0. Therefore contribution to the expectation values of physical quantities like the electric and the heat current come exclusively from the correlations. The most important is the quadratic one

C(r, t; r0_{, t}0_{) = hψ(r, t)ψ}∗_(r0_{, t}0_{)i ,} _(2.51)

called the correlation function of the order parameter. In particular the superfluid density is

h|ψ(r, t)|2_{i = C(r, t; r, t).} _(2.52)

A simple approximation which captures the most interesting fluctuations effects in the Gaussian approximation (see Refs. [35, 40] and Appendix B for details), in which the cubic term in the GL equation Eq. (2.12) |ψ|2_{ψ is replaced by a linear one 2 h|ψ|}2_{i ψ}

· ∂t−1 2D 2_{+ ε −} b 2 ¸ ψ = ζ, (2.53)

leading the “renormalized” value of the coefficient:

ε = −ah + 2h|ψ|2i. (2.54)

The formal solution of this equation is

ψ(r, t) = Z dr0 Z dt0_G 0(r, t; r0, t0)ζ(r0, t0), (2.55)

(45)

where G0 is the equilibrium Green’s function (GF) which satisfies · ∂t− 1 2D 2_{+ ε −} b 2 ¸ G0(r, t; r0, t0) = δ(r − r0)δ(t − t0). (2.56)

The easiest way to find G0 is by Fourier transformation, which immediately gives

· iΩ −1 2D 2_{+ ε −} b 2 ¸ G0(r, r0, Ω) = δ(r − r0), (2.57) where G0(r, r0, Ω) = Z t0 G0(r, t; r0, t0)e−iΩ(t−t 0₎ . (2.58)

By expanding G0 in term of the Landau eigenfunction one has

G0(r, r0, Ω) = X n ϕn(r)ϕ∗n(r0) iΩ + En , (2.59) where ϕn(r) satisfies " −1 2 µ ∂ ∂x − iby ¶₂ − 1 2 ∂2 ∂y2 + ε − b 2 # ϕn(r) = Enϕn(r). (2.60)

Equation (2.60) is solved exactly in Quantum mechanics, so one gets

G0(r, t; r0, t0) = √ b 4π2 Z Ω,ey0 G0(ey, ey0, Ω, ey0)e−i √ bey0(x−x0)_eiΩ(t−t0)_, _(2.61) where ey =√by , and ey0 = −kx/ √

b, kx is the x component of the vector momentum and

G0(ey, ey0, Ω, ey0) = µ b π ¶_1/2 exp£−(ey − ey0)2/2 − (ey0− ey0)2/2 ¤ X n 1 2n_n! Hn(ey − ey0)Hn(ey0− ey0) (iΩ + En) , (2.62)

Ginzburg-Landau 理論對於 Type-II 超導體的傳輸性質與漲落之研究

國 立 交 通 大 學

電子物理學系

博 士 論 文

Ginzburg-Landau 理論對於 Type-II 超導體的傳輸性質與漲落之研究

Ginzburg-Landau theory of transport properties and fluctuations in

Type-II superconductors

博 究 生：卜德廷

指導教授：儒森斯坦 教授

Ginzburg-Landau 理論對於 type-II 超導體的傳輸性質與漲落之研究

Ginzburg-Landau theory of transport properties and fluctuations in

type-II superconductors

博 究 生：卜德廷 Student：Bui Duc Tinh

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

Acknowledgements

FLUCTUATIONS IN TYPE-II SUPERCONDUCTORS

Abstract

Contents

Chapter 1

Introduction

1.1

Superconductivity

1.2

Fluctuation phenomena in superconductor

1.3

Phenomenology of fluctuation thermodynamics

and transport

Theory of Nernst effect in high-T

c

superconductor

2.1

Introduction

2.2

Heat transport and Nernst effect

2.3

Nernst effect in superconductor

2.4

The Ginzburg-Landau Model in 2D

2.4.1

Free energy

2.4.2

Relaxation dynamics and thermal fluctuations

2.4.3

The heat and the electric total and transport currents

2.5

The transverse thermoelectric conductivity in the

vortex liquid phase

2.5.1

Melting of the vortex solid, vortex glass and the range of

validity of the gaussian approximation

2.5.2

Magnetization in the vortex liquid within the Gaussian

approximation

2.5.3

Vortex liquid within the Gaussian approximation

國立交通大學

博士論文

博究生：卜德廷

指導教授：儒森斯坦教授

博究生：卜德廷 Student：Bui Duc Tinh

國立交通大學

電子物理學系

博士論文

_c