Phương pháp tiếp cận vi mô ginzburg landau đối với sự đồng tồn tại pha trong hệ nhiều hạt tt tiếng anh

INTRODUCTION In ferromagnets below the Curie temperature, Tc , the electron spins align to produce a net magnetization It was thought that superconductivity was incompatible with ferromagnetism for a long time This point of view derives from the microscopic theory of superconductors published in 1957 by Bardeen, Cooper, and Schrieffer (BCS) In the framework of the standard BCS theory, superconducting condensate is formed under the influence of an attractive force due to lattice vibrations which binds electrons with antiparallel spins in singlet Cooper pairs When magnetic impurity atoms are put in conventional superconductor, the local field around impurity atoms prevents the formation of singlet Coopers [12] This causes a rapid drop in the superconducting phase transition temperature Tsc However, around 1980, it was realized that under special conditions superconducting order could coexist with antiferromagnetic order [64], where adjacent electron spins arranged into an opposite configuration For instance, in heavy fermion antiferromagnets, the itinerant magnetic moments have almost no impairing effect on singlet Cooper pairs, because the average exchange interaction is zero The first observation of zero electrical resistance in the ferromagnet state of HoMo6 S8 was published by Lynn et al [42], followed by Genicon et al [22] Subsequently, neutron scattering experiments carried out on singlet crystals of HoMo6 S8 by Rossat et al [52] confirmed further the observations above They found that when T < 0, 60K, with slow cooling, a ferromagnetic phase appeared The magnetic field strength in the ferromagnetic phase increases and then reaches saturation at about T = 0, 4K, indicating that below this temperature, the thermal fluctuations are small This material has a superconducting phase transition at T = 1, 82K and a magnetic phase transition at T = 0, 67K, in the vicinity of the superconducting reentry to the normal conducting state [42] The discovery of the first superconducting ferromagnet (Tsc < Tc ) UGe2 [54] in the year 2000 came as a big surprise In this material, a superconducting transition occurs at a temperature Tsc deeply submerged in the ferromagnet state, ie, below the Curie temperature Tc , without excluding the ferromagnetic order (we call it superconducting ferromagnet) Since then, three other superconducting ferromagnets have been discovered: UIr [4], URhGe [7] and UCoGe [30] These materials have a common feature that the ferromagnetic order is decided by the uranium 5f magnetic moments and has a strong itinerant characteristic In addition, superconductivity appears together with an unstable magnetism The coexistence of superconductivity and ferromagnetism in these materials can be understood as spin fluctuation models: in the vicinity of the ferromagnetic quantum critical point, critical magnetic fluctuations can mediate superconductivity by pairing electrons into spin-triplet Cooper pairs [18, 41], that is, the equalspin pairing (ESP) states (L = 1, Sz = 1; L = 1, Sz = −1 and L = 1, Sz = 0) In recent years, many evidences have shown that such anomalous pairing mechanism exists in superconducting ferromagnetic materials [53, 20, 45] Following the new findings obtained by the foreign research groups, in Vietnam Prof Cat Do Tran was the first one carrying out the theory of superconducting and ferromagnetic phases in UGe2 based on the cohesive properties of the energy region structure research and he also obtained interesting results [15, 14] Members of the strong correlation system group, Center for Theoretical Physics, Institute of Physics have also begun studying the coexistence of magnetic and superconducting phases in UGe2 since 2006 and obtained initial results for FM1 and FM2 phases in UGe2 [65] After that they proposed a superconducting mechanism based on the exchange of magnetic excitations due to crystal field separation of the uranium local levels [66] With the discovery of superconducting ferromagnetic materials, a new research topic in the field of magnetism and superconductivity has been outlined Studying superconducting ferromagnetic materials will help us to unravel how magnetic fluctuations can stimulate superconductivity, a property that is a central theme running through an increasingly diverse family of materials for instance heavy fermion superconductors, high Tsc copper superconductors and the recently discovered superconductors whose main component is FeAs [33] This novel point of view can play a key role in the creation of new superconducting materials However, the problem of understanding the true nature and mechanism of coexistence of superconductivity and ferromagnetism is very complicated, a complete solution is still not available It has been proposed that the combination of horizontal and vertical spin fluctuations may play an important role and will be included in the study of the problem [62] Abrikosov [2] and Mineev et al [44] mentioned that s wave superconductivity may be the result of the electronic interaction mediated by ferromagnetically aligned localized moments In recent years, beside the experimental investigations examining the dependence of the phase transition on the applied pressure and the magnetic field, the theoretical works of research groups in Germany, USA, Russia, Japan, Bulgaria concentrated on finding the phase transition mechanism, the natures of phases and the dependences of the temperature of the phase transition and of the spontaneous magnetization moment on the parameters of the materials Different mechanisms, such as coupled charge density waves and spin density waves [67, 68, 13], magnon exchange [34], screened phonon interactions [55], d-electron exchange [59], etc., have been proposed However, so far the nature of the superconducting phase (singlet or triplet) is unclear, the mechanism of magnetic order is also controversial (due to the localized moment or the itinerant electronics), the cause of superconductivity has not been determined clearly (due to phonon or magnon exchange or charge wave) Especially, the dependence of phase transition temperature and magnetization on pressure has not been studied theoretically In summary, the theory of coexistence of superconducting and ferromagnetic phases is still a challenge for physicists [16, 38] Therefore, we chose this research direction as the main topic with the title of the thesis: “Ginzburg-Landau micro-approach to phase coexistence in many particle systems” The above consideration motivates transforming the fermionic field theory to an effective one based on the coupling fields which are expressed in terms of order parameter fields for different channels The main purpose of the thesis is formulation of a multi-component Ginzburg-Landau (GL) functional which can describe the coexistence of many phases In our research, through HS transformation, a microscopic Hamiltonian will be split into possible channels, then we can get a functional which only depends on order parameters Thereby, we will arrive at a generic performance of GL functional for three order parameter system through calculations based on Green function Based on the specific problems of ferromagnetic superconductivity of the U-based heavy-fermion system, we will draw the formal performance for GL functional, and then using that one to study the coexistence of magnetism and superconductivity in UGe2 system, at once create graphs showing the dependence between order parameters and denoting equilibrium phase domains The thesis will be divided into chapters: Chapter 1, overview In this chapter, the first section gives an overview of the coexistence of magnetic and superconducting orders in heavy fermion systems The next section is the theory of ferromagnetism and superconductivity theory The final section presents the Ginzburg-Landau theory of phase transition Chapter 2, Ginzburg-Landau micro approaches In this chapter, section 2.1 provides an overview of the Green function method and its application for ferromagnetic and superconducting systems In Section 2.2, we will develop the BCS superconducting problem by using functional integral method to calculate the distribution function of the system, establishing the one-component Ginzburg-Landau energy functional for BCS superconductivity Chapter 3, establishing the multi-component Ginzburg-Landau energy functional In this chapter, the first part of building a model of three order parameters and setting the Ginzburg-Landau energy function for three order parameters The following part we build some simple models corresponding to each separate channel Chapter 4, the coexistence of magnetic and superconducting orders in the heavy fermion compounds In this chapter, we firstly present a micro approach to derive the two-component GL energy functional Then basing on the specific problem of ferromagnetic superconductors of the U-based heavy-fermion system, we will draw an official representation for the GL functional with two ferromagnetic and superconducting order parameters, and then use it to study the coexistence of ferromagnetic and superconducting phases in UGe2 system OVERVIEW In this chapter we present an overview of coexistence of the magnetic and superconducting orders in heavy fermion systems, introduce superconductivity and the formation of superconductivity, magnetism of materials, and present the Ginzburg-Landau theory on phase transition 1.1 Overview of coexistence of magnetic and superconducting orders in the heavy fermion system 1.1.1 The phenomenon of coexisting magnetic - superconducting phase in heavy fermion materials Competition of superconducting (SC) and magnetic orders in heavy fermion systems [54, 29, 60, 48, 61, 50, 67] has been one of the central issues for the condensed matter community in recent decade In particular one usually is interested in materials showing the coexistence of SC and antiferromagnetic (AF) [48, 61] or ferromagnetic (FM) [54, 29] orders in uranium-based heavy fermions inter-metallic compounds Experimental evidences for non-phase separated coexistence of ferromagnetic and superconducting orders has recently been found in UGe2 [54, 29, 50, 67] They are in favor of the point of view that the ferromagnetism and superconductivity are caused by 5f -electrons in the same band, and magnetic fluctuations induceding pairing are a possible mechanism This indicates that the attractive effective interaction between the strongly renormalized heavy quasiparticles in UGe2 is not provided by the electron - phonon interaction as in ordinary superconductors, but rather is mediated by electronic spin fluctuations In the vicinity of a ferromagnetic quantum critical point, critical magnetic fluctuations can mediate superconductivity by pairing the electrons in spin-triplet Cooper pairs These spintriplet Cooper pairs have quantum states with parallel electron spins and therefore they can survive in the presence of magnetic moments 1.1.2 Experimental studies about the heavy fermions compounds The experimental researches about UGe2 [54, 29, 63, 25] show that at ambient pressure UGe2 is an itinerant ferromagnet having Curie temperature Tc = 52K, and the spontaneous moment µs = 1.4µB/U − atom An easy axis is the a-axis in orthorhombic crystal [67] When the pressure increases the system passes through two quantum phase transition, one is from ferromagnetic phase to FS phase at P = 1GP a, and the other from ferromagnetic phase to paramagnetic phase at higher pressure Pc = 1.6GPa The superconducting phase exists entirely within the ferromagnetic domain at low temperature and pressure interval between 1.0 ÷ 1.6 GPa with a maximum Tsc = 0.8K near 1.2 GPa In ferromagnetic domain, there are two distinct ferromagnetic phases usually denoted by FM2 (highly polarized phase, µ = 1.5µB) and FM1 (weakly polarized phase, µ = 0.9µB) The line Tx separate two phases ends at Tx = Tsc and the pressure Px = 1.2 GPa The order of transition from FM1 to paramagnetic changes from second to first at the tricritical point Tcr on the T (P ) diagram, and at critical pressure Pc The more P increases, the more both TF M and TF S drop down to disappear almost simultaneously around P ∼ 1.7GPa (see Fig 1.1) The experimental researches about CeRhIn5 [27, 35, 69, 49, 8, 22, 23] also show that at ambient pressure, the AFM order appears at TN = 3.8K with a staggered moment of about 0.8 µB By application of pressure, at Tc < TN the system can be tuned through a quantum phase transition on cooling from AF to AF+SC When pressure increases, the Neel temperature slightly increases and has a smooth maximum around 0.8 GPa Then TN reduces as P approaches critical pressure Pc at which superconductivity sets in, where the superconducting transition temperature Tc = TN The antiferromagnetic order vanishes near Pc∗ = 1.95 GPa due to a first order transition from AF+SC to SC anomaly The coexistence of AF and SC is found in a narrow P range of 1.6-1.95 GPa As P is further increased (P above Pc∗ ) a nice superconducting anomaly appears, in the other words, if Tc > TN the ground state is purely superconducting in a large pressure region from about 1.95 to GPa (see Fig 1.2) 1.2 Ferromagnetism in metals Figure 1.1: Phase diagram of UGe2 determined by magnetization measurements under pressure Tc is the Curie temperature and Tx locates the phase transition between two ferromagnetic phases FM1 and FM2 with different polarization Tsc is the superconducting transition temperature [63, 25] 1.1.3 Theoretical studies about the heavy fermions compounds According to previous theories, magnetism is induced by the spin moments of localized 4f, 5f electrons whereas superconductivity comes from the Cooper pairs formed by conduction electrons Such a discovery, together with a number of reliable experimental data about the coherence length and the superconducting gap [54, 29, 50, 7], leads to the conclusion that 4f -electrons from Ce-atoms and 5f -electrons from U-atoms are responsible for both magnetism (AFM or FM) and SC The Cooper pairs in these metallic compounds belong to a spin-triplet, and magnetic-fluctuation induced pairing is a possible mechanism In recent years, beside the experimental investigations that examined the dependence of the phase transition on the applied pressure and the magnetic field, other theoretical research concentrated on finding the phase transition mechanism, the natures of phases and the dependences of the temperature of the phase transition and of the spontaneous magnetization moment on the parameters of the materials Different mechanisms, such as coupled charge density waves and spin density waves [67, 13], magnon exchange [34], electron interaction mediated by ferromagnetically aligned localized moments [2, 44], screened phonon interactions [55], d-electron exchange [59], M-trigger [58, 57, 56], the multiband model [1, 19, 31, 5], etc., have been proposed These theoretical works have tackled this important issue and provided invaluable information about the interplay between AFM, FM and conventional SC, and unconventional SC in the coexistence states 1.2 Ferromagnetism in metals 1.2.1 Ferromagnetic order in local magnetic moment systems 1.2.1.1 Exchange Interactions Heisenberg model To lead to the concept of exchange interaction, we consider the Helium problem By exchange interaction between electrons, we can explain the nature of ferromagnetism The fraction of the exchanged energy which depends on the orientation of the electron spins equal H=− Jij Si Sj , (1.1) i=j Jij is the exchange integral between the electrons of atoms i and j in the crystal at 0K, if Jij > then the spin magnetic moments of the electrons in i and j atoms are oriented in the same direction and the object is 1.2 Ferromagnetism in metals Figure 1.2: P −T phase diagram of CeRhIn5 at zero magnetic field determined from specific heat measurements with antiferromagnetic (AF, blue) and superconducting phases (SC, yellow) When Tc < TN a coexistence phase AF+SC exist When Tc > TN the antiferromagnetic order is abruptly suppressed A coexistence phase AF+SC exists below Pc∗ [69, 23] a ferromagnet If Jij < 0, then the spin magnetic moments of the electrons in the i and j atoms are in the opposite direction and the object is a antiferromagnet The ferromagnetic sample that the exchange interaction is described by this Hamiltonian operator call the Heisenberg ferromagnetic sample 1.2.1.2 Weiss molecular field theory In the Heisenberg Hamiltonian (1.1) with Si Sj = S(S + 1), we write Si = Si + (Si − Si ), average (1.2) fluctuations This led to the mean field Hamiltonian having the form  H∼ = Jij Si Sj −  ij i Si HW ef f , Jij Sj  = E0 − Si 2 j (1.3) i where HW = j Jij Sj is called the Weiss molecular magnetic field or the effective magnetic field Calculating Sj in the effective magnetic field, it can infer kB Tc = 2S (S + 1) Jij (1.4) j This formula gives the relationship between the Curie temperature TC and the exchange integral Jij If Jij = then TC = 0, it means that the object is not a ferromagnet Thus, the nature of ferromagnetism is due to exchange interaction, with Jij > 1.2.2 The magnetic order of the itinerant spins system 1.2.2.1 The Hubbard model The Hubbard model is defined by the Hamiltonian c†jσ clσ + U H = −t jl σ nj↑ nj↓ − µ j (nj↑ + nj↓ ) j (1.5) 1.2 Ferromagnetism in metals The first term is the kinetic energy: It describes the destruction of an fermion of spin σ on site l and its creation on site j The symbol jl emphasizes that hopping is allowed only between two sites which are adjacent The second term is the interaction energy It goes through all the sites and adds an energy U if it finds the site is doubly occupied The final term is a chemical potential which controls the filling The Hubbard Hamiltonian is exactly solvable in the limits U = and t = The non-interacting limit corresponds to band theory in the tight-binding approximation, while the t = limit corresponds to the atomic limit The noninteracting limit U = In the limit U = 0, the Hamiltonian reduces to the kinetic energy term ξk c†k,σ ck,σ , H0 = (1.6) k,σ where ck,σ is the Fourier transform of cj,σ and d ξk = − µ = −2t k cos kν − µ, (1.7) ν=1 is the dispersion of the free fermions moving in the lattice The density of states reads δ( − N (ξ) = k ), with = ξ + µ, k ∈ [−2dt, 2dt] (1.8) k The local limit t = For vanishing hopping amplitude t = 0, the Hamiltonian reduces to [U nj↑ nj↓ − µ (nj↑ + nj↓ )] H= (1.9) j We deduce the partition function α| e−βH |α = + 2eβµ + e2βµ−βU , Z= (1.10) α the energy as E = Z −1 α| He−βH |α = + 2eβµ + e2βµ−βU −1 U e2βµ−βU (1.11) α and the mean number of spin-σ fermions per site is nσ = βµ e + e−β(U −2µ) Z (1.12) 1.2.2.2 Stoner’s criterion for ferromagnetism Stoner developed a very simple picture of ferromagnetism based on the competition between the kinetic energy cost of making the up and down spin electron numbers different and the associated potential energy gain The basic idea is the following: Consider a system with density of states N (E) ) and both up and down spin electrons filling the energy levels up to the same maximum called the ‘Fermi level’ EF The density of up and down electrons is equal We’ll call it n Let’s compute the change in energy which results from a reduction in the density of down spin electrons by δn and at the same time an increase the number of up spin electrons by δn The result of this process is δE = δP + δK = (−U + (δn)2 )(δn)2 = (−U N (EF ) + 1) N (EF ) N (EF ) (1.13) if U N (EF ) > the total energy change δE < 0, so it is favorable to have the up and down electron densities different and hence favorable to have ferromagntism This is called the Stoner criterion 1.3 Superconductivity and BCS theory 1.3 Superconductivity and BCS theory 1.3.1 The history of superconductivity • Superconducting phenomena were discovered in 1911 at the laboratory of Heike Kammerlingh Onnes [47] in Leiden, Netherlands Onnes and colleagues found that the resistance of the mercury sample suddenly dropped to an un measurable value, meaning it became truly zero at low temperatures So they called superconductivity the perfect electrical conductivity below the critical temperature TC , a property that properly represents the name of the phenomenon • One of the two experiments, which greatly contributed to the proper understanding of the phenomenon and to the formulation of the correct phenomenological and later microscopic theory of superconductivity, is that discovered the Meissner effect in 1933 by Meissner and Ochsenfeld [43] When a superconductor is cooled in a small magnetic field, the flux is spontaneously excluded as it becomes superconducting The Meissner effect demonstrates that a superconductor is, in essence a perfect diamagnet • Heinz and Fritz Londons were the first to understand the real importance of the Meissner and Ochsenfeld observation and soon proposed a phenomenological theory (in 1937) which was able to account for electrodynamic properties of superconductors However, Heinz and Fritz London are only concerned with superconducting phenomena without making any effort to describe the reason Instead they found an equation for the λ depth penetration of superconductors [40] But their results were overestimated by empirical values, and so their assumptions were rejected In 1950 Fră ohlich proposing that the interaction between electrons and lattice vibrations was responsible for the superconductivityn [21], at the same time isotope effect was discovered In fact there were two reports [17, 51], both obtained by Physical Review on the same day 24 March 1950 and published in the same issue of the journal in section Letters to the Editor One by E Maxwell from NBS, Washington D.C and other by C.A Reynolds and Colleagues from Rutgers University, New Jersey Both groups were aware of their work and mutually cited each others work It is important to note the nice agreement of those very precise and carefull measurements The superconducting transition temperature of natural mercury with average atomic weight 200.6 was found to be TC = 4, 156 K in one the papers [17] and TC = 4, 177 K for 198 Hg The other group [51] reported TC = 4, 150 K for natural mercury and TC = 4, 143 K for the sample consisting of the 202 Hg isotope The experimental results within each series of isotopes can be matched to the relationship M 1/2 TC = const • For decades the full understanding of quantum mechanics about superconductivity progressed rather slowly, always after experiment, but finally a breakthrough was achieved in 1950 when the GL theory was proposed on the ground of phenomenology, starting with the general Landau theory of second phase transition as we will see in more detail in section 1.4 • Seven years later a microscopic theory, called the BCS theory, was developed by Bardeen, Cooper and Schrieffer [10, 9] They found that the appearance of superconductivity was aided by the formation of Cooper pairs, a molecular type of loosely coupled two electrons The attractive interaction needed to bond the electrons was found to originate in the electron-phonon interaction Cooper pairs are no longer described by fermi statistics and as bosons they can occupy a coherent quantum state described by a macro wave function Ψ We will outline the main points of BCS theory in section 1.3.2 • It is important to note that two years after proposed the BCS theory, Gor’kov shows that the phenomenological GL theory can actually be deduced precisely from the BCS theory and its coefficients can be thus related microscopic material parameters such as Fermi velocity vF and state density at Fermi N (0) In addition Gor’kov proves that the thermodynamic quantities of the two theories, that is, the BCS gap parameter, and the GL wave functions, Ψ are related by a ratio constant and Ψ can be regard as the wave function of Cooper pair in a cubic structure • Another important great advance was made by Abrikosov in 1957 [3], who realized that the GL equations allowed the existence of superconductors with values of negative energy of superconducting -normal interface He named this new class II superconductors, the materials that create the interface as much as possible, resulting in the penetration of magnetic flux in the sample in the smallest units that are quantum mechanically permitted - The magnetic flux quantum Φ0 = hc 2e Each individual magnetic flux tube has circulating hypercurrents that shield the magnetic field and prevent it from spreading to the rest of the sample These circulating supercurrents are the reason why this magnetic structure is called vortex Because of the negative surface energy, the vortices repel each other at all distances and as a result form a characteristic network in the form of a triangle (called the Abrikosov lattice) • In 1979, Frank Steglich and his colleagues observed superconducting under the temperature TC ≈ 0, 5K in CeCu2 Si2 [60] This material is not an conventional metal in its normal state Instead it is a heavy metal 1.3 Superconductivity and BCS theory fermion Electrons at the Fermi energy level have strong orbital-f properties of Ce The strong Coulomb repulsion between the electrons in the shell -f results in a high effective mass m∗ me at the Fermi energy level, so it is named as above From then on, superconductivity was found in many other heavy fermion compounds • Also in 1979, D Jérome et al [32] (Klaus Bechgaard’s group) observed superconducting in an organic salt called (TMTSF)2 PF2 with TC = 1, 1K The superconducting phenomenon has since been found in various organic materials with a maximum transition temperature of about 18K • In 1986, J G Bednorz and K A Müller [11] observed superconductivity in La2−x Bax CuO4 with TC approximately over 35K In subsequent years, many another superconductors based on the same type of nearly flat surfaces CuO2 has been discovered The record phase transition temperature for copper ions and all superconductors is TC = 138K for Hg0.8 Tl0.2 Ba2 Ca2 Cu3 O8+δ at normal pressure and TC = 164K for HgBa2 Ca2 Cu3 O8+δ under high pressure 3− • In 1991, A F Hebard [26] and colleagues found that carbon compound (fullerite) (K − )3 C60 became superconducting in below TC = 18K TC in this class has been pushed to TC = 33K with Cs2 RbC60 at normal pressure and TC = 38K for Cs3 C60 under high pressure • In 2001, Nagamatsu and colleagues reported superconductivity in MgB2 [46] with TC = 39K • The most recent series of discoveries began in 2008, when Kamihara et al [33] observed superconductivity with TC ≈ 4K in LaFePO, a other layering substances While this result has added a new class of materials based on F e2+ to list superconductors, it has not caused much disturbance due to the low critical temperature However, in 2008, Kamihara and colleagues found superconductivity with TC ≈ 26K in LaFeAsO1−x Fx Shortly thereafter, the maximum critical temperature TC in this iron pnictide layer was pushed up to 55 K Superconducting phenomena were also observed in many related material layers, some of which were oxygen free (for example, LiFeAs) and a number with pnictogen (As) replaced by a chalcogen (for example, FeSe) The general structural element is a flat, square Fe2+ with a pnictogen or chalcogen placed alternately above and below the center of the Fe squares This superconducting phenomenon is considered to be unconventional 1.3.2 BSC theory of Superconductivity 1.3.2.1 The BCS ground state The BCS ground state is a superposition of states constructed from Cooper pairs, each of which is made up of two electrons in the states |k, ↑ and |−k, ↓ uk + vk c†k,↑ c†−k,↓ |0 , |ψBCS = (1.14) k where |0 is the vacuum state without any electrons and uk , vk are as yet unknown complex coefficients satisfied 2 the constraint |uk | +|vk | = for all k Note that the occupations of |k, ↑ and |−k, ↓ are maximally correlated; either both are occupied or both are empty The coefficients uk , vk indicate the amplitudes that is not occupied, is occupied by the pair of electron with k, ↑and −k, ↓ 1.3.2.2 Mutual attraction through intermediate phonons H.Frohlich has shown that electrons can actually attract each other, when there is a deformable ion lattice, which in fact always exists as a background The H.Frohlich mechanism [21] is as follows: An electron attracts the ions which are closest to it The ions respond by moving, though only slightly towards it, thus creating a positive charge redundance around it We say that the electron make polarized lattice Another electron was attracted towards the polarization around the first electron, and in doing so, it was actually attracted to the first electron In physical parlance of many-particle system, this mutual attraction is considered to be mediated by the phonon, that is, by the exchange of virtual quantum oscillations of lattice An electron emits a phonon, which is absorbed by another electron The result is that the attraction through phonon exchange is maximized when the two electrons have equal and opposite momentum This gives the two electrons the maximum advantage of the polarization generated by the particle to the other 1.3 Superconductivity and BCS theory Figure 1.3: Hiệu ứng hút hai điện tử trao đổi phonon [28] 1.3.2.3 Cooper Pairs The BCS theory relies on the assumption that superconductivity arises when the attractive Cooper pair interaction dominates over the repulsive Coulomb force [36] A Cooper pair is a weak electron-electron bound pair mediated by a phonon interaction A “paired” electron is one with opposite momentum and spin that is attracted to this force The energy of the pair satisfies E=2 − ωc e−2/N0 V F 0 t for a ferromagnetic interaction, J0 < for an anti-ferromagnetic interaction Heisenberg Hamiltonian is now written in the following form H = −2 Jij Si Sj (3.18) ij In the ferromagnetic case (J0 > 0) spins will prefer to be aligned That happens when the overlap between the i and j orbitals is large Then (similar to the case of Hund’s rule) electron will tend to align their spin due to the Pauli principle However if we have opposite spins sitting in neighboring atoms then it can be energetically preferable for one to tunnel, which is only possible if the spins are reversed, hence in this situation J0 < 3.2.2.2 The Hubbard Model Perform Fourier transforms, Hamiltonian’s new representation of the system in the basis of Wannier functions takes the form Uii jj a†iσ a†i σ aj tii a†iσ σ − H= σ ajσ (3.19) ii jj ii where tii = N ke ik(Ri −Ri ) (3.20) k and Uii jj = dr dr φ∗Ri (r)φRj (r)Vσ1 σ2 σ1 σ2 (r − r )φ∗Ri (r )φRj (r ) (3.21) Far into the atomic limit, where the atoms are very well separated, and the overlap between neighboring orbitals is weak, the matrix elements tij is exponentially small in the interatomic separation In this limit, the “onsite” Coulomb or Hubbard interaction, Uiiii = U , i Uiiii a†iσ a†iσ aiσ aiσ = i U n ˆ i↑ n ˆ i↓ , generates the dominant interaction mechanism Taking only the nearest neighbor contribution to the hopping matrix elements, and neglecting the energy offset due to the diagonal term, the effective Hamiltonian takes a simplified form known as the Hubbard model, a†iσ ajσ + U H = −t ij n ˆ i↑ n ˆ i↓ (3.22) i where ij is a shorthand used to denote neighboring lattice sites 3.2.3 The Cooper channel Finally, Cooper pairing between fermions in the Cooper channel can cause superconducting phenomena Ignoring pairing terms by direct and exchange channels, the Hamiltonian of the system now takes the form H = H0 + H1C , (3.23) where H1C = σ1 ,σ2 ,σ1 ,σ2 k,k ,q ψσ† (k − q q q q † )ψσ2 (k + ) (VC )σ1 σ2 σ σ (k, k , q)ψσ2 (k − )ψσ1 (k + ) 2 2 (3.24) Depending on the spin correlation is singlet or triplet, we have the conventional or unconventional superconducting model 23 3.3 Discussion 3.2.3.1 The conventional superconducting model In case the Cooper pair is singlet, the Hamiltonian is rewritten as † k ckσ ckσ H= + k,σ where Vkk = Vkk = Vkl k,l Ω c†k↑ c†−k↓ c−l↓ cl↑ (3.25) d3 rV (r)ei(k−k )r , and the average field approximate generated as follows Ω −V if F < k < otherwise, F + ωc (3.26) where F is the Fermi energy and ωc is the cutoff frequency Equation (3.26) shows that we only need to consider the interactions that are allowed in a frequency range of metals, like the assumptions made in the Debye model 3.2.3.2 The unconventional superconducting model The general BCS theory is based on an extended form of microscopic interactions in which only the scattering of the electron pair with the total momentum is zero and the interaction between them is the attractive interaction The Hamiltonian can be written as follows † k ckσ ckσ H= + Vk,l;σ1 σ2 σ3 σ4 k,l σ1 ,σ2 ,σ3 ,σ4 k,σ c†kσ1 c†−kσ2 c−lσ3 clσ4 , (3.27) with a paired scattering matrix (VC )σ1 σ2 σ σ2 (k, k , q) ≡ Vk,l;σ1 σ2 σ3 σ4 = −kσ1 ; kσ2 | Vˆ |−lσ3 ; lσ4 (3.28) can be represented as (VC )σ1 σ2 σ σ2 (k, k , q) † = (VC ) (k, k , q) (iσσy )σ1 σ2 (iσσy )σ with (VC ) (k, k , q) = VC is the attractive interaction constant 3.3 Discussion 24 σ2 (3.29) THE COEXISTENCE OF FERROMAGNETIC AND SUPERCONDUCTING ORDERS IN HEAVY FERMION COMPOUNDS 4.1 Microscopic Derivation Of The Two-Component Ginzburg-Landau Functional 4.1.1 The Model Hamiltonian Our starting point is the grand partition function of the system (3.1) [Dψ] Dψ † Z=   exp −  β ψσ† (k, τ ) (∂τ + dτ σ kσ ) ψσ (k, τ ) + H1 ψ † , ψ (τ )   (4.1)  k For this problem, the generic effective two-body interaction term H1 ψ † , ψ can be broken down to a summation of two possible fermionic bilinear terms with arbitrary parameters {γi }, where i ∈ {d,C} as H1 ψ † , ψ = γd2 H1d ψ † , ψ + γC H1C ψ † , ψ (4.2) the values of the parameters γi should satisfy the identity γd2 + γC = (4.3) For definiteness, we take the interaction matrix in a simple form (V )σ1 σ2 σ σ2 (k, k , q) = (Vd ) (k, k , q)σ σ1 σ1 σ †σ2 σ (4.4) with a constant (Vd ) (k, k , q) = Vd We consider also superconducting (SC) interaction only in the triplet channel, i.e., (V )σ1 σ2 σ † σ2 (k, k , q) = (VC ) (k, k , q) (iσσy )σ1 σ2 (iσσy )σ (4.5) σ2 with a constant (VC ) (k, k , q) = VC Introducing the auxiliary fields M(k, q), σ1 σ2 (k, q) ( they are responsible for magnetism and superconductivity, respectively), The Gaussian integration over the Grassmann field can now be evaluated straight forwardly, giving the formal expression for grand partition function Z= W D [M, M∗ , ∆, ∆∗ ] exp {−S0 [M, M∗ , ∆, ∆∗ ]} exp ln det −1 [G0 ] (4.6) Expanding effective action in Eq (4.6) with respect to the Hubbad-Stratonovich auxiliary fields {M, ∆} to quartic order, only including terms allowed by symmetry of system and retains minimum numbers of the 25 4.2 The coexistence of superconductivity and ferromagnetic orders in UGe2 simplest terms to get the meaningful results, we will obtain G-L free energy functional with the participation of several order parameters describing relationship of density spin wave and superconductivity phases f (M, ∆) = 1 2 2 |M| + αf |M.σ| + βf |M.σ| + |d| + αs |∆| + βs |∆| Vd VC 2 +uf s (M.σ) |∆| + vf s |M.σ| |∆| (4.7) In Eq (4.7), the first three terms describe part of the free energy of a standard isotropic ferromagnet, next three terms describe the superconductivity for M = H = and last two that describe the interaction between the ferromagnetic order parameter M and the superconducting order parameter ∆ The microscopic expressions for the GL coefficients which are functions of temperature (and pressure etc.) are production of free Green functions of electrons and holes They can be summed over fermion Matsubara frequency ωn = (2n + 1)πT and wave vectors k on the basis of Taylor series expansion technique and the application of the residue theorem 4.2 The coexistence of superconductivity and ferromagnetic orders in UGe2 4.2.1 The Ginzburg-Landau energy functional for superconductivity and ferromagnetism Here, we are only interested in the uniform phases, i.e, order parameters d and M that not depend on the spatial vector x BecauseUGe2 is a ferromagnet that has an orthorhombic structure with magnetic moments oriented along one of the crystallographic axes If we choose a coordinate system x//b, y//c, z//a where the magnetic easy axis is the a-axis, then M = (0, 0, M ) Because of the pairbreaking effect of the strong exchange field M, only the Cooper pairs with parallel spins will survive In this case of equal-spin pairing, we can write vector d in the form d = (d1 , d2 , 0), implying that the Cooper pair spinorientation points to the M direction Then, we obtain the G-L energy functional of the triplet ferromagnetic superconductor as follows: fGL (M, φ1 , φ2 , θ) + αf |M| σ0 + βf |M| σ0 Vd = + φ2 + φ22 σ0 + αs VC φ21 + φ22 +βs φ21 + φ22 σ0 − 2φ1 φ2 sin θ.σz σ0 + 4φ21 φ22 sin2 θ.σ0 −4 φ21 + φ22 φ1 φ2 sin θ.σz +uf s |M| φ21 + φ22 σz − |M| φ1 φ2 sin θ.σ0 +vf s |M| φ21 + φ22 σ0 − |M| φ1 φ2 sin θ.σz (4.8) 4.2.2 The phase diagram from the Ginzburg-Landau energy functional approach Using the conditions of equilibrium of the phase which has √ coexistence of the ferromagnetic order and superconducting order (FS), given by sin θ = −1, φ1 = φ2 = φ/ 2, we can rewrite the GL free energy functional (4.8) in term of the reduced form as follows fGL (M, φ) = af M + bf bs M + as φ2 + φ4 + γ0 M φ2 + δ0 M φ2 2 (4.9) where + αf , Vd as = + 2αs , VC af = bf = 2βf , γ0 = 2uf s , bs = 8βs , δ0 = 2vf s (4.10) 26 4.3 Discussions Next, we redefine, for convenience, the free energy in Eq (4.9) in a dimensionless form as f = fGL / bf M04 , 1/2 where M0 = [Λf Tf /bf ] is the value of the magnetization M corresponding to the pure magnetic subsystem d = at T = P = and Tf = Tf (0) The order parameters assume the scaling m = M/M0 ϕ = 1/4 φ/ (bf /bs ) f= M0 , and as a result, the free energy becomes rϕ2 + 21 ϕ4 + tm2 + 12 m4 + γϕ2 m + δϕ2 m2 (4.11) Finally, the conditions of equilibrium and the stable phases for the UGe2 system which has the free energy given by Eq (4.11) are used to outline the phase diagram T − P (Fig 4.1), and the phase diagram which indicates the domains of stability for the N, FM and FS phases in the (t, r) plane, we obtain Figure 4.1: An illustration of T − P phase diagram of UGe2 calculated for Ts = 0, Tf = 52K, Pc = 1.6GP a, γ/κ = 0.1089, δ/κ = 0.1867 The FS phase domain is shaded The solid line shows the second order FM-FS phase transition Figure 4.2: Phase diagram in the (t, r) plane for γ = 0.49, δ = 0.84 The phase diagram for concrete parameters of γ and δ is shown in Fig 4.2 The domains of stability of the N, FM and FS phases are indicated DCB is the line of demarcation between two domains of stability of the FM and FS phases, where the curve DC (the dashed line, to the left of point C) is the second-order phase transition and the line CB is the first-order phase transition The phase transition between the N and the FS phases is first-order and goes along the equilibrium line BA The vertical dashed line coinciding with the r-axis above B, which is the line of demarcation between two domains of stability for the FM and N phases, indicates the N-FM phase transition to be of second order A and C are tricritical points of the phase transitions; B is the triple point 4.3 Discussions 27 CONCLUSIONS AND PROPOSALS Conclusions Specifically in the thesis, we have achieved the following results: Forming a Ginzburg-Landau microscopic approach to phase coexistence Developing the BCS superconducting problem by using functional integral method to calculate the distribution function of the system, from that establishing the one-component Ginzburg-Landau energy functional for BCS superconductivity From the general Hamiltonian of interacting many-particle system leading on the Ginzburg-Landau formulation for coexistence, we obtain a GL microscopic free energy functional containing three ordered parameters, which can describe the coexistence of different phases in a many-particle system Using the stationary conditions of the grand thermodynamic potential with respect to the parameters γi and the minimum energy conditions to establish the self-consistent system of equations These equations describe many-body relations between the physical quantities of the system where many kinds of fluctuations of corresponding order parameters are considered, leading to Hatree-Fock-Bogoliubov (HFB) approximation The BCS gap equation and magnetization equation containing the contributions of other channels are obtained straightforward from HFB approximation, resulting in a possible solution for the coexistence of many phases We have microscopically derived the Ginzburg-Landau free energy functional with two ferromagnetic and triplet superconducting order parameters Based on the specific problem of ferromagnetic superconductivity of the UGe2 system, we transform the correlation between the order parameters described in the two-component Ginzburg-Landau functional into the correlation between order parameters T and P (physical quantities can be obtained directly from the experimental phase diagrams) T − P diagram of UGe2 outlined on the basis of theoretical calculations and calculated numerically has a agreement with the main experimental findings Proposals for further research • In the section 4.2, we have used the two-component Ginzburg-Landau functional to study the coexistence of superconducting and ferromagnetic orders in UGe2 , T − P diagram of UGe2 outlined on the basis of theoretical calculations and shown in Fig.4.1 has a agreement with the main experimental findings However Pm corresponding to the maximum (found at ∼1.45 GPa ) is about 0.25 GPa higher than experimental data [29] If the experimental plots are accurate, this difference may result from not including contribution of anisotropy of the spin-triplet Cooper pair and crystal or from any effect which is outside the scope of our current model The relative importance of this effect on the phase diagram needs to be investigated • In the section 3.1.2, we have established the multi-component Ginzburg-Landau functional describing the correlation among the order parameters Also, we obtain the self-consistent system of equations which describes many-body relations between the physical quantities of the system where many kinds of fluctuations of corresponding order parameters are considered In order to continue this work, we will need to give step by step numerical solution so as to show the relationships between the order parameters and their concrete contribution in processes Results would illustrate the coexistence of many phases in many-body system 28 ... superconductivity theory The final section presents the Ginzburg- Landau theory of phase transition Chapter 2, Ginzburg- Landau micro approaches In this chapter, section 2.1 provides an overview of... Ek 2kB T 10 (1.27) 1.4 Ginzburg- Landau theory of phase transitions 1.4 Ginzburg- Landau theory of phase transitions 1.4.1 Landau Theory 1.4.1.1 Order parameter concept Landau introduced the order... quantum phase transition, one is from ferromagnetic phase to FS phase at P = 1GP a, and the other from ferromagnetic phase to paramagnetic phase at higher pressure Pc = 1.6GPa The superconducting phase