The role of electron-vibron interaction and local pairing in conductivity and superconductivity of alkali-doped fullerides
TThe role of electron-vibron interaction and local pairing in conductivity andsuperconductivity of alkali-doped fullerides
Konstantin V. Grigorishin ∗ Boholyubov Institute for Theoretical Physics of the National Academyof Sciences of Ukraine, 14-b Metrolohichna str. Kiev-03680, Ukraine.
We investigate the competition between the electron-vibron interaction (interaction with the Jahn-Teller phonons) and the Coulomb repulsion in a system with the local pairing of electrons on the3-fold degenerate lowest unoccupied molecular orbital (LUMO). The el.-vib. interaction and thelocal pairing radically change conductivity and magnetic properties of alkali-doped fullerides A n C ,which would have to be antiferromagnetic Mott insulators: we have shown that materials with n = 1 , A = K , Rb are conductors but not superconductors; n = 3 and A = K , Rb are conductorsand superconductors at low temperatures, but with A = Cs they are Mott-Jahn-Teller insulators atnormal pressure; n = 2 , A C analytically, which is the result of interplay between the local pairing, the el.-vib.interaction, Coulomb correlations, and formation of small radius polarons. PACS numbers: 74.20.Fg,74.20.Mn,74.25.Dw,74.70.WzKeywords: alkali-doped fullerides, electron-vibron interaction, Hund coupling, local pairing, Coulomb corre-lations, Holstein polaron, Mott-Jahn-Teller insulator
I. INTRODUCTION
Alkali-doped fullerides ( A n C with A = K , Rb , Cs and n = 1 . . .
5) demonstrate surprising properties. Simple bandtheory arguments predict that any partial filling between 0 and 6 electrons (between empty and full molecular orbital t u , respectively) should give a metallic behavior. In the same time these materials are characterized with a narrowconduction band W ∼ . . . . . eV and a strong on-site Coulomb repulsion U ∼ . . . . . eV . Moreover electrons on t u molecular orbital should be distributed according to Hund’s rule: spin of a molecule must be maximal. Thusalkali-doped fullerides should be antiferromagnetic Mott insulators (MI). In reality the properties of the alkali-dopedfullerides are in striking contradiction to the expected they. So A C and A C are nonmagnetic insulators. Thusmolecule C with additional electrons in LUMO does not have spin, that contradicts to Hund’s rule. Under pressurethese materials become metallic. A C and A C are conductors. A C are superconductors with A = K , Rb forwhich the critical temperatures are sufficiently high T c ∼ K . However for A = Cs the material is insulator, butit becomes superconductor under high pressure ∼ kbar . The corresponding phase diagram of A C is shown inFig.(1). These materials at low temperature are superconductors with dome-shaped T c versus lattice constant, andthey are conductors for higher temperature. However at large lattice spacing and at hight temperature these phasesare broken off with a Mott insulating phase. The insulating phase is magnetic Mott-Jahn-Teller (MJT) insulator(antiferromagnetic with T N = 46 K for the A15 structure and T N = 2 . K for the fcc structure or with a spin freezingonly below 10 K due to frustration of the fcc lattice), with the on-molecule distortion creating the ground state withspin S = 1 /
2, which produces the magnetism. Results of infrared spectroscopy [1–4] are interpreted [1, 5] as thatthe insulator-to-metal transition is not immediately accompanied by the suppression of the molecular Jahn-Tellerdistortions. The metallic state that emerges following the destruction of the Mott insulator is unconventional -sufficiently slow carrier hopping and the intramolecular Jahn-Teller (JT) effect coexists with metallicity. This JTmetallic state of matter demonstrates both molecular (dynamically JT-distorted C − -ions observed in [2, 3]) and free-carrier (electronic continuum) features. As the fulleride lattice contracts further, there is a crossover from the JTmetal to a conventional Fermi liquid state upon moving from the Mott boundary towards the under-expanded regime,where the molecular distortion arising from the JT effect disappear and the electron mean free path extends to morethan a few intermolecular distances. However, it should be noted, since the line of phase transition from JT metal toconventional metal is absent and the molecular distortions gradually increase as lattice expands, that the JT metalis not a phase, unlike the MJT insulator which is separated from the metal state by a line of the first kind phasetransition. Spin-lattice relaxation measurements [6, 7] show that insulators Na C and K C have a nonmagnetic ∗ Electronic address: [email protected] a r X i v : . [ c ond - m a t . s up r- c on ] M a y ground state and their low energy electronic excitations are characterized by a spin gap (between singlet and tripletstates of the Jahn-Teller distorted C − or C − ), furthermore, it has been evidenced very similar electronic excitationsin Na CsC and Rb C , coexisting with typically metallic behavior: the C − and C − would be formed within themetal on very short time scales (10 − s ) that do not imply static charge segregation. These facts speak abouta tendency to charge localization in the odd electron system and the dynamic JT distortions can induce attractiveelectronic interaction in these systems. Cs C exists in two polymorphs: an ordered A
15 structure and a merohedrallydisordered fcc one similar to K C and Rb C . Though the ground state magnetism of the Mott phases differs, theirhigh T paramagnetic and superconducting properties are similar, and the phase diagrams versus unit volume per C are superimposed [8–10]. Due to the crystal field, determined by the potential created by the neighboring Cs + ions,the energies of molecular distortions, that are equivalent in a free molecular ion, can differ in a crystal when pointedat different crystallographic directions. The crystal field influences the vibrational levels of C − molecular ions makingthe spectra temperature- and polymorph-dependent, the presence of temperature-dependent solid-state conformersvalidates the proof of the dynamic JahnTeller effect [3]. Thus the crystal field can additionally induce and stabilizethe JT distortions that much stronger manifests in the A
15 structure compared with the disordered fcc structure.
Figure 1: Experimental phase diagram of fcc-structured Rb x Cs − x C , as a function of volume per C experimentally obtainedin [1]. Within the metallic (superconducting) regime, gradient shading from orange to green schematically illustrates crossoverfrom the JTM to conventional metal. Lower panel: schematic depictions of the respective molecular electronic structure,intermolecular electron hopping, and JT molecular distortion for conventional metal, JT metal and MJT insulator in theirinterpretation. The mechanism of superconductivity of the alkali-doped fullerides has not been fully understood. The positivecorrelation between T c and the lattice constant found in K - and Rb -doped fullerides has been understood in terms ofthe standard BCS theory: the density of states of conduction electrons is ν ∝ /W , and T c = 1 . (cid:104) ω (cid:105) exp( − /λν ),thus the smaller W the large T c . Therefore superconductivity of A C is often described with Eliashberg theory interms of electron-phonon coupling and Tolmachev’s pseudopotential µ ∗ [11–13]. In the same time there is anotherapproach to describe phases of alkali-doped fulleride - the model of local pairing [14–18]. The experimental basis forthis hypothesis is the fact that the coherence length (size of a Cooper pair) in the superconducting alkali-doped fullerideis ∼ . . . nm , which is comparable with a size of a fullerene molecule C ∼ nm . Moreover, the Hubbard-like modelspredict that A C is an anti-ferromagnetic insulator, while it is known experimentally that there are no momentsin A C . The electrons in A n C relatively strong interact with Jahn-Teller intramolecular phonons with H g and A g symmetries. The interaction favors a low-spin state and might lead to a nonmagnetic insulator. At the same timethere is, however, a Hund’s rule coupling, which favors a high-spin state. Thus competition between the Jahn-Tellercoupling and the Hund’s rule coupling takes place. As a result each molecule C − in Cs C crystal (antiferromagneticinsulator) has spin S = 1 / /
2. Proceeding from these facts the local pairing model suggests that theelectron-vibron interaction (interaction of electrons with H g and A g intramolecular Jahn-Teller oscillations) favorsthe formation of a local singlet: √ (cid:80) m a + im ↑ a + im ↓ | (cid:105) , where the spin-up and spin-down electrons are situated on asite i in the same quantum state m (here | (cid:105) is the neutral C molecule, the quantum number m labels the threeorthogonal states of t u symmetry). The local singlet state competes with the normal state (high spin state) of twoelectrons a + im ↑ a + im ↑ | (cid:105) dictated by Hund’s rule. Using this assumption some important results have obtained. In awork [19] the density of states in a band originated from t u level has been calculated by applying the unrestrictedHartree-Fock approximation and the many body perturbation method. It has been found that A C and A C arenonmagnetic semiconductors and the band gaps in these materials are cooperatively formed by the electron-electronand electron-vibron interactions. On the other hand, it is difficult to predict within the model whether the followingmaterials A C , A C and A C are metallic or not, it has been concluded at least that the materials are on theborder of the metal-insulator transition. In the work [20] it is conjectured that the Mott-Hubbard transition takesplace for U/W ∼ √ N , where N is an orbital degeneracy (for alkali-doped fullerides N = 3) due to the matrix elementsfor the hoping of an electron or a hole from a site i to a nearby site j are enhanced as (cid:104) i | t ij a + imσ a jmσ | j (cid:105) = √ N t , thusthe degeneracy increases U c (critical value of the on-site Coulomb repulsion such that if U > U c then a material isMI). In the same time in [21] an analogous expression (cid:104) i | (cid:98) H | j (cid:105) = √ kt has been obtained, but the factor k dependson both the degeneracy and the filling n , so for n = 3 the enhancement is k = 1 .
73, for n = 2 , k = 1 .
57, for for n = 1 , k = 1 .
21. Thus the degeneracy contributes to the metallization of the systems. However, as pointed above,the conductivity depends on filling n radically (for odd n - metals, for even n - insulators), therefore the competitionbetween the Jahn-Teller effect and the Hund’s rule coupling must be accounted [14].In a work [15] it is shown that in A C the local pairing is crucial in reducing the effects of the Coulomb repulsion. So,for the Jahn-Teller H g phonons the attractive interaction is overwhelmed by the Coulomb repulsion. Superconductivityremains, however, even for U vib (cid:28) U , and T c drops surprisingly slowly as U is increased. The reason is as follows. Fornoninteracting electrons the hopping tends to distribute the electrons randomly over the molecular levels. This makesmore difficult to add or remove an electron pair with the same quantum numbers m . However as U is large U > W theelectron hopping is suppressed and the local pair formation becomes more important. Thus the Coulomb interactionactually helps the local pairing. This leads to new physics in the strongly correlated low-bandwidth solids, due tothe interplay between the Coulomb and electron-vibron interactions. In a such system the Eliashberg theory breaksdown because of the closeness to a metal-insulator transition. Because of the local pairing, the Coulomb interactionenters very differently for Jahn-Teller and non-Jahn-Teller models, and it cannot be easily described by a Coulombpseudopotential g − µ ∗ . Theoretical phase diagram for A C systems has been obtained with the DMFT analysis in[22, 23]. There are three phase: the superconducting (SC) phase at low temperature, the normal phase at more hightemperatures and the phase of paramagnetic MI at bigger volume per C − . In the same time the dome shape of theSC phase is absent in the theoretical diagram. In this model the s-wave superconductivity is characterized by a orderparameter ∆ = (cid:80) m =1 (cid:104) a im ↓ a im ↑ (cid:105) , which describes intraorbital Cooper pairs for the t u electrons, m and i are theorbital and site (=molecule) indices respectively (the site index in ∆ has been omitted, because ∆ does not depend ona site - the solution is homogenous in space). Superconducting mechanism is that in such a system we have U (cid:48) > U ( U (cid:48) is interorbital repulsion and U is intraorbital one) due to the el.-vib. interaction. Interesting observation is thatthe double occupancy (cid:104) n m ↑ n m ↓ (cid:105) on each molecule increases toward the Mott transition and it jumply increases in apoint of transition from the metal phase to the MI phase. Conversely, the double occupancy (cid:104) n m ↑ n m (cid:48) ↓ (cid:105) and spin S per molecule decrease toward the Mott transition and they jumply decrease (to S = 1 /
2) in the point of transitionfrom the metal phase to the insulator phase. In the same time the local pairs are not bipolarons because in metallicphase the polarons (polaron band) are absent up to the Mott transition and stable electron pairs are not observedabove T c .The local pairing hypothesis is confirmed with quantum Monte Carlo simulations of low temperature properties ofthe two-band Hubbard model with degenerate orbitals [18]. It have been clarified that the SC state can be realizedin a repulsively interacting two-orbital system due to the competition between the intra- and interorbital Coulombinteractions: it must be U < U (cid:48) for this. The s-wave SC state appears along the first-order phase boundary betweenthe metallic and paired Mott states in the paramagnetic system. The exchange interaction J destabilizes the SCstate additionally. In [24] the phase portrait of A C has been obtained using the DMFT in combination with thecontinuous-time quantum Monte Carlo method. In the theoretical diagram there are the dome shaped SC regionand spontaneous orbital-selective Mott (SOSM) state, in which itinerant and localized electrons coexist and it isidentified as JT metal by the authors. The SOSM state is stabilized near the Mott phase, while SC appears in thelower- U region. The transition into the SOSM and AFM phases is of the first order. In the same time, as discussedabove, the line of phase transition from JT metal to conventional metal is absent in the experimental phase diagram(the molecular distortions gradually increase as lattice expands), hence the JT metal is not a phase, unlike the MJTinsulator which is separated from the metal state by a line of the first kind phase transition. In the theoretical phasediagram the characteristic vertical cutoff of the SC and metallic phases (at low temperature) by MJT insulator phaseis absent or weakly expressed (a hysteresis behavior is observed near the Mott transition point), the SC phase extendsfar into the region of large U/W where the SOSM phase occurs.In the present work we are aimed to find conditions of formation of the local pairs on fullerene molecules, then,based on the local pairing hypothesis, considering the Coulomb correlations and the JT-distortion of the molecules, topropose a general approach to description of the properties of alkali-doped fullerides A n C ( A = K , Rb , Cs , n = 1 . . . A C , conductivity of A n C with n = 1 , , n = 2 , A C with fcc structure which should be close to the experimental phase diagram. II. LOCAL PAIRING
Due to the quasispherical structure of the molecule C the electron levels would be spherical harmonics with theangular momentum l , however the icosahedral symmetry generates the splits of the spherical states into icosahedralrepresentation [25–27]. Fig.2 shows molecular levels close to the Fermi level. The LUMO is the 3-fold degenerate t u orbital (i.e. it can hold up to six electrons). It is separated by about 1.5 eV from the highest occupied molecular orbital(HOMO) and by about 1.2 eV from the next unoccupied level (LUMO+1). The alkali-metal atoms give electrons tothe empty t u level so that the level becomes partly occupied. Hamiltonian of the system can be written in a formof three-orbital Hubbard Hamiltonian with the Hund coupling [18, 28] and the electron-vibron (Jahn-Teller phonons)interaction: (cid:98) H = (cid:88) ij (cid:88) m (cid:88) σ ( t ij + ( ε m − µ ) δ ij ) a + imσ a jmσ + U (cid:88) i (cid:88) m (cid:88) σ n im ↑ n im ↓ + ( U (cid:48) − J ) (cid:88) i (cid:88) m
5) is given by the matrixes V (1) = − − V (2) = √ − V (3) = √ V (4) = √ V (5) = √ . (9)Vibrational energies for the A g and H g modes are within the limits ω = 200 . . . cm − [11, 21]. Figure 2: The molecular levels of C close to the Fermi level. In a substance A C the atoms of alkali metal A = K , Rb , Cs giveelectrons to the LUMO of the fullerene molecule (red color). Right panel: corresponding band structure (density of state as afunction of energy E schematically). According to the model of local pairing for superconductivity of A C the interaction of electrons with A g and H g intramolecular oscillations favors the formation of a local singlet [14–17]:1 √ (cid:88) m a + m ↑ a + m ↓ | (cid:105) , (10) Figure 3: The configurations of two electrons on the 3-fold degenerate orbital. Ground state corresponds to configuration withparallel spins. The intraorbital configuration has the largest energy.Figure 4: The configurations of three electrons on LUMO of a molecule C . Ground state corresponds to configuration withparallel spins. The configuration with a local pair (red color) has the largest energy. where the spin-up and spin-down electrons have the same m quantum number. Here | (cid:105) is the neutral C molecule,the quantum number m labels the three orthogonal states of t u symmetry. In contrast, the normal state (high spinstate) of two electrons is 1 √ (cid:88) m
0. Thus the ground state of the system is a state | m σ, m σ, m σ (cid:105) , in order to form a local pair(10) the energy E − E = U − U (cid:48) + 3 J must be expended. Therefore to study only the local pairing it is sufficientto measure the Coulomb energy of the local pairing configuration from the energy of the ground state (Hund’s ruleconfiguration | ↑↑↑(cid:105) ). Thus in this approach two electrons in the state | m ↑ m ↓(cid:105) on a fullerene molecule ”interact”with energy U − U (cid:48) +3 J which is the Hund coupling, but Coulomb correlation effects between electrons on neighboringmolecules are neglected by the averaging. Then we can reduce the Hamiltonian (1) to a form: (cid:98) H eff = 12 ( U (cid:48) − J ) (cid:88) i (cid:104) n i (cid:105) ( (cid:104) n i (cid:105) −
1) + V (cid:88) (cid:104) ij (cid:105) (cid:104) n i (cid:105)(cid:104) n j (cid:105) (12)+ (cid:88) ij (cid:88) m (cid:88) σ ( t ij + ( ε m − µ ) δ ij ) a + imσ a jmσ + ( U − U (cid:48) + 3 J ) (cid:88) i (cid:88) m n im ↑ n im ↓ + (cid:98) H el − vib + (cid:98) H vib , where (cid:104) n i (cid:105) is an average occupation number of a site i . Thus the effective Coulomb repulsion U − U (cid:48) + 3 J ∼ . eV ,which is much smaller than the on-site Coulomb repulsion U ∼ . eV , resists to the local pairing, unlike the usualHolstein-Hubbard model without degeneration.We can eliminate the vibron variables using perturbation theory. Let the molecular vibrations occur with a certainfrequency ω . Since a vibron is localized on a molecule then el.-el interaction mediated by the exchange of a vibronhas a form: (cid:98) H el − el = (cid:88) i (cid:88) m,m (cid:48) (cid:88) ν λ ν V ( ν ) mm (cid:48) V ( ν ) mm (cid:48) a + im ↑ a im (cid:48) ↑ a + im ↓ a im (cid:48) ↓ ( − i ) (cid:90) + ∞−∞ (cid:68) (cid:12)(cid:12)(cid:12) (cid:98) T (cid:8)(cid:0) b νi ( t ) + b + νi ( t ) (cid:1) (cid:0) b νi + b + νi (cid:1)(cid:9)(cid:12)(cid:12)(cid:12) (cid:69) dt, (13)where λ = λ A g , λ ν = λ H g for ν = 1 . . . (cid:98) T is a Bose time-ordering operator, b ( t ) = e iω b + bt be − iω b + bt is anannihilation (creation) phonon operator in Heisenberg representation, | (cid:105) is a zero-phonon state. Expression underintegral is propagation of a vibron in a time t . The corresponding processes are shown in Fig.(5). Then (cid:98) H el − el = (cid:88) i (cid:88) m,m (cid:48) (cid:88) ν λ ν V ( ν ) mm (cid:48) V ( ν ) mm (cid:48) a + im ↑ a im (cid:48) ↑ a + im ↓ a im (cid:48) ↓ ( − i ) (cid:90) + ∞−∞ (cid:2) θ t e − iω t + θ − t e iω t (cid:3) dt = − ω (cid:88) i (cid:88) m,m (cid:48) (cid:88) ν λ ν V ( ν ) mm (cid:48) V ( ν ) mm (cid:48) a + im ↑ a im (cid:48) ↑ a + im ↓ a im (cid:48) ↓ , (14)where θ t = 1 if t ≥ θ t = 0 if t <
0. The same result we have in frequency representation of the vibron’s propagator: (cid:98) H el − el = (cid:88) i (cid:88) m,m (cid:48) (cid:88) ν λ ν V ( ν ) mm (cid:48) V ( ν ) mm (cid:48) a + im ↑ a im (cid:48) ↑ a + im ↓ a im (cid:48) ↓ − i πω (cid:90) + ∞−∞ ω ω − ω + 2 iδω dω = − ω (cid:88) i (cid:88) m,m (cid:48) (cid:88) ν λ ν V ( ν ) mm (cid:48) V ( ν ) mm (cid:48) a + im ↑ a im (cid:48) ↑ a + im ↓ a im (cid:48) ↓ . (15)At nonzero temperature we should pass to complex time it = → τ , where τ = − /T . . . /T . This is the same that topass to complex frequency ω → iω n , where ω n = πnT . Then Eq.(15) takes a form: (cid:98) H el − el = − (cid:88) i (cid:88) m,m (cid:48) (cid:88) ν λ ν V ( ν ) mm (cid:48) V ( ν ) mm (cid:48) a + im ↑ a im (cid:48) ↑ a + im ↓ a im (cid:48) ↓ Tω ∞ (cid:88) n = −∞ ,n (cid:54) =0 ω ( πnT ) + ω = − ω (cid:20) coth (cid:16) ω T (cid:17) − Tω (cid:21) (cid:88) i (cid:88) m,m (cid:48) (cid:88) ν λ ν V ( ν ) mm (cid:48) V ( ν ) mm (cid:48) a + im ↑ a im (cid:48) ↑ a + im ↓ a im (cid:48) ↓ . (16)The addendum with n = 0 is excluded from the sum, because it corresponds to scattering of electron on the thermalphonons, which do not influence on the pairing of electrons [33]. We can see that at T → λ ω (cid:16) − Tω (cid:17) , that is at increasing of temperature the effectiveness of el.-vib. interaction is decreasing. At T → ∞ ( T (cid:29) ω ) the el.-el. coupling via vibrons is λ T → m that corresponds to diagonal elements of the matrixes (cid:98) V ( ν ) : U mmmmvib = 2 ω (cid:20) coth (cid:16) ω T (cid:17) − Tω (cid:21) (cid:88) ν λ ν V ( ν ) mm V ( ν ) mm (17) Figure 5: interaction between two electrons on a site i mediated by the exchange of a vibron localized in this molecule: thecase (a) corresponds to a direct process ω (cid:80) ν λ ν V ( ν ) mm V ( ν ) mm a + im ↑ a im ↑ a + im ↓ a im ↓ , the case (b) corresponds to an exchange process ω (cid:80) ν λ ν V ( ν ) mm (cid:48) V ( ν ) mm (cid:48) a + im ↑ a im (cid:48) ↑ a + im ↓ a im (cid:48) ↓ , where m (cid:54) = m (cid:48) . The picture (c) corresponds to the shift of electron’s energy due tothe el.-vib. interaction on a site. and between different orbitals m and m (cid:48) that corresponds to the nondiagonal elements: U mm (cid:48) mm (cid:48) vib = 2 ω (cid:20) coth (cid:16) ω T (cid:17) − Tω (cid:21) (cid:88) ν λ ν V ( ν ) mm (cid:48) V ( ν ) mm (cid:48) . (18)For simplicity we suppose that the interaction energies are the same for all orbitals m : U mmmmvib = U vib - direct(intraorbital) interaction, U mm (cid:48) mm (cid:48) vib = J vib - exchange (interorbital) interaction. It should be noticed that J vib > (cid:80) ν λ ν V ( ν ) mm V ( ν ) m (cid:48) m (cid:48) a + im ↑ a im ↑ a + im (cid:48) ↓ a im (cid:48) ↓ which corresponds to the direct interorbital interaction U (cid:48) vib ∝ (cid:80) ν λ ν V ( ν ) mm V ( ν ) m (cid:48) m (cid:48) . However, using the matrixes (8,9)we can see that A g and H g phonons give contribution to the interaction U mmm (cid:48) m (cid:48) vib with different signs: V mm V m (cid:48) m (cid:48) = 1, (cid:80) ν =1 V ( ν ) mm V ( ν ) m (cid:48) m (cid:48) = − /
2, unlike the intraorbital and the exchange interorbital interactions. Hence it is possible tochoose such values of the coupling constants λ A g and λ H g so that U (cid:48) vib = 0 (for example, λ H g = 2 λ A g for the modelwith the same frequency ω ). Thus this interaction can be much weaker than U vib and J vib , that is confirmed bynumerical calculation for vibron-mediated interactions in Cs C [34]. Then the Hamiltonian (12) takes a form: (cid:98) H eff = (cid:88) ij (cid:88) m (cid:88) σ ( t ijmm + ( ε m − µ ) δ ij ) a + imσ a jmσ + ( U − U (cid:48) + 3 J − U vib ) (cid:88) i (cid:88) m n im ↑ n im ↓ − J vib (cid:88) i (cid:88) m (cid:48) (cid:54) = m a + im ↑ a im (cid:48) ↑ a + im ↓ a im (cid:48) ↓ , (19)where we have omitted the constant contribution in energy (the first and the second terms in Eq.(12)) and thevibrons’ energy (cid:98) H vib . If each molecule is isolated, that is t ij = 0, and the el.-vib. interaction is stronger than theHund coupling, that is U − U (cid:48) + 3 J − U vib − J vib <
0, then the anti-Hund distribution of electrons over orbitalsoccurs (the low-spin state (10)). Turning on the hopping t ij (cid:54) = 0 between molecules the electrons are collectivized andthey aspire to distribute randomly over the molecular levels. Then the local pairing (10) is determined by presenceof the anomalous averages (cid:104) a im ↓ a im ↑ (cid:105) and (cid:104) a + im ↑ a + im ↓ (cid:105) , which are determined self-consistently over the entire system.To find the anomalous averages we should distinguish them in Eq.(19), then the Hamiltonian takes a following form: (cid:98) H eff = (cid:88) ij (cid:88) mm (cid:48) (cid:88) σ (cid:0) t mmij + ( ε m − µ ) δ ij (cid:1) a + imσ a jmσ + (cid:88) i (cid:88) m (cid:104) ∆ + m a im ↑ a im ↓ + ∆ m a + im ↓ a + im ↑ (cid:105) + ( U vib − U + U (cid:48) − J ) (cid:88) i (cid:88) m (cid:104) a + im ↑ a + im ↓ (cid:105)(cid:104) a im ↓ a im ↑ (cid:105) + J vib (cid:88) i (cid:88) m (cid:48) (cid:54) = m (cid:104) a + im (cid:48) ↑ a + im (cid:48) ↓ (cid:105)(cid:104) a im (cid:48) ↓ a im (cid:48) ↑ (cid:105) (20)where ∆ m = U vib − U + U (cid:48) − JN (cid:88) i (cid:104) a im ↓ a im ↑ (cid:105) + J vib N (cid:88) i (cid:88) m (cid:48) (cid:54) = m (cid:104) a im (cid:48) ↓ a im (cid:48) ↑ (cid:105) ∆ + m = U vib − U + U (cid:48) − JN (cid:88) i (cid:104) a + im ↑ a + im ↓ (cid:105) + J vib N (cid:88) i (cid:88) m (cid:48) (cid:54) = m (cid:104) a + im (cid:48) ↑ a + im (cid:48) ↓ (cid:105) (21)is the order parameter. N is the number of lattice sites (number of the molecules). Eq.(21) means that the orderparameter in such system is determined by the local pairing. It should be noticed that the condensates (cid:104) a im ↓ a im ↑ (cid:105) fordifferent orbitals m have the same phase because J vib >
0. In this model the exchange energy 3 J and the difference U − U (cid:48) ∼ J resists the attraction energies U vib , J vib . As indicated above the exchange energy J is much less thanCoulomb repulsion U, U (cid:48) . Thus for the local pairing a weaker condition U vib (cid:38) J than U vib (cid:38) U must be satisfied.The average number of electrons per site (cid:88) m (cid:104) n m (cid:105) = 1 N (cid:88) i (cid:88) m (cid:88) σ (cid:104) a + imσ a imσ (cid:105) (22)is determined by the position of the chemical potential µ , and (cid:104) . . . (cid:105) = Ξ − T r . . . exp( − (cid:98) H/T ) denotes averagingprocedure where Ξ is a partition function.The effective Hamiltonian (12) does not account Coulomb correlations between electrons on neighboring sites,therefore to study conduction or insulation of the material we should proceed from the full Hamiltonian (1). Theon-site Coulomb repulsions
U, U (cid:48) , the on-site exchange interaction energy J and the Coulomb repulsion betweenneighboring sites V determine the change of energy at transfer of an electron from a site to a nearby site. This processis shown in Fig.6a. We can see that the energy change in this process is∆ E Hund = E − E = U + 4 J − V, (23)where E is energy of an initial electron configuration (the Hund’s rule) of neighboring sites, E is energy of theconfiguration if to transfer an electron from the site to another. A band conductor becomes insulator if an electrondoes not have enough reserve of kinetic energy (which is a bandwidth W ) to overcome the Coulomb blockade on a site:∆ E > W . In the absence of long-range order all numerical and analytical calculations indicate that the Mott-Hubbardtransition should occur in the region 0 . W < U c < . W [35], so in Hubbard-III-like analytical calculation of thesuperconducting critical temperature in the presence of local Coulomb interactions a critical value U c = W can beused [36]. Thus we can assume that a band conductor becomes the Mott insulator if∆ EW (cid:62) . (24)For example, let us consider Rb C . According to [30] W = 0 . eV , U = 0 . eV , J = 34 meV , V = 0 . eV , then∆ E/W = 1 .
73. This means that Rb C would have to be the Mott insulator. However Rb C is a conductor and, atlow temperatures, is a superconductor even. As discussed above, the results of [20, 21] state that the degeneracy andthe filling contribute to the metallization of the systems due to the matrix elements for the hopping of an electron ora hole from a site i to a nearby site j are enhanced as (cid:104) i | t ij a + imσ a jmσ | j (cid:105) = √ kt where k >
1. However, it should benoted, that in a single-orbital system the bandwidth W is determined by the hopping as W = 2 z | t | , where z is numberof the nearest neighbors. As stated above in multi-orbital system the hopping is renormalized as t → √ kt , hence0the observed bandwidth W has to be determined with the renormalized hopping. Therefore the degeneracy cannotchange the criterium (24) which is determined with the energetic balance. The degeneracy and the filling essentiallyinfluence on the metal-insulator transition in a multi-orbital system but due to configuration energy in ∆ E as will bedemonstrated below.Accounting of the el.-vib. interaction can change the situation. As we can see from Eq.(19) for 3-fold degeneratelevel t u each pair obtains energy − U vib − J vib due to interaction within own orbital (for example m = 1) 1 ↔ − U vib and due to interorbital coupling with other two orbitals 1 ↔ , ↔ − J vib for each.Formation of the local pair is possible if electron configuration with the pair has energy which is less than energy ofelectron configuration according to the Hund’s rule (see Fig.4): U − U (cid:48) + 3 J − U vib − J vib <
0. Then the change ofenergy at transfer of an electron from a site to the nearby site is∆ E anti-Hund = E − E = U − V − U vib − J vib , (25)as shown in Fig.6b. Here we can see more favorable situation for conductivity because ∆ E in this case is less than ina case of the Hund’s rule configuration (23) due to configuration (exchange) energy 4 J and the el.-vib. interaction.However in the normal state (metallic) the anomalous averages are absent: (cid:104) a im ↓ a im ↑ (cid:105) = 0 and (cid:104) a + im ↑ a + im ↓ (cid:105) = 0. Thismeans the local pair configuration is absent at the initial stage and we have the transfer of an electron according toHund’s rule but with el.-vib. interaction∆ E Hund = U + 4 J − V − U vib − J vib , (26)where the local pair is formed on the second site - Fig.6a. But ∆ E Hund > ∆ E anti-Hund due to configuration(exchange) energy 4 J . Then it can be that ∆ E Hund /W > E anti-Hund /W <
1. Hence transition fromSC state to the normal state would be transition to MI. However the normal state is conducting due to followingmechanism. Let ∆ E Hund /W > U − U (cid:48) + 3 J − U vib − J vib <
0, hence on the isolated molecule the local pairingconfiguration is set. Then we obtain situation as in Fig.6b where ∆ E anti-Hund /W <
1, that allows an electron istransferred on the nearby molecule. This makes the normal state to be conducting. Thus if the energy is such that∆ E anti-Hund /W <
1, then the material becomes conductor and superconductor at low temperatures. In Fig.6bwe can see formation of configurations C − and C − with zero spins in the process of charge transferring. Theseconfigurations give a gain in JT energy (all electrons are in the pairs with energy − U vib − J vib each) but increasethe Coulomb energy of a crystal. Therefore these configurations would be formed within the metal on very shorttime scales that do not imply static charge segregation, that corresponds to the results of spin-lattice relaxationmeasurements [6, 7]. As it will be demonstrated in Section IV for the even systems, i.e. A n C with n = 2 ,
4, theconfigurations C − and C − are stabilized that leads to insulation of these materials.It should be noted that the local pair is not a local boson. Following [37] if the size of a local pair a p is much largerthan the mean distance R p between the pairs then the bosonization of such local pairs cannot be realized due to theirstrong overlapping. Thus for fermionic nature of the pair it should be a p /R p >
1. Using the uncertainty principle, thesize of the local pair is defined as a p ( T ) = (cid:16) (cid:126) | ∆ | (cid:17) (cid:112) ε F / m , where the energy gap | ∆ | plays role the binding energyin a pair. The size is compared with R p = (3 / πn p ) / , where n p is the density of the pairs (half density of particleswhich is determined by Fermi energy ε F = (cid:126) m (cid:16) π NV (cid:17) / ⇒ R p ∼ (cid:112) V /N ∼ (cid:126) / √ ε F m ). Then we have a p R p ∼ ε F | ∆ | ∼ W | ∆ | (cid:29) Cooper pairs ∼ crossover from BCS to BEC (cid:28) compact bosons (27)Maximal T c of alkali-doped fullerides is 35 K , then | ∆ | ≈ . T c = 62 K (cid:28) W ≈ . eV . Thus the size of a pair is largerthan average distance electrons and the pair is smeared over the crystal. That is in the metallic phase the local pairshave fermionic nature. At the border of transition to the MJT insulator the bosonization of local pairs occurs due toCoulomb blockage of hopping of electron between neighboring molecules. Thus electron configuration with the localpairs is a dynamical configuration of all molecules unlike the statical one with a compact local pair on each moleculefor the MJT insulator state.Since the local pairs smeared over the crystal the anomalous averages (cid:104) a i ↑ a i ↓ (cid:105) exist in momentum space too: (cid:104) a − k ↓ a k ↑ (cid:105) , and the BCS-like theory can be applied for description of the superconducting state of alkali-doped ful-lerides. Following a work [38], we can make transition from the site representation (20) into the reciprocal (momentum)space using relations: a k mσ = 1 √ N (cid:88) j e i kr j a jmσ , a jmσ = 1 √ N (cid:88) j e − i kr j a k mσ , (28)1 Figure 6: The transfer of a charge from a site to a nearby site. E and E are the energies of the electron configurationsbefore and after the transfer. (a) - the transfer of an electron between sites with Hund’s electron configuration without el.-vib.interaction, (b) - the transfer of an electron between sites with the local pairs configuration. then we obtain the effective Hamiltonian like Hamiltonian of a multi-band superconductor: (cid:98) H eff = (cid:88) m (cid:88) k (cid:88) σ ξ m ( k ) a + k mσ a k mσ + (cid:88) k (cid:88) m (cid:104) ∆ + m a k m ↑ a − k m ↓ + ∆ m a + − k m ↓ a + k m ↑ (cid:105) , (29)with ξ m ( k ) = ε m ( k ) − µ , and two last terms in Eq.(20) have been omitted as a constant. The homogeneous equilibriumgaps are defined as ∆ m = U vib − U + U (cid:48) − JN (cid:88) k (cid:104) a − k m ↓ a k m ↑ (cid:105) + J vib N (cid:88) k (cid:88) m (cid:48) (cid:54) = m (cid:104) a − k m (cid:48) ↓ a k m (cid:48) ↑ (cid:105) ∆ + m = U vib − U + U (cid:48) − JN (cid:88) k (cid:104) a + k m ↑ a + − k m ↓ (cid:105) + J vib N (cid:88) k (cid:88) m (cid:48) (cid:54) = m (cid:104) a + k m (cid:48) ↑ a + − k m (cid:48) ↓ (cid:105) , (30)and the average number of electrons per site is (cid:88) m (cid:104) n m (cid:105) = 1 N (cid:88) k (cid:88) m (cid:88) σ (cid:104) a + k mσ a k mσ (cid:105) (31)Equations (30) and (31) should be solved self-consistently. It is easy to find that (cid:104) a − k m ↓ a k m ↑ (cid:105) = ∆ m E m tanh E m T (32)and (cid:104) a + k mσ a k mσ (cid:105) = 12 (cid:18) − ξ m ( k ) E m tanh E m T (cid:19) , (33)where E m = (cid:112) ξ m ( k ) + ∆ m . Thus a system with the local pairing (cid:104) a im ↓ a im ↑ (cid:105) is equivalent to a multi-band super-conductor with a continual pairing (cid:104) a − k m ↓ a k m ↑ (cid:105) in each band. The multi-band theory [39–41] can be mapped ontoan effective single-band problem [42]. Thus the three-orbitals t u system can be reduced to an effective single-band2superconductor, the more so that the condensates (cid:104) a − k m ↓ a k m ↑ (cid:105) for different orbitals m have the same phase because J vib >
0. If to suppose the dispersion law of electrons in t u conduction band ξ m ( k ) is the same for all orbitals m : ξ = ξ = ξ = ξ , then from Eq.(30) we obtain a simple equation for critical temperature: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( U vib − U + U (cid:48) − J ) ζ − J vib ζ J vib ζJ vib ζ ( U vib − U + U (cid:48) − J ) ζ − J vib ζJ vib ζ J vib ζ ( U vib − U + U (cid:48) − J ) ζ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0= ⇒ ( U vib + 2 J vib − U + U (cid:48) − J ) ζ = 1 , (34)where ζ = 1 N (cid:88) k ξ ( k ) tanh ξ ( k )2 T c (35)In the alkali doped fulerides the conduction band is narrow W = 0 . . . . . eV and energies of vibrons are large ω ∼ . eV , thus almost all electrons take part in el.-vib. interaction (2 ω ∼ W ), unlike electrons in conventionalmetals where only electrons near Fermi surface interact via phonons because ω (cid:28) W there. Hence the final resultsare weakly sensitive to distribution of the density of states in conduction band, that is some averaged density canbe used in this case, unlike conventional conductors where the results strongly depend on the density on Fermi level.Thus we can suppose the density of states in the conduction band is a constant ν = ν if − W/ < ξ < W/
2, otherwise ν = 0. Since (cid:80) k (cid:104) n ( k ) (cid:105) = V (cid:82) ν ( ξ ) dξ it can be seen from Eq.(31) that3 = (cid:88) m (cid:104) n m (cid:105) = 2 VN (cid:88) m (cid:90) − W/ ν dξ = ⇒ ν = NV W . (36)Then Eq.(34) is reduced to a form 1 = ( g − µ c ) (cid:90) Ω − Ω ξ tanh ξ T c dξ, (37)where the integration is restricted by energy Ω = min( ω , W/
2) proceeding from the rectangular approximation forthe density of states in conduction band, narrowness of conduction band and large vibron’s energy, for example for Cs C at normal pressure the bandwidth is W = 0 . eV < ω ≈ . eV . The coupling constant g is determinedwith el.-vib. interaction U vib + 2 J vib , and the Coulomb pseudopotential µ c is determined with the Hund coupling U − U (cid:48) + 3 J : g = U vib + 2 J vib W = (cid:101) U vib W , µ c = U − U (cid:48) + 3 JW . (38)The therm (cid:101) U vib ≡ U vib + 2 J vib corresponds to the energy of attraction in a pair as discussed above. The bandwidthof alkali-doped fullerides is W ∼ . eV , the energy gap is ∆ ≈ T c ∼ . . . K (cid:28) ε F = W/ > ε F and the change of the chemical potential plays important role in formationof SC state [43]. On the other hand the vibrational energies for the A g and H g modes are ω ∼ . eV ∼ ε F thatmeans Tolmachev’s weakening of the Coulomb pseudopotential by a factor ln ε F ω does not take place.It should be noticed that the effect of weak renormalization of electron’s mass due to el.-vib. interaction, which issimilar to the effect in ordinary metals due to el.-phon. interaction, should be absent in alkali-doped fullerides. Asshown in [44] the renormalization is consequence of el.-phon. interaction and of the fact that the total momentum ofthe system P tot = p + (cid:88) q b + q b q (cid:126) q (39)is a constant of the motion. Here p is the momentum of an electron, q is the wave vector of a phonon. The electronenergy is renormalized as E = E − αω + p m
11 + α/ , (40)3where α ∼ λ ω . However in alkali-doped fullerides electrons interact with internal oscillations (vibrons) of the fullerenemolecules. Each vibron is localized on own molecule and cannot propagate to nearby molecules, unlike phonons inmetals which propagate throw the system as waves. This means that the wave vector q is not quantum number forvibrons and Eq.(39) does not have physical sense. Then the electron’s momentum p should conserves separately.Moreover unlike conventional metals, where U/W (cid:28)
1, the alkali-doped fullerides are strongly correlated systems,where
U/W ∼
1, therefore the renormalization of the density of state is caused by the Coulomb correlations atpresence of the el.-vib. interaction, that will be discussed in the next section.In the same time in our case the shift of the electron’s energy − αω is a consequence of a process of el.-vib.interaction on a cite like the el.-el. interaction via vibrons considered above. This process is shown in Fig.(5c).Since a vibron is localized on a molecule we can write the energy shift using Eq.(6) and the averaging analogously toEqs.(13,15): (cid:98) H el. − vib. = (cid:88) i (cid:88) m,m (cid:48) (cid:88) ν λ ν V ( ν ) mm (cid:48) V ( ν ) mm (cid:48) a + im ↑ a im ↑ i π (cid:90) + ∞−∞ ω ω − ω + 2 iδω ε − ω − ε m (cid:48) + iδ dω = ε = ε m (cid:48) − (cid:88) i (cid:88) m,m (cid:48) (cid:88) ν λ ν ω V ( ν ) mm (cid:48) V ( ν ) mm (cid:48) a + im ↑ a im ↑ . (41)Expressions under the integral are propagator of a vibron and propagator of an electron located on a molecular level ε m (if we suppose ε m > δ → +0, if ε m < δ → − ε = ε m . Thus interaction of an electron with the vibron field on a molecule shifts the electron’senergy as − (cid:88) m (cid:48) (cid:88) ν λ ν ω V ( ν ) mm (cid:48) V ( ν ) mm (cid:48) , (42)whose modulus is the binding energy of an electron with deformation of a molecule if the electron is localized on it. III. PHASE DIAGRAMA. Influence of Coulomb correlations
The BCS formula (37) do not account Coulomb correlations between electrons on neighboring sites because it hasbeen obtained using the effective Hamiltonian (19). For studying of these phenomena we should proceed from thefull Hamiltonian (1). The correlations are determined by the fact that in order to transfer an electron from a siteto a nearby site we have to change the energy of the configuration as in Fig.6b. On the other hand the possibilityof this transferring determines metallic properties of the material - Eqs.(23,24,25). The Coulomb correlations can beaccounted with Gutzwiller-Brinkman-Rice approach [45] by the following manner. Quasiparticle states are determinedwith the pole expression of an electron propagator: G ( k , ε ) = Zε − ξ ( k ) + iγ k , (43)where 0 < Z ≤ n k at the Fermi surface, in other handthe function Z determines intensity of the quasiparticle peak at ω = 0 of a spectral function A ( ω ) = (cid:80) k A ( k , ω ),where A ( k , ω ) = − π Im G ( k , ω ) ≈ Zπ γ k ( ω − ξ ( k )) + γ k . For noninteracting electrons Z = 1. At transition from metal stateto MI state the peak disappears and in its place an energetic gap appears separating two Hubbard subbands. Thevanishing of Z therefore marks the metal-insulator transition [46]. Such Coulomb correlations redistribute the densityof state in a conduction band. However in alkali-doped fullerides the el.-vib. interaction takes the whole bandwidth2 ω ∼ W hence the effective density of state is determined by the bandwidth ν F ∼ /W up to the appearance of theHubbard subbands. In the Brinkman-Rice approach the renormalization parameter Z is obtained as Z = 1 − (cid:18) ∆ E ∆ E c (cid:19) = 1 − (cid:32) U − V − (cid:101) U vib W (cid:33) , (44)where ∆ E c is a critical energy change at transfer of an electron from a site to a nearby site. The last part of thisformula is written for our model - Eqs.(23,24,25). Then the condition (24) is Z ≤
0. Using the propagator (43) we4can obtain normal and anomalous Green functions from Gorkov’s equation: G ( ε n , ξ ) = i Z · ( iε n + ξ )( iε n ) − ξ − Z | ∆ | (45) F ( ε n , ξ ) = i Z ∆( iε n ) − ξ − Z | ∆ | , (46)where ε n = πT (2 n + 1). Then the self-consistency condition for the order parameter is∆ = ( g − µ ) T ∞ (cid:88) n = −∞ (cid:90) Ω − Ω dξiF ( ε n , ξ ) = ⇒ g − µ ) Z (cid:90) Ω − Ω ξ tanh ξ T c dξ, (47)where Ω = min( ω , W/
2) as discussed above. This formula for the critical temperature T c , unlike the formula (37),accounts the Coulomb correlations through the renormalization parameter Z . For uncorrelated metal Z = 1, at thetransition to MI Z = 0. Thus Coulomb correlations suppress the critical temperature. Table I: Calculated in [30] bandwidth W , cRPA screened Coulomb parameters U , U (cid:48) , J , V and dielectric constant ε for thecompounds with fcc -lattice: K , Rb and Cs in superconducting phases with maximum T c (at pressure 7 kbar ), in the vicinityof the metal-insulator transition (2 kbar ) and in the antiferromagnetic insulating phase (normal pressure), respectively. At thebottom of the table the Coulomb barrier U − U (cid:48) + 3 J for the local pairing is calculated. K C Rb C Cs C (7 kbar ) Cs C (2 kbar ) Cs C volume per C ( A ) 722 750 762 784 804 W ( eV ) 0.502 0.454 0.427 0.379 0.341 U ( eV ) 0.82 0.92 0.94 1.02 1.08 U (cid:48) ( eV ) 0.76 0.85 0.87 0.94 1.00 J ( meV ) 31 34 35 35 36 V ( eV ) 0.24 − − − − ε cRPA U − U (cid:48) + 3 J ( eV ) 0.153 0.172 0.175 0.185 0.188 The RPA screened parameters of Coulomb interactions
U, U (cid:48) , J, V for the family A C have been calculated in[30] by the constrained RPA method (cRPA). The results are presented in Tab.I. The unscreened (bare) parametersare U = 3 . eV , U (cid:48) = 3 . eV , J = 96 meV , V = 1 . eV . From the table we can see that the Coulomb parametersare functions of volume per C molecule: the screening becomes less effective when the lattice is expanded and theparameters aspire to the bare values. The screening can be effectively described with the dielectric constant definedas [30]: ε cRPA = lim Q → lim ω → ε − cRPA ( q , ω ) , (48)with ω being the frequency and Q = q + G , where q is a wave vector in the first Brillouin zone and G is a reciprocallattice vector. Thus the screened Coulomb interaction and the bare one are connected as U cRPA (cid:39) U bare ε cRPA . (49)Here the static dielectric constant (48) can be used since the plasmon energy in A C is ∼ . eV [47] and the vibronenergies are 0 . eV . . . . eV . Indeed, calculation in [22] shows the results of U ( ω ), U ( ω ), J ( ω ) are almost flat in thefrequency region where the vibron-mediated interactions U vib ( ω ) and J vib ( ω ) are active ( ω (cid:46) . eV ). Furthermore,the values of the cRPA interactions differ by less than 15% from the ω = 0 value up to, at least, ω = 3 eV which ismuch larger than the bandwidth ∼ . eV .Following to [34], the change of the potential ∆ V SCF , on which electrons are scattered, is given by a sum of thechange in the ionic potential ∆ V ion and the screening contribution from the Hartree and exchange channels, that5can be reduced to ∆ V SCF = ∆ V ion /ε . Since the electron-phonon coupling λ represents the scattering of electrons by∆ V SCF , the screening process for el.-vib. coupling can be decomposed in the very same way as that of ∆ V SCF : λ cRPA = λ bare ε cRPA . (50)In the same time the experimentally observed vibron frequencies in K C [48] differ little from the vibron frequenciesin C . Calculations in [34] show that the screening has weak effect on the frequencies. Apparently, the oscillations ofa fullerene molecule are determined with internal elastic constants and little depend on the external environment (therelationship ω RPA = ω bare /ε is correct in the ”jelly” model for metals only). Moreover, in the case of alkali-dopedfullerides, the el.-vib. coupling of the individual mode is not large, while the accumulation of their contributions leadsto the total el.-vib. coupling of g ∼ . . . .
1. Therefore, we do not expect a large difference between bare ω and thescreened frequencies. Since (cid:101) U vib ∝ λ ω then (cid:101) U cRPA vib (cid:39) (cid:101) U bare vib ε cRPA . (51)Thus we can see that as the crystal lattice expands (the bandwidth W decreases) the density of state in the conductionband increases - Eq.(36) and the el.-el. interaction via vibrons increases too - Eq.(51). In the same time the Coulombbarrier U − U (cid:48) + 3 J for the pairing enlarges - Eq.(49), Tab.(I), but slower than (cid:101) U cRPA vib . Thus as volume per C enlarges then T c increases. However, in the same time, the Coulomb correlations are amplified (the bandwidth W narrows and the on-site Coulomb repulsion grows), hence the renormalization parameter Z decreases - Eq.(44). Thissuppresses the critical temperature - Eq.(47). At some critical value of W , where Z = 0, transition to MI state occursand T c = 0 in this point. Thus we have a dome shaped line T c which is shown in Fig.7 as the line (a): for weaklycorrelated regime ( Z → T c increases with decreasing of W (with enlarging of volume per molecule C ), for stronglycorrelated regime ( Z → T c decreases with decreasing of W .We could see that the effectiveness of el.-vib. interaction is decreasing as temperature rises - Eqs.(17,18). So forsmall T we have U vib ∝ λ ω (cid:16) − Tω (cid:17) . Thus the function ∆ E = U − V − (cid:101) U vib ( T ) is increasing with temperature, thenat some temperature T MI the criterion (24) of transition from the metallic state to the Mott insulator state will besatisfied: ∆ EW = U − V − (cid:101) U vib ( T MI ) W = 1 . (52)We can see that at decreasing of W the temperature T MI drops. This dependence is schematically shown in Fig.7 asthe line (b). It should be noted that at nonzero temperatures the transition between metallic and insulator phases isblurred because temperature is larger than the energetic gap between Hubbard subbands. Hence the line separatingmetal and Mott insulator phases at nonzero temperature T MI is more crossover than phase transition. B. Influence of Jahn-Teller deformations
According to JT theorem any non-linear molecular system in a degenerate electronic state will be unstable andwill undergo distortion to form a system of lower symmetry and lower energy thereby removing the degeneracy. Thusa fullerene molecule with partially filled t u orbital ( C n − , n = 1 . . .
5) must be distorted (to D h symmetry) due tothe el.-vib. interaction and the degeneracy of electronic state must be removed. In the same time the molecules arecombined into a crystal and their electrons are collectivized. The collectivized electrons interact with the moleculardistortions and formation of polarons (bound state of an electron with induced deformation by it) can take place.According to Eq.(42) the binding energy is ∼ λ ω ∼ U vib . Then according to the uncertainty principle the relation U vib lv F ∼ (cid:126) occurs, where l is localization radius of an electron, v F is Fermi velocity. Following to [49] the electroncan be localized in the induced deformation if the localization radius is less than average distance between electrons (cid:104) r (cid:105) , which determines Fermi energy ε F = mv F = (cid:126) m (cid:16) π NV (cid:17) / ⇒ (cid:104) r (cid:105) ∼ (cid:112) V /N ∼ (cid:126) / √ ε F m . Then the condition l = (cid:104) r (cid:105) is U vib ∼ ε F = W . (53)6 Figure 7: Theoretical phase diagram of alkali-doped fullerides as a function of volume per C . A line (a) is the criticaltemperatures T c described with Eq.(47). The line separates the SC phase and the normal metallic phase. A line (b) is thecritical temperatures T MI described with Eq.(52). The line separates the normal metallic phase and the MI phase. A line(c), described with Eq.(59), separates the metallic phase (normal and superconducting) and the MJT insulator. Lower panel:corresponding schematic depictions of the electronic structure and JT molecular distortions for the metallic state and MJTinsulator. Both in the metallic state and in the MI state the molecules are distorted due to el.-vib. interaction. As the latticeexpands the distortions enlarge. If the localization radii of electrons on neighboring molecules overlap, then the electrons arecollectivized and the metallic state takes place (four molecules are pictured for example). If the localization radius is smallerthan intermolecular distance, then electrons can be localized by the Coulomb correlations and the MI state occurs. Thus we have lattice of JT distorted molecules with collectivized electrons if U vib (cid:28) W . As lattice expands thedistortions of the molecules increase but electrons cannot be localized because the localization radius is larger thanaverage distance between electrons (and intermolecular distance). This state can be called the Jahn-Teller metal whichis observed in [1]. When the localization radius becomes equal and less than the average distance between electrons,then formation of polaron of small radius (the Holstein polaron) occurs [50]. Such polaron can be localized on a siteby the Coulomb blockade ∼ U and we obtain the Mott-Jahn-Teller insulator. Since the electrons form the pairedstates on each molecule (the local pairs), which, in turn, are locked on the molecules by the Coulomb correlations,then the bosonization of Cooper pairs occurs. These configurations (JT metal and MJT insulator) are schematicallyillustrated in the lower panel of Fig.7.The Hamiltonian (1,6) is similar to the Holstein-Hubbard Hamiltonian: (cid:98) H = − t (cid:88) (cid:104) ij (cid:105) (cid:88) σ a + iσ a jσ − µ (cid:88) i ( n i ↑ + n i ↓ ) + U (cid:88) i n i ↑ n i ↓ + λ (cid:88) i ( n i ↑ + n i ↓ − (cid:0) b i + b + i (cid:1) + ω (cid:88) i b + i b i , (54)Using Lang-Firsov canonical transformation e S (cid:98) He − S [51–53], where S = λω (cid:80) i ( n i ↑ + n i ↓ ) (cid:0) b + i − b i (cid:1) , and projectiononto the subspace of zero phonons, H LF = (cid:68) (cid:12)(cid:12)(cid:12) e S (cid:98) He − S (cid:12)(cid:12)(cid:12) (cid:69) , the Hamiltonian (54) is diagonalized: (cid:98) H LF = − te − λ ω (cid:88) (cid:104) ij (cid:105) (cid:88) σ a + iσ a jσ − µ eff (cid:88) i ( n i ↑ + n i ↓ ) + (cid:18) U − λ ω (cid:19) (cid:88) i n i ↑ n i ↓ , (55)where µ eff = µ − λ ω . (56)7 Table II: Vibrational energies A g and H g of C (from [21]). ω M A g (1) A g (2) H g (1) H g (2) H g (3) H g (4) H g (5) H g (6) H g (7) H g (8) K
717 2117 393 624 1024 1116 1585 1801 2053 2271 meV
62 182 34 54 88 96 137 155 177 196
Thus we obtain effective el.-el. interaction reduced to BCS-like interaction (point, nonretarded): U − λ ω . Thehopping is renormalized as t → te − λ ω , thus the collapse of the conduction band to the narrow polaron band occurs.This polaron is a statical deformation of a molecule, where an electron is in potential well of depth ∼ λ /ω and itsupports the deformation by own field. From Eq.(56) we can see that the chemical potential µ = W/ λ ω .If µ eff <
0, then the Fermi level falls below the bottom of conduction band, hence the localization of an electron ona site by formation of local deformation (the Holstein polaron) becomes possible. Then the condition of collapse ofconduction band is λ ω = W . (57)This equation can be rewritten as U vib = W , where U vib = λ ω is the attraction energy in a pair, that correspondsto Eq.(53) obtained from the uncertainty principle. When deriving Egs.(37,38), we have seen that the multi-bandsystem can be reduced to an effective single band superconductor. Thus for the case of interaction with A g and H g phonons the condition (57) will have a form: (cid:101) U vib W ⇒ g = 1 . (58)Formation of the JT deformation leads from the Hund’s electron configuration | ↑ , ↑ , ↑(cid:105) to the local pair configuration | ↑↓ , ↑ , (cid:105) . As we have seen above in order to make the local pair on a site we have to overcome the configurationalCoulomb barrier U − U (cid:48) + 3 J - Fig.(4). If the JT energy ∝ λ /ω is less than Hund’s coupling ∝ J , that, as ithas been shown in [54], the JT effect is completely suppressed. In formation of configuration with a local pair threeelectrons take part, hence the energy ( U − U (cid:48) + 3 J ) / (cid:101) U vib − U − U (cid:48) + 3 J W ⇒ g − µ c = 1 . (59)The polaron narrowing of the conduction band enhances Coulomb correlations in the already highly correlated sys-tem: W e − λ ω (cid:28) W < U , that must turn the material to the Mott insulator. Above we could see that as the crystallattice expands (volume per C increases) the bandwidth W decreases and the el.-vib. coupling (cid:101) U vib enlarges. Thusincreasing the volume per the molecule we reach the border (59). This border is shown in Fig.7 as the line (c). Thenthe conduction band collapses and the material becomes Mott-Jahn-Teller insulator. Since a Mott insulator is anti-ferromagnetic at half-filled conduction band [55, 56], hence the MJT-insulating phase of A C should be magneticallyordered at low temperature. C. Calculation of the phase diagram
Using obtained results (47,52,59), values of bandwidth W and Coulomb parameters U , U (cid:48) , J , V from Tab.I wecan calculate T c , T MI , border of the collapse of conduction band (59) and the Mott parameter ∆ EW with ∆ E fromEqs.(23,25) for A C (where A = K , Rb , Cs and the substance with cesium is considered at normal pressure, 2 kbar and 7 kbar with fcc structure). However we must know parameters λ ν it order to calculate the el.-el. coupling viavibrons (cid:101) U vib - Eqs.(17,18). We can use an adjustable parameter: for K C the critical temperature is T c = 19 K , then8 Table III: Calculated characteristics of A C (where A=K,Rb,Cs at normal pressure and
A=Cs at pressures 7 kbar , 2 kbar ) asfunction of volume per C neglecting thermal expansion of the lattice: the Mott parameters corresponding to Hund’s rulewithout el.-vib. interaction U +4 J − VW and to presence of the local pairs on sites U − V − (cid:101) U vib W (anti-Hund’s rule), renormalizationparameter Z , the el.-vib. coupling constant λ (for K C the constant is an adjustable parameter - bold font), energy of attractionin the local pair (cid:101) U vib at T = 0, el.-el. attraction constant g at T = 0, el.-el. repulsion constant µ c ,the parameter g − µ c (atthe border of collapse of conduction band it is equal to 1), the critical temperature T c (in brackets the values of T c are givenwhich would be if Ω = ω is supposed), the temperature of the Mott transition T MI (for K C this temperature is so high thatmakes no sense). For Cs C at normal pressure the parameters, which would be if the conduction band does not collapse, areshown in italic. K C Rb C Cs C (7 kbar ) Cs C (2 kbar ) Cs C volume per C ( A ) 722 750 762 784 804 ∆ EW = U +4 J − VW experimental T c ( K ) 19 29 35 26 - λ ( meV ) (cid:101) U vib ( eV ) 0.34 0.41 0.44 0.50 0.55 g µ c g − µ c ∆ EW = U − V − (cid:101) U vib W Z calculated T c ( K ) 19 26 43 32 (33) ( ) T MI ( K ) - 450 400 215 we choose the parameter of el.-vib. coupling λ so that the critical temperature calculated from (47) would be equalto the experimental one. For other materials we calculate the parameter λ using Eqs.(50,51), for example λ ( Rb ) = λ ( K ) ε ( Rb ) ε ( K ) ⇒ (cid:101) U vib ( Rb ) = (cid:101) U vib ( K ) ε ( Rb ) ε ( K ) . (60)Moreover we must known the vibron frequency ω . In a fullerene molecule an electron interacts with A g and H g vibrational modes presented in Tab.II. Then the parameter (cid:101) U vib can be calculated with a following manner: (cid:101) U vib = 2 λ (cid:88) M = A g ,H g ω M (cid:20) coth (cid:16) ω M T (cid:17) − Tω M (cid:21) (61)Here the coupling constants λ ν has been replaced by effective coupling constant λ : (cid:80) ν λ ν (cid:104) V ( ν ) mm V ( ν ) mm + (cid:80) m (cid:48) (cid:54) = m V ( ν ) mm (cid:48) V ( ν ) mm (cid:48) (cid:105) → λ , because we use λ as an adjustable parameter. The integral(47) can be cut off by the largest vibron energy ω = ω ( H g (8)) = 0 . eV , that is confirmed by numerical calculationfor vibron-mediated interactions in Cs C in [34] where ω ≈ . eV , or it can be cut off by the half-bandwidth W/ ω > W . Results of the calculations, neglecting thermal expansion of the lattice , are presented in Tab.(III) andFig.8. From the table we can see that without the el.-vib. interaction we have ∆ EW >
1, hence all materials would beMott insulators. However the el.-vib. interaction and the local pairing change the relation as ∆ EW <
1, hence thesematerials becomes conductors, in the same time the Coulomb correlations enlarges as the lattice constant increases.The parameter g − µ c at volume per C A becomes 1, hence material Cs C at pressure 2 kbar is near theborder of collapse of conduction band, and at normal pressure the material is Mott-Jahn-Teller insulator. As notedabove at the border of collapse the localization radius becomes equal to the average distance between electrons, thus9 Figure 8: Results of calculations for T c (circle markers), T MI (square markers) and the border of collapse of conduction band(vertical line) from Tab.III. Thus we have a phase diagram of of alkali-doped fullerides as a function of volume per C andtemperature. formation of polaron of small radius occurs. Such polaron can be localized on a site by the Coulomb blockade U .It should be noted that since ω ≈ W/ Cs C at pressure 2 kbar and normal pressure, then the cutting off theintegral (47) by W/ ω does not influence on the results significantly. From Fig.8 we can see that thecalculated phase diagram of alkali-doped fullirides is quantitatively close to experimental phase diagram in Fig.1. IV. CONDUCTIVITY OF A n C WITH n = 1 , , , For materials A n C , where n = 1 , , ,
5, we can suppose that the el.-el. interaction (cid:101) U vib via vibrons is approximatelythe same for these materials and is equal to the value in A C ∼ . eV , it is analogously for Coulomb U, U (cid:48) , V andexchange J interactions. The charge transfer in these materials is shown in Fig.9. • A C . In order to form a local pair we have to transfer an electron from a site to a nearby site containinganother electron. For this it is necessary to make a positive work U − V − (cid:101) U vib >
0. Thus formation of thepairs is energetically unfavorable, i.e. the pairs are unstable. In the same time ∆ EW = U − V − (cid:101) U vib W <
1, becausewe create the pair on a neighboring molecule. Thus this material is a conductor due the el.-vib. interaction butit is not a superconductor. • A C . Two electrons are in the paired state on a site because the energy of the state is U − U (cid:48) + J − (cid:101) U vib < break a pair.In this case ∆ EW = U (cid:48) − U − V + (cid:101) U vib W >
1. Thus the transfer of a electron is blocked by Coulomb interaction and theel.-el. attraction via vibrons (cid:101) U vib . Then the pair is compact and it could be transferred but ∆ EW = U (cid:48) − VW > C − does not have spin. • A C . Like the previous material all electrons are in the paired state on a site because the energy of this state isnegative. We can transfer the charge from a site to a site by transferring of an electron with breaking of the pairor by transferring of the compact pair. In these cases ∆ EW is U (cid:48) − U − V + (cid:101) U vib W > U (cid:48) − VW > C − does not have spin. • A C . In order to transfer an electron from a site to a nearby site we have to spend such energy that ∆ EW = U − U (cid:48) − V − (cid:101) U vib W <
1. This process forms the pair, but 3 U − U (cid:48) − V − (cid:101) U vib > Figure 9: The charge transfer in materials A n C ( n = 1 , , , hence the Cooper pairs are not stable like in A C . Thus this material is conductor due to the el.-vib. interactionbut it is not superconductor.As expected, since the system is particle-hole symmetric, therefore the properties for n = 1 , n = 5 , A n C with n = 1 , , n = 2 , V. RESULTS
We have considered the problem of conductivity and superconductivity of alkali-doped fullerides A n C ( A = K , Rb , Cs , n = 1 . . .
5) while these materials would have to be antiferromagnetic Mott insulators because the on-site Coulombrepulsion is larger than bandwidth: U ∼ eV > W ∼ . eV , and electrons on a molecule have to be distributed overmolecular orbitals according to the Hund’s rule. We have found important role of 3-fold degeneration of LUMO ( t u level), small hopping between neighboring molecules and the coupling of electrons to Jahn-Teller modes (vibrons).The el.-el. coupling via vibrons U vib cannot compete with the on-site Coulomb repulsion U (cid:29) U vib , but it can competewith the Hund coupling U − U (cid:48) + 3 J ≈ J ∼ U vib (where the exchange energy is much less than direct Coulombinteraction J (cid:28) U ). This allows to form the local pair (10) on a molecule. Formation of the local pairs radicallychanges conductivity of these materials: they can make ∆ EW <
1, where ∆ E is the energy change at transfer of an1electron from a site to nearby site, while without the el.-vib. interaction we have ∆ EW > A n C with n = 1 , , n = 2 , A C which is result of interplay between the el.-el. coupling via vibrons, the Coulomb blockade on a site and the hoppingbetween neighboring sites. It should be noted that the size of a pair is larger than average distance between electrons(between molecules) and the pair is smeared over the crystal. That is in the metallic phase the local pairs havefermionic nature. At the border of transition to Mott-Jahn-Teller insulator the bosonization of local pairs occurs dueto Coulomb blockage of hopping of electron between neighboring molecules. Thus system with the local pairing canbe effectively described by BCS theory. In such system we have the effective coupling constant as g − µ c > g is determined with the el.-vib. interaction and µ c is determined with the Hund coupling - Eq.(38)) unlike usual metalsuperconductors where g − µ ∗ c > µ ∗ c is a Coulomb pseudopotential with Tolmachev’s reduction).Since A C is a strongly correlated system, i.e. U/W ∼
1, then the equation for the critical temperature (47), unlikeordinary BCS equation, accounts inter-site Coulomb correlations by renormalization parameter (44), which is Z = 0at transition to Mott insulator. The correlations are amplified when the conduction band compressions and theysignificantly suppress the critical temperature T c . In the same time, as the crystal lattice expands (the bandwidth W decreases) the density of states in the conduction band increases as 1 /W and the el.-vib. interaction intensifiesdue to weakening of the screening. As a result the depending of T c on the volume per a molecule has a dome shape- Figs.(7,8), unlike the theoretical phase diagram calculated by DMFT method in [22, 23]. Thus el.-vib interactionensures conductivity and superconductivity of alkali-doped fullerides. However the effectiveness of el.-vib. interactionis decreasing as temperature rises due to a vibron propagator ω ( πnT ) + ω . Hence at some temperature T MI the criterionof transition from the metallic state to the Mott insulator state will be satisfied - Eq.(52), and the material becomesMott insulator.As the bandwidth decreases the collapse of the conduction band to the narrow polaron band occurs when thecondition (59) is satisfied. This Holstein polaron is a statical JT deformation of the molecule, where an electronsupports the deformation by own field. According to JT theorem a fullerene molecule with partially filled t u orbitalmust be distorted due to the el.-vib. interaction and the degeneracy of electron state must be removed. In thesame time the molecules are combined into a lattice and their electrons are collectivized. As the lattice expands thedistortions of the molecules increase, but electrons cannot be localized because the localization radius is larger thanaverage distance between electrons (and intermolecular distance), i.e. the electrons are collectivized. This state canbe called the Jahn-Teller metal which is observed in [1]. When the localization radius becomes equal and less thanthe average distance between electrons, then formation of polaron of small radius occurs. The polaron narrowing ofthe conduction band enhances Coulomb correlations in already strongly correlated system, that must turn the alkali-doped fullerides to the Mott insulator, and the material becomes Mott-Jahn-Teller insulator. We have demonstratedthat border of the collapse of conduction band vertically cuts off the SC and metallic phases (at low temperature) -Figs.(7,8), unlike the theoretical phase diagram calculated in [24].Thus we have three phases of A C illustrated schematically in Fig.(7): superconductor, metal and MJT insulator,which are separated by lines T c , T MI and the border of collapse of conduction band. Calculated phase diagram inFig.8 of alkali-doped fullerides is quantitatively close to experimental phase diagram in Fig.1. We have illustratedthat K C , Rb C are conductors (and superconductors) but Cs C is MJT insulator at normal pressure, at 2 kbar itis superconductor on the border with MJT insulator, and at 7 kbar it is superconductor with maximal T c . Thus wehave shown that superconductivity, conductivity and insulation of alkali-doped fullerides have common nature: thelocal pairing due to interaction with the Jahn-Teller phonons. The proposed model does not account influence of thecrystal field, therefore we consider materials only with the merohedrally disordered fcc structure unlike the ordered A
15 structure where the effect of crystal field should be stronger.
Acknowledgments
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