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THE PSEUDOGAP IN THE BOSON–FERMION MODEL FOR HIGH-Tc SUPERCONDUCTIVITY
Pergamon
PII: S0022-3697(98)00103-6
J. Phys. Chem Solids Vol 59, No. 10–12, pp. 1759–1763, 1998 0022-3697/98/$ - see front matter 䉷 1998 Elsevier Science Ltd. All rights reserved
THE PSEUDOGAP IN THE BOSON–FERMION MODEL FOR HIGH-T c SUPERCONDUCTIVITY J. RANNINGER Centre de Recherches sur les Tre`s Basses Tempe´ratures, Laboratoire Associe´ a´ l’Universite´ Joseph Fourier, Centre National de la Recherche Scientifique, BP 166, 38042 Grenoble Ce´dex 9, France Abstract—High-T c superconductors are considered as composite systems made up of itinerant electrons and localized bipolarons trapped in unstable ligand configurations, responsible for triggering a charge exchange between the two species. As a consequence, the intrinsically localized bipolarons become itinerant upon lowering the temperature and eventually condense into a superfluid state. Concomitantly a pseudogap in the density of states of the electrons opens up, deepens upon decreasing the temperature and finally develops into a true gap as superconductivity sets in. The photo-emission spectral properties exhibit a large, broad, incoherent contribution coming from the phonon shake-off of the bipolarons. 䉷 1998 Elsevier Science Ltd. All rights reserved Keywords: high-T c superconductors, pseudogap, Boson–Fermion model
The importance of electronic and magnetic correlations in superconducting materials with high transition temperatures (T c) has been discussed widely since their discovery over 10 years ago. By contrast, the dielectric properties of these materials have received little attention so far. The aim of the present note is to focus on precisely these properties. Because of the complexity of the problem we shall neglect the effect of electron correlations, although in the final picture they are expected to play an essential role. The first indications for polaronic charge carriers and local lattice instabilities came from Raman and infrared measurements of samples with photo-induced charge carriers [1], photoconductivity [2] and optical conductivity [3]. Recent extended X-ray absorption fine structure (EXAFS) measurements [4] added further support in favour of the existence of dielectric subsystems in these materials. As chemists have pointed out for some time, they contain ligand environments of the cations which have tendencies to ambiguous valence configurations [5], a fact which favours valence fluctuations accompanied by sizeable local deformations of specific molecular units. It has been noticed that it is precisely these molecular units (such as the O–Cu þ –O dumb-bell, respectively the quadratic Cu 2þO 4 configurations in cuprates with chains; the octahedral Bi 5þO 6, respectively pyramidal Bi 3þO 5 oxygen environments of bismuth in the bismuthates; and the diamond sp 3, respectively graphite sp 2 configurations in fullerenes) which are known as the location of small polaron, respectively bipolaron formation. These units have been identified as polaronic centres by the anomalous temperature dependence of the frequency shift and linewidth of the local modes that are associated with their dynamic deformations linking the metastable ligand configurations. Moreover, resonant
Raman scattering on these local modes shows a sizeable number of overtones [6], which are equally present in tunnelling spectra [7] and are clear-cut indications for strong and local electron coupling. The theoretical picture we propose here consists of an electronic subsystem (such as the CuO 2 planes) which is in contact with a subsystem containing the local atomic clusters and which, owing to their dielectric properties, form the seat of polaronic charge carriers. As soon as the system is doped into the metallic regime we assume that a dynamic charge transfer sets in between the two subsystems. The electrons, upon arriving at the polaronic sites, will be converted into small polarons or bipolarons. Elementary polaron theory, as well as the dependence on laser intensity of the Raman scattering cross-section and very recent femtosecond pulsed experiments [8], permit us to study the relaxation behaviour of such polaronic charge carriers. These measurements indicate that they exist preferentially in the form of bipolarons rather than single polarons. Hence we shall assume that the charge transfer between the electronic and dielectric subsystems occurs in the form of a transfer of a pair of itinerant electrons into a localized bipolaron and vice versa. In order to formulate such a scenario let us consider the simplest version of it, consisting of effective sites on which these charge-transfer processes take place. We thus are led to the following model: X X c†j þ d, j cj, j H ¼ (zt ¹ m) c†j, j cj, j ¹ t
1759
j, j
j, d, j
i Xh ÿ X † þ DB ¹ 2m bj bj þ v c†j, ↑ c†j, ↓ bj þ b†j cj, ↓ cj, ↑ j
X ¹l j
j
X Mh i 2 X˙ j þ q20 Xj2 b†j bj Xj þ j 2
ð1Þ
1760
J. RANNINGER
where X j denote the local ligand deformations, M their effective mass and q 0 their vibrational frequency. j indicates the effective site index and d the vectors linking such neighbouring sites. The hopping integral for the electrons is denoted by t and the level of the localized electron pairs by D B. l refers to the coupling strength between those localized electron pairs and the local lattice deformations. This model represents a generalization of the Boson–Fermion model (BFM) which was introduced by us many years ago [9] in order to study the intermediary coupling regime of electron–lattice coupled systems. Recent numerical studies of polaron systems [10] give some microscopic foundations for such a scenario. The original BFM is obtained from the above expression for eqn (1) when setting l equal to zero. In order to extract the essential physics contained in this model, we begin by studying this model in the atomic limit; a procedure which has previously contributed largely to clarify such questions as the driving mechanism for the pseudogap in the original BFM [11]. For that purpose let us consider the above Hamiltonian for a single site and diagonalize it. The corresponding eigenstates and eigenvalues are given by:
ÿ with GD⬍ F qn beingEgiven by the Fourier transform of † G⬍ F ¼ cj, j (t)cj, j (0) ⌰(t). A straightforward calculation yields: 1 X ¹ bð0 þ lÉq0 Þ þ e ¹ bð20 þ E0 þ lÉq0 ¹ lX0 =2Þ I(q) ¼ e Z l ÿ 1 X 2 l ⫻ d q ¹ 0 þ e ¹ ba ula Z a ¼ ⫾, m, l 2 ÿ hf(X)lF(X)iula d q þ 0 ¹ la þ mÉq m 2 1 X þ e ¹ bð0 þ E0 þ lÉq0 ¹ lX0 =2Þ vm a Z a ¼ ⫾, m, l 2 hf(X ¹ X0 )lF(X)ivma l ÿ ⫻ d q ¹ 0 ¹ E0 þ m ð4Þ a ¹ lÉq þ lX0 =2 where
Xh 1 þ 2e ¹ b0 þ 2e ¹ bð0 ¹ lX0 =2 þ E0 Þ
Z¼
l
þ e ¹ bð20 ¹ lX0 =2 þ E0 Þ þ e ¹ bð þ ¹ lÉq0 Þ l
þ e ¹ bð ¹ ¹ lÉq0 Þ ÿ e ¹ blÉq0 l
l0i 䊟 l0) 䊟 lf(X)il
¼ 0 þ lÉq0
l ↑ i 䊟 l0) 䊟 lf(X)il , l ↓ i 䊟 l0) 䊟 lf(X)il ul⫿ l
↑ ↓ i 䊟 l0) 䊟
lF(X)iul⫿ þ vl⫿ l0i
ð5Þ
¼ 0 þ lÉq0
䊟 l1) 䊟 lF(X)ivl⫿
¼ l⫿
(2)
l ↑ i 䊟 l1) 䊟 lf(X ¹ X0 )il , l ↓ i 䊟 l1) 䊟 lf(X ¹ X0 )il
¼ E0 þ 0 þ lÉq0 ¹ lX0 =2
l ↑ ↓ i 䊟 l1) 䊟 lf(X ¹ X0 )il
¼ E0 þ 20 þ lÉq0 ¹ lX0 =2
which are listed in order of increasing number of particles per site. ul⫿ and vl⫿ are constants depending on the various parameters of the model. l⫿ denote the energies of the bonding and antibonding local two-particle states. lf(X)i l and lf(X ¹ X 0)i l denote the lth excited harmonic and displaced harmonic oscillator state with a displacement X0 ¼ l=Mq20 . lF(X)iul⫿ and lF(X)ivl⫿ represent respectively the lth excited exact and non-harmonic oscillator states for the atomic-limit problem with two charge carriers per site. l0i, l ↑ i and l ↑ ↓ i denote the various Fermionic states, while l0) and l1) denote the states with zero and one Boson per site, respectively. To keep the algebra to a minimum we shall begin by considering the symmetric case, i.e. ¼ zt ¹ m ¼ 0 and E 0 ¼ D B ¹D 2m ¼ 0,Ewith the number of Fermions per spin † given D Eby cj, j cj, j ¼ 1=2 and the number of Bosons by † bj bj ¼ 1=2. The photo-emission cross-section for the electrons is then determined by: 1 ÿ q ¼ q þ id I(q) ¼ ¹ Im G⬍ p F n
(3)
denotes the partition function. Approximating the oscillator states via lF(X)iula ⯝ aula lf(X)il þ bula lf(X ¹ X0 )il
(6)
lF(X)ivla ⯝ avla lf(X ¹ X0 )il þ bvla lf(X)il with coefficients a and b which in general satisfy a ⬎ b, and using the Glauber coherent-state representation of the displaced oscillator states, we find m hf(X)lF(X)iula
2
l
⯝ bula e ¹ a Pul,am þ aula dm, l
and l hf(X
2
l
¹ X0 )lF(X)iua, m ⯝ bvla e ¹ a Pvl,am þ avla dm, l
where Pul,am, va denote polynomials of order min[l,m]. We thus notice that the photo-emission spectrum consists of: l
l
1. a coherent part which is essentially due to electrons that do not participate in the exchange processes with the bipolarons (the first term on the right-hand side of eqn (4)); and
Pseudogap in the Boson–Fermion model
2. an incoherent part due to the electrons that do participate in such an exchange processes and hence show phonon shake-off signatures which are typical for polaronic charge carriers (the last two terms on the right-hand side of eqn (4))†.We notice that as the temperature is lowered, the spectral weight of the coherent part tends to zero. This foreshadows the pseudogap features that are inherent in this scenario and which we have discussed in great detail in the past on the basis of the original BFM, which neglects the coupling of the Bosons to the lattice [13, 14]. In order to explore this feature further, the itinerancy of the electrons must be taken into account. This can be handled in various approximate schemes. The simplest way is to incorporate the itinerancy into the atomic-limit results, presented above, by means of a Hubbard I type approximation. This leads straightforwardly to a broadening of the coherent d-function peak and a smearing of the series of d-functions of the incoherent part of the photo-emission spectrum. A more controlled approach consists in a diagrammatic self-consistent conserving scheme for deriving the Fermion and Boson Green’s functions. Both of these procedures have been followed by us [11, 13] for the simpler case of the original BFM, which treats the bipolarons simply as Bosons without any internal phonon structure. For that case, the simplest conserving diagrammatic Born–Green approach to this problem leads to the following set of equations for the Fermion and Boson self-energies: ÿ v2 X ÿ ÿ SF k, qn ¼ q, qm GF q ¹ k, qm ¹ qn GB q, qm Nb (7) ÿ ÿ v2 X ÿ SB q, qm ¼ ¹ G q ¹ k, qm ¹ qn GF k, qn Nb k, qn F with G B(q,qm ) ¼ [iqm ¹ E 0 ¹ S B(q,qm )] ¹1 and G F(k,q n) ¼ [iq n ¹ k ¹ S F(k,q n] ¹1 representing the fully selfconsistently determined Fermion and Boson one-particle Green’s functions. This set of equations has been studied for bare one- and two-dimensional densities of states [13] for the Fermions. Preliminary results for the three-dimensional and infinite dimensional case confirm the qualitative features of the earlier calculations, i.e. the appearance of a pseudogap in the density of states of the Fermions which, upon lowering the temperature, deepens and finally opens up into a true gap when the system becomes superconducting. Treating the generalized BMF equation (eqn (1)) within the same diagrammatic approximation changes †A detailed discussion on the temperature dependence of this incoherent contribution and its relation to the closing of the pseudogap can be found in some recent work [12].
1761
the above expression for the Fermion self-energy in the following way: ÿ ÿ 1X ÿ 2 SF k, qn SF k, qn ⇒ e ¹ a SF k, qn ¹ N k 2 X ÿ 2 v e¹a GF k, qm þ qm⬘ ¹ qn þ 2 2 2N b k, q, qm , qm⬘ ⬁ ÿ X a2l ⫻ GB q, qm l ¼ 1 l! hÿ ¹1 ÿ ¹ 1i ⫻ iqm⬘ ¹ lq0 ¹ iqm⬘ þ lq0
ð8Þ
This clearly indicates, looking at the last term in eqn (8), how the series of d-functions describing the lattice contributions to the incoherent part of the spectrum gets smeared out by the convolution of the phonon states with those of the itinerant Fermions. As regards the low-frequency properties of the system close to the Fermi level, it is adequate to consider the original BFM (which neglects any coupling of the Bosons to the lattice degrees of freedom) as long as the frequency of the external perturbation is small enough not to excite the internal phononic structure of the Bosons. Therefore, in discussing the appearance of the pseudogap in the Fermionic density of states, we shall concentrate henceforth on the physics of the original BFM. In the present paper we report on the numerical solutions of eqns (7), carried out for a one-dimensional (1D) BFM with a representative choice of parameters: D B ¼ 0.4, v ¼ 0.1 in units of the bare bandwidth, 4t, and a total average occupation of charge carriers per site equal to unity. The integration in wavevector space was carried out on a grid of up to 2050 wavevectors in the Brillouin zone. The frequency integration was done by two methods: summing Matsubara frequencies on the imaginary axes and integrating along a path close to the real axis. We present here only results concerning the 1D BFM because of their much better quality compared with the 2D case. Qualitatively, however, the results are the same and do not depend significantly on dimensionality. The manifestation of the pseudogap can be traced back to the onset of itinerancy of the intrinsically localized Bosons and their approach to condensation. The Bosons acquiring itinerancy is due to a highly non-linear coupling between the localized Bosons and the itinerant Fermions, which manifests itself in a gradual increase in itinerancy (decrease in effective Boson mass and increase of the Boson lifetime) as the temperature is reduced, as can be seen from Fig. 1. This evolution of the spectral properties of the Bosons is accompanied by a simultaneous evolution of the spectral properties of the Fermions, which result in the formation of a pseudogap as the temperature is reduced to below a certain characteristic temperature, T *. This pseudogap, close to the Fermi level, is due to a splitting of the bare Fermion
1762
J. RANNINGER
Fig. 1. Evolution of the Boson dispersion qBq for a one-dimensional BFM as the temperature (in units of the bandwidth, 4t) is lowered.
Fig. 2. The Fermionic spectral function as a function of wavevector k in units of the Fermi vector k F for a one-dimensional BFM for a temperature T ¼ 0.001 (in units of the bandwidth, 4t).
Pseudogap in the Boson–Fermion model
Fig. 3. Temperature evolution of the intensity of the photo-emission spectrum for a one-dimensional BFM for k ¼ k F and various temperatures: T ¼ 0.007 (solid line); T ¼ 0.01 (dashed line); and T ¼ 0.02 (dotted line) (in units of the bandwidth, 4t).
dispersion into bonding states, involving Fermion pairs and Bosons, and non-bonding states that are essentially overdamped single Fermion states. This feature has been discussed above on the basis of the atomic-limit calculation of the generalized BFM. The spectral functions of the Fermions, ¹ p1 Im GF (k, q), illustrated in Fig. 2 for various k vectors, clearly bear this out in the three-peak structure, already seen in the atomic limit. The evolution with temperature of these spectral properties of the Fermions can be seen from the corresponding angle resolved photoemission spectroscopy (ARPES) spectral intensity, I(k,q) ¼ ¹ Im G F(k,q)n F(q ¹ m). n F denotes the Fermion distribution function. In Fig. 3 we plot this ARPES lineshape for various temperatures and the Fermi wavevector k F(T), which has been determined as that Z wavevector for which the distribution function of the Fermions, nFk ¼ dqI(k, q)nF (q), shows an inflection point; this is the standard experimental procedure to determine k F. This temperature dependence of the Fermi vector arises naturally in any system comprising two subsystems, such as the BFM, because of the inevitable redistribution of relative occupation of the two types of charge carrier. From the plot in Fig. 3 we clearly see how, upon reducing the temperature, the spectral weight is removed from the Fermi level, showing the opening of the pseudogap in the density of states of the Fermions. The consequences of this opening up of a pseudogap and of the related destruction of the Fermi liquid properties have been explored [14] in connection with the anomalous temperature behaviour of the specific heat, the nuclear magnetic resonance relaxation rate, the optical conductivity and the spingap features in the magnetic susceptibility [15]. The agreement with experiment is generally satisfactory. Clearly there remains the very important question concerning the anisotropy of several properties and in particular of the pseudogap. In the different versions of the Boson–Fermion models studied
1763
so far, this question cannot be tackled, since we assume a system with effective lattice sites. Calculations are in progress where we consider a lattice composed of two sublattices, where one of them houses the itinerant electrons and the other one the localized bipolarons. An anisotropy then appears naturally in the gap, showing pseudogap behaviour together with polaron-induced, incoherent, high-energy contributions along the directions linking two ajacent Fermionic and Bosonic sites. Along directions linking neighbouring Fermionic sites, the spectral properties of the Fermions remain quasi freeFermion-like, unaffected by the presence of the Bosons and hence of the lattice.
Acknowledgements—I would like to thank my collaborators, J.-M. Robin, A. Romano, T. Domanski and P. Devillard, with whom most of this work has been caried out over the past few years. Part of it represented the Ph.D. thesis of J.-M. Robin. I am particularly grateful to P. Devillard for permitting me to publish some of his recently obtained numerical results on the Fermionic spectral properties.
REFERENCES 1. Talliani, C., Pal, A. J., Ruani, G., Zamboni, R., Wei, X. and Vardeny Z., in Electronic Properties of High T c Superconductors and Related Compounds (Solid State Science Series), ed. H. Kuzmany, M. Mehring and J. Fink. Springer Verlag, Berlin, 1990. 2. Yu, G., Lee, C. H., Mihailovic, D., Heeger, A. J., Fincher, C., Herron, N. and McCarron, E. M., Phys. Rev. B, 1993, 48, 7545. 3. Mihailovic, D., Foster, C.M., Voss, K. and Heeger, A.J., Phys. Rev. B, 1990, 42, 7989; Bi, X.-X. and Ecklund, P., Phys. Rev. Lett., 1993, 70, 2625. 4. Ro¨hler, J., Phase separation, electronic imhomogeneities and related mechanisms for high T c superconductors. Proc. Int. School of Solid State Physics, Erice, 1995. 5. Simon, A., Chemie unserer Zeit, 1988, 22, 1; Wilson, T.A., J. Phys. C, 1987, 20, L911. 6. Calvani, P., Capizzi, M., Lupi, S., Maselli, P., Paolone, A. and Roy, P., Phys. Rev. B, 1996, 53, 2756. 7. Aoki, R., Murkami, H., Shirai, M., Svitunov, V.M., Dyachenko, A.I. and Afanassyev, D.N., in Spectroscopic Studies of Superconductors, ed. I. Bozovic and D. van der Marel. SPIE, Bellingham, 1996. 8. Stevens, C. J., Smith, D., Chen, C., Ryan, J. F., Podobnik, B., Mihailovic, D., Wagner, G. A. and Evetts, J. E., Phys. Rev. Lett., 1997, 78, 2212. 9. Ranninger, J. and Robaszkiewicz, S., Physica B, 1985, 135, 468. 10. de Mello, E. V. L. and Ranninger, J., Phys. Rev. B, 1997, 55, 14872. 11. Domanski, T., Ranninger, J. and Robin, J. M., Solid State Commun., 1998, 105, 473. 12. Ranninger, J. and Romano, A., Preprint, 1998. 13. Ranninger, J., Robin, J.-M. and Eschrig, M., Phys. Rev. Lett., 1995, 74, 4027; Ranninger, J. and Robin, J.-M., Solid State Commun., 1996, 98, 559. 14. Ranninger, J. and Robin, J.-M., Phys. Rev., 1996, B53, R11961. 15. Ranninger, J. and Robin, J.-M., Physica C, 1997, 274, 304.
PII: S0022-3697(98)00103-6
J. Phys. Chem Solids Vol 59, No. 10–12, pp. 1759–1763, 1998 0022-3697/98/$ - see front matter 䉷 1998 Elsevier Science Ltd. All rights reserved
THE PSEUDOGAP IN THE BOSON–FERMION MODEL FOR HIGH-T c SUPERCONDUCTIVITY J. RANNINGER Centre de Recherches sur les Tre`s Basses Tempe´ratures, Laboratoire Associe´ a´ l’Universite´ Joseph Fourier, Centre National de la Recherche Scientifique, BP 166, 38042 Grenoble Ce´dex 9, France Abstract—High-T c superconductors are considered as composite systems made up of itinerant electrons and localized bipolarons trapped in unstable ligand configurations, responsible for triggering a charge exchange between the two species. As a consequence, the intrinsically localized bipolarons become itinerant upon lowering the temperature and eventually condense into a superfluid state. Concomitantly a pseudogap in the density of states of the electrons opens up, deepens upon decreasing the temperature and finally develops into a true gap as superconductivity sets in. The photo-emission spectral properties exhibit a large, broad, incoherent contribution coming from the phonon shake-off of the bipolarons. 䉷 1998 Elsevier Science Ltd. All rights reserved Keywords: high-T c superconductors, pseudogap, Boson–Fermion model
The importance of electronic and magnetic correlations in superconducting materials with high transition temperatures (T c) has been discussed widely since their discovery over 10 years ago. By contrast, the dielectric properties of these materials have received little attention so far. The aim of the present note is to focus on precisely these properties. Because of the complexity of the problem we shall neglect the effect of electron correlations, although in the final picture they are expected to play an essential role. The first indications for polaronic charge carriers and local lattice instabilities came from Raman and infrared measurements of samples with photo-induced charge carriers [1], photoconductivity [2] and optical conductivity [3]. Recent extended X-ray absorption fine structure (EXAFS) measurements [4] added further support in favour of the existence of dielectric subsystems in these materials. As chemists have pointed out for some time, they contain ligand environments of the cations which have tendencies to ambiguous valence configurations [5], a fact which favours valence fluctuations accompanied by sizeable local deformations of specific molecular units. It has been noticed that it is precisely these molecular units (such as the O–Cu þ –O dumb-bell, respectively the quadratic Cu 2þO 4 configurations in cuprates with chains; the octahedral Bi 5þO 6, respectively pyramidal Bi 3þO 5 oxygen environments of bismuth in the bismuthates; and the diamond sp 3, respectively graphite sp 2 configurations in fullerenes) which are known as the location of small polaron, respectively bipolaron formation. These units have been identified as polaronic centres by the anomalous temperature dependence of the frequency shift and linewidth of the local modes that are associated with their dynamic deformations linking the metastable ligand configurations. Moreover, resonant
Raman scattering on these local modes shows a sizeable number of overtones [6], which are equally present in tunnelling spectra [7] and are clear-cut indications for strong and local electron coupling. The theoretical picture we propose here consists of an electronic subsystem (such as the CuO 2 planes) which is in contact with a subsystem containing the local atomic clusters and which, owing to their dielectric properties, form the seat of polaronic charge carriers. As soon as the system is doped into the metallic regime we assume that a dynamic charge transfer sets in between the two subsystems. The electrons, upon arriving at the polaronic sites, will be converted into small polarons or bipolarons. Elementary polaron theory, as well as the dependence on laser intensity of the Raman scattering cross-section and very recent femtosecond pulsed experiments [8], permit us to study the relaxation behaviour of such polaronic charge carriers. These measurements indicate that they exist preferentially in the form of bipolarons rather than single polarons. Hence we shall assume that the charge transfer between the electronic and dielectric subsystems occurs in the form of a transfer of a pair of itinerant electrons into a localized bipolaron and vice versa. In order to formulate such a scenario let us consider the simplest version of it, consisting of effective sites on which these charge-transfer processes take place. We thus are led to the following model: X X c†j þ d, j cj, j H ¼ (zt ¹ m) c†j, j cj, j ¹ t
1759
j, j
j, d, j
i Xh ÿ X † þ DB ¹ 2m bj bj þ v c†j, ↑ c†j, ↓ bj þ b†j cj, ↓ cj, ↑ j
X ¹l j
j
X Mh i 2 X˙ j þ q20 Xj2 b†j bj Xj þ j 2
ð1Þ
1760
J. RANNINGER
where X j denote the local ligand deformations, M their effective mass and q 0 their vibrational frequency. j indicates the effective site index and d the vectors linking such neighbouring sites. The hopping integral for the electrons is denoted by t and the level of the localized electron pairs by D B. l refers to the coupling strength between those localized electron pairs and the local lattice deformations. This model represents a generalization of the Boson–Fermion model (BFM) which was introduced by us many years ago [9] in order to study the intermediary coupling regime of electron–lattice coupled systems. Recent numerical studies of polaron systems [10] give some microscopic foundations for such a scenario. The original BFM is obtained from the above expression for eqn (1) when setting l equal to zero. In order to extract the essential physics contained in this model, we begin by studying this model in the atomic limit; a procedure which has previously contributed largely to clarify such questions as the driving mechanism for the pseudogap in the original BFM [11]. For that purpose let us consider the above Hamiltonian for a single site and diagonalize it. The corresponding eigenstates and eigenvalues are given by:
ÿ with GD⬍ F qn beingEgiven by the Fourier transform of † G⬍ F ¼ cj, j (t)cj, j (0) ⌰(t). A straightforward calculation yields: 1 X ¹ bð0 þ lÉq0 Þ þ e ¹ bð20 þ E0 þ lÉq0 ¹ lX0 =2Þ I(q) ¼ e Z l ÿ 1 X 2 l ⫻ d q ¹ 0 þ e ¹ ba ula Z a ¼ ⫾, m, l 2 ÿ hf(X)lF(X)iula d q þ 0 ¹ la þ mÉq m 2 1 X þ e ¹ bð0 þ E0 þ lÉq0 ¹ lX0 =2Þ vm a Z a ¼ ⫾, m, l 2 hf(X ¹ X0 )lF(X)ivma l ÿ ⫻ d q ¹ 0 ¹ E0 þ m ð4Þ a ¹ lÉq þ lX0 =2 where
Xh 1 þ 2e ¹ b0 þ 2e ¹ bð0 ¹ lX0 =2 þ E0 Þ
Z¼
l
þ e ¹ bð20 ¹ lX0 =2 þ E0 Þ þ e ¹ bð þ ¹ lÉq0 Þ l
þ e ¹ bð ¹ ¹ lÉq0 Þ ÿ e ¹ blÉq0 l
l0i 䊟 l0) 䊟 lf(X)il
¼ 0 þ lÉq0
l ↑ i 䊟 l0) 䊟 lf(X)il , l ↓ i 䊟 l0) 䊟 lf(X)il ul⫿ l
↑ ↓ i 䊟 l0) 䊟
lF(X)iul⫿ þ vl⫿ l0i
ð5Þ
¼ 0 þ lÉq0
䊟 l1) 䊟 lF(X)ivl⫿
¼ l⫿
(2)
l ↑ i 䊟 l1) 䊟 lf(X ¹ X0 )il , l ↓ i 䊟 l1) 䊟 lf(X ¹ X0 )il
¼ E0 þ 0 þ lÉq0 ¹ lX0 =2
l ↑ ↓ i 䊟 l1) 䊟 lf(X ¹ X0 )il
¼ E0 þ 20 þ lÉq0 ¹ lX0 =2
which are listed in order of increasing number of particles per site. ul⫿ and vl⫿ are constants depending on the various parameters of the model. l⫿ denote the energies of the bonding and antibonding local two-particle states. lf(X)i l and lf(X ¹ X 0)i l denote the lth excited harmonic and displaced harmonic oscillator state with a displacement X0 ¼ l=Mq20 . lF(X)iul⫿ and lF(X)ivl⫿ represent respectively the lth excited exact and non-harmonic oscillator states for the atomic-limit problem with two charge carriers per site. l0i, l ↑ i and l ↑ ↓ i denote the various Fermionic states, while l0) and l1) denote the states with zero and one Boson per site, respectively. To keep the algebra to a minimum we shall begin by considering the symmetric case, i.e. ¼ zt ¹ m ¼ 0 and E 0 ¼ D B ¹D 2m ¼ 0,Ewith the number of Fermions per spin † given D Eby cj, j cj, j ¼ 1=2 and the number of Bosons by † bj bj ¼ 1=2. The photo-emission cross-section for the electrons is then determined by: 1 ÿ q ¼ q þ id I(q) ¼ ¹ Im G⬍ p F n
(3)
denotes the partition function. Approximating the oscillator states via lF(X)iula ⯝ aula lf(X)il þ bula lf(X ¹ X0 )il
(6)
lF(X)ivla ⯝ avla lf(X ¹ X0 )il þ bvla lf(X)il with coefficients a and b which in general satisfy a ⬎ b, and using the Glauber coherent-state representation of the displaced oscillator states, we find m hf(X)lF(X)iula
2
l
⯝ bula e ¹ a Pul,am þ aula dm, l
and l hf(X
2
l
¹ X0 )lF(X)iua, m ⯝ bvla e ¹ a Pvl,am þ avla dm, l
where Pul,am, va denote polynomials of order min[l,m]. We thus notice that the photo-emission spectrum consists of: l
l
1. a coherent part which is essentially due to electrons that do not participate in the exchange processes with the bipolarons (the first term on the right-hand side of eqn (4)); and
Pseudogap in the Boson–Fermion model
2. an incoherent part due to the electrons that do participate in such an exchange processes and hence show phonon shake-off signatures which are typical for polaronic charge carriers (the last two terms on the right-hand side of eqn (4))†.We notice that as the temperature is lowered, the spectral weight of the coherent part tends to zero. This foreshadows the pseudogap features that are inherent in this scenario and which we have discussed in great detail in the past on the basis of the original BFM, which neglects the coupling of the Bosons to the lattice [13, 14]. In order to explore this feature further, the itinerancy of the electrons must be taken into account. This can be handled in various approximate schemes. The simplest way is to incorporate the itinerancy into the atomic-limit results, presented above, by means of a Hubbard I type approximation. This leads straightforwardly to a broadening of the coherent d-function peak and a smearing of the series of d-functions of the incoherent part of the photo-emission spectrum. A more controlled approach consists in a diagrammatic self-consistent conserving scheme for deriving the Fermion and Boson Green’s functions. Both of these procedures have been followed by us [11, 13] for the simpler case of the original BFM, which treats the bipolarons simply as Bosons without any internal phonon structure. For that case, the simplest conserving diagrammatic Born–Green approach to this problem leads to the following set of equations for the Fermion and Boson self-energies: ÿ v2 X ÿ ÿ SF k, qn ¼ q, qm GF q ¹ k, qm ¹ qn GB q, qm Nb (7) ÿ ÿ v2 X ÿ SB q, qm ¼ ¹ G q ¹ k, qm ¹ qn GF k, qn Nb k, qn F with G B(q,qm ) ¼ [iqm ¹ E 0 ¹ S B(q,qm )] ¹1 and G F(k,q n) ¼ [iq n ¹ k ¹ S F(k,q n] ¹1 representing the fully selfconsistently determined Fermion and Boson one-particle Green’s functions. This set of equations has been studied for bare one- and two-dimensional densities of states [13] for the Fermions. Preliminary results for the three-dimensional and infinite dimensional case confirm the qualitative features of the earlier calculations, i.e. the appearance of a pseudogap in the density of states of the Fermions which, upon lowering the temperature, deepens and finally opens up into a true gap when the system becomes superconducting. Treating the generalized BMF equation (eqn (1)) within the same diagrammatic approximation changes †A detailed discussion on the temperature dependence of this incoherent contribution and its relation to the closing of the pseudogap can be found in some recent work [12].
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the above expression for the Fermion self-energy in the following way: ÿ ÿ 1X ÿ 2 SF k, qn SF k, qn ⇒ e ¹ a SF k, qn ¹ N k 2 X ÿ 2 v e¹a GF k, qm þ qm⬘ ¹ qn þ 2 2 2N b k, q, qm , qm⬘ ⬁ ÿ X a2l ⫻ GB q, qm l ¼ 1 l! hÿ ¹1 ÿ ¹ 1i ⫻ iqm⬘ ¹ lq0 ¹ iqm⬘ þ lq0
ð8Þ
This clearly indicates, looking at the last term in eqn (8), how the series of d-functions describing the lattice contributions to the incoherent part of the spectrum gets smeared out by the convolution of the phonon states with those of the itinerant Fermions. As regards the low-frequency properties of the system close to the Fermi level, it is adequate to consider the original BFM (which neglects any coupling of the Bosons to the lattice degrees of freedom) as long as the frequency of the external perturbation is small enough not to excite the internal phononic structure of the Bosons. Therefore, in discussing the appearance of the pseudogap in the Fermionic density of states, we shall concentrate henceforth on the physics of the original BFM. In the present paper we report on the numerical solutions of eqns (7), carried out for a one-dimensional (1D) BFM with a representative choice of parameters: D B ¼ 0.4, v ¼ 0.1 in units of the bare bandwidth, 4t, and a total average occupation of charge carriers per site equal to unity. The integration in wavevector space was carried out on a grid of up to 2050 wavevectors in the Brillouin zone. The frequency integration was done by two methods: summing Matsubara frequencies on the imaginary axes and integrating along a path close to the real axis. We present here only results concerning the 1D BFM because of their much better quality compared with the 2D case. Qualitatively, however, the results are the same and do not depend significantly on dimensionality. The manifestation of the pseudogap can be traced back to the onset of itinerancy of the intrinsically localized Bosons and their approach to condensation. The Bosons acquiring itinerancy is due to a highly non-linear coupling between the localized Bosons and the itinerant Fermions, which manifests itself in a gradual increase in itinerancy (decrease in effective Boson mass and increase of the Boson lifetime) as the temperature is reduced, as can be seen from Fig. 1. This evolution of the spectral properties of the Bosons is accompanied by a simultaneous evolution of the spectral properties of the Fermions, which result in the formation of a pseudogap as the temperature is reduced to below a certain characteristic temperature, T *. This pseudogap, close to the Fermi level, is due to a splitting of the bare Fermion
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J. RANNINGER
Fig. 1. Evolution of the Boson dispersion qBq for a one-dimensional BFM as the temperature (in units of the bandwidth, 4t) is lowered.
Fig. 2. The Fermionic spectral function as a function of wavevector k in units of the Fermi vector k F for a one-dimensional BFM for a temperature T ¼ 0.001 (in units of the bandwidth, 4t).
Pseudogap in the Boson–Fermion model
Fig. 3. Temperature evolution of the intensity of the photo-emission spectrum for a one-dimensional BFM for k ¼ k F and various temperatures: T ¼ 0.007 (solid line); T ¼ 0.01 (dashed line); and T ¼ 0.02 (dotted line) (in units of the bandwidth, 4t).
dispersion into bonding states, involving Fermion pairs and Bosons, and non-bonding states that are essentially overdamped single Fermion states. This feature has been discussed above on the basis of the atomic-limit calculation of the generalized BFM. The spectral functions of the Fermions, ¹ p1 Im GF (k, q), illustrated in Fig. 2 for various k vectors, clearly bear this out in the three-peak structure, already seen in the atomic limit. The evolution with temperature of these spectral properties of the Fermions can be seen from the corresponding angle resolved photoemission spectroscopy (ARPES) spectral intensity, I(k,q) ¼ ¹ Im G F(k,q)n F(q ¹ m). n F denotes the Fermion distribution function. In Fig. 3 we plot this ARPES lineshape for various temperatures and the Fermi wavevector k F(T), which has been determined as that Z wavevector for which the distribution function of the Fermions, nFk ¼ dqI(k, q)nF (q), shows an inflection point; this is the standard experimental procedure to determine k F. This temperature dependence of the Fermi vector arises naturally in any system comprising two subsystems, such as the BFM, because of the inevitable redistribution of relative occupation of the two types of charge carrier. From the plot in Fig. 3 we clearly see how, upon reducing the temperature, the spectral weight is removed from the Fermi level, showing the opening of the pseudogap in the density of states of the Fermions. The consequences of this opening up of a pseudogap and of the related destruction of the Fermi liquid properties have been explored [14] in connection with the anomalous temperature behaviour of the specific heat, the nuclear magnetic resonance relaxation rate, the optical conductivity and the spingap features in the magnetic susceptibility [15]. The agreement with experiment is generally satisfactory. Clearly there remains the very important question concerning the anisotropy of several properties and in particular of the pseudogap. In the different versions of the Boson–Fermion models studied
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so far, this question cannot be tackled, since we assume a system with effective lattice sites. Calculations are in progress where we consider a lattice composed of two sublattices, where one of them houses the itinerant electrons and the other one the localized bipolarons. An anisotropy then appears naturally in the gap, showing pseudogap behaviour together with polaron-induced, incoherent, high-energy contributions along the directions linking two ajacent Fermionic and Bosonic sites. Along directions linking neighbouring Fermionic sites, the spectral properties of the Fermions remain quasi freeFermion-like, unaffected by the presence of the Bosons and hence of the lattice.
Acknowledgements—I would like to thank my collaborators, J.-M. Robin, A. Romano, T. Domanski and P. Devillard, with whom most of this work has been caried out over the past few years. Part of it represented the Ph.D. thesis of J.-M. Robin. I am particularly grateful to P. Devillard for permitting me to publish some of his recently obtained numerical results on the Fermionic spectral properties.
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