Dispersive Hybridization Correlation and Magnetism in a Two - Band Model

We study a twoband Hamiltonian for hybridized and correlated bands of sc and bcc symmetry, the latter case approximating a twodimensional band. Magnetic phase diagrams are obtained, showing in the bcc case a region of rapid phase change with filling which is absent in the sc case. We consider a twoband system described by the b z -2(A,+B,) following Hamiltonian in the site representation: (3) c (A: &g) + (B: EE) + 4A, Bo bb +b H = C ( t r ~ k ~ ~ u + tq biu jr) + -2 (4 + c:) Ga2 G~~ (4) i,i,q + (t$u&bj. + h.c.) d = 2 { 3 , (EC -A:) +A, (EP B:) (a, b), (i, j) and (a, 7) label, respectively, the two types of orbitals, the sites and the spin. The remaining notation is standard. Diagonalization of H in the reciprocal (k) space is performed by treating the onsite correlation (Uaa, ubb, Uab) and exchange (J) terms in the unrestricted HartreeFock approximation, i.e. by allowing for nonvanishing offdiagonal expectation values (aiubk) = (b&,ukU). The ree en function equationof-motlon technique yields analytically the renormalized energies for para-(P), ferr;(F), and antiferro-(A) magnetic-phases. By introducing the Fourier transforms of the hopping amplitudes t?;, t tb , t:, i.e. EL, EL, -yk we can write the renormalized energies for the P and F phases as: In (3-6) the spin label is actually irrelevant. The quantities Ga = UaaSa + 5sb /2 and Gb = ubbsb + JSa/2, with Sa and sb the amplitudes of the antiferromagnetic moments, e.g.: E' = $ { E ~ + E ~ + A , + B , T We take Q half a reciprocal lattice vector, yielding a twosublattice antiferromagnetic structure. To study [(&; + A, B,)' + 4 (% + cu)2] 'I2} (2) concrete examples, we assume the intersite hybridization of the form yk = ro + r s k where ro is a dispersionless past, while sk is the same canonical dispersion where relation assumed to describe the bare bands: B, = ubb (ni-,) + 2uab (na) ; with a shift with respect to the origin of the C, -2Uaa (a&,bk,> . energy. The dispersive character of the hybridization has rarely been taken into account in this type of studFor the A case, the energies (spin independentlare ies. However, the approximation of assuming the same the solution of E$ + b ~ i + C E ~ + dEk + e = 0 dispersion relation for and x, while justified as a whose coefficients are: mean to obtain analytics results, has no pretention to Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988822 C8 72 JOURNAL DE PHYSIQUE be soundly realistic. The density .of states (DOS) can be obtained analytically from the renormalized energies, as discussed in 111. The auantitiesto be evaluated. which characterize each phase, are n: = (a$,ak,) ; . n: (biUbiu) ; (b~pak,) = (ai,bkr) and the amplitude of the magnetic moments. The bare DOS (corresponding to sk) we consider in the present work are s-type, of simple (sc) and body centered (bcc) cubic symmetry. The first one represents a generic featureless 3-dimensional DOS, while the second one, with its logarithmic singularity in the middle, is an approximation to the Zdimensional tetragonal DOS of interest for the properties of the magnetic compounds related to the high Tc superconductors. Due to the large number of parameters in the model, and to lack of space, we shall discuss here only the effects of varying I? and the numbers (n) of electrons in the band. A more complete study will be given elsewhere. The chosen values of the parameters, in arbitrary units of energy, are: wa = 2; wb = wa/2; A, = 0; ab = 0.1; uaa = wal2; bb U = 0.8 w b ; uab = 0.4 Uaa; J = 0.1 Uaa. They represent a wider moderately correlated band (a) and a narrower, strongly correlated one (b). For the hybridization term, we take I?o = 0, because its effect just adds to C, a constant term, and we vary the amplitude I' of the dispersive part. It can be shown [I] that in the P phase, no ap due to hybridization opens provided f E I'/ ? WaWb < 1. As we are not interest in such effects, we take f _< 0.9. The system of coupled integral equations for evaluating the various expectation values, the Fermi level (EF) , and finally the ground state energy (EGs) was solved numerically by iteration. The interplay of hybridization and correlation can deeply modify the DOS already in the P state, by introducing peaks in a featureless bare DOS (sc case for large f ) . Additionally, the relative position of the two bands can change, so that, in the present case, while the narrower bare P band (b) is above the wider one (a), the renormalized P DOS for n > 2 shows the narrower band below the wider one. When magnetic effects add to the interactions, very large modifications of the magnetic DOS with respect to the P DOS are usually observed. This effect is striking in the A case. With few exceptions, the four A subbands are much narrower than in the corresponding P or F phases, and they exhibit high peaks. In many cases, some of the A subbands are so narrow that they can be considered as quasi atomic localized levels. The features of the A DOS are more sensitive to the values of the model parameters than the P or F DOS. Such properties of the DOS's in the different phases imply that the phase transitions may cause a strong modification of the DOS. This is expected for transitions to the A phase, but it is a rather unfamiliar concept for P + F transitions. As a consequence, the extension of simple singleband results (like StonerWohlfarth) to two hybridized and correlated band may be misleading, in particular when local properties of the P DOS around EF are used to predict the transitions. Indeed, the magnetic phase diagrams (Fig. 1) show the F phase to set in for n 5 0.5 in both cases, i.e. when EF is in a region of low P DOS. The F phase is stable up to n close to 2, the exact value depending on the DOS and on f . For 2 5 n _< 3 in the sc case, the A phase is steadily stable in a range of n values which shrinks as f grows. In contrast, the bcc case shows a zone (hatched in Fig. 1) where the stable phase varies very rapidly with n. For example, when = 0.1 one has the sequence P -+ F -+ P -+ A -, P -+ A -r F + A as 2 < n < 3. For larger n the P phase is stable in both cases. To explain this peculiarity of the bi-dimensional case, let us recall that EGS = El + Es, where El is the sum of products of the average energy of each nonempty subband times the subband occupancy, while Es (= 0 in P state) explicitly depends on s"(~). USUally (Ell >> (Es(. For n small, the presence of narrow peaks and gaps in the A DOS causes E? > E',~. 4 When n e 2, correlation effects raise E: and El to values close to E f . For 2 < n < 3, E? grows with n less quickly than E: or E:, due both to the narrowness of A subbands, and to the pinning of EF by the A DOS peaks. This suggests that the difference between threeand twodimensional phase diagrams for 2 < n < 3 has to be traced to the presence of high peaks in the P or F DOS of bcc type. Their effect on El is hence similar to that of the -4 DOS peaks. Then the relatively small, and strongly n-dependent difference E.$ ESPyF becomes relevant to determine the stability. The sc DOS, on the contrary, has high peaks only in the A phase, so that its range of stability is mainly fixed by E? E:' ", which has a more regular n-dependence than the difference in Es. BCC DOS ; f f J FP .3