Physica C 171 (1990) 395-400 North-Holland Energy spectrum of a hole in a two-dimensional antiferromagnet A.V. Sherman Institute of Physics, Estonian Academy of Sciences, Riia 142, 202400 Tartu, USSR Received 18 June 1990 Revised manuscript received 3 September 1990 The t-J model of a hole in a spin-½ Heisenberg antiferromagnet on a square lattice is considered. In the spin-wave approxima- tion the lowest eigenenergies and eigenfunctions are calculated as functions of the wave vector for u/t<200, where U is the Hubbard energy, t is the hopping matrix element. The results obtained reveal the formation of ferromagnetic clusters around the hole starting from U/t~ 10 in the top of the band, and at larger values of U/t, in other regions of the band. 1. Introduction Interest in strongly correlated electronic systems has essentially grown after Anderson [ 1 ] made a suggestion that the Hubbard model can explain the superconducting transition in planar-cuprate per- ovskites. One of the essential problems here is the description of hole movement in an antiferromag- netic plane modelling CuO2 planes of these crystals. Recently, a number of papers devoted to this prob- lem has appeared [2-5 ]. These papers, dealing with slightly different models (all being consequences of the Hubbard model at U>> t, the limit presumably realized in cuprate perovskites [ 6 ] ), allow one to get a general impression about the character of the en- ergy spectrum of the system at moderate values of U/t. However, the mapping of a more realistic two- band Hubbard model to the considered single-band one [ 7 ] and the available values of parameters for La2CuO4 [8,9 ] indicate that the effective values of U/t may be larger than those used, e.g. in [5] (by accepting J=4t2/U.~O.1 eV [9], where J is the ex- change constant (see eq. ( 1 ) ), and, taking into ac- count the mapping, t ~ 0.3-1 eV [ 8 ], one obtains for the effective value of the parameter in this crystal 12< U/t<40). Besides, in the papers cited above, the value of the total spin in the obtained lowest states was not determined. It is known [ 10 ] that at large values of U/t this quantity in the ground state may be larger than ½ (for one hole in a lattice with an even number of sites), which is connected with the for- mation ofa ferromagnetically ordered region around the hole (ferromagnetic cluster). In case of a small hole concentration this circumstance may have def- inite consequences for the carrier transport and mag- netic properties of crystals [ 11 ], and from this point of view it is of interest to determine the parameter regions where such clusters exist. The aim of this paper is to obtain the transformed form of the t-J Hamiltonian that is convenient for calculations in the presence of a few holes and to cal- culate eigenenergies and eigenfunctions as functions of wave vector k in the case of one hole for U~t < 200. The obtained results indicate that the minimum of the excitation band is positioned in the point (n/2, n/2) of the Brillouin zone for U/t<70 and in the point (0, 0) for larger U/t. In the spin-wave ap- proximation used in this paper, the z component of the total spin Sz may be taken as an indicator of the ferromagnetic cluster formation. It is shown that these clusters (the lowest states at a given value of k with Sz > 1.5) appear near U/t= 10, at the begin- ning near the top of the band and at larger values of U/t, in other regions of the band. Near the band minimum Sz> 0.5 when U/t> 70. With the growth of U/t, Sz grows, acquiring half-integer values, which indicates the enlargement of the cluster size, and at U/t= 200 it achieves 3.5 near the band bottom. The results of the calculations are shown in figs. 2 to 4. 0921-4534/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)