Z. Phys. B - Condensed Matter 53, 221-231 (1983) Condensed f,?'äfliif;' Matter (Q Springer-Verlag 1983 Rectangular Lattice Gas Model with Competing Interactions and the Two-Dimensional ANNNI Model in a Field P. Rujan Institute for Theoretical Physics, Eötvös University, Budapest, Hungary W. Selke Institut für trestkörperforschung der Kernforschungsanlage Jülich, Federai Republic of Germany G.V. Uimin L.D. Landau Institute for Theoretical Physics, Chernogolovka, USSR Received July 18, 1983 A lattice gas model with short range competing interactions for adsorption on (110) surlaces of fcc crystals, in particular for O/Pd(110), as well as its Ising analog, the two- dimensional ANNNI model with antiferromagnetic axial nearest and next-nearest neighbour interactions in a field, are studied using the free lermion approximation and Monte Carlo techniques. The phase diagrams display dilferent commensurate phases and incommensurate regions. Static and dynamic aspects of topological defects (walls and dislocations) characterising the incommensurate structures are investigated. 1. Introduction Currently commensurate (C) and incommensurate ANNNI model with competing axial antiferromag- (IC) structures are investigated extensively, both ex- netic interactions (Jr,Jr<0) in a lield (the three- perimentally and theoretically [1]. On the theoreti- dimensional version, -Ir>0, is considered in Ul). cal side it was found that rather simple models give The study is motivated by work of Ertl and Küppers rise to a remarkable richness of phenomena. For on the adsorbed overlayer O/Pd(110) [8]. Indeed, example, the three-dimensional axial next nearest the model, they suggested in that context, is just the neighbour Ising (or ANNNI) model [2] exhibits not lattice gas analog of our model. Experimentally, only an incommensurate ordering t3], but also a O/Pd(110) displays different commensurate phases, sequence of inlinitely many distinct commensurate (.2x1) aI coverages around Il2 and (3 x 1) at co- phases [2] and a special multicritical point, the Lif- verages around 1/3 ((3 x 1) denotes a situation, where shitz point. (In the d-dimensional ANNNI model the oxygen is adsorbed predominantly on sites of nearest neighbour interactions tn (d- 1) dimensional parallel rows separated by two mostly empty rows. planes, .Io, and between adjacent planes, ,.Ir, are aug- The adsorption lattice is a rectangular one in the mented by competing next-nearest neighbour inter- case of O/Pd(l10). In between these two C phases actions, Jr, acting parallel to a single axis.) Although IC structures are observed. In the corresponding the ANNNI model seems to be too simple to de- theoretical analysis [8], however, no attempt has scribe specific experiments quantitatively, it repro- been made to establish the phase diagram ol the duces crucial qualitative leatures observed in modu- lattice gas model: only a few Monte Carlo (MC) lated structure materials [4]; in particular, it pro- equilibrium configurations and MC data of the vides a useful framework to study complicated phase structure lactor at a lew temperatures and coverages diagrams in magnetic systems, such as the cerium were presented. In view of recent elaborate investi- monopnictide compounds CeSb [5] and CeBi [6]. gations on the two-dimensional ANNNI model In this article we shall study the two-dimensional without lield [9-12] as weli as on lattice gas models 222 P. Rujan et al.: Rectangular Lattice Gas Model with Competing lnteractions : ("/1 with competing interactions [13] it seems to be fea- /{ -L s",rs"+ 1,y * /2s,.ys"1z,y sible to analyse that lattice gas model and its Ising analog, the ANNNI model in a field, in much more iJoS',rS,,r* 1+I1S",y) (1) detail. Our aim is to determine their phase diagrams Ising spins, are located on a and to relate the results to current concepts on two- where the S,,r: t1, rectangular lattice with the lattice constants a, in r- dimensional C and IC structures (of course, it is Jo refers coupling between hoped that the results of our study may also stimu- direction, and b. to the along the y-direction; and J, are nea- late further experiments on O/Pd(110)). In particu- neighbours J, neighbour exchange interac- lar, we shall pay attention to the existence and ex- rest and next nearest (i.e. tent of an IC phase with algebraic decay of cor- tions along the r-direction Jo, Jr, "/r denote interactions between up to geometrically third near- relations, the floating phase [14, 9] (in contrast to est neighbours appropriate choices of a and b). disordered IC structures with exponentially decaying lor In the following we set The analogous lattice correlations). It has been argued convincingly that c:1. gas has been considered some years ago there is no direct transition from the (2 x 1) phase to model [8] in order to describe experimental findings on the such a floating phase [14, 15]. Various, mutually adsorbate system O/Pd(110). As usual for (110) sur- exclusive, theoretical suggestions have been put for- faces of fcc substrate crystals the adsorption lattice ward on the melting of the (3 x 1) phase ((3 x 1) indeed, rectangular one; this case the two phases are realized in the three-state chiral Potts is, a in lattice constants are ax2.5A and bx3.6A. The cor- model [16-19] as well as in other adsorption sys- responding Hamiltonian is tems like H/Fe(110) [13]): Second order transitions of conventional 18] or "chiral" type from //' (to, [17, [20] : - c'.', c, t .r * uoz c^., c"- .:., the (3 x 1) phase into the disordered phase have been ,-I. (3 1) advocated. It has also been claimed that the x *€ro C*,rC*,r* r f fui*e) C".n). Q) phase should always melt into a floating phase [21], unless the transition is of first order. Even the exis- The occupation number C,,, takes on the values 0 tence of a commensurate (3 x 1) phase has been and 1 depending on whether the site I = (x, y) is questioned [22]. One might hope that these con- empty or occupied; hence flicting statements can be clarified in the lramework C*,r:(1 (3) of our fairly simple model. Indeed, we shall present -5,,)12 some evidence on these subtle questions, although e denotes the binding energy of the adatoms, O, to the main emphasis will be on the gross features of the substrate, Pd, and p is the chemical potential. the phase diagrams. The coverage ol the overlayer is given by the In Sect. 2 we shall define the models, both tL ANNNI model in a lield and its lattice gas analog. c:l t <c,> (4) The ground states are determined and exact properties L,?'' briefly of the one-dimensional version of the model are where the brackets denote a thermal average; L is mentioned. In the next section results of a Monte the number of lattice sites. From experiments on in Carlo study are presented. Phase diagrams, both O/Pd(110) the pair interactions are estimated to be the temperature-field and temperature-coverage t8l plane, are shown. Kinetic aspects of walls and dislo- cations and the role of these topological defects in e o, :0.3 so, ( 0 (5) characterising the IC structures and in driving the re- transitions between the various phases are discussed. i.e. both interactions along the x-direction are pulsive. ero is found to be attractive In Sect. 4 the free lermion approximation [14] is applied to the ANNNI model in a field. Results are cro:-0'3c0,. (6) compared to the ones obtained in the Monte Cario study. - The last section provides a short summary. However, a precise determination of the adatom in- teractions is hardly possible within this model, be- cause additional pair and triple interactions play an important role, see U3,231 and below. Therefore we 2. The Models used the parameters of (6) only in a comparatively small number of Monte Carlo runs. In general we 2.1. Hamihonians put ero: -tor. By that choice the main features of The two-dimensional ANNNI modei in a field is the phase diagram are not changed; as discussed described by the Hamiltonian below. a 223 P. Rujan et al.: Rectangular Lattice Gas Model with Competing Interactions Using (3) one can easily transform the two Hamil- ground state under a change of the sign of the field; tonians, (1) and (2), into each other. In particular, (5) in particular (3 x 1)* goes over into (3 x 1) . - At corresponds to non-zero temperatures the model with Jt,Jr<O can be solved easily in the limit of vanishing coupling r:JzlJr:0.3, Jr,Jz<O. (1) Jo, i.e. for the spin -j Ising chain with nearest and next nearest neighbour interactions in the presence ero/e :JolJt, coupling is positive Because of o, the "In of a field. Correlation functions and thermodynamic (ferromagnetic). use either or We shall Jo: -0.3Jr quantities can be expressed in terms of the eigen- Jo: *Jr. Another important identity relates the - values of the transfer matrix [13, 16, 25]. ln partic- magnetization, nr, ular, we calculated the magnetization and the tL asymptotic behaviour of the correlation function of ,,:1 rs.> (8) spins at sites I and r>1, L,?'t " i+r, (sis'+,) a as '11 nia' (10) of the Ising model to the coverage 0 of the lattice -1n2 gas model where rr is the magnerizarion; ,4 is an amplitude. The correlation length, (, and the phase, c1, are de- 0:(1-m)12. (e) termined by the two largest eigenvalues. Results for J2lJr--0.3 are shown in Fig.2. The step-like be- 2.2.