PHYSICAL REVIEW B VOLUME 47, NUMBER 4

Theory of adsorbate-induced surface reconstruction on W(100)

15 JANUARY 1993-II

Kari KankaalaCenter for Scientific Computing, P.O. Box 40, SF 02I-OI Espoo, Finland

and Department ofElectrical Engineering, Tampere University of Technology, P.O. Box 692, SF 33101-Tampere, Finland

Tapio Ala-NissilaPhysics Department, Box 1843, Brown University, Providence, Rhode Island 02912;

Department ofElectrical Engineering, Tampere University of Technology, P.O. Box 692, SF 33IOI-Tampere, Finland;and Research Institute For Theoretical Physics, University ofHelsinki,

Siltavuorenpenger 20 C, SF-00170 Helsinki, Finland

See-Chen YingPhysics Department, Box 1843, Brown University, Providence, Rhode Island 02912

(Received 8 January 1992; revised manuscript received 31 August 1992)

We report results of a theoretical study on an adsorbate-induced surface reconstruction. Hydrogenadsorption on a W(100) surface causes a switching transition in the symmetry of the displacements of theW atoms within the ordered c(2X2) phase. This transition is modeled by an effective Hamiltonian,where the hydrogen degrees of freedom are integrated out. Based on extensive Monte Carlorenormalization-group calculations we show that the switching transition is of second order at high tem-

peratures and of first order at low temperatures. This behavior is qualitatively explained in terms of anXYmodel where there is an interplay between fourfold and eightfold anisotropy fields. We also comparethe calculated phase diagrams with a simple mean-field theory.

I. INTRODUCTION

Structural phase transitions have been observed on anumber of surfaces. One of the most extensively studiedsystems undergoing a surface reconstruction is the cleanW(100) surface (see, e.g. , Refs. 1 —3 and referencestherein). It has a very rich phase diagram with a recon-structive phase transition from a (1X1) phase into ac (2X2) phase. ' The microscopic structure of thelow-temperature phase was unraveled early by Debe andKing, and it has since then been accepted that this phaseinvolves alternate displacements of the W atoms alongthe (11) direction in the surface plane. A number offirst-principles total-energy calculations (e.g. , Refs. 7 —9)have verified the inherent instability of the ideal (1X 1)phase toward this reconstruction, and the Debe-Kingstructure is found to have the lowest energy. There wassome dispute on the nature of the high-temperaturephase, and thus also on the details of the structural phasetransition between these two phases. Earlier, there wasboth experimental" ' and theoretical' evidencewhich was interpreted in favor of an ordered high-temperature phase where the tungsten atoms would vi-brate about their ideal bulk positions. The other possibil-ity was that the tungsten atoms also would be displacedfrom their bulk positions in the high-temperature phasebut the displacements would point into random direc-tions, thus forming a disordered phase. ' Recent experi-mental ' ' and theoretical' studies have corroborat-ed the disordered nature of the high-temperature phase,and it is now generally accepted that the transition is ofan order-disorder type instead of a displacive one.

While the clean surface of W(100) has been studied ex-tensively both experimentally and theoretically, theinhuence of adsorbates on the reconstruction has beenstudied to a far less extent and in most cases a thoroughunderstanding of these phenomena is missing. Amongthe adsorbates, the adsorption of hydrogen has been stud-ied in most detail. ' ' ' ' It was found that smallamounts of hydrogen enhance the inherent instability ofthe clean surface, and increase the critical temperature ofthe order-disorder transition. ' The structure with ad-sorbed hydrogen was initially believed to be the same asthe clean surfa-ce reco-nstructed c (2X2) phase or verysimilar to it. ' ' ' It was later shown' and verified '

that the hydrogen-adsorbed surface has a different struc-ture when the coverage exceeds 0=0.1. Namely, theclean c(2X2) has a p2mg symmetry whereas thehydrogen-rich (coverage 8)0. 1) c(2X2)-H phase has ac2mm symmetry. The symmetry change was cleverlyverified by GriSths, King, and Thomas in a low-energyelectron-diffraction (LEED) spot intensity study. Theystudied the difference in the relative intensities of twohalf-order beams, namely ( —,', —,

') and ( —,', —,

') at a normal in-

cidence. The difference between these two beams is finitein the reconstructed clean surface p2mg phase, and itvanishes when the entire surface has switched into thec2mm phase. This symmetry switching was initiated at acoverage of 0=0.04 and completed at 0=0.16. Similarsymmetry switching has been observed in an infraredspectroscopy study. The c2mm phase is interpreted asdue to a structure in which the W atoms have alternatedisplacements along the ( 10) or (01) directions.

The hydrogen-induced switching of the W(100) surface

2333

2334 KARI KANKAALA, TAPIO ALA-NISSILA, AND SEE-CHEN YING 47

between these two structures is the main subject of thepresent work. Our study is motivated by theoreticaland experimental indications that the switching transi-tion could be of first order at low temperatures, instead ofbeing an XY-like continuous transition. We will first

briefly discuss the theoretical model for this work. Forthe clean surface, we adopt a lattice-dynamical Hamil-tonian which has recently been shown to describe thecritical properties of the W(100) surface very accurate-ly. ' It corresponds to an XY model with an intrinsiccubic anisotropy field. To this model then, we add theshort-range interactions of hydrogen with its nearest-neighbor W atoms. The hydrogen degrees of freedom arethen explicitly integrated out leaving an effective Hamil-tonian. This new Hamiltonian is shown to contain an-isotropy fields of all orders allowed by symmetry, and weargue that the nature of the switching transition at lowtemperatures is dictated by the interplay betweeneffective fourth- and eighth-order anisotropy fields. Wethen proceed to calculate the details of the phase dia-gram, using first simple mean-field arguments to locatethe line of switching transitions on the (T,p, ) plane. Tostudy the nature of the transition in detail, we present re-sults of extensive Monte Carlo renormalization-groupcalculations. These results show that at high tempera-tures the transition is continuous, while below a mul-ticritical point T, it becomes discontinuous. Finally, wesummarize our results and discuss the relevance of ourwork to experimental studies.

II. THEORETICAL MODELS

As discussed above, it has been experimentally verifiedthat hydrogen induces a switching transition where the Hatoms adsorbed on bridge sites pin the displaced W atomstoward themselves, resulting in displacement orientationsalong either ( 10) or (01) direction, instead of theclean-surface (11) direction. This reconstruction hasbeen subject to theoretical investigations (e.g., Refs.3,8,9,26,27,29—31) but to a far less extent than the cleanW(100) surface.

The models discussed below, as well as our model, be-long to the class of so-called lattice-dynamical modelswhich are written in terms of lattice displacements.When studying the driving forces for the transition, mi-croscopic details have to be included in the model. How-ever, lattice-dynamical models are usually sufficient tostudy the critical properties of the systems under con-sideration. The role of the detailed electronic degrees offreedom in these models has been discussed by variousauthors. '

A. Lattice-dynamical models

To study the effect of an adsorbate on the switchingtransition on the W(100) surface, Lau and Ying suggest-ed a lattice-dynamical model Hamiltonian

H =H"""+H'

In their work, the clean-surface term H"""is of the typesuggested by Fasolino, Santoro, and Tosatti, with thedetails left unspecified. The crucial point is that for theclean W(100) surface, it has the symmetry of an XY mod-el with cubic anisotropy. The intrinsic anisotropy of theclean surface is such that the W atoms are displacedalong the (ll) direction in the reconstructed c(2X2)phase. The adsorbate part H' in Eq. (1) was chosen tobe

H' =—,' g J, n, n + g n, v(R;+u; —R; ), (2)

which describes the effects of hydrogen on the structuraltransition. In Eq. (2), n, is ,an occupation number, andJ,-. ' is an interaction term between hydrogen atoms. Thesecond summation describes the interaction between hy-drogen and tungsten atoms as a sum of pair potentials U

where R; is the coordinate of a hydrogen adsorption siteand R s denote the ideal bulk positions of the surfacetungsten atoms. The addition of u; describes the spon-taneous displacements of the surface atoms.

Lau and Ying started from Eq. (2) and found that theadsorbate induces a cubic anisotropy field that opposesthe intrinsic clean-surface one. This adsorbate-inducedanisotropy will then increase in magnitude with coverageleading to the change of sign of the total anisotropy field.Hence the switching occurs from the phase in which Watoms are displaced along the ( 11 ) directions to a phasein which they are displaced along the (10) directions,favored by the opposing cubic term. This conclusion wassupported by electronic band-structure calculations,where it was suggested that even at low hydrogen cover-ages it would be energetically favorable for the surfaceatoms to align along the (10) or (01) directions in thelow-temperature c(2X2) phase with hydrogen atoms onbridge sites. Fasolino, Santoro, and Tosatti suggestedthe switching to occur at a coverage of8=0.044=(a /g'), where a is the lattice constant and g isthe surface coherence length. This value for switchingagrees with LEED studies.

Ying and Roelofs ' developed the model further bychoosing for the clean-surface Hamiltonian

u +—u + 8/ou u2Bi 4 i 4 ix iy

+C i ( u /» up» + u /» u/ » + u

/yu

Jy+ u

/yu / y )

(3)

where u; is the displacement of the surface W atom at theith lattice site, and u;„and u; are the x and y com-ponents of the displacement, respectively. 2 and 8 arecoefficients of a simple double-well potential, C, is thenearest-neighbor interaction term, and h4 is the cleansurface cubic anisotropy field. For the clean W(100) sur-face h 4 & 0, which aligns the displaced W atoms along the( 11) directions. Summation i in Eq. (3) goes over all lat-tice sites.

Using (2) and (3), Ying and Roelofs obtained an

47 THEORY OF ADSORBATE-INDUCED SURFACE. . . 2335

effective H' to describe the hydrogen effects by integrat-ing out the hydrogen degrees of freedom in the mean-fieldapproximation, and working with a constant coveragedescription. However, this description breaks down ifthe transition happens to be of first order as we shall dis-cuss below. Also, the approximation of retaining onlythe lowest-order anisotropy field is not justified for strongH-W coupling, as addressed below. In a closely relatedcomputer study, Sugibayashi, Hara, and Yoshmori usedan extended anisotropic XY model to describe the cleansurface, with a two-body lattice-gas model for the adsor-bate part. They presented results from their computersimulations for the switching transition, but were notable to conclude the nature of the transition at low tem-peratures. Their results seem to indicate, however, thatthere is a phase coexistence region at low temperaturesbetween the (11) and (10) phases, which would bestrong evidence for a first-order transition.

B. New model for H/W(100)

To be able to study a possible first-order regime, and togive a more accurate description of the low-temperaturephase, we have undertaken simulations of a new lattice-dynamical model. The clean-surface contribution in ourmodel is that described by Eq. (3), to which we have add-ed an additional eightfold anisotropy field:

g ~+ g 4+ 8$ Og 2 g yA 82 ' 4 ' 4

1

+C, (u;„uj +u;„uz„+u; u.

+u; uk )+hscos(8$, )' . (4)

with H' =g; n; (V, .—p), where the summation i goesover all lattice sites and i' over all bridge sites. As usualP=1/(ktt T), n;. is the actual hydrogen occupation num-ber at a bridge site i, V,' is the interaction potential be-tween hydrogen and surface tungsten atoms, p is thechemical potential, and u,. is the displacement vector of atungsten atom. Next, we sum over all hydrogen degreesof freedom to obtain

Z =Tr(„)exp[ —P(H' )j,where H' in an e+ectiue Hamiltonian, which can be ob-tained in a straightforward fashion as

H' =H""" ktt T g lnI1+exp—[ —P( V, —JLt)]I .

The summation i' goes over all bridge sites. They canalso be labeled as i '= (i, v), in which there are two bridgesites v associated with each lattice site i. The interactionV; is obtained by expanding a pairwise interaction poten-

where P; =arctan(u, /u; ) is the displacement angle. Tocalculate the total surface-adsorbate partition function,we will make the simplifying assumption of neglectingdirect H-H interactions. Thus, we can write

Z =Tr(„) („)exp[ P(H"""+H'—)],

3.&eh

IUiy

@=2 x(

Uly U 1~ xUix

Xk

= &10&

FICx. 1. Here we show schematically the relative positions ofW and H atoms on the W(100) surface. The solid dots (~ )

denote the ideal W atom positions, and the vectors u; denote thepositions of the displaced W atoms. The hydrogen atoms resideon the bridge sites ( X )i'=(i, v), where i denotes the tungsten-atom lattice sites, and v=1,2 the two bridge sites related toeach lattice site. The H atoms feel the W atom potential of thenearest sites only, e.g., V;~= —au; +auq, cf. Eq. (7). We havealso depicted the magnitudes of the displacements and the lat-tice constant.

in which the summation i, v goes over all bridge site andthe average ( ) denotes an average over configurations.Since there are two bridge sites for each W, the maximumallowed coverage is normalized to the value of two.

C. KS'ect of the anisotroyy fieldsin the model for H/W(100)

We can predict qualitatively the behavior described byH' if we express it entirely in terms of the displacementangles P; for fixed amplitudes instead of the displacementvectors u;. If we retain only the isotropic nearest-neighbor coupling terms and the leading anisotropy fieldsin the on-site potential, we then obtain a simplified Ham-iltonian

Hg i =K g cos(P; P~ ) h4(T, p) —g c.os4$;—

+ h s( T,p) g cos8$, .

tial to lowest order to the W-atom displacements. Thisgives us V; „=g a,. u, where the components of a aregiven by a;&&=( —a, O), a;&2=(a, O), a;2&=(0, —a),a;zz=(O, a), and ui =(uj„,ui~), and a is an effective in-teraction parameter between hydrogen and tungstenatoms. The relative positions of H and W atoms as wellas displacement vectors are illustrated in Fig. 1.

The Hamiltonian (7) is written in terms of the chemicalpotential whereas the experiments are carried out at aconstant coverage. The description of the adsorbate interms of the chemical potential rather than the coverageallows us to better investigate the possibility of both afirst- and second-order switching transition. The cover-age can be calculated and is given by the expression

0= 1

, , exp[P( V, „—p)]+ 1 ) '

2336 KARI KANKAALA, TAPIO ALA-NISSILA, AND SEE-CHEN YING 47

The anisotropy fields h~(T p)=h4(T p)+h~ andhs(T, p)=hs(T, p, )+h8 where superscripts 0 and a referto clean-surface- and adsorbate-induced terms, respec-tively. This Hamiltonian is an XY model with both four-fold and eightfold anisotropies.

The form of H' together with the dependence of h4and h8 on T and p has important consequences for theexpected behavior of the H/W(100) system, sincerenormalization-group analysis tells us that the fourfoldfield is relevant at all temperatures T(T, . For smallvalues of p, h~(T, p) (0, which favors the orientation ofthe W atoms along the (11) directions. However, sinceh ~( T,p) is a monotonically increasing function of p (for afixed T), there will be a point (p, , T, ) where

h4(p, , T,~ ) =0. Beyond this, h4( T,p) )0, and theminimum energy state is now given by W atoms dis-placed along the (10) directions. This simple analysispredicts that the switching transition in the (T,p) planeis determined by the condition h~(p, , T,„)=0, and it isan XY transition of Kosterlitz- Thouless type. Physi-cally, for the clean surface, the intrinsic fourfold fieldh4 &0, indicating a preference of the displacements forthe W atoms along the ( 11) directions. As H is ad-sorbed, it sits on the bridge sites and the H-W interactionenergy is minimized if locally the displacements of thenearest-neighbor W atoms switch and point along the(10) or (01) directions. This local distortion has afinite range. Globally, the average h4(T, p) is still nega-tive for low hydrogen coverages. However, the effectiveh 4( T,p) now increases with the coverage or the chemicalpotential, rejecting these local distortions around each Hatom.

To understand the nature of the switching transition inmore detail, we must also consider the effect of the eight-fold anisotropy field hs(T, p) which is induced by the H-W interaction term in (7). It is well known that the eight-fold field is relevant at low temperatures but above someT, & Tz it becomes irrelevant. When the eightfold fieldis negative, it favors orientations P = n (n /4),n =0, . . . , 7, while for h&(T, p) )0 the preferred orienta-tions are P=vr!8+n(n/4), n =0, . . .. , 7. This is demon-strated in Fig. 2. Consider first a fixed, negative h8. Theeightfold field favors the same directions which are alsofavored by both positive and negative h4(T, p). Thus,when h~(T,„,p,„)~0,from the negative side a relevanth8( T,p) field keeps the system at the orientation P =~/4.However, as soon as the fourfold field is positive andfinite, it dictates again the orientation of the systemwhich suddenly switches from P=m/4 to /=0. Thus,the transition is expected to be of first order at low tem-peratures when the eightfold field is relevant and nega-tive. However, the situation is changed above T„whereh s becomes irrelevant. Then, at h4( T,„,p, ) =0 the sys-tern is pure XY model. The phase boundary thus has amulticritical point T„where T, and T, coincide forsome p. This is precisely the behavior for our choice ofH"""as in (4) with a zero intrinsic eightfold field. Thereason is that the hydrogen-induced fourfold and eight-fold fields are of the same sign, and thus compatible witheach other.

h8&0

h8&0

hg&0

(b)

FIG. 2. (a) The eight directions favored by the positive andnegative fourfold-anisotropy fields. The dotted directions arefavored by a negative h4 and the solid lines show directionsfavorable for a positive h4. A negative eightfold field h 8 favorsall these directions. (b) The eight directions favored by a posi-tive eightfold field. Compared to (a), these directions are rotat-ed by ~/8.

For the case of a positive hs(T, p), which competeswith the fourfold field, more complicated behavior canresult below T, . Then we expect the phase boundaryseparating the ( 11 ) and ( 10) phases to open up into twoIsing-like transitions with an intermediate phase in be-tween. ' ' This is a subject of a separate study.

It should also be noted here that this discussion offourfold and eightfold fields is, strictly speaking, onlyqualitative for H/W(100). As we discuss in the Appen-dix, higher-order fields are comparable in magnitude withh4 and h8 for a realistic H-W coupling. Therefore, thefollowing mean-field and Monte Carlo renormalization-group analyses are based on the full effective Hamiltonianof Eq. (7). In addition, although the switching transitionoccurs at h4(T, p) =0, this fourfold field should be a re-normalized effective field in which the short-wavelengthdetails have been averaged away. Thus, even if we areable to extract correctly the bare h4 field in the Hamil-tonian, it still would not allow us to locate the switchingpoint exactly. This is one more reason why the numericalwork is absolutely necessary. The discussion above, how-ever, is valuable in that it nicely brings about the physicalideas behind the change in the order of the switchingtransition.

III. MEAN-FIELD ESTIMATEAND THE CHOICE OF PARAMETERSFOR THE SWITCHING TRANSITION

Before setting out to an extensive numerical study ofour effective Hamiltonian, it is useful to study the mean-field solution for the switching transition. The mean-fieldapproximation is exact at T =0 and thus at least at lowtemperatures it should yield reasonable results. Athigher temperatures, the temperature Auctuations affectthe transition, making the switching occur at a lower

THEORY OF ADSORBATE-INDUCED SURFACE. . . 2337

coverage or chemical potential than the mean-field theorypredicts.

The energy difference between the two phases that wedenote by P =a./4 and / =0 (N is the number of tungstenatoms), is

SZ= H" (t= ——H"(y=o)4

where

(10)

and

H P= —=2Nh +Nh —k T g ln 1+exp . —P V P= ——p . +N +——2Ceff ~ 0 0 B4 4 8 B V, l 4 1

V, l

H' (/=0) =Nba —k&T g ln(1+exp[ —P[ V, , (/=0) —p]] )+N +——2CiV, l

(12)

This can be simplified by noting that there are only fourdifferent kinds of energy changes related to the bridgesites (total of 2N) that can take place:

ho N„ 11+exp[ —P( —~2a —8)]

2 1+exp(Pp)

N 1+exp[ —P( —&2a —p) ]2 1+exp[ —P(2a —p)]

Nk T 1

1+exp[ —P(&2a —p)]2 1+exp(PM)

Nk 1

1+exp[ P(V 2a p—)]—2 1+exp[ —P( —2a —p) ]

(13)

In order to study in more detail the low-temperature be-havior of the system, we set AE =0 and seek solutions toit. These solutions become exact at T =0 in the absenceof Auctuations.

At T =0, we get b,E/N =—,'p, „+2h4+a=0 in the in-

terval —2a &p, & —~/2a. The solution for p,, is then

p, =2( —2h~ —a), where p, is the value for the chemi-cal potential where the switching between (11) and(10) phase occurs. Thus, by fixing the value of a and h4we can compare the prediction of the mean-field theorywith the experimentally observed values of coverage forthe transition. From the result we can also deduce a con-straint between the fourfold anisotropy field h 4 and theinteraction parameter a as 4~h4~ &(2—v'2)a which re-stricts the choice of parameters, as will be discussedbelow.

In our model, there is also a solution for a higherchemical potentia1 which corresponds to the unphysicalsituation of switching back to the ( 11 ) phase at a highercoverage. This is due to our omission of direct interac-tions between hydrogen adatoms on adjacent bridge sites.A direct repulsion between the adatoms on these siteswould eliminate this unphysical transition. Experimen-tally, many complicated structures ' have been observedat high hydrogen coverages and these are outside thepresent scope of study.

We have worked with two different sets of parameters.

The first set (set I) is very close to that of Sugibayashi,Hara, and Yoshimori and was chosen so that directcomparison could be made with their results. We areparticularly interested in answering the question whetherthe transition is of first order at low temperatures. Theirresults indicate this possibility but the conclusions wereunclear. A notable point in this set of parameters is thatthe clean-surface fourfold field h4 is very weak comparedto other similar studies' or total-energy calcula-tions. '

The other set of parameters (set II) is more consistentwith experimental work and total-energy calculations. AHamiltonian based on this set of parameters has beensuccessfully used to explain many clean-surface phenome-na. ' ' However, to prevent a switching into displace-ments along directions other than (10), we need to setthe eightfold field h s of Eq. (4) to a negative value.

In what follows, we work with dimensionless displace-ments u and temperature T defined as u =u, u, T=T, T,where T, and u, are the scale factors. The numericalvalues for the first set of parameters areA =Au, /T, = —2, B=Bu, /T, =8.8, Ci=C, u, /T,=1.5, h4= —0. 1, a=4. 5, and h8=0. The numericalvalues for set II are A = Au, /T, = —10,B =Bu,"/T, =40, and C, =C, u, /T, =3.75, h 4

= —1.85,cx = 17, and h &

= —0.6. The displacement amplitudeu =Q(4C i

—A ) /(B + 8h & ) = 1 for parameter set I and isset to the value 1 for the second set. The critical temper-ature for the clean surface (p= —oo ) is about Tc—- 1.3for the first parameter set and about T~ =2.3 for the oth-er set. The experimentally observed values for T&-—230K and u =0.2 A yield the scale factors u, =0.2 andT, =175 for the first set and T, =100 for the second set.For simplicity, from here on throughout the paper wewill always use the scaled values for the parameters, e.g. ,Tfor T.

The choice of a=4. 5 in the first set of parameters fol-lows Roelofs and Ying. In their approach, the effectiveHamiltonian was obtained by keeping terms only tofourth order in the displacements of the %' atoms. Thevarious parameters including a were then determined bycomparing theoretical results with the experimental data.

2338 KARI KANKAALA, TAPIO ALA-NISSILA, AND SEE-CHEN YING 47

As we show in the Appendix, the higher-order terms arequantitatively important and cannot be neglected.Indeed, the estimate for the clean-surface anisotropy h 4in Ref. 3 is about an order of magnitude smaller than thecurrent first-principles results. Thus, the value of +=4.5can also no longer be viewed a reliable estimate. In thesecond set of parameters, the anisotropy field is muchstronger and close to the first-principles result. As aconsequence, the constraint (2 —&2)u )4~ h 4 ~

impliesthat a much larger value of a is required. Our choice ofa = 17 yields a switching coverage in agreement with theexperimental observation. It is also consistent with theinfrared vibrational spectroscopy data when the newlarger value of the anistropy field is taken into account.

Finally, we should note that a smaller value of h4makes the transition more XY like and thus more difficultto analyze numerically, whereas a larger value would de-crease the fluctuations and result in more Ising-like be-havior in the vicinity of the transition with smaller finite-size effects. Larger absolute values of the fourfold field

h~ also hold the system, preferably in the (11) direction,and thus would move the switching to occur at a higherchemical potential. However, the parameters within themodel cannot be changed arbitrarily, as discussed above,and the value of h 4 imposes a restriction on the value ofthe interaction parameter a. Increasing h4 means in-creasing a, and this leads the switching to occur approxi-mately at a constant coverage over a large temperaturerange. The qualitative behavior of these two parametersets is the same both in the ( T,p) and in ( T, g) planes (thelatter is not sketched here ).

IV. NUMERICAL RESULTS

To study the nature of the switching transition for thefull Hamiltonian of Eq. (7), we have carried out extensiveMonte Carlo renormalization-group (MCRG) simula-tions. By studying the cumulants of the order parame-ter, we have been able to determine the order of the tran-sition both at low and high temperatures. These resultsare supported by our additional studies of the order pa-rameter, its distribution function, coverage, and the ob-servation of strong hysteresis at low temperatures, as wewill elucidate below.

In the simulations, we have used the standard Metrop-olis updating scheme. They were carried out using a non-conserved order parameter and Glauber dynamics. Theamplitudes of the tungsten-atom displacements u; wereheld fixed, as it has been shown recently' ' that dis-placement amplitude fluctuations are not important forthe critical behavior of this system. We studied mainlytwo system sizes, namely 64X64 and 32X32. Computa-tional time involved with larger systems becomes ratherformidable. The averages were computed over 50000configurations for parameter set I, and over 20000 for setII.

In the MCRG studies, the lattice is divided into blocksof size L, and we then study the cumulants of the mo-ments of the order-parameter distribution function. Forthis, we used two order parameters, namely

(14)

(15)

where u„; and u; are the x and y components of the dis-placement vector at the ith lattice site, and i and i arethe x and y coordinates of the ith lattice site in a block of

(i +i ) .size L. The coefficient ( —1) " ' is a phase factor. ( )denotes an average over configurations, and the summa-tion goes over all lattice sites. In the (11) phase both(14) and (15) have finite values, whereas in the transitionto ( 10 ) ( (01 ) ) phase ( u, ) ( ( u~ ) ) vanishes. There is nopreference for one phase over the other. By studying thebehavior of the cumulants of these average displacementcomponents, we can probe the order of the switchingtransition.

The block cumulants used are defined by

U~, =1— (16)

V„=1— (17)

where i refers to x and y. The variation of these two cu-mulants as a function of the block size L, depicts a flowdiagram analogous to that of a renormalization-groupmethod. It can be shown that these cumulants ap-proach zero above Tz as the block size increases. BelowT~, both these cumulants tend to nonzero valuesU~ =U*

3 Vg V J5 and at a second-order tran-sition they tend to nontrivial values.

We also monitored the combined order parameter ofEqs. (14) and (15):

(@)= g( —1) ' 'u„;u, .L

(18)

For the ordered ( 11) phase this order parameter has afinite value, and in the (10) phase it vanishes.

If the transition is of first order, one of the block cumu-lants is expected to flow toward the ordered state value of3

at al 1 chemical potential s . As the transition occurs sud-

denly between two ordered phases, one of the order-parameter components does not vanish at the transition((u ) for the (01) phase and (u„) for the (10) phase)but abruptly changes from one ordered state value to theother. The other component vanishes suddenly at thetransition ( ( u„) for the ( 01 ) phase and ( u ) for the( 10 ) phase).

One should note that if the transition is very weakly offirst order, it is very difFicult to distinguish between it anda second-order transition. This is true in our case in thevicinity of the multicritical point where the order of thetransition changes.

In the following, we will present numerical data onlyfor one parameter set (set I, Figs. 3 and 4) but will showthe phase diagram for both parameter sets in Fig. 5. Thereason for this is that results for both sets are qualitative-ly very similar, and the quantitative differences can be

THEORY OF ADSORBATE-INDUCED SURFACE. . . 2339

ULx

T= 1.4Q,x64

~ p =-14& p=-13o p=-12~ p=- 11.75+ p=- 11.5op=-11

(aj

0,7 „—

06 '—

0.5 +g.0.4

0.3

0.2

0,1

0.1 0.2

0.7 1,—

0.6

0.5

0.4

0.3

0.2

0.1

0.1I

0.2

FIG. 3. MCRG cumulant behavior typical(a) for a second-order transition at T=1.4,and (b) for a nrst-order behavior at low tem-peratures (T=0.5) for parameter set I. Seetext for details.

Uix

T=0.5 ~ p=-90o p=-89o p=-8.8~ p=-8.7

ULy

0.7 „—0.6

0.50.40.3

020.1

I

0.1

I

0.2

0.7 }I

0.6

0.5

0.4

03 -"0.2

0,1

9 l i

0.1 0.2

seen in the respective phase diagrams. For parameter setI, the switching occurs at approximately 0=0.02, whichis lower than the experimentally observed value. For setII, we reproduce the switching at approximately 8=0.1,which is in good agreement with experiments.

A. High-temperature MCRG results

Figures 3(a) shows the behavior for parameter set I athigh temperatures (T=1.4), and confirms our predic-tions for the second-order transition. The switching tran-sition in this case is from the ( ll) state into the (01}state, i.e., (u„) vanishes. The cumulants UL„of u„showbehavior typical for a second-order transition: in the(11) phase (p( —12) the cumulants approach the fixedpoint of UL„=0.67, and in the (01) phase (p) 11.S) thecumulants approach zero. We estimate the transition tohappen in the region where chemical potential is—12 &p & —11. Inside this range of chemical potential,the cumulants approach a nontrivial 6xed point(p= —11.S). The probleinatic behavior of the cumulantsat p= —14 is attributed to large fluctuations at this tem-perature and the vicinity of two phase transitions (order-disorder transition and switching).

When studying the cumulants of ( u~ ), we see that in

the ordered phase they approach the value of ULy 0 67,and then near the transition they How to a nontrivialfixed point at pH[ —12, —11.S]. At larger values of pthe cumulants again approach the ordered state value.The errors have been taken from the results from thelargest lattice sizes.

Based on these results, we conclude that the switchingtransition at T, =1.4 occurs at p,„H[—12, —11.S],and it is of second order. Data for the parameter setII are very similar, and result in a switching transi-tion for T =2.4 ( (Tc, high temperature) atp, E [—28.4, —28.0].

B. I om-temperature MCRG results

In Fig. 3(b) we depict the behavior of the cumulants atT, =0.5, again for the parameter set I. At chemical po-tential values p& —8.8, the cumulants of UL all ap-proach the ordered phase value. At p & —8.8, the cumu-lants vanish. The cumulant behavior of Uly shows nochange in behavior when passing through the transition,but the cumulants approach the ordered-state value of0.67 at all chemical potentials. We interpret this as evi-dence of a first-order phase transition into the (10)

2340 KARI KANKAALA, TAPIO ALA-NISSILA, AND SEE-CHEN YING 47

0.2

T=1.0 xT=1.2 ~T=10 ~T= 0.8 +

T= 0.5 ~

T=02 o t

r

o/ /

I

l

X/

//

very pronounced at T=0.2. Similar behavior was alsoobserved for the order parameter. The jump in coveragewas very clear in simulations with set-II parameters inthe first-order regime, and more gradual at higher tem-peratures, reconfirming the qualitative similarity in simu-lations for both parameter sets.

In addition, we have observed strong hysteresis andphase coexistence (as determined from the order-parameter distribution function) at low temperatures tofurther support the scenario for a first-order transition.Hysteresis studies have not been performed for thesecond parameter set.

D. The phase diagram

Based on the mean-field theory and our numerical re-sults, we sketch phase diagrams for both parameter setsin Figs. 5(a) and 5(b) for the line of switching transitions

1.6

-12 -11 -10 -9 -8 -7(

-6

'1.4

FIG. 4. Coverage [Eq. (g)] increases as a function of thechemical potential. At elevated temperatures the incrementtakes place smoothly, whereas at lower temperatures an abruptincrease is seen. The latter would indicate a possible first-ordertransition at low temperatures as discussed in the text. The cal-culation is performed for parameter set I.

0.8

0.6—(10)

phase at this temperature. The strange behavior of ULyat p= —8. 8 is probably due to strong metastable effectscombined with slow dynamics of the system at these lowtemperatures.

Based on similar arguments, the switching for set IIoccurs for T =0.6 at p,„H [ —26. 7, —26.4]. In this casethe metastability effects seem to be less severe, and theaccuracy of simulations is thus better seen at relativelylow temperatures.

0

2.2

1.8

T

1.4

I

—13I

-12 -10

C. Results from other quantities

Further qualitative support for the MCRG results canbe found in our coverage studies. In Fig. 4 the coverage,as in Eq. (8), is plotted as a function of temperature fordifferent values of the chemical potential. If the transi-tion were of first order a discontinuity in the real cover-age would be observed at the transition. However, as wework with a finite system, we observe only a very sharprise in coverage at the transition.

In the case of a second-order transition, we would ex-pect the coverage to increase gradually when the transi-tion region is passed. The data in Fig. 4 are in qualitativeagreement with the prediction of a first-order transitionat low temperatures and a second-order transition athigher temperatures. The abrupt increase in coverage is

1.0

0.6

0.2

-30I

-29I

-28I

-27 -26

FIG. 5. The phase diagrams for H/%(100) in the (T,p)plane, based on numerical simulations of our model Hamiltoni-an for parameter sets I (a) and II (b). The solid dots denote theMCRG results. The switching transition changes the order atT„above which the transition is of second order (solid line),and below which it is assumed to be of first order (dashed line).The dotted line shows the mean-field result. The solid line be-tween the ordered and disordered phases in (a) is only a guide tothe eye. Tz for the clean surface in (b) is -2.4.

47 THEORY OF ADSORBATE-INDUCED SURFACE. . . 2341

as induced by hydrogen adsorption on the W(100) sur-face. The temperatures for the (1X1)~c(2X2) transi-tion both for the clean-surface (p= —~ ) and for the lowhydrogen coverages have been determined from MCRGstudies. The solid order-disorder line is only a guide tothe eye.

In both Figs. 5(a) and 5(b), the dashed line denotes themean-field solution. The solid line is the part of theswitching transition where it is expected to be of secondorder, and the dotted line depicts the first-order switch-ing transition. The agreement with the mean-field line isvery good at low temperatures where the role of fluctua-tions is not important. At higher temperatures the devia-tions are due to temperature fluctuations which cause thesystem to undergo a phase transition at lower chemicalpotentials than our mean-field results would imply. Themean-field result for the parameter set II at higher tem-peratures deviates particularly strongly from the MCRGdata.

The whereabouts of the multicritical point T„wherethe transition changes its order, could not be accuratelypinpointed by our simulations. The locations of the mul-ticritical points depicted in Fig. 5 are only schematic.We expect it to be somewhere between 0.5 & T, & 1.0 forset I, and 1.0 & T, & 1.5 for set II. The error bars showndepict the variance of numerical results from a series ofsimulation runs.

V. SUMMARY AND DISCUSSION

To summarize, we have developed a model Hamiltoni-an to describe the adsorbate-induced effects on theW(100) surface. The clean-surface part consists of alattice-dynamical Hamiltonian, which describes the(I X I)~c(2X2) transition as studied by Han andYing. ' ' To include the hydrogen-induced effects, wehave considered a simple model of interactions betweenW and H atoms, and integrated out the hydrogen degreesof freedom to obtain an effective Hamiltonian. This leadsto an XF model with anisotropy fields of all orders al-lowed by symmetry. In particular, the interplay betweenthe fourth- and eight-order anisotropy fields indicatesthat the switching transition should be first order at lowtemperatures and of second order at high temperatures.This prediction was confirmed by Monte Carlorenormalization-group calculations, and further corro-borated by studies of other quantities. We also used theMCRG studies together with a simple mean-field theoryto map out the line of switching transitions on the (T,p)plane.

We have carried out simulations with two ratherdifferent sets of parameters, and obtained qualitativelysimilar behavior for both sets. With set I, we have beenable to reproduce results similar to those of Sugibayashi,Hara, and Yoshimori, and reinterpret their findings interms of first- and second-order transitions. The transi-tion coverage is somewhat lower than experimentally ob-served. The second parameter set is more in agreementwith total-energy calculations, and the results are qualita-tively the same as for the first set. Both these parametersets reveal qualitatively similar behavior. Thus, regard-

less of the strength of the fourfold anisotropy field in anexperimental sample, a change in the order of the switch-ing transition should be observed as a function of temper-ature.

Experimental evidence from an infrared spectroscopy(IR) study supports the scenario for a second-order tran-sition at room temperature. In their study, Arrecis, Cha-bal, and Christman observe only one peak in their IRspectra over the coverage range 9& [0.044, 0.22]. Ifthere were coexisting phases present, this would show asadditional peaks in the spectra corresponding to thedifferent symmetries of the phases. The LEED study ofGriSths, King, and Thomas was performed at lower tem-peratures (T=200 K) although the temperature was notheld constant. Their data show indications of a coex-istence region between 8& [0.05,0. 16] which could be in-terpreted in favor of a first-order transition. The temper-ature range in Ref. 24 is somewhat higher than our pre-dictions but is in qualitative agreement with our work.Very recently, Okwamoto has presented results ofmean-field calculations on the model of Lau and Ying,showing the switching transition to be of first order at alltemperatures below the order-disorder line. However, itshould be noted that as the cubic anisotropy field van-ishes at the transition, Okwamoto's method is equivalentto treating the pure XY model with a mean-field theory.In this case it is well known that the mean-field treatmentdoes not predict the order of the transition correctly asthe angular spin fluctuations destroy conventional long-range order.

The major drawbacks in this study were the slow dy-namics of the model and the finite-size effects. These to-gether made it impossible for us within given computertime to locate the multicritical point more accurately. Inaddition, the adsorbate-induced part generates all al-lowed anisotropy fields, which complicates the analysis.From an experimental point of view, however, the loca-tion of the multicritical point may vary from sample tosample, depending on the possible intrinsic eightfoldfield. Thus a more accurate theoretical determination ofT, may not prove to be necessary after all.

ACKNOWI. EDGMENTS

Thanks to Mike Kosterlitz for many useful discussions,and to Wone Keun Han for useful advice in simulations.S.C.Y. and T.A-N. were supported in part by an ONRcontract. K.K. was supported in part by a contract be-tween the Center for Scientific Computing and theAcademy of Finland„and grants from the FinnishAcademy of Sciences and the Finnish Cultural Founda-tion. K.K. also acknowledges the hospitality of BrownUniversity where part of this work was carried out. Weacknowledge the computational resources provided bythe Center for Scientific Computing, Finland, TampereUniversity of Technology, Finland, and Brown Universi-ty, U.S.A.

APPENDIX

In this appendix, we will discuss the hydrogen-inducedanisotropy fields in more detail. We shall first write aseries expansion for the part of the Hamiltonian which is

2342 KARI KANKAALA, TAPIO ALA-NISSILA, AND SEE-CHEN YING

due to the adsorbate. We shall then expand the H-W in-teraction potential in terms of the anisotropy fields, anddiscuss the convergence of this expansion.

We start from the adsorbate-induced part of Eq. (7)

and denote fugacity by y=exp(P}Lt, ). The summation igoes over all lattice sites and v over both bridge sites oneach lattice site. Notation is in all cases equivalent tothat in the text.

H' H'—""=—k T gin(1+e '" "} (Al)

r'= —k~ T g y exp( —P V; ) — exp( —2P V, )+2

(A2)

(V;„)"= —k~TQ yg( —P)"i, v n

2 (V }ll

g( —2P)" ' +2 n! (A3)

, (V;„)"= —g (—p)"

i, v, n

2n 2 3n 3

+ + 0 ~ ~

2 3(A4)

(V; )"( P)n

—i ' ~ y k( 1)(k +1) k(n—1)

n1I, V, fl k=1

net' ( V, ,}"= —g ( —p)" ' ', F„(y,IJ) .i, v, n

(A5)

(A6)

Next, we study the effect of the H-W interaction terms g; „(V~ „) which generate all the anisotropy fields induced bythe hydrogen degrees of freedom. Due to symmetry arguments, the odd powers of u vanish. Let us now for the sake ofargument consider the term in Eq. (A5) for n =4:

g(V, ) = g g(a, „u,)g(a, „.u„)g(a; u )g(a;,„.u„) (A7)

which reduces to

g(V, ) = g [a,„u, ]I, V, J

(AS)

when we consider only the on-site terms.On the other hand, based on symmetry arguments we can readily write the expansion for (Al) in a general form when

we consider contributions only from the on-site terms:

H'~ H"""=g—g h4k(T, p, )cos(4k/;),i k=0

(A9)

where P; is the displacement angle. The terms where k = 1 and 2 correspond to the analysis in Sec. II C.When we now expand Eq. (AS) in terms of the displacement angle P, and note that u = ( u„,u~ ), ~

u~

= u o,u =uocosP, u =uosinP, and ~a; .

~

=a, we get a contribution for the fourfold anisotropy field as4

kg T aup[y —Sy +27y —64y + ] . (A 10)

4!4 k~ T

12495k~ T au,12!1024 k~ T F,z(p, T)k .

where F„(p,T) is defined by (A5) and (A6).To study analytically the interplay between the cubic and eighth-order anisotropy fields, and the vanishing of the to-

tal h4(T, p, ) at the switching transition, one must first ensure the overall convergence condition expI —P( V; —p)] ~ 1.

However, it is important to notice that this is not the total adsorbate-induced field h 4( T,p). Namely, the evaluation ofhigher powers of ( V~ )" in Eq. (A5) results in contributions not only to h„', but also to lower-order anisotropy fields.These contributions can be of the same order of magnitude as the leading terms, which severely hampers the analysis ofthese expansions. Thus, for example, for the coefficient of the fourfold anisotropy field we obtain

6 8k~ T aup 3k, T au, 7k' T aup

h 4( T,p) = — F4(p, T) F—6(p, T) , F—s(p,T)

1015k& T auo

Fio(C T}— (Al 1)10!32 k~ T

47 THEORY OF ADSORBATE-INDUCED SURFACE . . ~ 2343

Unfortunately, for the values we have used for the parameters a, p, and T, this condition is not usually met in the vicin-ity of the switching transition line. We have also verified this numerically. In addition, for the parameters that we use,the term auo/(k&T) is never less than l for physically reasonable temperatures, thus further hampering the conver-gence of (All). The additional effect of the more complicated off-site terms, which involve products of cosines atdifferent lattice sites, is also diFicult to determine accurately, but it seems evident that they should be included in aquantitatively accurate calculation.

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