Theory and Monte Carlo simulation of adsorbates on corrugated surfaces

Surface Science Letters 284 (1993) L449-L454 North-Holland

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Surface Science Letters

Theory and Monte Carlo simulation of adsorbates on corrugated surfaces

Eduard Vives 1,, and Per-Anker Lindg~rd Riso National Laboratory, DK-4000 Roskilde, Denmark

Received 26 August 1992; accepted for publication 29 December 1992

Phase transitions in systems of adsorbed molecules on corrugated surfaces are studied by means of Monte Carlo simulation. Particularly, we have studied the phase diagram of D 2 on graphite as a function of coverage and temperature. We have demonstrated the existence of an intermediate y-phase between the commensurate and incommensurate phase stabilized by defects. Special attention has been given to the study of the epitaxial rotation angles of the different phases. Available experimental data is in agreement with the simulations and with a general theory for the epitaxial rotation which takes into account defect stabilized situations.

The ordering in frustrated systems represents a challenging unsolved problem in several cases. Surface reconstructions, intercalated compounds or adsorbed monolayers on for example graphite are cases of frustrated systems in which the pair interaction is incompatible with the corrugation potential - either with respect to the distances or the symmetry [1]. 3D examples are charge density waves and fluxlattices in superconductors. We specifically study the case of adsorbed monolayers and by means of Monte Carlo (MC) simulations we demonstrate a number of features of defect-induced ordering. Molecular dynamics simulations have previously been done [2]. The experimental systems we have in mind are, in particular, D E and H 2 adsorbed on graphite [3-5] at low temperatures and with variable coverage, p. The experimentally determined phase diagram for D 2 (fig. la) reveals several unexpected phases showing epitaxial rotation angles, which bear no resemblance with existing theoretical predictions [6].

We have used the following simple model [7] for N interacting classical spheres, on a hexago-

1 On leave from: Dept. E.C.M., Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Catalonia, Spain.

* Supported by Ministerio de Educaci6n y Ciencia.

nal corrugated substrate with lattice parameter a and periodic boundary conditions. The Hamilto- nian is H = H 0 + Uc, where:

n0 = E (i,j>

= E 4e((or/rij)12--(or/rij)6), (i,j)

UC = E E UHK ciQHK'rj, ( 1 ) j HK

where rj are the positions of the particles, % = [r i - r j l , cr and e are the parameters of the Lennard-Jones potential VLj(r) having a minimum at r 0 = 1.122¢r (here r 0 = 1.46a, a = 2.456

for graphite) and UHK are the coefficients of the Fourier expansion of the corrugation potential with QnK being the reciprocal lattice vectors of the substrate. By Fourier transforming the Hamiltonian we notice how the frustration arises:

. =

+ ~ UnKt$(q - Qttr)P(q)l dq, HK ]

~(q) = ~_, eiq'r~ lT'q= fVLs(r ) e iq'r dr, (2) J

where S(q)= (I~(q)l 2) is the structure factor.

0039-6028/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

L450 E. I/ives, P.-A. Lindg~rd / Theory and Monte Carlo simulation of adsorbates on corrugated surfaces

30

,,, 20 r'r"

UJ ,', 1 0 -

5

0 ~ I I I I I I

,.6 i bl

' 4 1.4 ~ MC

J 1.2 ', l

i

0.8 I

0.6 - ~

O.4 i i

m'

1.8 1.4 1.0 0.8 0.4 COVERAGE p

Fig. 1. (a) Experimental phase diagram for D 2 on graphite from ref. [5]. (b) Corresponding simulated phase diagram. Data has been obtained from the temperature dependence of the energy at constant coverage (0, e) and from the structure

factor determinations at constant temperature ([]).

a) 002, ,, . . . . . . . , , r . . . . . . . . ., 7 / b) / / /

/ i x / x

/ qo2 " $ Ool * - - - J - - / / / o , , / /

Ooo Qlo ', Qoo Q~o Fig. 2. (a) Reciprocal space diagram showing a number of substrate reciprocal vectors QHr, adsorbate reciprocal vectors qhk and Q ~ positions (×) . 4' is the epitaxial rotation angle

(between ql0 and Qll, thin dotted line). The dot-dashed lines correspond to the directions of the maximum ridges of the anisotropic diffuse scattering due to elastic vibrations in the system. (b) Description of the structure factor peaks of the y-phase. The (©) indicates the dominant peaks and (©) the

weaker modulation peaks, see also fig. 5.

formly over the available area, leading to a decreasing I ql0t for decreasing coverages. We have indeed found in the simulations that I ql0l de- creases linearly with V~ (see fig. 3a) towards the value Q ~ . This is in good agreement with experiments [4,9]. For a general p the energy is not optimum with respect to H 0 and if t qhk I ~ [ Q10 1, no advantage can be taken from the corrugation

The dominant term in the corrugation potential expansion is Q~0, and we neglect the influence of smaller higher-order components. 12q has a main minimum ring [8] with a radius qm. This is slightly smaller than q0 = J qtoJ 0, where qt0 is the princi- pal reciprocal lattice vector for the adsorbate, and the index zero indicates the hexagonal structure with the lattice constant equal to r 0, and which has the coverage po = 3 ( a / ~ ) 2, here P~0 = 1.19. We can write q0 = (a~/3/ro)Qv~, where Q v~ = 4~r/3a corresponds to the perfect commensurate v ~ × v~- structure with coverage p = 1. Fig. 2a shows the first two reciprocal space unit cells of the substrate, the epitaxial rotation angle t~, the minimum ring qm and a number of qhk and QHr (adsorbate and substrate reciprocal wave vectors respectively). At coverages less than P0 the particles (at least for a range of coverages) might distribute themselves more or less uni-

o , , o a> r i

0.70 i J o

c~ 0.60 o .

G"

-0-

r 15

10 f b)

5 -

0

[ ~ .L J ' , _ ~ _ _ _ L ~ _ _ J

I J ' i I ] J

o,.xo i I : IT m

1.00 1.10 1.20 1.30

Fig. 3. Behavior of the main Bragg peak ql0 of the adsorbate as a function of the square root of the coverage ~ for T = O . 3 e / k n. The module Iqt01 is shown on top and the epitaxial rotation angle ~b (O) below. Experimental data for D 2 on graphite (<3) is also shown [3,4]. The straight line on the top panel is the expected linear behavior. The lines on the bottom panel are the predictions for cases (i) and (ii) of eq. (3). The thin dashed line corresponds to the finite size effects

eq. (4).

E. Vives, P.-A. Lindg~rd / Theory and Monte Carlo simulation of adsorbates on corrugated surfaces LA51

potential U c. Therefore, for a perfect (i.e., undis- torted), incommensurate, hexagonal structure, there is no selection mechanism for a preferred epitaxial rotation angle 4).

Disorder - of any kind - produces diffuse scattering extending ~(q) in all the reciprocal space [10], i.e., also yielding a finite value of t~(Q10). Thus, an imperfect structure is able to pick up energy from the corrugation potential 8-functions in eq. (2). The predominant diffuse scattering occurs around the largest Bragg peaks of the adsorbate structure (low indices h, k). The maximum in the anisotropic diffuse scattering occurs along ridges of decaying intensity from the peak at qhk, in directions which we indicate by dq. On this basis we can distinguish the following two possible cases of stabilization:

(i) Elastic stabilization. Elastic deformations of the adsorbate give anisotropic diffuse scattering oriented according to the adsorbate symmetry, i.e., along dq II q~0 (q~0 means the ql0 star), indi- cated with dot-dashed lines in fig. 2a around the qn peak. Consequently 4) will be determined by the anisotropic diffuse broadening of that Bragg peak which is closest to the Q10 position. For the interesting coverage range this is qll, and we limit ourselves to this case. This elasticity mechanism is, in essence, the Novaco-McTague-Vil- lain-Shiba theory [6] the former of which were worked out for small deformations. Thus, in the presence of the corrugation potential, fluctua- tions (thermal or quantum) will lift the infinite rotational degeneracy of the ground state and select only two domain - or twin - possibilities with +4). We shall call this the two domain 1-q structure, where q = q~0 is that characterizing one of the twins with the epitaxially rotated, hexagonal (RIC) structure. This mechanism ex- plains the epitaxially rotated structures observed for the heavy rare gas adsorbates which have a preferred interparticle distance (r 0) close to that of the v~ x v~- structure. However, the systems of light molecules (D 2 and H e) have larger corrugation potential and smaller r 0, between that corresponding to v~- x ~ and 1 × 1 structures. The existing theories [6], which assume elastically deformed uniform structures cannot explain the observed epitaxial rotation angles.

(ii) Defect stabilization. For variable coverage we must further consider regular defects, i.e., vacancies or interstitials in the adsorbates. The diffuse scattering from these has the symmetry of the substrate and gives rise to epitaxial rotation determined by the diffuse scattering around the Bragg peak qhk closest to Q~0, but now with maximum intensity oriented according to dq l[ Q~0.

Fig. 2a shows the situation corresponding to case (i) in which the maximum diffuse scattering is pointing towards the Q10 position in a direction parallel to ql0. This gives maximum energy gain from the substrate potential for case (i). A similar picture can be drawn for case (ii). Analysis of fig. 2a, demonstrates that the following geometrical condition yields a minimum for H by relating the mismatch and the epitaxial rotation angle 4):

ql0 tan(/3) Qlo ~/3-[COS(q~) tan(/3) - sin(~b)] ' (3)

with the two possibilities /3 --- 30 ° + 4) and 60 ° for cases (i) and (ii) respectively, and case (0) 4) = 0, [11]. This minimization is approximate since we are taking into account only the main symmetry directions and not the exact shapes of the function t;(q) which will depend on details like the elastic constants or the exact correlations between vacancies. It is interesting that the same conditions were found by Grey and Bohr [12,13] by considerations of the moire patterns and by making use of a simple symmetry principle. They demonstrated that numerous experimental find- ings were in agreement with eq. (3) and also that calculations for finite clusters with chosen sym- metries did fit into the picture. Since finite size gives rise to diffuse scattering with the selected shape symmetry, the finite size dependence is also contained in the above more general theory, which is based on minimization of energy. The tendency for orienting the first Bragg peak according to the modulation induced by the substrate was pointed out in connection with the D 2 experiments [3,4]. Here, the theoretical founda- tion is given (eq. 3). An interplay between the mentioned mechanisms will result in intermediate values of 4). We should also mention the coinci- dence lattice theory [14], developed for the case

L452 E. I/ires, P.-A. Lindg~rd / Theory and Monte Carlo simulation of adsorbates on corrugated surfaces

of high-order commensurate structures for which there exists indices h, k and H, K for which qhk = QnK" This gives a set of stable, discrete points in the (m, ~b) plane which fall close to the three lines here calculated. The reason is in that case different, since the stabilization is depending on contribution from potential expansion terms UHx with high values of H and K.

For systems with a substantial amount of defects we must further consider two cases, examples of which can be seen in the simulated real space pictures figs. 4d-4f. They are (a) of non- uniform density, where the defects are immiscible and precipitate out from essentially defect free domains and concentrate in the domain bound- aries, and (b) of uniform density, where the defects are miscible and may even form superstruc- tures. The latter is found both in the simulations and experimentally for D e on graphite and be- longs to the case (ii) of eq. (3). In case (a) it is clearly difficult to give a unique relation between an observed epitaxial angle and the nominal coverage, since the angle is in this case determined

by the local density of the ordered domains and by the irregular shape of the domains.

We have performed MC simulations on large systems with 8100 substrate positions and N ranging from 2700 (p = 1), corresponding to a perfect v~ × x ~ - structure [7] up to 4000 particles (v/p - = 1.22). We have chosen parameters close to those expected for D 2 on graphite [15] (tr = 1.3a and Uto/e = - 1/3). Details of the choice of parameters and simulations will be given elsewhere. The obtained phase diagram is shown in fig. lb. Data have been obtained from temperature scans at constant coverage. There is good agreement with the experimental phase diagram, fig. la. The simulated linear variation of the first Bragg peak, corresponding to an expansion with decreasing coverage, is shown in fig. 3a. Below, (3b), the epitaxial rotation angle is compared with the theoretical calculation (eq. 3), and the experimental points for D 2 [3,4]. We notice a good agreement for regions I and II between both simulations, experiments and the defect stabilized epitaxial angle, whereas there are deviations from the the-

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Fig. 4. Top panel, intensity plots of the two-dimensional structure factors S(q). Bottom panel, small fractions (1t%) of the corresponding real space snapshots (o particles, • substrate wells) obtained by Monte Carlo simulation for the square root of the coverages ~ = 1.051 (a,d), V~ = 1.127 (b,e) and y/p = 1.212 (c,O and temperature T = 0.3~/k n. We have chosen a logarithmic color scale (increasing intensity from blue to red) in order to emphasize the diffuse scattering. The shown reciprocal space cells are identical to those discussed and defined in fig. 1. The ~%functions at Q]0 and Qm can hardly be seen in this representation;

see fig. 5.

E. Hves, P.-A. Lindg~rd / Theory and Monte Carlo simulation of adsorbates on corrugated surfaces IA53

oretical predictions in region III. Intensity maps of the simulated structure factor and fractions of the corresponding real space structures are shown in fig. 4.

Region I (case (0)) is a commensurate phase with qll = Q10, yielding a perfect non-rotated v~- x v~ structure with additional density concen- trated in meandering domain walls (figs. 4a and 4d). On our ideal surface, we do not find evi- dence for a striped phase. Region III (case(i)) is an incommensurate, rotated two-domain 1-q structure (RIC). The obtained epitaxial rotation angles differ from the experimental and the ones calculated from eq. (3). They can be understood by observing that the real space pictures (similar to fig. 40 show that domains elongated in the high symmetry directions are predominant. The domain shape dependent diffuse scattering around q02 enables energy minimization, when q02 falls on the line joining Q01 and Qn. This gives the following condition (dashed line in fig. 3b) for the epitaxial rotation when such elongated domains are present:

qlo v~ Q,---o = 4 cos(60 ° - ~b)" (4)

The experimental equilibration time is larger than that practicable in the simulations (~ 105 MC steps per particle), therefore we expect the effect of the finite domain sizes and irregular shapes to be less important in the experimental situation.

In region II (case(ii)), at high defect concentra- tions a miscible, defect phase occurs (fig. 4e), called the y-phase by Cui et al. [16]. The 3,-phase has a structure, which simultaneously contain both of the +~b 1-q domains (similar to fig. 4f), but with ~b given by case (ii) of eq. (3). Therefore, with respect to the dominant first Bragg peak there is for decreasing coverage a transition between a twinned, two domain 1-q structure (RIC) to a single domain 2-q structure, the y-structure [8]. The particle positions are strongly relaxed (see figs. 4e and 40. Experimentally it has so far not been possible to go beyond q ~ 1.75Q¢~, therefore, the dramatic disappearance of the sec- ond harmonic peak S(qo2) has not been observed,

-3.0 ~ , ~ ,

~ . - 3 . 1

v - 3 . 2 . • > -

- 3 . 3 " ¢z: " ' - 3 . 4 Z • • •

L U - 3 . 5 ° ° " , , ° •

10 0 x I= ~ I i I r

10-3

1 . 0 0 1 . 1 0 1 . 2 0

Fig. 5. Top panel, energy per particle. Bottom panel, intensity of the characteristic peaks of S(q) as a function of V~, obtained by MC simulation for T = 0.3e/k B. Peak intensity at Qlo(*), Qvf3-(×), q2o(e) of fig. 2a, and at the satellites 1

(o) and 2 ((3) of fig. 2b.

as the coverage is decreased. The 2-q phase is a defect modulated hexagonal phase with a funda- mental reciprocal vector qy (see fig. 2b) and with modulation vectors including both Q10 and Q01. With variable coverage I q~l varies as V/p -. Close to V/-p -= 1.145 (qv = 9/4Qv ~) the perfect 4× 4 structure is stabilized and further small satellites are found in the simulation - in agreement with the observations of the 8-phase for D 2. A con- ceivable devil's staircase of commensurate structures with even larger unit cells is difficult to reveal both in simulations and experiments. We have here for the first time demonstrated the real space structure (fig. 4e) for the y-phase and we can corroborate one of the experimentally de- duced possibilities [4]. The intensity of the characteristic peaks in S(q) together with the calculated energy per particle is shown in fig. 5. In spite of the available equilibration time, yielding extended regions of coexisting phases, we find good agreement with the number of phases observed by specific heat measurements for D 2 [5] except for the e-phase. We notice for decreasing coverage the steadily increasing intensity of the Q10 peak, which makes it possible for the system to gain energy by expanding beyond the

L454 E. ~ves, P.-A. Lmdg&d / Theory and Monte Carlo simulation of adsorbates on corrugated surfaces

Lennard-Jones optimum coverage P0; and a small additional increase in the y-phase. This shows that the corrugation potential acts as a negative pressure.

Summarizing, we have identified the RIC- phase as a twinned 1-q two domain structure and the y-phase as a 2-q modulated single domain structure. Within this phase a further modulated perfect 4 × 4 structure is found. For low coverages (fig. 4a) we obtain an S(q) very similar to that observed by LEED [3] in the so called a- phase region. We identify this as an inhomoge- neous phase of domains of f3- × x/~ phases, sep- arated by domain walls with the structure of the higher density y-phase (fig. 4d). Close agreement is obtained with the rich phase diagram of D 2 on graphite. The present study may be of importance for a structural understanding and the rich one recently determined for 4He on graphite [17]. A general agreement is also found for the simpler phase diagram for H 2 by simply taking into account an expected larger renormalization of the corrugation potential (Ulo/E = -1 /6 ) . We con- clude that the essential elements of the statistical behavior of light molecules are included in the discussed purely classical model. This model might be of interest also for application to the rare gases. By analyzing a frustrated system in which interparticle interactions compete with a fixed single site particle potential, we have found that it is essential to consider imperfect structures, where the imperfections are of both elastic nature as considered by previous theories, and of defect nature such as vacancies, interstitials and finite size effects. A general theory for epitaxial rotation including these effects has been given.

We acknowledge fruitful discussions with J. Bohr and P, Harris. E.V. acknowledges the hospi- tality of Ris0 Physics Department.

References

[1] K.J. Strandburg, Rev. Mod. Phys. 60 (1988) 161. [2] F.F. Abraham, Phys. Rev. Lett. 50 (1983) 978,

J.D. Fan, O.A. Karim, G. Reiter and S.C. Moss, Phys. Rev. B 39 (1989) 6111.

[3] J. Cui and S.C. Fain, Phys. Rev. B 39 (1989) 8628. [4] H. Freimuth, H. Wiechert, H.P. Schildberg and H.J.

Lauter, Phys. Rev. B 42 (1990) 587. [5] H. Freimuth and H. Wiechert, Surf. Sci. 178 (1986) 716. [6] A.D. Novaco and J.P. McTague, Phys. Rev. Lett. 38

(1977) 1286; J. Villain, Phys. Rev. Lett. 41 (1978) 36; J.P. McTague and A.D. Novaco, Phys. Rev. B 19 (1979) 5299; H. Shiba, J. Phys. Soc. Jpn. 46 (1979) 1852; 48 (1980) 211.

[7] E. Vives and P.-A. Lindg~trd, Phys. Rev. B 44 (1991) 1318.

[8] The 2D-Fourier transformation of the Lennard-Jones potential is mathematically ill-defined. In our simulations we have truncated the r ~ 0 divergency at r 1 = 2~/'3-a/3 which also ensures that two D 2 molecules cannot be in the same graphite unit cell.

[9] Experimentally it is found that at coverages slightly big- ger than 1.0, I ql01 does not follow the linear behavior observed here, but shows a step due to the presence of a coexistence region. The neutron satellites and the dependence of I qa0] on density are consistent with a striped phase [4]. We can but barely resolve such satellites [18], they are not evident from fig. 4a.

[10] A detailed classification of defects and the diffuse scattering can be found in: M.A. Krivoglaz, Theory of X-Ray and Thermal-Neutron Scattering by Real Crystals (Plenum, New York, 1969).

[11] Two extra cases (iii)/3 = 60°+ ~b and (iv)/3 = 30 ° can be justified considering the elasticity of the substrate which reverses the argument presented here. As pointed by Bohr and Grey [13], all the 4 cases can also be justified from finite size effects which produce diffuse scattering oriented according to the finite domain shape which usually follows a high symmetry direction of the adsorbate or substrate.

[12] F. Grey and J. Bohr, Phase Transitions on Surface Films, NATO ASI Series, Ed. H. Taub (Plenum, New York, 1990).

[13] J. Bohr and F. Grey, Condensed Matter News 1 (1992) 12.

[14] C.R. Fuselier, J.C. Rauch and N.S. Gillis, Surf. Sci. 92 (1980) 667; D.L. Doering and S. Semancik, Surf. Sci. 175 (1986) L730.

[15] I.F. Silvera, Rev. Mod. Phys. 52 (1980); A.D. Novaco, Phys. Rev. Lett. 60 (1988) 2058.

[16] J. Cui, S.C. Fain, H. Freimuth, H. Wiechert, H.P. Schild- berg and H.J. Lauter, Phys. Rev. Len. 60 (1988) 1848.

[17] D.S. Greywall and P.A. Busch, Phys. Rev. Lett. 67 (1991) 3535.

[18] E. Vives and P.-A. Lindg~rd, Phys. Rev. B. (1993), to he published.

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Theory and Monte Carlo simulation of adsorbates on corrugated surfaces