Incommensurate sandwiches in displacive surface reconstructions

Surface Science 178 (1986) 483-495 North-Holland, Amsterdam

483

INCOMMENSURATE SANDWICHES IN DISPLACIVE SURFACE RECONSTRUCTIONS A. FASOLINO

and E. TOSATTI

International

for Advanced Studies, Strada Costiera II, I-34014 Trieste, Italy

Received

School

14 March

1986; accepted

for publication

16 June 1986

The study of the phonon spectrum of a surface undergoing a continuous displacive phase transition between two commensurate phases, reveals that the details of the transition may often be complicated. Typically, a thin region of incommensurability is predicted to appear, sandwiched between the two commensurate phases. This phenomenon is caused by indirect coupling of the soft mode to acoustical surface phonons of long but finite wavelength. We discuss first this type of incommensurate instability under general conditions. Then, as a direct application, we predict the occurrence of an incommensurate sandwich at the c(2X2)(11) + c(2~2)(10) transition of H/W(OOl). We also discuss a possible relevance to the incommensurate phase recently reported by Rocca et al. on C/Ni(OOl).

1. Introduction In surface reconstruction problems, one finds frequent examples of transitions between different commensurate phases. The transition may be displacive, order-disorder (lattice gas) or of other type. It may be driven by temperature, or by some external agents, such as adsorbate coverage. Hydrogen coverage, to mention an example, is capable of causing transitions between 1 X 1, 2 X 1 and 3 X 1 in a typical lattice gas system like H/Fe(llO) [1,2], as well as a c(2 X 2)(11) --, c(2 X 2)(10) transition in a typically displacive system, such as H/W(OOl) [3,4]. In the former (lattice gas) system, the appearance of incommensurate regions in between two commensurate phases has already been noted, and explained in terms of competing interactions [2]. Here we report on a study of the nature of commensurate-commensurate transitions in displuciue systems. By investigating the behavior of the soft phonon modes, we have uncovered an apparently general mechanism that will give rise to an intermediate incommensurate region, sandwiched between two commensurate phases, in the process of transforming from one into the other. This mechanism does not occur in either a purely two-dimensional system, or 0039-6028/86/$03.50 0 Elsevier Science,Publishers (North-Holland Physics Publishing Division)

B.V.

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sandwtches

W

Fig. 1. Dispersion of the two coupled surface soft phase mode and acoustical mode. When the &oft phase mode frequency is lowered and reaches the critical value wi. due to the y-dependent coupling between them, the acoustical mode goes soft at a finite wavevector 4:.

in a three-dimensional bulk system. It is characteristic of the surface, the semi-infinite crystal below the first layer playing a crucial role. Any approaching continuous displacive transition involves a certain amount of accompanying elastic distortion. In lattice-dynamical terms, this is signalled by the fact that while some “optical” mode is going soft, it cannot avoid interacting with an acoustical mode of the semi-infinite crystal. The coupling is zero for q = 0, and increases for increasing q. In a bulk case this increase is only linear with q, and the resulting coupled soft mode remains at q = 0. In a surface case, however, we argue that the coupling increases faster than q, and this causes the soft mode to move to q # 0, as sketched in fig. 1. This leads to the conclusion that in this situation an accompanying incommensurate modulation is the most effective in relieving the elastic stress at the transition. We note here that the soft surface mode discussed in this paper is, in reality, a soft surface resonance, due to weak hybridization with the degenerate continuum of bulk modes. However, it can be checked by, e.g., a direct slab

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phonon calculation, that a small amount of hybridization has a negligible effect in the present context. It was shown for example in ref. [5] that the critical surface resonances connected with the c(2 X 2) displacive reconstruction of W(OO1) are still highly localized on the first layer, and behave in all respects like true surface modes in the neighborhood of 4 = 0. Therefore, we shall in the following use quite liberally the term “surface modes” to include such surface resonances. In section 2, we set the frame, by means of phenomenological Landau expansions, for a continuous transition where the order parameter magnitude remains finite across a transition, which affects only its phase. In section 3, we examine the alternative possibility of a transition where the order parameter magnitude itself goes to zero at the transition. In section 4 we consider the qualitative nature of the actual rearrangement of atomic positions and the soliton-like defects to be expected at the incommensurate instability. As a direct application of the present theory, we consider in section 5 the transition, caused by a modest increase of hydrogen coverage I?,, between the c(2 x 2)(11) and c(2 x 2)(10) reconstructions of H/W(OOl). Here, an incommensurate region is predicted to occur, “sandwiched” between the two commensurate phases, as 6, is varied continuously. Finally, section 6 is devoted to a discussion of the possible relevance of the mechanism discussed in section 3 to the recently discovered incommensurate phase of C/Ni(lOO).

2. Case 1: surface incommensurability

generated by a soft phase mode

Let us consider a displacive phase transition on a crystal surface, where the order parameter 1c,has more than one component. To be specific let us take a two-component case, tJ = p exp(i9). The distorted phase is then characterized by two soft modes, corresponding to amplitude and phase modulations [6]. In such a system one can consider two types of phase transitions, which we call 1 and 2. Case 1, which we consider in this section, occurs when the phase mode alone goes soft, while the amplitude (p) remains non-zero. In the other type, case 2, to be dealt with in the next section, also the amplitude vanishes at the transition, where both amplitude and phase modes go soft. Physically, case 1 necessarily corresponds to a transition between two surface structures characterized by identical unit cells: only the symmetry of the actual distortion changes, not its translational periodicity. In case 2, on the other hand, the surface cell may be different on the two sides of the transition. In both cases 1 and 2 we speak of a commensurate-commensurate transition, provided the unit cell is a (small) multiple of the ideal surface cell. We proceed in this section to examine further case 1, characterized by just one soft phase mode. In the absence of any coupling to the other degrees of

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scmdwiche.s

freedom, the free energy change across the transition phenomenological Landau expansion F= f(r)

/

can be described

by the

d'rf(r). = - +mwip’ (r) + ap4(r)

+ +bp4(r)

++J1[vp(r)]2++J2p2(r)[V~(~)]2+

sin228(r) ....

(2)

corresponding to an XY model with cubic anisotropy. For a uniform distortion (no gradients), a mean field minimization of this free energy yields (p)2=mwi/4a and (sin228) =0 if b>O, and mw$‘4(ajb1/4) and (sin226) = 1 if b < 0. For b > 0, the non-zero component of the order parameter is p cos 8, while for b < 0 it is p cos(9 + a/4). In terms of these parameters, the amplitude mode frequency is tit = 2w,$, and the phase mode frequency is wi = (b/2a) wi for b > 0 and wz = [ ) b (/2( a - 1b I/4)] wi. Hence. a soft phase mode is expected when, as a function of temperature or of another external parameter, the cubic anisotropy parameter b changes sign. Our main observation is that close to the actual transition (b = 0) this scheme is invariably too simple. In fact every surface has acoustical modes, either as proper surface modes or as part of a continuum of projected bulk modes. In the strict q -+ 0 limit these modes coincide with the three bulk acoustical modes. At a general q-point in the Brillouin zone, the soft phase mode and at least one acoustical mode have the same symmetry, and then couple with one another. For q along a high symmetry direction (for simplicity) we need to consider only, e.g., shear horizontal modes *. The soft mode becomes a mixed mode, corresponding to the new Landau expansion (valid for b > 0)

f = - imwip2 + up4 + abp4 sin228 + $J,p’ ~0~~98’~ + iJ+f2 + fhp cos 9( u’p sin 9 - up6’ cos a),

(3)

where all derivatives are meant along the chosen direction. Here, a single additional (harmonic) acoustical degree of freedom u is assumed instead of the whole multiplicity of bulk projected modes. This simplification is acceptable if c’ is taken to be the mode of lowest frequency, which is the most important in driving the instability. The coupling to the soft phase mode occurs via the (“Lifshitz”) terms of the third line. The amplitude gradients are omitted. The * The actual symmetry and nature of the relevant acoustical modes (which couple to the mode and will drive the incommensurate instability) depend upon the nature of the phase In eq. (1) and (2) 6 describes the orientation of the displacement vector within the surface plane. Hence a modulation of 6 couples to shear horizontal modes. However, situations could occur where. e.g.. 9 could describe an orientation in the sagittal (Y. 2) There, the other type of modes, in particular the Rayleigh wave, is involved instead.

phase mode. (x. r) other plane.

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sandwiches

coupling between u and p sin 9 is proportional to the order parameter p cos 9, for in the absence of distortion, the two modes do not belong to the same wavevector. Correspondingly, at a given k-space point q close to q = 0 we have, instead of one soft phase mode, the dynamical matrix describing the coupled phase mode-acoustical mode pair. Within mean field, we set (p cos 8) = (p), and write Euler-Lagrange minimum equations for the variables u, p sin 6, yielding

0; + J2q2

iVp)q

D, =

(4)

J3q2

-ix<&

All we have said so far applies to a bulk, as well as to a surface. From this point onwards, there is however a crucial difference. In the bulk case, the coupling h is essentially q-independent. In that case, the softening of or is accompanied by a concurrent softening of the acoustical sound velocity J3 -+ J3 - (h,‘~~)~. The resulting elastic distortions are well known for the bulk case [7]. The basic difference of the surface case, lies in the fact that the coupling h is itself strongly q-dependent. In fact, we expect, quite generally, that X=X(q)

(6)

= (q/l(y2X,

where I, and x are constants. The reasoning leading to this expectation is quite simple. Close to the transition, both the soft phase mode, and the acoustical mode are exponentially localized near the surface, in the form 1;1/2 exp( -z/l,,) and q112 exp( - qz) respectively. The soft mode penetration depth I, is of the order of the interatomic distance, hence its amplitude is essentially concentrated on the first layer or two. Conversely, the acoustical mode penetration q -’ is large, and the amp litude or the first layer is small. The coupling X between them must be roughly proportional to their overlap on the first layer, whence the limiting form (6). The effects of introducing the q-dependent coupling (6) are quite dramatic. The sound velocity does not go soft any longer. Instead, we obtain incommensurability, as in fig. 1, at a wavevector

(7)

q; = x2/2 J2 J310. Incommensurability a critical value

sets in when the decreasing

phase mode frequency

reaches

(8)

w; = x2/2 J3 J;/210. All of the above

is obtained

by solving

det(D, - 1w2) = 0 and looking

for

488

Fig. 2. Case 1: behavior of the lowest mode frequency wavevector as a function of the surface phase mode frequency for three different, increasing values of the coupling coefficient x. The solid line is obtained for b > 0. the dashed line for h < 0.

w = 0, do/dq = 0. As wp is decreased below its critical value (8) all the way to zero, the lowest mode frequency goes imaginary (a “growing” mode), while its wavevector changes only marginally, as shown in fig. 2. This indicates that for a whole range of wP below WE the commensurate phase (6) = 0 is replaced by an incommensurate phase. At tip2 = 0, a switch occurs from (9) = 0 to (9) = ~,/4, and a symmetric reasoning applies. In conclusion, the switching between the two phases (8) = 0 and (9) = 71/4 does not take place abruptly. Instead, it should occur across two intermediate incommensurate phases. We mention two, and not one, because the incommensurate instability of the (19) = 0 phase and that of the (8) = n/4 phase are generally distinct. This point will appear clearer in section 4, where morphology and domain walls will be described. Before moving on to consider other details and consequences of the above, a comment is in order on the physical mechanism which causes what would be an elastic instability in a bulk case to turn into an incommensurate instability on a surface. In a bulk both the soft mode and the acoustical modes extend uniformly over all space, and that implies a resulting q -+ 0 instability, also uniform. In presence of a surface, in particular with a surface-localized soft mode it is clearly not possible to drive an accompanying bulk acoustical distortion. The acoustical mode must also be surface localized and this can only be the case for an acoustical node at q f 0.

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3. Case 2: surface incommensurability generated by soft amplitude and phase modes We start by referring back to the free energy expansion (2). Case 1, just discussed, correspond to the cubic anisotropy parameter b 3 0 while wa and thus the average distortion magnitude (p) remains fiite. We now discuss case 2, where conversely b remains finite, but wO,and thus (p) tends to zero at the transition. This is the case, for example for a reconstructed + unreconstructed transition. At such a transition, both amplitude (0: = 2~;) and phase mode frequency (~i=(b/2a)wi for b>O, [lbl/2(alb1/4)]w$ for b 0), D =

4

(b/h)@,” + J2q2

i( mw~/4n)1’2q

(9)

J3q2 1* I -i( m0~/4a)1’2q The coupled modes are now to be studied as a function of IJ~, for we -+ 0. The crucial difference from case 1 arises from the fact that in this limit the soft mode frequency and its coupling to the acoustical modes tend to zero simultaneously. Hence, right at the transition threshold the coupling vanishes,

Fig. 3. Case 2: behavior of the lowest mode frequency wavevector as a function of 0: for three increasing values of the coupling coefficient x.

and no incommensurability is possible, i.e. q. = 0. However. it is still possible to find an incommensurate instability at a finite oO. where the coupling is 9() finite. At some finite value of tioY the fastest growing mode wavevector jumps from zero (for wa < GJ~) to a finite value. Approximate formulae for 9; and ti; are

Both a;;, and q; depend strongly on the coupling coefficient h, as illustrated in fig. 3. As shown there. the behavior is reversed with respect to case 1: in&ommensurability may only occur some distance away from the ideal transition point. where LJ, = wP = 0 (rather that only close to wP = 0 as in case 1). For weaker and weaker coupling strength h, the minimum value 9; increases. Clearly, when q; becomes large, comparable with the zone size ~/cr, the present analysis is no longer applicable, and we expect no incommensurability.

It is clear from the preceding sections that our main instruments in arriving “sandwiched” phase is simply the detection of an at the incommensurate instability of each of the two “parent” commensurate phases. In this respect. it has remained so far unspecified what the detailed morphological, energetic and statistical mechanical properties of the resulting incommensurate state might be. While each of these items would require and deserve a separate study by itself, we are already in a position to point out a few expectations, based on our limited present state of knowledge. Specifically we can argue about morphology starting from a study of the unstable phonon eigenvector at q. (the “fastest growing mode”), and looking for soliton, or domain wall, properties along similar lines as discussed for bulk cases by Heine and McConnell [8]. At finite temperatures, the evolution of the incommensurate phase into either a “floating” or an outright disordered phase, can finally be expected in analogy with other 2R incommensurate cases [Qf.

We study the nature of the soft mode eigenvector the incommensurate instability satisfies

rp, = 0.

uyq which at the onset of

(12)

By writing out uq as a linear combination of the soft mode eigenvector $q and of the acoustical eigenvector Ok uq = al& + PO,

(13)

with jru12+ I@j2= 1, we get from eq. (4) p/o = -if2J2/J,)“2.

(14)

Thus, in the incommensurate soft mode the soft (optical) phase mode and the acoustical components are mixed, in quadrature [S], in a relative proportion which is controlled by the relative dispersion coefficient J. I

4.2. Solitons It is well known from many examples that the actual distortion that arises after an incommensurate instability is actually more complicated than sug-

r

Fig. 4. (a) spatial dependence of the two components (u: acoustical, 4: phase mode) of the soft mode eigenvector at the onset of incommensurability. (b) Qualitative picture of the incommensurate phase at T= 0. Almost commensurate regions C are separated by soliton-antisoliton pairs, with an acoustically distorted phase in between. The width ratio A/C is similar to the amplitude ratio /3/a.

gested by the soft mode eigenvector, which is the result of a linearization. In fact, the starting hamiltonian (1) is highly nonlinear, and will in general give rise to soliton-like solutions. These solitons correspond to abrupt, rather then diffuse, distributions of strains in the system, which prefers to remain basically “commensurate” over a whole region, and then slip into another commensurate region through narrow sohtons. A pictorial idea of how this might be realized in the present case is shown in fig. 4. The incommensurate phase at T = 0 is generally speaking either a striped phase or a 2D domain wall lattice [lo] (commensurate regions C separated by antiphase domains A, each of which is made up of a soliton-antisoliton pair with an “acoustically” distorted phase in between). If the solitons were noninteracting, the relative extension A/C would closely reflect / /3/a I’. This ratio can however change substantially due to soliton-soliton interactions, which for a finite q. may be far from negligible. 4.3. Finite temperature

behacior

The present discussion is entirely restricted to T = 0. At finite temperatures. an incommensurate phase in 2D does not retain a conventional long-range order, but becomes instead a “floating phase” with power-law decay of correlations. The incommensurate phases discussed here are no exceptions, and this should be kept in mind when looking for experimental realizations. The theory of the 2D commensurate-incommensurate transitions, in particular, is very extensively covered in the literature [9-131 and should apply to the present case as well.

5. An application: the (II} + (10) transition of H/W(OOl) The (001) surface of tungsten offers an attractive example of displacive commensurate-commensurate transition as a function of hydrogen coverage. The clean surface has a c(2 x 2) reconstruction below 250 K, characterized by in-plane zig-zag atomic displacements along the (11) direction [3]. Upon gradual hydrogen coverage, the reconstruction does not disappear, but is rather seen to switch from (11) to (10) displacements f4]. This corresponds to “pinching” pairs of surface W atoms together by the H atoms. which adsorb on bridge sites. Fasolino, Santoro and Tosatti 1141 have established an effective hamiltonian which provides a semimicroscopic connection with a phenomenological Landau description similar to that of eqs. (1 and 2) for the displacive transition of W(OO1). Lau and Ying [15] and Roelofs and Ying [16] have shown how (annealed) hydrogen coverage is capable of renormalizing the Landau parameters, in particular of reversing the sign of b from b > 0. corresponding to a (ll} distortion to b < 0, corresponding to a (10) distor-

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b-0 111,

I,

Cl01

I,

Fig. 5. Schematic phase diagram of H~~~l~

as a function of hydrogen coverage i&.

tion. They have pointed out, in particular, that right at b = 0 the character of the phase transition should become of the Kosterlitz-Thouless type, as appropriate for an XY model without any cubic anisotropy. This model also leads to expect power-law correlations at low temperatures, in place of long-range correlations typical of 1b 1 f 0. This (11) -+ (10) transition provides a str~~tfo~ard application of case 1. In the c(2 X 2)(11) distorted surface, both phase and amplitude modes occur at q = 0 [S]. The phase mode frequency should decrease with increasing coverage, which causes ] b 1 to decrease. When wP approaches zero, the additional coupling of the soft phase mode to acoustical modes of section 2, not considered in previous work [14,16], is important, and must be included. The result is that, before the b = 0 point is reached, a commensurate-incommensurate transition is expected on each side of b = 0. That is to say, a (11) -+ in~~ensurate transition is expected for low but increasing hydrogen coverage (9i = 0.05 monolayers), and another (10) -+ incommensurate is expected for larger but decreasing hydrogen coverage (9, = 0.15 -monolayers). The nature of both incommensurate phases at finite temperature should be, as discussed in the previous section, that of floating phases, with power-law correlations. The resulting room temperature phase diagram of H/W(OOl) for example is sketched in fig. 5. The phase 11 consists of (11) domains separated by hydrogen-rich domain walls. In the phase 12, conversely, the hydrogen is concentrated inside the (10) domains. The intermediate situation between 11 and 12, finally, is not accessible to the present analysis, and is probably disordered. Thus, the details of the phase diagram in the region of the (11) + (10) switch are predicted to be substantially richer than supposed hitherto. We are not aware of any existing evidence for these sandwiched phases at the (11) ---*(10) transition of H/W(OOl). However, their existence, at least in some temperature range, seems to us unescapable, and may well be worth a new investigation aimed at their detection.

6. Case 2: possible relevance for C/Ni(lOO) In the preceding section we have discussed a possible realization of case 1. Now we would like to discuss other possible candidates particularly for case 2

incommensurate phases. This is generally problematic, however. The main reason is that strictly speaking a type 2 situation can only be realized in a transition where the order parameter, and with it the soft mode frequency, tends to zero in a mean field like fashion. However, mean field theory is generally a very bad description for any phase transition dynamics, particularly a two-dimensional transition at that. In this sense, the modefs of sections 2 and 3 are undoubtedly exceedingly crude. They can at best be used as qualitative guidelines, when one is confronted with a realistic situation. Nonetheless, we wish to discuss and argue briefly on the possible relevance of case 2 to the 1 X 1 ---f incommensurate transition recently discovered by Rocca on C/Ni(lOO). Upon carbon coverage the Ni(lOO) surface, develops an incommensurate phase [17] close to a c(2 X 2). followed by a p(2 x 2) commensurate phase at larger coverages. We argue that, in analogy with 0 and S adsorbates [18]. the Ni( 100) surface develops, in presence of carbon, a tendency to become c(2 X 2). Once in the c(2 X 2) state, however, the resulting folded phase mode near q = 0 (which did not exist in the 1 X 1 phase) can couple strongly to the acoustical modes, causing the c(2 x 2) to become incommensurate. Thus, the actual c(2 x 2) C/Ni(lOO) might never be realized, even though it is crucial to the understanding of the incommensurate phase. Interestingly, the p(2 x 2) state of C/Ni(lOO) has also been ascribed to a zone-boundary instability of the same, ubiquitous but unobservable c(2 x 2) phase 1191. Based on our model results of section 3, a narrow c(2 X 2) region could in fact be realized at very low carbon coverages. In summary, we have presented physical reasoning and model calculations indicating how incommensurate phases can creep in between or near displacive commensurate-commensurate phase transitions of reconstructed surfaces. As possible applications, we have pin-pointed the (11) -j (10) transition of H/W(OOl)c(2 x 2) and the so far unexplained 1 x 1 + incommensurate transition of C/Ni(lOO). It is hoped that the present work will stimulate further experimental work aimed at clarifying the validity of our mechanism. Acknowledgments We are grateful to C.Z. Wang for many useful discussions and to M. Rocca for bringing to our attention and for clarifying C/Ni(lOO).

about W(OO1) the physics of

References [l] R. Imbihl. R.J. Behm, IL Christmann. G. Ertl and T. Matsushima. Surface Sci. 117 (19X2) 257.

A. Fasolino, E. Tosatti / Incommensurate [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15] [16] [17] [18]

[19]

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