Scattering of atoms from a rough surface: Non-Markovian effects

385

Surface Science 137 (1984) 385-396 North-Holland, Amsterdam

SCATTERING OF ATOMS FROM NON-MARKOVIAN EFFECTS Andrea

A ROUGH

SURFACE:

C. LEVI

di Scienze Fisiche dell’(lniuersitb and Gruppo Narionale Via Dodecaneso 33, I I61 46 Genoua, Italy

di Struttura

Received

1983

Isiituto

13 June 1983; accepted

for publication

1 November

della Materla del CNR,

Scattering intensities of atoms reflected by a stepped surface are calculated (using the hard wall model and the eikonal approximation) assuming the surface to be above the roughening transition and taking into account the non-Markovian (logarithmic) increase of the mean square height difference between two points on the surface. The non-Markovian effects are shown to change the incoherent peaks drastically, causing them to be infinite in height, to be exceedingly narrow within a sizable range of perpendicular momentum transfer, and to possess a set of similar, typically star-shaped, equal-intensity curves with no tendency to approach circles towards the top of a peak.

1. Introduction A central problem in statistical-mechanical surface theory, as well as in crystal growth theory, is related to the two possible states of the surface - the smooth state and the rough state - and to the smooth-rough (or roughening) transition between them [l]. This transition, although well defined theoretically, and proven explicitly to exist at least in one model (by Van Beijeren [2] in the FSOS model [l] on the basis of Lieb’s mathematical results for the ice and ferroelectric problems [3]), is rather elusive experimentally: experiments directly relevant for the smooth-rough transition have been performed only by Jackson and Miller on the growth of NH,Cl and C,Cl, from vapour [4] and, more recently, by Avron et al. and by Balibar on solid helium growing from its liquid [5,6]. Atom-surface scattering might be a sensitive tool to probe the state of a surface. Indeed both Lapujoulade and Lejay [7] and Poelsema et al. [8], for copper and platinum respectively, have shown the characteristic effects of a random step distribution on helium scattering from surfaces. The specular and diffraction peaks are broadened and the intensity falling into a finite detector decreased; both effects, moreover, oscillate in a characteristic way as a function of the angle of incidence, according to whether the interference between atoms scattered from two terraces separated by a unit step is constructive or 0039-6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

386

A. C. Leui / Scarterrng of UIIO~Sfrom rough surface

destructive. Very recently Lapujoulade has presented evidence indicating that a sudden increase in the slope of diffraction intensities as a function of temperature for various crystal faces may be related to roughening transitions 191. although for faces with high Miller indices more complicated transitions may take place, as shown by Marchenko [lo]. A stochastic theory of atom scattering from stepped surfaces was presented by Lapujoulade [II]. Recently Spadacini and Tommei have given a more complete treatment of the problem 112,131, and in the present work I shall use many of their notations and results. It should be noted that Spadacini and ‘Tommei describe the rough surface as a Markov process in two dimensions, while the present paper is an attempt to go beyond such Markovian assumption. This attempt is motivated by the fact that statistical-mechanical theories of the smooth-rough transition [I] give definite answers to questions concerning the mean square height difference (MSHD) between two points on the surface. and these answers are in contrast with the Markovian assumption. The MSHD is found to grow only logarithmically with increasing distance, while it should be linear in the distance in any Markovian theory. This has a strong effect on the scattering pattern, for example the incoherent lines are infinite in height (under ideal conditions) rather than finite, as shown below. 2. Heuristic considerations Atomic scattering calculations, even in the elastic case, are far from simple if they are to be carried out exactly. In the present article, however, I shall (a} assume that the potential may be treated as a corrugated hard wall, and (b) use the eikonal (Kirchhoff) approximation. Of course, these simplifications may be unrealistic, but the purpose of this article is to point out qualitative features of the scattering pattern, rather than to predict quantitatively experimental peak shapes for particular systems. With these assumptions, the (elastic) scattering intensity of atoms may be written 111,121 as

XL;*

C

exp[iQ-(&--I,,)]

(exp[ -irt(m,--m,)])

da,

(1)

where Aq, q = (Q, q,), is the momentum transfer to the solid, L, and L, are lattice vectors varying over a two-dimensional lattice enclosed in a large square of side L,, .9(q) is an atomic form factor, and if h is the height of a unit step then m,h and m,h (m1.2 integers) are the heights of the surface at positions L, and L, respectively and n = - hq,.

A. C. Leai / Scarrerrng of aroms /mmrough surjace

387

Assumption (a) is reasonable except for metals (see also section 4). As for assumption (b), it is well known that the exact theory may be cast in a form similar to the eikonal approximation. except that in the eikonal integral a source function S(R) is introduced. In the eikonal approximation S(R) reduces either to a constant or (in the more refined version considered here) to a quantity proportional to the local component k,, of the incident wave vector. If. however, S(R) is assumed to be periodic with the lattice periodicity. then (1) holds even though F(q) may be quite different from its eikonal counterpart. Although the periodicity will ultimately be lost when the step density becomes very large because of multiple scattering between such steps. the deviations from periodicity will probably remain small even for considerably rough surfaces. (Even in the extremely rough case, T = 2T,. considered below, the probability of “non-stepping” is still about 70% while the probabilities of “stepping” up or down are 15% each.) The average L,.(9)

= (exp[-iq(m,

- m,)l>

(2)

occurring in (1) depends on n and on the distance L = L, - L,. In the rough state E,(n) vanishes at infinity (unless 11 is a multiple of 277) while in the smooth state E,_(q) tends to Ip(n where P(q)

= (exp( -imn)>.

(3)

Coherent scattering, proportional to Ifi(q)I*, may only occur from a smooth surface and the disappearance of coherent scattering is one of the possible definitions of the roughening transition. In the rough state, on the other hand. only differences in height are meaningful, so letting m = m2 - WI, and introducing the conditional probability f,(m. L) that the surface is at height mh at L if it is at height 0 at 0 E,_(n)=CL’z(m, m

L) exp(-im-rl)

(4)

and the quantity

.~(Q.v)=CexdiQ-L) 1. =F.P,(m.

E,_(v) L) exp(iQ.L-imq)

(5)

contains all the statistical information about the system and is proportional to the (purely incoherent) scattering intensity. The second derivative E;_‘(O)equals E;_‘(O) = - zmzP2(m, m

L) = -g,_,

(6)

388

A. C. Levr / Scattering

of atoms from rough surface

where h2g, is the MSHD and g, has been carefully studied (especially in its asymptotic behaviour). It turns out that g, behaves asymptotically as [l]

”

_

.rl&! 7r

a’

where a is the lattice spacing, and A equals 2/71. at the transition and increases with temperature (for instance in van Beijeren’s model [2] A = 4/n at 2Tn [1,3]). A Gaussian approximation for EL gives EL(q)=

+f

exp[-tg,(17-2kv)*],

(8)

k=--Xi

where the sum over k takes care of the periodicity with respect to 7. This Gaussian approximation requires g, to be large enough, hence IL] to be considerably larger than a. Then applying (7), EL(,n) ~

y

(1/iLl)(a/2r)l?-21n)2.

(9)

k=-cc

Substituting this expression one gets for a square lattice

into (S), and replacing

the sums by integrals,

This naive derivation is only formal, and in fact the integral is not well behaved (except for 1 < p < 2, a rather restricted range). However, as will be confirmed below, the dependence on QP2’fi (or more generally ]Q - Gle2+p) is significant and shows that the incoherent peaks, centred at the diffraction angles, have infinite height. Before turning to a more precise calculation, it may be expedient to elucidate the non-Markovian nature of this process. Consider paths over the lattice leading from 0 to L, and let 1 be the minimum number of paces in one such path. Then (7) may be replaced (equivalently to logarithmic accuracy) by gt

= Aln r

(12)

I.

It can be immediately shown that, parallel to a crystal axis, the MSHD g, is related to G,, the correlation of steps at a distance I from each other, by g, =

i

G,,-,,,.

(13)

k,k’=l

Then the logarithmic

behaviour

(120 for g, leads to a behaviour

in - lm2 for G,.

389

A. C. Levi / Scarrering of atoms from rough surface

This means that the probability, if a step is up, that a second step at a distance I from the first is also up is = 1 - B/I’. This is a strongly non-Markovian behaviour for two reasons. Firstly, the Markovian correlations decrease faster (typically, exponentially) with distance, rather than as I-*. Secondly, and more fundamentally, in the case of three steps, if the above-said conditional probability for the first and second step is f - A and for the second and third it is f - p, then for the first and third a Markovian theory gives a conditional probability larger than i (i.e. f + 2Xp), while the present situation gives less than i (i.e. i - Xp/(A + p + 2JXp)).

3. Calculation A calculation of Y’(Q, 77) is relatively easy if it is assumed that EL(n) depends only on 1, rather than on the full vector L. For example in the case of a square lattice, letting u = aQ,, u = aQ,:

sinp?exp(iu~)E,(v)

~(Q,~)=l-cosp~cosuIm

I=0

-sin

u f exp(iul) I=0

In the Markovian E,(,V) =

.

E,(q)

approximation

(14)

i [ll]

[ens

(15)

where R(n)

= WI)

(16)

is the characteristic function of the transition probability for nearest neighbours. A natural generalisation of (15), if R(n) is a positive function, is: WI)

=

mdl xt

07)

where X,, in order to satisfy (6) must be A, = &5/g,.

(18)

Numerical checks show (17) to be a reasonable approximation. Some (not very strong) deviations occur only near n = (2n + 1)~ (see appendix). The next problem is to evaluate g,. Extensive Monte Carlo simulations have been performed for this quantity [15]; moreover there is at least one case where g, has been obtained analytically. This is Sutherland’s calculation [14] of correlation functions in the ice model of Lieb [3] at twice the transition

A.C. Levi / Scattering

390

temperature, interpretation

G,= From

of atoms from rough surface

which can be applied to rough surfaces according to van Beijeren’s [Z]. Sutherland’s result (parallel to a (110) crystal axis) is if f is even,

0, -4/71’1’,

if! is odd.

this g, can be immediately

obtained:

gI=I-(8/12*)h,,

(20)

Asymptotically g, - (4/~‘)(ln

1 -t 1 + C + In 2)

(22) fC is Euler’s constant). Formula (17) cannot be applied directly, since in van Beijeren’s model R(q) = cos 71. However une can consider even 1 only (physically this amounts, for a bee crystal, to taking into account only atoms at the corners of a cube and ignoring those at the centre, a very reasonable description since the atoms at the centre are commonly considered as belonging to a different layer); then I ~1/2,g,-(4~/n2)(ln~~1~C-i-21n2),andR(~)=~ + 2,/n’ + ($ - 2/7r2 > cos q (see appendix) is positive. AsYmptotically h, - a(ln I + fi>, where (IL= 8/(n2 - 4) = 1.36295 used for E,(q), the sum fm E exp(iuf)

and @ = 1 + C-t- 2 In 2 = 2.96351.

(23) If (17) is

Eifq)

may be approximated, exp( -p/L> T(l -P) where p = --a! In

R;

for u small, by cos(aP/2)

P-I+?

and y(Q,

7) by

where tan ‘p = (Q,, - G,.)/(Q.,

- G,) and the angular

(24)

factor k(cp, p) equals

and varies from 1 for ‘p = 0” to ~2 - p/z for Q)= 45”. The incoherent peaks are divergent at the centre and the constant intensity curves have a star shape given bY

(27)

A. C. Levi / Scarrering of atomsfrom roughsurface

391

Fig. 1 shows such constant intensity curves for the case p= 4 (i.e. R = 0.693, n = 1.604). The qualitative picture is clear, even though the precise shape of the line rests on the assumption that EL(q)depends on f only, which is not exact. More important and reliable is the following consequence of the nonMarkovian theory: the overall width of the peak is very narrow, compared to the Markovian case [16]. Indeed, if the width Qis defined by half the intensity of the peak being in the range ]Q - G] < g, then Qarexp(

-y)=exp(s),

(28)

which becomes exceedingly small when R approaches 1. As shown in fig. 2, this implies essentially vanishing width (i.e. S-function peaks) for a considerable range of n within each period of 277. It should be noted that the values of p occurring in (11) and (25) agree.

4. Discussion

For comparison, height [11,12]:

in the Markovian

model the incoherent peak is finite in

(l-R*)* ~(Q~~)=(1-2Rcosp+R2)(1-2Rcoso+R2)'

(29)

(30) and the constant intensity curves are given by

a2(Q-G)2=2(1-R)2 R sin*2cp

(31)

where y is the intensity relative to that for Q = G. This is a circle for y very close to 1 and becomes star-like (i.e. acquires an outward curvature for cp= 45”) only when y becomes less than a (see fig. 3). More important, the overall width of the peak is roughly proportional to 1 - R in the Markovian case, i.e. essentially sinusoidal as a function of n, in contrast with the finite ranges of 11corresponding to vanishing width illustrated in fig. 2. In principle, the difference between (27) and (31) should be experimentally observable, although the experiment might be a very delicate one, requiring very high resolution in both energy and angle of the atomic beam and also a small detector. Similarly the 1 dependence (i.e. dependence on the angle of incidence) of the incoherent peak width should be experimentally accessible,

A. C.

392

Leoi

/

Scairering of atoms

fromrough surface

Fig. 1. Equal intensity curves in the incoherent peak for a square lattice (van Beijeren’s model for the (001) surface of a bee crystal). T = 2 Ta, n = 1.604 + 2 n r (R (7) = 0.693, a = i). The intensities at two neighbouring

.7 r

lines differ by a factor

Width

2fi

a

Fig. 2. Width of the incoherent peak as a function of n. Full curve: no instrumental broadening. Dotted curve: instrumental width $ of maximal theoretical width. The ranges of n where the theoretical width is negligible are marked by thicker curves. Model and temperature as for fig. 1.

A. C. Levi / Scattering

of atoms from rough surface

393

aQx

Fig. 3. Equal intensity curves in the incoherent lines correspond to y = 0.75, 0.5, 0.25 and 0.1.

peak for a square lattice (Markovian

model). The

and the finite ranges where the width is negligible should be a very characteristic feature of a non-Markovian rough surface. No such effect is apparent in the experiments so far [7,8,11], but one may argue that no genuine rough surface in thermal equilibrium has been investigated with atomic beams yet, and that the non-thermal roughness caused by etching or mechanical treatments may be reasonably assumed to be Markovian. In the case of metal surfaces, the softness of the potential cannot be ignored. An effect of softness is an enhancement of intensities for beams emerging with k, < Ik,=l: this may lead to a loss of symmetry in the equal intensity curves appearing in figs. 1 and 3. It should be noted that, although the explicit values (23) for CYand p are only valid for Van Beijeren’s model at T = 2T,, the qualitative behaviour described may be expected to hold in the whole rough range T > TR [3,15]. Unfortunately, the presence of infinite peaks even in the rough range, not so different from the true diffraction peaks of the smooth range, may cause the roughening transition to be elusively difficult to observe directly by atom scattering. ‘The present calculations are in the spirit of the roughening transition theory, where attention is paid to defects (vacancies, adatoms, steps), assuming the cell

A.C. Levi / Scattering of atoms from rough surface

394

structure of the crystal to be preserved and ignoring vibrations. An alternative point of view is possible, the surface melting theory, such as appears in the numerical simulations of 3roughton and Woodcock 1171 and in the theoretical study of Pietronero and Tosatti 1181. Surface melting theories describe three-dimensional melting as starting from the surface, a very old idea [19], and focus the attention on atomic displacements (both perpendicular and parallel) at the surface, as being extremely anharmonic and leading ultimately to a catastrophe (a phase transition). The role of defect formation is typically ignored in these theories. It shoutd be observed that defects are rather difficult to obtain from simulations since they require a very long time to be created [20]. In fact, roughening transition and surface melting are opposite approximations to the true transition occurring in reality. The relationship between these several transitions is being explored.

Acknowledgment Continuing discussion and cooperation with Drs. Renato Spadacini and Giuliana Tommei are gratefully acknowledged. The author is also grateful to Professor Harry Suhl for ilIu~nating discussions during the author’s visit at La Jolla in summer 1982.

Appendix The expression for R(q) is obtained as follows. By renormalizing to comer atoms only, as indicated in section 3, let a new argument ?j = 2q be introduced; similarly E,( $) = E,,(n), g, = g2,, and the quantity to be evaluated should in fact be written I?(e). Then

where in P(m, E) the first argument denotes the height m and the second distance I. On the other hand g2 = 8P(2,2) = 2 - S/n2 so that P(2,2)

= a - l/P*,

P(O,2) = 1 - 2P(2,2)

ri(Tj)=f+2/m*+(f-2/B2)

= f + 2/r?,

cos?j.

O~tting the tildes the result given in the text is obtained. In order to check the approximation (17), one may go further paces on the lattice. The exact result for E,(q) is E&I)

= P(O,4) -t 2P(2,4)

the

cos 2~ f 2P(4,4)

= 1 - +g, sin211 + 16P(4,4)

sin4q.

cos 471

considering

4

of atoms from rough surface

A.C. Levi / Scattering

Now g, = 4 - 224/9~* is 1 - (2 - 112/9n2)

and P(4,4) is positive,

so that a lower bound

395

for Ed(q)

sin*v.

To obtain an upper bound, it may be observed that the conditional probabilities P(l,l(O,O) = 0.5, P(2,211,1) = 0.297358, P(3,312,2) = 0.159263 are decreasing, so that a good conjecture is P(4,413,3) < 0.159263 whence P(4,4) < 0.003771 and an upper bound for Ed(q) is 1 - (2 - 112/9a*)

sin*q + 0.06034 sin4n.

On the other hand the approximation

(17) gives

E;Prr”“(q)=[l--[l-$)

u=2==1.242803.

sin’q)‘,

Table 1 gives the lower and upper bounds and the approximation for E,(,q)= E2(ji) as a function of q = 277. It is seen that (17) is a reasonable approximation, lying up to < = 110” within the accepted bounds and never deviating far from the upper bound even for larger values of e ($ is called TJin the text).

Table

1

Lower and upper bounds li (de@ 0

and the approximation

Lower bound

for Ed(q)

Upper bound

= I?~(+) as a function off= Approximation

1

1

1

10

0.99439

0.99439

0.99439

20

0.97771

0.97777

0.97776

30

0.95049

0.95076

0.95073

40

0.91354

0.91437

0.91428

50

0.86799

0.86991

0.86974

60

0.81522

0.81899

0.81869

70

0.75684

0.76337

0.76293

80

0.69462

0.70492

0.70436

90

0.63044

0.64553

0.64494

100

0.56627

0.58705

0.58657

110

0.50405

0.53122

0.53104

120

0.44566

0.47961

0.47995

130

0.39290

0.43361

0.43467

140

0.34735

0.39439

0.39630

150

0.31040

0.36292

0.36570

160

0.28317

0.33993

0.34347

170

0.26650

0.32593

0.32999

180

0.26089

0.32123

0.32548

27

A. C. Levi / Scattermg

of atomsfromroughsurface

References [l] J.D. Weeks, in: Ordering in Strongly Fluctuating Condensed Matter Systems. Ed. T. Riste (Pienum. New York, 1980) p. 293. [2] H. van BeiJeren, Phys. Rev. Letters 38 (1977) 993. [3] E.H. Lieb and F.Y. Wu, in: Phase Transitions and Critical Phenomena, Vol. 1, Eds. C. Domb and MS. Green (Academic Press, London, 1972) p. 331. [4] K.A. Jackson and C.E. Miller, J. Crystal Growth 40 (1977) 169. [5] J.E. Avron, L.S. Balfour, C.G. Kuper, J. Landau, S.G. Lipson and L.S. Schulman, Phys. Rev. Letters 45 (I 980) 814. [6] S. Balibar, presented at 3me Rencontre de Mecanique Statistique. Paris, 1983. [7] J. Lapujoulade and Y. Lejay, J. Physique Lettres 38 (1977) L303; J. Lapujoulade and Y. Lejay; in: Proc. 7th Intern. Vacuum Congr. and 3rd Intern. Conf. on Solid Surfaces, Vienna, 1977. Eds. R. Dobrozemsky et al. (Berger, Vienna, 1977) p. 1373. [8] B. Poelsema. R.L. Palmer, G. Mechtersheimer and G. Comsa, Surface Sci. 117 (1982) 60. 191 J. Lapujoulade, presented at 3me Rencontre de Mecaniyue Statistique, Paris, 1983: J. Lapujou~ade, J. Perreau and A. Kara, Surface Sci. 129 (1983) 59. [lo] V.I. Marchenko, JETP Letters 35 (1982) 567. [Ill J. Lapujoulade, Surface Sci. 108 (1981) 526. [12] R. Spadacini and G.E. Tommei, Surface Sci. 133 (1983) 216. (131 G.E. Tommei. A.C. Levi and R. Spadacini. Surface Sci. 125 (1983) 312. (141 B. Sutherland, Phys. Letters 26A (1968) 532. [15] W.J. Shugard, J.D. Weeks and G.H. Gilmer, Phys. Rev. Letters 41 (1978) 1399. 1161 I am indebted to R. Spadacini for this observation. 1171 J.O. Broughton and L.V. Woodcock, J. Phys. Cl1 (1978) 2743. [18] L. Pietronero and E. Tosatti, Solid State Commun. 32 (1979) 255. [IS] See B. Mutaftschiev, in: Interfacial Aspects of Phase Transformations. Ed. B. Mutaftschiev (Reidel, Dordrecht, 1982). [20] L. Pietronero. private communication.