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Disordered clean surfaces and adsorbates investigated with atom-surface scattering
312
Surface Science 125 (1983) 312-316 North-Holland Publishing Company
DISORDERED CLEAN SURFACES AND ADSORBATES INVESTIGATED WITH ATOM-SURFACE SCATTERING G.E. TOMMEI,
A.C. LEVI and R. SPADACINI
Istituto di Scienre Fisiche dell’llnioersitir and Gruppo Narionale di Struttura 5 Viale Benedetto XV, I-16132 Genoua, Italy Received
13 May 1982; accepted
for publication
della Mater-in de1 CNR,
25 July 1982
In this paper two different physical situations are considered which can be treated with the same method: a fluid adsorbate (disordered in the x, y plane) and a clean surface with random steps (disordered in the z direction). The hard corrugated wall model is used in the eikonal approximation; the differences between the two cases arise only from the different statistical properties of the two physical situations. The differential scattering probability is evaluated. For the fluid adsorbate the latter splits into a coherent (purely specular) contribution and an incoherent one (which is, in fact, weakly inelastic and related to classical diffusion on the surface). For stepped “rough” surfaces only incoherent scattering is present and the differential scattering probability for hexagonal lattices is given.
In the present paper two different physical situations are considered: (a) a surface covered with an adsorbate in a fluid phase (this case has been studied by Ibafiez et al. [I] and ourselves [2]; (b) a clean surface with random steps (a problem studied by Lapujoulade [3]). Both cases present disorder; in the first case the disorder is in the x,y plane while in the second the disorder is in the z direction. In both cases the situation may be described by a shape function l(R)presenting a certain amount of randomness. The same mathematical description will be applied, as far as possible, to both cases: the only differences arising from the different statistical properties of the models appropriate to describe the fluid adsorbate and the stepped surface respectively. The hard corrugated wall (HCW) model will be used in the eikonal approximation (this is sufficient at a modest level of accuracy). Thus the differential scattering probability may be written:
(1) 0039-6028/83/0000-0000/$03.00
0 1983 North-Holland
313
G. E. Tommei et al. / Disorderd clean surfaces and adsorbates
where L2 = S is the area of the surface. In both cases the corrugation may be written as a sum of shape functions referring to the individual scattering atoms:
S(R)=C[Z(R-R,) +‘jl
(2)
2
.i
where the randomness lies in the coordinates (Rj, zj) of the atomic centres (in Rj for fluid adsorbates, in z, for stepped surfaces). If such coordinates were known (deterministic case), the scattering integral occurring in (1) would become:
~=Cexp[iQ.(Rj-R,)][~(Q,O)-exp(iq,z,)~(Q,q,)] t [%*(Q,O> -exp(-iq,z,) +4m2 s(Q){
9*(Q,q,)]
S - 2N g(O,O) - 2 Rez
[exp(iq;r,)5(0,qZ)]),
(3)
i where S(Q,qz)
=//exp[iQ.R
+ iq,Z(R)]
d2R.
(4)
(The integral in (4) is extended to an atom, or to a unit cell, and has the physical meaning of an atomic form factor.) The statistical problems are solved by taking averages of Llover the appropriate distribution of atomic centres. (a) Random (or fluid) ahorbate. In this case, zj may be taken to vanish and the only statistical quantity is the average zexp]iQ.(R,-R,)]=N[l+Z’d’(~)]. iJ The structure
(5)
factor ECd’(Q), defined
by:
ECd’( Q) = N-’ c exp[iQ . (R, - R,)j,
(6)
i*j
may be expressed in terms of the radial distribution two-dimensional fluid of density N/S = p: B’“‘(Q)=p//[g(R)-I] = S(Q)
=
‘nc
g(R)
of the
exp(iQ.R)d2R+4m2pS(Q) - 1 + 47~‘~ S(Q).
In this case the coherent scattering is given by: dP.
function
kk-d2 4m2]ki,1q,’
scattering
(7) is purely
specular,
PI~(Q,qz) - ~(Q4)12 S(Q) d0.
while the incoherent
(8)
G. E. Tommei et al. / Disorderd clean surfaces and adsorbates
314
For hard disks g(R) = g(R) is isotropic and has been evaluated many years ago by Monte-Carlo methods [4]. In ref. [2] the quantity S(Q), proportional to the scattering intensity, is evaluated and plotted for a reduced density pa2 = 0,199 (where a is the disk radius). The whole problem can be reformulated in a time-dependent theory, where atomic diffusion over the surface is taken into account [2]. In this case the incoherent scattering turns out to be weakly inelastic, with an energy exchange of the order of ADQ’, where D is the diffusion coefficient. (b) Random steps. In this case I, = m,h, where h is the height of a unit step and mj a discrete stochastic variable, while R, is a fixed lattice site position. Coherent scattering is given by: dP
kkd2
=
c S(Q-
c”h @21k,lqfG
G) I&l2 I~(Cq,)12dQ,
where
Now modern statistical-mechanical theories of surfaces [5] recognize two possible states of the surface: a smooth state at low temperatures and a rough state at higher temperatures separated by a smooth-rough (or roughening) transition at a temperature T,. In the present context, the rough state (which will be exclusively treated here) may be identified as that state where & = 0 for all G, while in the smooth state & = 1 except in the immediate vicinity of T,. In the rough state the coherent scattering vanishes exactly. The incoherent scattering is given by d Pine =
k(k, .qj2
4m2W,,lqf
dtiz
x {m-
ii
exp[iQ.(R,-R,)] IP12}l~(Qd12.
(10)
where the term ]p]’ can again be ignored for a rough surface. The relevant statistical quantity in (10) is S = N-‘x
exp[iQ.
(Rj-
R,)]expliqzh(m,
- m,)],
where m, may be considered as a discrete stochastic process. Such process is “stationary” in the R-plane and uniform in the z direction (the latter property is true in the rough state, but false in the smooth state or in epitaxial growth). This implies that the conditional probability P2( m,, L21m,, L ,) of having a height m2 h at the lattice site L,, if m, h is the height at L ,, depends only on the differences L = L, - L, and m = m2 - m,, and will henceforth be indicated by P,(m,L).
G.E. Tommei et al. / Disorderd clean s&aces
and adsorbates
315
Then, in the rough state, S may be rewritten as S =C CP,(m,L) L
exp(iQ.L
+ imq),
(11)
m
where q= q,h, It seems reasonable to assume that mj is a Markov process in two dimensions; then, generalizing the Smoluchowski equation [6], P2( m, L) can be constructed from the conditional probability Q(ml0) between two nearest neighbours. For lattices with unit vectors of the same absolute value, the result is that the Fourier transform of P2 may be expressed in terms of the Fourier transform of Q, i.e.: m=+m (12) C P,(m,L) exp(imn) = R’(q), PW==-C0 where .C=fCU R t rt) = C Q(@) exp(isrt), z=-Cc
(13)
and I is the number of elementary lattice vectors which connect 0 and L. Thus S =CR’(?f) L
exp(iQ*L).
114)
For a square lattice, an (essentially one-dimensional~ formula - obtained by Lapujoulade [3] - is recovered (if Q is restricted to Q(mlO)=&8_.,,+
$blo+
(1 -e)S,,,
where c is the probability to find a unit step between two nearest nei~bours~. More interesting is, therefore, the hexagonal lattice where several new effects take place. In fact, in the hexagonal lattice, any atom has six nearest neighbours and, moreover, two nearest neighbours may be nearest neighbours of each other. This fact leads to some limits in the choice of the possible Q’s and to a different form of S: S=([R3(17)-
l]“-
[3 + 2( cosu+cosu+COS(U-u))]
x {ER2(rl) - 2R(q) cosu-l- l][R2(?l)-2R(~)cosu+ x[R2(r))-2R(q)cos(u-v)fl]}-‘,
R*(q) [R(q)-
I]“)
1] (1%
where u and 1)are the components of the parallel momentum transfer along the principal reciprocal lattice directions. A more detailed account of the case of random steps will appear elsewhere f71.
316
G.E. Tommei et al. / Disorderd ciean surfaces and adsorbates
References [I] [2] [3] [4]
J. IbLfiez, N. Garcia, J.M. Rojo and N. Cabrera, Surface Sci. 117 (1982) 23. A.C. Levi, R. Spadacini and G.E. Tommei, Surface Sci. 121 (1982) 504. J. Lapujoulade, Surface Sci. 108 (1981) 526. W.W. Wood, in: Physics of Simple Liquids, Eds. H.N.V. Temperley, J.S. Rowlinson and G.S. Rushbrooke (North-Holland, Amsterdam, 1968) p. 115. [5] H.J. Leamy, G.H. Gilmer and K.A. Jackson, in: Surface Physics of Materiats, Ed. J.M. Blakely, Academic Press, New York, 1975) p. 121. [6] Ming Chen Wang and G.E. Uhlenbeck, Rev. Mod. Phys. 17 (1945) 323. (71 R. Spadacini and G.E. Tommei, to be published.
Surface Science 125 (1983) 312-316 North-Holland Publishing Company
DISORDERED CLEAN SURFACES AND ADSORBATES INVESTIGATED WITH ATOM-SURFACE SCATTERING G.E. TOMMEI,
A.C. LEVI and R. SPADACINI
Istituto di Scienre Fisiche dell’llnioersitir and Gruppo Narionale di Struttura 5 Viale Benedetto XV, I-16132 Genoua, Italy Received
13 May 1982; accepted
for publication
della Mater-in de1 CNR,
25 July 1982
In this paper two different physical situations are considered which can be treated with the same method: a fluid adsorbate (disordered in the x, y plane) and a clean surface with random steps (disordered in the z direction). The hard corrugated wall model is used in the eikonal approximation; the differences between the two cases arise only from the different statistical properties of the two physical situations. The differential scattering probability is evaluated. For the fluid adsorbate the latter splits into a coherent (purely specular) contribution and an incoherent one (which is, in fact, weakly inelastic and related to classical diffusion on the surface). For stepped “rough” surfaces only incoherent scattering is present and the differential scattering probability for hexagonal lattices is given.
In the present paper two different physical situations are considered: (a) a surface covered with an adsorbate in a fluid phase (this case has been studied by Ibafiez et al. [I] and ourselves [2]; (b) a clean surface with random steps (a problem studied by Lapujoulade [3]). Both cases present disorder; in the first case the disorder is in the x,y plane while in the second the disorder is in the z direction. In both cases the situation may be described by a shape function l(R)presenting a certain amount of randomness. The same mathematical description will be applied, as far as possible, to both cases: the only differences arising from the different statistical properties of the models appropriate to describe the fluid adsorbate and the stepped surface respectively. The hard corrugated wall (HCW) model will be used in the eikonal approximation (this is sufficient at a modest level of accuracy). Thus the differential scattering probability may be written:
(1) 0039-6028/83/0000-0000/$03.00
0 1983 North-Holland
313
G. E. Tommei et al. / Disorderd clean surfaces and adsorbates
where L2 = S is the area of the surface. In both cases the corrugation may be written as a sum of shape functions referring to the individual scattering atoms:
S(R)=C[Z(R-R,) +‘jl
(2)
2
.i
where the randomness lies in the coordinates (Rj, zj) of the atomic centres (in Rj for fluid adsorbates, in z, for stepped surfaces). If such coordinates were known (deterministic case), the scattering integral occurring in (1) would become:
~=Cexp[iQ.(Rj-R,)][~(Q,O)-exp(iq,z,)~(Q,q,)] t [%*(Q,O> -exp(-iq,z,) +4m2 s(Q){
9*(Q,q,)]
S - 2N g(O,O) - 2 Rez
[exp(iq;r,)5(0,qZ)]),
(3)
i where S(Q,qz)
=//exp[iQ.R
+ iq,Z(R)]
d2R.
(4)
(The integral in (4) is extended to an atom, or to a unit cell, and has the physical meaning of an atomic form factor.) The statistical problems are solved by taking averages of Llover the appropriate distribution of atomic centres. (a) Random (or fluid) ahorbate. In this case, zj may be taken to vanish and the only statistical quantity is the average zexp]iQ.(R,-R,)]=N[l+Z’d’(~)]. iJ The structure
(5)
factor ECd’(Q), defined
by:
ECd’( Q) = N-’ c exp[iQ . (R, - R,)j,
(6)
i*j
may be expressed in terms of the radial distribution two-dimensional fluid of density N/S = p: B’“‘(Q)=p//[g(R)-I] = S(Q)
=
‘nc
g(R)
of the
exp(iQ.R)d2R+4m2pS(Q) - 1 + 47~‘~ S(Q).
In this case the coherent scattering is given by: dP.
function
kk-d2 4m2]ki,1q,’
scattering
(7) is purely
specular,
PI~(Q,qz) - ~(Q4)12 S(Q) d0.
while the incoherent
(8)
G. E. Tommei et al. / Disorderd clean surfaces and adsorbates
314
For hard disks g(R) = g(R) is isotropic and has been evaluated many years ago by Monte-Carlo methods [4]. In ref. [2] the quantity S(Q), proportional to the scattering intensity, is evaluated and plotted for a reduced density pa2 = 0,199 (where a is the disk radius). The whole problem can be reformulated in a time-dependent theory, where atomic diffusion over the surface is taken into account [2]. In this case the incoherent scattering turns out to be weakly inelastic, with an energy exchange of the order of ADQ’, where D is the diffusion coefficient. (b) Random steps. In this case I, = m,h, where h is the height of a unit step and mj a discrete stochastic variable, while R, is a fixed lattice site position. Coherent scattering is given by: dP
kkd2
=
c S(Q-
c”h @21k,lqfG
G) I&l2 I~(Cq,)12dQ,
where
Now modern statistical-mechanical theories of surfaces [5] recognize two possible states of the surface: a smooth state at low temperatures and a rough state at higher temperatures separated by a smooth-rough (or roughening) transition at a temperature T,. In the present context, the rough state (which will be exclusively treated here) may be identified as that state where & = 0 for all G, while in the smooth state & = 1 except in the immediate vicinity of T,. In the rough state the coherent scattering vanishes exactly. The incoherent scattering is given by d Pine =
k(k, .qj2
4m2W,,lqf
dtiz
x {m-
ii
exp[iQ.(R,-R,)] IP12}l~(Qd12.
(10)
where the term ]p]’ can again be ignored for a rough surface. The relevant statistical quantity in (10) is S = N-‘x
exp[iQ.
(Rj-
R,)]expliqzh(m,
- m,)],
where m, may be considered as a discrete stochastic process. Such process is “stationary” in the R-plane and uniform in the z direction (the latter property is true in the rough state, but false in the smooth state or in epitaxial growth). This implies that the conditional probability P2( m,, L21m,, L ,) of having a height m2 h at the lattice site L,, if m, h is the height at L ,, depends only on the differences L = L, - L, and m = m2 - m,, and will henceforth be indicated by P,(m,L).
G.E. Tommei et al. / Disorderd clean s&aces
and adsorbates
315
Then, in the rough state, S may be rewritten as S =C CP,(m,L) L
exp(iQ.L
+ imq),
(11)
m
where q= q,h, It seems reasonable to assume that mj is a Markov process in two dimensions; then, generalizing the Smoluchowski equation [6], P2( m, L) can be constructed from the conditional probability Q(ml0) between two nearest neighbours. For lattices with unit vectors of the same absolute value, the result is that the Fourier transform of P2 may be expressed in terms of the Fourier transform of Q, i.e.: m=+m (12) C P,(m,L) exp(imn) = R’(q), PW==-C0 where .C=fCU R t rt) = C Q(@) exp(isrt), z=-Cc
(13)
and I is the number of elementary lattice vectors which connect 0 and L. Thus S =CR’(?f) L
exp(iQ*L).
114)
For a square lattice, an (essentially one-dimensional~ formula - obtained by Lapujoulade [3] - is recovered (if Q is restricted to Q(mlO)=&8_.,,+
$blo+
(1 -e)S,,,
where c is the probability to find a unit step between two nearest nei~bours~. More interesting is, therefore, the hexagonal lattice where several new effects take place. In fact, in the hexagonal lattice, any atom has six nearest neighbours and, moreover, two nearest neighbours may be nearest neighbours of each other. This fact leads to some limits in the choice of the possible Q’s and to a different form of S: S=([R3(17)-
l]“-
[3 + 2( cosu+cosu+COS(U-u))]
x {ER2(rl) - 2R(q) cosu-l- l][R2(?l)-2R(~)cosu+ x[R2(r))-2R(q)cos(u-v)fl]}-‘,
R*(q) [R(q)-
I]“)
1] (1%
where u and 1)are the components of the parallel momentum transfer along the principal reciprocal lattice directions. A more detailed account of the case of random steps will appear elsewhere f71.
316
G.E. Tommei et al. / Disorderd ciean surfaces and adsorbates
References [I] [2] [3] [4]
J. IbLfiez, N. Garcia, J.M. Rojo and N. Cabrera, Surface Sci. 117 (1982) 23. A.C. Levi, R. Spadacini and G.E. Tommei, Surface Sci. 121 (1982) 504. J. Lapujoulade, Surface Sci. 108 (1981) 526. W.W. Wood, in: Physics of Simple Liquids, Eds. H.N.V. Temperley, J.S. Rowlinson and G.S. Rushbrooke (North-Holland, Amsterdam, 1968) p. 115. [5] H.J. Leamy, G.H. Gilmer and K.A. Jackson, in: Surface Physics of Materiats, Ed. J.M. Blakely, Academic Press, New York, 1975) p. 121. [6] Ming Chen Wang and G.E. Uhlenbeck, Rev. Mod. Phys. 17 (1945) 323. (71 R. Spadacini and G.E. Tommei, to be published.