Distribution of domain sizes during overlayer growth

L467

Surface Science 159 (1985) L467-L473 North-Holland, Amsterdam

SURFACE

SCIENCE

LETTERS

DISTRIBUTION

OF DOMAIN

J.M. PIMBLEY

* and T.-M. LU

Center for Integrated Electronics New York 12180-3590, USA

SIZES

DURING

and Physics Departmeni,

OVERLAYER

Rensselaer

Polytechnic

GROWTH

Institute,

Troy,

and G.-C. WANG

**

Solid State Diuision, Oak Ridge National L.aboratoty ***, Oak Ridge, Tennessee 37830, USA Received

13 December

1984; accepted

for publication

The distribution of antiphase boundaries two-dimensional lattice gas phase is described of antiphase boundaries with different widths used to describe the growth of oxygen p(2 direction on a W(112) surface after quenching

18 March

1985

created in an overlayer after quenching from a by Markovian disorder. We show that many types are possible. This one-dimensional distribution is Xl) antiphase domains along the doubly-spaced from a disordered, lattice gas state.

Recently, the kinetics of domain growth in two-dimensional systems has attracted much theoretical [l-6] and experimental [7-111 attention. The time evolution of a nonconserved, order-disorder transition has been described by Lifshitz [12] and by Allen and Cahn (LAC) [13]. This theory assumes that after the system is quenched from a disordered to an ordered state domains separated by antiphase boundaries are formed. The degree of ordering of the individual domains is close to the equilibrium value. These domains will grow to reduce the curvature of the boundaries. In this letter we construct a simple, one-dimensional microscopic model to describe the distribution of domain or island sizes [14] during the growth of an overlayer. One-dimensional models are useful for estimating feature (i.e., domain) sizes in lieu of more precise, but prohibitively difficult, two-dimensional models. This model is used to describe our recent low-energy electron * Also, General Electric Research and Development Center, Schenectady, New York, U.S.A. ** Present address: Physics Department, Rensselaer Polytechnic Institute, Troy, New York 12180-3590, USA. *** Research sponsored by the Division of Materials Sciences, US Department of Energy under contract DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc.

J.M. Pimbley et al. / Distribution of domain sizes

L468

diffraction (LEED) experiments: the growth of the p(2 x 1) oxygen antiphase domains on a W(112) surface [7]. Oxygen adsorption at low temperature ( - 170 K) formed an immobile lattice gas. The system is then “up-quenched” to an elevated temperature where the p(2 X 1) structure was created. Our basic assumption is that, after a quench, antiphase boundaries are generated at random positions and obey Markovian disorder. This assumption implies physically that we neglect the possibility of boundary-boundary interactions, i.e., the occurrence of a boundary is independent of proximity of neighboring boundaries. (However, this does not imply that we neglect the boundary energy itself.) The boundaries may possess different phases (to be explained later) and annihilation of boundaries occurs during the coarsening process. The density of boundaries is then reduced and the average domain size grows. In order to describe the above statements quantitatively, let us consider a one-dimensional cut along the doubly-spaced direction in the oxygen p(2 x 1) structure on W(112) [7]. The substrate unit is a and the overlayer unit 2~. For simplicity, we shall assume that the adsorbed oxygen atoms sit on top of the tungsten atoms [15]. Let y be the probability of stepping down from an overlayer atom to a substrate atom in going from one lattice site to the adjacent one. Therefore, the probability of an overlayer atom going to an adjacent overlayer atom is 1 - y. Since the coverage is l/2, the probability of stepping up from a substrate atom to an overlayer atom in going from one lattice site to the adjacent site is also y. The y for each site is independent and thus defines a Markovian disorder [16]. One can show that the p(2 X 1) domain size distribution Pn( N) of such “Markov Chains” obeys a geometric distribution function: w-yl

P,(N)=Y

-y”),

(1)

where N is the number of overlayer atoms in a domain in the direction of a. This number of overlayer atoms is therefore a measure of the diameter of the two-dimensional domains. The sum of P,(N) is normalized to unity, i.e., f

P,(N)=l,

N=l

and the average domain

E= f N=l

size is given by

Np,(N)=J-

1 -y2’

Therefore y = 1 implies a perfect p(2 X 1) structure with an infinite domain size (corresponding to time t = XI), and y = l/2 implies a disordered random lattice gas having an average domain size of 4/3 atoms (corresponding to t = 0). For any fixed value of y, there are more domains smaller than F than there are those larger than N in this geometric distribution of domain sizes.

IA69

J. M. Pimbley et al. / Distribution of domain sires

The LAC kinetic theory of domain growth is cast in terms of antiphase boundary motion. In our model many types of antiphase boundaries with different widths na, where n = 1, 3, 4, 5,. . . , are possible. They occur with the probability of

Pa(n)

=

1 -Y,

n= 1,

y*(l--y)“-‘,

n=3,4,5

1

,....

Fig. la illustrates a possible diagram of one portion of the overlayer. Two domains with sizes N = 3 and 4, and three antiphase boundaries with widths n = 1, 3 and 4 are shown. These boundaries give n7~ phase difference in the ($0) beam in a LEED experiment. Notice that

C (n - 2) Pa(n)= Pa(l),

n=3

which assures the conservation of coverage (= i) for any value of y. Therefore, a fraction (1 + y)-’ of the boundaries are of n = 1 type. The n = 3 type occurs N=4

I

a X

N=3

X

r-l

x x

X

X

X

X

x

x

X

X

X

l eeeeeeeeeeeeeeeeeeeeeeeeee

n=l

n=3

N=2

b

n=4

N=4

N=3

n

r-l

r-7

x

xxxx

xxx

x

x x

l eeeeeeeeeeeeeeeeeeee

n=

3

n=4

n=2

Fig. 1. (a) A portion of the p(2 X 1) overlayer with two domains of size N = 3 and 4 and three boundaries of width n =l, 3 and 4: (x) overlayer atom; (0) substrate atom; (t) position of antiphase boundary. (b) A portion of the (1 x 1) overlayer with island sizes N = 2, 3 and 4, and “sea” sizes of n = 3 and 4: (t) position of “sea”. For the case of Si on Si(lll), each overlayer lattice actually represents two atoms and each boundary is of double atomic step height.

J.M. Pimbley et al. / Distribution

L470

ofdomainsizes

with

the next highest probability. During the coarsening process, adjacent n = 3 types of boundaries can annihilate and disappear altogether yielding larger domains. Likewise, adjacent n = 1 and n = 4 types of boundaries can combine to form an n = 3 type boundary, and so on. The diffraction problem of the above distribution of domain sizes and boundaries can be solved exactly (kinematic scattering) and the LEED angular profile can be written in closed form [17]: n = 1 and

I(S)

=

1 - p2 1 + p2 - 2p cos(S,,a)

’

(4

where S and S,, are the momentum transfer and the momentum transfer parallel to the surface, respectively, and p = 1 - 2y. Eq. (4) gives a smooth broadening (approximately Lorentzian near the profile peak) of the half-order beams. As the average domain size increases, the profile becomes sharper. The width of the profiles is determined by y only. (Nongeometric distributions having a preferred domain size occur in the presence of strong boundary-boundary interactions and generally yield a split, or shouldered, structure in the half-order beams [11,17]. Qualitatively, this is consistent with our observation of the diffraction angular profiles during the growth of oxygen antiphase domains on a W(112) surface. The half-order angular profiles first appeared to be faint and broad and are not elongated; as the time elapsed, the profiles sharpened. In order to test the model quantitatively, a large number of (to) beam angular profiles were measured at different up-quenching temperatures ranging from 266 to 333 K at the final stages of growth. All profiles were obtained at approximately 300 s after quenching. At this time, the growth came to a halt, representing a quenched-in, metastable state [18]. Figs. 2a and 2b show two representative angular profiles for the up-quenching temperatures of 274 and 291 K. The normal incidence electron beam energy is 59 eV. The squares are the experimental results and the solid curves are the best fit of eq. (4) (after convoluting with the LEED instrument response function) [7,19]. The fitting appears to be good and the y values extracted from the two profiles are y = 0.952 -t_0.003 and 0.974 f 0.004, which correspond to the average domain sizes of N= 10.6 + 0.7 and 19.1 k 3.0 in the units of 2a (= 5.472 A). The estimated error in the determination of the average domain size mainly comes from the uncertainty in the measurement of the angular profile ( + 5% error in the profile width). The domain size distributions can be obtained from eq. (1) and are shown in fig. 3. Since the profiles are not elongated, we do not expect elongated domains. The distribution is therefore a measure of the domain diameter. At this point we should comment on the inherent assumptions of our model. The exclusion of multiple scattering (i.e. kinematic scattering) allows us to

J.M. Pimbley et al. / Distribution

W(112)

T=274K

L471

of domain sires

p(2x1)-0

a

Fig. 2. LEED (f0) beam angular profiles for the O/W(112) system at a half monolayer coverage measured at t = 300 s after up-quenching the low temperature immobile gas to final temperatures of (a) T = 274 K and (b) T = 291 K. Squares: experimental values; solid curves: model calculation based on a Markovian disorder theory.

derive closed-form results for the diffracted intensity (eq. (4)). This neglect of multiple scattering is reasonable since the experimental profiles are symmetric within experimental error. Asymmetric angular profiles are one manifestation of multiple scattering [20]. Also the exclusion of two-dimensional correlations

0.1

1

0

0

20

40 DOMAIN

60 SIZE

80

100

N

Fig. 3. The oxygen ~(2x1) domain size distribution extracted from the LEED angular shown in fig. 2. (The dashed curve and the solid curve correspond to fig. 2a and respectively.)

profiles fig. 2b,

J.M. Pimbley et al. / Distribution ofdomainsizes

u-72

in our one-dimensional model may produce some error in the mean domain sizes calculated in the preceeding paragraph. A two-dimensional analysis is more desirable but much more complicated [21]. Interpretation of LEED data with a one-dimensional model is a compromise in that the analysis is tractable but the observations (i.e. domain shape and mean domain size) are only semi-quantitative. There is also evidence for a Markovian disorder during the growth of overlayer islands in a phase separation experiment. Gronwald and Henzler [lo] have recently performed two-dimensional phase separation experiments of Si chemisorbed on a Si(ll1) surface using a LEED technique. Again, the adsorption (half a monolayer of Si) has taken place at low temperature where an immobile random gas was formed. They then annealed the system to an elevated temperature, and islands of (1 x 1) structure were created. (These islands actually contain two atomic layers.) We interpret this result as an overlayer phase condensation in a two-phase coexistence region, where the (1 x 1) islands coexist with the “sea” of random gas [22]. The order parameter associated with this phase transition is conserved and is determined by the coverage. Fig. lb illustrates schematically a possible arrangement. If y is independent for each site, one again has a Markovian chain. However, the distribution of island sizes in this case is given by P(N)

= (1 - y)N-‘y.

(5)

The average island size is l/y. The diffraction problem (including the substrate scattering) of this geometric distribution of island sizes has been solved exactly by Lent and Cohen [23] and by us [17,24]. The integral order beam angular profile shows a central peak sitting on a Lorentzian shape background profile. This was observed in the Si/Si(lll) experiment [lo]. In contrast, the angular profile obtained in a two-dimensional phase separation experiment of W/W(llO) [ll] clearly showed a shouldered structure, which is an indication of a nongeometric (therefore, non-Markovian) distribution of islands [11,17,22]. We have also found recently that the geometric distribution we employ here is inadequate for the O/W(112) system with oxygen coverage not equal to one-half. Physical interpretation and improved modeling of this situation are discussed elsewhere [25]. In summary, we have shown that the distribution of domain sizes during the growth of an overlayer structure, specifically O/W(112) with half a monolayer of oxygen, can be described by a simple Markovian disorder. LEED angular profiles calculated from this distribution are consistent with the experimental results obtained from the overlayer oxygen antiphase domain growth at one-half coverage on the W(112) surface. The LEED profiles sharpen in time after quenching from a disordered state and the degree of this sharpening is

J.M. Pimbley et al. / Distribution

temperature-dependent. and antiphase boundary

We interpreted annihilation.

of domain sizes

these results in terms of domain

L473

growth

We thank C.S. Lent and P.I. Cohen for bringing our attention to the relationship between Markov Chains and the geometric distribution function. We would also like to thank M.G. Lagally and D. Saloner for invaluable discussions.

References [l] P.S. Sahni, G. Dee, J.D. Gunton, M. Phani, J.L. Lebowitz and M. Kalos, Phys. Rev. B24 (1981) 410. [2] S.A. Safran, Phys. Rev. Letters 46 (1981) 1581. [3] P.S. Sahni and J.D. Gunton, Phys. Rev. Letters 47 (1981) 1754. [4] P.S. Sahni, G.S. Crest and S.A. Safran, Phys. Rev. Letters 50 (1983) 60. (51 P.S. Sahni, G.S. Crest, M.P. Anderson and D.J. Srolovitz, Phys. Rev. Letters 50 (1983) 263. [6] A. Sadiq and K. Binder, Phys. Rev. Letters 51 (1983) 674. [7] G.-C. Wang and T.-M. Lu, Phys. Rev. Letters 50 (1983) 2014; G.-C. Wang and T.-M. Lu, Phys. Rev. B28 (1983) 6795. [8] P.K. Wu, J.H. Perepezko, J.T. McKinney and M.G. Lagally, Phys. Rev. Letters 51 (1983) 1577. [9] M.G. Lagally, G.-C. Wang and T.-M. Lu, in: Ordering in Two Dimensions, Ed. S.K. Sinha (North-Holland, New York, 1980) p. 113; M.G. Lagally, G.-C. Wang and T.-M. Lu, in: Chemistry and Physics of Solid Surfaces, Vol. 2, Ed. R. Vanselow (CRC Press, Boca Raton, FL, 1979) p. 153. [lo] K.D. Gronwald and M. Henzler, Surface Sci. 117 (1982) 180. [ll] P. Hahn, J. Clabes and M. Henzler, J. Appl. Phys. 51 (1980) 1079. [12] I.M. Lifshitz, Zh. Eksperim. Teor. Fiz. 42 (1962) 1354 [Soviet Phys-JETP 15 (1962) 9391. (131 S.M. Allen and J.W. Cahn, Acta Met. 27 (1979) 1085. [14] We use “domain” when referring to an antiphase p(2X 1) structure and “island” for a (1 X 1) structure. [15] This assumption affects neither the physics of the problem nor the diffracted angular profile which we shall discuss later. [16] M. Iosifescu, Finite Markov Processes and Their Applications (Wiley, Bucharest, 1980). [17] J.M. Pimbley and T.-M. Lu, J. Appl. Phys. 57 (1985) 1121. [18] The long-time peak intensity did not reach the well-annealed equilibrium value. [19] T.-M. Lu and M.G. Lagally, Surface Sci. 99 (1980) 685, and references therein. [20] H. Jagodzinski, W. Moritz and D. Wolf, Surface Sci. 77 (1978) 233; C.B. Duke and A. Liebsch, Phys. Rev. B9 (1974) 1150. [21] J.M. Pimbley and T.-M. Lu, J. Appl. Phys., in press. [22] T.-M. Lu, G.-C. Wang and M.G. Lagally, Surface Sci. 92 (1980) 133. [23] C.S. Lent and P.I. Cohen, Surface Sci. 139 (1984) 121. [24] J.M. Pimbley and T.-M. Lu, J. Vacuum Sci. Technol. A2 (1984) 457. [25] .J.M. Pimbley, T.-M. Lu and G.-C. Wang, to be published.