MONTE CARLO STUDY OF EPITAXIAL OVERLAYER WITH SUBSTANTIAL LATTICE MISMATCH

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MONTE CARLO STUDY OF EPITAXIAL OVERLAYER WITH SUBSTANTIAL LATTICE

MISMATCH

T. Miyasaki, K. Aizawa, H. Aoki, C. Itoh, M. Okazaki

To cite this version:

T. Miyasaki, K. Aizawa, H. Aoki, C. Itoh, M. Okazaki. MONTE CARLO STUDY OF EPITAXIAL

OVERLAYER WITH SUBSTANTIAL LATTICE MISMATCH. Journal de Physique Colloques, 1987,

48 (C5), pp.C5-199-C5-202. �10.1051/jphyscol:1987540�. �jpa-00226744�

JOURNAL DE PHYSIQUE

Colloque C5, supplbment au nOll, Tome 48, novembre 1987

MONTE CARL0 STUDY OF EPITAXIAL OVERLAYER WITH SUBSTANTIAL LATTICE MISMATCH

T. MIYASAKI, K. AIZAWA, H. AOKI(' ) , ^C. ITOH( ) and M. OKAZAKI Institute of Materials Science, University of Tsukuba,

Ibaraki 305, Japan

Abstract. The atomic configuration of an epitaxial overlayer on a substrate with substantial lattice mismatch is studied by the Monte Carlo method. Covering a wide range of the ratio of intra- and inter-layer interactions, we have clarified a succession of several phases.

Circular boundary conditions are also exanlined to elucidate island/domain structures.

1. Introduction

Although epitaxial growth technique is a powerful approach towards the fabrication of novel crystal structures, heterostructures and superlattices made by the conventional molecular-beam epitaxy are primarily limited to combinations of similar materials like GaAs/AlGaAs, in which the lattice mismatch is as small as -0.1%. Recent new techniques, especially the van der Waals epitaxy[l], enable us to fabricate more versatile heterostructures. In a typical van der Waals epitaxy, transition-metal dichalcogenide(MX,) heterostructures like MoS,/NbSe, have lattice mismatch of -lo%, and LEELS and RHEED studies[l] indicate an interesting situation that the monolayer of NbSe,, although basically retains its own lattice constant, interacts with MoS, substrate to make the crystal axes of both layers aligned.

Bearing this type of systems in mind, we have studied an epitaxial monolayer/substrate systems with a substantial lattice mismatch. To systematically investigate a variety of possible situations, the ratio of intra-layer interaction and inter-layer interaction is varied.

The atomic configurations have been determined by the Monte Carlo method.

2. Model

We take a simple model consisting of two layers of triangular lattices with different lattice constants(Fig. 1). For MX,/M'X', system, this amounts to considering only the chalcogen atoms a t the interface. In the case of van der Waals epitaxy, the intra-layer interaction is covalent, which is described here by a potential,

vcov

^? *bij -

where k is af'force constant for the bending of covalent bond angles, and rii is the distance of nearest-neighbow atoms within each layer with an unperturbed distance r,. The inter-layer van der Waals interaction is described by

VvdW =

zn

^-

^{I ,}

where ri,,, is the distance between atoms in different layers. Since the strength of VvdW may

("present address : Department of Physics. University of Tokyo. Hongo, Tokyo 113, Japan (I'present address : Seiko-Epson Co. Ltd. 3-3-5 Owa. Suwa. Nagano 392. Japan

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987540

OVERLAYER

k--- . ? a

JOURNAL DE PHYSIQUE

L

SUBSTRATE I+-- a 0 4

Fig. 1. The model system with a top view of the undistorted system for ao2/a0,=1.2.

be represented by its curvature, 4 d2V,,/dr2, at the minimum, R , = Z ' / ~ ~ , of the Lenard-Jones potential, the ratio of the inter-layer and intra-layer interactions becomes

@ / k . We take a van der Waals radius R0=3.66L and r,substrate=3.16i, bulk values for MoS2[2], with the force constant k = 1 . 4 ~ 1 0 ~ d ~ n / c m [ 3 ] . Typical MX, systems have @/k-0.2.

Here we take the ratio of the lattice constants of overlayer and substrate to be 1.2 (20%

lattice mismatch). System size dependence is studied in both periodic and circular boundary conditions. The atomic configuration is obtained by the standard Monte Carlo method[4] at T=300K with typically 4x10' Monte Carlo steps. We allow lattice distortions within each layer, and the equilibrium layer distance is determined separately.

3. Results and discussions

The results(Fig. 2) for the equilibrium atomic configurations show that there exist various phases according as $ / k is changed. Namely the atomic displacements exhibit, as @/k is increased, (a)glassy flow, (b)array of local expansion/compressions and (c)array of vorteces.

These features arise because the lattice mismatch makes the system a kind of frustrated systems, in which the intra- and inter-layer interactions con~pete. This is particularly manifest in the glassy regime or vortex regime, in which an array of vorteces and antivorteces occurs. In the box boundary condition, in which we confine the atoms in a circular potential well, the results for atomic displace~nents are qualitatively the same. The important observation in this boundary condition, however, is that the overall crystal axis of the overlayer is rotated relative to substrate(Fig. 3). The angle of rotation, 8 , decreases with the size of the system as 8=3O0 (number of atoms=217, size-3&), 8" (562, 6 1 i ) , 2" (1072, 8 3 i ) for smaller interlayer interactions in agreement with experimental results for MX, systems. For larger interlayer coupling (@/k=2.0), 6 is shown to be small even for small sizes. Since the box boundary condition may be thought of as simulating island or domain structures, we can predict that islands or domains of varying sizes in the overlayer assume similar behaviour. In the regimes (b) and (c), the centres of compression or vortex are distributed quasi-periodically. This results in a structure factor with characteristic satellites(Fig. 4). These results can be confirmed by diffraction studies or scanning tunneling microscopy. In conclusion, there are interesting features in the epitaxy of systems with various ratios of intra-layer and inter-layer interactions. These features should be reflected in electronic and phonon structures. The authors are grateful to Professor A. Koma for valuable discussions and for showing experimental results.

SUBSTRATE OVERLAYER

(a) 'P/k = 0 . 2

. . . ~

L

,

-

...

f d-. r r a . . 1 r r r t t f , , , t

, + - . - - + - . . - a * , - . P A . . . .

Fig. 2. The equilibrium atomic configuration for @/k=0.2(a), 0.5(b), 2.0(c) with N=976 atoms in the periodic boundary condition. Displacenlent vectors of atoms from the perfect triangular lattice are shown with exaggerated arrow lengths for the substrate(1eft) and overlayer(right).

References

[I] A. Koma, K. Sunouchi and T. Miyajima, Microelectronic Engirleering Z(1984) 129; in Proc. 17th Int. C o n f . on Physics of Semiconductors ed. by J.D. Chadi and W.A.

Harrison (Springer, 1985) p.1465.

12) J.L. Verble, T.J. Wietling and P.R. Reed, Solid State Commun. ll(1972) 941.

C5-202 JOURNAL DE PHYSIQUE

(a) N = 217

Fig. 3. The equilibrium atomic configuration for N=217(a), 562(b), 1072(c) atoms for @/k=0.2 in the circular box boundary condition. The inset shows the corresponding total

5 -

as a function of the angle of rotation of the overlayer relative to substrate -660

0 10 2 0 3 0 with arrows indicating the positions of

Relative rotation (deg) minima.

Fig. 4. The structure factor is shown by contours for the configuration of the overlayer in the Monte Carlo result for $/k=2.0 with N=1072 in the circular boundary condition.

(31 N. Wakaba~ashi, H.G. Smith and R.M. Nicklow, Phys. Rev. B 12(1975) 659.

141 K. Binder(ed), Application o f the Monte Carlo Method i n Statistical Physics (Springer, 1984).