Molecular dynamics simulations of InGaAs/GaAs nanotubes synthesis
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Molecular dynamics simulations of InGaAs/GaAs nanotubes synthesis
A.V. Bolesta, I.F. Golovnev, and V.M. Fomin
Institute of Theoretical and Applied Mechanics SB RAS, Novosibirsk, 630 090, Russia
The results of InGaAs/GaAs nanotubes synthesis on atomic scale using molecular dynamics are presented. On the basis of a comparison with mechanics of continua it is shown that the radius of nanotubes obtained by molecular dynamics calculations are less than elasticity theory predicts. The discrepancy exceeds 30% for thinnest heterostructures. The proposed simple discrete model produces estimation of the correction to the elasticity theory formula due to discreteness of the matter. Though this correction also predicts decrease for equilibrium heterostructure radius of curvature, its value is not sufficient for explanation of molecular dynamics results. A comparison of the potential energy distribution with elasticity theory predictions shows the importance of heterostructure surface layer contribution to this discrepancy.
At present wide development in the nanotechnology research is governed by both microelectronics demands and possibility to fabricate novel materials with unique mechanical and chemical properties. Synthesis of nano-sized cylindrical structures — nanotubes — is one of the directions in nanotechnology. One of the methods for nanotubes production is proposed and realized in the Institute of Semiconductor Physics SB RAS [1−4]. The main point ofthe method is that the multi-layer ideal crystal strained heterostructure is formed on the substrate by molecular beam epitaxy. Then sacrificial layer is removed by lateral selective etching with upper layers (double layer usually) being removed. Lattice mismatch induces the appearance of moment of force that results in heterostructure rolling-up to form nano-sized cylindrical tube-scroll. The estimation connecting the nanotube diameter, lattice mismatch and layer thickness was given. Theoretical consideration of this process in the framework of continuum mechanics  provides improved formula for nanotube diameter. Morozov underlines that the question of mechanical parameter determination (Young's modulus, Poisson’s ratio, bending stiffness factor) at the nanoscale remains opened . At this scale the discreteness of the matter is considerable, and fundamental problem of continuum mechanics applicability to the processes at nanoscale becomes particularly actual. Thus there exists a necessity of nanotube synthesis research at the atomic level with further comparison with elasticity theory and experimental results.
2. Research method
Tersoff potential function  is widely used for molecular dynamics simulations of the processes in crystals with diamond/zincblende structure . The choice of constants for the Tersoff function defines an applicability of this potential to investigation of one or another phenomena. A set of parameters we make use of in the present work was proposed in  for InGaAs system. The characteristic feature of this parameterization is the sufficiently precise description of crystal lattice elastic properties.
An external interaction potential has been added to the potential energy of the system for simulation of the interaction with the sacrificial layer:
W (r, t) = k- (t)(r — r-o)2, (1)
where kt (t) is the linear function decreasing during etching:
ko (ts +At -1)
t & lt- t s
, ts & lt- t & lt- ts + At,
tS +At & lt- t,
ts is the beginning and At is the average duration of etching for the sarificial layer atom with number i.
The equations of motion were integrated by the velocity Verlet algorithm . Figure 1 schematically shows the initial configuration for molecular dynamics calculations. Atoms are placed in zincblende structure nodes with lattice constant being aGaAs. Periodical boundary conditions were applied
© A.V. Bolesta, I.F. Golovnev, and V.M. Fomin, 2004
Fig. 1. Initial configuration
in 7-axis direction to simulate sufficiently long nanotube synthesis. Figure 2 shows the results for 2ML InAs + 2ML GaAs rolling-up simulation.
Fig. 2. Double layer 2MLInAs + 2MLGaAs heterostructure rolling-up to form a nanotube
3. Equilibrium radius of curvature
The radius of curvature for a double layer heterostructure can be calculated by the minimization of potential energy density in the elasticity theory framework : n (e, k) =
0 Ei (e-eo -Kz)2dz | ^(e-Kz^dz I
1 -v? 0 1 -V2 J'
where e0 is the initial strain of the lower layer, K = 1/R is the curvature. As a result, the expression for equilibrium radius of curvature Rel takes the form :
KD — L
Aa K1L2 + K2L1
jj J-v2 ,
3(1 -v 2)'-
2(1 -v j) j = 1, 2,
K — Ki + K2, L — Li ~Lj, D — Di + Di.
The comparison of our results with (4) shows that molecular dynamics values for radius of curvature are less than
Fig. 3. Relative difference between molecular dynamics simulations RMD and elasticity theory results (4) for «ML InAs + «ML GaAs heterostructure radius of curvature. The results of simple discrete model (6) are given for comparison
those predicted by elasticity theory. For thinnest heterostructures the difference exceeds 30%, and this discrepancy disappears with increasing a thickness of layers (Fig. 3, Table 1). Comparing our molecular dynamics results with experimental data for very thin heterostructures (Table 1) we conclude that experimentally measured values for nanotube radius seem to differ from those predicted by elasticity theory to a greater extent. It is possibly connected with insufficiently accurate description of InGaAs surface energy by the potential function , which should be improved. Also we should mention that not all of given in Figure 3 heterostructures could be realized in experiment. There exists a maximum thickness of InGaAs layer for epitaxial film growth depending on gallium content and, hence, on lattice mismatch. Increasing the thickness of InGaAs layer results in dislocation nucleation due to internal stress and, consequently, in the change of growth regime. Nevertheless, the calculations for sufficiently thick heterostructures are also of interest for comparison with mechanics of continua. Under zero temperature conditions they remain free of defects, and by increasing thickness of layers we can observe a change from a few monolayers to thick heterostructure which can be described satisfactorily by elasticity theory.
For estimation of the correction to the elasticity theory formula due to the discreteness of the matter we propose a simple discrete model as follows. We consider separately deformation of each monolayer and ignore the change in elastic constants due to the surface. Thus integration in (3) can be changed to summation in the following way:
E (e-Kz)2dz n E (e-каг/2)
Heterostructure radius of curvature: Rd is the result of elasticity theory application (4), Rmd is the result of molecular dynamics simulations and Rexp is the experimental value 
Heterostructure Rel, nm Rmd, nm Rexp, nm 
2ML GaAs + 1ML InAs 6.5 4.4 1. 5
2ML GaAs + 1ML In08Ga0. 2As 8.2 5.2 5
Fig. 4. An increase of potential energy per atom as a result of 20ML InAs + 20ML GaAs (a) and 2ML InAs + 2ML GaAs (b) heterostructure rolling-up in comparison with elasticity theory expression (7)
where h = na/ 2 and a/ 2 is the distance between monolayers. Then minimization ofpotential energy density results in the correction to (4):
AR* _ AR* - Rel _
4(K1h -K2h2)2 + 16K1K2(h1 + hS
The correction (6) predicts a decrease for radius of curvature as compared to elasticity theory expression (4), however the value of (6) is insufficient to explain the difference with molecular dynamics results (Fig. 3).
Since the estimation of correction (6) to the elasticity theory formula due to the discreteness of the matter does not explain the difference between molecular dynamics and elasticity theory results in full measure, the influence of free surfaces in the heterostructure should be estimated. In this connection our interest is in constructing the distribution of potential energy density along Z-axis in the direction normal to the interface. In the view of elasticity theory an increase of potential energy per atom has to be equal to:
AUel = U (e, k) — U (0, 0) =
E 2,((e~e0 -Kz)2 -e0) ^ z & lt- 0,
& quot-'- E (7)
-----^-(e-Kz)2 V2, z & gt- 0
. 2(1 -v 2)
with V1 and V2 being volume size to one atom below and above the interface correspondingly, in a state with minimum potential energy according to Eq. (3). Figure 4 shows the comparison of molecular dynamics results with (7) for sufficiently thick and thin heterostructure. The difference can be seen only for one or two monolayers near the both of free surfaces. Thus, the presence of free surfaces results in the significant correction in elasticity theory formula (4) for a few monolayer thick heterostructures with high volume fracture of atoms located near the surface.
On the basis of molecular dynamics research of InGaAs/ GaAs nanotubes synthesis it was shown that the radius of simulated nanotubes is less than that predicted by elasticity theory. The discrepancy exceeding 30% for a few monolayer thick heterostructures. With increasing the thickness of heterostructure the difference between molecular dynamics and continuum mechanics results disappears. A simple discrete model has been proposed which provides the correction to the elasticity theory formula due to the matter discreteness. Though this correction also predicts a decrease of radius for heterostructure in equilibrium, its value is not sufficient to explain molecular dynamics results. By comparison of potential energy distribution with elasticity theory it has been shown that the determining factor here is the presence of surface layers with high volume fracture of atoms in conditions different from those in the bulk.
This work was supported by President of RF (Grant No. NSh-2282. 2003. 1), RFBR (Grant No. 02−01−371), and Academician Lavrentev Grant of SB RAS for young scientists.
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