Vibrational modes of silicon nanostructures

Solid State Communications, Vol. 96, No. 4, pp. 231-235, 1995 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098/95 $9.50 + .@I

0038-1098(%)00343-6

VIBRATIONAL

MODES OF SILICON NANOSTRUCTURES

Xiaodun Jing, N. Troullier & James R. Chelikowsky Department

of Chemical Engineering and Materials Science, Minnesota Supercomputer of Minnesota, Minneapolis, MN 55455, U.S.A.

Institute, University

K. Wu and Y. Saad Department of Computer Science, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN 55455, U.S.A. (Received 28 February 1995; in revisedform

6 April 1995 by S.G. Louie)

We present a method for predicting the vibrational modes of small semiconductor clusters. We employ ab initio pseudopotentials and apply a higher-order finite difference procedure to solve the KohnSham equations. We predict the vibrational modes of small silicon clusters (Si,, n = 4-7) based on their ground state structures. Our calculated vibrational modes agree very well with experimental data, and with other theoretical calculations based on quantum chemistry and tight binding methods. This comparison confirms the accuracy of the finite difference procedure for calculating not only the first order derivative of the energy, but the second derivatives as well. It also validates the accuracy of pseudopotential-local density calculations for the ground state structures for Si clusters.

VIBRATIONAL modes are quite difficult to determine, especially for clusters with more than a few atoms. Before the modes can be calculated, it is necessary to determine the ground state structure of the cluster of interest. As the cluster size increases, it rapidly becomes more difficult to determine ground state structures. A multitude of “isoenergetic” structures may exist with very different electronic and structural properties. It becomes virtually impossible to make a complete inventory of all possible topologically distinct clusters and determine by direct calculation which is the lowest energy structure. Moreover, even if the structure is given, vibrational modes may depend sensitively on the second order derivatives of the total energy with respect to atomic coordinates. The calculation of second order derivatives can be quite sensitive to the choice of basis and potential. We have developed a method [ 1,2] for determining the electronic structure of localized systems such as clusters which addresses many of these issues. Our

method combines a higher order finite difference procedure [3] for solving the Kahn-Sham equations [4] with a pseudopotential description [5] of the ioncore potential. The Kohn-Sham equations are solved directly on a simple grid in real space. Since the pseudopotential approximation results in slowly varying wavefunctions, the kinetic energy operator converges quickly within a high-order finite difference expansion. This “higher-order finite difference”pseudopotential (FDP) method has been used to calculate the structural properties of diatomic molecules and small silicon clusters [l, 2, 61. This method possesses a number of obvious advantages. Unlike traditional solid state electronic structure methods, this method does not utilize a supercell geometry and plane waves to describe localized systems. Numerous plane waves are required for a converged basis when one attempts to calculate the properties of a localized system as the plane waves must not only describe molecular orbitals, but also the “vacuum” region of the supercell. Another disadvantage of plane waves is

231

232

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MODES OF SILICON NANOSTRUCTURES

the use of fast Fourier transforms (FFTs). Typically, FFTs can consume a large fraction of the computational time in a plane wave-supercell calculation. Also, FFTs can present communication obstacles when dealing with massively parallel computer architectures. Compounding the theoretical problems in determining the electronic and structural properties of a cluster is the lack of direct experimental probes. Cluster assembled matter, unlike crystalline matter, is not amenable to an X-ray structural analysis, nor is it amenable to scanning tunnelling microscopy as are solid surfaces. Recently, photoemission and Raman spectroscopies have been applied to silicon clusters. However, no direct information on structural properties is obtained with either of these spectroscopies. In the case of photoelectron spectroscopies, the problem is made more difficult by the necessity of charging the clusters. The structure and electronic properties for charged clusters can be very different than that for neutral clusters. Here we focus on comparing vibrational modes with those obtained from Raman spectra of silicon clusters deposited on an inert substrate. This approach has been used in conjunction with quantum chemistry techniques to obtain invaluable information on the ground state structures of clusters [7]. Details of the finite-difference pseudopotential method have been presented elsewhere [ 1, 2, 61. As such, here we outline the essential features of our approach. We use the method of Troullier and Martins [5] to construct ab initio pseudopotentials for silicon. The exchange-correlation potential of Ceperley and Alder [8] was used in this construction. A pseudopotential cutoff radius of 2.5a.u. was used for both the s and p valence electronic states. With respect to the finite difference grid, a grid spacing, h, of h = 0.70a.u. was used. We use a spherical domain with radius R,,, to confine the wave functions of atoms in a cluster. Rmax is chosen so that no atom contained within the domain is within a distance of N 7 a.u. of the domain boundary. The wave functions outside this domain vanish. It is important to insure that R,,, is sufficiently large to permit the wave function to decay properly. If the domain is not large enough, it can affect the accuracy of vibrational frequency, especially for the breathing mode. The kinetic energy operator is expanded up to twelfth order in h in the higher order finite difference expansion [l, 31. To obtain the vibrational spectra of a cluster, we need to calculate the dynamical matrix, Mia,j,, which is either the second order derivative of total energy vs displacement or equivalently the first order derivative

Vol. 96, No. 4

of force vs displacement,

where mi is the mass for atom i, E is the total energy of the system, I;i* is the force on atom i in the direction cr. RF is the a component of coordinate for atom i. In our previous work [6], we have derived finite difference expressions for the interatomic forces using the Hellmann-Feynman theorem. This force calculation has yielded very accurate results [6]. Using this formalism, we calculate the dynamical matrix elements by calculating the first order derivative of force vs atom displacement numerically. To accelerate the convergence of the interatomic forces, we have included a “correction” term to account for any errors in obtaining the self-consistent field. This approach by Chan et al. has been discussed elsewhere [9]. In Fig. 1, we illustrate the ground state structures of Si clusters, Si,, 4 5 n 5 7. These structures were determined via a simulated annealing performed with Langevin dynamics [6]. Sib merits special note. This St

S’s

$+ 2.31

2.22

2.40

2. 1-m.11 --.. .

S4j (a)

9, lb)

2.25 .-• --1.1 2.7l-

l

-1.. c ‘

+

Fig. 1. Ground state structures of small Si clusters, from which the vibratiogal modes were calculated. The bond lengths are in A.

VIBRATIONAL

Vol. 96, No. 4

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MODES OF SILICON NANOSTRUCTURES

FSX

FSZ

FIX -1

.o ’

-3

1

-2

I

I

I

-1

0

SRlX

( 10-Z

1

I

I

I

2

3

4

a-u.)

Fig. 2. The force on atom 1, F,X, (solid line), and on atom 5 along x direction Fc, (dashed line), z direction F