NANO LETTERS
Critical Thickness and Radius for Axial Heterostructure Nanowires Using Finite-Element Method
2009 Vol. 9, No. 5 1921-1925
Han Ye,†,‡ Pengfei Lu,†,‡ Zhongyuan Yu,*,†,‡ Yuxin Song,§ Donglin Wang,†,‡ and Shumin Wang*,§ Institute of Optical Communication and Optoelectronics, Beijing UniVersity of Posts and Telecommunications, Beijing 100876, People’s Republic of China, Key Laboratory of Optical Communication & LightwaVe Technologies, Ministry of Education, Beijing UniVersity of Posts and Telecommunications, Beijing 100876, People’s Republic of China, and Photonics Laboratory, Department of Microtechnology and Nanoscience, Chalmers UniVersity of Technology, 41296 Gothenburg, Sweden Received January 7, 2009; Revised Manuscript Received March 1, 2009
ABSTRACT Finite-element methods are used to simulate a heterostructured nanowire grown on a compliant mesa substrate. The critical thickness is calculated based on the overall energy balance approach. The strain field created by the first pair of misfit dislocations, which offsets the initial coherent strain field, is simulated. The local residual strain is used to calculate the total residual strain energy. The three-dimensional model shows that there exists a radius-dependent critical thickness below which no misfit dislocations could be generated. Moreover, this critical thickness becomes infinity for a radius less than some critical values. The simulated results are in good agreement with the experimental data. The critical radius from this work is smaller than that obtained from previous models that omit the interaction between the initial coherent strain field and the dislocation-induced strain field.
Semiconductor heteroepitaxy has been pervasively applied to fabricate electronic and optoelectronic devices. The properties and performance of these devices are dependent on the stress state and the defect structure in the epitaxial thin film.1 Tremendous interest exists in the possibility of growing high-quality heterostructure nanowires without dislocations, such as GaAs, InAs, InP and GaP.2-5 This onedimensional (1D) heterostructure can accommodate much larger misfit strain, because it has free surfaces not only on the top but also at the side, which could allow efficient lateral strain relaxation.6 Thus, the axial and radial heterostructure nanowires7-9 have been proposed as promising nanoscale building blocks for future devices with superior properties. Nanowire lasers, resonant tunneling diodes, and singleelectron tunneling diodes have already been demonstrated.10-12 Over the years, papers have been published on theoretical studies of the critical thickness (hc) on patterned sub* Authors to whom correspondence should be addressed. E-mail:
[email protected],
[email protected]. † Institute of Optical Communication and Optoelectronics, Beijing University of Posts and Telecommunications. ‡ Key Laboratory of Optical Communication & Lightwave Technologies, Ministry of Education, Beijing University of Posts and Telecommunications. § Photonics Laboratory, Department of Microtechnology and Nanoscience, Chalmers University of Technology. 10.1021/nl900055x CCC: $40.75 Published on Web 03/30/2009
2009 American Chemical Society
strates.13-17 Luryi and Suhir13 first proposed using a nanostrip substrate and predicted that the critical thickness can reach infinity when the strip width is below a critical value. The critical thickness can also be enhanced by growing a heterostructure on a compliant substrate when part of the strain energy can be stored in the compliant substrate. Very recently, Ertekin et al.18 and Glas6 studied the critical dimensions for strained axial heterostructures in free-standing nanowires using analytic approaches. A critical radius below which the critical thickness reaches infinity has been found. In some treatments, an energy balance model that followed that of People and Bean (PB)19 was employed, where the linear density of dislocation energy is converted to an area density by dividing an average spacing distance between the two neighboring dislocations. However, a simple energy balance between the strain and the dislocation energy assumes that the strain is completely released by misfit dislocations, and it is difficult to motivate such a sudden increase of misfit dislocations at the critical thickness. In other treatments, the total energy (sum of the strain and the dislocation energy) was minimized, with respect to the residual strain, to obtain the critical thickness. This scheme has been used to study the critical thickness in twodimensional (2D) lattice mismatched systems,20 leading to
Table 1. Material Parameters Used in the Calculation
Figure 1. Schematic showing a circular heterostructure nanowire with a radius (r).
the result of the Matthews and Blakeslee (MB) model.21 The dislocation energy consists of two parts: the elastic energy and the dislocation core energy. For a single screw or edge dislocation in a homogeneous crystal, the logarithmic-form elastic energy is the integration over the local dislocationinduced strain energy that can be expressed analytically.1 The sum of the initial strain energy and the dislocation energy, as employed in most previous models,14,19 implies the neglect of interaction between the initial strain field and the dislocation-induced strain field. In a relaxed heterostructure, the dislocation-induced strain field offsets the misfit strain and reduces the overall strain energy. Therefore, only the dislocation core energy should be considered separately while the elastic energy is included in the overall strain energy of the system. In this work, we focus on axial nanowire heterostructures where materials are stacked along the wire axis. We calculate the strain field before and after the introduction of a pair of orthogonal misfit dislocations using the finite-element method (FEM). The FEM is widely used to simulate stress and strain fields22-24 and is often used to check the validity of the analytic strain field.25,26 The interaction between the initial coherent strain field and the strain field induced by the first pair of misfit dislocations is included. Comparing the overall energy stored in the nanowire with and without dislocations, we obtain a radius-dependent critical thickness. The results are compared with previous theoretical and experimental values. This model provides a framework to determine the structural feasibility of various coherently strained axial heterostructure nanowires. The nanowire with an axial heterostructure considered in this work is shown in Figure 1. In our simulation, the bulk substrate presented at the base of the nanowire is assumed to be rigid without deformation. This is a good approximation when the height of the compliant mesa substrate is thick enough, as will be shown late. For simple geometric description of the orthogonal dislocations, we take the Descartes coordinate. The x- and z-axes correspond to the [110] direction and the [001] direction of a zincblende structure, respectively. The nanowire is oriented along the z-axis and is comprised of an elastical epilayer and a compliant mesa substrate, with the interface at z ) 0. Two 1922
parameter
value
lattice constant, a (Å) Young’s moduli, E (GPa) shear modulus, µ (GPa) Poisson’s ratio, υ density, F (× 103 kg/m3)
6.0583 - 0.405x 51.4 + 34.5x 19 + 13.8x 0.35-0.04x 5.68-0.37x
misfit dislocations lie along the x- and y-directions and intersect on the wire axis. The calculations are presented here for the specific InxGa1-xAs/GaAs system, which is a prototype latticemismatched heterostructure, but can be extended to any lattice-mismatched material system with some care. Simulations are performed in the framework of linear isotropic elasticity, taking different Young’s moduli (E) and Poisson’s ratios (ν) for the film and the substrate. The E- and ν-values for InxGa1-xAs are interpolated using respective single-crystal data, assuming that the Vegard’s law is valid, as shown in Table 1. We first consider the coherent film growth without interfacial dislocations in a free-standing nanowire. The film and substrate are treated as continua. To simulate a strained epilayer, the coherency is imposed at the interface through lattice misfit, fm: fm )
af - as af
(1)
where af is the lattice constant of the film and as is the lattice constant of the substrate. The lattice parameter of the film is a function of indium fraction (x). In the case of the InxGa1-xAs/GaAs system, the misfit is a positive quantity. The film is compressive after the strain is imposed. Correspondingly, the mesa substrate is tensile. Note that the compliant mesa substrate is assumed in this calculation, because the radius is finite. Under this assumption, the lattice mismatch is shared in the epilayer and the mesa substrate, because the compliant substrate has flexibility to coherently accommodate the misfit. As an example, a strained layer of In0.5Ga0.5As is grown on a GaAs mesa substrate of 50 nm in height. We take nanowire radius r ) 30 nm, which is of the order of typical radii of current nanowires, based on the vapor-liquid-solid growth. The distribution of strain field component, xx, is shown in Figures 2a and b. For the symmetry, yy has a similar distribution. It can be seen that the epilayer is in compression and the GaAs mesa substrate is in tension. Figure 2b shows the normal strain component, xx, along the z-direction at the center of the nanowire. The xx-value approaches zero when the film reaches a thickness on the order of its diameter. Thus, the epilayer recovers to the strainfree state and will retain it for the subsequent growth. Also, the xx-value is negligible at the interface between the compliant mesa substrate and the rigid substrate, confirming the validity of the previous assumption of neglecting any deformation in the rigid substrate. When the mesa height is
