Analytical Study of Elastic Relaxation and Plastic Deformation in

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Analytical Study of Elastic Relaxation and Plastic Deformation in Nanostructures on Lattice Mismatched Substrates Xu Zhang,†,‡ Vladimir G. Dubrovskii,*,†,§ Nickolay V. Sibirev,† and Xiaomin Ren‡ †

St. Petersburg Academic University, Khlopina 8/3, 194021 St. Petersburg, Russia State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, P.O. Box 66, #10 Xitucheng Road, Haidian District, Beijing 100876, China § Ioffe Physical Technical Institute of the Russian Academy of Sciences, Politekhnicheskaya 26, 194021 St. Petersburg, Russia ‡

bS Supporting Information ABSTRACT: In view of a continuously growing interest in monolithic integration of dissimilar semiconductor materials in a nanostructure form, we present an analytical study of elastic strain energy in nanostructures of different isotropic geometries grown on lattice mismatched substrates. An analytical solution for the elastic stress field is derived, which is not restricted by the small aspect ratio or particular geometry. It is shown that, at a large enough aspect ratio, the relaxation of displacement fields with the vertical coordinate is exponential. This allows us to find an analytical expression for the strain energy density. The cases of rigid and elastic substrates are considered simultaneously. By minimizing the total energy, we obtain the elastic energy density as a function of aspect ratio for given nanostructure geometry. Our data indicate that the elastic energy is highly dependent on the island shape, with the relaxation becoming faster as the contact angle increases. We also present a simple expression for the elastic relaxation where the fitting coefficient depends on the geometry. A dislocation model is then considered to analyze the competition between the elastic and dislocation energies. We calculate the critical radius below which the plastic deformation is energetically suppressed. Our results demonstrate a good quantitative correlation with the available experimental data on IIIV semiconductor nanowires grown on silicon substrates.

’ INTRODUCTION Laterally confined nanostructures (NSs) such as quantum dots (QDs),1 nanowires (NWs),2 and nanoneedles (NNs)3 offer exciting opportunities for band gap engineering and monolithic integration of dissimilar semiconductors in lattice mismatched material systems. While lattice mismatch makes the growth of high quality two-dimensional (2D) layers impossible, an energetically favorable relaxation of elastic stress at free sidewalls of three-dimensional (3D) NSs126 enables a radical increase of the critical thickness for plastic deformation via the formation of misfit dislocations. Relaxation of elastic stress has been known for a long time as the major driving force for a spontaneous formation of coherent StranskiKrastanow QDs.1,49 The extreme case of elastic stress relief is realized in a cylindrical NW with the diameter being dictated by a small footprint of a metal catalyst particle.2,1013,1722 Unlike in the case of pyramidal QDs, theory predicts that the critical thickness for plastic relaxation in a NW should increase infinitely when its lateral dimension is smaller than a certain critical diameter at a given lattice mismatch.10 Indeed, the Au-assisted metal organic chemical vapor deposition (MOCVD)11 and molecular beam epitaxy (MBE)13 have enabled the fabrication of coherent, straight IIIV NWs on Si(111) substrates, with a critical diameter of only 2426 nm for r 2011 American Chemical Society

the 11.6% lattice mismatch in the InAs/Si system. A large progress has been achieved in the synthesis of self-catalyzed, high quality GaAs, InP, InAs, and other IIIV NWs on specially treated Si substrates.1722 Catalyst-free GaN NWs were also obtained by molecular beam epitaxy directly on Si(111) substrates.23 Recently, Chang-Hasnain and coauthors have demonstrated a new growth mechanism resulting in catalyst-free, single crystalline wurtzite GaAs NNs with 69° hexagonal pyramid shape and an ultrasharp 24 nm tip, obtained by MOCVD on Si3 and sapphire24 substrates. The latter case relates to an extremely large mismatch of 46%. In ref 16 it was hypothesized that the NNs were initiated as nanoclusters whose nucleation was driven by a large lattice mismatch between GaAs and the substrate. IIIV NWs and NNs grown on Si can comprise optically bright heterostructures,22,25 which is of paramount importance for monolithic integration of optoelectronics with existing Si technologies. Obviously, further progress in the synthesis of highly anisotropic NSs with the desired properties on lattice mismatched substrates requires the development of relevant theoretical models. Received: August 8, 2011 Revised: September 26, 2011 Published: October 05, 2011 5441

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Crystal Growth & Design Theoretical analysis of elastic energy stored in coherent strained NSs of different geometries is very important for better understanding of their growth and physical properties. In particular, the elastic energy relaxation as a function of the aspect ratio, elastic constants, and lattice mismatch enters the nucleation barrier that determines the probabilities of 3D island formation and therefore the resulting morphology of NS ensembles.1416 Analytical or numerical calculations of elastic energy are also required for the correct determination of critical thickness for plastic deformation and the critical dimensions for the coherent growth of NWs.2,1113 There has been large interest in the past in the elastic energy of self-assembled StranskiKrastanow QDs, whose aspect ratio is typically small. For example, Tersoff and Tromp4 proposed the small slope approximation to study the elastic energy of an elongated ridge in 3D space, with the interfacial force density being proportional to the derivatives of local height with respect to 2D coordinates. Shchukin and coauthors5,9 used the small slope interpolation to compute the strain energy of a pyramidal island, and the energy of elastic interaction between such islands. The surface traction was obtained through the 2D Green tensor, which could only be justified for flat enough islands. Ratsch and Zangwill6 employed the reiterative 2D approach in the framework of the FrenkelKontorova scheme.27 It was assumed that each monatomic layer induces a rigid sinusoidal potential acting upon the upper layer, while the relaxation of lateral strain at the edges of each FrenkelKontorova chain propagates into the underlying layer. As a result, the lattice mismatch gradually decreases with the height. Gill and Cocks7 and Gill8 developed the analytical models for the elastic energy relaxation beyond the small slope interpolation that would work for the QDs with a high aspect ratio. However, their results become less accurate as the contact angle of island facets approaches 90 deg. Therefore, results of refs 7 and 8 cannot describe highly anisotropic NSs such as NWs and NNs having typical aspect ratios on the order of 10 or more.2,3,1026 Despite a large interest in the field, there have been relatively few relevant analytical studies of for the elastic energy of NWs and NNs grown on lattice mismatched substrates. Glas10 presented the analytical fit to the finite element calculations in the case of a mismatching layer seated on top of a dissimilar cylinder. More recently, Glas28 also obtained a more complex fit to the finite element calculations of the elastic energy relaxation in a uniformly strained cylinder standing on an infinite foreign substrate. This formula has been found to fit well with our numerical results for an isotropic full cone,16 but with different fitting parameters. Shi and Wang26 used a very interesting approach based on the adaptation of the StranskiKrastanow growth mode to the heteroepitaxial growth of NWs. Their strain versus dislocation model enables the identification of different growth modes. In particular, it shows that NWs could be grown coherently on the substrate only when their sizes are smaller than the critical size, while a defective intermediate structure is necessary for growing NWs larger than the critical size. Some of the approximations discussed above will be presented in the next section. The aim of this paper is an analytical study of elastic energy relaxation in the NSs of different 3D isotropic geometries beyond the limit of a small aspect ratio. The paper is organized as follows. In Section II, we consider the general properties of elastic energy relaxation and present the known results relevant for the further analysis. Section III presents the formulation of the 2D isotropic

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Figure 1. Model NS geometries: cylinder (a), full cone (b), truncated cone (c), and reverse truncated cone (d).

problem. In Section IV, we derive a general analytical solution for the strain and stress components and therefore for the elastic energy density in the cases of rigid and elastic substrate, with an exponential decay of radial and vertical displacements with the vertical coordinate. In Section V, we modify the results of ref 10 to obtain the dislocation energy of the NSs having different geometries. Section VI contains numerical results for the elastic energy and the critical dimensions for plastic deformation in different NSs, their comparison with the existing models, finite element calculations and the available experimental data for different IIIV NWs grown on lattice mismatched substrates, and a discussion.

’ EXISTING MODELS FOR ELASTIC RELAXATION Elastic relaxation (or relative strain energy), in a 3D NS with height H, base radius R (see Figure 1) and aspect ratio β = H/(2R) is generally defined as z(β) = W(β)/W2D. Here, W(β) is the elastic energy of NS and W2D = EVε02/(1 v) is the elastic energy in a uniformly strained 2D layer of the same volume V on a semi-infinite rigid substrate with lattice misfit ε0, E is the Young’s modulus, and ν is Poisson’s ratio of the NS material. Regardless of NS shape, the function z(β) obeys the following asymptotic behaviors at small and large β10,16 βf0

βf∞

zðβÞ f 1; zðβÞ f 1=ðAβÞ

ð1Þ

As given by these expressions, the relaxation must equal one at β f 0 (i.e., in a 2D thin film case) and scale as 1/β at β f ∞, since no additional strain energy is generated when the island reaches a height on the order of R and completely recovers its strain-free state. The coefficient A depends on the Poisson’s ratio but not on the Young’s modulus or mismatch. We now present some of the known analytical approximations for the elastic relaxation mentioned in the Introduction, which are relevant for the foregoing analysis (for more details, see ref 16). With regard to the two asymptotes given by eq 1, the simplest formula for z(β) can be chosen in the form zðβÞ ¼

1 1 þ Aβ

ð2Þ

In the case of mismatched axial heterostructure in a cylindrical NW,10 the constant A equals approximately 27.4 at ν = 1/3. 5442

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Crystal Growth & Design

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The RatschZangwill formula,6 which is proved to be a good approximation for rectangular QDs, is given by 1 expð ΛβÞ zðβÞ ¼ Λβ

following equations of equilibrium:29 ∂σ rr ∂σrz σ rr σθθ ∂σ rz ∂σ zz σrz þ þ ¼ 0; þ þ ¼0 ∂r ∂z r ∂r ∂z r ð8Þ

ð3Þ

with the relaxation coefficient Λ = 3π for a cubic material. The GillCocks expression,7 obtained beyond the assumption of identical elastic constants of the island and the substrate, in the case of an isotropic full cone takes the form 1 þ 6β 1 þ 2bβð1 þ 6β2 Þ þ ð16 10kÞβ2

Combination of eqs 68 results in the well-known equations for the displacement field given by30 1 ∂g 1 ∂g þ Δuz ¼ 0; þ Δuz ¼ 0 1 2v ∂z 1 2v ∂z

2

zðβÞ ¼

ð4Þ

Here, the coefficient k t [2(1 2ν)]/[3(1 ν)] accounts for the compressibility of NS material and the coefficient b t 1.059[E(1 νs2)]/[Es(1 ν)] contains the elastic constants of the substrate Es and νs. The Glas fit28 to the finite element calculations of the elastic relaxation of a uniformly strained cylinder standing on an infinite foreign substrate, is written as p1 þ ð1 p1 Þ expð p3 βÞ ð5Þ zðβÞ ¼ 1 þ p2 β The pk are ν-dependent fitting coefficients such that p1 = 0.557, p2 = 10.15, and p3 = 9.35 for a cubic material at ν = 1/3. We do not consider here some of earlier results obtained within the frame of small slope approximation4,5 that are relevant for flat QDs having aspect ratios β e 0.2 but not applicable for taller NSs.

’ FORMULATION OF 2D ISOTROPIC PROBLEM Typical 2D isotropic geometries describing NWs (cylinder, truncated cone, reverse truncated cone), NNs (cone), and QDs (cone, truncated cone) are presented in Figure 1. For a 2D isotropic NS of any shape, the displacement field is angular independent. In the case of the circular plate under a radial force, the 2D radial displacement is presented as ur = B/r + Cr, with r being the distance from the origin, while uθ = 0.2933 If there is no hole in the center, the displacement should remain finite when the radius tends to zero, yielding B = 0 and ur = Cr. In a 3D case without a volume force, the displacement field caused by a 2D stress force acting upon the NS-substrate heterointerface takes the form:33 ur = Brf(z). The function f(z) describes the relaxation with the distance from the interface z; its form will be considered below. In the cylindrical coordinates r, θ, and z, the tangential component uθ vanishes in view of 2D isotropy, while ur and uz are independent of θ. The displacement strains εij are determined by the corresponding derivatives of displacement field ui as follows ∂ur ur ∂uz ; εθθ ¼ ; εzz ¼ ; εrz ∂r r ∂z 1 ∂ur ∂uz þ ¼ ; εrθ ¼ εzθ ¼ 0 2 ∂z ∂r

Δg ¼ 0

g ¼

∂ur ur ∂uz þ þ ∂r r ∂z

’ ANALYTICAL SOLUTION FOR ELASTIC STRESS In order to obtain an analytical approximation for the z dependence of displacement fields, we introduce the normalized variables by definition ðF, ςÞ ¼ ðr=R, z=HÞ, ðuF , uς Þ ¼ ður , uz Þ=R Rescaling eq 10 and 11 for g in terms of these variables results in ! 1 ∂2 g ∂2 g 1 ∂g þ þ ¼ 0; ∂F2 F ∂F 4β2 ∂ς2 ∂uF uF 1 ∂uς g ¼ þ þ 2β ∂ς ∂F F For tall enough NSs with 2β . 1, these equations are simplified to ∂2 g 1 ∂g ∂ur ur ¼ 0; g ¼ þ þ ∂r 2 r ∂r ∂r r

with the elastic constants μ = E/[2(1 + ν)] and λ = (Eν)/[(1 2ν)(1 + ν)]. Furthermore, the stress components must satisfy the

ð12Þ

In the same approximation, eqs 9 are reduced to 1 1 2v 1 1 2v

∂g ∂2 ur 1 ∂ur ur þ 2 þ ¼ 0; ∂r r ∂r r 2 ∂r ∂g ∂2 uz 1 ∂uz þ 2 þ ¼0 ∂z r ∂r ∂r

ð13Þ

Obviously, any z-independent g is the solution to the first eq 12. Let us now consider g(z) in the form g(z) = const (1 2ν) exp(αz/R), where α is a constant. The first eq 13 is satisfied when ur ¼ Br expð αz=RÞ

ð7Þ

ð11Þ

and Δ = (∂2/∂r2) + [(1/r)(∂/∂r)] + (∂2/∂z2) is the angular independent 3D Laplace operator.

ð14Þ

with arbitrary constant B, that is, when the decrease of radial displacement with z is exponential. The second eq 13 is satisfied with

According to Hooke’s law,29,30 the elastic stress components are given by σij ¼ 2μεij þ λδij εkk

ð10Þ

Here,

εrr ¼

ð6Þ

ð9Þ

uz ¼

1 ðCαr 2 þ SR 2 Þ expð αz=RÞ 4 R

ð15Þ

where S is a constant. Below we will use eqs 14 and 15 for any β, assuming that the exponential approximation is also good for flat islands. This assumption will be checked for validity by the direct comparison with numerical calculations. 5443



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Since the displacement fields in both radial and vertical directions decay exponentially with z, the boundary conditions of free lateral surfaces, σrr(r = R(z),z) = 0 and σrz(r = R(z),z) = 0 (where R(z) describes the NS sidewalls) cannot be exactly satisfied. Instead, we proceed by using the minimization of total elastic energy of given NS to obtain the unknown coefficients α, C, and S.31 Likewise in ref 31, it can be shown that the minimum elastic energy approach can be used to best reflect the exact displacement fields that satisfy the above boundary conditions. The comparison between the two approaches will be presented elsewhere. The nonzero components of strain field are now readily obtained from eqs 6: εrr ¼ B expð αz=RÞ; εθθ ¼ B expð αz=RÞ, εzz

Here, χ = (1/2) tanh(aθ) is the index describing the singularity at the island edge, a = 1.41 for a cubic NS having the identical elastic constants with the substrate, and δ = 0.114/χ + 0.638 is the exponent introduced to ensure that fr(0) = 0. With the surface traction given by eq 22, the strain energy of the substrate is readily obtained from eqs 20 and 21 in the form Wsub ¼

α ðCαr 2 þ SR 2 Þ ¼ expð αz=RÞ; εrz 4 R2 1 rαð2B CÞ expð αz=RÞ ¼ 4 R

Integration in eq 20 is performed over the entire substrate area outside the NS base. In an isotropic case, the radial surface traction can be approximated by the numerical fit of Gill:8 " # r χ r δ fr ðrÞ ¼ c 1 1 ð22Þ R R

ð16Þ

F¼

σrr ¼

The elastic energy density is given by30 wðr, zÞ ¼

1 εij σij 2

ð18Þ

Thus, integrating the density over the entire NS volume gives the total elastic energy stored in the NS: Z W ¼ d3 xwðr, zÞ ð19Þ For an NS with the lattice constant l grown on a substrate with the lattice constant l0, the lattice mismatch parameter is defined as ε0 = (l l0)/l0. For a rigid substrate, the constant B must equal ε0, and the total elastic energy Wtot = W. The constants α, C, and S are then obtained by the minimization of total energy W defined by eq 19 for a given NS geometry. For an NS grown on an elastic substrate, the total elastic energy contains the contribution from the substrate. Elastic energy of the substrate is induced by the radial surface traction forceB(r f B) which is singular at the NS edge and is highly dependent on the NS contact angle θ (the latter relates to the aspect ratio as θ = arctan (2β) in the case of full cone geometry). Generally, the elastic energy of the substrate is given by32 Z Z 1 2 Wsub ¼ r d2 B r 0 fi ð B r ÞGij ð B r B r 0 Þfj ð B r 0 Þ ð20Þ d B 2 where the Green’s tensor on the surface is defined as5,8,30,32

" # ðB r B r 0 Þi ð B r B r 0 Þj 1 þ v ð1 vÞδij Gij ð B r B r Þ¼ þ v πE j B r B r 0j jB r B r 0 j3 0

ð21Þ

ð23Þ

Here, Es and νs are the elastic constants of the substrate; other parameters are defined as

The elastic stress components are easily found from eqs 7: 1 ð8λBR 2 λCα2 r 2 λαSR 2 þ 8μBR 2 Þ expð αz=RÞ; 4 R2 2 2 2 2 2 1 ð8λBR λCα r λαSR þ 8μBR Þ σ θθ ¼ expð αz=RÞ; 4 R2 2 2 2 2 2 2 1 ð8λBR λCα r λαSR 2μα Cr 2μαSR 2 Þ σ zz ¼ 4 R2 1 μrαð2B CÞ expð αz=RÞ expð αz=RÞ; σ rz ¼ ð17Þ 2 R

Es ε20 πR 3 F ð1 v2s Þ 3 J

6 6 , ð1 χÞð2 χÞð3 χÞ ð1 þ δÞð2 þ δÞð3 þ δÞ

J ¼ 1:059 þ 41:25χ3

ð24Þ

The work done by the surface traction relaxes the strain at the NS base from the mismatch strain ε0 to a lower value of ε0 ε. In the case of elastic substrate, the radial component of displacement field should be therefore changed to ur = (ε0 ε)r exp(αz/R). The total elastic energy of the system is given by Wtot = W + Wsub. The constant ε is obtained by minimizing the total energy, that is, from the condition ∂Wtot/∂ε = 0 at ε = ε*. Substitution of this ε* into the total energy Wtot and global minimization of the latter enables, as above, to find the unknown parameters α, C, and S for given NS geometry. In particular, for a cylindrical NW geometry at the aspect ratio of 5 we obtain the following numbers: α = 5.23, C = 0.103, and S = 0.0336.

’ CRITICAL DIMENSION FOR PLASTIC DEFORMATION We now turn to the analysis of plastic deformation via the formation of misfit dislocations. This will enable the comparison of the energies of coherent and dislocated state and the analysis of the critical dimensions for the onset of plastic deformation depending on the lattice mismatch. To preserve at most a 2D isotropy of the problem, we consider a single pair of orthogonal dislocations placed at the heterointerface of radius R. According to Ovid’ko34 and Glas,10 the energy of dislocation line per unit length can be approximated as Eð1 v cos2 θÞb2 h̅ ln þ 1 wd ¼ 2 b 8πð1 v Þ

! ð25Þ

Here, B b is the Burgers vector, θ is the angle between the dislocation line and its Burgers vector, and b = |b B| is taken as the core cutoff radius for the calculation of elastic energy. Since the distance to the nearest free surface varies along the dislocation line, we use the effective distance h defined as10 h = h if h e ηR, h = ηR if h g ηR, with η = 2/π. The total dislocation energy of a single pair of orthogonal dislocations at the heterointerface is given by Wd = 4Rwd. 5444



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Figure 2. Relative strain energy z(β) in a cylindrical NW on a rigid (dashed line) and elastic (dashdotted line) substrates, compared to the Glas formula given by eq 5 at p1 = 0.557, p2 = 10.15, and p3 = 9.35 (solid line).

With the simplest approximation of the elastic stress relaxation given by eq 2 (this choice will be justified in the next section), the elastic energy can be put in the form We ¼

Figure 3. Relative strain energy in the cylinder, cone, truncated cone, and reverse truncated cone (solid lines) and their fits by the formula Z(β) = 1/(1 + Aβ) with A = 15, 5.5, 8, and 50, respectively (dotted lines).

dimension of the form 2πða2 þ a þ 1Þ α2 b2ef f αbef f ε0 Rc 3A 4

E E πR 2 hε20 ða2 þ a þ 1Þ z ðβÞV ε20 = ð1 vÞ ð1 vÞ ð1 þ AβÞ 3

ð26Þ Here, the total volume of a NS is represented by V = (a2 + a + 1)πr02h/3, with a being equal to 1 and 0 for a cylinder and a full cone, respectively. For a more complex geometry, a is just the ratio of the upper radius to the base radius, with a < 1 corresponding to a truncated cone and a > 1 to a reverse truncated cone. From eqs 25 and 26, the excess energy of the system with a dislocation pair with respect to a fully coherent state can be written down as

! " Er0 h α2 b2ef f αbef f ða2 þ a þ 1Þ πr0 h z ε ΔWðR, hÞ ¼ 0 2R 3 1v 4R 2 R !# h̅ ðR, hÞ þ 1 ð27Þ þ C ln b

Here, C = (1 v cos2 θ)b2/[2π(1 + v)], α = 4/π, for pure edge (θ = π/2, beff = b) and 60° dislocations (θ = π/3, beff = b/2). Both types of dislocations pertain to a face-centered cubic crystal. Obviously, a coherent state is stable at ΔW > 0, whereas at ΔW < 0 the introduction of dislocations is energetically preferred. Therefore, the equation ΔW(R,h) = 0 defines the critical thickness hc for the onset of plastic deformation as a function of the base radius R. Below hc, the coherent state is energetically preferred, while exceeding hc leads to the plastic deformation. As shown, for example, in ref 10 in the case of axial NW heterostructure and will be demonstrated in the next section in the case of NW on a lattice mismatched substrate, the critical thickness increases infinitely at a certain critical dimension Rc. Below this Rc, a coherent growth of infinitely long NS is possible. From eq 1, βZ(β) f 1/A when β f ∞ regardless of particular NS geometry. Using this asymptote in eq 27, we readily obtain the transcendent equation for the critical

!

βRc þ 1 ¼0 þ C ln b

ð28Þ This result generalizes the corresponding formula of ref 10 to include different NS geometries.

’ RESULTS AND DISCUSSION Relative strain energies z(β) = W(β)/W2D for the four geometries shown in Figure 1 were computed at different aspect ratios β by the integration of strain energy density as given by eqs 18 and 19. In the case of elastic substrate, the strain energy of the latter was included as given by eqs 23 and 24. The minimization of total strain energy was performed as described above for each β. Figure 2 shows the resulting z(β) in the cylindrical NS geometry, computed in the case of a cubic NS material having identical elastic constants E and ν with the substrate. The computations were performed separately for the rigid and elastic substrates. As expected, both curves rapidly decay as the aspect ratio increases, so that about 80% of the 2D elastic energy is relieved already at β = 0.2. In other words, the effect of elastic relaxation at free lateral surface is huge.10 As is seen from Figure 2, the relaxation on the elastic substrate is faster than on the rigid one, which is indeed a natural and anticipated result. It is also seen that the Glas fit given by eq 5 with coefficients p1 = 0.557, p2 = 10.15, and p3 = 9.35 is excellent for a rigid substrate. As discussed above, the simplest possible expression for the elastic relaxation is given by eq 2. Because of its simplicity, such a formula enables many important physical characteristics to be obtained such as stress-driven nucleation barriers1416 in an analytical, physically transparent form. We therefore have performed the calculations of z(β) for different NS geometries (in the case of a rigid substrate) and fitted the results by eq 2. The best fits were obtained with the fitting coefficient A = 5.5 for the full cone, 8 for the truncated cone (with the 70° contact angle at the base), 15 for the cylinder, and 50 for the reverse truncated cone (with the 110° contact angle at the base), Figure 3 showing 5445



Figure 4. Relative strain energy in different NS geometries (dashed lines), compared to the Glas formula 5 for the cylinder at p1 = 0.557, p2 = 10.15, and p3 = 9.35, the RatschZangwill formula 3 for the rectangular island at Λ = 3π and the GillCocks formula 4 for the full cone at E = Es, all calculated for cubic materials.

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Figure 6. Critical radius for plastic deformation in different NS geometries, plotted against the lattice mismatch, in the case of 60° dislocations with b = 2 nm. The dots represent experimental results of refs 3 (triangles) and 13 (squares).

Figure 5. Strain energy density of cylindrical 20 nm radius NW as a function of z at ε0 = 0.01, 0.03, 0.05, and 0.07 (lines). Results of corresponding finite elements calculations are shown by dots.

the corresponding curves. As follows from the figure, the simple fits are excellent for the full and truncated cones, the typical model geometries for the StranskiKrastanow QDs in different material systems.7,8 The full cone is also a good model geometry in the case of GaAs NNs on silicon and sapphire substrates.3,16 The correspondence becomes worse for the cylinder and reverse truncated cone, with the numerical relaxation decreasing more slowly in the beginning and becoming steeper toward the larger β than that given by eq 2. We can also conclude that the elastic relaxation depends dramatically on the contact angle at the NS base, with the fitting coefficient A increasing by more than 6 times when the contact angle changes from 70° to 110°. Figure 4 presents the comparison of our results obtained for different NS geometries with the known approximations for z(β) discussed in Section II, in the case of a cubic NS and identical elastic constants of the substrate and the NS. All calculations were performed on a rigid substrate. It is seen that the Gill Cocks expression7 predicts a slower decay than our result for the

Figure 7. Radius dependences of critical thickness in cylindrical NWs. Curves at 4%, 8.1%, and 11.6% relate to GaAs, InP, and InAs NWs on Si(111) substrates, respectively.

identical full cone geometry. The RatschZangwill formula6 for a rectangular island is asymptotically close to our result for the truncated cone; however, the decay of our curve at small β < 0.5 is noticeably steeper. The Glas formula for the mismatching cylinder28 is very close to our curve, as we saw earlier in Figure 2. Analytical solution for the stress and strain fields presented in Section IV has also been directly compared to the results of finite element calculations of elastic energy density, Figure 5 showing the corresponding results. The elastic strain density at the center of cylindrical NW (i.e., at r = 0) was computed with the following parameters: E = 90 GPa, ν = 0.3, and R = 20 nm, with different lattice mismatches from 0.01 to 0.07. The corresponding finite element calculations (shown by dots in Figure 5) show excellent correspondence with analytical results. Overall, the data presented in Figure 5 again demonstrate a very efficient strain 5446


Crystal Growth & Design relaxation in a 20 nm wide NW, with the strain-free state completely recovering already at z = 10 nm. The critical dimensions for plastic deformation in NSs of different geometries, calculated as a function of lattice mismatch from eq 28 (in the case of 60° dislocations), are shown in Figure 6. Squares and triangles correspond to the experimental critical radii for different IIIV NWs grown on lattice mismatched substrates by the Au-assisted MBE13 and MOCVD.2,11 It is seen that our calculations at A = 15 are in a good quantitative correlation with the experimental data. It is noteworthy that the experimental critical dimensions are very close for NWs obtained by two different epitaxy techniques. This fact supports the validity of our equilibrium approach and shows that the kinetic growth effects do not significantly influence the critical dimension for plastic deformation. Figure 7 shows the radius dependence of the critical thickness for plastic deformation in a cylindrical NW seated on a mismatched rigid substrate, for different values of ε0 at fixed v = 1/3 and b = 2 nm, in the case of 60° dislocations. As discussed, the critical thickness tends to infinity at the critical radius Rc, below which an infinitely long coherent NW can be grown. The lines in Figure 7 separate the thickness-radius plane into two domains, the elastic domain (left and below) and the plastic domain (above and right). The values of lattice mismatch correspond to the case of GaAs NWs on the Si(111) substrate (ε0 = 4%), InP NWs on the Si(111) substrate (ε0 = 8.1%), and InAs NWs on the Si(111) substrate (ε0 = 11.6%). As for general tendencies regarding the elastic stress relaxation and the critical base dimensions for plastic deformation, we would like to point out the following. As is seen directly from Figures 3 and 6, the strain relief is most efficient in a reverse tapered cone. Such geometry is experimentally observed in some Ga-catalyzed GaAs NWs grown on Si substrates, most probably due to the sidewall nucleation from the droplet surrounding the NW top.21 Reverse tapering during IIIV NW growth on Si substrates can be therefore considered as the major advantage: it does not extend the dimension of base heterointerface which would remain dislocation-free if its initial diameter is smaller than critical and, on the other hand, enables an increase in the material volume toward the NW top which is crucial for the optical gain.25 Cylindrical NWs exhibit less efficient strain relaxation. The critical base dimensions for their plastic relaxation are on the order of several tens of nanometers (see Figures 6 and 7), which is within the range of diameters of IIIV NWs typically grown on Si.2,1113,1722 This explains the presence of some misfit dislocations at the bases.13,17 Conical geometry is the worst regarding the elastic relaxation; that is, the critical diameter for the formation of dislocations at their base is the smallest and rapidly decreases as the taper angle increases. However, GaAs NNs grown on roughened Si3 and planar sapphire24 substrates emerge as tiny nanoislands only a few nanometers in diameter, which is why they should not contain dislocations initially. The energetically preferred aspect ratio on the order of 10 is acquired at the short scale of the nucleation stage16 and is maintained throughout the subsequent growth stage by growing in a core shell fashion.35 Sooner or later, such growth would develop a dislocation pair at the base, and relevant NN theory must take this into account. This problem will be considered elsewhere. To conclude, we have presented the analytical solution for the elastic energy density which is valid for a much wider range of aspect ratios and NS geometries than the known models. The minimization of total elastic energy allows us to find the

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unknown coefficients entering the analytical solutions. Not being restricted to the small slope interpolation, the results obtained can be applied to different NSs such as QDs, NWs, and NNs. It is shown that the elastic energy depends strongly on the NS geometry, with the relaxation becoming faster as the contact angle of facet planes increases. The critical thicknesses and the critical dimensions for plastic deformation via the formation of misfit dislocations have been considered. Solutions obtained for different NS geometries have been checked for consistency by the comparison with the existing models, the finite element calculations, and the available experimental data. We now plan to apply the obtained results to the modeling of growth and crystal structure of different NSs in lattice mismatched material systems, whose physical properties are greatly influenced by the elastic relaxation. Overall, our results reveal a possibility of growing high quality NSs on lattice mismatch substrates, which confirms the importance of highly anisotropic NSs for monolithic integration of dissimilar semiconductor materials, for example, of IIIV NWs and NNs with Si.

’ ASSOCIATED CONTENT

bS

Supporting Information. (1) Scanning electron microscopy (SEM) images of InAs, InP, and GaAs NWs grown by Auassisted MBE on lattice mismatched substrates and their measured critical dimensions. (2) SEM and transmission electron microscopy (TEM) images of self-catalyzed GaAs NW on Si(111) substrate exhibiting the reverse tapering. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]ffe.ru.

’ ACKNOWLEDGMENT This work was partially supported by the National Basic Research Program of China (No. 2010CB327600), the National “111” Project (No. B07005), the National High Technology Research and Development Program of China (No. 2007AA03Z418) and the National Natural Science Foundation of China (No. 90201035), Ministry of Education and Science (Contract Nos. 02.740.11.0383, Nos. 16.740.11.0019 and 14.740.11.0592), the scientific program of Russian Academy of Sciences “Fundamental aspects of nanotechnologies and nanomaterials”, few grants of Russian Foundation for Basic Research and FP7 projects SOBONA and FUNPROBE. N.V.S. acknowledges the support of the Council of the President of Russian Federation. ’ REFERENCES (1) Bimberg, D.; Grundmann, M.; Ledentsov, N. N. Quantum Dot Heterostructures; Wiley & Sons: New York, 1999. (2) Chuang, L. C.; Moewe, M.; Crankshaw, S.; Chase, C.; Kobayashi, N. P.; Chang-Hasnain, C. Appl. Phys. Lett. 2007, 90, 043115. (3) Moewe, M.; Chuang, L. C.; Crankshaw, S.; Chase, C.; ChangHasnain, C. Appl. Phys. Lett. 2008, 93, 023116. (4) Tersoff, J.; Tromp, R. M. Phys. Rev. Lett. 1993, 70, 2782. (5) Shchukin, V. A.; Bimberg, D.; Munt, T. P.; Jesson, D. Phys. Rev. B 2004, 70, 085416. (6) Ratsch, C.; Zangwill, A. Surf. Sci. 1993, 293, 123. (7) Gill, S. P. A.; Cocks, A. C. F. Proc. R. Soc. A 2006, 462, 3523. (8) Gill, S. P. A. Appl. Phys. Lett. 2006, 89, 203115. 5447



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Analytical Study of Elastic Relaxation and Plastic Deformation in

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