Comparison of Modeling Strategies for the Growth of Heterostructures

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Comparison of modeling strategies for the growth of heterostructures in III-V nanowires Frank Glas Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.7b00732 • Publication Date (Web): 28 Jul 2017 Downloaded from http://pubs.acs.org on August 4, 2017

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Crystal Growth & Design

Comparison of modeling strategies for the growth of heterostructures in III-V nanowires Frank Glas∗ Centre for Nanoscience and Nanotechnology, CNRS, Université Paris-Sud, Université Paris-Saclay, Route de Nozay, 91460 Marcoussis, France E-mail: [email protected]

Abstract We present two models of the vapor-liquid-solid growth of nanowires of alloyed compound semiconductors. These models are tested against experiments on axial heterostructures in self-catalyzed (Al,Ga)As nanowires. They make use of the available bulk thermodynamic functions for this system. With a growth rate set by the group V dynamics, the crucial question is to determine the relationship between the compositions of liquid catalyst nanoparticle and solid. The first model assumes the equilibrium relationship. It predicts heterostructure profiles in excellent agreement with the experiments. The second model acknowledges that nanowiress grow via nucleation at the solid-liquid interface. We find the size and composition of the critical nucleus at the saddle point of the surface describing the work of formation of the nucleus. Assuming a fixed nucleus edge energy, the critical composition is virtually independent of the As concentration in the liquid and therefore close to the equilibrium composition. This model thus predicts equally well the interface profiles. With a composition-dependent edge energy, the profiles may differ significantly. We clarify why the effects of group

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III and group V atoms may be decoupled and why the reservoir effect is weak. Finally, we discuss how these findings could be extended to other systems.

Introduction In principle, forming quantum size nanostructures is easier in nanowires (NWs) than in structures based on planar substrates. For instance, after a NW stem of material A has been grown, switching the external fluxes to grow material B and then returning to A should produce an axial insertion of B into A, its diameter being determined by the initial stem diameter and its length by the time of exposure to the B fluxes. On the one hand, no strain is needed to form narrow insertions, as opposed to the standard Stranski-Krastanov mode of formation of ”self-assembled” quantum dots on planar substrates 1 (which moreover invariably produces a broad size distribution). On the other hand, provided the NW is narrow enough, widely misfitting materials can be stacked without generating misfit dislocations. 2 In practice however, forming sharp interfaces in axial NW heterostructures may prove difficult because, even if the externally-controlled growth fluxes may be switched abruptly, some A material is usually stored in the system and will be fed to the growing NW when only B is aimed at. Depending on the growth method, there are several possible manifestations of this ”reservoir effect”. 3 The most widely discussed occurs in the vapor-liquid-solid (VLS) growth mode, when the NW elements transit from the vapor phase to the solid NW via a liquid catalyst particle sitting at the top of the NW. Several strategies have been developed to circumvent this type of reservoir effect. An obvious mean to reduce the amount of NW constituents stored in the droplet during growth is to use elements having low solubility in the catalyst. This is clearly interesting when the latter is based on an element not present in the NW, for instance a metal such as gold. However, even for the self-catalyzed VLS growth of III-V NWs (the catalyst is then based on a group-III NW element, the archetypical case being the Ga-based growth of GaAs NWs),


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this idea may be applied by forming heterostructures involving a switch (or a change of concentration) between group V elements. This is because group V elements such as As and P have low solubility in group III liquids and are present at low concentration even during NW growth: for instance, the concentration of As that mediates the Ga-catalyzed growth of GaAs NWs is only of the order of a percent. 4,5 In the present work, we consider the a priori trickier case of the self-catalyzed growth of III-V NWs with heterostructures formed by alternating group III elements. We treat specifically the case of the (Al,Ga)As system but we argue that our methods and conclusions should apply to other similar systems. Growing an AlAs insertion on top of a GaAs selfcatalyzed NW might seem very difficult: since one starts from a nearly-pure Ga droplet, Ga may be expected to be incorporated in the insertion for a very long time even though only Al is provided. Even if an (Al,Ga)As alloy insertion is aimed at, Al will remain a minority element in the droplet long after Ga has been turned off. Actually, things are not that bad and sharp interfaces may be obtained even in such a system. 6 As argued previously, the reason is that the incorporation of Al in solid (Al,Ga)As from the liquid is very effective: one needs only a very low Al concentration in the droplet to form Al-rich alloys. 6 Another difficulty for controlling axial heterostructures in NWs is that their growth rate, which determines their length, may not depend on a simple fashion on the external fluxes. A reason for this is the multiplicity of pathways available to NW elements (direct impingement on droplet, surface diffusion, re-emission, desorption...). Modeling is thus essential for the prediction of the dimensions of the heterostructure formed in given growth conditions. Reports on modeling the formation of axial heterostructures in NWs remain scarce. Up to now, simple schemes have been applied. A basic choice consists in assuming that the incorporation of the various species in the solid is proportional to their concentrations in the liquid; for NWs of ternary compound semiconductors, the rates of incorporation of the two pairs are taken as proportional to the products of the concentrations of the constituents of the pair. 7–9 Alternatively, the incorporation may be assumed to be proportional to the



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difference of chemical potential for the element (case of group IV semiconductors) 10 or for the pair 11 between liquid and solid. For the Au-catalyzed systems, the chemical potentials are calculated for the purpose of modeling due to lack of published thermodynamic functions. A limitation of these works is that they do not consider nucleation, which is widely believed to mediate the formation of each NW monolayer (ML). 12 In the present work, we investigate how these limitations may be overcome. On the basis of our detailed experiments on the growth of heterostructures in the self-catalyzed (Al,Ga)As system and of our first attempts at modeling their formation, 6 we develop two models for the formation of heterostructures. Namely, we compare an ”equilibrium model” to two variants of a nucleation-based model. The equilibrium model proves able to produce interface profiles that match very closely our experiments. However, we can justify it theoretically only on a qualitative basis. On the other hand, NWs obviously grow out of equilibrium. More precisely, most advanced growth modeling acknowledges that NWs develop via the successive nucleation and growth of monolayers at the top of the NW 12–15 and this has now been confirmed by direct observation. 16,17 It thus seems crucial to treat the formation of axial heterostructures in NWs in a nucleation framework, which has not been done so far and, more specifically, in a two-dimensional nucleation scheme. Our models both rely on a detailed thermodynamic assessment of the bulk system 18 and, as a consequence, do not involve any arbitrary parameter. In the next section, we show how this assessment can be used for our purpose.

Thermodynamics We consider the growth of the solid from the liquid phase in the (Al,Ga)As system at temperature T . The stoichiometric pseudo-binary Alx Ga1−x As solid is defined by its AlAs content x whereas the ternary liquid must be specified by two independent atomic concentrations, e.g. yAl and yAs for Al and As, with yGa = 1 − yAl − yAs .


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For growth to happen, the liquid must be supersaturated with respect to the solid, and quantification of this supersaturation is of paramount importance. 4,19,20 In the present case, liquid supersaturation is measured by two differences of chemical potential, corresponding to the two pairs that can be incorporated in the solid, namely:

∆µAl,As = µLAl + µLAs − µSAl-As

(1)

∆µGa,As = µLGa + µLAs − µSGa-As

(2)

where µLi (yAl , yAs , T ) is the chemical potential of atom i in the liquid and µSi-j (x, T ) that of the i-j pair in the solid (here, we omit the dependence on pressure). For the liquid, we have: ∂g L ∂g L − yAs ∂yAl ∂yAs L L ∂g ∂g = g L − yAl − yAs ∂yAl ∂yAs L ∂g ∂g L = g L − yAl + (1 − yAs ) ∂yAl ∂yAs

µLAl = g L + (1 − yAl )

(3)

µLGa

(4)

µLAs

(5)

where g L (yAl , yAs , T ) is the Gibbs free energy per atom, each partial derivative being taken L with all other parameters fixed. Introducing the excess Gibbs energy Em (the difference

between actual and ideal liquids), 21 we have:

pL pL pL L + yGa gGa + yAs gAs + Em + kB T (yAl lnyAl + yGa lnyGa + yAs lnyAs ) g L = yAl gAl

(6)

with gipL (T ) the atomic Gibbs energy of pure liquid i (i.e. its chemical potential). Hence:



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L

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L

pL L m m µLAl = gAl + Em + (1 − yAl ) ∂E − yAs ∂E + kB T lnyAl ∂yAl ∂yAs L

(7)

L

pL L m m = gGa + Em − yAl ∂E − yAs ∂E + kB T lnyGa ∂yAl ∂yAs

µLGa

L

(8)

L

pL L m m µLAs = gAs + Em − yAl ∂E + (1 − yAs ) ∂E + kB T lnyAs ∂yAl ∂yAs

(9)

These chemical potentials are then easily calculated for any value of yAl and yAs as follows. The chemical potentials of the pure liquids are given by Dinsdale (with respect to their atomic enthalpies in their conventional reference structure at T0 = 295.16 K). 19,22 In turn, the comprehensive thermodynamic assessment of the bulk Al-As-Ga system by Li et al. 18 yields the excess Gibbs energy as a Redlich-Kister polynomial, namely:

L Em

= yAl yGa

2 ∑

LkAl,Ga

(yAl − yGa ) + yAs yGa k

1 ∑

LkAs,Ga (yAs − yGa )k + yAl yAs L0Al,As

k=0

k=0

+yAl yAs yGa L0Al,As,Ga

(10)

where the L symbols represent temperature-dependent interaction coefficients, which are listed in Ref. 18 Note that this expression goes well beyond the expression for a regular liquid, which would only contain pair products of concentrations (summations reduced to k = 0 and no triple product). Let us now consider the solid. The general expressions of the chemical potentials are similar but simpler, since the solid is effectively a binary compound: S ∂Em + kB T lnx ∂x

(11)

S ∂Em + kB T ln (1 − x) ∂x

(12)

S µSAl-As = µAlAs Al-As + Em + (1 − x) S µSGa-As = µGaAs Ga-As + Em − x

GaAs where µAlAs Al-As and µGa-As are the chemical potentials of the Al-As and Ga-As pairs in pure S the excess Gibbs energy, in the expression of which AlAs and GaAs, respectively, and Em


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Li et al. only retain the zeroth-order Redlich-Kister term. 18 At variance with the liquid, the solid is thus a regular solution with interaction parameter Ω (equal to the Redlich-Kister coefficient) and:

2 µSAl-As = µAlAs Al-As + Ω (1 − x) + kB T lnx

(13)

2 µSGa-As = µGaAs Ga-As + Ωx + kB T ln (1 − x)

(14)

Finally, using Eqs. (7)-(10) and (13),(14) in Eqs. (1),(2) yields the differences of chemical potential for the two III-V pairs. The thermodynamic functions of the liquid derived by Li et al. involve a rather large set of parameters [namely 7; see Eq. (10)], which are clearly listed by the authors 18 and therefore not repeated here. For the regular solid, the single interaction parameter has the rather high value Ω = 2187 cal/mol. Note that there are considerable discrepancies in the literature as regards the interaction parameter of the pseudo-binary AlAs-GaAs solid solution, when treated as regular. Models that account only for the contribution of bond distortions yield values very close to zero because of the very low lattice mismatch between AlAs and GaAs. However, models that include the ’chemical’ contribution (electronic redistribution due to covalent binding) yield values of the same order as for strongly mismatched systems. The thermodynamic functions of Li et al. 18 pertain to bulk phases. The modification of the chemical potentials in the liquid due to the small size of the droplet could easily be included. However, it is extremely small in the case of self-catalyzed GaAs NWs 4 and hence also here since, as will be discussed in the next section, the droplet is nearly pure Ga and the volumes of the Al-As and Ga-As pairs are very close. We thus dispense with this modification.



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Experimental basis and governing equations Experimental basis We first recall some results about the self-catalyzed growth of III-V NWs, that will substantiate the hypotheses on which our models are based. (i) Self-catalyzed NWs of pure GaAs tend to reach a steady-state growth regime characterized by constant radius and growth rate. 23,24 The latter depends only on the As flux 25,26 and is fixed by the As concentration yAs in the liquid droplet. 4 yAs is of the order of 1% so that the droplet consists of nearly pure Ga. 4 (ii) Under a given As flux, the steady-state growth rate of self-catalyzed Alx Ga1−x As NWs is virtually independent of the Al concentration x in the solid. 27 (iii) Unless x is very close to 1, yAl

Comparison of Modeling Strategies for the Growth of Heterostructures

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