Polytype Attainability in IIIâV Semiconductor Nanowires - American

Article pubs.acs.org/crystal

Polytype Attainability in III−V Semiconductor Nanowires Jonas Johansson,*,† Zeila Zanolli,‡,§ and Kimberly A. Dick†,∥ †

Solid State Physics and NanoLund, Lund University, Box 118, S-22100 Lund, Sweden Peter Grünberg Institut (PGI) & Institute for Advanced Simulation (IAS), Forschungszentrum Jülich, D-52425 Jülich, Germany § European Theoretical Spectroscopy Facility (ETSF) ∥ Polymer & Materials Chemistry, Lund University, Box 124, S-22100 Lund, Sweden ‡

ABSTRACT: We propose a model that explains the phenomenon of polytypism in metal particle-seeded III−V semiconductor nanowires. The model is based on classical nucleation theory, utilizing the axial next-nearest-neighbor Ising (ANNNI) model to account for interlayer interaction up to the third nearest-neighboring layer. We investigate the limits of polytypism by varying the ANNNI interaction parameters. These calculations lead to attainability diagrams, which show the regions in interaction energy space where certain polytypes can be attained given that the supersaturation is precisely tuned. We calculate the values of the ANNNI interaction parameters for six common III−V materials from first principles by means of the projector-augmented wave method. We discuss our calculated values in view of previous results. Using these calculated values in our nucleation model, our analysis suggests that besides the commonly observed 3C (zinc blende) and 2H (wurtzite) polytypes the higher order polytypes 4H and 6H can also be attained, in agreement with experimental observations.

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INTRODUCTION One of the major challenges for successful commercial applications of III−V semiconductor nanowires in electronics and optoelectronics is the control of their crystal structure. In bulk, all III−V semiconductors, except nitrides, crystallize in the zinc blende structure (the 3C polytype in the Ramsdell notation,1 which we will use in this investigation). However, when these materials are grown as nanowires, they often show features of polytypism,2 which is a kind of polymorphism where the polymorphs differ in the stacking sequence of the constituting layers. Metal-particle seeded nanowires often form in a predominantly hexagonal phase, from a more or less pure wurtzite structure (2H polytype) to structures with inclusions of higher order polytypes, such as 4H and 6H. Frequently, the crystal structure is most easily characterized as a random mixture of cubic and hexagonal phases. The fabrication of nanoscale optoelectronic devices requires precise control of the crystal structure, since stacking faults and polytype mixing can be detrimental to electronic properties.3,4 Moreover, the possibility to controllably fabricate the same material in different polytypes enables bandgap engineering using a single material. In a few materials, such as InAs and GaAs, the crystal structure can be experimentally tuned between 3C and 2H,5,6 and crystal phase quantum dots of one polytype in the other have been demonstrated.7−9 The possibility of also fabricating 4H and 6H in III−V materials would further increase the possibilities of single material band gap engineering, which provides an additional degree of freedom for device fabrication and functionalization of nanowires. Even if these polytypes have occasionally been observed (see ref 10 and references therein), systematic and © 2015 American Chemical Society

reproducible fabrication of 4H, 6H, and other higher order polytypes has not been achieved in III−V materials. There have been several attempts to explain the 3C−2H polytypism in nanowires, from equilibrium calculations balancing surface energy and bulk cohesive energy11−13 to various approaches to nucleation calculations.14−17 Only a couple of attempts have been made to explain the occurrence of higher order polytypes. Those are the thermodynamic equilibrium calculations by Dubrovskii et al.12 and the combinatorial approaches by Johansson et al.10 Yet another approach was made by Panse et al.18 They used the axial nextnearest-neighbor Ising (ANNNI) model to discuss the tendency for polytypism in III−V semiconductors. This model has been successfully used to explain the polytypism in SiC.19 Panse et al. were the first to calculate the interlayer interaction parameters of the ANNNI model for III−V semiconductors. By plotting the interaction parameters in the ANNNI phase diagram, they could indeed confirm that the 3C polytype is very stable for III−V materials.18 In the current investigation, we take a nucleation theoretical approach, including the ANNNI model, to explain the occurrences and limits of polytypism in metal particle-seeded III−V nanowires, including polytypes up to 6H. We present updated first-principles calculations of the ANNNI interaction parameters and compare these to the previously calculated values.18,20 Inserting the calculated interaction parameters into the nucleation model offers an explanation to the observations Received: September 15, 2015 Revised: November 19, 2015 Published: November 24, 2015 371

DOI: 10.1021/acs.cgd.5b01339 Cryst. Growth Des. 2016, 16, 371−379

Crystal Growth & Design

Article

of the nucleus along the triple phase boundary and the curved edge in the interior of the seed−nanowire interface, the step energy is given by

of higher polytypes in nanowires of III−V materials. We also discuss the role of the metal particle as a limiting factor for polytype control.

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Γq = (2γq + πγ0)h

where γq is the edge step energy for q stacking, γ0 is the interior step energy, and h is the height of the bimolecular step. The chemical potential difference measures the supersaturation during growth and is generally expressed by a Δμ = kBT ln aeq (6)

THEORETICAL MODEL

In order to describe the formation of higher order polytypes, interaction between the stacked layers, which goes beyond nearestneighbor interactions, must be taken into account. For this purpose, we use the ANNNI model. A good introduction to the ANNNI model and its application to polytypism in SiC is given in ref 19. This model has recently also been applied to III−V nanowires in order to demonstrate that the stacked layers in a seemingly randomly stacked nanowire are correlated.21 In addition, a certain degree of interlayer correlations in nanowires was found by Schroth et al. using X-ray characterization and Markov chain modeling.22 In the ANNNI model, a stacking sequence is treated as a sequence of generalized spins and the total energy, NE, of a system consisting of N stacked layers can be written as N

NE = NE0 −

where a and aeq are the activities of the limiting specie in the seed particle during growth and at equilibrium. To account for the size dependence of the seed particle, a Gibbs−Thomson term, proportional to the inverse of the radius of the seed particle, is sometimes added to the right-hand side of eq 6.15,23 The interface energy, σq, entering eqs 3 and 4 is calculated using the ANNNI model. We consider interactions of the nucleating layer with up to the third nearest-neighboring layer, leading to an interface energy of the form

3

∑ ∑ Ji snsn+ i n=1 i=1

σq = σ0 − J1s1s2 − J2 s1s3 − J3s1s4

(1)

Figure 1. Schematic of the interlayer interactions.

(2)

where ω* is an effective attachment frequency of III−V atomic pairs to the critical nucleus, n is the concentration of the limiting specie (group III or group V atoms) in the seed particle, kB is Boltzmann’s constant, and T is temperature in kelvin. For our approximation, using semicircular 2D nuclei (see more details in refs 16 and 23), the Zeldovich factor24,25 is given by

Zq =

⎞ 1 s ⎛ Δμ ⎜ − σq⎟ ⎠ 2kBT Γq ⎝ s

To construct explicit expressions for the necessary σq, we need the stacking sequences represented by generalized spin sequences for the considered polytypes. These are

3/2

(3)

and the nucleation barrier is given by ΔGq* =

3C: ... ↑ ↑ ↑ ...

(8a)

2H: ... ↑ ↓ ...

(8b)

4H: ... ↑ ↑ ↓ ↓ ...

(8c)

6H: ... ↑ ↑ ↑ ↓ ↓ ↓ ...

(8d)

That is, 3C is represented by purely cubic (c) stacking. To account for interactions up to J3, we describe this as stacking of one cubic layer on top of two cubically stacked layers, c-on-cc, or ccc for short. The interface energy corresponding to this is denoted σccc. Three such stacking events are needed to form one unit of 3C (which consists of three layers). In a similar fashion, one unit of 2H results from two consecutive stacking events of the kind hexagonal on two hexagonal layers (h-on-hh or hhh with interface energy σhhh). One unit of the

Γ q2 2π(Δμ/s − σq)

(7)

where σ0 is the energy of the stacked layer if no interactions are present. J1, J2, and J3 are the interaction parameters accounting for the interaction between the stacked layer and its nearest neighbor, next nearest neighbor, and third nearest neighbor, respectively. The schematics are given in Figure 1. The generalized spin, sn, account for the stacked layer (n = 1), nearest neighbor (n = 2), next nearest neighbor (n = 3), and third nearest neighbor (n = 4).

where E0 is the energy per layer if there is no interaction between the layers, Ji is the interaction energy between the ith neighboring layers, and the generalized spin sm = ±1, depending on the stacking orientation of layer m on layer m − 1. It is common to include interaction parameters up to J3, and this is enough to describe polytypes up to 6H. We will use the ANNNI model to describe 3C, 2H, 4H, and 6H polytypism. The stacking sequences of these polytypes in the ⟨111⟩B growth direction, which is the most relevant one for III−V nanowires, are represented by ABC for 3C, AB for 2H, ABAC for 4H, and ABCACB for 6H. The generalized spin sm = +1 (also denoted ↑) if layer m on layer m − 1 is one of the following combinations: B-on-A, C-on-B, or A-on-C, that is, if the stacking is cubic (c). In the other three cases, the stacking is hexagonal (h), and sm = −1 (↓). Different polytypes represent different sequences of generalized spins and thus give different explicit total energy expressions when the respective generalized spins are applied to eq 1. If the total energies for the considered polytypes are obtained by, e.g,. first-principles calculations, then the respective total energy expressions can be used to compute the interaction parameters (the Ji’s). This has recently been done for III−V materials.18 Our approach is quite different; we use the ANNNI model to describe the interface energy contribution in our classical nucleation model for nanowire growth.16,23 The rate of nucleating a specific layer on top of a certain specified stacking sequence (resulting in stacking sequence q) is given by

Iq = ω*nZq exp(−ΔGq*/kBT )

(5)

(4)

In eqs 3 and 4, s denotes the area of a molecular site on the {111} interface, Γq is the step energy, Δμ is the chemical potential difference, and σq is the interface energy. For our geometry, with the straight edge 372



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polytype 4H is constructed by the four stackings: h-on-ch, c-on-hc, hon-ch, and c-on-hc. Thus, we need the interface energies σhch and σchc. Along the same lines, one unit of 6H is constructed by six stackings: hon-cc, c-on-hc, c-on-ch, h-on-cc, c-on-hc, and c-on-ch. For this, we need the interface energies σhcc, σchc, and σcch. In order to construct the stacking probabilities, we also need σchh and σhhc. By applying the sequences given in eqs 8 to eq 7, we can explicitly express the eight interface energies

σccc = − J1 − J2 − J3

(9a)

σhhh = J1 − J2 + J3

(9b)

σhch = J1 + J2 − J3

(9c)

σchc = − J1 + J2 + J3

(9d)

σhcc = J1 + J2 + J3

(9e)

σcch = − J1 − J2 + J3

(9f)

σchh = − J1 + J2 − J3

(9g)

σhhc = J1 − J2 − J3

(9h)

pP =

Icuv Icuv + Ihuv

(10)

phuv =

Ihuv Icuv + Ihuv

(11)

p(k) = pPk (1 − pP ),

2 ρ2H = (phhh )N /2

(12b)

2 2 N /4 ρ4H = 2(phch pchc )

(12c)

2 2 2 N /6 ρ6H = 3(phcc pchc pcch )

(12d)

(14)

∞

fP (M ) =

∑ p(k) = pPM

(15)

k=M

which we readily obtain by evaluating the geometric sum.

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CALCULATION OF ANNNI INTERACTION PARAMETERS

The computation of the ANNNI interaction parameters (J1, J2, and J3) for various technologically important III−V materials (GaSb, InSb, GaAs, InAs, GaP, and InP) has been performed from first principles using the ABINIT26 implementation of density functional theory. The advantage of this approach is that, given the ideal geometric structure of the various polytypes, their structural properties and total energies are computed without any arbitrary parameter. For the present calculations, the local density approximation (LDA) of the exchangecorrelation functional in the Perdew−Wang parametrization27 has been used. The electron−ion interaction has been modeled by the projector-augmented wave method (PAW).28 For Ga and In, the semicore d electrons were included among the valence states, while for the other elements, only the external s and p shells were included. The convergence of total energies and of ANNNI parameters with respect to the energy cutoff of the plane-wave basis has been tested for each individual compound since this parameter is material-dependent. We used 30 hartree for GaSb and InSb, 26 hartree for GaAs and InAs, 34 hartree for GaP, and 18 hartree for InP. Calculations have been performed for the hexagonal polytypes (2H, 4H,and 6H) in their primitive cell and for the zinc blende in the (nonprimitive) cell oriented along the [111] direction, as illustrated in Figure 2. Total energy convergence below 1 meV is obtained by sampling the Brillouin zone with a shifted 12 × 12 × N Monkhorst−Pack grid with N = 8, 6, 4, and 3 for 2H, 3C, 4H, and 6H, respectively. A metallic occupation of the electronic levels with electronic temperature smearing of 0.005 hartree has been used to account for the metallic character of some compounds (InAs, InSb, and InP) in LDA.29 Full optimization of cell geometry and atomic positions has been performed while simultaneously taking into account the symmetry of the system, which is F4̅3m (T2d) for the 3C phase and P63mc (C46v) for the hexagonal phases (2H, 4H, and 6H). The optimization of the internal-cell parameters, i.e., the deviations from ideal tetrahedral structures ε and δ in Figure 2, is included in this full structural optimization. Initial structures, i.e., unit cell and ideal atomic positions, are defined as in refs 18 and 20 with internal-cell parameters set to zero. For the relaxed structures, forces on atoms are smaller than 5 ×

where the two general indices u and v denote c or h and the nucleation rate is given by eq 2 and depends on the interface energies expressed in eqs 9. Note that the two probabilities given in eqs 10 and 11 are complementary to each other, pcuv + phuv = 1, meaning that the stacking on top of any sequence, uv, is either cubic or hexagonal. These probabilities are the building blocks to construct the polytype formation probabilities, which, according to the discussion immediately following eqs 8, we express as (12a)

k = 0, 1, 2, ...

Now, the fraction of at least a number of M (N layer thick) segments of polytype P in a row, that is, at least MN uninterrupted layers of polytype P, in a wire is given by

and

3 N /3 ρ3C = (pccc )

(13)

where P and Q indicates one of the polytypes (3C, 2H, 4H, and 6H) and the summation in the denominator runs over all four polytypes. The model for polytypism that we have introduced here is an approximation, as only four polytypes can be the fate of N stacked layers. In real experiments, the polytypes are not crystallographically pure but contain a certain amount of stacking defects, which might be considered more or less equally distributed among the polytypes. Given the polytype formation probabilities, it is possible to estimate the probability that a given length (a multiple of N layers) of a polytype will form. Since the nucleation of layers can approximately be described as a Poissonian process (see ref 16 and references therein), the formation of polytypes should also be Poissonian, meaning that the number of polytype formation events in nonoverlapping time intervals is statistically independent. Then, the distribution of polytype segments will follow the geometric probability distribution, according to which the probability of forming exactly k (N layers long) segments of polytype P before another polytype forms is given by

where we have put σ0 = 0, since this term is the energy of a noninteracting layer and does not contribute to the interface energy. For the step energy, eq 5, we use a simpler model. According to our previous work,16 the interior step energy, γ0, is the one for 3C. For the edge step energy, we do not use any extended interactions but differ only between cubic and hexagonal stacking. We set the edge step energy for a cubic layer γc = γ0 and let γh/γc < 1.10 The next step is to express the probabilities of the stackings described above. We do this using nucleation rates so that the probability, e.g., for c-on-cc is given by the nucleation rate for c-on-cc divided by the sum of the two nucleation rates: c-on-cc and h-on-cc. Explicitly, these probabilities are written as

pcuv =

ρP ∑Q ρQ

where N denotes the number of layers in the polytypic fragment. The factors 2 and 3 in eqs 12c and 12d are degeneracy factors accounting for the 2- and 3-fold degeneracy of these polytypes as compared to 2H and 3C.10 For the ease of comparison, all of the polytypic fragments have the same number of layers. We choose N = 6, corresponding to one unit of 6H, two units of 3C, three units of 2H, and one and a half units of 4H. The final polytype formation probabilities are given by the normalization of the expressions in eqs 12 373



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−1 erg/cm2, the probability for 4H is dominating in a small supersaturation interval. However, this probability decreases as J3 is decreased from 1 to −1 erg/cm3. At the same time, the probability for 6H is increasing. At J2 = J3 = −1 erg/cm2 all four polytypes are, in principle, attainable, even if it could be experimentally unfeasible to fabricate 4H material since its maximum probability is only about 0.4 and the supersaturation interval where it dominates is very small. At J2 = 0 and the highest values of J3, only 3C and 2H can form. However, as J3 is decreased to −1 erg/cm2, 6H is attainable with a significant formation probability of about 0.8, albeit in a quite narrow supersaturation interval. At J2 = 1 erg/cm2, only the 3C and 2H polytypes are dominating for all of the investigated values of J3. Next, with Figure 3 as a starting point, we set out to investigate the locations of the borders between the regions where the 4H and 6H polytypes start or cease to dominate. The resulting diagram shows the polytypes that are attainable as a function of supersaturation for a certain choice of J2 and J3, given that J1 is known. Such an attainability diagram is shown in Figure 4. Here, we see that in order for higher polytypes, such as 4H and 6H, to form J2 needs to be smaller than a certain J3dependent value. In order for all four polytypes to be attainable, both J2 and J3 need to be negative and J3 needs to be smaller than J2 by a certain amount (see Figure 4). In order to model realistic systems, the J1, J2, and J3 ANNNI parameters have been computed from first principles for a set of III−V semiconductors. The importance of performing a full structural optimization of the polytypes has been stressed in refs 18 and 20. However, in those works, the structural optimization has been performed using a three-step procedure, which leads to uncertainties on the internal-cell parameters for the 6H polytypes. In the present work, instead, optimization of cell geometry and atomic positions has been performed simultaneously, which results in higher accuracy. This directly affects the quality of the computed total energies and, hence, the ANNNI parameters, which are obtained from total energies via eqs 16. Our relaxed lattice constants, internal-cell parameters, total energies, and ANNNI parameters are collected in Tables 1−3. The computed lattice constants (Table 1) can be compared to experiment (room temperature) for the 3C polytype, which is the stable phase for the investigated bulk materials,30 and to the lattice constants measured in nanowires in the 2H and, when available, 4H phases.31−34 The usual underestimation of lattice constant provided by LDA gives lattice parameters that are smaller by Δa/a = 0.67% for GaSb, 0.37−0.44% for InSb, 0.78−0.82% for GaAs, 0.41−0.44% for InAs, 0.86−0.90% for GaP, and 0.48−0.52% for InP. These discrepancies are typical for III−V materials computed in LDA, and our lattice constants agree well with previously obtained ones (see, for example, refs 18 and 20). The main difference on the structural properties concerns, instead, the internal-cell parameters for the 6H polytypes. While the order of magnitude and sign of internal parameters (Table 2) for the 2H and 4H polytypes are the same as those in refs 18 and 20, this is not always the case for the 6H ones. The discrepancy can be attributed to the relaxation method employed in refs 18 and 20, which is an approximation with respect to the simultaneous structural optimization performed in the present work. In general, the difference in the total energy of each phase with respect to the 3C polytype (ΔE(2H), ΔE(4H), ΔE(6H); Table 3) presented in this work agrees well with refs 18 and 20. The respective values differ by less than 1 meV per cation−

Figure 2. Ball-and-stick models of the four investigated polytypes: 2H, 3C, 4H, and 6H. Group III elements (cations) are in orange, and group V elements (anions) are in green. The stacking of the cation− anion pairs, indicated with capital Latin letters, is along the [111] direction for the cubic phase and along the [0001] direction for the hexagonal phase. The cells are primitive for the hexagonal polytypes buts not for the zinc blende. The atomic positions are indicated by reduced coordinates and internal cell parameters (ε and δ). 10−4 eV/Å and maximum stress is smaller than 0.0027 GPa. In this framework, the total energy of each polytype has been obtained and the ANNNI parameters have been computed according to refs 18 and 20 as

J1 =

ΔE(2H ) ΔE(4H ) 3 + − ΔE(6H ) 2 2 4

(16a)

J2 = −

ΔE(2H ) ΔE(4H ) + 24 22

(16b)

J3 = −

ΔE(4H ) 3 + ΔE(6H ) 2 4

(16c)

where ΔE is the total energy difference of polytypes with respect to the 3C phase.

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RESULTS AND DISCUSSION We start by a general investigation of the Ji dependence of the model outlined in the previous section. We calculate the polytype formation probabilities according to eq 13 for a constant value of J1 = 10 erg/cm2 and let J2 and J3 take all combinations of the values −1, 0, and 1 erg/cm2. The value of J1 has the correct order of magnitude for III−V materials, and J2 and J3 are typically 1 to 2 orders of magnitude smaller.18,19 For the step energies, we chose the values that we previously used to model Au-seeded GaAs, γc = 119.3 erg/cm2 and γh/γc = 0.4.10 The temperature is 500 °C. In Figure 3, the polytype formation probabilities are plotted as functions of supersaturation per area, Δμ/s, for these nine combinations of J2 and J3. For all of the investigated values of the interaction parameters, the 3C and 2H polytypes are always dominating at low and high supersaturations, respectively. This is in agreement with experiments and previous theories.2 For J2 = 374



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Figure 3. Polytype formation probabilities for J1 = 10 erg/cm2, γc = 119.3 erg/cm2, and γh/γc = 0.4. The values of J2 and J3 are −1, 0, and 1 erg/cm2. All nine permutations are investigated.

Table 1. Relaxed Lattice Parameters in Å for the Various Polytypesa 2H

3C

4H

6H

compound

a

2c/p

a

2c/p

a

2c/p

a

2c/p

GaSb GaSb exp InSb InSb exp GaAs GaAs exp InAs InAs exp GaP GaP exp InP InP exp

4.2646

7.0376

7.0122

4.2754

7.0049

7.4923 7.5221 6.5181 6.5701 6.9962 7.0250 6.2811 6.3353 6.7685 6.8013

6.9900 7.0390 7.4534 7.4813 6.4762 6.5278 6.9664 6.9955 6.2404 6.2937 6.7436 6.7766

4.2726

4.5512 4.5712 3.9520 3.9845 4.2556 4.2742 3.8072 3.8419 4.1193 4.1423

4.2816 4.3105 4.5644 4.5813 3.9664 3.9975 4.2660 4.2836 3.8209 3.8541 4.1282 4.1498

4.5570 4.5753 3.9584 3.9900 4.2602 4.2780 3.8136

7.4726 7.5029 6.4968 6.5482 6.9813 7.0086 6.2597

4.5593

7.4663

3.9610

6.4895

4.2620

6.9764

3.8158

6.2533

4.1234

6.7554

4.1249

6.7515

a

For comparison, the experimental lattice parameters for the 3C (bulk; ref 30) and, when available, for nanowires in the 2H and 4H phases are reported. Measured lattice constants for InAs and InSb (2H and 4H) are taken from ref 31, InP (2H phase), from ref 32, GaP (2H phase), from ref 33, and GaAs (2H and 4H), from ref 34.

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Table 2. Computed Cell Internal Parameters (in Units of 10−4) 2H

4H

6H

compound

ε(1)

ε(1)

δ(2)

ε(2)

ε(1)

δ(2)

ε(2)

δ(3)

ε(3)

GaSb InSb GaAs InAs GaP InP

−12.272 −9.268 −11.095 −6.638 −12.392 −6.702

6.010 3.290 3.842 2.523 2.968 1.676

9.737 4.516 5.289 2.314 4.921 1.639

−1.524 −3.710 −4.132 −3.942 −4.234 −3.904

5.986 3.529 4.151 2.632 3.497 1.870

8.924 4.376 5.008 2.556 4.799 1.936

3.540 0.719 1.070 −0.136 0.704 −0.575

2.492 0.741 0.857 −0.109 0.627 −0.254

−1.611 −2.620 −2.880 −2.741 −3.011 −2.698

between 2H and 3C as the supersaturation is changed. In InSb, this was observed by Mandl et al.35 during MOVPE growth of In-seeded InSb nanowires. During the growth, the crystal structure evolved from 3C to 2H via a segment of 4H. At the same time, the concentration of Sb in the In−Sb seed particle increased, likely resulting in a supersaturation high enough for 4H and then for 2H.35 Similar observations of 4H segments between 3C and 2H at heterointerfaces in Sb-containing materials have been reported for gold-seeded GaAs1−xSbx/ GaAs36 and InAs1−xSbx/InAs10 heterostructure nanowires. Other observations of 4H in transition regions between 3C and 2H were made in gold-seeded GaAs/GaP nanowire heterostructures. The 4H polytype segments were observed in both the GaAs and GaP parts of the structure.37 The occurrence of 4H at heterostructure transitions and under varying degrees of supersaturation indicates that the polytype 4H is an intermediate structure between 3C and 2H, forming at intermediate supersaturation. Indeed, the hexagonality in 4H is 50%, and our calculations also show that it will form at intermediate supersaturation (if it is attainable at all) (see Figure 3). The reason that 4H is observed in a heterostructure transition region is that the supersaturation changes continuously with time, and at a certain time, the supersaturation passes through the required interval for 4H formation. Since this interval can be quite narrow, it would be difficult to set the relevant experimental parameters to grow 4H in a deterministic manner. However, mostly by chance, the correct supersaturation for 4H or 6H is sometimes attained. Thus, there are also a few examples of 4H and 6H in single material nanowires, which are not related to heterostructures. Both polytypes 4H10,38 and 6H10,39 have been observed in gold-seeded GaAs nanowires, and mixtures of these polytypes have been reported for selective-area-grown InAs nanowires.40,41 Extended segments of 4H were also reported in Cu-seeded InAs nanowires with a small diameter.42 We can conclude that there appears to be more experimental observations of 4H than of 6H in nanowires, which is in contrast to the theoretical predictions we presented above for the six different materials. The most significant reason for the discrepancy between experiment and theory is probably the values of the interlayer interaction parameters. The values used above are the values extracted from total energies of the respective materials in bulk. We expect the situation to be different for metal-particle-seeded nanowire growth. In this case, the relevant interaction parameters are those that account for the interaction between the growing layer and the three layers directly underneath. Since the growing layer is in direct contact with the metal particle, it is reasonable to expect that these uppermost interaction parameters, relevant for polytypism, are significantly perturbed compared to the values in Table 4 due to interactions with the metal.

Table 3. Total Energy (ΔE) of the 2H, 4H, and 6H Polytypes with Respect to the 3C Phase for Various III−V Compounds Computed from First Principlesa compound

ΔE(2H)

ΔE(4H)

ΔE(6H)

J1

J2

J3

GaSb InSb GaAs InAs GaP InP

24.8 21.6 23.1 17.6 18.0 11.4

11.2 10.5 10.5 8.6 7.3 5.1

7.2 6.9 6.8 5.7 4.6 3.3

12.61 10.82 11.72 8.84 9.19 5.79

−0.62 −0.15 −0.50 −0.13 −0.82 −0.30

−0.20 −0.04 −0.18 −0.03 −0.21 −0.08

a

The corresponding ANNNI interaction parameters (Ji) as obtained from eqs 16. Units are meV per cation−anion pair.

anion pair for all of the compounds except for 2H-GaP (1.9 meV), 4H-GaP (1.4 meV), and 6H-InSb (1.1 meV). The corresponding ANNNI parameters (reported in Table 3) also show good agreement. The larger differences are found for InAs (∼1 meV per cation−anion pair), GaP (∼0.8 meV per cation− anion pair), and, in general, for the J3 parameter (∼0.3−0.4 meV per cation−anion pair). As the next step, we use the calculated interaction parameters to evaluate the possibility of forming higher polytypes in nanowires of a few specific III−V materials. The interlayer interaction parameters together with step energies are listed in Table 4. By considering these interaction parameters in view of Table 4. Interaction Parameters J1, J2, and J3 Computed from First-Principles and Step Energies γ0 Estimated Using the Method by Voronkov47,48 GaSb InSb GaAs InAs GaP InP

J1/erg cm−2

J2/erg cm−2

J3/erg cm−2

γ0/erg cm−2

12.55 9.54 13.57 8.91 11.44 6.22

−0.62 −0.13 −0.58 −0.13 −1.02 −0.32

−0.20 −0.04 −0.21 −0.03 −0.26 −0.09

60.0 27.8 119.3 61.0 125.4 64.2

Figure 4, it is obvious that it should be possible to achieve polytypism beyond just 3C−2H in these materials. We verify this by calculating the polytype probability as a function of supersaturation using the materials’ parameters listed in Table 4. According to our material-specific calculations (example shown for GaP in Figure 5), besides 3C and 2H, 6H and 4H are attainable in GaAs, GaP (Figure 5), and InP, whereas the only higher polytype attainable in GaSb, InSb, and InAs is 6H. This does not fully match the experimental observations since 4H has also been observed in these materials, which is described below. Experimental observations of higher polytypes such as 4H and 6H in nanowires typically occur in a transition region 376



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In order to estimate how much the interaction parameters must be perturbed in order to change the polytypism properties, we take GaAs as an example and calculate the values of the maximum dominating probabilities for 4H and 6H as a function of the interaction parameter J1, keeping J2 and J3 constant (using the values for GaAs in Table 4). Using eq 15, we see that if the probability of forming at least 10 (six-layer) segments of 6H before any other polytype forms is higher than 0.5 then p6H needs to be larger than 0.93, and this corresponds to J1 ≈ 9 erg/cm2. Requiring an equally long uninterrupted sequence of 4H (15 four-layer segments) would correspond to J1 ≈ 3 erg/cm2. This is quite low, but J1 ≈ 9 erg/cm2 seems quite reasonable (compare values for other materials in Table 4). Thus, it seems plausible that the metal particle could influence the interaction parameters of the layers in the semiconductor close to the metal to such a degree that the polytypism features of the nanowire change. We note that the effect of independently changing the interaction parameters J2 and J3 can be qualitatively assessed by choosing a point of reference in Figure 4 and moving vertically and horizontally, respectively, in the neighborhood of this point. From the estimations above, we learn that the interaction parameters have to be significantly changed from their bulk values in order to favor the formation of higher order polytypes. In summary, for the formation of extended segments of higher order polytypes, two requirements have to be met: (i) a specific seed particle metal−semiconductor combination with favorable interaction parameters (see Figure 4) and (ii) supersaturation in a narrow interval, favoring either 4H or 6H (see Figure 3). We believe that the occurrence of 4H, often observed close to the heterointerface during growth of axial heterostructures, is an effect of requirement (ii) being fulfilled for a very short time only, that is, when the varied supersaturation passes through the interval where 4H or 6H dominates. The extended

Figure 5. Polytype formation probability for GaP as a function of supersaturation.

segments of 4H observed in Cu-seeded InAs nanowires42 could be an effect of requirement (i) being fulfilled to a higher extent than in the above case. In this case, the 4H probability peak is high and the supersaturation interval where 4H dominates can also be broader than in the more common Au-seeded heterostructure case, which makes this growth slightly less sensitive to the exact value of the supersaturation, even if the growth parameter window still is small. We conclude this section by discussing some properties known to affect the polytype selection in nanowires and relate them to our model. As can be seen in eq 4, the nucleation barrier depends on the step energy, Γq, the chemical potential difference, Δμ, and the interface energy term, σq. It is wellknown that the smaller step (or surface) energies of 2H, as compared to those of 3C, in combination with a sufficiently high Δμ favors the formation of 2H over 3C.14−16 The wetting angle of the seed particle during growth somewhat complicates this picture. During nucleation of a new layer, there is a small change in the seed particle surface area, which is proportional to the wetting angle.15 As an effect of this, the surface or step energy term, Γq, will depend on the wetting angle.15 In ref 43, it is argued that a smaller wetting angle increases the probability for twinning, in other words, forming hexagonal layers in 3C. A larger wetting angle, on the other hand, is reported to favor nucleation in the center of the growth interface, leading to 3C without planar twin defects.43 Moreover, for thermodynamic stability of vapor−liquid−solid growth of nanowires, a stability criterion involving the wetting angle must be fulfilled.44,45 We have not considered the wetting angle dependence in the current investigation, which focuses on the role of the interface energy. However, for more quantitative calculations, a wetting angle dependence in Γq could easily be incorporated. By changing the metal−semiconductor materials system, both Γq and σq would change. The change in σq would be due to the change in the interaction parameters, J1, J2, and J3, close to the growth interface. A reversed polarity of the nanowire, that is, growth in ⟨111⟩A instead of the much more common ⟨111⟩B,43,46 should also lead to altered J1, J2, and J3 close to the

Figure 4. Polytype attainability diagram for J1 = 10 erg/cm2, γc = 119.3 erg/cm2, and γh/γc = 0.4. The areas indicate regions where the labeled polytypes can be attained, meaning that they each dominate in a certain supersaturation interval. Note that 2H and 3C are always attainable for the investigated parameters. 377



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growth interface. This is because of the different interactions of A-polar layers with the metal compared to those of B-polar layers with the metal. Here, it is interesting to note that in the traditional equilibrium thermodynamics use of the ANNNI model18,19 the total energies of the different polytypes do not depend on polarity. This is because when no surfaces or interfaces are considered, A-polar material is just B-polar material upside down. In addition to the interlayer correlations influencing the interface energies, there could be similar correlations affecting the edge step energies,21 which we have not considered in this investigation due to the lack of data. Such correlations could either strengthen or weaken the tendency for higher order polytype attainability.

CONCLUSIONS We have proposed a model that explains the often observed polytypism in metal-particle-seeded III−V nanowires. The model is based on classical nucleation theory but incorporates ANNNI with three interlayer interaction parameters as a means to calculate the interface energies relevant for the stacking of layers in polytypes 3C, 6H, 4H, and 2H. The polytype formation probabilities are estimated by the ratios of nucleation rates. By keeping one interaction parameter fixed and varying the other two, we map out the regions in interaction energy space where different ranges of polytypes are attainable as a function of supersaturation. Such attainability diagrams outline the polytype that one could expect (or not expect) in a given growth experiment, provided that the relevant interlayer interaction parameters are known. On a more concrete level, the model explains the formation of the 3C polytype (zinc blende) and the 2H polytype (wurtzite) at low and high supersaturations, respectively. In addition to this, it explains the occurrence of the higher order polytypes, 6H and 4H, which can form at intermediate supersaturation. We present updated first-principles calculations of the ANNNI interlayer interaction parameters of the materials GaSb, InSb, GaAs, InAs, GaP, and InP. By inserting these in our model, we conclude that, besides 3C and 2H, the higher polytypes 6H and 4H can, indeed, also form in these materials, in agreement with experimental observations. AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +46 46 2221472. Fax: +46 46 2223637. Notes

The authors declare no competing financial interest.

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REFERENCES

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ACKNOWLEDGMENTS

This work was funded by NanoLund (the Center for Nanoscience at Lund University). In addition, we acknowledge funding from the Swedish Research Council (VR), the Knut and Alice Wallenberg Foundation (KAW), and the EU ITN NanoEmbrace (grant agreement no. 316751). Z. Zanolli acknowledges EU support under a Marie-Curie fellowship (PIEF-Ga-2011-300036) and computational resources from the PRACE-3IP project (FP7 RI-312763) and the JARA-HPC project (no. 8215) at Forschungszentrum Jülich. 378



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