# Recent Progress in Computational Materials Science for Semiconductor Epitaxial Growth

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## Abstract

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## 1. Introduction

## 2. Computational Methods

_{gas}[5,6]. The adsorption-desorption behavior of In atom and As molecule on these surfaces can be determined by comparing μ

_{gas}with the adsorption energy E

_{ad}. An impinging atom (molecule) can adsorb on the surface if the free energy of the atom (molecule) in the gas phase is larger than its adsorption energy. In contrast, an impinging atom (molecule) desorbs if its gas-phase free energy is smaller than the adsorption energy. The μ

_{gas}can be computed using quantum statistical mechanics as functions of temperature and pressure. The adsorption energy can be obtained using ab initio calculations. Exemplifying In atom and As

_{4}molecule, the chemical potential μ

_{gas}of ideal gas is given by the following Equations:

_{In}= −k

_{B}T(k

_{B}T/p

_{In}× g × ζ

_{trans}),

_{As4}= −k

_{B}T(k

_{B}T/p

_{As4}× g × ζ

_{trans}× ζ

_{rot}× ζ

_{vib}).

_{B}is the Boltzmann’s constant, T the gas temperature, g the degree of degeneracy of electron energy level, p the beam equivalent pressure (BEP) of In atom or As molecule (As

_{4}), and ζ

_{trans}, ζ

_{rot}, and ζ

_{vib}are the partition functions for translational, rotational, and vibrational motions, respectively. In general, both μ

_{In}and μ

_{As4}decrease with temperature and increases with BEP. The adsorption energy E

_{ad}is obtained by

_{ad}= E

_{total}− E

_{substrate}− E

_{atom},

_{total}is the total energy of the surface with adatoms, E

_{substrate}the total energy of the surface without adatoms, and E

_{atom}is the total energy of isolated atoms. We note that the Gibbs free energy of formation vibrational contribution is very small compared with the energy difference between a given structure and the ideal surface. The gas-phase entropy difference is also considerably larger than the surface entropy change, when the temperature or pressure is varied. Therefore, only the entropic effects of the gas phase are considered throughout our theoretical approach. Using the chemical potential and the adsorption energy, the adsorption and desorption behaviors on the surfaces under growth conditions are obtained as functions of T and p, i.e., the structure corresponding to adsorbed surface is favorable when E

_{ad}is less than μ

_{gas}, whereas desorbed surface is stabilized when μ

_{gas}is less than E

_{ad}. Based on these results, we obtain adsorption-desorption boundaries for In and As and surface phase diagrams of the InAs WL.

_{gas}of Ga atom using Equation (1) decreases with temperature and increases with BEP. By comparing the adsorption energy of Ga atom (−3.3 eV) using Equation (3), the μ

_{gas}of Ga atom becomes lower than the adsorption energy when temperatures is higher than 1000 K for Ga-BEP at 1.0 × 10

^{−5}Torr. This suggests that the critical temperature for Ga adsorption is ~1000 K for Ga-BEP at 1.0 × 10

^{−5}Torr. The surface phase diagram for Ga adsorption obtained by the comparison of μ

_{gas}with E

_{ad}thus suggest that the Ga-droplet appear under low temperature and high Ga-BEP conditions, while GaAs-(4 × 2)β2 surface is stabilized under high temperature and low Ga-BEP conditions. The calculated results are consistent with the observation of Ga-droplet under ~900 K during the MBE growth of GaAs under Ga-rich conditions and Ga desorption above ~970 K after turning off the Ga flux, suggesting that ab initio-based approach that incorporates the free energy of the gas phase is feasible for investigating surface structures, while previous calculations [2,3,4] are unable to incorporate temperature and pressure.

^{−3}Ry/Å. The valence wave functions are expanded by the plane-wave basis set with a cutoff energy of 16 Ry. We take the model with slab geometries of six atomic layers for the InAs(111)A and InP(111)A and eight atomic layers for the InAs(001), WZ-InP($1\overline{1}00$) and WZ-InP($11\overline{2}0$) with artificial H atoms [13] and a vacuum region equivalent to nine atomic layer thickness. The k-point sampling corresponding to 36 points in the irreducible part of the 1 × 1 surface Brillouin zone, which provides sufficient accuracy in the total energies, is employed. The computations are carried out using Tokyo Ab initio Program Package [12].

_{2D-coherent}(h) = γ + Mε

_{0}

^{2}h/2,

_{2D-MD}(h) = γ + E

_{d}/l

_{0}− E

_{d}

^{2}/2(Mε

_{0}

^{2}l

_{0}

^{2}h),

_{2D-SFT}(h) = γ + Mε

_{0}

^{2}h/2 + E

_{SFT}/A

_{unit},

_{SK-coh}(h) = γ(1 + β) + M(1 − α)ε

_{0}

^{2}h/2,

_{0}the intrinsic strain (=0.072) [14], h the layer thickness, E

_{d}the dislocation formation energy, l

_{0}the average dislocation spacing (=58.76 (Å) [14]), E

_{SFT}the SFT formation energy, A

_{unit}the area of 14 × 14 planar unit cell used in the system energy calculations for the 2D-MD with l

_{0}and the 2D-SFT, and β and −α are the effective energy increase in surface energy of the epitaxial layer and the effective decrease in strain energy due to SK-island formation.

_{d}, and E

_{SFT}, we employ a simple formula for estimating system energy E for computational models such as the 2D-coherent, the 2D-MD, and the 2D-SFT as follows.

_{0}+ ∆E

_{SF},

_{0}= 1/2 × ∑

_{i,j}V

_{ij},

_{ij}= Aexp[−β(r

_{ij}− R

_{i})

^{γ}][exp(−θr

_{ij}) − B

_{0}exp(−λr

_{ij})G(η)/Z

_{i}

^{α}],

_{SF}= K [3/2 × (1 − f

_{i}) × Z

_{b}

^{2}/r

_{bb}− f

_{i}× Z

_{i}

^{2}/r

_{ii}].

_{0}is the cohesive energy estimated by Kohr–Das Sarma-type empirical interatomic potential V

_{ij}within the second-neighbor interactions [15]. The potential parameters A, β, R

_{i}, γ, θ, B

_{0}, λ, η, and α are determined by reproducing equilibrium interatomic bond lengths, elastic stiffness for zinc blende structure, and relative stability among zinc blende, rocksalt, and CsCl structures [16]. Stacking-fault energy ∆E

_{SF}is described as the summation of electrostatic energies consisting of repulsive interaction between covalent bond charges Z

_{b}(=−2) and attractive interaction between ionic charges Z

_{i}(=±3 for III–V compound semiconductors) depending on ionicity f

_{i}(=0.298 for InAs) [17]. The value of coefficient K is determined to be 8.7 (meV·Å) by reproducing energy difference 25.3 (meV/atom) between diamond and hexagonal structures for C with f

_{i}= 0 obtained by ab initio calculations. In the system energy calculations, the lattice parameter a of InAs is fixed to be that of GaAs(111) for the 2D-coherent and the 2D-SFT or relaxed value of a for the 2D-MD, and the lattice parameter c and the atomic positions are varied to minimize the system energy in a 14 × 14 planar unit cell with increase of layer thickness h.

_{ad}(x) = exp{−Δμ(x)/k

_{B}T}/[1 + exp{−Δμ(x)/k

_{B}T}],

_{diff}(x→x’) = Rexp{ΔE(x→x’)/k

_{B}T},

_{de}(x) = Rexp[−{E

_{de}(x) − Δμ(x)}/k

_{B}T],

_{ad}(x) and μ

_{gas}at adsorption site x, R is diffusion prefactor taken to be k

_{B}T/πћ [19], ΔE(x→x’) is the local migration barrier involving the adatom hopping from site x to x’ and E

_{de}(x) is desorption energy at site x. The probability for surmounting the activation energy is reduced (or enhanced) by a weighting function of exp{Δμ(x)/k

_{B}T}, which corresponds to the local-thermal equilibrium desorption probability. In the simulation procedure, we adopt extensive surfaces as growth substrates which are 30,000 × 30,000 times as large as the surface unit cells. The atoms and molecules with pressure p are supplied from gas phase to these surfaces at an interval of Δt

_{gas}per surface area that is derived from Maxwell’s law of velocity distribution given by

_{gas}= (2mk

_{B}T/p)

^{1/2}

_{diff}(x→x

_{i}) = P

_{diff}(x→x

_{i})/{Σ

_{j}P

_{diff}(x→x

_{j}) + P

_{de}(x)},

_{de}(x→x

_{i}) = P

_{de}(x→x

_{i})/{Σ

_{j}P

_{diff}(x→x

_{j}) + P

_{de}(x)},

_{j}P

_{diff}(x→x

_{j}) + P

_{de}(x)},

_{diff}(x→x

_{i}) is the relative diffusion probability from site x to another site x

_{i}and K

_{de}(x) is the relative desorption probability at site x. The sum is carried out all over the neighboring site x

_{j}. Events for adatoms are taken place with a time increment Δt expressed in Equation (18).

## 3. Hetero-Epitaxial Growth of InAs on GaAs

#### 3.1. Surface Reconstructions on InAs(111)A WL

_{In}), As-adatom, and As-trimer (T

_{As}) WL surface structures considered in this study. Figure 2 shows the calculated surface phase diagrams for (a) InAs(111)A WL surface (InAs(111)-WL) and (b) fully relaxed (FR) surface without interface structure of InAs/GaAs (InAs(111)-FR) as functions of temperature T and BEP of As

_{4}molecules p

_{As4}. Here, we assume that the BEP of In atom p

_{In}in gas phase is p

_{In}~10

^{−7}Torr. The (2 × 2) with T

_{As}is stabilized only at low temperatures, while the (2 × 2) with V

_{In}appears at high temperatures. The stable temperature ranges for the (2 × 2) with V

_{In}(beyond 550–600 K) on the InAs(111)-WL are consistent with MBE growth temperature range of 723–773 K (black area) at which the (2 × 2) with V

_{In}is observed experimentally [20,35,36]. It should be noted that the InAs(111)-WL with As-adatom surface newly appears between stable regions of the (2 × 2) with V

_{In}and the (2 × 2) with T

_{As}, where the As adatom stably resides in the interstitial site bonding with three substrate In atoms. The large decrease in the charge density is found at surface As atoms on the In-vacancy surface [31]. This is because the electronegativity of In on the surface is smaller than that of Ga at the interface [37]. This results in breaking the electron counting model (ECM) [38,39] to destabilize the In-vacancy surface. The As-adatom surface makes up this deficiency in charge density, where the charge density around the As adatom is transferred to the substrate In atom to strengthen the interatomic bond between the In and As atoms. Therefore, the As-adatom surface appears as a stable phase between V

_{In}and T

_{As}on the InAs(111)-WL, while the InAs(111)-FR without interface Ga does not favor the As-adatom surface.

#### 3.2. Growth Process on InAs(111)A WL

_{In}of (a) the InAs(111)-WL and (b) the InAs(111)-FR as functions of T and p

_{In}. These figures reveal that In adsorption on the In-vacancy surface does not occur without simultaneous As adsorption under conventional growth temperatures in the range of 723–773 K (black area). This is because In adsorption energies on the In vacancy surface dramatically decrease after As adsorption such as −1.76 eV without As to −4.17 eV with As on the InAs(111)-FR and −1.46 eV to −3.27 eV for the InAs(111)-WL. These findings thus imply that In adsorption is promoted only when As atoms are adsorbed on the InAs(111)A-(2 × 2) surface and the InAs growth proceeds with the adsorption of As atoms. This reveals that the self-surfactant effect is crucial for the growth processes similarly to the GaAs growth [40,41]. Moreover, it is found that relative stability among the surface lattice sites and adsorption energies for In adatom do not strongly depend on the layer thickness [31]. Therefore, it is expected that the strain in the WL does not significantly affect the adsorption-desorption transition curve and the kinetic behavior of In adatom. Considering the fact that the misfit dislocation is formed at 4–7 ML of InAs [23,24], these results suggest that strain accumulated in the WL is not significant for the hetero-epitaxial growth on the InAs(111)A to keep 2D growth at the initial growth stage.

#### 3.3. Strain Relaxation on InAs(111)A WL

_{MD}between the 2D-coherent and the 2D-MD as a function of layer thickness h [42]. Here, the MD core with five- and seven-member rings (5/7 core) are inserted at the interface between InAs and GaAs(111), as shown in Figure 4a. The 5/7 core is often found in transmission electron microscopy (TEM) observations [43], and is recognized to be the stable core structure in the MD formation energy calculations for compound semiconductors [44,45,46]. This reveals that the ∆E

_{MD}changes its sign from positive to negative at 7 ≤ h ≤ 8 ML, where the strain in InAs thin layers is relaxed to stabilize the 2D-MD. The critical layer thickness 7 ≤ h ≤ 8 ML for the MD generation agrees well with its value of h about 7 ML estimated from People-Bean’s formula to minimize energy in the entire crystal at thermodynamic equilibrium [47]. Using the calculated results shown in Figure 4a, the parameter values used in Equations (4) and (5) are determined to be M = 2.63 × 10

^{10}(N/m

^{2}) and E

_{d}= 0.675 (eV/Å).

_{SFT}between the 2D-coherent and the 2D-SFT as a function of layer thickness is shown in Figure 5 [42] along with the schematic of the SFT consisting of the face with stacking-fault and the ridge corresponding to stair-rod dislocation along the (110) direction, similarly to the SFT in Si [48]. The ∆E

_{SFT}with negative at h ≥ 4 ML suggests that the SFT formation acts as a strain relaxation mechanism near the surface as well as the MD formation at the interface in InAs/GaAs(111)A system. The ∆E

_{SFT}results from the competition between energy profit in the face region and energy deficits in the face and the ridge regions. The face region dramatically decreases the system energy due to strain relaxation, inducing upward displacements of atoms in the SFT that overwhelm the energy deficit due to the stacking-fault formation. In the ridge region, however, the stair-rod dislocation increases system energy due to its energetically unfavorable dimers along the ridge line, shown in Figure 5b. The calculated results shown in Figure 5b approximately determine the parameter value of E

_{SFT}= 0.014h − 0.0011h

^{2}(eV) as a function of layer thickness h.

#### 3.4. Hetero-Epitaxial Growth of InAs on GaAs(111)A

_{0}/(1 − h

_{c}

^{MD}/h) approaching the average dislocation spacing l

_{0}(=58.76 Å) with increase of h, where h

_{c}

^{MD}is estimated by E

_{d}/(Mε

_{0}

^{2}l

_{0}) [21]. Figure 6 implies that the InAs growth on the GaAs(111)A proceeds along the lower energy path from the 2D-coherent (h ≤ 4 ML) to the 2D-MD (h ≥ 7ML) via the 2D-SFT (4 ML ≤ h ≤ 7 ML). This is consistent with STM observations, where faulted triangle domains containing stacking-faults appear with the MD network at 5 ML on the InAs/GaAs(111) [21]. Furthermore, a similar process was found in molecular dynamics simulations for (111)-oriented heteroepitaxial Al films forming a local disorder zone like the SFT near surface layers followed by the MD nucleation [49]. Experimental results, however, indicate that strain is gradually relieved beyond 1–2 ML, contradicting our calculated results with 4 ML [21,22]. This discrepancy can be interpreted by considering the fact that our computational model is not optimized for 2D-SFT. The 14 × 14 unit cell used in this study is not set for the SFT, but for the MD with geometrically optimized dislocation spacing. If the SFT spacing is optimized, the ∆E

_{SFT}becomes lower to give smaller layer thickness for the SFT formation. Moreover, employing the MD consisting of faulted and unfaulted domains observed by STM [21], further strain relaxation may occur at smaller layer thickness [23]. Consequently, our calculated results suggest that the first strain relaxation occurs near the surface due to the SFT formation before the MD formation at the interface. Assuming γ = 42 (meV/Å

^{2}) for the InAs(111)A, β = 0.084, and α = 0.748 extracted from the previously reported results for the InAs/GaAs(001) [50], the calculated free energy difference ∆F between the 2D-coherent and the SK-coherent is also shown in Figure 6. Although the ∆F for the SK-coherent becomes negative at 5 ML ≤ h ≤ 6 ML that is energetically competitive with the 2D-MD, the SK-coherent does not appear due to the 2D-SFT preceding the SK-coherent.

#### 3.5. Surface Reconstructions on InAs(001) WL

_{0.375}Ga

_{0.625}As interface reasonably appears after 0.625 ML InAs deposited on the GaAs(001)-c(4 × 4)α. This is consistent with experimental findings where 2/3 ML InAs deposition creates the (2 × 3) surface with In

_{1/3}Ga

_{2/3}As [26,51,52,53]. Further deposition with 0.708 ML InAs realizes the (2 × 4) surface that is also consistent with the RHEED observations where the (2 × 4) appears supplying ~1.3–1.4 ML InAs on the GaAs(001)-c(4 × 4)α [26]. Here it should be noted that the desorption of 0.375 ML As is indispensable to realize the (2 × 4) surface during the structural change from the (2 × 3) to the (2 × 4). Figure 8a shows the calculated surface phase diagram for the InAs(001)-(2 × 3) and (n × 3) WL surfaces (n = 4, 6, and 8) as functions of T and p

_{As4}. Here the (2 × 3) surface is fully covered by As-dimers on the surface while one As-dimer is missing every n As-dimers on the (n × 3) surfaces to approach satisfying the ECM at n = 8. This implies that the (2 × 3) surface is unstable at growth conditions, since desorption energy of As-dimer on the (2 × 3) surface is very small such as 1.12 eV to easily desorb from the WL surface to change its structure from the (2 × 3) to the (8 × 3) even at low temperatures. Figure 8b shows the calculated surface phase diagram for the InAs(001)-(2 × 4) WL surfaces as functions of T and p

_{As4}. The phase boundary between the (001)-(2 × 4)α2 and the (001)-(2 × 4)β2 is ranging from 470 to 600 K. The stable temperature range of these structures is consistent with experimental conditions during MBE growth [52,54]. This reveals that (2 × 4)α2 is stable at the conventional growth conditions such as T~700–750 K and p

_{As4}~10

^{−7}–10

^{−6}Torr.

#### 3.6. Growth Process on InAs(001) WL

_{In}. This indicates that In adsorption does not occur on the (2 × 3) but is allowed on the (4 × 3), the (6 × 3), and the (8 × 3) at growth conditions. This is because the adsorption energy for In atom is very large such as −1.68 eV on the (2 × 3) in contrast with −3.87 eV on the (4 × 3), −3.66 eV on the (6 × 3), and −3.33 eV on the (8 × 3). This is because the stable adsorption site of In on the (2 × 3) is around the center position between upper As-dimer and lower As-dimer quite different from the missing As-dimer site on the (n × 3) as shown in Figure 9a. Figure 9b shows the calculated adsorption-desorption boundaries for As dimer on the stable In-adsorbed InAs(001)-(n × 3) WL surfaces as functions of T and p

_{As4}. This implies that In adsorption does not induce As desorption, indispensable for InAs growth to change the surface structure from the (2 × 3) to the (2 × 4)α2, on the (4 × 3) and (6 × 3). Although As dimer desorption can be found on the (8 × 3) shown in Figure 9b schematically, it should be noted that newly appeared missing dimer is quickly occupied by In adatom to form the surface structure equivalent to the In-adsorbed (4 × 3) also schematically shown in Figure 9b. Moreover, it is found that further In adsorption also no longer occurs to prevent InAs growth on these (n × 3) WL surfaces.

_{(4×3)}= 0.70 and f

_{(6×3)}= 0.30 can be favorably compared with f

_{(4×3)}= 0.83 and f

_{(6×3)}= 0.17 estimated by directly counting the number of the units from Figure 2 in Ref. 27. Therefore, the (4 × 3) with three In-As dimers is most likely to appear among the (n × 3) WL surfaces during MBE growth. In order to realize the (2 × 4) α2 on the (4 × 3), further 0.5 ML In adsorption is necessary. It should be noted, however, In atoms are no longer incorporated on the (4 × 3) unless the strain is relaxed similar to the (2 × 4) α2 [33]. These results suggest that strain relaxation might occur at InAs coverage less than about 0.9 ML, contradicting well-known SK growth mode, to continue InAs growth.

#### 3.7. Strain Relaxation on InAs(001) WL

_{MD}between the energies for the 2D-coherent and the 2D-MD with 5/7-core changes its sign from positive to negative at h~1.3 ML, where the strain in InAs thin layers is relaxed to stabilize the 2D-MD with 5/7-core [55]. Using the calculated results shown in Figure 11b, the parameter values used in the free energy formula are determined to be M = 5.72 × 10

^{11}(N/m

^{2}) and E

_{d}= 1.51 (eV/Å). These values easily lead the layer thickness for the first MD generation h

_{c}

^{2D−MD}using the relationship of h

_{c}

^{2D−MD}= E

_{d}/(Mε

_{0}

^{2}l

_{0}) [14]. The calculated results suggest that the first MD formation occurs at 0.5 ML and the MD spacing l

^{2D−MD}gradually decreases with increase of the layer thickness according to l

^{2D−MD}= l

_{0}/(1 − h

_{c}

^{2D−MD}/h) to effectively relax the strain accumulated in the InAs layers. Therefore, the MD formation is one of possible mechanisms of eliminating lattice strain to proceed the growth on the InAs(001)-(n × 3) WL surfaces.

#### 3.8. Hetero-Epitaxial Growth of InAs on GaAs(001)

^{2}) for the InAs(001), β = 0.076, and α = 0.188 extracted from the previously reported results for the InAs/GaAs(001) [50], Figure 12 shows the calculated free energy differences ∆F between the 2D-coherent and various growth modes such as the 2D-MD and the SK-coherent as a function of layer thickness h. It is found that the ∆F for the 2D-MD becomes negative at 0.46 ML, while negative ∆F appears at 0.75 ML for the SK-coherent. This suggests that the initial 2D coherent growth mode changes into the 2D growth mode with MD formation. Therefore, strain relaxation occurs at early growth stage before the SK island formation. This is not consistent with well-known QD formation mechanism due to the SK island formation but is consistent with the fact that strain accumulated in InAs(001)-(n × 3) WL should be relaxed at InAs coverage less than about 0.9 ML to continue InAs growth as pointed out in Section 3.6. Although further careful investigations incorporating surface reconstructions as a submonolayer phenomenon are necessary for confirming the strain relaxation for the InAs/GaAs(001), the calculated results for the InAs/GaAs systems suggest that the strain relaxation due to the MD formation to produce the 2D growth is competitive with the SK-island formation inducing the 3D growth during MBE growth.

#### 3.9. Quantum Dot Formation of InAs

_{d}/l

_{0}). Using the MD formation energy E

_{d}= 0.675 (eV/Å) for InAs/GaAs(111)A and 1.51 (eV/Å) for InAs/GaAs(001) obtained by our empirical potential calculations, Figure 13 shows the growth mode boundaries for the InAs/GaAs system depending growth orientations as functions of β/α and surface energy γ. This reveals that the stable region of the 2D-MD in the InAs/GaAs(111)A is larger than that in the InAs/GaAs(001) because of its smaller MD formation energy. Furthermore, the larger the β/α and γ, the more favourable the 2D-MD. This is because the energy increase in surface energy γβ tends to overwhelm the decrease in strain energy −Mαε

_{0}

^{2}h/2 due to SK-island formation. According to these facts, the surface energy γ in addition to E

_{d}is a crucial factor for determining the growth mode, since the change in γ is related not only to γ itself but also to β/α through γβ. Figure 13 also includes the boundary denoted by dotted line for the InAs/GaAs(110) with E

_{d}= 0.96 (eV/Å) obtained by ab initio calculations [14]. The InAs/GaAs(110) system has been extensively examined using RHEED, TEM, and STM for a wide range of film thickness where the growth follows a layer-by-layer mode irrespective of thickness and strain relaxation occurs solely by the formation of MD [20,56]. However, self-assembled InAs QD has been realized on GaAs(110) using MBE with buffer layer insertion [57] and MOVPE with optimizing growth temperature [58], where the strain reduction is considered to be a crucial origin of the QD formation. This novel QD formation can be interpreted by considering the change in γ. It is known that the strain reduction lowers the surface energy by 10–20 (meV/Å

^{2}) [50]. Assuming β/α = 0.2 and γ = 50 (meV/Å

^{2}) (denoted by closed diamond in Figure 13), the 2D-MD is favourable without strain relaxation while decrease in γ due to strain relaxation with the buffer-layer insertion easily changes favourable growth mode from the 2D-MD to the SK-coherent found in the (110). Although the dependence of β/α and γ on strain relaxation should be rigorously investigated using ab initio calculations, Figure 13 gives an insight on the growth modes among different oriented InAs/GaAs systems such that (001), (110), and (111)A favours the 3D growth, the 2D and the 3D growths depending on the strain relaxation, and the 2D growth, respectively. Consequently, the QD formation on the InAs/GaAs can be qualitatively interpreted by considering γ and E

_{d}.

## 4. Growth Processes of InP NWs

#### 4.1. Nanowire Growth on the Basis of Classical Nucleaton Theory

^{2}Δμ

_{liquid-solid}/2 + πr

^{2}σ

_{int}/2 + rΓ

_{step},

_{liquid-solid}is chemical potential difference (per unit area) between liquid and solid phases, σ

_{int}is the interface energy per area between nucleus and nanowire top layer, and Γ

_{step}is the contribution of the step energy caused by a two-dimensional nucleus. According to Equation (19), the critical radius for nucleation r* satisfy the condition, ∂G(r)/∂r = 0. Therefore, r* and activation energy for nucleation ΔG* are written as

_{step}/π(Δμ

_{liquid-solid}− σ

_{int}),

_{step}

^{2}/(2πΔμ

_{liquid-solid}).

_{WZ}, is also written as

_{WZ}= Γ

^{WZ}

_{step}

^{2}/{2πΔμ

_{liquid-solid}− σ

^{WZ}

_{int}}},

^{WZ}

_{step}is the contribution of the step energy caused by two dimensional growth with WZ structure, and σ

^{WZ}

_{int}is the interface energy for WZ structure. The WZ structure is formed satisfying ΔG* < ΔG*

_{WZ}, and the boundary between ZB and WZ structure in NWs are obtained using the step energy difference ΔΓ

_{step}= Γ

_{step}− Γ

^{WZ}

_{step}, as

_{step}/Γ

_{step})

^{−2}= (1 − σ

^{WZ}

_{int}/Δμ

_{liquid-solid})

^{−1}

_{step}and/or Δμ

_{liquid-solid}is large. Figure 16 shows the estimated phase diagram for crystal structure of InP NWs as functions of Δμ

_{liquid-solid}and ΔΓ

_{step}/Γ

_{step}using the energy difference between ZB and WZ structures in bulk InP (6.8 meV/atom) [86] for σ

^{WZ}

_{int}. Actually, this figure manifests that the WZ structure is stabilized if Δμ

_{liquid−solid}is large and ΔΓ

_{step}/Γ

_{step}takes positive value. Similar results have been obtained if we take more sophisticated nucleation model into account to evaluate the free energies [88,89,91]. These results suggest that the stability of the step and driving force for nucleation during the VLS growth are crucial for the formation of NWs with WZ structure.

#### 4.2. Effect of Side Facets on Adsorption–Desorption Behaviors at Top Layers in InP NWs

_{2}pressure. The pressure of In is taken to be $3.3\times {10}^{-3}$ Torr, which corresponds to a typical growth condition [90,91]. These surface phase diagrams for the adsorption P atoms exhibits little difference in the adsorption probability between 3In-ZB-P and 3In-WZ-P. The small energy difference is due to similar bonding configurations, where both 3In-ZB-P and 3In-WZ-P form nine In-P bonds regardless of the adsorption site. These results imply that the WZ structure as well as the ZB structure can be formed when nucleation occurs away from the NW side facets.

#### 4.3. Nanowire Growth Processes under Epitaxial Growth Condirions

_{gas}in Equations (1) and (2) under the growth condition (B) which are −2.5 and −1.2 eV for In and P, respectively. There are four adsorption sites for an In atom labeled 1, 2, 3, and 4 in Figure 23a. Three of them (sites 1, 2, and 3) are symmetrically equivalent and correspond to the adsorption sites for the ZB structure, and the other (site 4) corresponds to the adsorption site for the WZ structure. The adsorption energy difference among the adsorption sites for the WZ and ZB structure is negligible (0.06 eV). The positive values of adsorption energy (0.60 and 0.66 eV) indicate that the adsorption hardly occurs under typical growth conditions [90,91]. The calculated adsorption probability estimated using Equation (12) of an In atom under the growth condition (B) is found to be less than 0.03%. However, even if the adsorption probability is 0.03%, the adsorption occurs. This is because of the stabilization of the P trimer. The P trimer on (2 × 2) surface hardly desorbs under typical growth conditions. Indeed, the MC simulations for adsorption, migration and desorption of an In atom on the P trimer surface demonstrate that the adsorption of an In atom occurs even under the growth conditions (A) and (B), and surface lifetime and diffusion length of an In adatom are relatively large. The relatively large values in lifetime and diffusion length is due to low migration barriers on the (2 × 2) surface with P trimer less than 0.13 eV.

^{−4}%) is much lower than that of an In atom. From these results, the adsorption of an In atom is obviously feasible compared with that of a P atom on the (2 × 2) surface with P trimer. It is thus concluded that the crystal growth on the (2 × 2) surface starts with the adsorption of an In atom. Further investigations for the adsorption of In and P atoms on the In-adsorbed surfaces clarify that there are two types of plausible growth processes depending on the growth conditions.

_{In}= 0). There are three symmetrically equivalent and stable adsorption sites for an In atom on the surface with θ

_{In}= 1/4, which correspond to the WZ structure. The calculated adsorption energy is −0.54 eV. The surface with θ

_{In}= 2/4 represents the surface after adsorption of an In atom at the site 1 on the surface with θ

_{In}= 1/4. After the formation of the surface with θ

_{In}= 2/4, the adsorption of an In atom with calculated adsorption energy of −0.06 eV subsequently occurs at one of the two symmetrically equivalent. The surface after the adsorption of a P atom is the $(2\times 2)$ surface with In vacancy surface which is well known as a stable surface satisfying the ECM [38,39]. It should be noted that this In vacancy surface has an InP monolayer belonging to the WZ structure. In order to verify whether this process is likely under the growth conditions, we have calculated phase diagrams as functions of temperature and pressures. Figure 25a illustrates the adsorption–desorption behavior of an In atom on the surface with θ

_{In}= 1/4 shown in Figure 24. This calculated phase diagram manifests that the adsorption of an In atom easily occurs and stabilizes the surface under both the growth conditions (A) and (B). The adsorption of an In atom at θ

_{In}= 1/4 originates from the fact that the In-adsorbed surface corresponding to θ

_{In}= 2/4 satisfies the ECM [38,39]. As shown in Figure 25b, the adsorption of an In atom occurs on the surface with θ

_{In}= 2/4 under the growth condition (B) (red triangle), while the growth condition (A) is located near the boundary between adsorption and desorption (red circle). However, the adsorption possibility of an In atom under the growth condition (A) is relatively high (15%). The adsorption of an In atom certainly occurs even under the growth condition (A). In addition, once the adsorption of an In atom occurs, a P atom adsorbs easily and stabilizes the surface as shown in Figure 25c. The growth process shown in Figure 24 indeed occurs under both the growth conditions (A) and (B), suggesting that an InP monolayer belonging to the WZ structure is formed.

_{In}= 0). There are two symmetrically equivalent adsorption sites for a P atom on the surface with θ

_{In}= 1/4, which are located at near P lattice sites, whose adsorption energy is −0.02 eV. The P-adsorbed surface with ${\theta}_{\mathrm{In}}$ = 1/4 represents the surface after adsorption of a P atom at the site 2. The adsorption of an In atom subsequently occurs after the formation of the P-adsorbed surface. Although there are three adsorption sites for an In atom on the P-adsorbed surface, these three sites are no longer equivalent, so that calculated adsorption energies at the sites 1, 2, and 3 are different with each other (−0.07, −0.05, and 0.37 eV, respectively). The positive value at the site 3 indicates that the adsorption of an In atom hardly occurs. Consequently, this growth process splits into two sub-processes with almost the same probabilities after the adsorption of a P atom. One of the two sub-processes starts with the adsorption of an In atom at the site 1 which corresponds to the adsorption site for the WZ structure. The adsorption of an In atom (the calculated adsorption energy is −2.00 eV) at site 1 easily occurs on the surface with θ

_{In}= 2/4 in this sub-process, leading to the formation of the In vacancy surface which has an InP monolayer belonging to the WZ structure. The other sub-process begins with the adsorption of an In atom at the site 2, which corresponds to the adsorption site for the ZB structure, on the P-adsorbed surface with θ

_{In}= 1/4. The calculated adsorption energy of an In atom at site 1 on the surface with θ

_{In}= 2/4 is −1.95 eV, resulting in the formation of the In vacancy surface which has an InP monolayer belonging to the ZB structure.

_{In}= 1/4 in the growth process shown in Figure 26. The phase diagram indicates that a P atom can adsorb on the surface with ${\theta}_{\mathrm{In}}$ = 1/4 only under the growth condition (B). Therefore, this growth process can occur only under the growth condition (B). The adsorption of a P atom on the surface with θ

_{In}= 1/4 under the growth condition (B) is predominant for the growth process shown in Figure 26, which is quite different from that in the growth process shown in Figure 24. This is because the P-adsorbed surface satisfies the ECM [38,39], and is stabilized only for low temperatures and high P pressures. Although the adsorption probability of a P atom under the growth condition (A) might not be zero, the growth process shown in Figure 24 must be dominant under the growth condition (A) as shown in Figure 25a). The growth process shown in Figure 26 occurs only under the growth condition (B), leading to the formation of InP layers with both WZ and ZB structures. Both WZ and ZB structures can be grown with the same probability under growth condition (B).

_{re}(P), on the surface with θ

_{In}= 1/4 as functions of temperature and pressure estimated by the MC simulations. Here, we note that P

_{re}(P) + P

_{re}(In) = 100%, where P

_{re}(In) is relative adsorption probabilities of an In atom. Because of high V/III ratio, the adsorption of a P atom on the surface at θ

_{In}= 1/4 is dominant under the growth condition (B) compared with that of an In atom, leading to the growth process shown in Figure 26. From these results, it can be concluded that the growth process shown in Figure 24 is dominant for high temperatures and low P pressures, while the growth process shown in Figure 26 is dominant for low temperatures and high P pressures. Hence, it can also be concluded that InP layers grown for high temperatures and low P pressures (low V/III ratio) take the WZ structure, while those grown for low temperatures and high P pressures (high V/III ratio) include both the WZ and ZB layers with rotational twins. These results are consistent with the experimental result [91]. It is thus concluded that adsorption of P atoms depending on temperature and pressure is an important factor determining crystal structures of InP layers on InP(111)A surface. If the nucleation occurs on the InP(111)A top layer of InP NWs far from side facets, the growth processes depending on the growth conditions could strongly affect the structural stability of InP NWs. We believe that this scenario is applicable to clarify the crystal structure of InP NWs on InP(111)A substrates depending on growth conditions [91].

#### 4.4. Effects of Growth Condition on InP NW Shape

_{diff}, respectively, for an In adatom on $(1\overline{1}00)$ and $(11\overline{2}0)$ surfaces obtained by the MC simulations. Here, surface lifetime $\tau $ is defined by the time during the migration, and diffusion length L

_{diff}is the distance between the adsorption site and the desorption site. Furthermore, In pressure is fixed at 3.3 × 10

^{−3}Torr, which corresponds to the experimental growth conditions [90,91]. It is found that once the adsorption of an In atom occurs, the In adatom can migrate a certain distance (101–103 nm) with certain life time of 10

^{−8}–10

^{−6}s under the experimental growth conditions. The values of τ and L

_{diff}are long enough to grow at relatively low temperatures. Furthermore, both τ and L

_{diff}on $(1\overline{1}00)$ surface are larger than those on $(11\overline{2}0)$ surface, although there is little orientation dependence in the adsorption energy. This is because the migration barriers on $(1\overline{1}00)$ surface (0.27 eV) is smaller than that on $(11\overline{2}0)$ surface (0.32 eV) as shown in Figure 29. For a P atom, the calculated adsorption sites are the same as those for an In atom, and calculated adsorption energies on $(1\overline{1}00$) and $(11\overline{2}0)$ surfaces are 0.36 and 0.43 eV, respectively. These positive values indicate that the adsorption of a P atom does not occur even at 0 K. This is because the P-adsorbed surfaces have more excess electrons, which does not satisfy the ECM [38,39], compared to the In-adsorbed surfaces. Hence, the growth on both $(1\overline{1}00)$ and $(11\overline{2}0)$ surfaces starts from the adsorption of an In atom. In order to proceed the lateral growth on the surfaces, either In adatoms on the surface encountering each other caused by migration or additional adsorption on the In-adsorbed surfaces is necessary before vaporization of adsorbed In adatoms.

_{re}

^{(1–100)}(P) and P

_{re}

^{(11–20)}(P), respectively, which are percentages calculated by n

_{P}/(n

_{In}+ n

_{P}). Here, n

_{In}and n

_{P}are the number of the formation of In and P adatoms using MC simulations, respectively, on the In-adsorbed surfaces during the MC sampling (n

_{In}+ n

_{P}= 1000). Both P

_{re}

^{(1–100)}(P) and P

_{re}

^{(11–20)}(P) increase with temperature up to 550 °C, because In adatoms hardly encounter each other as seen in the decrease in τ and L

_{diff}in Figure 30. On the In-adsorbed $(1\overline{1}00)$ surface, P

_{re}

^{(1–100)}(P) shifts to decrease around 550–600 °C. This is because a P atom hardly adsorbs beyond 600 °C. On the In-adsorbed $(11\overline{2}0)$ surface, a P atom as well as an In atom hardly adsorbs beyond 600 °C. These results demonstrate that the probability of P/In atoms is higher, indicating that the adsorption of a P atom is dominant on the In-adsorbed surfaces. Although adsorption probability of an In atom on the In-adsorbed $(1\overline{1}00$) surface is higher than that of a P atom as shown in Figure 32, P atoms frequently attach on the surface owing to high V/III ratio, as seen in the estimated $\Delta {t}_{\mathrm{gas}}$ using Equation (15) shown in Table 2. It is found that in most cases, the adsorption of a P atom occurs on the In-adsorbed surfaces. From the calculated results by the MC simulations, we can deduce the initial growth process, such as a sequence of adsorption, migration, and desorption of In atoms, which iteratively occur on $(1\overline{1}00)$ and $(11\overline{2}0)$ surfaces, and the adsorption of P atom supplied from gas phase with In adatoms. Therefore, there are two reasons for the growth rate difference between $(1\overline{1}00)$ and $(11\overline{2}0)$ surfaces. One is the longer τ and L

_{diff}of In adatoms on $(1\overline{1}00)$ surface as shown in Figure 30 because In adatoms are necessary for the adsorption of a P atom. The other is higher adsorption probability of a P atom on the In-adsorbed $\{1\overline{1}00\}$ surface compared with that on $(11\overline{2}0)$ surface, as shown in Figure 29b and Figure 31b. The latter is more important than the former, since adsorption behavior of P atoms is sensitive around the critical temperature and pressure. It is expected that at 600 °C P atoms tend to adsorb on the In-adsorbed $(1\overline{1}00)$ surface whereas P atoms tend to desorb on the In-adsorbed $(11\overline{2}0)$ surface. The difficulty in P adsorption on the In-adsorbed $(11\overline{2}0)$ surface beyond 600 °C causes the growth rate difference. If we consider successful epitaxial growth of semiconductor NWs, the growth in the axial direction should be faster than that in the lateral directions. The difference in growth rate in InP.

_{diff}of an In adatom on the (111)A surface are longer than those on other surfaces. The remarkable values in τ and L

_{diff}on the (111)A surface originates from lower migration barriers on the (111)A, as shown in Table 3. Furthermore, the calculated adsorption energies of an In and a P atom on an In-adsorbed (111)A surface are lower than those on the In-adsorbed $(1\overline{1}00)$ and $(11\overline{2}0)$ surfaces.

## 5. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Schematics of top view of InAs(111)A-(2 × 2) with (

**a**) In-vacancy; (

**b**) As-adatom; and (

**c**) As-trimer surfaces considered in this study. Blue and orange atoms denote In and As, respectively.

**Figure 2.**Calculated surface phase diagrams for (

**a**) InAs(111)A WL surface and (

**b**) fully relaxed surface without interface structure of InAs/GaAs as functions of temperature and BEP of As

_{4}. Black area denotes the temperature range of the conventional MBE growth.

**Figure 3.**Calculated adsorption-desorption boundaries for In atom without (dotted line) and with simultaneous As adsorption (solid line) on the (2 × 2) with In-vacancy of (

**a**) the InAs(111)-WL and (

**b**) the InAs(111)-FR as functions of temperature and BEP of In. Black area denotes the temperature range of the conventional MBE growth.

**Figure 4.**(

**a**) Schematic of the 2D growth with dislocation (5/7-ring core structure) network formation at the interface; (

**b**) Calculated energy difference between the 2D coherent growth (2D-coherent) and the growth with MD (2D-MD) as a function of layer thickness h.

**Figure 5.**(

**a**) Schematic of the 2D growth with formation of stacking-fault tetrahedron (SFT) consisting of ridge and face; (

**b**) Calculated energy difference between the 2D coherent growth (2D-coherent) and the growth with SFT (2D-SFT) as a function of layer thickness h.

**Figure 6.**Calculated free energy differences ∆F for various growth modes for the InAs/GaAs(111)A as a function of layer thickness h.

**Figure 7.**Schematic of structural change during MBE growth from initial GaAs(001)-c(4 × 4) to InAs(001)-(2 × 4)α2 via InAs(001)-(2 × 3). Green, blue, and orange atoms denote Ga, In, and As, respectively.

**Figure 8.**Calculated surface phase diagrams for InAs(001) WL surfaces such as (

**a**) (2 × 3) and (

**b**) (2 × 4) as functions of temperature and BEP of As

_{4}. Black square denotes the t conventional MBE growth conditions.

**Figure 9.**Calculated adsorption-desorption boundaries for (

**a**) In atom and (

**b**) As-dimer on the InAs(001)-(n × 3) WL surfaces as functions of temperature and BEP of In and As

_{4}, respectively.

**Figure 11.**(

**a**) Schematic of the 2D growth with dislocation (8-ring core and 5/7-ring core) formation at the interface; (

**b**) Calculated energy for the 2D coherent growth (2D-coherent) and the 2D growth with MD (2D-MD with 8-ring and 2D-MD with 5/7-ring) as a function of layer thickness h.

**Figure 12.**Calculated free energy differences ∆F for various growth modes for the InAs/GaAs(001) as a function of layer thickness h.

**Figure 13.**Calculated growth mode boundaries as functions of β/α and γ for the InAs/GaAs systems with different orientations. Open square and open triangle denote the results for the (001)- and the (111)A-oriented InAs/GaAs, respectively. The growth mode boundary (dotted line) and the data (closed diamond) for the InAs/GaAs(110) are also shown in this figure.

**Figure 14.**Schematics of crystal structures of III-V compound semiconductors. Large and small circles represent cations and anions, respectively. Stacking sequences of (

**a**) zinc blende (ZB) and (

**b**) wurtzite (WZ) structures are $\cdots ABCABC\cdots $ and $\cdots ABAB\cdots $, respectively, while (

**c**) rotational twins are formed in semiconductor nanowires (NWs). Dashed line denote a twin plane.

**Figure 15.**Schematics of InP NWs consisting of (

**a**) $\{1\overline{1}00\}$ and (

**b**) $\{11\overline{2}0\}$ side facets. Cross-sectional views and side views of NW facets are also shown. Dashed red lines in the top views represent unit cells. Large gray and small yellow circles represent In and P atoms, respectively.

**Figure 16.**Phase diagram for crystal structure of InP NWs as functions of chemical potential difference between liquid and solid phases Δμ

_{liquid-solid}and normalized step energy difference between ZB and WZ structures ΔΓ

_{step}/Γ

_{step}, using the energy difference between ZB and WZ structures in bulk InP (6.8 meV/atom) [87] for interface energy per area between nucleus and nanowire top layer for WZ structure σ

^{WZ}

_{int}. White and grey areas denote the stable regions for WZ and ZB structures, respectively.

**Figure 17.**Top views of (

**a**) 3In-P occupying the ZB lattice sites (3In-ZB-P) and (

**b**) 3In-P occupying the WZ lattice sites (3In-WZ-P) on the InP(111)A surface. Large and small circles represent In and P atoms, respectively. Blue and green circles denote the In adatoms that occupy the lattice sites of the ZB and WZ structures, respectively. Orange circles represent the P adatoms. Reprinted with permission from [95]. Copyright (2011) by the Japan Society of Applied Physics.

**Figure 18.**Calculated phase diagrams of (

**a**) 3In-ZB-P and (

**b**) 3In-WZ-P adsorbed surfaces as functions of temperature and P

_{2}pressure. In pressure is taken to be $3.3\times {10}^{-3}$ Torr. Colored regions correspond to the atom-adsorbed regions. Reprinted with permission from [95]. Copyright (2011) by the Japan Society of Applied Physics.

**Figure 19.**Top views of (

**a**) 4In-2P occupying the ZB lattice sites (4In-ZB-2P) and (

**b**) 4In-2P occupying the WZ lattice sites (4In-WZ-2P) near the side facet. There is a two-coordinated In adatom indicated by the red circle in 4In-ZB-2P, whereas all the In adatoms in 4In-WZ-2P are three-coordinated. The three-coordinated In adatom at the side facet in 4In-WZ-2P is due to the formation of In-P bonds with the P atom in the second layer. Reprinted with permission from [95]. Copyright (2011) by the Japan Society of Applied Physics.

**Figure 20.**Top views of (

**a**) 4In-ZB-2P and (

**b**) 4In-WZ-2P adsorbed surfaces in the side area. There is a two-coordinated In adatom indicated by the red circle in 4In-ZB-2P, whereas all the In adatoms in 4In-WZ-2P are three-coordinated. The three-coordinated In adatom at the side facet in 4In-WZ-2P is due to the formation of In-P bonds with the P atom in the second layer. Reprinted with permission from [96]. Copyright (2011) by the Japan Society of Applied Physics.

**Figure 21.**Calculated phase diagrams of (

**a**) 4In-ZB-2P and (

**b**) 4In-WZ-2P adsorbed surfaces as functions of temperature and P

_{2}pressure. In pressure is taken to be $3.3\times {10}^{-3}$ Torr. Colored regions correspond to the atom-adsorbed regions. Reprinted with permission from [95]. Copyright (2011) by the Japan Society of Applied Physics.

**Figure 22.**Top view of the $(2\times 2)$ surface with P trimer surface. Red dashed lines correspond to the (2 × 2) unit cells. Large and small circles represent In and P atoms, respectively. Symbols of W and Z stand for In lattice sites in the WZ and ZB structures, respectively. Reprinted with permission from [97]. Copyright (2013) by Elsevier.

**Figure 23.**Contour plots of adsorption energies for (

**a**) In and (

**b**) P atoms on the P trimer surface. These contour plots are indicated by colors with respect to μ

_{gas}under the growth condition (B) which is displayed in green region. Atoms tend to adsorb (desorb) in blue (red) colored regions under the growth condition (B). Encircled numbers represent adsorption sites. The coverage of In (θ

_{In}) and a kind of atom for contour plots are denoted at the lower left and right, respectively. Dotted white lines in (

**a**) denotes a possible migration path of In adatom. Reprinted with permission from [98]. Copyright (2013) by Elsevier.

**Figure 24.**Growth process which is plausible under the growth conditions (A) and (B). The adsorption of a P atom occurs on the surface with θ

_{In}= 3/4. This growth process results in formation of an InP monolayer belonging to the WZ structure. Reprinted with permission from [97]. Copyright (2013) by Elsevier.

**Figure 25.**Calculated phase diagrams as functions of temperature and pressure in the growth process shown in Figure 21. Panels (

**a**) and (

**b**) show adsorption–desorption behavior of an In atom on the surface with θ

_{In}= 1/4 and 2/4, respectively. Panel (

**c**) illustrates adsorption–desorption behavior of a P atom on the surface with θ

_{In}= 3/4. Regions with hatched lines correspond to atom-adsorbed regions. Red circles and triangles represent the growth conditions (A) and (B), respectively. Reprinted with permission from [97]. Copyright (2013) by Elsevier.

**Figure 26.**Growth process which is feasible only under the growth condition (B). The adsorption of a P atom occurs on the surface with θ

_{In}= 1/4. This growth process splits into two sub-processes after the adsorption of a P atom, and results in the formation of an InP monolayer belonging to the WZ or ZB structure. Reprinted with permission from [97]. Copyright (2013) by Elsevier.

**Figure 27.**Calculated phase diagrams functions of temperature and pressure for adsorption–desorption behavior of a P atom in the growth process shown in Figure 26. Red circle and triangle represent the growth conditions (A) and (B), respectively. Reprinted with permission from [98]. Copyright (2013) by Elsevier.

**Figure 28.**Relative adsorption probabilities of a P atom, ${P}_{\mathrm{re}}(\mathrm{P})$, on the surface with θ

_{In}= 1/4 as functions of temperature and pressure. It should be noted that P

_{re}(P) + P

_{re}(In) = 100%. In pressure is fixed at 3.3 × 10

^{−3}Torr which corresponds to both the growth conditions (A) and (B). Triangle represents the growth condition (B). Reprinted with permission from [97]. Copyright (2013) by Elsevier.

**Figure 29.**Adsorption sites and migration barriers for an In atom on (

**a**) $(1\overline{1}00)$ and (

**b**) $(11\overline{2}0)$ surfaces. Encircled numbers and contour plots represent adsorption sites and migration barriers, respectively. Large gray and small yellow circles represent In and P atoms, respectively. Reprinted with permission from [99]. Copyright (2013) by Elsevier.

**Figure 30.**Temperature dependence of (

**a**) surface lifetime $\tau $ and (

**b**) diffusion length ${L}_{\mathrm{diff}}$ for an In adatom on $(1\overline{1}00)$ and $(11\overline{2}0)$ surfaces. In pressure is fixed at $3.3\times {10}^{-3}$ Torr, which corresponds to the experimental growth conditions [90,91]. Reprinted with permission from [99]. Copyright (2013) by Elsevier.

**Figure 31.**Calculated adsorption sites and migration barriers on the In-adsorbed $(1\overline{1}00)$ surface for (

**a**) In and (

**b**) P atoms. Encircled numbers and contour plots represent adsorption sites and migration barriers, respectively. Large gray and small yellow circles represent In and P atoms, respectively. Reprinted with permission from [99]. Copyright (2013) by Elsevier.

**Figure 32.**Calculated phase diagrams as functions of temperature and pressure on the In-adsorbed $(1\overline{1}00)$ surface for adsorption of (

**a**) In and (

**b**) P atoms. Regions with hatched lines correspond to atom-adsorbed regions. Red circles, squares, and triangles represent the growth conditions for the high temperature and low V/III ratio, the intermediate temperature and V/III ratio, and the low temperature and high V/III ratio, respectively. In pressure is fixed at 3.3 × 10

^{−3}Torr which corresponds to the experimental growth conditions [90,91]. Reprinted with permission from [99]. Copyright (2013) by Elsevier.

**Figure 33.**Calculated adsorption sites and migration barriers on the In-adsorbed $(11\overline{2}0)$ surface for (

**a**) In and (

**b**) P atoms. Encircled numbers and contour plots represent adsorption sites and migration barriers, respectively. Large gray and small yellow circles represent In and P atoms, respectively. Reprinted with permission from [100]. Copyright (2013) by Elsevier.

**Figure 34.**Calculated phase diagrams as functions of temperature and pressure on the In-adsorbed $(11\overline{2}0)$ surface for adsorption of (

**a**) In and (

**b**) P atoms. Regions with hatched lines correspond to atom-adsorbed regions. Red circles, squares, and triangles represent the growth conditions for the high temperature and low V/III ratio, the intermediate temperature and V/III ratio, and the low temperature and high V/III ratio, respectively. In pressure is fixed at 3.3 × 10

^{−3}Torr which corresponds to the experimental growth conditions [90,91]. Reprinted with permission from [99]. Copyright (2013) by Elsevier.

**Figure 35.**Phase diagram for growth modes of InP NWs as functions of temperature and pressure. Red circle, square, and triangle represent the experimental growth conditions for the high temperature and low V/III ratio, the intermediate temperature and V/III ratio, and the low temperature and high V/III ratio, respectively [90,91].

**Figure 36.**Surface phase diagrams for H-adsorbed (

**a**) AlN and (

**b**) GaN with different orientations as functions of temperature and pressure.

**Figure 37.**Calculated band structure and imaginary part of dielectric function of (

**a**) GaN and (

**b**) ZnO NWs with diameter of 1.3 nm. Energies are measured from the highest occupied state. Solid and dashed lines in dielectric function represent the imaginary part of dielectric function polarized along the nanowire axis and that in the orthogonal plane.

**Table 1.**Relative adsorption probabilities of a P atom on the In-adsorbed $(1\overline{1}00)$ and $(11\overline{2}0)$ surfaces, P

_{re}

^{(1–100)}(P) and P

_{re}

^{(11–20)}(P), respectively, obtained by MC simulations. It should be noted that P

_{re}

^{(1–100)}(In) + P

_{re}

^{(1–100)}(P) = P

_{re}

^{(11–20)}(In) + P

_{re}

^{(11–20)}(P) = 100%, where P

_{re}

^{(1–100)}(In) and P

_{re}

^{(11–20)}(In) are relative adsorption probabilities of an In atom on the In-adsorbed surfaces. Here, we assume that pressure is 0.1 Torr.

Temperature (°C) | P_{re}^{(1–100)}(P) (%) | P_{re}^{(11–20)}(P) (%) |
---|---|---|

500 | 67.3 | 72.3 |

550 | 72.1 | 77.1 |

650 | 69.1 | 76.9 |

**Table 2.**Values of

**Δt**for In and P obtained by Equation (15). Here, we assume that In and P pressures are 3.3 × 10

_{gas}^{−3}and 0.1 Torr, respectively.

Temperature (°C) | Δt_{gas} for In (s) | Δt_{gas} for P (s) |
---|---|---|

500 | 4.53 × 10^{−13} | 1.11 × 10^{−14} |

550 | 4.67 × 10^{−13} | 1.15 × 10^{−14} |

650 | 4.81 × 10^{−13} | 1.18 × 10^{−14} |

**Table 3.**Values of τ and L

_{diff}obtained by MC simulations, and migration barriers of In adatom on bare (111)A, $(1\overline{1}00)$, and $(11\overline{2}0)$ surfaces using Figure 23 and Figure 29. Here, we assume that temperature and In pressure are 600 °C and 3.3 × 10

^{−3}Torr, respectively. The lowest values for migration barriers are taken on each surface.

Surface Orientation | τ (×10^{−7} s) | L_{diff} (×10^{2} nm) | Migration Barrier (eV) |
---|---|---|---|

(111)A | 22.4 | 16.3 | 0.13 |

$(1\overline{1}00)$ | 8.57 | 4.36 | 0.27 |

$(11\overline{2}0)$ | 6.97 | 2.38 | 0.32 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Ito, T.; Akiyama, T.
Recent Progress in Computational Materials Science for Semiconductor Epitaxial Growth. *Crystals* **2017**, *7*, 46.
https://doi.org/10.3390/cryst7020046

**AMA Style**

Ito T, Akiyama T.
Recent Progress in Computational Materials Science for Semiconductor Epitaxial Growth. *Crystals*. 2017; 7(2):46.
https://doi.org/10.3390/cryst7020046

**Chicago/Turabian Style**

Ito, Tomonori, and Toru Akiyama.
2017. "Recent Progress in Computational Materials Science for Semiconductor Epitaxial Growth" *Crystals* 7, no. 2: 46.
https://doi.org/10.3390/cryst7020046