Zeldovich Nucleation Rate, Self-Consistency Renormalization, and

Subscriber access provided by - Access paid by the | UCSF Library

Article

Zeldovich nucleation rate, self-consistency renormalization and crystal phase of Au-catalyzed GaAs nanowires Vladimir G. Dubrovskii, and Jurij Grecenkov Cryst. Growth Des., Just Accepted Manuscript • Publication Date (Web): 24 Nov 2014 Downloaded from http://pubs.acs.org on November 25, 2014

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Crystal Growth & Design is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Crystal Growth & Design

Zeldovich nucleation rate, self-consistency renormalization and crystal phase of Au-catalyzed GaAs nanowires

V.G. Dubrovskii1-3,* and J. Grecenkov1 1

2

St. Petersburg Academic University, Khlopina 8/3, 194021, St. Petersburg, Russia

Ioffe Physical Technical Institute RAS, Politekhnicheskaya 26, 194021, St. Petersburg, Russia 3

ITMO University, Kronverkskiy pr. 49, 197101 St.Petersburg, Russia *e-mail: [email protected]

ABSTRACT: We present a self-consistent model for the Zeldovich nucleation rate that determines the nucleation probabilities, growth rates and even the preferred crystal structure of Au-catalyzed III-V nanowires fabricated by the vapor-liquid-solid growth method. The obtained expression accounts for the nucleation kinetics in ternary Au-III-V alloys and shows that the nucleation rate in vapor-liquid-solid nanowires is proportional to the As concentration, As diffusion coefficient in the droplet and the activity of solid GaAs. The leading exponential term of the nucleation rate is modified due to the self-consistency renormalization. As a result, the behavior of the effective nucleation barrier versus Ga concentration is changed significantly with respect to the commonly used expression. This strongly affects the values of Ga concentrations during growth which are obtained within the self-consistent approach with the known nanowire elongation rates. In turn, the renormalized nucleation rates change the predictions regarding the zincblende-wurtzite phase transitions in III-V nanowires. In particular, our calculations show why the Aucatalyzed GaAs nanowires grown by molecular beam epitaxy at 550oC are predominantly wurtzite, while the high temperature hydride vapor phase epitaxy at 715oC yields pure zincblende crystal structure. We also obtain useful estimates for the As diffusion coefficients in ternary Au-Ga-As liquids at different conditions.

1 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 30

INTRODUCTION Perhaps the most important result of classical nucleation theory is the Zeldovich nucleation rate (ZNR), showing how many stable nuclei would emerge per unit time in a given system at the known supersaturation1. This ZNR is widely used in modeling the size distribution of different “clusters” such as droplets and three-dimensional (3D) or two-dimensional (2D) surface islands2-9. In the case of semiconductor nanowires (NWs) fabricated by the metalcatalyzed vapor-liquid-solid (VLS) method10, the ZNR is thought to determine the probability of 2D nucleation from a supersaturated liquid alloy of a metal catalyst with the growth constituencies2. In this way, the ZNR controls the vertical growth rate of NWs2,11-13, nucleation statistics in VLS NWs14-17, morphology of the growth interface18,19 and even the preferred crystal structure of Au-catalyzed III-V NWs which can be either cubic zincblende (ZB) or hexagonal wurtzite (WZ)20-34. Since most NWs grow in the so-called mononuclear mode with only one island emerging in each NW monolayer (ML)2, we need to consider the chain of individual nucleation events. This simplifies modeling due to the absence of any collective effects or ensembles of nuclei. On the other hand, theoretical description of nucleation probabilities is complicated by the fact that the state of a nano-sized catalyst is extremely sensitive to the influx and the nucleation-mediated sink of semiconductor material, the latter being described by a position-dependent ZNR. Despite significant progress toward understanding the VLS growth and tailoring the crystal structure of III-V NWs in different epitaxy techniques2,20-34, the ZNR in VLS NWs remains generally unknown. The only case where the ZNR has been accurately calculated is the Ga-catalyzed VLS growth of GaAs NWs12. When III-V NWs are obtained using group III catalysts (such a method is often called the self-catalyzed VLS growth12,25,35 ), the liquid alloy in the droplet is binary (e.g. Ga-As). This allows for a comprehensive modeling of Ga-catalyzed VLS growth based entirely on the kinetics of As species12, while the crystal structure of Ga-


Page 3 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


catalyzed GaAs NWs is predominantly ZB for energetic reasons25. Clearly, the situation becomes much more complex when NWs are catalyzed by Au and the liquid alloys are ternary (e.g. Au-Ga-As). The calculations of Glas36 provide the chemical potential of Au-catalyzed III-V NWs as a function of concentrations of the group III and V atoms in the droplet and the growth temperature. Under steady state conditions, the unknown group V concentration can be expressed through much more transparent influx and desorption rate of the group V element37. These results determine with high precision the driving force for 2D nucleation in VLS NWs, while the nucleation energies can be obtained within macroscopic approach2,11-15,20,21,27,30,34,37. However, the energetic and kinetic pre-factors of the ZNR in Au-catalyzed VLS NWs have not been studied so far. The important effect of ZB-WZ polytypism in Au-catalyzed III-V NWs appears to be closely related to the position-dependent ZNRs of ZB versus WZ nuclei. It has been argued that the bulk energy difference between the WZ and ZB phases can be more than compensated by a lower surface energy of relevant sidewall planes of WZ NWs20,21,27 . Glas et al.20 established a kinetic approach based on comparing the nucleation barriers of WZ and ZB 2D nuclei at the triple phase line (TPL), with the lower nucleation barrier dictating the preferred crystal phase. This view has recently been generalized to include the center (C) nucleation of ZB nuclei in Garich liquid alloys34. The modified phase diagrams qualitatively explain most of the experimentally observed trends such as (i) higher chemical potential favoring the WZ phase during Au-catalyzed molecular beam epitaxy (MBE) of GaAs NWs20,22,23 ; (ii) pure ZB phase of Au-catalyzed GaAs and other III-V NWs at high material inputs employed in metal organic chemical vapor deposition (MOCVD)30-32 and hydride vapor phase epitaxy (HVPE)33,34; (iii) predominantly ZB structure of Ga-catalyzed GaAs NWs25; and (iv) qualitatively different results on the WZ structure being favored at higher group V fluxes in Au-catalyzed MBE24 and chemical beam epitaxy26 but showing an opposite behavior in MOCVD29,38,39. Although this approach works qualitatively well, a better quantitative correspondence requires a comparison of 3 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 30

the position-dependent ZNRs with the correct pre-factors rather than the nucleation barriers alone. As regards the normalization of the ZNR, it has been pointed out (see, e.g., Ref. [3], Ch. 7 and references therein) that the use of macroscopic formation energy of a nucleus is selfinconsistent as it does not give correctly the concentration of free “monomers” in the quasiequilibrium size distribution. The corresponding self-consistency (SC) renormalization of the ZNR changes the Zeldovich expression by several orders of magnitude and introduces a new exponential factor which depends on the surface energy. Such a renormalization is particularly important for the correct determination of the growth rates of Au-catalyzed VLS NWs and therefore the preferred crystal structure, because the droplet surface energy is a function of the group III concentration. Consequently, in this work we derive a self-consistent expression for the ZNR in the case of 2D nucleation from a ternary Au-Ga-As alloy and then apply the result in crystal phase modeling. It will be shown that the SC renormalization changes drastically the calculated group III concentrations related to the measured NW growth rates in a given epitaxy technique. In turn, this strongly affects the predictions on the preferred crystal structure of Aucatalyzed GaAs NWs.

NUCLEATION RATE OF III-V ISLANDS GROWING FROM A TERNARY LIQUID When nuclei are single-component 2D surface islands fed from a metastable “sea” of surface adatoms, the ZNR is given by1-8

J=

F ′′(ic ) n1W + (ic )e − Fc . 2π

(1)

Here, Fc ≡ F (ic ) is the nucleation barrier in kBT units (with T as the surface temperature and kB as the Boltzmann constant), defined as the point of maximum of the island formation energy F (i )


Page 5 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


as a function of the number of adatoms i in the nuclei, ic is the number of adatoms in the critical nucleus for which the formation energy reaches its maximum, n1 is the adatom concentration, W + (ic ) is the adatom attachment rate to the critical nucleus, and F ′′(ic ) is the second derivative of

the formation energy F (i ) at i = ic . Equation (1) contains both thermodynamic [ Fc , ic , F ′′(ic ) ] and kinetic [ n1 , W + (ic ) ] parameters which correspond to a given supersaturation ζ . The latter is generally defined as ∆µ = ln(ζ + 1) , where ∆µ is the difference of chemical potentials between a metastable adatom phase and a 2D solid layer (here and below we measure chemical potentials in kBT units). For a dilute adatom sea, supersaturation equals ζ = n1 / n1e − 1 in view of

∆µ = ln(n1 / n1e ) for a perfect gas, with n1e as the equilibrium adatom concentration at the surface temperature T .

Figure 1. Comparison of nucleation of a single-component 2D island from a metastable adatom sea with the chemical potential difference ∆µ = ln(n1 / n1e ) and diffusivity D (a) and nucleation of a 2D GaAs island from a non-stoichiometric Au-Ga-As droplet having the chemical potential ∆ µ = µ 3L + µ 5L − µ 35S , As concentration c5 and As diffusivity D5 (b).

Let us now see how the described picture for elemental 2D liquid-solid transition changes when rather from an adatom sea, 2D islands grow from a 3D metastable liquid Au-III-V alloy, as illustrated in Fig. 1. For the transition from liquid to solid III-V pairs, we first need to define the 5 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 30

notions of “monomer” in the mother phase before the transition, the ground state after the transition and the corresponding chemical potential difference. As in Ref. [36], the chemical potential per pair in an Au-III-V liquid is measured with respect to the stoichiometric III-V solid (denoted below as µ 35S ): ∆ µ = µ 3L + µ 5L − µ 35S , with µ 3L , µ 5L as the chemical potentials of group III and V atoms in the liquid state, respectively. Therefore, the monomer in this case should represent two group III and group V atoms dissolved in liquid gold which undergo the transition to the stoichiometric III-V solid state. The size i in Eq. (1) should now refer to the number of III-V pairs in the solid. Clearly, our liquid monomers should not have any surface energy since group III and V atoms in the droplet are spatially separated. If we consider regular triangle nuclei (with side r ) of a III-V material which emerge from a metastable Au-III-V liquid, the macroscopic expression for the formation energy is therefore given by2

F (i) = −i∆µ + ai1 / 2 , a = 2 × 33 / 4

γ eff k BT

(Ω35h)1 / 2 .

(2)

Here, Ω35 is the elementary volume per III-V pair in the solid, γ eff is the effective surface energy of the island lateral surface and h is the height of a solid ML. The first, negative term in Eq. (2) describes thermodynamically favorable lowering of the system energy by the monomer assembling into a nucleus, while the second, positive term stands for the surface energy of the nucleus lateral facets having the surface area 3hr , with r = ( 2 / 31 / 4 )( Ω 35 / h )1 / 2 i1 / 2 . Maximizing this F (i ) readily yields

ic =

2∆µ 3 a2 a2 ′ ′ F ( i ) = ; ; , F = c c a2 4 ∆µ 2 4 ∆µ

(3)

the well-known results used extensively in modeling of NW growth12,14,20,21,34. Very importantly, the surface energy constant a is determined by the ratio of the characteristic surface energy per 6 ACS Paragon Plus Environment

Page 7 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


elementary area to thermal energy and therefore is much larger than unity for all the systems of interest – otherwise, 2D islands could not exist because their boundaries would be destroyed by thermal fluctuations. In view of a >> 1 , the ZNR is extremely sensitive to supersaturation via the exponential factor exp(− Fc ) = exp(−a 2 / 4∆µ ) , while the pre-exponential factor in Eq. (1) is often assumed constant. This gives the simplified expression J = J 0 exp(− Fc ) , where J 0 can be treated as independent of supersaturation in the first approximation2. Despite its clear physical meaning and simplicity, Eq. (2) has one important drawback as F (i ) does not equal zero at i = 1 . This makes the whole theory self-inconsistent – indeed, if we

first consider a simplified case of elemental transition from adatoms to 2D islands with ∆µ = ln(ζ + 1) , the quasi-equilibrium size distribution neq (i ) = n1 exp[ − F (i )] [relating to a given

adatom concentration n1 = n1e (ζ + 1) ] does not give correctly this adatom concentration at i = 1 . Therefore, it has been proposed (see the discussion by Kashchiev in Ref. [3], Ch. 7 and references therein, or Eq. (3) of Ref. [4]) to use the renormalization F~ (i ) = F (i ) − F (1) = −i∆µ + ai1 / 2 + ∆µ − a ,

(4)

~ which corrects the macroscopic formation energy to acquire the property F (1) = 0 . Obviously,

this renormalization does not affect ic and F ′′(ic ) in Eqs. (3), while the effective nucleation barrier is changed to

a2 F~c = + ∆µ − a . 4∆µ

(5)

Since a >> 1 as discussed earlier, the additional factor exp(a) , appearing in Eq. (1) after the SC renormalization of the ZNR, increases the corrected ZNR by several orders of magnitude. It seems logical to introduce the same SC renormalization for the VLS growth of III-V compounds. Re-formulation of the SC criterion in this case is rather straightforward – we request 7 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 30

that the re-normalized macroscopic free energy of forming a monomer is zero and thus the ~ condition F (1) = 0 applies also for III-V pairs. We note that our liquid monomers are not identical

to bound Ga-As pairs on the solid substrate which are nuclei of size one (because the difference of their chemical potentials equals ∆µ ), while in elemental growth from the adatom sea nuclei of size one are the adatoms themselves. Thus, the chemical potential difference ∆µ in Eq. (4) for F~ (1) can be further discussed. However, the shift of the formation energy by ∆µ is usually much

smaller than by the surface energy constant a and the latter is required to eliminate the unphysical surface energy of two spatially separated atoms in the droplet. Therefore, we will use the SC formula for the free energy of forming a 2D III-V island from liquid group III and V atoms given by

a2 F~c = + (1 − χ )∆µ − a . 4∆µ

(5a)

Here, χ = 0 or 1 depending on whether the chemical potential difference between the liquid monomer and the solid III-V pair is disregarded or included. We now investigate how the kinetic coefficients in Eq. (1) change in the case of Aucatalyzed VLS growth of III-V NWs as schematized in Fig. 1. We assume that a droplet is so small that the concentrations of both group III and V atoms dissolved in Au are spatially uniform in the entire liquid volume. The lateral island surface is considered as a linear sink for III-V pairs which should arrive to the island through its perimeter in stoichiometric one-to-one proportion. Due to the known low solubility of highly volatile group V elements in metals, the relative atomic concentrations of group III ( c3 ) and group V ( c5 ) atoms in liquid should obey the strong inequality c5 > ic where dF / di ≅ −∆µ . On the other hand, in the diffusion-transport model2, the growth rate of supercritical islands would be di dn h dc1 . = 3rD 1 = 3rD dt dx Ω35 dx

(8)

This would pertain if islands were growing from a dilute adatom sea having the diffusion coefficient D , the surface coverage c1 and on the surface with the elementary site area Ω35 / h . Here, dc1 / dx denotes the concentration gradient at the island boundary. In the perfect gas approximation, we have simply ∆µ = ln(c1 / c1e ) and hence dc1 / dx = c1e d [exp(∆µ )] / dx . For III-V growth, we use the same expression with ∆ µ = µ 3L + µ 5L − µ 35S - indeed, no growth should occur when ∆µ = 0 . Of course, the equilibrium coverage c1e should be changed to exp( µ 35S ) , which is the equilibrium activity of a III-V solid. Overall, these considerations yield the substitution of dc1 / dx to d [exp( µ 3L + µ5L )] / dx in Eq. (8). Finally, within our “lattice gas” model, the chemical

potential changes abruptly from µ 3L + µ 5L in a spatially uniform liquid to µ 35S in the reference solid state, over the elementary distance ( Ω 35 / h )1 / 2 . This allows us to write approximately



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1/ 2

d µ3L + µ5L  h   e ≅  dx  Ωs 

(e

µ3L + µ5L

)

− e µ35 . S

Page 10 of 30

(9)

We now use this in Eq. (8) instead of dc1 / dx . It is also clear that the diffusion transport from liquid to solid should be limited by group V atoms, which are a thinner and slower diffuser, so the diffusion coefficient D should be put to D5 . Thus, we arrive at

(

)

di h µ3L + µ5L 1 / 2 = 2 × 33 / 4 D5 e i 1 − e − ∆µ , dt Ω 35

(10)

where the i1/ 2 size dependence shows that the growth rate is proportional to the island perimeter 3r , as usual in the ballistic growth regime 2. Comparing Eqs. (10) and (7) and then using Eq. (3)

for ic , we find:

W + (i ) = 2 × 33 / 4 D5

W + (ic ) = 33 / 4 D5

h µ3L + µ5L 1 / 2 e i ; Ω35

h µ3L + µ5L a , e Ω 35 ∆µ

(11)

where the last expression is what is required in Eq. (1). Using Eq. (3) for F ′′(ic ) , Eq. (5a) for the nucleation barrier, Eq. (6) for the effective surface concentration of III-V “monomers” and Eq. (11) for W + (ic ) , we get the final result for the ZNR in the case of III-V islands growing from a metastable ternary Au-III-V liquid with a given ∆µ :

J=

2

 h  µ 35S + χ∆µ 1 / 2  a2   e . ∆µ exp a − D5c5  4∆µ  π   Ω35 

33 / 4

(12)

This expression gives the normalization with the SC renormalization included. It is seen that the ZNR is proportional to the group V concentration, the diffusion coefficient of group V atoms in a 10 ACS Paragon Plus Environment

Page 11 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


ternary liquid, and contains the additional exp( a + µ 35S + χ ∆µ ) exponential term due to the combined effect of the SC renormalization and the attachment rate being proportional to the activity of Au-III-V liquid alloy as given by Eq. (11). The ∆µ 1 / 2 factor before the exponent holds only when the attachment rate is proportional to the island perimeter (i.e., in the ballistic exchange regime for the critical nucleus), which is essential to reduce one power of ∆µ in the Zeldovich factor

F ′′(ic ) in Eq. (1) (Ref. [8]). For example, if the attachment rate were

independent of the island size, the resulting expression for the nucleation rate would contain the factor ∆µ 3 / 2 instead of ∆µ 1 / 2 . Equation (12) can be recommended within a wide range of conditions and for different island shapes other than regular triangle, which would change only the geometrical coefficient 33 / 4 .

THEORETICAL ANALYSIS Let us now analyze two important particular cases of the obtained expression for the ZNR. First, we again consider growth from a diluted adatom sea with the concentration n1 where

∆µ = ln(ζ + 1) ,

ζ = c1 / c1e − 1 = n1 / n1e − 1 ,

Ω35 / h = σ is

the

elementary

surface

area,

exp( µ 35S ) = n1eσ and χ = 0 . This reduces Eq. (12) to

J=

n1e (ζ + 1) ln1 / 2 (ζ + 1)

τ

π

  a2  , exp a − 4 ln(ζ + 1)  

(13)

with

τ −1 =

33 / 4 n1eσ σ , D= tD tD

(14)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 30

defining the characteristic time τ for the island growth and t D for the diffusion hops. This is exactly the result given in Ref. [2] for a one-component 2D system with the SC renormalization included. Second, Glas et al.12 have recently studied the case of Ga-catalyzed GaAs NWs limited entirely by the kinetics of As species, based on the equation

 a2   . J = A(T )c5 ∆µ 1 / 2 exp −  4∆µ 

(15)

The authors assumed that the pre-factor A depends on temperature but neither on ∆µ nor c5 . Comparing this to Eq. (12) at χ = 0 , we find 2

 h  a + µ35S  e A= D5  . π  Ω35  33 / 4

(16)

For a binary Ga-As liquid and at a low As concentration c5 (dL / dt ) ZB or, from Eq. (17), JWZ > J ZB . When J is taken in the form of Eq. (15) with A independent of c3 , and all the factors except the leading exponential are neglected, this is reduced to the Glas condition for the nucleation barriers 13 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

∆µ > ∆µ c =

ψ WZ 1 − (aWZ / aZB )

2

=

ψ WZ WZ 1 − (γ eff / γ effZB ) 2

.

Page 14 of 30

(18)

The Glas condition shows that the WZ structure forms when the surface energy of WZ island is lower than that of ZB one, and requires a high enough chemical potential (larger than the critical value ∆µc ) to overcome the bulk energy difference between the WZ and ZB structuresψ WZ . The effective surface energies of ZB islands depend on the island position: either at the TPL where the vapor, liquid and solid phases meet or away from it (C). For WZ islands, we consider only the TPL position since it is known that the C nucleation of the WZ structure is always suppressed relative to ZB. For the TPL position, we use the known expressions20: WZ γ effZB = (γ ZB − γ LV sin β ) x + γ SL (1 − x ) ; γ eff = (γ WZ − γ LV sin β ) x + γ SL (1 − x ) , while for the (ZB,C)

nucleation scenario γ effZB = γ SL . The γ ZB , γ WZ are the surface energies of relevant ZB and WZ NW sidewall facets, γ LV is the liquid-vapor surface energy of the droplet, γ SL is the surface energy of the lateral solid-liquid interface of a 2D island, β is the contact angle of the droplet at the NW top and x = 1/ 3 is the fraction of the island perimeter at the TPL. Thus, the Glas condition actually gives the two branches of the critical chemical potential

∆µc which are shown in Fig. 2 for the following parameters of Au-catalyzed GaAs NWs: ψ WZ = 24 meV/pair41, γ WZ = 1.30 J/m2 for the lowest energy (1 1 00) WZ side facets and γ ZB = 1.543 J/m2 for the lowest energy (110) ZB side facets42, and β = 110o. The γ LV (c3 ) dependence is taken as the linear approximation between pure liquids ( γ Ga = 0.675 J/m2; γ Au = 1.248 J/m2 at 600oC (Ref. [43]), however, the temperature dependences of these surface energies is almost negligible). The γ SL (c3 ) is taken as the linear approximation between 1.0 J/m2 at c3 = 0.2 (Ref. [20]) and 0.123 J/m2 for a pure Ga droplet12. It is seen that WZ structure is prevalent at the topleft corner of the (∆µ , c3 ) plane, while the ZB structure forms when the liquid chemical potential


Page 15 of 30

is outside this corner. At low c3 , the critical chemical potential is determined by the competition between the two TPL nucleation scenarios. At large c3 , the TPL nucleation is suppressed on surface energetic grounds (due to a low surface energy of liquid Ga) actually regardless of the growth conditions, and the winning nucleation scenario is ZB islands forming away from the TPL. This picture can be considered temperature-independent due to a weak temperature dependence of the surface energies of interest.

400

Critical chemical potentials (meV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


350 300

WZ TPL to ZB C WZ TPL to ZB TPL WZ TPL to ZB C renormalized WZ TPL to ZB TPL renormalized

250

Wurtzite

200 150 100

Zincblende

50 0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Ga concentration c3 Figure 2. WZ “corner” at the top-left of the (c3 , ∆µ ) plane obtained with and without the SC renormalization for the parameters of Au-catalyzed GaAs NWs.

Now, if the SC renormalization is included, the exponential terms in the ZNR which

(

)

should be compared for the WZ and ZB structures are exp a − a 2 / 4∆µ . This leads to the modified condition for the preferred WZ structure of the form 2 aWZ a2 − aWZ < ZB − aZB 4(∆µ −ψ WZ ) 4∆µ

(19)

for both ZB nucleation scenarios. Resolving the quadratic equation for ∆µ , we obtain 15 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

∆µ >

B B 2 − 4C − , 2 2

Page 16 of 30

(20)

where 2 aZB aZB + aWZ ψ WZ . + ψ WZ ; C = B= 4 4(aZB − aWZ )

(21)

The second root of the quadratic equation is always so large that the upper limit for ∆µ is not relevant. The condition for the WZ phase formation given by Eq. (20) is distinctly different from the Glas condition defined by Eq. (18) and is reduced to the latter only when 16ψ WZ /[( a ZB + aWZ )(1 − ( aWZ / a ZB ) 2 )] > 4ψ WZ . The modified critical chemical

potentials are also shown in Fig. 2 for the same parameters of Au-catalyzed GaAs NWs as before. In is seen that the SC renormalization generally narrows the WZ corner so that the first ZB-to-WZ phase transition at low c3 occurs later and the reverse WZ-to-ZB transition at high c3 relates to a lower Ga concentration than in the Glas model. However, the overall effect of the SC renormalization on the position of the WZ corner is relatively small. Figure 3 shows the liquid-solid chemical potentials of 20 nm radius Au-catalyzed GaAs NWs calculated in the case of MBE at 550oC (Ref. [44]) and HVPE at 715oC (Ref. 34]) VLS growths with the parameters summarized in Table 1. The calculations are based on the expression of Ref. [37]:

1  ω v (1 + ε ) − ω3dL / dt  S ∆µ = µ3L + ln  5 5  − µ35 . 0 n  v5 

(22)

Here, the chemical potential of As in steady state is expressed through the direct atomic flux of As atoms into the droplet v5 , re-emitted As flux ( εv5 ), the NW elongation rate dL / dt and the characteristic As desorption flux v50 , with n being the number of As atoms in a given arsenic vapor. The coefficient ω5 = 1 / sin 2 β in MBE and 1 for vapor phase epitaxy, ω3 = 1 in MBE and 16 ACS Paragon Plus Environment

Page 17 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(1 + cos β ) / 2 in vapor phase epitaxy. The chemical potential of Ga atoms, µ 3L , is calculated from the model of Glas36, with the parameters v5 , v50 , µ 35S of Refs. [34,45-48], while the NW growth rates were directly measured. The MBE NWs have the modest growth rates of the order of 1 nm/s and are known to be predominantly WZ44, while the VLS-HVPE NWs growing at a much higher elongation rate of 28 nm/s are pure ZB regardless of their radius33,34. Within our structural model, the preferred crystal structure is now determined by the actual Ga concentrations during growth which relate to the measured growth rates dL / dt (Ref. [34]). A very important effect of the SC renormalization is seen immediately when we imply the self-consistency condition for the determination of the Ga concentration according to

dL 3 3 2 3 = R h∑η k J k . dt 2 k =1 Here,

the

factors

(23)

η k = 2 rc( k ) / R for

the

TPL

nucleation

position,

with

rc( k ) = (1 / 31 / 4 )( Ω 35 / h )1 / 2 [ a /( ∆µ − ψ k )] as the critical radii. These geometrical factors account for

the reduction in the available nucleation area to the TPL “ring” of the critical radius21. The ZNRs for the three nucleation scenarios k = (WZ,TPL), (ZB,TPL) and (ZB,C) relate to different γ eff( k ) in Eq. (2) for ak and different ψ k = ψ WZ for the WZ structure and ψ k = 0 for the ZB structure. Using Eq. (12) at χ = 0 , one obtains 2

 h  µ 35S +ψ k   ak2  e Jk = D5  c5 ( ∆µ − ψ k )1 / 2 exp ak − . 4( ∆µ −ψ k )  π  Ω35   33 / 4

(24)

Thus, we equate the observed NW elongation rate to the sum of the three possible nucleationmediated growth rates corresponding to different crystal phases and nucleation positions. This allows us to obtain the Ga concentrations during growth, which are shown by points in Fig. 3 with and without the SC renormalization.



Table 1. Growth parameters of Au-catalyzed GaAs NWs Growth method

T ( C) o

As

dL / dt

vapor

(nm/s)

(nm/s)

ε

v5

µ 35S

P50( n )

D5

(meV)

(Pa)

( m2/s)

v50 (nm/s)

MBE

550

As2

1

1

3.25

1.1 × 104

-1615

16

1.4 × 10-11

HVPE

715

As4

28

6.2 × 104

3.25

8.5 × 1014

-1820

2800

8.1 × 10-10

800

600

∆µ (meV/pair)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 30

Wurtzite o

MBE, T=550 C

o

HVPE, T=715 C

400

∆µC

∆µTPL

200

Zincblende 0 0.0

0.2

0.4

0.6

0.8

1.0

Ga concentration c3 Figure 3. Chemical potentials during Au-catalyzed VLS growth of GaAs NWs by MBE and HVPE versus Ga concentration, compared to the two critical chemical potentials. The diamond symbols show the calculated Ga concentrations for the parameters given in Table 1 without the SC renormalization of the ZNRs, while the circle symbols correspond to the ZNRs with the SC renormalization included.

Thus, the effect of the SC renormalization on the preferred crystal phase is very substantial. Indeed, the huge exponential factor exp(a ) in Eq. (12) increases the ZNRs by several orders of magnitude and therefore lowers down the calculated values of c3 for a given NW growth rate. Consequently, the “operating points” on the (c3 , ∆µ ) plane are shifted toward the WZ corner in both cases. For VLS-HVPE GaAs NWs, the Ga concentration remains within the 18 ACS Paragon Plus Environment

Page 19 of 30

ZB domain due to a much larger material input, although the chemical potential curve itself is lower than in the MBE case because of a much higher growth temperature36. This correlates with our previous results34 obtained without the ZNR renormalization and shows that 2D islands prefer to form in the C position due to a low surface energy of Ga-rich droplets. In MBE case, our calculations without the SC renormalization yield the ZB structure which is qualitatively incorrect, while the renormalization retains the structure to the experimentally observed dominant WZ phase at a modest Ga concentration of about 0.4. This value is sensitive to the As flux and would increase for lower v5 , which might explain the discrepancy with the measured Ga concentrations after growth (about 0.5 in Ref. [49] and more than 0.6 in Ref. [50]). We note that our results qualitatively correlate with the observations of Ref. [50], where the transition from Au-catalyzed to pseudo Ga-catalyzed VLS growth (at high Ga concentrations more than 0.7) led to the appearance of ZB structure of GaAs NWs.

1.0 Wurtzite percentage

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


o

HVPE, 715 C o MBE, 550 C

0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

Ga concentration c3 Figure 4. WZ percentages versus Ga concentration in Au-catalyzed MBE and HVPE growths of GaAs NWs. The diamond and circle symbols show the Ga concentrations relating the measured NW growth rates without and with the ZNR renormalization, respectively.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 30

In the first approximation, without accounting for a more delicate effects of correlations between the stacking sequences and the formation of higher order polytypes51,52, the WZ percentage in NWs can be obtained as

pWZ =

(dL / dt )WZ ,TPL . dL / dt

(24)

Here, (dL / dt )WZ ,TPL is the term in the right-hand side of Eq. (23) corresponding to the WZ phase formation via the (WZ,TPL) nucleation scenario, and the denominator sums up the three nucleation scenarios to yield the appropriately normalized probability. These graphs are shown in Fig. 4 for the same parameter values as in Fig. 3. It is seen that the curves of pWZ are rather different from a simplified picture shown in Fig. 3 and tend to zero at a lower Ga concentrations below 0.5, while the curves in Fig. 3 cross with ∆µC at c3 ≅ 0.6. This effect is associated with the geometrical pre-factors that reduce the TPL nucleation area to the ring of the critical radius. In summary, we have presented a new expression for the ZNR in Au-catalyzed VLS III-V NWs which takes into account the nucleation kinetics in a ternary Au-III-V alloy and the SC renormalization. It has been shown that the kinetic pre-factors contain the As diffusivity in liquid gold rather than that of Ga, because the As is a diluted and slow diffuser, with typical diffusion coefficients between 10-11 and 10-9 m2/s in the temperature range 550-715oC. The obtained ZNR is useful for growth and structural modeling of III-V NWs, and explains fairly well the experimental data on the preferred crystal structure of Au-catalyzed GaAs NWs fabricated by MBE and HVPE techniques. A more detailed analysis must include a study of the liquid-solid surface energies versus the group III concentration, which is solely lacking in the field of NW growth. We also plan to apply the obtained results to other Au-catalyzed III-V NWs synthesized by different growth techniques and under varying conditions such as temperature and V/III flux ratio.


Page 21 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Acknowledgement VGD gratefully acknowledges financial support of the Russian Science Foundation under the grant 14-22-00018. JG thanks FP7 project NANOEMBRACE (Grant Agreement 316751).

REFERENCES (1) Zeldovich, J. Sov. Phys. JETP (Eng. Transl.) 1942, 12, 525. (2) Dubrovskii, V.G. Nucleation theory and growth of nanostructures. Springer: Heidelberg – New York – Dordrecht – London, 2014. (3) Kashchiev, D. Nucleation: Basic Theory with Applications. Butterworth Heinemann: Oxford, 2000. (4) Kashchiev, D. J. Chem. Phys. 2008, 129, 164701. (5) Dubrovskii, V.G.; Nazarenko, M.V. J. Chem. Phys. 2010, 132, 114507. (6) Kukushkin, S.A.; Osipov, A.V. Prog. Surf. Sci. 1996, 51, 1. (7) Kuni, F.M.; Shchekin, A.K.; Grinin, A.P. Physics Uspekhi 2001, 171, 345. (8) Markov, I. Crystal growth for beginners. World Scientific: New Jersey – London – Singapore – Hong Kong, 2003. (9) Dubrovskii, V.G.; Sibirev, N.V.; Zhang, X.; Suris, R.A. Cryst. Growth Des. 2010, 10, 3949. (10) Wagner, R.S.; Ellis, W.C. Appl. Phys. Lett. 1964, 4, 89. (11) Dubrovskii, V.G.; Sibirev, N.V.; Cirlin, G.E.; Harmand, J.C.; Ustinov, V.M. Phys. Rev. E

2006, 73, 021603. (12) Glas, F.; Ramdani, M.R.; Patriarche, G.; Harmand, J.C. Phys. Rev. B 2013, 88, 195304. (13) Kashchiev, D. Cryst. Growth Des. 2006, 6, 1154. (14) Glas, F.; Harmand, J.C.; Patriarche, G. Phys. Rev. Lett. 2010, 104, 135501. (15) Dubrovskii, V.G. Phys. Rev. B 2013, 87, 195426.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 30

(16) Sibirev, N.V.; Nazarenko, M.V.; Zeze, D.A.; Dubrovskii, V.G. J. Cryst. Growth 2014, 401, 51. (17) Glas, F. Phys. Rev. B 2014, 90, 125406. (18) Gamalski, A.D.; Voorhees, P.W.; Ducati, C.; Sharma, R.; Hofmann, S. Nano Lett. 2014, 14, 1288. (19) Wen, C.-Y.; Tersoff, J.; Hillerich, K.; Reuter, M.C.; Park, J.H.; Kodambaka, S.; Stach, E.A.; Ross, F.M. Phys. Rev. Lett. 2011, 107, 025503. (20) Glas, F.; Harmand, J.C.; Patriarche, G. Phys. Rev. Lett. 2007, 99, 146101. (21) Dubrovskii, V.G.; Sibirev, N.V.; Harmand J.C.; Glas, F. Phys. Rev. B 2008, 78, 235301. (22) Dubrovskii, V.G.; Sibirev, N.V.; Cirlin, G.E.; Bouravleuv, A.D.; Samsonenko, Yu.B.; Dheeraj, D.L.; Zhou, H.L.; Sartel, C.; Harmand, J.C.; Patriarche, G.; Glas. F. Phys. Rev. B 2009, 80, 205305. (23) Shtrikman, H.; Popovitz-Biro, R.; Kretinin, A.; Heiblum, M. Nano Lett. 2009, 9, 215-219. (24) Fakhr, A.; Haddara, Y.M.; LaPierre, R.R. Nanotechnology 2010, 21, 165601. (25) Dubrovskii, V.G.; Cirlin, G.E.; Sibirev, N.V.; Jabeen, F.; Harmand, J.C.; Werner, P. Nano Lett. 2011, 11, 1247-1253. (26) Husanu, E.; Ercolani, D.; Gemmi, M; Sorba, L. Nanotechnology 2014, 25, 205601. (27) Johansson, J.; Karlsson, L.S.; Dick, K.A.; Bolinsson, J.; Wacaser, B.A.; Deppert, K.; Samuelson, L. Cryst. Growth Des. 2009, 9, 766. (28) Dick, K. A.; Caroff, P.; Bolinsson, J.; Messing, M. E.; Johansson, J.; Deppert, K.; Wallenberg, L. R.; Samuelson, L. Semicond. Sci. Technol. 2010, 25, 024009. (29) Lehmann, S.; Wallentin, J.; Jacobson, D.; Deppert, K.; Dick, K.A. Nano Lett. 2013, 13, 4099. (30) Joyce, H. J.; Wong-Leung, J.; Gao, Q.; Tan, H. H.; Jagadish, C. Nano Lett. 2010, 10, 908. (31) Huang, H. ; Ren, X. ; Ye, X. ; Guo, J. ; Wang, Q. ; Yang, Y. ; Cai, S. ; Huang, Y. Nano Lett. 2010, 10, 64.


Page 23 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(32) Ren, X.; Huang, H.; Dubrovskii, V.G.; Sibirev, N.V.; Nazarenko, M.V.; Bolshakov, A.D.; Ye, S.; Wang, Q.; Huang, Y.; Zhang, X.; Guo, J.; Liu, X. Semicond. Sci. Technol. 2011, 26, 014034. (33) Ramdani, M.R.; Gil, E.; Leroux, Ch.; André, Y.; Trassoudaine, A.; Castelluci, D.; Bideux, L.; Monier, G.; Robert-Goumet, C.; Kupka, R. Nano Lett. 2010, 10, 1836-1841. (34) Gil, E.; Dubrovskii, V.G.; Avit, G.; André, Y.; Leroux, C.; Lekhal, K.; Grecenkov, J.; Trassoudaine, A.; Castelluci, D.; Monier, G.; Ramdani, M.R.; Robert-Goumet, C.; Bideux, L.; Harmand, J.C.; Glas, F. Nano Lett. 2014, 14, 3938. (35) Spirkoska, D.; Arbiol, J.; Gustafsson, A.; Conesa-Boj, S.; Glas, F.; Zardo, I.; Heigoldt, M.; Gass, M. H.; Bleloch, A. L.; Estrade, S.; Kaniber, M.; Rossler, J.; Peiro, F.; Morante, J. R.; Abstreiter, G.; Samuelson, L.; Fontcuberta i Morral1, A. Phys. Rev. B 2009, 80, 245325. (36) Glas, F. J. Appl. Phys. 2010, 108, 073506. (37) Dubrovskii, V.G. Appl. Phys. Lett. 2014, 104, 053110. (38) Lehmann, S.; Jacobsson, D.; Deppert, K.; Dick, K. Nano Res. 2012, 5, 470. (39) Assali, S.; Zardo, I.; Plissard, S.; Kriegner D.; Verheijen M.A.; Bauer, G.; Meijerink,A.; Belabbes, A.; Bechstedt, F.; Haverkort, J. E. M.; Bakkers, E.P.A.M. Nano Lett. 2013, 13, 1559. (40) André, Y.; Lekhal, K.; Hoggan, P.; Avit, G.; Cadiz, F.; Rowe, A.; Paget, D.; Petit, E.; Leroux, C.; Trassoudaine, A.; Ramdani M.R.; Monier, G.; Colas, D.; Ajib, R.; Castelluci, D.; Gil, E. J. Chem. Phys. 2014, 140, 194706. (41) Yeh, C.Y.; Lu, Z.W.; Froyen, S.; Zunger, A. Phys. Rev. B 1992, 46, 10086. (42) Sibirev, N.V.; Timofeeva, M.A.; Bolshakov, A.D.; Nazarenko, M.V.; Dubrovskii, V.G. Phys. Solid State 2010, 52, 1531. (43) Jasper, J.J. J. Phys. Chem. Ref. Data 1972, 1, 841. (44) Soshnikov, I.P.; Cirlin, G.E.; Tonkikh, A.A.; Samsonenko, Y.B.; Dubrovskii, V.G.; Ustinov, V.M.; Gorbenko, O.M.; Litvinov, D.; Gerthsen, D. Phys. Sol. State 2005, 47, 2213. (45) Stringfellow, G.B. J. Phys. Chem. Solids 1972, 33, 665. (46) Dinsdale, A.T. CALPHAD 1991, 15, 317. (47) Ansara, I.; Chatillon, C.; Lukas, H.L.; Nishizawa, T.; Ohtani, H.; Ishida, K.; Hillert, M.; Sundman, B.; Argent, B.B.; Watson, A.; Chart, T.G.; Anderson, T. CALPHAD 1994, 18, 177. 23 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 30

(48) Shawarz R.I. Thin Solid Films 1980, 66, L3. (49) Harmand, J. C.; Patriarche G., Péré-Laperne N., Mérat-Combes, M.-N.; Travers L.; Glas, F. Appl. Phys. Lett. 2005, 87, 203101. (50) Soda, M.; Rudolph, A.; Schuh, D.; Zweck, J.; Bougeard, D.; Reiger, E. Phys. Rev. B 2012, 85, 245450. (51) Johansson, J.; Bolinsson,J.; Ek, M.; Caroff, P.; Dick, K.A. ACS Nano 2012, 6, 6142. (52) Priante, G.; Harmand, J.C.; Patriarche, G.; Glas F. Phys. Rev. B 2014, 89, 241301(R).


Page 25 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


For Table of Contents Use Only

Zeldovich nucleation rate, self-consistency renormalization and crystal phase of Aucatalyzed GaAs nanowires V.G. Dubrovskii and J. Grecenkov We derive a new expression for the Zeldovich nucleation rate in Au-catalyzed III-V nanowires, which is proportional to the group V concentration and group V diffusion coefficient. The leading exponential term of the nucleation rate is modified due to the self-consistency renormalization. The obtained results allow for a fully self-consistent modeling of vapor-liquid solid growth and crystal structure of III-V nanowires as demonstrated by our calculations.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

Page 26 of 30

n1

(a)

D  eff 3L



L 5



S 35

(b)

c5

D5

ACS Paragon Plus Environment

Ga As

Page 27 of 30

400

Critical chemical potentials (meV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42


350 300

WZ TPL to ZB C WZ TPL to ZB TPL WZ TPL to ZB C renormalized WZ TPL to ZB TPL renormalized

250

Wurtzite

200 150 100

Zincblende

50 0 0.0

0.1

0.2

0.3

0.4

0.5

Ga concentration c3


0.6

0.7

0.8


800

600

 (meV/pair)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

Page 28 of 30

Wurtzite o

MBE, T=550 C

o

HVPE, T=715 C

400

C

TPL

200

Zincblende 0 0.0

0.2

0.4

0.6

Ga concentration c3 ACS Paragon Plus Environment

0.8

1.0

Page 29 of 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42


o

HVPE, 715 C o MBE, 550 C

0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

Ga concentration c3


0.8

1.0



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

Page 30 of 30

0.8 0.6 0.4 0.2

MBE, 550oC HVPE, 715oC

0.0 0.0

0.2

0.4

0.6

Ga concentration c3 ACS Paragon Plus Environment

0.8

1.0

Zeldovich Nucleation Rate, Self-Consistency Renormalization, and

Recommend Documents