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Development of growth theory for vapor-liquid-solid nanowires: contact angle, truncated facets and crystal phase V. G. Dubrovskii Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.7b00076 • Publication Date (Web): 28 Mar 2017 Downloaded from http://pubs.acs.org on April 1, 2017

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Crystal Growth & Design

Development of growth theory for vapor-liquid-solid nanowires: contact angle, truncated facets and crystal phase V. G. Dubrovskii1,2,3* 1

2

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

Ioffe Physical Technical Institute of the Russian Academy of Sciences, Politekhnicheskaya 26, 194021 St. Petersburg, Russia 3

ITMO University, Kronverkskiy pr. 49, 197101 St. Petersburg, Russia

ABSTRACT: This paper tries to circumvent the fundamental uncertainty in the contact angle of droplets catalyzing vapor-liquid-solid nanowires. Based entirely on surface energetic grounds, our model predicts the following behavior. The initial contact angle of the surface droplets is smaller than for developed nanowires. At the beginning, the energetically preferred growth configuration is inward tapered nonwetted side facets such that the contact angle increases and the radius decreases with length. Such isotropic growth continues until a steady state contact angle is reached which is usually larger than 90o (111o for the plausible parameters of Au-catalyzed GaAs nanowires) but smaller than the Nebol’sinShchetinin angle. This value determines the contact angle for stable vertical nanowires at a fixed liquid volume. If the latter keeps increasing for kinetic reasons, the contact angle also increases to the second critical point (128o for our parameters) where vertical interface is transitioned to a wetted truncated as in Tersoff’s picture. This wetted configuration is stable against the droplet sliding down. The crystal phase is predominantly wurtzite for smaller contact angles and planar growth interfaces according to Glas et al., and pure zincblende for larger contact angles and truncated growth interfaces according to Tersoff et al.

*[email protected]

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INTRODUCTION Since the early stage of semiconductor nanowire research1,2 and over the recent boost

whereby nanowires have become most promising building blocks for fundamental nanoscience and nanotechnology3-5, such structures have mainly been grown by the vapor-liquid-solid (VLS) method1. Essentially, the VLS technique uses liquid droplets of a metal catalyst that are formed initially on a substrate surface and then promote the vertical growth of nanowires. The steady state growth geometry, as it was first drawn by Wagner and Ellis1 and later by Givargizov2, assumes a time-independent droplet volume, shape, vertical nanowire sidewalls and planar growth interface with the nanowire edges pinned to the droplet surface at the triple phase line. For cylindrical nanowire geometry, the droplet must be a spherical cap with the contact angle β to the horizontal, and even for hexahedral nanowire this shape does not change too much6. Such geometry was used for the growth and structural modeling of different semiconductor nanowires, in particular, in connection with a very interesting effect of wurtzite/zincblende (WZ/ZB) polytypism of III-V nanowires7-11. Only recently, using in situ high-resolution growth monitoring in a transmission electron microscope (TEM), it has been found that some elemental as well as III-V nanowires exhibit wetted truncated facets at the growth interface12-14. This truncated geometry is of paramount importance from many viewpoints and in particular because it completely eliminates the possibility to form the WZ phase in III-V nanowires12. The energetically preferred growth configurations of the droplet with respect to the nanowire top are highly sensitive to the contact angle7-9,13-21. It also enters the conditions governing the crystal phase of III-V nanowires, in both planar7 and non-planar geometries of the growth interface (in the latter case, this is simply related to whether the top facet is truncated or not13). Quite surprisingly, almost all growth theories developed so far treat the contact angle as a free parameter7-14 that depends, for example, on the gallium flux in Au-catalyzed VLS growth of III-V nanowires13. The only expression given for the contact angle by Nebol’sin and Shchetinin18 is based on equilibrium considerations of the droplet shape and under the constraint of a fixed 2 ACS Paragon Plus Environment

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droplet volume, the condition which is never satisfied in layer-by-layer growth of narrow nanowires of interest (see Ref. [11] for a detailed review). The persisting “mystery” of the contact angle is unacceptable given its fundamental importance for growth and physical properties of nanowires11. Another issue is related to the very early stage of the VLS process. Usually, the initial contact angle of the droplet resting on a substrate surface is smaller than that for a developed nanowire and hence the bottom part of nanowire tapers to transfer one geometry to the other7,11,15,16. Most droplets observed on tops of long nanowires have the contact angles of 90o or larger7,10-14. However, it is poorly known why and how this tapering goes and when it stops, partly due to the mentioned uncertainty in the steady state contact angle. The existing models for the initial nanowire shapes are based either on fully isotropic geometries without crystal facets15 or equilibrium thermodynamic considerations16.

The VLS growth start is also

important for understanding the incubation times that are required to nucleate anisotropic nanowires and critically influence the collective properties within the resulting nanowire ensembles (such as the length distributions)17. Furthermore, the Glas condition for the triple phase line nucleation7, which is also the necessary condition for the WZ phase formation in III-V nanowires with planar growth interfaces, is identical to the Nebol’sin-Shchetinin condition for stable VLS growth of cylindrical nanowires18. When it is broken, as in Ga-catalyzed GaAs nanowires19, the droplet should slide down along the vertical sidewalls to finally acquire a non-spherical shape19, which is not often seen experimentally. Generally, it remains unclear how self-catalyzed III-V nanowires can grow if their group III droplets are unstable on the nanowire tops. The model of Tersoff12, supported by the in situ TEM data12-14, only partly resolves this issue by introducing a wetted truncated edge because, if the droplet randomly shifts down and reach the vertical part of the nanowire, it may continue sliding down according to Ref. [19]. Consequently, herein we present a model that answers the following questions: i) What is the steady state contact angle of the droplet on top of vertical VLS nanowire? 3 ACS Paragon Plus Environment


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ii) What determines the initial radius of vertical nanowire tops? iii) Why and how the nanowire geometry changes at the beginning of growth? iv) When vertical facets are transitioned to truncated at large contact angles, why this truncated configuration is stable? Answering these questions is the really new aspect of this work. Additionally, our model contains the result of Ref. [12] on oscillatory behavior of the truncated growth interface. It refines the results of Tersoff et al.13 regarding pure ZB phase of III-V nanowires with truncated growth interfaces catalyzed by large contact angle droplets (such as III-rich Au-III-V droplets) and the dominant WZ phase of vertical nanowires otherwise, showing why the wetted truncated configuration is stable against the droplet sliding down. Our model uses only one crystallographic assumption on the three possible types of facets restricting the nanowires (vertical, narrowing and widening), which can easily be modified to account for more facets and hence more complex nanowire shapes. We do not introduce any additional parameters and use only the surface energies of different interfaces (unfortunately, some of them are unknown or poorly known, particularly for the solid-liquid interfaces). We show that the Nebol’sinShchetinin formula for the steady state contact angle18, although cannot be rigorously justified, gives a reasonable estimate for the true value.

NEBOL’SIN-SHCHETININ CONTACT ANGLE

The Nebol’sin-Shchetinin expression for the steady state contact angle of the droplet β * seated on top of cylindrical nanowire with planar growth interface18, is given by cos β * = −γ SL / γ LV , where γ SL is the surface energy of the solid-liquid interface under the droplet

and γ LV is the liquid-vapor surface energy of the droplet. It suggests that droplet is always more than a hemisphere and corresponds to the balance of horizontal forces acting upon the triple phase line. This formula has been used extensively in modeling the VLS nanowire growth and its stability11,16,20,21. Interestingly, it was justified in Ref. [22] for Si nanowires with sawtooth 4 ACS Paragon Plus Environment

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faceting of the sidewalls, although on completely different grounds. Original consideration of Ref. [18] was the minimization of the reduced surface energy of a VLS system related to the droplet only, G0 = γ LV 2πR 2 /(1 + cos β ) + γ SLπR 2 , with R as the nanowire radius, and under the condition of a fixed liquid volume V in the non-wetting configuration. Supporting information (SI) 1 shows that doing so is generally incorrect because varying the droplet base radius R necessarily requires a modification of the nanowire sidewalls, which can be done in many ways and changes the surface energy of the growth interface. Same applies to derivation of Ref. [18] for the contact angle of inward tapered conical nanowires. Generally, we can conclude that there is no equilibrium contact angle corresponding to the minimum of the reduced surface energy that includes only droplet but ignores the nanowire sidewalls. In addition, considering the VLS growth kinetics of small nanowires at a constant liquid volume is incorrect for monolayers, because their formation is known to have the mononuclear character whereby the equivalent monolayer volume is instantaneously removed from the droplet in each VLS growth pulse7,8,11,12.

Figure 1. Cylindrical and inward tapered conical nanowires before (top) and after (bottom) instantaneous formation of a monolayer in non-wetting (1) and wetting (2) configurations of spherical cap droplet. Mononuclear growth proceeds at a fixed number of semiconductor atoms or pairs in the VLS system (

N = const ). Surface energies of forming the monolayers are insensitive to the pre-history of nanowires, i.e., whether the lower part is vertical or tapered. However, if truncated facets are preferred, the faceting will cut through a thickness of material much larger than a single monolayer. 5 ACS Paragon Plus Environment


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GROWTH PICTURE

We claim that, if the allowed side facets of a nanowire are given (for example, defined by equilibrium shapes of the crystal) and under the assumption of a fixed liquid volume during vertical growth on a large time scale (that is, not for nanowire monolayers), both the steady-state contact angle and the nanowire radius are exactly known. To prove that, we consider angularisotropic geometries shown in Fig. 1, with a possibility to form either vertical or two truncated facets, one of which is narrowing and the other widening. This simplifies the approach of Refs. [15,23] where the initial nanowires were allowed to have non-vertical growth directions or form side facets of any orientations. Although it is not critical, in the following we restrict ourselves to the case of two symmetrical non-vertical facets making the angles θ = ±α to the vertical, as (111)A and (111)B facets of ZB III-V nanowires ( α =19.5o)7. The surface energies of the initial VLS system, in addition to the previously introduced γ SL and γ LV , involve those of the solidvapor interfaces, denoted γ 0V for vertical facets and γ θV for tapered facets. The symmetrical case corresponds to γ αS = γ −αS . For sufficiently narrow nanowires of interest, the VLS growth proceeds in the mononuclear regime7-11,24 in which the nucleation of a monolayer takes much shorter time than the refill stage. Therefore, we consider the surface energies ∆Gs of forming a monolayer of height h either in the non-wetting mode (1) where the droplet stays on the nanowire top or in the wetting mode (2) where the droplet surrounds a monolayer, under the assumption of a fixed number of semiconductor atoms or pairs N in the initial droplet and in the droplet plus monolayer after the monolayer formation. The liquid volume V in such process is not conserved. In the wetting mode, we need to introduce three more surface energies, namely, those of the liquid-solid interfaces for vertical ( γ 0 L ) and tapered ( γ θL ) facets. The symmetrical case corresponds to

γ αL = γ −αL . The minimum of the surface energies of forming the allowed facets will determine the actual growth mode, i.e., vertical or tapered and wetted or not. 6 ACS Paragon Plus Environment

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Taking the normalized surface energies F = ∆Gs /(2πRh ) , the result for the non-wetting mode (1) is given by Fθ(1) =

Ω γ θV − γ LV L sin β − (γ SL + γ LV cos β ) tan θ . cos θ ΩS

(1)

This is reduced to F0(1) = γ 0V − γ LV

ΩL sin β ΩS

(2)

for vertical facets. These expressions are derived in SI 2 in the case of circular nanowire crosssections. The Ω L and Ω S denote the elementary volumes per atom or pair in the liquid and solid phases, respectively. In the wetting mode, the results change to

Fθ( 2 ) =

Ω  γ θL − γ LV  L − 1 sin β − γ SL tan θ , cos θ  ΩS 

(3)

Ω  F0( 2 ) = γ 0 L − γ LV  L − 1 sin β (4)  ΩS  When γ θV > γ 0V cos θ , vertical facets in contact with vapor are less energetically costly with respect to tapered, meaning that the nanowire sidewalls themselves are stable against faceting.

Figure 2. Surface energies of forming the nanowire monolayers with vertical ( θ = 0 ) and tapered side facets ( θ = 19.5o for widening and -19.5o for narrowing facet) for the parameters of Au-catalyzed GaAs nanowires. Solid lines relate to the non-wetting and dashed to the wetting modes. The chart illustrates



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possible growth morphologies. Point I represents the initial condition, a droplet resting on the substrate with a contact angle, say, 60o. The lowest surface energy mode at the beginning of growth is forming inward tapered non-wetted facets, which explains why nanowires can start and why their contact angle increases. If the droplet volume is fixed from the very beginning or stabilizes early, the nanowire radius will decrease accordingly. After crossing point II, the surface energy of forming tapered monolayers becomes higher than that of vertical non-wetted monolayers. This point is stable, because any random change to a smaller contact angle (by making a widening facet) will immediately introduce the opposite facet and the contact angle will retain to point II. Any narrowing facet will also shrink and disappear. Therefore, the steady state contact angle β * in stable vertical growth is obtained from the condition

Fθ(1) ( β * ) = F0(1) ( β * ) for narrowing facet ( θ > 0) . The growth interface of the upper part of the nanowire will always remain planar, and the crystal structure of GaAs will be predominantly WZ according to Glas7. This situation is idealized because the droplet volume can be changed kinetically in the initial stage, particularly for group III rich alloys. If the droplet volume keeps increasing, say, due to excessive gallium influx, the contact angle will further increase to point III after which the surface energy of the wetted truncated facets becomes lower. In this case, however, truncated facet will not grow infinitely, but rather to a finite size according to Tersoff et al12. The droplet will newer slide downward because the surface energy of the wetted vertical sidewalls lies higher (in other words, the Nebol’sin-Shchetinin condition for stable VLS growth in vertical direction18 is still satisfied). This suppresses the Dubrovskii scenario with a non-spherical droplet shape19 that would correspond to point V. The top part of the nanowire should be pure ZB according to Ref. [13]. Overall, the WZ phase should dominate for β between ~ 111o and ~ 128o for these parameters, while larger β favor pure ZB phase.

Figure 2 shows the curves of different surface energies for the typical parameters of Aucatalyzed GaAs nanowires as an example25: γ LV = 1.0 J/m2, γ 0V = 1.3 J/m2, γ SL = 0.6 J/m2,

γ θV = 1.25 J/m2 and γ θL = 0.7 J/m2 for both tapered facets and Ω L / Ω S = 0.883. Alternatively, we can use the lower γ 0V value of ~ 0.7 given in Ref. [26] and decrease the solid-liquid surface energy according to Ref. [27], yielding a similar behavior. The chart illustrates the evolution of the nanowire morphology, starting from the substrate and increasing the contact angle by forming narrowing non-wetted facets. This process yields the conical shape of the bottom part of the nanowire, as observed experimentally7,11, and continues until the surface energy of tapered monolayers becomes equal that of vertical non-wetted monolayers. If the droplet volume


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stabilizes before that point, the steady state contact angle relates to Fθ(1) = F0(1) for positive θ and is given by cos β * = −

γ SL γ θV − γ 0V cos θ + γ LV γ LV sin θ

(5)

For the energetically preferred vertical facets (a positive second term), this decreases the Nebol’sin- Shchetinin value of the contact angle, which is given by the first term of this formula and relates to the case γ θV = γ 0V cos θ where the sidewall surface energy is insensitive to the facet orientation. This contact angle corresponds to stable vertical growth without changing the droplet volume. The radius of vertical part of the nanowire is given by R* = R0 [ f ( β 0 ) / f ( β )]1 / 3 , where

R0 and β 0 are

the

initial

radius

and

contact

angle

and

f ( β ) = (1 − cos β )(2 + cos β ) /[(1 + cos β ) sin β ] .

STABLE TRUNCATED FACET If the droplet volume is allowed to increase above point II in Fig. 2, as in group III rich

alloys with gold or self-catalyzed III-V nanowires13,27-31, the facets will remain vertical between points II and III, while after point III the introduction of truncated wetted facet is preferred. Now, the question is to which extent this facet will grow (because continuing growth of the facet would destroy the VLS configuration by piercing the droplet). According to Ref. [12], answering that cannot be done by considering the surface energies alone but requires introduction of a volume term in the formation energy related to chemical potential. Within our model, rather than considering instantaneous formation of monolayers at a fixed number of semiconductor atoms in the system, we study the interface stability by comparing the difference of the total free energies of forming vertical non-wetted nanowire section of height y > h and the truncated wetted section of the same height and base radius. Both processes should now be considered at a fixed liquid volume V = const , because the formation of any facet larger than one monolayer is no longer instantaneous as it requires refill each time before nucleation of the next monolayer. The model 9 ACS Paragon Plus Environment


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geometry is shown in Fig. 3. Denoting the two free energies as ∆G (1) and ∆G ( 2 ) , respectively, and taking the difference, the free energy of forming truncated wetted facet relative to vertical non-wetted is given by

γ sin β  tan θ 2 ∆G ( 2 ) − ∆G (1)  γ θL   = − γ 0V + γ LV sin β − γ SL tan θ  y +  ∆µ − LV y .  2πR R  cos θ    2

(6)

Here, ∆µ is the chemical potential difference that drives the nanowire growth per unit volume of the solid. The details of calculations are given in SI 3.

Figure 3. Illustration of the growth modes with vertical non-wetted facets (1) and truncated wetted facets (2) at a fixed liquid volumeV = const .

Thus, we arrived at the same result as in Refs. [12,13]. The normalized free energy difference between truncated and vertical facets has the form g = c1 y + c2 y 2 , where the sign of c1 depends on the contact angle. This property is central for obtaining the growth diagrams such

as shown in Fig. 2. Comparing Eq. (6) to Eqs. (2) and (3), we can see that c1 = Fθ( 2 ) − F0(1) and hence the condition c1 < 0 is equivalent to Fθ( 2 ) < F0(1) . The sign of c2 should be positive for droplets whose size is large enough to overcome the Gibbs-Thomson effect8 – otherwise, the VLS growth is not possible (actually, this remark equally applies to all our previous considerations). As discussed by Tersoff et al.12, there is a stable size of truncated facet at c1 < 0 which oscillates in synchronization with chemical potential due to nucleation antibunching12,32,33. Crystallization cannot start on the truncated facet and hence the triple phase line nucleation is 10 ACS Paragon Plus Environment

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suppressed, leading to ZB phase of III-V NWs12. Most importantly for our analysis, the truncated geometry occurs whenever Fθ( 2 ) is smaller than F0(1) , i.e. for contact angles larger than point III in Fig. 2. The droplet will never slide downward along the vertical sidewalls, because the surface energy F0( 2 ) for that process is larger. This proves the stability of the truncated geometry against the droplet sliding down, at least for the parameters considered. With our model parameters, the outward faceting (corresponding to the green curve in Fig. 2) has no influence on the morphology due to a higher surface energy of forming such facets. It can be shown, however, that if tapered facets have lower surface energy compared to vertical ones, we can observe the sawtooth faceting of the nanowire sidewalls as described in Ref. [22] and often seen in ZB III-V nanowires11. This requires the presence of a closed loop on the morphology diagram corresponding to the two alternating widening and narrowing facets, perhaps including some wetting geometries within a range of contact angles. Of course, one can argue that the surface energy values depend on the droplet composition26,34 and the crystal phase7-9, while we have used the same numbers for different conditions. We believe, however, that the presented picture gives a good estimate for the VLS growth behavior and the crystal phase trends, which can be further refined for particular material systems and growth conditions. It is noteworthy that the morphology trends such as shown in Fig. 2 actually depend on the surface energy ratios rather than their absolute values. This fact can broaden the applicability range of the composition and phase independent diagrams.

CONCLUSIONS

In summary, we have obtained a new expression for the steady state contact angle of stable vertical nanowires with planar growth interfaces, which can be used for the analysis and modeling of the VLS growth rates, morphologies and crystal structures of semiconductor nanowires. Our value is different from the Nebol’sin-Shchetinin angle and is smaller than the latter if vertical facets have lower surface energy than tapered. The steady state contact angle in 11 ACS Paragon Plus Environment


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our picture depends not only on the surface energies of the solid-liquid and liquid-vapor interfaces, but also on the surface energies of vertical nanowire sidewalls and inward tapered sidewalls in contact with vapor, as well as on the orientation of tapered sidewalls from which the VLS growth has transitioned to vertical anisotropic. It has been confirmed that planar growth interfaces are energetically preferred only within a range of contact angles if the droplet volume is allowed to vary during growth, while larger contact angles yield stable wetted truncation of the growth front, whose amount of truncation oscillates in synchronization with chemical potential. The truncated geometry is stable against the droplet sliding down because truncated facet has a lower surface energy than vertical. These two geometries lead to different crystal phases of III-V nanowires, with the WZ phase being preferred for smaller and the ZB phase for large contact angles. This explains, for example, pure ZB phase of Au-catalyzed GaAs nanowires grown by hydride vapor phase epitaxy down to 5 nm radius10, where high material inputs yield gallium-rich catalyst alloys. We have shown how and under which conditions the nanowires can start from the substrate and shrink their top radius by introducing inward tapered facets. Our model contains earlier results of nanowire modeling within a broader picture. It is not restricted to particular material system and should work equally well for elemental, II-VI and oxide nanowires grown by different epitaxy techniques and on different substrates. The necessary refinements require knowing the surface energy values for the allowed side facets as functions of the composition and crystal phase. Overall, the presented approach should be useful for fundamental understanding as well as tailoring the properties of VLS nanowires to the desired values.

SUPPORTING INFORMATION

Details of calculations for the Nebol’sin-Shchetinin contact angle (SI 1), surface energy differences of forming vertical and tapered monolayers in instantaneous nucleation process (SI 2), and Eq. (6) on a longer time scale under the constraint of a fixed droplet volume (SI 3).


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ACKNOWLEDGMENT

The author gratefully acknowledges financial support of the Russian Science Foundation under the Grant 14-22-00018.

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For Table of Contents Use Only Development of growth theory for vapor-liquid-solid nanowires: contact angle, truncated facets and crystal phase V. G. Dubrovskii

Synopsis Our model circumvents fundamental uncertainty in the contact angle of the droplets catalyzing the VLS growth of nanowires, gives a refined growth picture with vertical non-wetted or truncated wetted facets being preferred for small and large contact angles, respectively. It describes the transitions between different growth modes that influence the resulting morphology and crystal phase of III-V nanowires.


Development of Growth Theory for VaporâLiquid ... - ACS Publications

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