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Phase Field Model for Morphological Transition in Nanowire Vapor-Liquid-Solid Growth Yanming Wang, Paul C McIntyre, and Wei Cai Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.7b00197 • Publication Date (Web): 10 Mar 2017 Downloaded from http://pubs.acs.org on March 20, 2017

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Crystal Growth & Design

Phase Field Model for Morphological Transition in Nanowire Vapor-Liquid-Solid Growth Yanming Wang,∗,†,‡ Paul C. McIntyre,‡ and Wei Cai∗,†,‡ †Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA ‡Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305, USA E-mail: [email protected]; [email protected] Abstract The vapor-liquid-solid (VLS) method is widely used for nanowire (NW) synthesis. However growth of irregular NW geometries is often observed, and the mechanisms responsible are not fully understood. In this paper, we present a multi-phase field model for studying NW morphological transition during the VLS growth. Introducing the chemical potential and an external perturbation force by means of the method of Lagrange multipliers, the position of the catalyst droplet and the volumetric growth rate of the NW can be precisely controlled in the model. This allows us to capture the morphology of NWs growing along metastable orientations and simulate the formation of complex NW structures. Additionally, with the model’s ability to constrain the NW shape, the interface free energy change along designated droplet moving path can be evaluated, from which a free energy landscape for the catalyst droplet on top of a NW pedestal can be obtained. ACS Paragon Plus Environment

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Introduction

Semiconductor nanowires (NWs) are considered as promising components of the next-generation electronic and optoelectronic devices, for their special one-dimensional nanostructure. 1 Vaporliquid-solid (VLS) growth has been widely used for nanowire synthesis. 2,3 However, several fundamental questions regarding the VLS growth still await answers. Irregular nanowire geometries, either created intentionally or formed unexpectedly, have been reported, such as kinks, 4–6 branches 7 and helices. 8 Several hypotheses have been proposed to explain the formation of these NW anomalies: they may be related to capillary instability of the liquid catalyst droplet (e.g. unpinning and breakup) and NW growth orientation selection (e.g. dependence on NW radius, vapor pressure and temperature). 6,9,10 Due to the limited time resolution and specialized operation condition of the current in-situ characterization techniques, it is difficult to distinguish the controlling mechanism from dynamic observation of the NW morphological transition during the VLS process. Therefore, modeling and simulations can serve as important complements, in order to establish a full understanding of the NW growth mechanisms. As the surface energies are believed to play an important role in the NW VLS growth, 11,12 the phase field model, which has been successful in modeling the evolution of interfaces, can be an effective approach to studying VLS growth. The existing phase field models have shown the ability to capture reasonable wetting behaviors of the VLS catalyst-nanowire system at both equilibrium 13 and steady state growth conditions. 14 In order to study structural transitions, e.g. NW kinking, it is necessary to simulate nanowire growth and morphology evolution under conditions in which the catalyst droplet position is metastable or unstable. For this purpose, we have recently developed a three-dimensional multi-phase field model and applied it for studying NW kinking problems. 6 In this paper, we will describe this phase field model in detail, as well as several extensions and further applications to NW VLS growth. The paper is organized as follows. In Section 2, we describe the model formulation and discuss the algorithms for introducing an ACS Paragon Plus Environment

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external force, modifying the chemical potential and constraining the NW shape. Section 3 demonstrates the applications of the model, including a series of NW growth along various orientations, NW growth with oscillating wire diameter and a 2D free energy landscape for studying droplet capillary instability on top of a NW pedestal. A brief summary is given in Section 4.

2

Phase field model formulation

2.1

Free energy functional

The model presented here is based on our previous 3D multi-phase field model for NW VLS growth. 14 The fundamental degrees of freedom of our model are the phase fields φi (x), where i = L, S, V for liquid, solid and vapor phases, respectively. By convention, at every point x, φi (x) = 1 means point x is in the ith phase, φi (x) = 0 means it is not in the ith phase, and φi (x) ∈ (0, 1) means point x is on the phase boundary. Additionally, the constraint φS (x) + φL (x) + φV (x) = 1 is applied at every point in the simulation domain. The total free energy of the system, expressed as a functional of the phase fields, is given by Equation 1.

F [φL (x), φS (x), φV (x)] =

X Z

fi (φi (x), ∇φi (x)) d3 x

i=L,S,V

+ + +

Z

[αx (x − x0 ) + αy (y − y0 )] C(φL ) d3 x

Z

P φ2V (x) φ2L (x) φ2S (x) d3 x

Z

2 3 PS (φS (x) − φorig S (x)) d x

(1)

where fi is the local free energy density of phase i, fi (φi , ∇φi ) = Ui φ2i (1 − φi )2 + ǫ2i |∇φi |2 + µi ρi C(φi ) ACS Paragon Plus Environment

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(2)


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The first term in Equation 1 is the sum of the spatial integral of the free energy density for each phase field over the entire simulation domain. Equation 2 gives the explicit expression of the free energy density function. It consists of a double well potential for stabilizing the phase field in the bulk region, a gradient term to penalize abrupt changes in the phase field, and a chemical potential term that provides the driving force for NW growth. In Equation 2, Ui and ǫi are energetic parameters, 14 whose relation to interfacial energies will be discussed in Section 2.2. µi and ρi specify the chemical potential and the atomic density of the material species in each phase i respectively.† C(φi ) = [tanh(10 φi − 5) + 1] /2 is a function, whose value ranges from 0 and 1 and with zero derivative at φi = 0 and 1, preventing the shift of the minimum of φi when µi is non-zero. 14 The second term in Equation 1 introduces a horizontal external force applied on the liquid phase as a perturbation to the system. Assuming the x-y plane is the horizontal plane, αx and αy specify the x and y components of a uniformly distributed body force density. x − x0 and y − y0 provide the in-plane distance between x = (x, y) and a reference point (x0 , y0 ). C(φL ), with the same function form as C(φi ) in Equation 2, limits the region of the body force to the liquid phase only. The third term in Equation 1 penalizes the co-existence of all the three phases at the same point x to prevent a third phase component from being trapped at the two-phase interface region. Introduction of this high order term may modify the line tension of the triple junction, but it has been shown that its effect on the equilibrium wetting geometry of the three-phase system is negligible. 14 The fourth term in Equation 1 is a quadratic function that penalizes the deviation of the current solid phase field φS from a pre-set solid shape describedc by φorig S , if a positive pre-factor PS is used. The purpose of this term is to preserve the shape of certain metastable nanowire configurations, while the liquid and vapor phases are evolving. †

It should be noted that the model discussed in this paper applies to a single-component nanowire (e.g. Ge), and that the chemical potential µ is the chemical potential of the component in each of the three phases (vapor, Au-Ge liquid and solid). It would be necessary to introduce additional chemical potentials in order to represent growth of multicomponent nanowires.

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2.2

Interface anisotropy

In Equation 2, we let ǫi be a function of the local orientation of the phase field, which satisfies the cubic symmetry as shown in Equation 3. 15 ǫi (n) = ǫi0 · 1 + ǫi1 · (n2x n2y + n2y n2z + n2z n2x ) + ǫi2 · (n2x n2y n2z ) + ǫi3 · (n2x n2y + n2y n2z + n2z n2x )2

(3)

For each phase field i, ǫi0 , ǫi1 , ǫi2 , ǫi3 and Ui can be determined based on the experimental measurements of the interfacial energies for various orientations. The relations among the energetic parameters (Ui and ǫi (n)), the surface energy of i-j interface σij and the interface thickness ξij can be described by Equation 4 and 5. In our model ǫ2i (n) is always anisotropic. However, the sum ǫ2i (n) + ǫ2j (n) can be isotropic for an isotropic interface (such as the vaporliquid interface).

2.3

q 1 (Ui + Uj ) · ǫ2i (n) + ǫ2j (n) 3s ǫ2i (n) + ǫ2j (n) = 2 Ui + Uj

σij =

(4)

ξij

(5)

Equation of motion

The equation of motion of this phase field model, adopted from Steinbach’s formulation, 16 has the form of the Ginzburg-Landau (GL) equation. 17 The time evolution of the phase field i is governed by Equation 6.


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( ) dφi (t) δF 1 δF δF δF δF GL + K0GL − = −Γim (x) Kim − + dt δφi δφm 2 δφi δφm δφn ( ) δF 1 δF δF δF δF GL + K0GL − − + − Γin (x) Kin δφi δφn 2 δφi δφn δφm δF δF δF − [1 − Γim (x) − Γin (x)] K0GL 2 − + δφi δφm δφn

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(6)

where i, m, n = S, V, L and m 6= i, n 6= i, m 6= n. Coefficients K GL control the interface GL GL kinetics. Kim and Kin are associated with the change of the phase field component φi at i-m

and i-n interfaces respectively, and K0GL is the kinetic coefficient for the phase field evolution involving the third phase component at the two phase interface regions. For simplicity, we GL GL GL is set to a value smaller than K0GL , which suppresses set KLS = KLV = K0GL , while KSV

the NW sidewall deposition to match the experimental observations. Γ(x) is a function designed to detect the type and range of the interfaces, 14 so that the kinetic coefficients can be correctly assigned to the corresponding interfaces. From Equation 6, it is clear that the driving force for the system evolution comes from the variational derivative difference between the phases, e.g.,

δF δφi

−

δF . δφm

The expression for the variational derivative of F with

respect to φi is given by Equation 8. δF ∂fi ∂fi = −∇ δφi ∂φi ∂(∇φi ) = Ui

4φ3i

−

6φ2i

(7)

+ 2φi − ∇ ·

2ǫ2i (n)∇φi

∂ǫi (n) ∂ 2 2(∇φi ) ǫi (n) − ∂xj ∂φi,j j=1,2,3

+ µi ρi C ′ (φi ) + [αx (x1 − x0 ) + αy (x2 − y0 )] + P

X

∂C(φL ) ∂φi

∂ ∂ 2 (φS − φorig φ2V φ2L φ2S + PS S ) ∂φi ∂φi

(8)

where x1 ≡ x, x2 ≡ y, x3 ≡ z, φi,j ≡ ∂φi /∂xj , and adopting the same function form of C(φi ) in Section 2.1, C ′ (φi ) =

dC(φi ) dφi

= 5 − 5 tanh2 (10φi − 5). ACS Paragon Plus Environment

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2.4

Control of growth rate and direction

The effective volume Ωi of each phase i can be defined by Equation 9,

Ωi ≡

Z

C(φi ) d3 x

(9)

Then the horizontal center of mass position (X, Y ) of the liquid phase, for example, its x component with respect to the reference position x0 is given by Equation 10,

X≡

R

(x − x0 ) C(φL ) d3 x ΩL

(10)

With the quantities defined above, the behaviors of the NW VLS system can be measured and tracked during the simulation. For instance, the volumetric growth rate of the NW can be calculated by

dΩS . dt

can be evaluated by

q

The horizontal velocity of the droplet during the kinking process dX 2 dY 2 + . From Equations 1 and 2, the volume changes of the dt dt

phases are associated with the chemical potential µ. And the droplet displacement can be induced by a horizontal external force (when αx and αy are non-zero). This indicates that the

NW growth rate and the horizontal velocity of the droplet can be controlled by dynamically adjusting the parameters µL , µV , αx and αy .‡ In our model, growth rate control is achieved by a predictor-corrector approach. At the beginning of each simulation step, we start with the unperturbed time derivative of each phase field φi , denoted as Gi ({φj }; [µ∗L µ∗V αx∗ αy∗ ]), where the parameters with superscript asterisks ∗ are assigned arbitrary values (e.g. they can all be set to 0). At the end of this simulation step, we expect to obtain the true time derivative Gi ({φj }; [µL µV αx αy ]), whose ‡

Because it is the difference between chemical potentials, and not their absolute values, which provides the driving force for growth, for simplicity, µS is always set to zero as a reference in our model.


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expression is given by Equation 11 with extra correction terms.

Gi ({φj }; [µL µV αx αy ]) = Gi ({φj }; [µ∗L µ∗V αx∗ αy∗ ]) + ∆µL +∆µV

∂G∗i ∂µL

∂G∗i ∂G∗i ∂G∗i + ∆αx + ∆αy ∂µV ∂αx ∂αy

(11)

where ∆µL ≡ µL − µ∗L , ∆µV ≡ µV − µ∗V , ∆αx ≡ αx − αx∗ , ∆αy ≡ αy − αy∗ and G∗i is abbreviated from Gi ({φj }; [µ∗L µ∗V αx∗ αy∗ ]). To determine the values of ∆µL , ∆µV , ∆αx and ∆αy , an equation set can be established, which links the local modification on the phase field to the macroscopic constraints of the system. For example, if the designated liquid volume change per timestep ∆t is set to ∆ΩL , Equation 12 should be satisfied, Z

CL′

·

G∗L

∂G∗L ∂G∗L ∂G∗L ∂G∗L + ∆µL + ∆µV + ∆αx + ∆αy ∂µL ∂µV ∂αx ∂αy

d3 x =

∆ΩL ∆t

(12)

If we set the reference droplet position at x0 = y0 = 0, Equation 13 sets the target of the droplet displacement along x-direction in the current step to be ∆x, Z

CL′

·

G∗L

∂G∗L ∂G∗L ∂G∗L ∂G∗L + ∆µL + ∆µV + ∆αx + ∆αy ∂µL ∂µV ∂αx ∂αy

· (x − X0 ) d3 x = Ω0L

∆x (13) ∆t

where Ω0L and X0 are the volume and the x component center of mass position of the liquid droplet from the previous simulation step. Therefore, considering the constraints on the change rate of the phase volumes (∆ΩL , ∆ΩV , and ∆ΩS = −(∆ΩL + ∆ΩV )) and the liquid droplet center of mass displacements (∆x and ∆y), a matrix equation in the form of Ax = b can be written, with A, x and b given explicitly in Equation 14, 15 and 16. The solution x to this equation set determines the values of ∆µL , ∆µV , ∆αx and ∆αy respectively, for the


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current time step. 

A=

 R ∂G∗  CL′ ∂µLL V      ∗ R  ′ ∂GV  CV ∂µL V      R ∗ R  ′ ∂GL  V CL ∂µL (x − X0 ) V     R  R ∂G∗ CL′ ∂µLL (y − Y0 ) V V

∂G∗

∂G∗

       ∗ ∗ R R R  ′ ∂GV ′ ∂GV ′ ∂GV  C C C V ∂µV V ∂αx V ∂αy V V V      ∗ ∗ ∗ R R ′ ∂GL ′ ∂GL ′ ∂GL CL ∂µV (x − X0 ) V CL ∂αx (x − X0 ) V CL ∂αy (x − X0 )       ∗ ∗ ∗ R R  ∂G ∂G ∂G ′ ′ ′ L L L CL ∂µV (y − Y0 ) CL ∂αx (y − Y0 ) CL ∂αy (y − Y0 ) V V R

CL′ ∂µVL

V

x=

 

R

∆µL

∆µV

V

CL′ ∂αxL

∆αx

R

∆αy

T

b=

3.1

V

CL′ ∂αyL

(14)

(15)







3

∂G∗



  R   CL′ G∗L − ∆ΩL /∆t V           R   ′ ∗   C G − ∆Ω /∆t V V V V    −      R    ′ ∗ 0  V CL GL (x − X0 ) − ΩL ∆x/∆t           R  ′ ∗ 0 CL GL (y − Y0 ) − ΩL ∆y/∆t V

(16)

Applications of the phase field model Model parameters for Au-Ge system

Germanium (Ge) NW has generated much interest for its higher intrinsic carrier mobility and reasonable compatibility to the current Si-based devices. 18 Gold (Au) is a commonly used catalyst for synthesizing Ge NWs, and it can induce the VLS growth at sub-eutectic temperature. 19,20 Therefore, we choose to demonstrate the applications of the model on the ACS Paragon Plus Environment

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Au catalyzed Ge NW VLS growth. Setting the thickness of each i-j interface ξij ≈ 12 Å, the key parameters of the model are listed in Table 1, reproducing the interfacial energies of the Au-Ge system and their crystalline anisotropy as reported in literature. 21–23 Additionally −3

for simplicity, we adopt the approximation that ρV = ρL = ρS = 0.0442Å

for Au-Ge

system. 14 With the above parameters, our model captures reasonable facets with clear edges of an equilibrium Ge nano-crystal particle surrounded by either vapor or liquid, as shown in Figure S1 in Supporting Information. 3

Table 1: Parameters of the phase field model. Ui , P and PS are in unit of eV/Å . ǫ2i0 is in 3 GL unit of eV/Å. K0GL and KSV are in unit of Å /(eV · s). h is in unit of Å.

UL 0.00672

3.2

US

UV

ǫ2L0

ǫ2S0

0.0105 0.0227 0.184 0.561

h

ǫL1

ǫL2

2.5

-0.608

0.692

ǫL3

ǫS1

ǫ2V0

P

PS

K0GL

GL KSV

0.874

0.5

0.05

106

10

ǫS2

ǫS3

ǫV1

ǫV2

ǫV3

0.134 7.184 -7.482

-1.502

0.758 -1.202

-0.207

Nanowire growth with controlled orientations

Many experimental studies have been conducted on controlled growth of NW along specific orientations. 18,24–26 However, the existing phase field model in literature is only capable of simulating NW growth along stable orientations. With this new model, by constraining the center of mass position of the droplet, the NW can be induced to grow along any designated direction. Such constrained simulations can provide new insights on NW VLS growth along metastable orientations. Figure 1 provides 3D and cross sectional views of the vertically grown NW along [110], [111] and [100] orientations respectively, given the same catalyst liquid droplet volume Vdroplet of 1.92×104 nm3 (assuming the droplet has a perfect spherical shape, its diameter Ddroplet ≈ 33 nm). At this droplet size, both experiments and simulations 6 show that the h111i direction ACS Paragon Plus Environment

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[110] NW

[100] NW

[111] NW

{100}

{100} {111}

{111} {111}

Figure 1: Simulation snapshots of NW grown along [110], [111] and [100] orientations in 3D and cross sectional views. is a stable growth direction for Ge NWs. For [111] growth, our model predicts the cross sectional shape of the wire as an irregular hexagon consisting of three long edges and three short edges. This three-fold symmetry is consistent with the experimental observation. 27 The liquid-solid interface contains a large (111) facet on top with three inclined {111} ledges symmetrically wet by the droplet, which is similar to the TEM observation in literature. 28 For [110] growth, the liquid-solid interface shows a hip-roof like structure, consisting of two large {111} facets and two small {100} facets, also well capturing the observations in experiments. 24 The combination of the {111} and {100} surfaces have much lower average interfacial energy compared with a flat {110} surface. Therefore, though the decomposition of the (110) solid/liquid growth facet creates more interface area, it is energetically preferred. The sidewalls of [110]-oriented NWs are smooth with an irregular hexagonal cross-section, bounded by four {111} facets and two {100} facets. For [100] growth, the morphology of the liquid-solid interface contains a square {100} facet in the center with each edge connected to a {111} facet. The sidewalls of the (100) ACS Paragon Plus Environment

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NW have a four-fold symmetry. The {100} surfaces are smooth, as can be observed from the 3D view. However, the side walls with an average {110} orientation have a higher surface energy if they are flat. To avoid this high energy state, these side walls are predicted to spontaneously roughen into a corrugated form, which consists of small {111} facets. It appears that the sidewall facets of the [100] NW are not as clear as those for the [111] and [110] NWs. This indicates that the driving force for side wall faceting for [100] growth is not as strong as for the other two growth orientations. Increasing the solid-vapor interface GL kinetic coefficient KSV from 10 Å3 /(eV· s) to 50 Å3 /(eV· s) enhances the side wall faceting

(see Figure S2 in Supporting Information), but also leads to more pronounced side wall GL deposition and hence tapering of the NW. For simplicity, KSV is kept at the lower value of

10 Å3 /(eV· s) in the following discussions. The average sidewall surface energies σ ¯SV of NWs growing along different orientations at the same volumetric rate of 2 × 103 nm3 /s and droplet volume of 1.92×104 nm3 are calculated and compared in the first three rows of Table 2. Consistent with our observations of the NW sidewall morphologies, the h110i NW with four {111} sidewall facets has the lowest average side wall surface energy σ ¯SV . The h100i NW has the highest σ ¯SV , because its sidewall consists of four {100} planes. Given the same liquid droplet volume, the NW’s effective diameter D is slightly different, due to the orientation dependent contact geometry. Table 2 lists the values of σ ¯SV /D for the three NWs as well as ∆µSV , the chemical potential difference between the solid and vapor phases for achieving the same NW volumetric growth rate. From these data it is clear that larger σ ¯SV /D requires larger ∆µSV to reach the same growth velocity. In other words, given the same chemical potential driving force ∆µSV , NWs with smaller sidewall surface energy σ ¯SV are expected to have a larger growth rate. This is consistent with the Gibbs-Thompson effect, in which the chemical potential driving force for initializing NW growth is expected to be proportional to σ ¯SV /D. 29 Applying the same constraints, phase field simulations are performed to grow NWs with a smaller droplet volume of 7.61×103 nm3 . The above calculations are repeated, which generate the data in ACS Paragon Plus Environment

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the last three rows of Table 2, for h110i, h100i and h111i orientations respectively. For all three growth orientations, the estimated σ ¯SV values are very close to those for larger Vdroplet , but the D values are decreased due to the reduction of the droplet volume, resulting in the increasing of the σ ¯SV /D. As expected, the values of ∆µSV are increased, which is again consistent with the Gibbs-Thompson effect that a higher chemical potential driving force is required for thinner NWs to achieve the same growth rate of thicker wires. Table 2: Average sidewall surface energy (¯ σSV ), effective NW diameter (D), the ratio of σ ¯SV to D and the chemical potential difference between solid and vapor phases ∆µSV required for NWs growing p along h110i, h100i and h111i orientations to reach the same growth rate. D is defined as 2 A/π, where A is the NW’s cross section area for different growth directions. Vdroplet (nm3 )

1.92×104

7.61×103

3.3

NW orientation σ ¯SV (J/m2 ) D (nm) σ ¯SV /D (eV/nm2 ) ∆µSV (eV/nm3 ) h110i

1.128

38.260

0.184

0.82

h100i

1.184

35.253

0.210

0.95

h111i

1.154

34.180

0.211

0.99

h110i

1.115

28.079

0.248

1.16

h100i

1.174

27.008

0.272

1.33

h111i

1.155

26.538

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Nanowire growth with diameter oscillation

The phase field model can simulate the formation of complex NW structures in the VLS process, e.g. vertical NW growth with periodic diameter oscillation. 30,31 For this study, the computation cell size is set to 75×75×110 nm3 , and the substrate orientation is set to (111). An initial growth simulation of the NW with conserved droplet volume is performed, resulting in a pyramidal shaped NW pedestal structure as shown in Figure 2(a). Keeping the NW growth rate the same, a subsequent simulation is performed with the liquid droplet volume ACS Paragon Plus Environment

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changing periodically at a constant expansion and shrinkage rate. Quantitively, the changes of the droplet volume Vd and the cross sectional area of the wire Ac during the vertical growth of the NW are described by Figure 3, which shows a clear correlation between Vd and Ac . The above simulation generates the NW configurations shown in Figure 2(b) and 2(c), consistent with the NW structures observed in experiments when the temperature and/or partial pressure of the precursor gas are changed intentionally. 32 A rough estimation based on the Kelvin equation shows that the expansion of the droplet volume in our simulation corresponds to the condition of doubling the partial pressure of Ge precursor in the gas phase. In reality, the adjustment of the experimental conditions alters the material’s solubility in the binary liquid droplet, which finally leads to the change of the droplet volume and the interfacial energies. Therefore, in a more sophisticated model, the inputs to the simulation, such as the changing rate of the droplet volume and the energetic parameters Ui and ǫi (n) for determining the interfacial energies, can be set as functions of both temperature and pressure.

(a)

(b)

(c)

Figure 2: Simulation snapshots in a time series from (a) to (c) show the NW VLS growth with periodically oscillated diameters. The NW configurations in (a), (b) and (c) correspond to h = 0 nm, h ≈ 21.5 nm and h ≈ 43 nm respectively in Figure 3.


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h (nm) Figure 3: The droplet volume Vd is plotted as a function of NW height h (Top). The cross sectional area of the wire Ac is plotted as a function of h (Bottom).

3.4

2D free energy landscape for NW kinking

We have applied the phase field model to study the droplet instability induced NW kinking and have estimated the free energy change along a specific droplet displacement path. 6 To deepen our understanding of kinking at the NW base, it is helpful to obtain a 2D free energy landscape for the droplet displacement, from which minimum energy paths can be revealed. For this purpose, an initial NW configuration is created, as shown at the bottom center in Figure 4(a). Subsequent simulations are performed with an external horizontal force applied on the droplet, forcing it to move along a designated direction at a constant velocity. Assuming the process of the droplet slipping off the NW top surface happens very fast and during this time the amount of NW growth can be neglected, in our simulation the shape of the NW is conserved, while the other phases are allowed to relax, in order to minimize the total free energy. Setting the initial center of mass position of the droplet as 0, the free energy change ∆F can be tracked and plotted as a function of droplet displacement, as shown in Figure 4(a). It clearly suggests when the droplet moves in opposite directions, the change in the free energy is highly asymmetric. Applying the same procedure, more simulations ACS Paragon Plus Environment

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4

Conclusion

We present a multi-phase field model that is capable of studying NW structural transition during the VLS process. The design of the model makes it possible to simultaneously control the spatial velocity of the catalyst droplet, the volumetric growth rate of the NW and the shape of the NW. This model successfully captures the morphologies of NWs growing along metastable orientations. The model can also be used to simulate the formation of complex NW structure, such as a NW with diameter oscillation that is comparable to the experimental observation. To study the NW kinking, the interface free energy changes due to the droplet displacement along different paths are estimated, from which a 2D free energy landscape for NW kinking on a pedestal is obtained. The findings discussed in this paper illustrates how the phase field model can be used to improve our understanding of the NW VLS growth process.

Supporting Information Equilibrium shapes of single crystalline germanium nanoparticles in vapor and in liquid predicted by the phase field model. Simulation snapshots of [100] NW growth with the GL increased to 50 Å3 /(eV· s). solid-vapor interface kinetic coefficient KSV

Acknowledgement This work is supported by National Science Foundation Division of Materials Research programs DMR-1206511 and DMR-0907642. The computer simulations are performed on Sherlock cluster operated by the Stanford Research Computing Center.


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[110] NW

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Phase Field Model for Morphological Transition in ... - ACS Publications

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