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Model of step propagation and step bunching at the sidewalls of nanowires
Journal of Crystal Growth 427 (2015) 60–66
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Model of step propagation and step bunching at the sidewalls of nanowires Sergey N. Filimonov, Yuri Yu. Hervieu n Department of Physics, National Research Tomsk State University, 634050 Tomsk, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 19 March 2015 Received in revised form 9 June 2015 Accepted 3 July 2015 Communicated by: K. Deppert Available online 14 July 2015
Radial growth of vertically aligned nanowires involves formation and propagation of monoatomic steps at atomically smooth nanowire sidewalls. Here we study the step dynamics with a step flow model taking into account the presence of a strong sink for adatoms at top of the nanowire and adatom exchange between the nanowire sidewall and surrounding substrate surface. Analytical expressions for velocities of steps propagating from the nanowire base to the nanowire top are obtained. It is shown that the step approaching the nanowire top will slow down if the top nanowire facet is a stronger sink for adatoms than the sidewall step. This might trigger bunching of the steps at the sidewall resulting in development of the pencil-like shape of nanowires such as observed in, e.g., the Au-assisted MBE growth of InAs. & 2015 Published by Elsevier B.V.
Keywords: A1. Growth models A1. Nanostructures A1. Surface processes A3. Molecular beam epitaxy
1. Introduction In recent years vertically aligned semiconductor nanowires have attracted great attention due to prospects of their use in microelectronics, optoelectronics, photovoltaics, nanosensing etc. [1–4]. Specific properties of nanowires are related to their highly anisotropic shape, which may vary a lot depending on the deposition conditions [4–6]. Moreover, even at the given deposition conditions the nanowire shape may change in course of the growth [7–10]. In general the nanowire shape is determined by the interplay of the lateral and axial growth of nanowires [7,11]. In that respect it is essential that the nanowire sidewalls remain atomically smooth while the nanowire is growing laterally. This indicates that the lateral growth of nanowires occurs in the layer-by-layer or stepflow fashion involving nucleation of new steps at the sidewall facets and their propagation along the facets. Monoatomic steps at the sidewalls of nanowires were directly visualized by transmission electron microscopy [12–14] and scanning tunneling microscopy [15,16]. Bearing in mind the step-flow mechanism of the radial growth of nanowires it is tempting to relate the experimentally observed transformations of the nanowire shape with a kinetic instability of the steps propagating along the sidewall facets. For instance, it is known that on vicinal surfaces the kinetic step bunching may lead to the surface faceting [17]. Similarly, the step bunching may be
n
Corresponding author. E-mail address: [email protected] (Yu.Yu. Hervieu).
http://dx.doi.org/10.1016/j.jcrysgro.2015.07.005 0022-0248/& 2015 Published by Elsevier B.V.
responsible for the formation of new facets at the nanowire sidewalls such as the {10–11} facets of pencil-like InAs nanowires grown by the Au-assisted MBE [9]. The transition from cylindrical to pencil-like shape of InAs nanowires was explained in earlier theoretical works [9,10] by the onset of the radial nanowire growth by nucleation and propagation of steps at some critical length of the nanowire. However dynamics of the steps at the sidewall was not considered explicitly within the models developed in [9,10]. Therefore it remains unclear why the appearance of the steps leads to the specific nanowire shape observed in the experiment. Intuitively, one would expect that the step velocity should be considerably higher than the axial growth rate of the nanowire, because the advance of the step by one atomic position requires formation of an atomic row along the step edge whereas the increase of the nanowire length by the same distance requires deposition of a whole atomic layer on the nanowire top facet. Therefore, one would expect the steps moving in the direction of the nanowire top to catch up and join the nanowire top facet leading to the thickening of the nanowire including its upper part near the catalyst droplet. However, in the aforementioned experiments on the InAs nanowires growth [9] the nanowire radius at the top did not change with time. In [18] the formation of tapered InP nanowires with non-tapered sections at the top was observed. These observations clearly indicates that for some reasons the steps slow down in the upper part of the nanowire instead of catching up the nanowire top facet. In the present work we model the axial and radial growth of nanowires treating explicitly dynamics of the steps nucleating sequentially at the nanowire base. This allows us to go beyond the
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earlier material conservation models [9,10] and reproduce different scenarios of the nanowire growth: thickening of nanowires while preserving their cylindrical shape and gradual transformation of the nanowire shape due to bunching of the steps at the nanowire sidewalls. We show that the step bunching might be conditioned by the slowing down of the leading step approaching the nanowire top due to the transfer of atoms from the step to the nanowire top with the later being a stronger sink for adatoms than the step itself. Results of numerical simulations of the step dynamics and of the corresponding evolution of the nanowire shape are in good agreement with the experimental observations of the shape transformations of the InAs nanowires [9].
2. Model Let us consider a cylindrical nanowire with k steps of the monoatomic height nucleated at the nanowire base and propagating from the base to the nanowire top (Fig. 1). The axisymmetric configuration of the steps shown in Fig. 1 is similar to that considered in the earlier model of filamentary crystal growth by Schwoebel [19] and resembles the “wedding cake” step configuration widely used to model growth and dissolution of mounds (see e.g. [20–23]). However, in contrast to the model of mounds, the steps in our model propagate towards the nanowire top without growing in the lateral direction. Considering the axial growth of the nanowire it is generally believed that under the catalyst droplet the top nanowire facet grows layer-by-layer by nucleation and spreading of two-dimensional (2D) islands [3,11,24–26]. The energy per unit length of a step in contact with a liquid phase is typically much smaller as compared to the energy of a step in contact with a gas phase. Moreover, the curvature of the step on the nanowire top may be negative if the 2D island nucleates as a circular step at the triple phase line [27] (this nucleus position favors formation of the wurtzite structure of III–V semiconductor nanowires [28]). This means that due to the Gibbs–Thomson effect the nanowire top facet should be a stronger sink for the sidewall adatoms than the sidewall step. This is again in contrast with the case of mounds where the Gibbs–Thomson effect generates the downhill flux of atoms from the upper (more curved) steps to the lower (less curved) steps [21–23] leading to shrinkage of the topmost terraces and, as a consequence, to gradual flattening of the mounds at the absence of the deposition flux. Concentration of adatoms on the substrate surface, ns , and on the terraces between the steps on the sidewall, ni , obey the steadystate diffusion equations [11,29] Ds d dns ns r þ J cos α ¼ 0; ð1Þ r dr dr τs 2
Df
d ni n þ J ω sin α i ¼ 0; τf dz2
ð2Þ
where Ds and Df are the surface diffusion coefficients on the substrate surface and on the sidewall, respectively, τs and τf are the respective life times of adatoms before desorption, J is the deposition flux, and α is the angle between the deposition flux direction and the nanowire axis (it is assumed that the nanowires grow normal to the surface). The coefficient ω is equal to 1=π in the case of molecular beam epitaxy considered in the present work. Index i in Eq. (2) enumerates the sidewall steps as sketched in Fig. 1. General solutions of Eq. (1) and (2) read
Fig. 1. Schematic view of a nanowire with steps at the sidewall.
respectively, I 0 ðr=λs Þ and K 0 ðr=λs Þ are the modified Bessel functions. The integration constants C 1 , C 2 , Ai and Bi are to be determined from the appropriate boundary conditions. Following [11,29] as the first boundary condition we request that the diffusion flux on the substrate surface vanishes at certain distance RW from the center of the nanowire base dns r ¼ RW ¼ 0: dr
ð4Þ
In the present paper we consider RW as a model parameter having a meaning of an effective radius of the feeding area around the given nanowire. Boundary conditions at the nanowire base follow from the conditions of balance of the diffusion fluxes and the resulting fluxes of adatoms crossing the boundary between the nanowire sidewall and substrate surface [30,31] s Ds dn ¼ ksf ns ðRk Þ kf s nk þ 1 ð0Þ; dr r ¼ Rk
dnf Df ¼ kf s nk þ 1 ð0Þ ksf ns ðRk Þ: dz z ¼ 0
ð5Þ
Here ksf and kf s are the rate constants of the transitions from the substrate surface to the sidewall surface and vice versa, and Rk is the radius of the nanowire base: Rk ¼ R þkh, where R is the radius of the nanowire top, and h is the height of the monoatomic step at the nanowire sidewall. The balance equations for the diffusion fluxes and the resulting fluxes of adatoms incorporating into the steps from the upper (-) and lower ( þ) terraces give us 2k boundary conditions of the form i þ 1 Df dndz ¼ β ½ni þ 1 ðli Þ n~ i ; z ¼ li
dn Df i ¼ β þ ½ni ðli Þ n~ i ; dz z ¼ li
ð6Þ
ð3Þ
where li is the distance from the nanowire base to the i-th step ði ¼ 1:::kÞ, β and β þ are the kinetic coefficients of the step [32], and n~ i is the adatom concentration in equilibrium with the i-th step. The last boundary condition represents the balance equation for the diffusion flux and the resulting flux of adatoms to the nanowire top [30,31] dn1 Df ¼ kf t n1 ðLÞ ktf C top ¼ kf t ½n1 ðLÞ n~ t ; ð7Þ dz z ¼ L
and λf ¼ ðDf τf Þ are the diffusion lengths of where λs ¼ ðDs τs Þ adatoms on the substrate surface and on the nanowire sidewall,
where L is the nanowire length, kf t and ktf are the rate constants of the transitions of adatoms from the sidewall facet to the nanowire
ns ðrÞ ¼ Jτs cos α þC 1 I 0 ðr=λs Þ þC 2 K 0 ðr=λs Þ; ni ðzÞ ¼ Jτf ω sin α þ Ai coshðz=λf Þ þ Bi sinhðz=λf Þ; 1=2
1=2
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top and in the opposite direction, respectively, and C top is the concentration of atoms on the top facet near its edge. The concentration n~ t ¼ ktf C top =kf t in the modified right part of Eq. (7) is the concentration of adatoms on the sidewall terrace in equilibrium with the nanowire top. Here for simplicity we neglect possible oscillations of the supersaturation in the droplet [24–26] and treat the nanowire top facet as a permanent sink for adatoms. For the rate constants of the adatom exchange between the nanowire top (t), its sidewall (f) and the substrate surface (s) we used standard Arrhenius expressions kαβ ¼ bα να expð Eαβ =kB TÞ, where Eαβ is the activation energy of the transition from α to β ðα; β ¼ s; f ; tÞ, να is the attempt frequency, kB is the Boltzmann constant, T is the substrate temperature, and bα is the interatomic distance at the given surface in the direction normal to its boundary. We also assumed for simplicity that the barrier for attachment of atoms to the steps equals to the surface diffusion barrier, so that β þ ¼ β ¼ Df =af , where af is the interatomic distance in the direction parallel to the step edge. The energy profile for an atom on the surface is shown schematically in Fig. 2. Typically, the nanowire radii are small (e.g. the radius of InAs nanowires at the onset of their lateral growth is about 15 nm [9]), therefore for the steps at the sidewall the Gibbs–Thomson effect has to be taken into account. Due to the Gibbs–Thomson effect the chemical potential along a circular step of a radius Rst is higher by ~ 1 Þ ¼ γ st Sf =kB TRst than the chemical potential along Δμ ¼ kB T lnðn=n an infinitely long straight step [32]. Here Sf is the area per atom of the sidewall facet, and γ st is the step formation energy per unit length. Thus for a single step at the sidewall of a nanowire of radius Rone has n~ 1 n~ ¼ n~ 1 exp½γ st Sf =kB TðR þ hÞ, where n~ 1 ¼ expð ΔE=kB TÞ=Sf is the concentration of the adsorption layer in equilibrium with an infinitely long straight step. Here ΔE is the depth of the potential energy trap for an atom at the kink site at the step edge relatively to the energy of an isolated adatom on the sidewall surface (Fig. 2). If there are several steps at the nanowire sidewall one should also take into account the pairwise entropic or elastic repulsion of the steps. Then the equilibrium adatom concentration on the i-th terrace ði ¼ 2; :::; k 1Þ is given by " # γ st Sf K K n~ i ¼ n~ 1 exp ; ð8Þ exp kB TRi ðli 1 li Þ3 ðli li þ 1 Þ3 where Ri ¼ R þ ih is the radius of the i-th step, K ¼ 2Sf A=kB T and A is the step–step repulsion parameter [33]. Interactions of the first ði ¼ 1Þ step with the nanowire top and the last ði ¼ kÞ step with the nanowire base were neglected. Concentration of adatoms on the sidewall surface in equilibrium with the nanowire top can be written as n~ t ¼ ktf C top = kf t ¼ ð1 þ σ top ÞC~ top expð ΔEf t =kB TÞbt νt =bf νf , where C~ top is the equilibrium concentration of atoms on the nanowire top (in the droplet), determined from the condition of the equilibrium between the respective infinite liquid and solid phases at a given temperature, σ top ¼ C top =C~ top 1 is the supersaturation at the boundary between the droplet and the nanowire sidewall, and
Esf
step
Efs
ΔEft
ΔE
ΔEtop
l L sidewall
ft
ð ΔEf t =kB TÞ ¼ n~ 1 Sf =St . Using this relation one gets n~ t ¼ ð1 þ σ top Þn~ 1 at νt =af νf . From the solution of the boundary problem (1)–(7) one finds the resulting fluxes of adatoms to the steps ðg i ¼ β þ ½ni ðli Þ n~ i þ β ½ni þ 1 ðli Þ n~ i Þ and to the nanowire top ðg top ¼ kf t ½n1 ðLÞ n~ t Þ. The step velocities and the rate of elongation of the nanowire are related to the fluxes as dli =dt ¼ Sf g i and dL=dt ¼ 2Ωg top =Rþ J top , where J top ¼ J Ωf ðα; β Þ is the contribution of direct adsorption of atoms onto the nanowire top, Ω is the volume per atom of the crystal, and f ðα; β Þ is a geometrical parameter related to the flux incidence angle α and to the contact angle β of the droplet [34]. The factor of 2=R in the expression for dL=dt reflects the fact that the total flux from the nanowire sidewall through the perimeter of the top nanowire facet 2π Rgtop corresponds to a covered area of π R2 .
3. Dynamics of a single step The solution of the boundary problem (1)–(7) in the case of a single step at the nanowire sidewall allows to write down the fluxes of adatoms incorporating into the step and into the nanowire top facet as g st ¼ Aσ S Bσ t þC st σ and ~ 1 is g top ¼ Bσ t þ C top σ f t , respectively. Here σ ¼ ðJ τf ω sin α=nÞ the supersaturation in the adlayer on the sidewall surface relatively to the step, σ f t ¼ ðJ τf ω sin α=n~ t Þ 1 is the supersaturation on the sidewall surface relatively to the nanowire top, ~ is the supersaturation at the step relatively to the σ t ¼ 1 ðn~ t =nÞ ~ 1 is the supersaturation on nanowire top, and σ s ¼ ðJ τs ξ cos α=nÞ the substrate surface relatively to the step, where ξ ¼ ksf =kf s exp½ðEf s Esf Þ=kB T is the parameter taking into account possible difference of the adsorption energies of adatoms on the substrate surface and on the nanowire sidewall [30,31]. The expressions for the coefficients A, B, C st and C top are given in Appendix A. Let us now analyze dynamics of the step in the limiting cases of strong and weak desorption of adatoms from the sidewall surface. For the case of strong desorption (short diffusion length λf of adatoms on the sidewall) and at a moderate supersaturation on the substrate surface ðσ S o o σ expðl=λf ÞÞ the velocity of the step at a distance l 4 4 λf 4 4 af is h i Sf Df n ðexpðL lÞ 1Þσ σ t dl ¼ Sf g st ≈ ; dt λf sinhðL lÞ þ 2af coshðL lÞ
ΔEsf substrate
ΔEf t ¼ Etf Ef t is the energy gain due to the transfer of an adatom from the sidewall surface to the nanowire top (Fig. 2). For C~ top one has C~ top ¼ expð ΔEtop =kB TÞ=St , where St is the area per atom of the top facet, ΔEtop is the depth of the trap at the kink site of an infinitely long straight step in contact with the liquid phase. It is convenient to express the concentration n~ t via the equilibrium concentration of adatoms on the sidewall face, n~ 1 . It is known that the work to transfer an atom from the kink site to the ambient gas phase taken with the negative sign is equal to the difference of chemical potentials of the ambient phase and that of the crystal and does not depend on the orientation of the surface and the presence of a liquid film on the surface [32]. Therefore ΔEtop þ ΔE ¼ ΔE (Fig. 2). Then it follows that C~ top exp
top
Fig. 2. Energy profile for an atom on the surface.
ð9Þ
where L ¼ L=λf and l ¼ l=λf . It is seen from this relation, that if the supersaturation σ t is negative or zero (the nanowire top is a poor sink for adatoms), the step velocity is positive independent on the distance between the step and the nanowire top. If, in addition, the nanowire is growing vertically manly due to supply of diffusing
S.N. Filimonov, Y.Yu. Hervieu / Journal of Crystal Growth 427 (2015) 60–66
adatoms from the sidewall surface then h i dL 2Ωg top 2ΩDf n ðcoshðL lÞ 1Þσ þcoshðL lÞσ t h i ≈ ≈ : dt R R λ sinhðL lÞ þ 2a coshðL lÞ f
ð10Þ
f
As seen dl=dt 4 dL=dt at σ t r0. Then the step will inevitably catch up the nanowire top and will merge with the top nanowire facet. As a result the nanowire increases its radius while preserving its cylindrical shape. When the sink for adatoms at the nanowire top is strong ðσ t 4 0Þ the step velocity vanishes at a certain critical distance to the nanowire top σt : ð11Þ ðL lÞcr ¼ λf ln 1 þ σ After approaching this distance, the step will move up in parallel with the nanowire elongation. Since, typically, λf =bf 4 4 1, it follows from Eq. (11) that at moderate σ the step cannot catch up the top even at small σ t . The slowing down of the step approaching the nanowire top might cause bunching of the steps nucleating sequentially at the nanowire base. In the opposite limit of no desorption (very long diffusion lengths of adatoms on the sidewall facets, λf , and on the substrate surface, λs , and high supersaturations σ and σ s ) one obtains the following approximation for the step velocity " # Df n~ σ t 1 dl R2 R21 J ω sin α W cot α þ L þ l : ð12Þ Sf dt ωR1 2 L l þ 2af Accordingly the vertical growth rate of the nanowire is approximated by Df n~ σ t 1 dL 2 J ω sin α ðL lÞ þ ð13Þ þ Jf ðα; βÞ: Ω dt R 2 L l þ 2af It follows from (12) that similarly to the case of intensive desorption the step velocity is positive if the nanowire top is a weak sink for adatoms (σ t r 0). Thereby, as seen from (12) and (13), dl=dt 4 dL=dt, i.e. the step moves faster than the nanowire top and will eventually catch up the top facet increasing the nanowire radius without changing its shape. In the case of a strong sink for adatoms at the nanowire top (σ t 40) the step velocity turns to zero at a certain critical distance ðL lÞcr from the top if " # af J ω sin α R2W R21 σt 4 cot α þ 2l ð14Þ Df n~ ωR1 It is essential that the right part of (14) depends on the step position and, hence, on the growth time. This makes the difference with the case of intensive desorption where due to a short length
63
of adatom surface diffusion on the sidewall surface, λf , the feeding area of the step is limited by λf and does not depend on l. Therefore, in the case of intensive desorption independent on l and on the presence of other steps at the sidewall, the leading step cannot catch up with the top even at relatively small positive σ t . Without desorption the feeding area behind the step increases with increasing l, so that the inequality (14) may be broken at sufficiently large l even if σ t ¼ 1. Therefore in the limit of no desorption the step should catch up the nanowire top and merge into the top facet. The above described scenario, however, might be altered by the appearance of other steps at the sidewall which will compete with the leading step for adatoms, thus decreasing velocity of the leading step.
4. Numerical modeling Numerical modeling of the nanowire growth was performed by integration of the equations for the step velocities and for the nanowire elongation rate. The elongation rate of the nanowire without sidewall steps was calculated using an expression obtained in [31] in the limit of very large λs and λf . The elongation rate of the nanowire with the sidewall steps and the corresponding step velocities were determined from the solution of the boundary problem (1)–(7). The corresponding expressions are listed in Appendix B. It was assumed that the first step is formed when the nanowire length reaches some critical value Ln , which might be determined experimentally by the onset of the radial growth of the nanowires. The moments of nucleation of the second, third, etc. steps were determined self-consistently as described in Appendix C. Whenever the leading step caught up with the nanowire top the radius of the nanowire was increased by one interatomic distance and integration of the equations of motion of the steps continued. Parameters of the model were chosen to simulate the Auassisted growth of InAs nanowires [9]. The step formation energy was approximated by γ st ¼ hγ f , where γ f is the surface energy per unit area of the sidewall nanowire facet, so that Sf γ st ¼ Ωγ f . For γ f we used the value of 5.7 eV/nm2 reported in [11,35] for the
InAsð1100Þface. Assuming the droplet contact angle β to be π =2 we put f ðα; βÞ ¼ ð1 þ cos αÞ=2. The critical length Ln ¼ 1500 nm, temperature T ¼ 683 К , incidence angle of the molecular beam α ¼ 331, deposition flux J= cos α ¼ 0:2 nm=s, and initial radius of the nanowire R ¼ 15 nm, were taken from the experiments on the growth of InAs nanowires [9]. For the structural parameters af , h, Sf ,St and Ω we used values of the experimentally observed wurtzite structure of InAs. The radius of the feeding zone of a
Fig. 3. Dependences of the nanowire height on the growth duration (red lines) and trajectories of the steps on the nanowire sidewall (black lines) at different supersaturations: (a) σ top ¼ 0:41 (σ t ¼ 0:001), (b) σ top ¼ 0:3 (σ t ¼ 0:08), and (c) σ top ¼ 0:1 (σ t ¼ 0:22). The insets show the nanowire shape evolution with the increasing number of steps formed at the nanowire base (the images are distorted in the horizontal direction for clarity). The magnified view of the area highlighted by the rectangle in (b) is shown in Fig. 4. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
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nanowire RW was set to 40 nm, which agrees with estimates made in [29]. We also set K ¼ 1 nm3 , νf ¼ 1013 s 1 , and at νt =af νf ¼ 1. Unknown parameters of the model are the surface diffusion barrier Es , the depth of the potential energy trap for an atom at the kink site ΔE, and the supersaturation at the nanowire top σ top . The parameters Es and ΔE can be combined into a single parameter Ekink ¼ Es þ ΔE, which is the activation energy of the detachment of an atom from the kink. Numerical modeling of the nanowire growth at Ekink ¼ 1:1eV has shown that at large σ top (at σ top Z 0:4 with the parameters used here) the effect of the slowing down of the step due to the transfer of atoms from the step to the nanowire top is small. Here formation of the steps at the sidewall results in increasing nanowire radius while its cylindrical shape is preserved (Fig. 3а). At small σ top a bunch of steps forms (Fig. 3b). As the nanowire is getting higher, the step bunch moves up keeping a roughly constant distance to the nanowire top. With decreasing σ top the distance from the bunch to the nanowire top increases (Fig. 3c), which is related to increasing supersaturation at the step relatively to the top, σ t , i.e. to a stronger effect of the step slowing down. The early stage of the step bunching is shown in Fig. 4. Partial dissolution of the leading step is conditioned by two factors: by the transfer of atoms from the leading step to the nanowire top and by decreasing adatom incorporation flux from the terrace behind the step due to shortening the distance to the second step approaching from the bottom. As the two steps get closer, the increasing step–step repulsion impels the leading step to start moving upwards again. As the next step approaches the first two steps the dissolution – growth cycle repeats. The atom transfer from the upper sidewall steps to the lower ones (because of the Gibbs–Thompson effect) plays a minor role here, however it shortens the average distance between the steps in the bunch and stabilizes the bunch at later stages of growth. At a given supersaturation σ top and a given critical length Ln , the formation and bunching of the steps is possible only in a certain range of the activation energies of the detachment of atoms from the kink site. For instance at σ top ¼ 0:1 the step bunching is observed at 1:25eV 4 Ekink 4 0:88 eV, while increasing the super saturation σ top to 0.3 shrinks this range to 1:2 eV 4 Ekink 4 0:83 eV. The upper bound of this interval arises from to the fact that with increasing Ekink the exchange flux between the step and the nanowire top decreases. This facilitates approach of the step to the nanowire top. At the same time at a given supersaturation σ t and fixed other model parameters, the activation energy Ekink determines the critical supersaturation σ n ð0Þ below which nucleation of new steps is impossible (see Appendix C). It follows from (C2) that at σ t ¼ 0 decreasing Ekink decreases σ n ð0Þ to zero. At
Fig. 4. Dependences of the nanowire height on the growth duration and trajectories of the steps on the nanowire sidewall in the early stage of step bunching (the magnified view of the area highlighted by the rectangle in Fig. 3b).
Fig. 5. Dependences of the nanowire height on the growth duration (red line) and trajectories of 111 steps on the nanowire sidewall (black lines) at σ top ¼ 0:05 ðσ t ¼ 0:26Þ. The inset shows the resulting shape of the nanowire (the image is distorted in the horizontal direction). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
σ t 4 0 the supersaturation σ n ð0Þ becomes negative below some
value of Ekink . This sets the lower bound for Ekink . Evidently, formation of the step bunch at the sidewall should be considered only as the first stage of the shape transformation of the nanowire. Indeed, as the number of steps at the sidewall increases, the velocity of the bunch becomes smaller and therefore the distance from the leading step of the bunch to the nanowire top increases. As a result the leading step may “detach” from the bunch so that at a later stage of the growth a set of separate step bunches may develop, which are characterized by decreasing number of the steps in the bunch as the bunch gets closer to the nanowire top (Fig. 5). At the same time it is well known that the step bunching may result in the formation of new stable facets on a vicinal surface [17]. The {10–11} oriented facets separated by flat areas were experimentally observed on the sidewalls of the InAs nanowires [9]. Kinetics of the facet formation and the growth at the faceted sidewalls of the nanowires require special consideration and are beyond the scopes of the present study.
5. Conclusions Transition from the cylindrical to pencil-like shape of nanowires may occur as a result of the bunching of the monoatomic steps nucleating at the sidewall nanowire facets and propagating from the nanowire base to the nanowire top. The necessary condition for the step bunching is slowing down of the leading step approaching the nanowire top. The step will slow-down when the detachment frequency of atoms from the step is substantially higher than the frequency of transitions from the nanowire top (from the catalyst droplet) to the sidewall facets. In this case the exchange flux between the step and the top facet is directed to the nanowire top and its magnitude increases as the step gets closer to the top. At certain distances between the step and the top the magnitude of the exchange flux becomes comparable with that of the flux of atoms adsorbing on the nanowire sidewalls and on the substrate surface and incorporating into the step. This should slow down the step. In the case of strong desorption of adatoms the feeding area of the step is limited by the adatom diffusion length prior to desorption. In this case the slowing down of the step is possible even at very small supersaturations at the step relatively to the nanowire top, σ t . When desorption of atoms is negligible, the flux of adatoms to the steps increases with increasing length of the nanowire, therefore a single step should unavoidably catch up with the nanowire top independent on σ t . However, appearance of
S.N. Filimonov, Y.Yu. Hervieu / Journal of Crystal Growth 427 (2015) 60–66
next steps at the sidewall reduces the flux of adatoms to the first step, so that at certain deposition conditions merging the step with the top facet might be impossible. Since both the supersaturation σ t and the supersaturation at the nanowire base, σ ð0Þ, increase with decreasing temperature, the formation of pensil-like nanowires should be more pronounced at lower temperatures, as observed in the experiment [9]. On the contrary, rising the temperature should favor formation of the cylindrical nanowires which diameter either does not change with time or increases due to occasional appearance of the steps that propagate along the nanowire sidewall without bunching. Numerical simulations of the nanowire growth with parameters and deposition conditions mimicking the growth of InAs nanowires induced by Au droplets have shown that bunching of the steps at the nanowire sidewall may occur at relatively small supersaturations in the droplet σ top (sufficiently large σ t ) and a relatively small energy of atom detachment from the kinks Ekink .
This study is supported by the Russian Foundation for Basic Research (Grant no. 13-02-12160).
Appendix A At the absence of extra barriers for attachment of adatoms to the step edge and for the transition of adatoms from the sidewall surface to the nanowire top, the coefficients in the expressions for the fluxes g st ¼ Aσ S Bσ t þ C st σ and g top ¼ Bσ t þ C top σ f t are given by ~ af 1 Df nΓ
; ð1 þ Γ 2 ΓÞsinh l þ Γð1 þ ΓÞcosh l ~ af 1 Df nΓ ; B¼ 2 ð1 þ Γ ÞsinhðL lÞ þ 2Γ coshðL lÞ ~ C st ¼ af 1 Df nΓ
Here L ¼ L=λf , l ¼ l=λf , Γ ¼ af =λf , and D k Γ ¼ f 1 þ sf λs Gðλs Þ ; af kf s Ds
af ðR2W R21 Þ
¼
h
σ 1 ¼ 1 n~n~1 ¼ 1 exp ðl "
K 3 1 l2 Þ
i
;
2Df τf R1 af ðR2W R21 Þ
:
#
In the case of k Z3 steps the expressions for the elongation rate of the nanowire and the step velocities have the form ~ σt σ1Þ 1 dL 2 J ω sin α Df nð ¼ U ðL l1 Þ þ þ Jf ðα; βÞ; L l1 þ 2af Ω dt R 2 ~ σ t σ 1 Þ Df nð ~ σ1 σ2Þ 1 dl1 J ω sin α Df nð ¼ ðL l2 Þ þ ; Sf dt L l1 þ 2af l1 l2 þ2af 2 ~ σ i 1 σ i Þ Df nð ~ σi σi þ 1Þ 1 dli J ω sin α Df nð ¼ ðli 1 li þ 1 Þ þ þ ; Sf dt li 1 li þ 2af li li þ 1 þ2af 2 i ¼ 2; :::; k 1; 1 dlk ¼ Sf dt
lk 1 þ lk þ
! ~ σk σk 1 Þ R2W R2k J ω sin α Df nð þ cot α : lk 1 lk þ 2af ωRk 2
ðB3Þ
" # n~ 1 K σ 1 ¼ 1 ~ ¼ 1 exp n ðl1 l2 Þ3 " # γ S 1 1 n~ K K σ i ¼ 1 ~i ¼ 1 exp st f þ n kB T Ri R1 ðli 1 li Þ3 ðli li þ 1 Þ3
i ¼ 2; 3; …; k 1;
ðA1Þ
"
#
γ S 1 1 n~ K þ σ k ¼ 1 ~k ¼ 1 exp st f : n kB T Rk R1 ðlk 1 lk Þ3
Appendix C ðA2Þ
where G ¼ UðR1 =λs Þ=U 0 ðR1 =λs Þ. Here UðxÞ ¼ K 1 ðRW =λs ÞI 0 ðxÞ þ I 1 ðRW =λs ÞK 0 ðxÞ is a combination of the modified Bessel functions, аnd U 0 ðxÞ is the derivative of UðxÞ, taken at x ¼ R1 =λs . In the limiting case of very weak desorption of adatoms from the substrate (λs =as 4 4 1) one can use the approximate expression 2ξDf τs R1
where,
Here,
ΓΓ sinh l þ cosh l 1
ð1 þ Γ 2 ΓÞsinh l þ Γð1 þ ΓÞcosh l ! Γ sinhðL lÞ þ coshðL lÞ 1 þ ; ð1 þ Γ 2 ÞsinhðL lÞ þ 2Γ coshðL lÞ h i af 1 Df n~ t Γ Γ sinhðL lÞ þ coshðL lÞ 1 : C top ¼ ð1 þ Γ 2 Þsinh l þ 2Γ coshðL lÞ
Γ
Similar expression was derived in [9] from the condition of the material balance of the incident atom flux at the surface and the flux of atoms incorporating into the nanowire top facet. The step velocity and the elongation rate of the nanowire in the case of a single step are giving by Eqs. (12) and (13) of Section 3. In the case of two steps the relevant expressions are: ~ σt σ1Þ 1 dL 2 J ω sin α Df nð ¼ U ðL l1 Þ þ þ Jf ðα; βÞ; L l1 þ 2af Ω dt R 2 ~ σ t σ 1 Þ Df nð ~ σ1 σ2Þ 1 dl1 J ω sin α Df nð ¼ ðL l2 Þ þ ; L l1 þ 2af l1 l2 þ2af Sf dt 2 ! ~ σ2 σ1Þ 1 dl2 R2 R22 J ω sin α Df nð ¼ l1 þ l2 þ W þ cot α ; ðB2Þ l1 l2 þ 2af Sf dt ωR2 2
γ S 1 1 K n~ þ σ 2 ¼ 1 ~2 ¼ 1 exp st f kB T R2 R1 n ðl1 l2 Þ3
Acknowledgments
A¼
65
ðA3Þ
We assumed that the new step is formed when the adatom concentration at the nanowire base nð0Þ reaches some critical value nn ð0Þ. If there are no steps at the sidewall and desorption of adatoms is negligible, adatom concentration on the sidewall at the nanowire base is given by the solution of the relevant diffusion problem [31] in the limit of large λs and λf : nð0Þ ¼
ðR2W R2 ÞJ cos α J ω sin α ðL þ af Þ þ ðL þ 2af ÞL þ n~ t 2RDf 2Df
ðC1Þ
Thus, the critical concentration nn ð0Þ is achieved at a critical length of the nanowire Ln , which obeys the following equation Appendix B In the limit of no desorption the elongation rate of the nanowire without sidewall steps is given by: " # 1 dL 2 R2 R2 ¼ U Lþ W cot α J ω sin α þ Jf ðα; βÞ: ðB1Þ Ω dt R ωR
σ n ð0Þ ¼
ðR2W R2 ÞJ cos α n J ω sin α n ðL þ af Þ þ ðL þ 2af ÞLn σ t ; 2RDf n~ 2Df n~
ðC2Þ
where σ n ð0Þ ¼ nn ð0Þ=n~ 1 is the supersaturation on the sidewall surface at the nanowire base in respect to a step of length 2π R1 . It worth to be mentioned that if σ t 40 and the nanowire length L is small, the supersaturation σ n ð0Þ is negative so that the formation
66
S.N. Filimonov, Y.Yu. Hervieu / Journal of Crystal Growth 427 (2015) 60–66
of new steps at the nanowire base is thermodynamically unfavorable. Moreover, as has been shown in [30], even if a step is formed by a chance at such unfavorable conditions, its growth rate will be negative and the step will dissolve. Thus, the presence of a strong sink for adatoms at the nanowire top prevents the lateral growth of relatively short nanowires. Similarly, behind the moving step the critical concentration n nn ð0Þ is achieved at some critical distance lk , which obeys the equation
σ n ð0Þ ¼
ðR2W R2k ÞJ cos α n J ω sin α n n ðlk þ af Þ þ ðlk þ 2af Þlk σ k ; 2Rk Df n~ 2Df n~
ðC3Þ
where k ¼ 1; 2; 3; …, Rk ¼ R þ kh and " # ðk 1ÞhSf γ st n~ k K σ k ¼ 1 ~ ¼ 1 exp n ðlk 1 lk Þ3 kB TðR þhÞðR þ khÞ Let us note that in the case of a single step ðlk ¼ l1 Þ σ k ¼ 0. It follows from (C2) and (C3) that if the nanowire top is a stronger sink for n adatoms than the sidewall step ðσ t 40Þ and R 4 4 h then Ln 4 l1 . In this case the second step forms before the first step will reach the nanowire top. Using (C2) and the experimentally measured value Ln , which is determined by the onset of the radial growth of the nanowires, one can determine σ n ð0Þ. Then solving (C3), one finds n the critical distance lk for nucleation of new steps. n
n
References [1] P. Yang, R. Yan, M. Fardy, Nano Lett. 10 (2010) 1529. [2] V.G. Dubrovskii, G.E. Cirlin, V.M. Ustinov, Semiconductors 43 (2009) 1539. [3] M. Meyyappan, M.K. Sunkara, Inorganic Nanowires: Applications, Properties and Characterization, CRC Press, Boca Raton, FL, 2009. [4] R.R. LaPierre, A.C.E. Chia, S.J. Gibson, C.M. Haapamaki, J. Boulanger, R. Yee, P. Kuyanov, J. Zhang, N. Tajik, N. Jewell, K.M.A. Rahman, Phys. Status Solidi (RRL) 7 (2013) 815. [5] M. Borgström, K. Deppert, L. Samuelson, W. Seifert, J. Cryst. Growth 260 (2004) 18.
[6] J.C. Harmand, M. Tchernycheva, G. Patriarche, L. Travers, F. Glas, G. Cirlin, J. Cryst. Growth 301–302 (2007) 853. [7] V.G. Dubrovskii, N.V. Sibirev, G.E. Cirlin, M. Tchernycheva, J.C. Harmand, V.M. Ustinov, Phys. Rev. E 77 (2008) 031606. [8] C. Chen, M.C. Plante, C. Fradin, R.R. LaPierre, J. Mater. Res. 21 (2006) 2801. [9] M. Tchernycheva, L. Travers, G. Patriarche, F. Glas, J.-C. Harmand, G.E. Cirlin, V.G. Dubrovskii, J. Appl. Phys. 102 (2007) 094313. [10] M.C. Plante, R.R. LaPierre, J. Appl. Phys. 105 (2009) 114304. [11] V.G. Dubrovskii, Nucleation Theory and Growth of Nanostructures, Springer, Heidelberg New York Dordrecht London, 2014. [12] S. Li, X. Zhang, L. Zhang, J. Phys. Chem. C 114 (2010) 10379. [13] M.C. Plante, R.R. LaPierre, J. Cryst. Growth 310 (2008) 356. [14] F. Furtmayr, M. Vielemeyer, M. Stutzmann, J. Arbiol, S. Estradé, F. Peirò, J.R. Morante, M. Eickhoff, J. Appl. Phys. 104 (2008) 034309. [15] E. Hilner, U. Håkanson, L.E. Fröberg, M. Karlsson, P. Kratzer, E. Lundgren, L. Samuelson, A. Mikkelsen, Nano Lett. 8 (2008) 3978. [16] M. Hjort, S. Lehmann, J. Knutsson, R. Timm, D. Jacobsson, E. Lundgren, K.A. Dick, A. Mikkelsen, Nano Lett. 13 (2013) 4492. [17] K. Yagi, H. Minoda, M. Degawa, Surf. Sci. Rep. 43 (2001) 45. [18] Y. Greenberg, A. Kelrich, Y. Calahorra, S. Cohen, D. Ritter, J. Cryst. Growth 389 (2014) 103. [19] R.L. Schwoebel, J. Appl. Phys. 38 (1967) 1759. [20] T. Michely, J. Krug, Islands, Mounds and Atoms:Patterns and Processes in Crystal Growth Far from Equilibrium, Springer, Heidelberg, 2004. [21] N. Israeli, D. Kandel, Phys. Rev. B 62 (2000) 13707. [22] P.-W. Fok, R.R. Rosales, D. Margetis, Phys. Rev. B 76 (2007) 033408. [23] P.-W. Fok, R.R. Rosales, D. Margetis, Phys. Rev. B 78 (2008) 235401. [24] C.-Y. Wen, J. Tersoff, M.C. Reuter, E.A. Stach, F.M. Ross, Phys. Rev. Lett. 105 (2010) 195502. [25] V.G. Dubrovskii, Phys. Rev. B 87 (2013) 195426. [26] F. Glas, Phys. Rev. B 90 (2014) 125406. [27] A.A. Golovin, S.H. Davis, P.W. Voorhees, J. Appl. Phys. 107 (2010) 024308. [28] F. Glas, J.-C. Harmand, G. Patriarche, Phys. Rev. Lett. 99 (2007) 146101. [29] V.G. Dubrovskii, N.V. Sibirev, R.A. Suris, G.E. Cirlin, V.M. Ustinov, M. Tchernysheva, J.C. Harmand, Semiconductors 40 (2006) 1075. [30] S.N. Filimonov, Yu.Yu. Hervieu, e-J. Surf. Sci. Nanotechnol. 12 (2014) 68. [31] V.G. Dubrovskii, Yu.Yu. Hervieu, J. Cryst. Growth 401 (2014) 431. [32] I.V. Markov, Crystal Growth for Beginners, Fundamentals of Nucleation, Crystal Growth and Epitaxy, Second ed., World Scientific, Singapore, 2003. [33] E.D. Williams, Surf. Sci. 299/300 (1994) 502. [34] F. Glas, Phys. Status Solidi B 247 (2010) 254. [35] N.V. Sibirev, M.A. Timofeeva, A.D. Bol’shakov, M.V. Nazarenko, V.G. Dubrovskii, Phys. Solid State 52 (2010) 1531.
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Model of step propagation and step bunching at the sidewalls of nanowires Sergey N. Filimonov, Yuri Yu. Hervieu n Department of Physics, National Research Tomsk State University, 634050 Tomsk, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 19 March 2015 Received in revised form 9 June 2015 Accepted 3 July 2015 Communicated by: K. Deppert Available online 14 July 2015
Radial growth of vertically aligned nanowires involves formation and propagation of monoatomic steps at atomically smooth nanowire sidewalls. Here we study the step dynamics with a step flow model taking into account the presence of a strong sink for adatoms at top of the nanowire and adatom exchange between the nanowire sidewall and surrounding substrate surface. Analytical expressions for velocities of steps propagating from the nanowire base to the nanowire top are obtained. It is shown that the step approaching the nanowire top will slow down if the top nanowire facet is a stronger sink for adatoms than the sidewall step. This might trigger bunching of the steps at the sidewall resulting in development of the pencil-like shape of nanowires such as observed in, e.g., the Au-assisted MBE growth of InAs. & 2015 Published by Elsevier B.V.
Keywords: A1. Growth models A1. Nanostructures A1. Surface processes A3. Molecular beam epitaxy
1. Introduction In recent years vertically aligned semiconductor nanowires have attracted great attention due to prospects of their use in microelectronics, optoelectronics, photovoltaics, nanosensing etc. [1–4]. Specific properties of nanowires are related to their highly anisotropic shape, which may vary a lot depending on the deposition conditions [4–6]. Moreover, even at the given deposition conditions the nanowire shape may change in course of the growth [7–10]. In general the nanowire shape is determined by the interplay of the lateral and axial growth of nanowires [7,11]. In that respect it is essential that the nanowire sidewalls remain atomically smooth while the nanowire is growing laterally. This indicates that the lateral growth of nanowires occurs in the layer-by-layer or stepflow fashion involving nucleation of new steps at the sidewall facets and their propagation along the facets. Monoatomic steps at the sidewalls of nanowires were directly visualized by transmission electron microscopy [12–14] and scanning tunneling microscopy [15,16]. Bearing in mind the step-flow mechanism of the radial growth of nanowires it is tempting to relate the experimentally observed transformations of the nanowire shape with a kinetic instability of the steps propagating along the sidewall facets. For instance, it is known that on vicinal surfaces the kinetic step bunching may lead to the surface faceting [17]. Similarly, the step bunching may be
n
Corresponding author. E-mail address: [email protected] (Yu.Yu. Hervieu).
http://dx.doi.org/10.1016/j.jcrysgro.2015.07.005 0022-0248/& 2015 Published by Elsevier B.V.
responsible for the formation of new facets at the nanowire sidewalls such as the {10–11} facets of pencil-like InAs nanowires grown by the Au-assisted MBE [9]. The transition from cylindrical to pencil-like shape of InAs nanowires was explained in earlier theoretical works [9,10] by the onset of the radial nanowire growth by nucleation and propagation of steps at some critical length of the nanowire. However dynamics of the steps at the sidewall was not considered explicitly within the models developed in [9,10]. Therefore it remains unclear why the appearance of the steps leads to the specific nanowire shape observed in the experiment. Intuitively, one would expect that the step velocity should be considerably higher than the axial growth rate of the nanowire, because the advance of the step by one atomic position requires formation of an atomic row along the step edge whereas the increase of the nanowire length by the same distance requires deposition of a whole atomic layer on the nanowire top facet. Therefore, one would expect the steps moving in the direction of the nanowire top to catch up and join the nanowire top facet leading to the thickening of the nanowire including its upper part near the catalyst droplet. However, in the aforementioned experiments on the InAs nanowires growth [9] the nanowire radius at the top did not change with time. In [18] the formation of tapered InP nanowires with non-tapered sections at the top was observed. These observations clearly indicates that for some reasons the steps slow down in the upper part of the nanowire instead of catching up the nanowire top facet. In the present work we model the axial and radial growth of nanowires treating explicitly dynamics of the steps nucleating sequentially at the nanowire base. This allows us to go beyond the
S.N. Filimonov, Y.Yu. Hervieu / Journal of Crystal Growth 427 (2015) 60–66
61
earlier material conservation models [9,10] and reproduce different scenarios of the nanowire growth: thickening of nanowires while preserving their cylindrical shape and gradual transformation of the nanowire shape due to bunching of the steps at the nanowire sidewalls. We show that the step bunching might be conditioned by the slowing down of the leading step approaching the nanowire top due to the transfer of atoms from the step to the nanowire top with the later being a stronger sink for adatoms than the step itself. Results of numerical simulations of the step dynamics and of the corresponding evolution of the nanowire shape are in good agreement with the experimental observations of the shape transformations of the InAs nanowires [9].
2. Model Let us consider a cylindrical nanowire with k steps of the monoatomic height nucleated at the nanowire base and propagating from the base to the nanowire top (Fig. 1). The axisymmetric configuration of the steps shown in Fig. 1 is similar to that considered in the earlier model of filamentary crystal growth by Schwoebel [19] and resembles the “wedding cake” step configuration widely used to model growth and dissolution of mounds (see e.g. [20–23]). However, in contrast to the model of mounds, the steps in our model propagate towards the nanowire top without growing in the lateral direction. Considering the axial growth of the nanowire it is generally believed that under the catalyst droplet the top nanowire facet grows layer-by-layer by nucleation and spreading of two-dimensional (2D) islands [3,11,24–26]. The energy per unit length of a step in contact with a liquid phase is typically much smaller as compared to the energy of a step in contact with a gas phase. Moreover, the curvature of the step on the nanowire top may be negative if the 2D island nucleates as a circular step at the triple phase line [27] (this nucleus position favors formation of the wurtzite structure of III–V semiconductor nanowires [28]). This means that due to the Gibbs–Thomson effect the nanowire top facet should be a stronger sink for the sidewall adatoms than the sidewall step. This is again in contrast with the case of mounds where the Gibbs–Thomson effect generates the downhill flux of atoms from the upper (more curved) steps to the lower (less curved) steps [21–23] leading to shrinkage of the topmost terraces and, as a consequence, to gradual flattening of the mounds at the absence of the deposition flux. Concentration of adatoms on the substrate surface, ns , and on the terraces between the steps on the sidewall, ni , obey the steadystate diffusion equations [11,29] Ds d dns ns r þ J cos α ¼ 0; ð1Þ r dr dr τs 2
Df
d ni n þ J ω sin α i ¼ 0; τf dz2
ð2Þ
where Ds and Df are the surface diffusion coefficients on the substrate surface and on the sidewall, respectively, τs and τf are the respective life times of adatoms before desorption, J is the deposition flux, and α is the angle between the deposition flux direction and the nanowire axis (it is assumed that the nanowires grow normal to the surface). The coefficient ω is equal to 1=π in the case of molecular beam epitaxy considered in the present work. Index i in Eq. (2) enumerates the sidewall steps as sketched in Fig. 1. General solutions of Eq. (1) and (2) read
Fig. 1. Schematic view of a nanowire with steps at the sidewall.
respectively, I 0 ðr=λs Þ and K 0 ðr=λs Þ are the modified Bessel functions. The integration constants C 1 , C 2 , Ai and Bi are to be determined from the appropriate boundary conditions. Following [11,29] as the first boundary condition we request that the diffusion flux on the substrate surface vanishes at certain distance RW from the center of the nanowire base dns r ¼ RW ¼ 0: dr
ð4Þ
In the present paper we consider RW as a model parameter having a meaning of an effective radius of the feeding area around the given nanowire. Boundary conditions at the nanowire base follow from the conditions of balance of the diffusion fluxes and the resulting fluxes of adatoms crossing the boundary between the nanowire sidewall and substrate surface [30,31] s Ds dn ¼ ksf ns ðRk Þ kf s nk þ 1 ð0Þ; dr r ¼ Rk
dnf Df ¼ kf s nk þ 1 ð0Þ ksf ns ðRk Þ: dz z ¼ 0
ð5Þ
Here ksf and kf s are the rate constants of the transitions from the substrate surface to the sidewall surface and vice versa, and Rk is the radius of the nanowire base: Rk ¼ R þkh, where R is the radius of the nanowire top, and h is the height of the monoatomic step at the nanowire sidewall. The balance equations for the diffusion fluxes and the resulting fluxes of adatoms incorporating into the steps from the upper (-) and lower ( þ) terraces give us 2k boundary conditions of the form i þ 1 Df dndz ¼ β ½ni þ 1 ðli Þ n~ i ; z ¼ li
dn Df i ¼ β þ ½ni ðli Þ n~ i ; dz z ¼ li
ð6Þ
ð3Þ
where li is the distance from the nanowire base to the i-th step ði ¼ 1:::kÞ, β and β þ are the kinetic coefficients of the step [32], and n~ i is the adatom concentration in equilibrium with the i-th step. The last boundary condition represents the balance equation for the diffusion flux and the resulting flux of adatoms to the nanowire top [30,31] dn1 Df ¼ kf t n1 ðLÞ ktf C top ¼ kf t ½n1 ðLÞ n~ t ; ð7Þ dz z ¼ L
and λf ¼ ðDf τf Þ are the diffusion lengths of where λs ¼ ðDs τs Þ adatoms on the substrate surface and on the nanowire sidewall,
where L is the nanowire length, kf t and ktf are the rate constants of the transitions of adatoms from the sidewall facet to the nanowire
ns ðrÞ ¼ Jτs cos α þC 1 I 0 ðr=λs Þ þC 2 K 0 ðr=λs Þ; ni ðzÞ ¼ Jτf ω sin α þ Ai coshðz=λf Þ þ Bi sinhðz=λf Þ; 1=2
1=2
62
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top and in the opposite direction, respectively, and C top is the concentration of atoms on the top facet near its edge. The concentration n~ t ¼ ktf C top =kf t in the modified right part of Eq. (7) is the concentration of adatoms on the sidewall terrace in equilibrium with the nanowire top. Here for simplicity we neglect possible oscillations of the supersaturation in the droplet [24–26] and treat the nanowire top facet as a permanent sink for adatoms. For the rate constants of the adatom exchange between the nanowire top (t), its sidewall (f) and the substrate surface (s) we used standard Arrhenius expressions kαβ ¼ bα να expð Eαβ =kB TÞ, where Eαβ is the activation energy of the transition from α to β ðα; β ¼ s; f ; tÞ, να is the attempt frequency, kB is the Boltzmann constant, T is the substrate temperature, and bα is the interatomic distance at the given surface in the direction normal to its boundary. We also assumed for simplicity that the barrier for attachment of atoms to the steps equals to the surface diffusion barrier, so that β þ ¼ β ¼ Df =af , where af is the interatomic distance in the direction parallel to the step edge. The energy profile for an atom on the surface is shown schematically in Fig. 2. Typically, the nanowire radii are small (e.g. the radius of InAs nanowires at the onset of their lateral growth is about 15 nm [9]), therefore for the steps at the sidewall the Gibbs–Thomson effect has to be taken into account. Due to the Gibbs–Thomson effect the chemical potential along a circular step of a radius Rst is higher by ~ 1 Þ ¼ γ st Sf =kB TRst than the chemical potential along Δμ ¼ kB T lnðn=n an infinitely long straight step [32]. Here Sf is the area per atom of the sidewall facet, and γ st is the step formation energy per unit length. Thus for a single step at the sidewall of a nanowire of radius Rone has n~ 1 n~ ¼ n~ 1 exp½γ st Sf =kB TðR þ hÞ, where n~ 1 ¼ expð ΔE=kB TÞ=Sf is the concentration of the adsorption layer in equilibrium with an infinitely long straight step. Here ΔE is the depth of the potential energy trap for an atom at the kink site at the step edge relatively to the energy of an isolated adatom on the sidewall surface (Fig. 2). If there are several steps at the nanowire sidewall one should also take into account the pairwise entropic or elastic repulsion of the steps. Then the equilibrium adatom concentration on the i-th terrace ði ¼ 2; :::; k 1Þ is given by " # γ st Sf K K n~ i ¼ n~ 1 exp ; ð8Þ exp kB TRi ðli 1 li Þ3 ðli li þ 1 Þ3 where Ri ¼ R þ ih is the radius of the i-th step, K ¼ 2Sf A=kB T and A is the step–step repulsion parameter [33]. Interactions of the first ði ¼ 1Þ step with the nanowire top and the last ði ¼ kÞ step with the nanowire base were neglected. Concentration of adatoms on the sidewall surface in equilibrium with the nanowire top can be written as n~ t ¼ ktf C top = kf t ¼ ð1 þ σ top ÞC~ top expð ΔEf t =kB TÞbt νt =bf νf , where C~ top is the equilibrium concentration of atoms on the nanowire top (in the droplet), determined from the condition of the equilibrium between the respective infinite liquid and solid phases at a given temperature, σ top ¼ C top =C~ top 1 is the supersaturation at the boundary between the droplet and the nanowire sidewall, and
Esf
step
Efs
ΔEft
ΔE
ΔEtop
l L sidewall
ft
ð ΔEf t =kB TÞ ¼ n~ 1 Sf =St . Using this relation one gets n~ t ¼ ð1 þ σ top Þn~ 1 at νt =af νf . From the solution of the boundary problem (1)–(7) one finds the resulting fluxes of adatoms to the steps ðg i ¼ β þ ½ni ðli Þ n~ i þ β ½ni þ 1 ðli Þ n~ i Þ and to the nanowire top ðg top ¼ kf t ½n1 ðLÞ n~ t Þ. The step velocities and the rate of elongation of the nanowire are related to the fluxes as dli =dt ¼ Sf g i and dL=dt ¼ 2Ωg top =Rþ J top , where J top ¼ J Ωf ðα; β Þ is the contribution of direct adsorption of atoms onto the nanowire top, Ω is the volume per atom of the crystal, and f ðα; β Þ is a geometrical parameter related to the flux incidence angle α and to the contact angle β of the droplet [34]. The factor of 2=R in the expression for dL=dt reflects the fact that the total flux from the nanowire sidewall through the perimeter of the top nanowire facet 2π Rgtop corresponds to a covered area of π R2 .
3. Dynamics of a single step The solution of the boundary problem (1)–(7) in the case of a single step at the nanowire sidewall allows to write down the fluxes of adatoms incorporating into the step and into the nanowire top facet as g st ¼ Aσ S Bσ t þC st σ and ~ 1 is g top ¼ Bσ t þ C top σ f t , respectively. Here σ ¼ ðJ τf ω sin α=nÞ the supersaturation in the adlayer on the sidewall surface relatively to the step, σ f t ¼ ðJ τf ω sin α=n~ t Þ 1 is the supersaturation on the sidewall surface relatively to the nanowire top, ~ is the supersaturation at the step relatively to the σ t ¼ 1 ðn~ t =nÞ ~ 1 is the supersaturation on nanowire top, and σ s ¼ ðJ τs ξ cos α=nÞ the substrate surface relatively to the step, where ξ ¼ ksf =kf s exp½ðEf s Esf Þ=kB T is the parameter taking into account possible difference of the adsorption energies of adatoms on the substrate surface and on the nanowire sidewall [30,31]. The expressions for the coefficients A, B, C st and C top are given in Appendix A. Let us now analyze dynamics of the step in the limiting cases of strong and weak desorption of adatoms from the sidewall surface. For the case of strong desorption (short diffusion length λf of adatoms on the sidewall) and at a moderate supersaturation on the substrate surface ðσ S o o σ expðl=λf ÞÞ the velocity of the step at a distance l 4 4 λf 4 4 af is h i Sf Df n ðexpðL lÞ 1Þσ σ t dl ¼ Sf g st ≈ ; dt λf sinhðL lÞ þ 2af coshðL lÞ
ΔEsf substrate
ΔEf t ¼ Etf Ef t is the energy gain due to the transfer of an adatom from the sidewall surface to the nanowire top (Fig. 2). For C~ top one has C~ top ¼ expð ΔEtop =kB TÞ=St , where St is the area per atom of the top facet, ΔEtop is the depth of the trap at the kink site of an infinitely long straight step in contact with the liquid phase. It is convenient to express the concentration n~ t via the equilibrium concentration of adatoms on the sidewall face, n~ 1 . It is known that the work to transfer an atom from the kink site to the ambient gas phase taken with the negative sign is equal to the difference of chemical potentials of the ambient phase and that of the crystal and does not depend on the orientation of the surface and the presence of a liquid film on the surface [32]. Therefore ΔEtop þ ΔE ¼ ΔE (Fig. 2). Then it follows that C~ top exp
top
Fig. 2. Energy profile for an atom on the surface.
ð9Þ
where L ¼ L=λf and l ¼ l=λf . It is seen from this relation, that if the supersaturation σ t is negative or zero (the nanowire top is a poor sink for adatoms), the step velocity is positive independent on the distance between the step and the nanowire top. If, in addition, the nanowire is growing vertically manly due to supply of diffusing
S.N. Filimonov, Y.Yu. Hervieu / Journal of Crystal Growth 427 (2015) 60–66
adatoms from the sidewall surface then h i dL 2Ωg top 2ΩDf n ðcoshðL lÞ 1Þσ þcoshðL lÞσ t h i ≈ ≈ : dt R R λ sinhðL lÞ þ 2a coshðL lÞ f
ð10Þ
f
As seen dl=dt 4 dL=dt at σ t r0. Then the step will inevitably catch up the nanowire top and will merge with the top nanowire facet. As a result the nanowire increases its radius while preserving its cylindrical shape. When the sink for adatoms at the nanowire top is strong ðσ t 4 0Þ the step velocity vanishes at a certain critical distance to the nanowire top σt : ð11Þ ðL lÞcr ¼ λf ln 1 þ σ After approaching this distance, the step will move up in parallel with the nanowire elongation. Since, typically, λf =bf 4 4 1, it follows from Eq. (11) that at moderate σ the step cannot catch up the top even at small σ t . The slowing down of the step approaching the nanowire top might cause bunching of the steps nucleating sequentially at the nanowire base. In the opposite limit of no desorption (very long diffusion lengths of adatoms on the sidewall facets, λf , and on the substrate surface, λs , and high supersaturations σ and σ s ) one obtains the following approximation for the step velocity " # Df n~ σ t 1 dl R2 R21 J ω sin α W cot α þ L þ l : ð12Þ Sf dt ωR1 2 L l þ 2af Accordingly the vertical growth rate of the nanowire is approximated by Df n~ σ t 1 dL 2 J ω sin α ðL lÞ þ ð13Þ þ Jf ðα; βÞ: Ω dt R 2 L l þ 2af It follows from (12) that similarly to the case of intensive desorption the step velocity is positive if the nanowire top is a weak sink for adatoms (σ t r 0). Thereby, as seen from (12) and (13), dl=dt 4 dL=dt, i.e. the step moves faster than the nanowire top and will eventually catch up the top facet increasing the nanowire radius without changing its shape. In the case of a strong sink for adatoms at the nanowire top (σ t 40) the step velocity turns to zero at a certain critical distance ðL lÞcr from the top if " # af J ω sin α R2W R21 σt 4 cot α þ 2l ð14Þ Df n~ ωR1 It is essential that the right part of (14) depends on the step position and, hence, on the growth time. This makes the difference with the case of intensive desorption where due to a short length
63
of adatom surface diffusion on the sidewall surface, λf , the feeding area of the step is limited by λf and does not depend on l. Therefore, in the case of intensive desorption independent on l and on the presence of other steps at the sidewall, the leading step cannot catch up with the top even at relatively small positive σ t . Without desorption the feeding area behind the step increases with increasing l, so that the inequality (14) may be broken at sufficiently large l even if σ t ¼ 1. Therefore in the limit of no desorption the step should catch up the nanowire top and merge into the top facet. The above described scenario, however, might be altered by the appearance of other steps at the sidewall which will compete with the leading step for adatoms, thus decreasing velocity of the leading step.
4. Numerical modeling Numerical modeling of the nanowire growth was performed by integration of the equations for the step velocities and for the nanowire elongation rate. The elongation rate of the nanowire without sidewall steps was calculated using an expression obtained in [31] in the limit of very large λs and λf . The elongation rate of the nanowire with the sidewall steps and the corresponding step velocities were determined from the solution of the boundary problem (1)–(7). The corresponding expressions are listed in Appendix B. It was assumed that the first step is formed when the nanowire length reaches some critical value Ln , which might be determined experimentally by the onset of the radial growth of the nanowires. The moments of nucleation of the second, third, etc. steps were determined self-consistently as described in Appendix C. Whenever the leading step caught up with the nanowire top the radius of the nanowire was increased by one interatomic distance and integration of the equations of motion of the steps continued. Parameters of the model were chosen to simulate the Auassisted growth of InAs nanowires [9]. The step formation energy was approximated by γ st ¼ hγ f , where γ f is the surface energy per unit area of the sidewall nanowire facet, so that Sf γ st ¼ Ωγ f . For γ f we used the value of 5.7 eV/nm2 reported in [11,35] for the
InAsð1100Þface. Assuming the droplet contact angle β to be π =2 we put f ðα; βÞ ¼ ð1 þ cos αÞ=2. The critical length Ln ¼ 1500 nm, temperature T ¼ 683 К , incidence angle of the molecular beam α ¼ 331, deposition flux J= cos α ¼ 0:2 nm=s, and initial radius of the nanowire R ¼ 15 nm, were taken from the experiments on the growth of InAs nanowires [9]. For the structural parameters af , h, Sf ,St and Ω we used values of the experimentally observed wurtzite structure of InAs. The radius of the feeding zone of a
Fig. 3. Dependences of the nanowire height on the growth duration (red lines) and trajectories of the steps on the nanowire sidewall (black lines) at different supersaturations: (a) σ top ¼ 0:41 (σ t ¼ 0:001), (b) σ top ¼ 0:3 (σ t ¼ 0:08), and (c) σ top ¼ 0:1 (σ t ¼ 0:22). The insets show the nanowire shape evolution with the increasing number of steps formed at the nanowire base (the images are distorted in the horizontal direction for clarity). The magnified view of the area highlighted by the rectangle in (b) is shown in Fig. 4. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
64
S.N. Filimonov, Y.Yu. Hervieu / Journal of Crystal Growth 427 (2015) 60–66
nanowire RW was set to 40 nm, which agrees with estimates made in [29]. We also set K ¼ 1 nm3 , νf ¼ 1013 s 1 , and at νt =af νf ¼ 1. Unknown parameters of the model are the surface diffusion barrier Es , the depth of the potential energy trap for an atom at the kink site ΔE, and the supersaturation at the nanowire top σ top . The parameters Es and ΔE can be combined into a single parameter Ekink ¼ Es þ ΔE, which is the activation energy of the detachment of an atom from the kink. Numerical modeling of the nanowire growth at Ekink ¼ 1:1eV has shown that at large σ top (at σ top Z 0:4 with the parameters used here) the effect of the slowing down of the step due to the transfer of atoms from the step to the nanowire top is small. Here formation of the steps at the sidewall results in increasing nanowire radius while its cylindrical shape is preserved (Fig. 3а). At small σ top a bunch of steps forms (Fig. 3b). As the nanowire is getting higher, the step bunch moves up keeping a roughly constant distance to the nanowire top. With decreasing σ top the distance from the bunch to the nanowire top increases (Fig. 3c), which is related to increasing supersaturation at the step relatively to the top, σ t , i.e. to a stronger effect of the step slowing down. The early stage of the step bunching is shown in Fig. 4. Partial dissolution of the leading step is conditioned by two factors: by the transfer of atoms from the leading step to the nanowire top and by decreasing adatom incorporation flux from the terrace behind the step due to shortening the distance to the second step approaching from the bottom. As the two steps get closer, the increasing step–step repulsion impels the leading step to start moving upwards again. As the next step approaches the first two steps the dissolution – growth cycle repeats. The atom transfer from the upper sidewall steps to the lower ones (because of the Gibbs–Thompson effect) plays a minor role here, however it shortens the average distance between the steps in the bunch and stabilizes the bunch at later stages of growth. At a given supersaturation σ top and a given critical length Ln , the formation and bunching of the steps is possible only in a certain range of the activation energies of the detachment of atoms from the kink site. For instance at σ top ¼ 0:1 the step bunching is observed at 1:25eV 4 Ekink 4 0:88 eV, while increasing the super saturation σ top to 0.3 shrinks this range to 1:2 eV 4 Ekink 4 0:83 eV. The upper bound of this interval arises from to the fact that with increasing Ekink the exchange flux between the step and the nanowire top decreases. This facilitates approach of the step to the nanowire top. At the same time at a given supersaturation σ t and fixed other model parameters, the activation energy Ekink determines the critical supersaturation σ n ð0Þ below which nucleation of new steps is impossible (see Appendix C). It follows from (C2) that at σ t ¼ 0 decreasing Ekink decreases σ n ð0Þ to zero. At
Fig. 4. Dependences of the nanowire height on the growth duration and trajectories of the steps on the nanowire sidewall in the early stage of step bunching (the magnified view of the area highlighted by the rectangle in Fig. 3b).
Fig. 5. Dependences of the nanowire height on the growth duration (red line) and trajectories of 111 steps on the nanowire sidewall (black lines) at σ top ¼ 0:05 ðσ t ¼ 0:26Þ. The inset shows the resulting shape of the nanowire (the image is distorted in the horizontal direction). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
σ t 4 0 the supersaturation σ n ð0Þ becomes negative below some
value of Ekink . This sets the lower bound for Ekink . Evidently, formation of the step bunch at the sidewall should be considered only as the first stage of the shape transformation of the nanowire. Indeed, as the number of steps at the sidewall increases, the velocity of the bunch becomes smaller and therefore the distance from the leading step of the bunch to the nanowire top increases. As a result the leading step may “detach” from the bunch so that at a later stage of the growth a set of separate step bunches may develop, which are characterized by decreasing number of the steps in the bunch as the bunch gets closer to the nanowire top (Fig. 5). At the same time it is well known that the step bunching may result in the formation of new stable facets on a vicinal surface [17]. The {10–11} oriented facets separated by flat areas were experimentally observed on the sidewalls of the InAs nanowires [9]. Kinetics of the facet formation and the growth at the faceted sidewalls of the nanowires require special consideration and are beyond the scopes of the present study.
5. Conclusions Transition from the cylindrical to pencil-like shape of nanowires may occur as a result of the bunching of the monoatomic steps nucleating at the sidewall nanowire facets and propagating from the nanowire base to the nanowire top. The necessary condition for the step bunching is slowing down of the leading step approaching the nanowire top. The step will slow-down when the detachment frequency of atoms from the step is substantially higher than the frequency of transitions from the nanowire top (from the catalyst droplet) to the sidewall facets. In this case the exchange flux between the step and the top facet is directed to the nanowire top and its magnitude increases as the step gets closer to the top. At certain distances between the step and the top the magnitude of the exchange flux becomes comparable with that of the flux of atoms adsorbing on the nanowire sidewalls and on the substrate surface and incorporating into the step. This should slow down the step. In the case of strong desorption of adatoms the feeding area of the step is limited by the adatom diffusion length prior to desorption. In this case the slowing down of the step is possible even at very small supersaturations at the step relatively to the nanowire top, σ t . When desorption of atoms is negligible, the flux of adatoms to the steps increases with increasing length of the nanowire, therefore a single step should unavoidably catch up with the nanowire top independent on σ t . However, appearance of
S.N. Filimonov, Y.Yu. Hervieu / Journal of Crystal Growth 427 (2015) 60–66
next steps at the sidewall reduces the flux of adatoms to the first step, so that at certain deposition conditions merging the step with the top facet might be impossible. Since both the supersaturation σ t and the supersaturation at the nanowire base, σ ð0Þ, increase with decreasing temperature, the formation of pensil-like nanowires should be more pronounced at lower temperatures, as observed in the experiment [9]. On the contrary, rising the temperature should favor formation of the cylindrical nanowires which diameter either does not change with time or increases due to occasional appearance of the steps that propagate along the nanowire sidewall without bunching. Numerical simulations of the nanowire growth with parameters and deposition conditions mimicking the growth of InAs nanowires induced by Au droplets have shown that bunching of the steps at the nanowire sidewall may occur at relatively small supersaturations in the droplet σ top (sufficiently large σ t ) and a relatively small energy of atom detachment from the kinks Ekink .
This study is supported by the Russian Foundation for Basic Research (Grant no. 13-02-12160).
Appendix A At the absence of extra barriers for attachment of adatoms to the step edge and for the transition of adatoms from the sidewall surface to the nanowire top, the coefficients in the expressions for the fluxes g st ¼ Aσ S Bσ t þ C st σ and g top ¼ Bσ t þ C top σ f t are given by ~ af 1 Df nΓ
; ð1 þ Γ 2 ΓÞsinh l þ Γð1 þ ΓÞcosh l ~ af 1 Df nΓ ; B¼ 2 ð1 þ Γ ÞsinhðL lÞ þ 2Γ coshðL lÞ ~ C st ¼ af 1 Df nΓ
Here L ¼ L=λf , l ¼ l=λf , Γ ¼ af =λf , and D k Γ ¼ f 1 þ sf λs Gðλs Þ ; af kf s Ds
af ðR2W R21 Þ
¼
h
σ 1 ¼ 1 n~n~1 ¼ 1 exp ðl "
K 3 1 l2 Þ
i
;
2Df τf R1 af ðR2W R21 Þ
:
#
In the case of k Z3 steps the expressions for the elongation rate of the nanowire and the step velocities have the form ~ σt σ1Þ 1 dL 2 J ω sin α Df nð ¼ U ðL l1 Þ þ þ Jf ðα; βÞ; L l1 þ 2af Ω dt R 2 ~ σ t σ 1 Þ Df nð ~ σ1 σ2Þ 1 dl1 J ω sin α Df nð ¼ ðL l2 Þ þ ; Sf dt L l1 þ 2af l1 l2 þ2af 2 ~ σ i 1 σ i Þ Df nð ~ σi σi þ 1Þ 1 dli J ω sin α Df nð ¼ ðli 1 li þ 1 Þ þ þ ; Sf dt li 1 li þ 2af li li þ 1 þ2af 2 i ¼ 2; :::; k 1; 1 dlk ¼ Sf dt
lk 1 þ lk þ
! ~ σk σk 1 Þ R2W R2k J ω sin α Df nð þ cot α : lk 1 lk þ 2af ωRk 2
ðB3Þ
" # n~ 1 K σ 1 ¼ 1 ~ ¼ 1 exp n ðl1 l2 Þ3 " # γ S 1 1 n~ K K σ i ¼ 1 ~i ¼ 1 exp st f þ n kB T Ri R1 ðli 1 li Þ3 ðli li þ 1 Þ3
i ¼ 2; 3; …; k 1;
ðA1Þ
"
#
γ S 1 1 n~ K þ σ k ¼ 1 ~k ¼ 1 exp st f : n kB T Rk R1 ðlk 1 lk Þ3
Appendix C ðA2Þ
where G ¼ UðR1 =λs Þ=U 0 ðR1 =λs Þ. Here UðxÞ ¼ K 1 ðRW =λs ÞI 0 ðxÞ þ I 1 ðRW =λs ÞK 0 ðxÞ is a combination of the modified Bessel functions, аnd U 0 ðxÞ is the derivative of UðxÞ, taken at x ¼ R1 =λs . In the limiting case of very weak desorption of adatoms from the substrate (λs =as 4 4 1) one can use the approximate expression 2ξDf τs R1
where,
Here,
ΓΓ sinh l þ cosh l 1
ð1 þ Γ 2 ΓÞsinh l þ Γð1 þ ΓÞcosh l ! Γ sinhðL lÞ þ coshðL lÞ 1 þ ; ð1 þ Γ 2 ÞsinhðL lÞ þ 2Γ coshðL lÞ h i af 1 Df n~ t Γ Γ sinhðL lÞ þ coshðL lÞ 1 : C top ¼ ð1 þ Γ 2 Þsinh l þ 2Γ coshðL lÞ
Γ
Similar expression was derived in [9] from the condition of the material balance of the incident atom flux at the surface and the flux of atoms incorporating into the nanowire top facet. The step velocity and the elongation rate of the nanowire in the case of a single step are giving by Eqs. (12) and (13) of Section 3. In the case of two steps the relevant expressions are: ~ σt σ1Þ 1 dL 2 J ω sin α Df nð ¼ U ðL l1 Þ þ þ Jf ðα; βÞ; L l1 þ 2af Ω dt R 2 ~ σ t σ 1 Þ Df nð ~ σ1 σ2Þ 1 dl1 J ω sin α Df nð ¼ ðL l2 Þ þ ; L l1 þ 2af l1 l2 þ2af Sf dt 2 ! ~ σ2 σ1Þ 1 dl2 R2 R22 J ω sin α Df nð ¼ l1 þ l2 þ W þ cot α ; ðB2Þ l1 l2 þ 2af Sf dt ωR2 2
γ S 1 1 K n~ þ σ 2 ¼ 1 ~2 ¼ 1 exp st f kB T R2 R1 n ðl1 l2 Þ3
Acknowledgments
A¼
65
ðA3Þ
We assumed that the new step is formed when the adatom concentration at the nanowire base nð0Þ reaches some critical value nn ð0Þ. If there are no steps at the sidewall and desorption of adatoms is negligible, adatom concentration on the sidewall at the nanowire base is given by the solution of the relevant diffusion problem [31] in the limit of large λs and λf : nð0Þ ¼
ðR2W R2 ÞJ cos α J ω sin α ðL þ af Þ þ ðL þ 2af ÞL þ n~ t 2RDf 2Df
ðC1Þ
Thus, the critical concentration nn ð0Þ is achieved at a critical length of the nanowire Ln , which obeys the following equation Appendix B In the limit of no desorption the elongation rate of the nanowire without sidewall steps is given by: " # 1 dL 2 R2 R2 ¼ U Lþ W cot α J ω sin α þ Jf ðα; βÞ: ðB1Þ Ω dt R ωR
σ n ð0Þ ¼
ðR2W R2 ÞJ cos α n J ω sin α n ðL þ af Þ þ ðL þ 2af ÞLn σ t ; 2RDf n~ 2Df n~
ðC2Þ
where σ n ð0Þ ¼ nn ð0Þ=n~ 1 is the supersaturation on the sidewall surface at the nanowire base in respect to a step of length 2π R1 . It worth to be mentioned that if σ t 40 and the nanowire length L is small, the supersaturation σ n ð0Þ is negative so that the formation
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of new steps at the nanowire base is thermodynamically unfavorable. Moreover, as has been shown in [30], even if a step is formed by a chance at such unfavorable conditions, its growth rate will be negative and the step will dissolve. Thus, the presence of a strong sink for adatoms at the nanowire top prevents the lateral growth of relatively short nanowires. Similarly, behind the moving step the critical concentration n nn ð0Þ is achieved at some critical distance lk , which obeys the equation
σ n ð0Þ ¼
ðR2W R2k ÞJ cos α n J ω sin α n n ðlk þ af Þ þ ðlk þ 2af Þlk σ k ; 2Rk Df n~ 2Df n~
ðC3Þ
where k ¼ 1; 2; 3; …, Rk ¼ R þ kh and " # ðk 1ÞhSf γ st n~ k K σ k ¼ 1 ~ ¼ 1 exp n ðlk 1 lk Þ3 kB TðR þhÞðR þ khÞ Let us note that in the case of a single step ðlk ¼ l1 Þ σ k ¼ 0. It follows from (C2) and (C3) that if the nanowire top is a stronger sink for n adatoms than the sidewall step ðσ t 40Þ and R 4 4 h then Ln 4 l1 . In this case the second step forms before the first step will reach the nanowire top. Using (C2) and the experimentally measured value Ln , which is determined by the onset of the radial growth of the nanowires, one can determine σ n ð0Þ. Then solving (C3), one finds n the critical distance lk for nucleation of new steps. n
n
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