Nanoscale Patterned Growth Assisted by Surface Out-Diffusion of

Article pubs.acs.org/crystal

Nanoscale Patterned Growth Assisted by Surface Out-Diffusion of Adatoms from Amorphous Mask Films in Molecular Beam Epitaxy S. C. Lee* and S. R. J. Brueck Center for High Technology Materials and Department of Electrical and Computer Engineering, University of New Mexico, 1313 Goddard SE, Albuquerque, New Mexico 87106, United States S Supporting Information *

ABSTRACT: Nanoscale patterned growth (NPG) of GaAs requires the suppression of nucleation on the amorphous SiO2 mask film, defined on a substrate by patterning. It is determined by the Ga adatom kinetics on SiO2, leading to desorption and surface outdiffusion (SOD) to the area beyond the mask. Their relative contributions to NPG are examined both theoretically and experimentally. From the rate equation of thin-film growth, a relationship between incident Ga flux and growth temperature for NPG is analytically derived as a function of the lateral dimension of the SiO2 mask film, LM. In the NPG by molecular beam epitaxy, LM is varied in the range of Ga adatom migration length. From comparison with the model, the activation energy of a Ga adatom for desorption, Edes, is found to be comparable to that for surface diffusion, Ediff, on the SiO2 mask. They are both in the range of 2.7−2.9 eV, lower than the Ga desorption energy from the GaAs substrate, confirming the validity of SOD and, as a result, NPG. This also implies that they are not clearly distinguishable on the amorphous surface, in contrast to crystalline surfaces, where Ediff < Edes, which is attributed to the random fluctuations in the potential lacking long-range order. SOD can induce an actual growth rate significantly enhanced from the nominal rate calibrated on an unpatterned wide area by the additional adatom diffusion flux across the substrate−mask boundary. Its role in controlling the shape and size of the nanostructures selectively grown on the adjacent substrate surface is addressed.

I. INTRODUCTION Recent progress in lithography technologies enables the exploration of epitaxial growth on nanoscale limited areas, where the lateral dimension of an amorphous mask, used to define the growth area, can be comparable to or less than the surface diffusion length of adatoms on the mask material under growth conditions. This new frontier in epitaxy has been referred to as nanoscale patterned growth (NPG).1 Some of the major results are patterned quantum dots,2 orientationdependent nucleation and incorporation,3 epitaxially grown semiconductor nanopillars and nanowires (NWs),4−6 and NPG with silica nanoparticles.7 Particularly, NPG of groups III−V NWs such as GaN and InAs on Si is attractive for nextgeneration electronic devices.8,9 A technological review of these growth modes has been recently published elsewhere.10 Amorphous dielectric films such as SiO2 and Si3N4 are widely used in NPG as masking materials because of their physical and chemical stabilities at typical growth temperatures. An obvious requirement for NPG is the realization of a selective growth mode, where nucleation and growth are suppressed on the amorphous masking film while epitaxial growth of semiconductor materials proceeds on the adjacent crystalline substrateunder identical atomic flux and temperature. Compared with crystalline surfaces, the study of adatom kinetics on amorphous surfaces remains relatively unexplored. Despite the technological progress of NPG, the related issues of surface diffusion and desorption of adatoms on amorphous films are not fully understood. In our previous work, selective © XXXX American Chemical Society

growth of GaAs, relying on a combination of desorption of adatoms from the mask surface and surface out-diffusion (SOD, i.e., surface diffusion of adatoms from a mask to a nearby substrate surface across the substrate−mask boundary), was demonstrated by lowering the Ga fluxes to ∼1013 atoms cm−2 s−1 at 630 °C from typical growth conditions (i.e., a flux of ∼1014−1015 atoms cm−2 s−1 at ∼580−600 °C) to reduce the interaction between adatoms on the mask film of lateral dimension, LM, comparable to the adatom migration length on the mask material.5,11 In SOD-assisted NPG, the adatoms migrating across the substrate−mask boundary can dramatically affect the growth of the nanostructures on a nearby substrate surface by adding to the effective adatom flux. For these reasons, SOD is highly important to shape and size control of nanostructures as well as to achieving a selective growth mode. The major purpose of this work is to analyze SOD in the NPG of GaAs on a SiO2-patterned substrate both theoretically and experimentally. In Section II, a model based on the general rate equation of thin-film growth is introduced to characterize SOD. The adatom surface diffusion and desorption on the amorphous SiO2 mask are primarily affected by substrate temperature, T, and incident Ga flux, F. SOD for a nucleationfree mask is examined in the equation and a relation between T and F for NPG by SOD is derived with LM as a parameter. In Received: January 21, 2016 Revised: April 7, 2016

A

DOI: 10.1021/acs.cgd.6b00111 Cryst. Growth Des. XXXX, XXX, XXX−XXX

Crystal Growth & Design

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Section III, experimental NPG is presented. Nanoscale interferometric lithography (NIL) and molecular beam epitaxy (MBE) are employed for mask fabrication and selective growth. SOD is explored with the variation of LM in MBE, which simplifies the modeling by eliminating any chemical reactions and catalytic effects. In Section IV, the experimental data are quantitatively compared with the F−T relation from the model. The activation energies of a Ga adatom for desorption from the mask, Edes, and for surface diffusion, Ediff, are comparable to each other, but both are lower than the Ga desorption energy from the GaAs substrate and clearly support SOD-assisted NPG. In contrast, this is very different from the typical results on crystalline surfaces, where Edes > Ediff, and is explained by the lack of long-range order on amorphous surfaces that exhibit a more random surface potential to adatoms. Detailed discussions of the role of SOD in NPG and associated shape/size control of epitaxial nanostructures are presented along with the experimental data. Finally, Section V provides the conclusions of this work with a summary.

II. A MODEL FOR NPG Three length parameters are employed for NPG modeling: LD, the surface diffusion length of an adatom on the mask material before desorption; LC, a critical length corresponding to the average steady-state inter-adatom distance on the mask during growth; and LM, the lateral mask dimension introduced in Section I. For selective growth, a mask film (typically an amorphous material) having a lateral dimension LM is fabricated on a semiconductor substrate, as shown in Figure 1a. A perfectly flat and clean surface is assumed. The thickness of the mask needs to be considered but is regarded as zero in this section and will be discussed later. Then, LD and LC of an adatom on the mask film are defined as12 LD =

τdesD

LC =

τcD ≈

Figure 1. (a) Schematic illustration of a single circular amorphous mask film fabricated on a semiconductor substrate. (b) SEM image of a selectively grown and an ungrown region of a SiO2-patterned GaAs(001) substrate in homoepitaxial NPG by MBE. Because of lateral overgrowth, the SiO2 masks in the selectively grown region were partly buried under the epilayer along their edges. During growth, a metal shadow mask was employed to cover the right side labeled “ungrown region” in the image. The apparent transition region roughly extends at least to ∼1 μm for the present growth conditions.

tion with a rate of ω1. The fourth term corresponds to nucleation and clustering by the incorporation of a single adatom into an i-atom-sized cluster. In this term, ωi (i = 2, ...) is the incorporation rate of a single adatom into a cluster consisting of i adatoms, and ni is a surface number density of iatom-sized clusters. These terms describe thin-film growth on an unpatterned surface. The last term in eq 3 is the diffusion flux density, which plays an important role for NPG in the presence of the substrate−mask boundary. The growth described by eq 3 with a nonzero diffusion flux of adatoms across the boundary corresponds to patterned growth for i = 1. Every ni for i ≥ 2 will have an equation similar to eq 3, and all of them ultimately describe the patterned thinfilm growth. There is, in principle, a similar set of equations for the growing film on the substrate beyond the mask. In this work, we assume that the adatom mobility and incorporation rate onto the growing nanostructures on such area are much faster than those on the mask so that they act as perfect sinks for adatoms, forcing the adatom concentration to be negligible on them, and use a boundary condition of n = 0 at the substrate−mask boundary. For simplicity, we neglect desorption of single atoms from existing clusters, which rarely occurs under the growth conditions employed for NPG. For the single circular SiO2 mask fabricated on a GaAs surface having LM as a diameter illustrated in Figure 1a, τdes and D in eq 3 correspond to the surface residence time of a single Ga atom on the SiO2 mask before desorption and the diffusion constant of Ga atoms on the SiO2 mask, respectively. Since the SiO2 film used in this work is amorphous, isotropic diffusion of Ga atoms with zero step density can be assumed. For each surface, the rate equation

(1)

1 n

(2)

−1

where τdes = ν0 exp(Edes/kBT) is the surface residence time of a single atom before desorption, ignoring any interactions between adatoms except for elastic collision and scattering, defined with a desorption rate constant, ν0, Edes, Boltzmann constant, kB, and growth temperature, T, and D = Cd exp(−Ediff/kBT) is a diffusion constant given with Ediff, a proportionality constant Cd. In eq 2, τc and n correspond to the average time interval between adatom collisions on the given surface, analogous to τdes, and the density of adatoms on an unit area. Two limiting conditions for NPG are identified: (1) selective growth depending entirely on desorption (independent of LM) and (2) selective growth depending on SOD from the mask area to the semiconductor nanostructure (independent of τdes). Both mechanisms contribute to NPG for the regime explored in this work. For thin-film growth on a f inite mask surface enclosed within a different (substrate) surface as shown in Figure 1, n satisfies the rate equation:13,14 ∞

n ∂n =F− − 2ω1n2 − n ∑ ωini + D∇2 n ∂t τdes i=2

(3)

where F is the incident atomic flux. The second term on the right represents the adatom desorption. The third term describes the consumption of single atoms through dimerizaB



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of eq 3 for all i’s with different parameters is required. The basic requirement of selective growth is the absence of nucleation or clustering on the SiO2 surface. That means ni = 0 for i ≥ 2 on the SiO2 surface for any t. The dimerization in eq 3 for compound semiconductors can include both Ga−Ga bonding leading to a Ga droplet on the SiO2 surface, not expected at the conditions for the NPG experiment, and also the pairing of Ga−As bonding units. Since such pairings eventually evolve to an immobile cluster, the rate equations of ni for i ≥ 2 are not considered in this work. Under these assumptions, eq 3 is simplified to ∂n n =F− + D∇2 n ∂t τdes

(4)

Equation 4, describing the density of adatoms on a nanoscale mask film in the absence of nucleation, is referred to as the NPG equation.15,16 The major concern for NPG is the consumption of all of the incident flux by desorption or SOD leading to the absence of nucleation on the mask film. In a more rigorous approach, as mentioned earlier, clusters of sizes smaller than the critical size for nucleation have to be retained in eq 4, and the growing film has to be treated more completely. Also, a dimer could also dissociate into two adatoms, or aggregate with an additional adatom into a trimer; the study of these processes is beyond the scope of this work. Nevertheless, as seen in Section IV, the results of this work confirm that monatomic Ga adatoms and their behavior described by eq 4 are enough for basic understanding of SOD-assisted NPG. Over the SiO2 mask, F, τdes, and D are assumed constant and independent of t. With the boundary condition of n(LM/2, t) = 0 at the substrate−mask boundary as mentioned earlier,17 the solution of eq 4 for circular symmetry can be written as18

Figure 2. Steady-state plots of (a) n(ρ) and (b) n(ρ)/n(0), normalized to the concentration at the center, versus ρ = 2r/LM along the variation of γ from 0.01 (desorption dominated) to 10 (SOD dominated), calculated from eq 5.

⎫ ⎧ ⎡ ⎛ 4D ⎪ 1 ⎞ ⎤⎪ ⎛ 2αj ⎞ ⎥⎬J0 ⎜ r ⎟ ⎢ −⎜ 2 αj2 + ⎟ n(r ; t ) = ∑ Aj ⎨ 1 exp t − ⎪ ⎪ ⎢⎣ ⎝ L M τdes ⎠ ⎥⎦⎭ ⎝ LM ⎠ ⎩ j

We take the most slowly varying term of eq 5 (j = 0) for further analysis. Then, the solution of eq 5 can be approximately written as18 ⎡ ⎛ t ⎞⎤ ⎛ 2α ⎞ n(r ; t ) ≈ Fτ ⎢1 − exp⎜ − ⎟⎥J0 ⎜ r ⎟ ⎝ τ ⎠⎦ ⎝ L M ⎠ ⎣

(5)

with Aj =

=

8Fτdes

(

2 2 LM J1 (αj) 1 +

4Dτdes 2 LM

)

αj2

8Fτdes

∫0

L M /2

∫ L 2 J 2 (α )[1 + γ(α /α )2 ] 0 M1

j

j

1

−1

−1

where 0 ≤ r ≤ LM/2 and τ is defined as τ = τdes + τc = τdes−1(1 + τdes/τc) = τdes−1(1 + γ), with τc ≈ LM2/4α2D (α1 = α = 4.8) and γ = τdes/τc. For t ≫ τdes, τc, the steady state of eq 6, the F−T relation for NPG shown in Figure 2 can be written in terms of Fc at a given T as

⎛ 2αj ⎞ rJ0 ⎜ r ⎟ dr ⎝ LM ⎠

L M /2

(6) −1

⎛ 2αj ⎞ rJ0 ⎜ r ⎟ dr ⎝ LM ⎠

Fc ≈

where the radial coordinate r varies from 0 to LM/2, and the αj’s are the zeroes of the zero-order Bessel function, J0 and γ = 4Dτdesα21/L2M. The details of the derivation of eq 5 are given in the Supporting Information. Figure 2 shows the steady-state variations of n(ρ) and n(ρ)/n(0) versus ρ = 2r/LM with the change of γ from 0.01 (desorption dominated) to 10 (SOD dominated). Figure 2b clearly reveals the relative change of n with γ. When NPG is governed by SOD, n(r) for r ∼ LM/2 (or ρ ∼ 1) is relatively low since the adatoms largely diffuse off from the mask at the substrate−mask boundary. On the other hand, n(r) becomes relatively constant over the mask if NPG is dominated by desorption. This is because most of the adatoms desorb from the mask surface before they reach the substrate− mask boundary (LD < LM).

⎛ 1 n̅ 4D ⎞ n ≈ n̅ ⎜ + 2 α 2⎟ = ̅ (1 + γ ) τ τdes LM ⎠ ⎝ τdes

(7)

from eq 6, where n̅ is the adatom density spatially averaged over the mask surface. Selective growth is then possible for a flux less than a critical flux (the highest flux for which NPG is achieved), Fc, of eq 7 at a given T and LM. From eq 7, the two extreme cases of NPG by perfect desorption or by complete SOD can be considered. The first case can be achieved only with desorption if LD < LC. Under this condition, the contribution of SOD to avoiding deposition on the mask becomes insignificant. Then Fc must be comparable to Fdes,c, given by 1 Fdes,c ≈ ∝ exp[− (2Edes − Ediff )/kBT ] 2 Dτdes (8) C



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Table 1. Summary of Experimental Results T (°C)

Fc (atoms cm−2 s−1)

Fdes (atoms cm−2 s−1)

Fdiff (atoms cm−2 s−1)

Fdiff/Fc

LM (nm)

630 615 600 595 570 570

× × × × × ×

× × × × × ×

5.0 × 1013 ∼0 2.7 × 1013 ∼0 2.1 × 1013 ∼0

0.31 ∼0 0.42 ∼0 0.64 ∼0

230−290 NA 170−220 NA 90−150 NA

1.6 6.5 6.5 3.2 3.3 1.2

14

10 1013 1013 1013 1013 1013

1.1 6.5 3.8 3.2 1.2 1.2

14

10 1013 1013 1013 1013 1013

III. EXPERIMENTAL SECTION

to suppress nucleation on the mask. Fdes,c in eq 8 is independent of LM and is solely determined by T.15 This means that eliminating deposition on the mask is always possible for a given F by lowering the incident flux. On the other hand, we can imagine a selective growth mode which can be mostly realized by SOD. Even if τdes is infinite, no deposition on the mask occurs if LM ≲ LC. Under this condition, Fc is given by Fdiff,c as Fdiff,c ≈ n ̅

Two different SiO2-patterned GaAs(001) substrates were used in MBE: one with unpatterned, wide-area SiO2 films for the exploration of selective growth by Ga desorption and the other with nm-scale patterns for SOD-assisted NPG. A 30- to 50-nm-thick SiO2 film was deposited with electron-beam evaporation. Large-area NIL and CF4 plasma etching were employed for photoresist patterning and its transfer into the underlying SiO2 film.15,20 Detailed fabrication processes have been reported elsewhere.1,10 Epitaxial growth of GaAs on these substrates was performed by MBE. Growth temperature, T, monitored by optical pyrometry was varied from 570 to 630 °C. Along with growth temperature, Ga flux, F, calibrated on a unpatterned GaAs substrate was increased from 1.2 × 1013 to 1.6 × 1014 atoms cm−2 s−1. Figure 1b shows a tilted scanning electron microscope (SEM) image of a 355 nm-period, 2D array of holes transferred from the photoresist pattern onto the SiO2 film by plasma etching and a selectively grown MBE GaAs epilayer. With respect to the dashed lines in Figure 1b, the right (left) side reveal a SiO2 post array of LM ≈ 200 nm fabricated on a GaAs(001) substrate (a ∼50-nm-thick GaAs epilayer selectively deposited on the SiO2patterned substrate) at T ≈ 600 °C with a deposition rate of ∼6.5 × 1013 atoms cm−2 s−1. The single circular mask on a substrate schematically illustrated in Figure 1a corresponds to one of the SiO2 masks forming the 2D array in Figure 1b. On the left of Figure 1b, the SiO2 post array was slightly buried into the GaAs epilayer but individual post surfaces were free from any GaAs since there was no direct nucleation on them. The right side was screened by a metal shadow mask during growth and no deposition occurred. At each T and F, the mask linear dimension was varied from ∼100 to 300 nm to measure LM for SOD-assisted NPG. All experimental results from both large scale selective growth and nm-scale NPG on SiO2-patterned substrates are summarized in Table 1. The temperatures and fluxes in Table 1 include measurement errors of about ±2 °C in T and ±5% in Fdes,c and Fc, respectively. During growth, the As/ Ga ratio in beam equivalent pressure was kept higher than 10 for all temperatures. The dimensions of the mask areas in Table 1 were obtained from top-down view SEM observations. The actual path length of an adatom for SOD should be the measured top-down LM plus the thickness of the mask. From the observed nonvertical sidewalls of SiO2 films formed by CF4 plasma etching, however, it can be assumed that ∼60% of the mask film thickness is reflected in the SEM measured LM of Table 1 and no additional correction for nonzero SiO2 film thickness is considered in this work.5 The details of the LM measurement were reported in our previous articles.1,15

4α 2D 4α 2D ≈ 2 2 LM LC2L M

⎧ 4α 2D ⎪ 4 ∝ exp( −Ediff /kBT ) if L M ≤ LC (9‐1) ⎪ LM 4 ⎪ ⎪ /L M ⎨ ≈ ⎪ 4α 2 D 4α 2 if LC ≤ L D (9‐2) ⎪ 2 2 ≈ 2 τdesL M ⎪ L DL M ⎪ ∝ exp( −E /k T )/L 2 ⎩ des B M

under LM ≲ LC ≲ LD. Here, it is assumed that n̅ = n in eq 2. Then, Fdiff,c is the flux below which selective growth can be achieved solely by SOD at a given LM. Two different cases are considered: eq 9-1, where Fdiff,c corresponds to the upper limit by LM ≲ LC and eq 9-2, where it is the lower limit by LC ≲ LD. It should be noted that both cases depend on LM, unlike Fdes,c of eq 8. In eq 9-1, as the other extreme case mentioned earlier, Fdiff,c ≈ Fc by the LM that excludes the contribution of desorption to NPG at the given condition since SOD occurs before desorption. In eq 9-2, however, Fdiff,c need not to be comparable to Fc since it is explicitly under consideration of desorption by the condition of LC ≲ LD that reduces the role of SOD in NPG.19 As mentioned in Section I, Fdiff,c could contribute to a GaAs nanostructure that is selectively grown at the nearby substrate surface beyond the mask, as another flux laterally supplied to it if most of the adatoms diffusing across the boundary by SOD participate its growth. Then, the effective incident flux for the epitaxial GaAs would be critically affected by finite Fdiff,c, depending on its relative significance. The two extreme cases described by eqs 8 and 9-1 can be characterized simply by LM ≫ LD for Fdes,c and LM ≪ LC for Fdiff,c. Equation 7 then shows the F−T relation for SOD-assisted NPG at a given LM. From eqs 6 and 7, Fc in eq 5 can be rewritten as Fc ≈ Fdes,c + Fdiff,c

IV. COMPARISON OF MODELING WITH EXPERIMENT In Figure 3, Fc and Fdes,c from Table 1 are indicated with solid squares and circles, respectively. The points for Fdes,c, the maximum flux for the suppression of nucleation in the desorption on the blanket SiO2 film, fit an Arrhenius dependence with

(10)

for LM ≈ LC ≈ LD, if desorption and SOD are taken as independent processes. Under this condition, both desorption and SOD contribute to the elimination of nucleation on the mask. We experimentally explore NPG at this regime by varying LM over a range of growth conditions. Equations 8−10 will be used for the comparison with experiment.

2Edes − Ediff = 2.5 ± 0.1 eV

for 570 − 630 °C

(11)

from eq 8. As seen in Figure 3 and Table 1, all of the Fc values measured with NPG are greater than the corresponding Fdes,c’s. At 570 °C, for example, Fc ≈ 3.3 × 1013 cm−2 s−1 whereas Fdes,c D



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Figure 4. Plot of LM2Fdiff,c (left axis) and LM4Fdiff,c (right axis) vs 1/T. The dashed lines individually fit the data points with the slopes indicated in the figure.

Figure 3. Plot of Fc, Fdes,c, FMEE, and FMEE′ vs 1/T. Here, FMEE is denoted with a black solid line, and FMEE′, indicated with a black dashed line, means the time-averaged FMEE (Ga flux) originally used in MEE. The red line fits Fdes,c, with the slope indicated in the figure.

eV referred above, can be regarded as an upper limit in the comparison with the experimental data. Ultimately, as mentioned earlier,18 the goal of eqs 9-1 and 9-2 is to replace LC that is not directly measurable, by LM and LD, respectively, which are controllable in or extractable from the experiment for the analysis of Edes and Ediff The activation energies from eq 9-2 raise issues as to the validity of the adatom kinetics on the amorphous surface that were assumed in the derivation of eq 9. Inherently, large fluctuations in the surface potential of an amorphous material are inevitable. The surface migration length of an adatom is directly correlated to the spatial scale of these fluctuations. A rigorous approach to the surface states of amorphous SiO2 is beyond the scope of this work. As a fundamental question, however, the definitions of Edes and Ediff in the model must be reexamined on an amorphous surface where there is no longrange ordering of the constituent atoms. While there is room for debate on its definition, as shown in Figure 5, Ediff on a crystalline surface is controlled by the variation of the surface potential between neighboring atoms that is an atomic level parameter of the given surface for hopping between sites. Measurements of the activation energy for the surface diffusion of a Ga adatom ∼1.2 eV and that for adatom formation ∼2.7

≈ 1.2 × 1013 cm−2 s−1 from the bold solid line. This means that SOD contributes to the suppression of nucleation on the mask films and allows higher fluxes at a given temperature. It should be noted that Fc was determined with different LM and it is not strictly appropriate to fit them with a single straight line in the semilog plot of Figure 3 although the alignment of these points is not too far from a simple exponential. A similar result has been obtained in a migration enhanced epitaxy (MEE) experiment.21 The relation between Ga flux (FMEE) and T for selective growth is indicated with a black line in Figure 3a. For the comparison with this work, FMEE was time-averaged and is labeled FMEE′, denoted by a black dashed solid line in Figure 3. It is interesting that all of the limiting Ga fluxes in Figure 3 cluster within an order of magnitude at a fixed T in spite of very different growth conditions. This means there is a certain upper limit on the Ga flux in MBE where selective growth only by Ga desorption from SiO2 mask surface is available. Also, such coincidence of the order of magnitude for the Ga flux for avoiding nucleation on the SiO2 is general evidence supporting the feasibility of selective growth, which has been thought to be unavailable for MBE. For quantitative analysis, Fdiff,c is estimated from eq 10 for each LM in Table 1. A plot of LM4Fdiff,c of eq 9-1 (right y-axis) and LM2Fdiff,c of eq 9-2 (left y-axis) vs 1/T is presented in Figure 4. First, Ediff is given as ∼4.4 eV with the upper-limited Fdiff,c from eq 9-1. In previous reports, Ediff ≈ 4.9 eV was obtained from the relation of LM ∝ (D /Fc)1/β with β ≈ 3.8.12,15 This is close to 4.4 eV of this work, which results in Edes,c ≈ 3.5 eV from eq 11 and contradicts the condition of Edes ≥ Ediff, imposed by their definitions. This is not physically valid and means that the Fdiff,c in eq 9 is not fully consistent with the experimental results. In the other case, LM ≤ LC < LD, from eq 9-2, as indicated in Figure 4 with left y-axis, Edes = 2.7 ± 0.1 eV and as a result Ediff ≈ 2.9 eV from eq 11. The Edes from this calculation is lower than the upper limit suggested from the desorption energy of a Ga atom from GaAs surface, 4.7 eV.22 At this condition, the NPG on an SiO2 film-patterned GaAs substrate allows finite deposition on a GaAs surface while preventing any nucleation on the SiO2 mask. Considering the experimental uncertainty, Edes ≈ Ediff for a Ga adatom on an amorphous SiO2 surface with a range of ∼2.7−2.9 eV from eq 11. Thus, Ediff ≈ 4.4−4.9 eV from eq 9-1 and previous work, which does not exceed the 4.7

Figure 5. Schematic illustrations of the surface potential of a GaAs (left) and an SiO2 (right) surface. A random (periodic) line in the right (left) means the surface potential for adatom diffusion on an amorphous (crystalline) surface. See the text for details. E



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eV on a crystalline GaAs surface have been reported, as indicated in Figure 5.23−25 On an amorphous structure schematically illustrated in Figure 5, on the contrary, the surface does not provide a periodic potential to adatoms but has randomized energy fluctuation due to the lack of ordering with spike potentials at imperfections or defects.26 As indicated in Figure 5, a large potential variation across the SiO2−GaAs interface is also expected. A simulation has found that the averaged activation energy for surface diffusion equals to ∼0.6−0.7 of that for desorption on a silica surface even without the consideration of the spiky potentials illustrated in Figure 5.27 This implies Ediff can be comparable to Edes on amorphous surfaces and supports the illustration in Figure 5. These fluctuating potentials would bounce, deflect, or trap an adatom and ultimately interrupt its surface diffusion but can be accommodated into Ediff by the spatial average over LM or LD. Accordingly, Ediff in eqs 8 and 9 can be regarded as the distance-averaged activation energies to reflect the spatial fluctuation of surface potential of the SiO2 surface. If Ediff is replaced by a distance-averaged value, then it is not atomistic. Such a distance-averaged Ediff may not be clearly distinguishable from the spatial fluctuation of Edes that must be also averaged along with Ediff for consistency. Depending on the details of the surface potential, therefore, Ediff on an amorphous surface could be imprecisely defined. From our experimental results, Edes ≈ Ediff ≈ 2.8 ± 0.1 eV for a Ga adatom on an amorphous SiO2 surface and this clearly allows SOD on a SiO2patterned GaAs surface. In Table 1, Fdiff,c/Fc increases with decreasing LM as the SOD becomes more effective. If most of the adatoms diffusing from the mask film to adjacent substrate surface across the substrate−mask boundary are incorporated into a nanostructure epitaxially grown on the substrate surface in the NPG with an Fc, the effective incident flux, F′, for the nanostructure becomes dependent on the ratio of the substrate area open for epitaxy to the whole area in a unit pattern, δ, and can be expressed as F ′ ≈ Fc +

Fdiff,c 1−δ Fdiff,c ≈ Fdes,c + δ δ

Figure 6. Tilted SEM image of homoepitaxial NPG on a SiO2patterned GaAs(001) substrate by MBE. The period and δ in eq 12 of the pattern fabricated into the SiO2 film were set to 355 nm and ∼0.04. The growth temperature and deposition amount calibrated on a unpatterned wide area were set to 630 °C, where Fdiff/Fc ≈ 0.31 in Table 1 and ∼200 nm per unit area in thickness, respectively. GaAs nanostructures grown individual openings show noticeable variation in shape and size. Particularly, two GaAs NWs grown along [111] have the length of ∼1.5−1.6 μm that is roughly ∼8 times the nominal deposition amount. It should be noted that the geometric shape variation in the opening areas inevitable at nm-scale also affects the initial faceting at the early stage of growth and ultimately causes large shape and size fluctuation of epitaxial nanostructures in continued growth, as observed in this figure.

Fdiff,c analytically. This can cause some fluctuation in Ediff. Another possible reason is uncertainty in the experimental data; Fc is relatively inaccurate compared with Fdes,c in measurement and L M in Table 1 includes ∼±30 nm fluctuation. This is because it depends not only T but also LM which has not been continuously varied. Also, the validity of eq 10 where Fdes,c and Fdiff,c are regarded simply as additive quantities to represent Fc must be reexamined. Finally, more rigorous boundary conditions may be required to describe NPG more precisely. Some additional issues should be addressed. First is LD of eq 1 that becomes insensitive to T because of Ediff ≈ Edes. At 600 °C, Fdiff,c/Fc ≈ 0.42 in Table 1, implying Fdes,c ≈ Fdiff,c. Then, LC ≈ LD ≈ LM where desorption and SOD are not clearly distinguishable, as discussed earlier. This implies LD ≈ 190− 200 nm over the range of T investigated. This is considerably smaller than the micrometer-scale Ga adatom migration length on a GaAs surface at similar temperatures. According to a theoretical study, the activation energy corresponding to Ediff on an amorphous surface is temperature- and diffusion distancedependent, unlike that on a crystalline surface.26 A crystalline surface keeps a certain reconstruction pattern for a finite range of T and be expected to have constant Edes and Ediff over that range.29 On the other hand, an amorphous surface has no definite surface reconstruction, and its E des and Ediff are affected by the randomized surface dangling bonds and may change along with T. The kinetics of surface adatoms on such amorphous surface have not been investigated comprehensively.26,27,30−32 For these reasons, the activation energies assumed “independent of T” in the range of the temperature of this work could be functions of T for a wider range. Ultimately, LD therefore would be temperature-dependent as T decreases and SOD would have more dependence on T. A second issue is the experimental limitation on the determination of NPG assisted by SOD in MBE. For T higher than 630 °C, Ga desorption occurs even from a GaAs surface, making it difficult to distinguish the elimination of nucleation exclusively by SOD from that by desorption. As seen in Figure 3, Fc would be merged to Fdes,c around T ≈ 650 °C above which SOD becomes negligible. For low T, on the other hand,

(12)

using eq 10. In eq 12, δ = 0 corresponds to the growth on an unpatterned blanket SiO2 film where the growth area becomes vanishingly small; δ = 1 means the growth on a SiO2-free substrate. If Fdiff,c = 0, selective growth entirely by desorption, F′ is identical to Fc regardless of δ, as expected. On the other hand, F′ ≈ Fc/δ if NPG is achieved mostly by SOD (Fdes,c ≪ Fdiff,c). In this case, δ is a critical parameter that determines F′ and as a result the effective growth rate of the nanostructure. As δ → 0, F′ is dramatically increased from Fc. This provides a partial answer to some of the GaAs NWs for which the length is much greater than the nominal deposition thickness calibrated on a unpatterned wide area.5 As an example, Figure 6 shows the NPG of GaAs NWs that reveal the apparent growth rate along [111] reaches ∼8 times the nominal growth rate calibrated in (001) for δ ≈ 0.04 at the identical T and F. In Table 1, Fdiff,c approaches Fc at low T and reduced LM, implying the significance of δ at this regime.28 Evidently, eq 12 confirms the importance of SOD in the shape and size control of nanostructures grown by NPG. There could be several reasons for the sources of Edes ≈ 2.7 eV < Ediff ≈ 2.9 eV in the model. One of them is the oversimplified Fdiff,c in eq 7. In this equation, only the leading term in the series given in eq 5 was considered to represent 27

F



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(4) Lee, S. C.; Dawson, L. R.; Brueck, S. R. J.; Jiang, Y.-B. Appl. Phys. Lett. 2005, 87, 023101. (5) Lee, S. C.; Dawson, L. R.; Brueck, S. R. J.; Jiang, Y.-B. J. Appl. Phys. 2005, 98, 114312. (6) Hersee, S. D.; Sun, X.; Wang, X. Nano Lett. 2006, 6, 1808−1811. (7) Lee, S. C.; Dawson, L. R.; Huang, S. H.; Brueck, S. R. J. Cryst. Growth Des. 2011, 11, 3673−3676. (8) Guo, W.; Zhang, M.; Banerjee, A.; Bhattacharya, P. Nano Lett. 2010, 10, 3355−3359. (9) Rieger, T.; Luysberg, M.; Schapers, T.; Grützmacher, D.; Lepsa, M. I. Nano Lett. 2012, 12, 5559−5564. (10) Xia, D.; Ku, Z.; Lee, S. C.; Brueck, S. R. J. Adv. Mater. 2011, 23, 147−179. (11) Lee, S. C.; Brueck, S. R. J. J. Appl. Phys. 2004, 96, 1214−1218. (12) Pimpinelli, A.; Villain, J. Physics of Crystal Growth: Cambridge University Press: Cambridge, UK, 1998; p 93. (13) Mutaftschiev, B. The Atomistic Nature of Crystal Growth; Springer-Verlag: Heidelberg, 2001; p273. (14) Lee, S. C. Nanoscale Patterned Growth by Molecular Beam Epitaxy and Its Applications; University of New Mexico, Albuquerque, 2002; pp 49−56. (15) Lee, S. C.; Brueck, S. R. J. Appl. Phys. Lett. 2009, 94, 153110. (16) A similar equation has been used in several articles to model epitaxial growth with adatom surface diffusion on a terrace defined by step edges under several different boundary conditions, referred to as step-flow growth. In this work, contrary to those models, eq 4 exactly describes NPG only under the assumption of the absence of nucleation on an amorphous mask film with zero step density and isotropic surface diffusion. (17) Khenner, M.; Braun, R. J.; Mauk, M. G. J. Cryst. Growth 2002, 241, 330−346. (18) Clemencon, A.; Guivarch, C.; Eury, S. P.; Zou, X. L.; Giruzzi, G. Phys. Plasmas 2004, 11, 4998−5009. (19) In eqs 9-2, LDLM in the denominator is between L2D for Fc ≅ Fdes,c of eq 8 and L2M for Fc ≅ Fdiff,c of eq 9-1. L2C under the given condition (LM ≲ LC ≲ LD) could provide a better approximation for the lower limit in the modeling as LM → LC. Nonetheless, LDLM can reflect Fdiff,c that is finite but less than Fc at the range of T of Table 1 shown later, in the comparison with experiment, because LC is not directly measureable. (20) Brueck, S. R. J. Proc. IEEE 2005, 93, 1704−1721. (21) Suzuki, K.; Ito, M.; Horikoshi, Y. Jpn. J. Appl. Phys. 1999, 38, 6197−6201. (22) Reithmaier, J. P.; Broom, R. F.; Meier, H. P. Appl. Phys. Lett. 1992, 61, 1222−1224. (23) Neave, J. H.; Dobson, P. J.; Joyce, B. A.; Zhang, J. Appl. Phys. Lett. 1985, 47, 100−102. (24) Garofalini, S. H.; Halicioglu, T.; Pound, G. M. Surf. Sci. 1982, 114, 161−170. (25) Tersoff, J.; Johnson, M. D.; Orr, B. G. Phys. Rev. Lett. 1997, 78, 282−285. (26) Avramov, I. J. Phys.: Condens. Matter 1999, 11, L267−272. (27) Stallons, J. M.; Iglesia, E. Chem. Eng. Sci. 2001, 56, 4205−4216. (28) In eq 12, the incorporation rate of Ga adatoms onto a nanostructure from the diffusion flux by SOD is assumed to be unity. For accurate calculation of F′, several additional contributions must be considered. Some of them are the desorption of the adatom from the mask surface, which is Lambertian and thus can partially contribute to nearby three-dimensional epitaxial nanostructures, and the actual cross section of a nanostructure to the normal incident flux, which is increased from the original opening area as growth proceeds. These were not included in eq 12 explicitly because of their significant complexity. Besides, the variation of the surface potential at the substrate-mask boundary in Figure 5 should affect SOD across it. (29) Vitomirov, I. M.; Raisanen, A. D.; Finefrock, A. C.; Viturro, R. E.; Brillson, L. J.; Kirchner, P. D.; Pettit, G. D.; Woodall, J. M. J. Vac. Sci. Technol. B 1992, 10, 1898−1903.

crystalline degradation is another limiting factor in epitaxy. At any T, epitaxy begins with single crystalline phase but undergoes a transition to amorphous or polycrystalline phase. For lower T, the transition occurs earlier and with poorer crystallinity and as a result different surface states restrict the variation of T for NPG.33 Practically, the control of F below ∼1013 atoms cm−2 s−1 is not reliable. This makes it difficult to observe NPG only by desorption at the conditions of lower F and T, which has an additional issue regarding the equilibrium partial pressure of Ga on a GaAs surface. These also restrict the variation of growth parameters and, as seen in Table 1, allow NPG only in the limited ranges of F and T at the experiment.

V. SUMMARY AND CONCLUSION The surface diffusion and desorption of the adatoms on amorphous SiO2 film has been investigated with homoepitaxial NPG on SiO2-patterned GaAs(001) substrates by NIL and MBE. From the rate equation simplified under the condition for the suppression of nucleation of GaAs on a SiO2 mask film by SOD and desorption, an equation for NPG providing a relation between Fc (≈ Fdes,c + Fdiff,c) and T has been derived for a given LM. From this equation, an analytical solution of NPG relying on SOD with Fdiff,c that depends on LM has been obtained at the steady state under the boundary condition deduced from NPG. The model provides Edes ≈ Ediff ≈ 2.8 ± 0.1 eV for the Ga adatom on an amorphous SiO2 surface. The resulting Edes ≈ Ediff is interpreted with the characteristics of amorphous materials that have indefinite surface reconstructions and surface potential fluctuating by the lack of long-range ordering, very different from crystalline surfaces. A finite Fdiff,c under this condition plays an important role in the shape and size control of epitaxial nanostructures beyond the substrate− mask boundary as an additional flux to them by SOD. These results should be applicable to the NPG of other group III−V semiconductors such as InAs and GaN that retain the adatom kinetics on SiO2 surface similar to GaAs.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.cgd.6b00111. Derivation of eq 5 (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: 505-272-7800. Fax: 505-2727801. Funding

This work was partially supported by the Defense Threat Reduction Agency. Notes

The authors declare no competing financial interest.

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REFERENCES

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(30) Scopigno, T.; Steurer, W.; Yannopoulos, S. N.; Chrissanthopoulos, A.; Krisch, M.; Ruocco, G.; Wagner, T. Nat. Commun. 2011, 2, 195. (31) Tiedje, T.; Ballestad, A. Thin Solid Films 2008, 516, 3705−3728. (32) Manaila, R.; Grigorovici, R. J. Non-Cryst. Solids 1987, 97−98, 195−198. (33) Eaglesham, D. I.; Gossmann, H.-J.; Cerullo, M. Phys. Rev. Lett. 1990, 65, 1227−1230.

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