Evolution of the Length and Radius of Catalyst-Free III-V Nanowires Grown by Selective Area Epitaxy.

We present a new model for the length and radius evolution of catalyst-free III-V nanowires grown by selective area epitaxy. We consider simultaneous axial and radial growth of nanowires, which is more typical for this technique compared to the vapor-liquid-solid growth of nanowires. Analytic expressions for the time evolution of the nanowire length and radius are derived, showing the following properties. As long as the nanowire length is shorter than the collection length of group III atoms on the sidewalls, the length evolves superlinearly and the radius evolves linearly with time. For longer nanowires, both the length and radius increase sublinearly with time. The scaling growth laws are controlled by a single parameter that depends on group V flux. The model fits well the data on the selective area growth of InAs and GaAs nanowires by different techniques. Overall, these results can be used for controlling the catalyst-free growth of III-V nanowires and their morphology, including ternary III-V material systems.

Introduction

Semiconductor nanowires (NWs) offer almost unlimited possibilities for the bottom-up design of their morphology, composition, and crystal phase, and enable monolithic integration of a wide range of optoelectronic III–V NW heterostructures with silicon electronic platforms.[1−3] Selective area epitaxy (SAE) of III–V NWs is a promising gold-free alternative to the vapor–liquid–solid (VLS) growth method.[3] The SAE growth of III–V NWs works equally well in molecular beam epitaxy (MBE) or in metal organic chemical vapor deposition (MOCVD) techniques. In this work, we try to understand the kinetics of the axial and radial growth of III–V NWs in the true SAE growth mode of catalyst-free NWs[4] on patterned substrates with regular arrays of lithographically defined pores.[4−9] Most importantly, we study theoretically the time evolution of the mean length and radius of SAE NWs versus the growth parameters and epitaxy technique. According to the experimental data,[5,7−9] both the length L and radius R of the SAE-grown NWs usually increase with the growth time, while the radius of Au-catalyzed VLS III–V NWs is fixed by the initial size of the growth seeds and stays constant in most cases.[10] Radial growth modifies very substantially the earlier models of VLS NWs at a time-independent radius R = const (see ref (10) for a review). We will show that, after a short incubation stage, the NW length and radius exhibit scaling power-law dependences on the growth time. The power exponents of these dependences are related to the growth conditions, the NW length, and the array pitch.

Model

The SAE growth and main model parameters are illustrated in Figure . We consider the group III and V atomic fluxes I in nm–2 s–1 (k = 3 for group III and 5 for group V atoms), arriving onto the top NW facet and its sidewalls with efficiencies χ and χ′, respectively. These χ are defined by the beam geometries in MBE or precursor pyrolysis efficiencies in MOCVD. Highly diffusive group III adatoms are able to migrate from the NW sidewalls to the top, with the corresponding diffusion current j3 giving the total number of the adatoms arriving to the top facet per unit time (in s–1). We neglect the desorption of group III adatoms from the top part of a NW on the time scale of interest (assuming sufficiently low substrate temperatures).[10] For highly volatile group V atoms such as arsenic and phosphorous, we neglect the surface diffusion but account for the desorption in the form of dimers (As2 or P2). The crystallization (or incorporation) rates of III–V pairs on the NW top and sidewalls are described by the quadratic terms, which are proportional to the product of the corresponding group III and V surface concentrations. This regular growth model assumes that the meeting of any two dissimilar adatoms on the surface immediately produces a stable III–V pair in the solid state.[11] These considerations yield the steady-state material balance equations for the surface concentrations of group III and V atoms on the NW top (n3, n5) and sidewalls (n3′, n5′) (in nm–2) of the form

Figure 1

Cylindrical NW with a uniform radius R from base to top, growing in a SAE mode without any droplet on top. Kinetic processes that govern the growth include impingement of group III and V atoms, desorption of group V dimers, crystallization of III–V pairs on the NW top facet and sidewalls, and the diffusion current j3 of group III atoms from the NW sidewalls to the top. The axial and radial NW growth rates are determined by the crystallization rates on the top NW facet and sidewalls, respectively.

Here, k5 and k5′ stand for the desorption rate constants of group V atoms from the NW top and sidewalls, whereas k35 and k35′ describe the crystallization rate constants of III–V pairs on the NW top and sidewalls, respectively. The net diffusion current j3 should be divided by the surface area of the top facet πR2 to give the diffusion-induced contribution into the total flux of group III atoms arriving to the top facet. The same current should be divided by the collection area 2πRL* to describe the corresponding decrease in the surface concentration of the group III adatoms per unit area of the NW sidewalls. The collection length for group III adatoms can equal either the entire NW length L or only the effective diffusion length λ3 at the NW top.[12−15] In directional deposition techniques such as MBE, this λ3 is usually limited by the shadowing effect,[16] with λ3 ≅ P cot anα3 related to the interwire spacing (pitch) P and the angle α3 of the group III beam with respect to the vertical.[17] Therefore, we will use the simplified expressionin what follows.

Finally, the axial and radial NW growth rates are given bywith Ω35 as the elementary volume of III–V pairs in solid. In the equation for the radial growth rate, the L*/L factor accounts for the fact that crystallization of III–V pairs occurs only at the NW areas exposed to vapor fluxes but contributes into the radial growth of the entire NW. This is valid under the assumption of the cylindrical NW having a uniform radius from base to top at any moment of time, consistent with the position-independent adatom concentrations and incorporation rates in eqs , 2, and 4. Tapering of NWs requires a substantial generalization of the model and will be presented elsewhere.

To obtain the analytical solutions to these kinetic equations, we use two additional simplifications. First, we assume that the SAE growth proceeds under group V rich conditions; otherwise, it would quickly transition to a self-catalyzed mode with a group III droplet on top.[4] Then, much more group V atoms should desorb than crystallize in solid, corresponding to negligible second terms in eqs and 2 for group V atoms. This yields the simple expressionswhich directly relate the group V concentrations to the input fluxes and temperature-dependent desorption rate constants. Introducing the effective lifetimes of the group III atoms on the NW top and sidewall facets by definitionsEquations and 4 are reduced to

Second, we should present the diffusion flux j3 as a function of n3. In the absence of desorption of a group III element, the simplest approximation readswith

Here, the ε parameter is given by the ratio of the group III adatom activities on the NW top and sidewalls, similar to that in refs (10)(13), and (18) for VLS NWs. The quantity 2πRL*χ3′I3 is the total number of group III atoms collected by the top part of the NW of length L* per unit time, which equals the direct diffusion current from the NW sidewalls to the top. The ε times the same quantity equals the reverse diffusion current from the NW top to the sidewalls. The resulting diffusion current given by eq equals zero under equilibrium conditions (at ε = 1).

We first use eq in eq for n3/τ5 and insert the result into eq for dL/dt, which yields the straightforward result for the axial growth rate. Using eq in eq for n3′/τ5′ gives n3′/τ5′ = εχ3′I3, and then eq for dR/dt yields the result for the radial growth rate. Thus, our final equations describing the simultaneous axial and radial growth of cylindrical NW have the formwith F3 = Ω35I3 as the deposition rate of group III atoms in nm/s. This model should be relevant for sufficiently long NWs, while for shorter ones, we should additionally account for a diffusion flux from the substrate surface.[14,18,19] While negligible for the growth on rough substrates with a parasitic layer,[13] this flux can be much more important in the case of atomically flat patterned SiO/Si(111) substrates and can be essential for understanding of the growth start.[4,6] This important question will be considered elsewhere. However, for NWs whose length is much larger than the group III diffusion length on the oxide surface, eq should give a reasonable approximation for the growth kinetics (this is guaranteed for the axial growth rate when L > λ3, and hence, the diffusion flux from the substrate no longer contributes into the NW elongation[12,18−21]).

Equation for dL/dt has a similar form as the conventional expression for the axial growth rate of VLS III–V NWs[10,18−21]or self-induced GaN NWs[12,22] under group V rich conditions. The parameter ε directs the diffusion flux of group III adatoms from the NW sidewalls to the top or in the opposite direction, depending on whether the adatom activity on the sidewall facets (χ3′I3τ5′) is larger or smaller than that on the top facet (χ3I3τ5), as in refs (13) and (18). However, there are two important differences. First, most growth models originally developed for VLS III–V NWs completely ignored the influence of a group V element, while our parameter ε contains explicitly the adsorption–desorption and crystallization rates of group V atoms. Second, the NW radius R increases due to incorporation of the material at the NW sidewalls, so eq for L(t) and R(t) should now be considered as the system of two connected differential equations.

Equation shows that ε decreases when k35′/k35 decreases (that is, when the crystallization rate on the NW sidewalls is much lower than on the top) or k5′/k5 increases (that is, when the desorption rate of group V atoms from the NW sidewalls is much higher than that from the top). This shows the importance of a catalyst droplet in enhancing the axial and suppressing the radial growth rate of VLS NWs, because the droplet should collect group V atoms more efficiently and at the same time, increase the crystallization rate of III–V pairs with respect to the vapor–solid growth. For the SAE NW growth, any differences in the crystallization or evaporation rates on the sidewalls and top facets should arise due to their different crystallographic structures, and hence, the ε parameter may be on the order of unity. We will therefore treat this ε as the free parameter of the model, whose value can be deduced from fitting the experimental data on the axial and radial NW growth rates under different conditions and in different material systems.

Results and Discussion

Equation for “short” NWs (L < λ3) is solved as follows. At L* = L, integration of eq for R readily yields R = R0 + F3χ3′εt, where R0 corresponds to the initial radius of the pore or pre-existing NW stem. Using dt = dR/(F3χ3′ε), eq for L can be put in the form

Integrating this with the initial condition L(R = R0) = L0, with L0 as the length of the initial NW stem at t = 0, we obtain

Therefore, the solutions can be presented aswith

The L0 can in principle be put to zero; however, we reiterate that our model may not be suited to describe the initial stage of NW growth evolving from the substrate surface. Also, some NWs can only be grown on pre-existing stems of a different material (for example, InAsSb NWs on InAs stems[5]), in which case L0 represents the length of the InAs stem.

For “long” NWs (L ≥ λ3), the solutions are obtained as follows. At L* = λ3, after dividing dL/dt by dR/dt, eq giveswith b = χ3/(χ3′λ3ε). Integrating this with the initial condition L(R*) = λ3, we get

Here, R* denotes the NW radius at the moment of time t* where L becomes equal to λ3. This expression is further reduced to L = λ3(R / R*) at R – R* ≪ λ3, which is the reasonable approximation unless the growth time is very long. Substitution of this power-law dependence to eq for R yields dR / dt = A / R, with A = F3χ3′λ3εR* / λ3. Upon integration with the initial condition R(t*) = R* and presenting both length and radius as functions of time, the resulting expressions take the formwith

Equation is similar to the one obtained previously for the self-induced GaN NWs,[12] with a short diffusion length λ3on the order of 40–50 nm.

With the known deposition rate F3 and geometry (the χ coefficients and the pore or stem radius R0), the obtained expressions are controlled by the single parameter ε. It determines the power exponents in the scaling power-law dependences of the length and radius on the growth time as well as the characteristic times t0 and t1 for the axial elongation and radial extension in different stages. After a short incubation stage (involving different types of incubation processes such as the NW nucleation on the substrate surface[3,10] or in the patterned holes,[6] diffusion of group III adatoms from the substrate surface to the NW top,[4,8,12,14] or shape transformation[22]), the NW radius evolves linearly with time as long as L < λ3, while the length increases superlinearly for ε < 2/3 or a > 1. The limiting case of the highest εmax = 2/3, or the lowest amin = 1, corresponds to the linear time dependence of the NW length. At L > λ3, both the length and radius increase sublinearly with time. At εmax = 2/3, the power exponent for both length and radius at L > λ3 equals 1/2. The standard VLS-like growth is recovered in the second limiting regime of the lowest εmin = 0, corresponding to a → ∞. In this case, the NW radius stays constant, while the length increases first exponentially and then linearly with time[13,14]These expressions are written for L0 = 0.

Typical time dependences of the NW lengths and radii for the fixed R0 = 50 nm, F3χ3′=2 nm/min, λ3 = 1000 nm, χ3 = χ3′, L0=0, and different ε are shown in Figure . It can be seen how the fast axial growth at a constant radius at ε = 0 is transitioned to a much slower one as the ε parameter increases, with the length and radius becoming comparable at ε = 0.64. The NW length increases superlinearly with time until it reaches the diffusion length of 1000 nm, after which the time evolution of length becomes sublinear. The further increase of ε up to its maximum value of 2/3 may lead to situations where the NW radius increases faster than the length so that NWs become nanodiscs, as in ref (5) for InAsSb NWs grown on InAs stems.

Figure 2

NW length (solid lines) and radius (dash-dotted lines) vs the growth time for different ε shown in the legend. These curves are obtained from eqs to 15 at R0 = 50 nm, F3χ3′=2 nm/min, λ3=1000 nm, χ3 = χ3′, and L0=0. The dotted line indicates the transition between different growth regimes occurring at L = λ3.

The evolution of NW length and radius shown in Figure qualitatively describes typical behaviors observed experimentally for different non-VLS III–V and III–N NWs in MBE and CBE.[7−9,12,22−25] To make our analysis more quantitative, we first consider the experimental data on SAE InAs NWs grown by MBE in patterned arrays on SiO2/Si(111) at 480 °C, with a fixed pore radius of 40 nm and variable pitches of the array.[8]Figure a, b show the time evolution of the NW length and diameter, respectively, for pitches varying from 250 to 3000 nm. The diameters given here were determined by taking the average values at the top and bottom of each NW, as some of the NWs were slightly tapered. As mentioned above, tapering is disregarded in our modeling. However, the model can be used for fitting the average diameters in the first approximation.

Figure 3

Time evolution of (a) length and (b) average diameter of SAE InAs NWs grown by MBE in patterned arrays on SiO2/Si(111) with different pitches from 250 to 3000 nm (symbols),[8] fitted by the model (lines). For the largest pitch of 3000 nm, the length evolution is slightly superlinear, and the diameter is linear in time in the absence of shadowing. Decreasing the pitch leads to the decrease of the indium collection length due to the shadowing effect, from more than 3500 nm for 3000 nm pitch down to 500 nm for 250 nm pitch. This leads to the sublinear behavior of both length and diameter after the NW length exceeds the collection length of indium.

In the MBE technique, the collection length of indium should be limited by shadowing, which is expected to be almost negligible for 3000 nm pitch. Therefore, we fit these data points by eq at L < λ3, yielding the linear time dependence of the NW diameter with R0=24 nm and t0 = 26 min. This corresponds to the effective deposition rate on the NW sidewalls χ3′F3≅1.45 nm/min. As discussed in detail in ref (8), the axial NW growth rate is faster at the beginning of growth, which can be attributed to the additional collection of indium from the oxide surface. Leaving aside this initial stage, we use eq for the NW length starting from L0=170 nm. The best fit to the length data for a 3000 nm pitch shown in Figure a is then obtained with Rc=17.5 nm and a=1.14, corresponding to ε=0.637. This large ε value yields only a slight superlinearity of the length versus time. The data for smaller pitches clearly show the sublinear behavior of the NW length and diameter, both decreasing with decreasing pitch. All sets of data for 250 to 1000 nm pitches are well fitted by eq at L > λ3 if we assume that the indium collection length decreases from ∼1.85 μm for a 1000 nm pitch to ∼0.5 μm for a 250 nm pitch, which seem plausible. Very importantly, all curves in Figure correspond to the same ε=0.637, related to a fixed V/III flux ratio,[8] with the characteristic times t1 in the range from 24 to 76 min. Overall, our model reproduces the data very well, explaining why the length and diameter are almost linear in time for the largest pitch and become more and more sublinear for smaller pitches after the NW length exceeds the collection length of indium. According to these results, the diffusion length of indium adatoms on the sidewalls of InAs NWs at 480 °C is more than 3.5 μm, and the indium collection on the sidewalls is limited only by the shadowing effect. This is consistent with the earlier observations for MBE-grown InP1–As NWs.[13]

Next, we consider the data of ref (9) for the length and diameter evolution of GaAs NWs grown by SAE MOCVD at 750 °C on patterned SiO2/GaAs substrates, with a 600 nm pitch and variable pore diameter from 125 to 225 nm. These curves, shown in Figure , feature an interesting nonlinear evolution of the NW lengths. Similar to Figure , the NW length first increases superlinearly with time, converging to a sublinear behavior for longer times. We speculate that this effect is due to a limited diffusion length of gallium on the NW sidewalls. At a high growth temperature of 750 °C, a fraction of gallium adatoms desorbs before reaching the NW top, corresponding to the transition from the growth regime at L < λ3 to the one at L > λ3. The pronounced superlinear increase of the NW length for shorter times requires sufficiently large a according to eq . The NW diameter in this stage is linear in time regardless of a. All theoretical curves shown in Figure correspond to a fixed a = 1.83 or ε = 0.522 at L0 = 0. Other parameters are summarized in Table and provide the excellent fits to the data. From the fits, we can deduce the values of F3χ3′=5.44 nm/min and χ3/χ3′=1.78, showing that the adsorption of gallium is better on the top NW facet than on the sidewalls. The observed decrease of the NW length for larger pore size is explained simply by the fact that the NW elongation is controlled by the surface diffusion of gallium, which contributes more into the axial growth rate of thinner NWs. Most importantly, the gallium diffusion length on the NW sidewalls at 750 °C appears very close to 2 μm in all cases, as shown in Figure .

Figure 4

Length and diameter evolution of SAE GaAs NWs grown by MOCVD in patterned arrays on SiO2/GaAs with different pore sizes from 125 to 225 nm (symbols), fitted by the model (solid lines for the length and dashed lines for the diameter) with the parameters summarized in Table . The time dependence of the NW length converges from superlinear to sublinear at around 2 μm, corresponding to the diffusion length of gallium adatoms on the NW sidewalls λ3 (shown by the shaded zone). The NW diameters increase linearly before and slightly sublinearly after 2 μm of length. The diffusion-like character of growth leads to the increase of length for thinner NWs. This is further demonstrated by the dash-dotted curve for a pore size of 75 nm.

Table 1

Parameters of GaAs NWs Used in Modeling

pore size 2R0 (nm)	ε	Rc/R0	λ3 (nm)	t0 (min)	t1 (min)	t* (min)
125	0.522	3.73	2100	22	44	44
150	0.522	3.73	2000	27	46	46.5
175	0.522	3.73	1950	37	47	57.2
200	0.522	3.73	1800	43	47	60
225	0.522	3.73	1900	53.5	47	69

In summary, the proposed approach allows one to treat simultaneously the axial elongation and radial extension of SAE III–V NWs. The most important parameter of the model, ε, determines the power exponents in the scaling time dependence of the NW length and average radius and can be finely tuned by the growth conditions including the V/III flux ratio and temperature. We plan to generalize the model by considering more accurately the initial stage of NW growth from the substrate surface, as well as spatially inhomogeneous incorporation rates and tapered NW geometries. The developed model yields simple analytical solutions with a minimum number of parameters and therefore can be useful for understanding and controlling the SAE growth in different material systems, including ternary III–V NWs such as InGaAs, InAsP, InAsSb, and their heterostructures. In particular, ref (26) gives an example of application of a similar growth model to SAE InAsSb NWs, where the ε parameter depends on the InSb fraction in a ternary NW. More studies of ternary III–V NWs grown in the SAE mode will be presented elsewhere.