Vapour-liquid-solid mechanism

In 1964, Wagner and Ellis25 demonstrated the growth of Si ‘whiskers’ upon vapour deposition of Si onto a Au droplet covered Si substrate. It was observed that each whisker (or indeed each NW) had a Au droplet at its tip following growth and its diameter (as small as ~ 100 nm) was equal to that of its droplet. They explained that this selective, highly directional seeded-growth was governed by the vapour-liquid-solid (VLS) mechanism - having a solid

substrate and eventual NW, a liquid droplet acting as a growth catalyst, and a vapour flux providing the growth material. This method is now in widespread use for NW growth.

Fig 3.2: Stages of VLS growth: a) Semiconductor deposition onto substrate coated with metal nano-droplets and incorporation of semiconductor into droplets, b) formation of

semiconductor-metal liquid alloy, its subsequent saturation, and precipitation of semiconductor to initiate VLS growth (inset: diffusion of material through droplet), c) continued VLS growth. d) If material impinges on sidewalls, VS lateral growth occurs.

3.2.2.1. Basic VLS processes. The main stages of VLS growth of a NW are shown in Fig

3.2a – c and now described in the context of the Au-Si system. A Au droplet (or catalyst) lies on a Si substrate which is heated to a temperature exceeding the Au-Si eutectic temperature (363°C)26 so that a liquid Au-Si alloy may form. A flux of Si vapour is then directed towards the substrate and adatoms arrive at the droplet either by: a) direct impingement; or b) surface

32 diffusion following arrival at the bare substrate (Fig 3.2a). The droplet is a preferred site for incorporation since it has a larger accommodation coefficient than the bare substrate as it consists of many atomic steps27. The droplet becomes saturated and Si precipitates out at the interface between the droplet and substrate, i.e. the liquid/solid interface (Fig 3.2b), initiating NW growth. Diffusivity of Si through the droplet to the liquid/solid interface is high since the droplet is liquid, and NW growth proceeds at this interface in a layer-by-layer fashion (Fig 3.2b inset), lifting the Au-Si droplet upwards so that it always remains at the NW tip. This process continues upon continued exposure to Si vapour (Fig 3.2c), and since the growth front is at the liquid-solid interface, the NW diameter is equal to that of the interface.

3.2.2.2 Lateral growth. In some cases, at later stages of NW growth, adatoms arriving on

the substrate or on the NW sidewalls may be incorporated onto the sidewalls via a standard vapour-solid (VS) growth mechanism, this occurring once the NW length exceeds surface diffusion lengths and adatoms are no longer able to reach the droplet. As a result, the diameter of the NW becomes greater than that of the liquid/solid interface, particular towards the base of the NW28.

3.2.2.3. ‘Nanowire root’ and ‘cooling neck’. Whilst the schematic in Fig 3.2c shows a

NW with a constant radius from substrate to droplet, VLS-grown NWs often have a shape like that shown in Fig 3.3a, with a tapered base (even in the case of no lateral growth) and tip, owing to the non-steady state mechanics at the initial and final stages of growth. An example of the tapered base, or ‘root’, is shown in the inset of Fig 3.3a29

. (This is the standard case, for a discussion of other NW shapes achieved by VLS growth, the reader is referred to Ref. 30). The root and neck features are now described.

Before NW growth initiates, the droplet may be modelled as a spherical cap (Fig 3.3b) with a contact angle, θ, defined by Young’s equation31:

where , and are the substrate-vapour, substrate-liquid and liquid-vapour surface energies respectively. However, as reported by Schmidt et al.29, 32, this contact angle changes during the initial phase of NW growth (Fig 3.3c) to account for the presence of the NW sidewall and is expressed as;

where is the sidewall-vapour surface energy and is the angle of inclination of the NW flank, equal to 0° prior to NW growth and then increasing to 90° for steady-state NW growth.

33 As increases during this initial stage, θ increases, resulting in a reduction of the liquid-solid interfacial area and a tapering of the NW until equilibrium and steady-state growth is reached. Note that for droplets with a diameter of just a few nm’s, a line tension term is included in Equations 3.1 and 3.2, its magnitude influencing the extent of the root29. For instance a large line tension results in a large amount of tapering. Line tension is ignored for larger droplets, but a root should still exist.

Fig 3.3: a) Typical NW shape including a tapered base, or ‘root’, and tapered cooling neck (inset: image taken from Ref. 29. b) and c) development of droplet and wire shape in the initial phases of NW growth, resulting in the formation of NW root.

Steady-state VLS NW growth ceases when vapour deposition is terminated, where upon the liquid droplet is saturated with residual semiconductor material (Fig 3.4a). Upon cooling, the solubility is reduced, and this residual material begins to precipitate out at the liquid-solid interface. This reduces the volume of the droplet and the liquid-solid interfacial area so the final stage of growth is tapered33, as shown in Fig 3.4b. (The contact angle between the droplet and NW will also change as the droplet composition changes). After cool-down there is therefore a ‘cooling neck’ and only traces of semiconductor remain in the droplet (Fig 3.4c).

Fig 3.4: Formation of a ‘cooling neck’ and purification of the droplet upon cool-down, see text.

34

3.2.2.3. Material requirements for VLS growth. Firstly, the minimum requirement for

VLS growth is for the droplet to be a liquid alloy at the growth temperature, this being dependent on the temperature-composition phase diagram of the materials. For Au-catalysed Si NW growth on a Si wafer supersaturation is achieved readily since both the substrate and the vapour contribute to the concentration of Si in the catalyst. In other cases however if the substrate and wire are of different materials there may be a longer nucleation delay while supersaturation is achieved from the vapour alone. In such a case (e.g. Au-catalysed growth of CdTe on Mo), the model describing the initial stages of NW growth is expected to be more complex.

Fig 3.5: Droplet-NW interface for standard VLS growth involving surface energies in cases of a) steady state, b) horizontal droplet shift and c) vertical droplet shift. d) Non-standard VLS growth whereby droplet wets the nanowire sidewalls due to excessive vertical droplet shift34.

Secondly, the droplet must be stable, this being determined by the balance of surface energies. In ‘standard’ steady-state NW growth the triple-phase line (TPL) between vapour, liquid and solid phases, as indicated in Fig 3.5a, is stable and is shifted upwards upon completion of each NW monolayer. Due to environmental perturbations however, the TPL can experience random shifts horizontally (Fig 3.5b) or vertically (Fig 3.5c) and a force is required in order to return the TPL to the steady-state position, this force being provided as long as certain surface energy inequalities are met34. In the horizontal case, a force, FH, will return the TPL to its steady-state position provided

which is always the case if one considers the following: The left hand side of the inequality is equal to the right hand side (RHS) prior to NW growth according to the model of a spherical cap on a surface (Equation 3.1) and during NW growth the RHS becomes smaller since becomes greater (Fig 3.3b and c) – even more so in the case of a horizontal shift of the

35 droplet. In the vertical case, a force FV, will return the TPL to its steady-state position

provided

which is the same as the Nebol’sin—Shchetinin inequality for standard VLS growth34, 35. If Equation 3.4 is not satisfied, the droplet slides downwards and wets the sidewall resulting either in a modified mode of VLS growth34 or droplet perforation and termination of NW growth. A separate phenomenon is that of droplet migration: Hannon et al.36 observed that some droplets gradually migrated from one NW to a neighbouring NW, resulting in the diameter of some NW’s decreasing over time and that of others increasing over time.

Further material properties desirable for VLS growth are: a) the solubility of the metal component in the liquid eutectic is greater than in the solid wire (i.e. the segregation

coefficient for the metal component in the semiconductor, k < 1); and b) the vapour pressure of the metal component must be smaller than that of the liquid eutectic so that the droplet does not evaporate or shrink in volume.

3.2.2.4 Growth rate (length and diameter). In order to understand the relationships

between NW length, l, radius, R, and growth time, t, during VLS growth, one should consider the possible adatom processes involved (see Fig 3.6), these being: 1) Direct adsorption on droplet surface; 2) desorption from droplet; 3) adsorption to sidewall and diffusion to droplet; 4) desorption from sidewall; 5) substrate-to-sidewall diffusion; 6) substrate-to-droplet

diffusion via sidewall; and 7) nucleation on substrate.

Fig 3.6: Possible adatom processes during VLS NW growth, modified from Ref. 37, see text.

For cases whereby the main contribution to NW growth is direct adsorption, NWs of greater diameter have greater elongation rates, dl/dt, owing to the increased adsorption

36 associated with the increased capture area. This is the assumption made by the Givargizov- Chernov38 theory, which also takes into account the Gibbs-Thomson effect; since smaller particles have greater curvature they have a higher vapour pressure which opposes the adsorption of adatoms.

Alternatively, for cases whereby the main contributions to NW growth are those of adatoms diffusing to the droplet from the substrate and sidewalls, NWs of smaller diameter have greater elongation rates. For GaAs NWs, Dubrovskii demonstrated that the adsorption- induced regime occurs for D < 50 nm, and the diffusion-induced regime for D > 50 nm39. Indeed, Dubrovskii et al. have published a series of pioneering articles focussed on

quantifying dl/dt, dR/dt and dl/dR for the case of diffusion-induced40-42 NW growth and also for more general cases that take into account all the kinetic processes43 indicated in Fig 3.6 as well as the Gibbs-Thomson effect37, 39, 44. For a diffusion-induced model41, dl/dt is given by;

√ √ ( )

where C0 is a coefficient accounting for the droplet shape, adsorption and desorption at the droplet and film growth on the substrate, is the vapour deposition flux, is the elementary volume in the NW, na is the adatom concentration on the sidewalls, z is the coordinate along the NW axis and is the diffusion coefficient. With na(z) obeying the form

(where is the adsorption from the vapour flux and is the effective adatom lifetime accounting for desorption and incorporation) and the boundary conditions of na(z = 0) = ns (i.e. the adatom concentration on the substrate) and

na(z = l) = nl being used, the solution to na(z) may be yielded and subsequently substituted into Equation 3.5 to obtain; √ ( )[ ⁄

where nf = is the effective adatom activity on the sidewalls, V = is the equivalent

deposition rate and λ = √ is the diffusion length.

This model may also account for the lateral extension of a NW, this becoming more significant as l increases beyond λ. The radial growth may be expressed by the following expression; √ * ⁄ ⁄ ( )+

37 The model assumes that R is constant along the NW length, however it is often observed that lateral growth occurs to a greater extent at the base of NWs resulting in the tapered morphology shown in Fig 3.2d. The model also assumes a hexahedral NW shape.