Fully Analytical Description for the Composition of Ternary Vapor

Article pubs.acs.org/crystal

Fully Analytical Description for the Composition of Ternary Vapor− Liquid−Solid Nanowires Vladimir G. Dubrovskii*,†,‡,§ †

St. Petersburg Academic University, Khlopina 8/3, 194021, St. Petersburg, Russia Ioffe Physical Technical Institute RAS, Politekhnicheskaya 26, 194021, St. Petersburg, Russia § ITMO University, Kronverkskiy pr. 49, 197101 St. Petersburg, Russia ‡

ABSTRACT: We present a fully analytical model that is capable of describing the steady state growth rates and compositions of Au-catalyzed ternary III−V nanowires growing from a quaternary alloy. We investigate some general features of this complex growth process and find out the reasons for the nanowire composition being different from the vapor content. The solid compositions are mapped out versus the vapor composition, nanowire radius, and V/III flux ratio in different regimes, both for group III- and V-based ternaries. We show how the nanowire composition can be tuned to a desired value by the group V flux and other growth parameters. Overall, our simple approach seems to be relevant to underline the major trends in the compositional control within vapor−liquid−solid III−V nanowires and may serve as the first step toward a more comprehensive understanding of quaternary solutions and ternary nanostructures.

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INTRODUCTION Vapor−liquid−solid (VLS) III−V nanowires (NWs) grown with a metal catalyst particle1 are interesting from a fundamental viewpoint as well as promising building blocks for scalable bottom-up nanoelectronics and nanophotonics.2,3 Highly desired bandgap tunability of such nanomaterials should be achieved through the fabrication of ternary III−V NWs with well-controlled composition. However, the compositional control becomes a very challenging task when, rather than directly from vapor, a nanostructure is grown from a nonstoichiometric quaternary liquid alloy4−16 such as an Au− In−Ga−As solution in the case of Au-seeded InGaAs NWs. A similar issue is encountered in doping of binary III−V NWs17−20 and even in a more simple case of Au-seeded NWs of elemental semiconductors.21 Because of a complex chemistry of quaternary alloys,22 uncertainties in determining the droplet compositions, different kinetic pathways of different elements into a nanophase and their bonding probabilities in the nanophase, there is a lack of theoretical understanding of the mechanisms that govern the formation of ternary VLS NWs. Indeed the standard approach based on the chemical potentials and macroscopic nucleation theory has only recently been applied for binary III−V NWs and ternary Au−III−V alloys and has shown the significance of the group V element23−27 which was commonly ignored in the past. Ternary VLS NWs as well as some other ternary nanostructures growing from liquid nanoparticles (for example, during droplet epitaxy28) add one more degree of freedom to the composition of both liquid and solid phases and thus are much more difficult to master within the standard growth theories. Even if the kinetic pathways for material transport and © XXXX American Chemical Society

thus the material influxes were exactly known for, say, In, Ga, and As species, there would be no guarantee that these atoms leave the droplet and incorporate into a nanostructure at the same rates. This brings additional complexity in understanding catalyzed ternary nanostructures compared to the self-induced ones.29 Recently, we proposed an irreversible growth model which describes well some experimentally observed trends of the growth rate and composition of Au-seeded InGaAs NWs versus the droplet diameter, V/III flux ratio, and temperature.16 However, this model is based on one central simplifying assumptionthe liquid and the solid compositions are considered identical. Here, we further develop our approach beyond this assumption by presenting a fully analytical description for the steady state growth rate and composition of Au-seeded ternary VLS NWs as functions of the vapor composition, size, and physically transparent kinetic constants. Ternaries based on mixing the group III or group V elements are considered simultaneously. We investigate the reasons for the NW composition being different from the vapor phase and find out the two physical effects that are responsible for it. Very importantly, the NW composition can be modulated by the group V flux. Finally, we map out the solid compositions against relevant experimentally controlled parameters and discuss different scenarios for the compositional control in ternary III−V NWs. Received: July 3, 2015 Revised: November 2, 2015

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DOI: 10.1021/acs.cgd.5b00924 Cryst. Growth Des. XXXX, XXX, XXX−XXX

Crystal Growth & Design

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SYSTEM OF GOVERNING EQUATIONS We consider a nanostructure growing at the rate G from a quaternary alloy characterized by the three atomic concentrations c0, c1, and c2. Here, c0 represents a single group V or group III element (e.g., As for InGaAs) and c1 and c2 stand for the two elements of the other group whose intermixing produces a ternary material (e.g., Ga and In for InGaAs). These concentrations should obey the normalization condition c0 + c1 + c2 + ccat = 1 including the catalyst concentration ccat (Au in the case of Au-catalyzed InGaAs NWs). For self-catalyzed NW growth from a group III melt, we have simply c0 + c1 + c2 = 1, and thus the number of unknowns is reduced from 3 to 2. If we assume the irreversible character of growth in which meeting two atoms of type 1 and 0 or type 2 and 0 immediately produces a stable solid III−V pair,16,26 the three concentrations obey the following balance equations in the steady state K1c0c1 = V1 − U1c1

(1)

K 2c0c 2 = V2 − U2c 2

(2)

G = c0(K1c1 + K 2c 2) = V0 − U0c0

(3)

simple consideration in terms of concentrations rather than chemical potentials. We also note an important difference between the atomistic growth picture adopted in this work and the nucleation models.26 Here, it is assumed that the alloy crystallization and the NW composition are simply determined by the formation of the two possible types of III−V pairs, described by the c0ci terms in eqs 1−3. On the other hand, in the nucleation approach we consider the island formation as being limited by meeting two group V atoms which almost immediately attach the two missing group III atoms (under Ga-rich conditions in the droplet), which is why the minimum island consists of two III−V pairs. Whenever it becomes more probable to form a stable III−V solid dimer, the next group V atom will form the second III−V monomer rather than attach to the first one, and hence, we arrive at the irreversible growth described by eqs 1−3. The atomistic growth does not require any island nucleation, and it seems impossible to introduce a group V-limited bonding regime into the irreversible equations. Indeed, writing the terms c20 instead of c0ci would leave no group III atoms in the equations, while they are certainly required to form a stoichiometric III−V solid. Finding the threshold growth parameters and bridging the two regimes requires a separate study. From very basic considerations, the atomistic growth is more anticipated in chemical epitaxies (such as metal organic chemical vapor deposition or hydride vapor phase epitaxy) with high material inputs, while the nucleation-limited regime is expected in molecular beam epitaxy and related growth techniques with modest material fluxes.

Here, Ki, Vi, and Ui are the kinetic coefficients that determine the incorporation rates from liquid into solid, material influxes, and outgoing fluxes, respectively, for each growth species.30 These quantities are considered known and will be discussed later for VLS NWs. The nucleation rates in irreversible growth are proportional to the bonding probabilities c0ci. The outgoing fluxes due to desorption or reverse diffusion from the droplet must be proportional to the concentrations ci. The first equation means that the incorporation rate of atoms of type 1 equalizes their total material flux; the second equation states the same for atoms of type 2, while the third equation implies that the total incorporation rate of the two types of atoms 1 and 2 equals the total material flux of atoms of type 0. This condition ensures stoichiometry of the solid alloy. On the other hand, the growth rate G is obtained by summing up the two incorporation rates producing bound 1−0 or 2−0 pairs within a nanostructure. In a more general case, the concentrations ci in eqs 1−3 should be changed to the corresponding activities exp(μi), with μi(c0,c1,c2) being the chemical potential of the ith element in the liquid phase in thermal units. Accordingly, the nucleation rates in the left-hand sides should become J(Δμ1) and J(Δμ2), with Δμi = μi + μ0 − μi0 for i = 1, 2 as the difference of the chemical potentials per pair i − 0 in the liquid and solid states due to crystallization (the driving force for the VLS growth), μi and μ0 as chemical potentials of the unbound atoms of types i and 0 in the solution and μi0 as the chemical potential of the bound i − 0 pair in the solid state.23 Such a form of the nucleation-mediated growth rate assumes that tiny 2D islands (emerging with the critical size30) are binaries which would mix to a ternary alloy in a later stage of their lateral growth. Alternatively, one can consider just one nucleation rate J(Δμ) of a ternary island with the chemical potential Δμ = (1 − x)μ1 + xμ2 + μ0 − μ120(x), where x is the solid composition of element 2 and μ120(x) ≅ (1 − x)μ10 + xμ20 is the x-dependent chemical potential of a ternary solid. Here we effectively assume that the compositions of liquid and solids are identical. In any case, chemical potentials μi(c0,c1,c2) are complex and not exactly known.22 Therefore, it seems reasonable to start with the approximation μi = ln ci which yields eqs 1−3 and allows for a much more

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ANALYTICAL SOLUTIONS Equations 1−3 contain three unknowns c0, c1 and c2, while the growth rate G is expressed through them with the known prefactors K1 and K2. We can introduce the solid composition x, the liquid composition y, and the effective “vapor” composition z (defined by the influxes V1 and V2) according to x=

K 2c 2 c2 V2 ;y= ;z= K1c1 + K 2c 2 c1 + c 2 V1 + V2

(4)

Clearly, the solid composition in our model is given by the ratio of the incorporation rates and equals the liquid composition only when K1 = K2, which seems to be a rare case. Exact solutions to eqs 1−3 are obtained as follows. First, we express the c0 through c1 and c2 from eq 3. Then we inject the result into eqs 1 and 2 and note that the more relevant variables are given by the “normalizes fluxes” of the ith atoms from liquid to solid a1 =

K1c1 Kc ; a2 = 2 2 U0 U0

(5)

In these variables, eqs 3 for the c0 and G and eq 4 for x become c0 =

V0 1 A ; G = V0(1 − A), A ≡ U0 1 + a1 + a 2

(6)

x=

a2 a1 + a 2

(7)

Equations 1 and 2 together with eq 6 for c0 can now be rearranged as a1(A + δ1) = v1 B


Crystal Growth & Design a 2(A + δ2) = v2

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(8)

with the parameters v1 =

UU UU V1 V ; v2 = 2 ; δ1 = 1 0 ; δ2 = 2 0 V0 V0 K1V0 K 2V0

(9)

Solutions to eqs 8 can be presented in several equivalent forms. We prefer to derive an explicit relationship between the solid composition x and the effective vapor composition z and to present all other quantities as functions of x only. Putting the ai as a1 = (1 − x)(1 − A)/A and a2 = x(1 − A)/A, we rewrite eqs 8 in the form (A + δ1)(1 − A)(1 − x) = (1 − z)(v1 + v2)A (A + δ2)(1 − A)x = z(v1 + v2)A

(10)

Figure 1. 3D graph of solid composition x versus the effective vapor composition z and the sink δ2.

Dividing the first eq 10 to the second one yields the relationship between z, x and A:

z=

xf (x) 1 − x + xf (x)

incorporates faster into the solid state (K2 > K1). This leads to its larger percentage in solid with respect to the first element. In the opposite case of δ2 > δ1, the incorporation of the second element is suppressed either because it is less soluble in the droplet (U2 > U1) or due to its lower coupling rate with the other group atoms (K2 < K1). Figure 2 shows 3D graphs of the solid composition x and the dimensionless growth rate g = G/(V1 + V2) versus the effective

(11)

where the function f(x) is given by f (x ) = 1 +

δ2 − δ1 δ1 + A(x)

(12)

Summing up eqs 10 yields the quadratic equation for A(x) whose solution is written as A=

1 ( B2 + 4C − B) 2

C = δ1(1 − x) + δ2x ; B = v1 + v2 − 1 + C

(13) (14)

Finally, the concentrations ci can be put as c1 =

G G (1 − x); c 2 = x K1c0 K 2c0

(15)

where, from eqs 6 and A = A(x), the right-hand sides depend only on x. The obtained solutions depend on the three control parameters v1 + v2 = (V1 + V2)/V0, δ1 and δ2. The first parameter gives the effective flux ratio. It can be put as v1 + v2 = 1/F53 * if 1 and 2 are the group III atoms and 0 is the group V atom, with F*53 as the effective V/III ratio, or as v1 + v2 = F*53 in the case of a ternary based on the group V intermixing. The δi parameters give the dimensionless sinks of atoms of type 1 and 2. They are of particular importance for solid composition since the latter differs from z only when there is an asymmetry δ1 ≠ δ2, or U1/K1 ≠ U2/K2. Whenever δ1 = δ2 ≡ δ and regardless of the absolute value of δ, the function f(x) in eq 12 equals one and thus x = z from eq 11. This generalizes the result of ref 16 where we assumed K1 = K2 (hence x = y) and the condition that gave x = z was reduced to U1 = U2. Figure 1 shows three-dimensional (3D) graph of the solid composition x versus the effective vapor composition z and the sink of the second element δ2 at the fixed v1 + v2 = 1/3 and δ1 = 1, obtained from eqs 11−14. Clearly, x = z at δ2 = δ1, so that the solid composition is entirely determined by the material influxes. Otherwise, the solid composition can noticeably exceed z when δ2 < δ1 or be much smaller than z when δ2 ≫ δ1. This can be well understood intuitively. Indeed, the condition δ2 < δ1 means that the second element composing a ternary is either more soluble in liquid (U2 < U1) or

Figure 2. 3D graphs of the solid composition (a) and the dimensionless growth rate g = G/(V1 + V2) (b) related to the *. effective vapor composition z and the effective V/III flux ratio F53 C



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Thus, we find out two reasons for the solid composition being different from that in vapor: (i) An obvious difference in diffusion transport of the two group III atoms from the NW sidewalls to the droplet,16 as given by the function α(R) eq 18. This difference is regulated mainly by the diffusion lengths λi and disappears at R → ∞ when χ1 = χ2. (ii) Much less obvious effect induced by the asymmetry of the sinks due to reverse diffusion fluxes and/or material incorporation into the NW. This difference is effective when ξ ≠ 1, or λ1K1/D1 ≠ λ2K2/D2. The δi coefficients in eqs 19 scale with the NW radius as R1/R, with a certain characteristic radius R1 which decreases for higher group V fluxes V0. Therefore, the difference between the solid and vapor compositions is expected to decrease for larger R and V0. Importantly, the solid composition can be tuned by varying the group V flux that influences both the surface diffusion lengths of the group III adatoms and the sinks δi, as we saw earlier in Figure 2. Figure 3 shows the NW composition versus the vapor composition for different NW radii for a model system with χ1

vapor composition z and the effective V/III flux ratio F*53 = V0/ (V1 + V2) for a ternary structure based on the group III intermixing at a fixed V1 + V2 and with variable group V flux V0. In this case, v1 + v2 = 1/F53 * , the δi scale with the flux ratio as δ(0) i /F* 53 and g = F* 53(1 − A) from the corresponding eq 6. The calculations were performed by means of eqs 11−14 with for δ(0) = 10 and δ(0) = 50 at F53 * = 1, the case relating to a 1 2 suppressed incorporation of the second element into a solid nanostructure. It is seen that the growth rate increases and the solid composition becomes closer to the vapor one for larger F53 *.

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TERNARY NANOWIRES BASED ON GROUP III MIXING We now consider the case of ternary NWs based on intermixing the two group III elements, i.e., when 1 and 2 represent the two group III atoms. The simplest approximation for the two group III influxes is written as16,27,30−32 ⎛ ⎛ 2λ ⎞ 2λ ⎞ V1 = V1g ⎜χ1 + φ1 1 ⎟ ; V2 = V2g ⎜χ2 + φ2 2 ⎟ ⎝ ⎝ R ⎠ R ⎠

(16)

Here, Vgi are the vapor fluxes, λi are the effective diffusion lengths, R in the NW radius, while χi and φi summarize the effects of cracking efficiencies in chemical epitaxies or geometry in molecular beam epitaxy.33 The kinetic prefactors of the reverse diffusion fluxes are given by30,34 U1 =

2hD1 2hD2 ; U2 = λ1R λ 2R

(17)

with h as the height of a monolayer and Di as the diffusion coefficients of the group III adatoms. Desorption of the group III atoms can usually be neglected at the typical growth temperatures.27,35 The incorporation rates Ki are assumed as radius-independent. This is consistent with our initial model in which the formation of the whole NW monolayer would require a number of pair meeting events (equal to the number of pairs in a monolayer). Finally, the U0 term for the group V atoms is considered as being limited by desorption with neglect of surface diffusion37−39 and hence independent of radius. Introducing the actual vapor composition zg = Vg2/(Vg1 + Vg2) and the actual V/III flux ratio F53 = V0/(Vg1 + Vg2), after simple calculations we get z=

δ1 =

zg α(R ) 1 − zg + zg α(R )

; α (R ) =

= χ2 = 2, φ1 = φ2 = 1, R1 = 15 nm, λ1 = 50 nm, λ2 = 200 nm, ξ = 0.2 and F53 = 10. These curves were obtained from eqs 11−14 and 18−20. Since element 2 is more diffusive than 1 and δ2 < δ1, as in the case of Au-seeded InxGa1−xAs NWs,16 its composition in solid is generally larger than in vapor. However, the difference between x and zg rapidly decreases with increasing the NW radius and becomes almost negligible for R = 200 nm.

χ2 R + 2φ2λ 2 χ1 R + 2φ1λ1

2hD1U0 R1 λ KD ; R1 = ; δ2 = ξδ1; ξ = 1 1 2 R λ1K1V0 λ 2K 2D1

v1 + v2 =

Figure 3. Indium composition in NWs versus the vapor composition in the model case of Au-seeded InxGa1−xAs NWs for different NW radii R, obtained with the model parameters listed in the inset. The indium content in NWs is always larger than in vapor due to its enhanced sidewall diffusion and more rapid incorporation rate with respect to Ga. The solid composition is greatest for the smallest NW radii but almost follow the vapor composition for R = 200 nm.

(18)

(19)

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⎛ ⎛ 2λ1 ⎞ 2λ ⎞⎤ 1 ⎡ ⎟ + zg ⎜χ2 + φ2 2 ⎟⎥ ⎢(1 − zg )⎜χ1 + φ1 ⎝ ⎝ F53 ⎣ R ⎠ R ⎠⎦

LIMITING CASE OF SMALL CONCENTRATIONS We now considered a particular case of the obtained expressions at x → 0 and z → 0, where both vapor and solid composition of a second element are small compared to the first one. This limiting case may describe a spatially homogeneous NW doping via the VLS mechanism, where the imposed stoichiometry of the solid alloy becomes unimportant. Of course, the approximation x → 0 does not apply to unusual situations such as B doping of InP NWs.20 In this case, boron was shown to significantly suppress the NW elongations rates, the effect which should be due to boron

(20)

The solid composition x is obtained from eq 11 where z is related to the vapor composition zg by eq 15. The δi coefficients in eq 12 for f(x) are given by eqs 19. Because of the radiusdependent diffusion fluxes of the group III adatoms, the control parameters v1 + v2, δ1 and δ2 become size-dependent. The z ratio is also radius-dependent and is generally different from the vapor composition zg due to different surface diffusivities of the two group III elements. D



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accumulation in the droplet. Such regimes should be studied within the general scheme. We also do not consider an interplay of different kinetic pathways of the dopant incorporation (in particular, a combination of the VLS and the vapor−solid growth regimes) or segregation effects leading to a heavily doped shell surrounding an underdoped core.21,40 If we assume that a dopant is diffusive and does not evaporate at a growth temperature, eqs 11 and 18 at x → 0, z → 0 and zg → 0 are reduced to β + A (R ) x = α (R ) 1 zg β2 + A(R )

v1 + v2 =

F53 χ0 + φ0(2λ 0 /R )

(23)

Now, the subscript “0” relates to a group III element. The solid composition is again obtained from eqs 11−14. However, there is a clear difference in the parameters defined by eqs 22 and 23 compared to ternary NWs based on the group III mixing [eqs 19 and 20]. In particular, the δ1 sink and the v1 + v2 influx now contain the diffusion term in the denominator, while the difference between δ2 and δ1 depends on the ratio of the desorption rates U2/U1. Figure 5 shows the radius-dependent graphs of the NW composition versus the vapor composition in the case of large

(21)

Here, the α(R) is given by eq 18, δi are obtained from eqs 19, and A(R) is given by eqs 13, 14 at x = 0. This expression shows very clearly the two effects that cause the deviation of x from zg as discussed above, which now enter the composition formula as the two multiplying factors. Whenever δ2 = δ1, the z/zg ratio equals α(R) and hence is determined entirely by the radius-dependent diffusion fluxes to the NW top. When δ2 ≠ δ1, there is also a difference in the atomic sinks in the droplet, which is described by the sizedependent function A(R). The curves shown in Figure 4 were

Figure 5. Solid composition versus vapor composition in the model case of Au-seeded InSbxAs1−x NWs for different NW radii.

R1 = 500 nm (due to enhanced desorption rate U1) and extremely small ξ = 10−4, with other relevant parameters listed in the inset. The strong inequality δ2 ≪ δ1 describes the situation where one group V element is much more soluble in the droplet than the other, as in the case of Sb in InSbAs NWs.41 The difference in solubility is so strong that the droplets are known to inflate when the As precursor is changed to the Sb one. With these parameters, Figure 5 shows that the solid composition of Sb is always larger than the vapor one. However, the radius dependence of the x(z) curves is nonmonotonicthinner NWs with R ≤ 75 nm and thick 200 nm radius NWs accommodate less Sb than 100 nm radius NWs.

Figure 4. Relative doping level x/zg versus the NW radius at the fixed λ1 = 400 nm for different λ2 (solid lines). Dash-dotted lines show the functions α(R) that describe the doping levels changing only due to different surface diffusivities of the two species.

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obtained for the model parameters χ1 = χ2 = 2, φ1 = φ2 = 1, R1 = 10 nm, λ1 = 400 nm, R1 = 15 nm, δ2 = (λ1/λ2)δ1, F53 = 10 and different λ2 changing from 50 to 350 nm. It is seen that the doping is more difficult for thinner NWs, while for thick enough NWs the doping level changes primarily due to different diffusivities of the two elements.

CONCLUSIONS Summarizing, we have developed a fully analytical model for the growth rates and compositions of Au-catalyzed ternary NWs and other ternary nanostructures growing from a quaternary liquid catalyst. This approach is based on the irreversible growth concept and allows one to easily introduce the asymmetry of the incorporation rates of the two growth species, which is not so obvious within the macroscopic nucleation theory. The model works equally well for ternaries based on the group III or V mixing. Despite its simplicity, our considerations reveal some general features of this complex growth process that are listed below. (i) Considering ternary VLS NWs based on the group III mixing, there are two reasons for the solid composition being different from the vapor content. The first reason is associated with different diffusivities of the two group III elements composing a ternary alloy. More diffusive elements will enter the NWs more easily. The second effect is due to different outgoing fluxes from the droplet and/or incorporation rates

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TERNARY NANOWIRES BASED ON GROUP V MIXING Finally, we consider Au-catalyzed ternary NWs based on mixing the two group V elements. Assuming, as above, that both group V species desorb from the droplet but do not diffuse on the NW sidewalls36−38 while the group III element is diffusive but does not desorb at a growth temperature, we can repeat the same procedure as before and get 2hD0U1 R1 KU ; R1 = ; δ2 = ξδ1; ξ = 1 2 δ1 = χ0 R + 2φ0λ 0 λ 0K1V0g K 2U1 (22) E



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from liquid into solid, lumped together in the δi parameters. Smaller reverse diffusion flux or faster coupling with the group V atoms will increase the content of this element in the solid state with respect to the other. (ii) The composition of ternary NWs is a function of the NW radius and becomes closer to the vapor content for thicker NWs in the case of group III-based ternaries. (iii) The NW composition can be modulated by the V/III flux ratio; larger V/III flux ratios usually lead to a smaller difference in the solid and vapor compositions. We have discussed only qualitative correlations of our model with some experimental data on different ternary NWs. A more detailed comparison requires a comprehensive study and will be presented elsewhere. From theoretical viewpoint, the next step would be an analysis of nonstationary effects and the properties of NW heterostructures that form by switching between the two vapor fluxes. In particular, accessing the NW interfacial abruptness would require a solution of nonstationary eqs 1 and 2 with the appropriate initial conditions, where the group V contribution is excluded by means of eq 3. Overall, we believe that our results will give useful guidelines for the compositional control and may serve as the first step toward a better understanding of ternary alloys in VLS NWs.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]ffe.ru. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The author gratefully acknowledges financial support of the Russian Science Foundation under the Grant 14-22-00018.

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