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Probability of twin formation on selfcatalyzed GaAs nanowires on Si substrate
Nanoscale Research Letters volume 7, Article number: 558 (2012)
Abstract
We attempted to control the incorporation of twin boundaries in selfcatalyzed GaAs nanowires (NWs). Selfcatalyzed GaAs NWs were grown on a Si substrate under various arsenic pressures using molecular beam epitaxy and the vaporliquidsolid method. When the arsenic flux is low, wurtzite structures are dominant in the GaAs NWs. On the other hand, zinc blende structures become dominant as the arsenic flux rises. We discussed this phenomenon on the basis of thermodynamics and examined the probability of twinboundary formation in detail.
Background
IIIV compound semiconductor nanowires (NWs) have been attracting significant attention as fundamental structures of novel optical and electronic devices. Especially, vaporliquidsolid (VLS) NWs grown on Si substrates have been investigated for optoelectronic integrated circuits [1, 2].
Previously, we succeeded in growing selfcatalyzed GaAs NWs on Si substrates using molecular beam epitaxy (MBE)VLS method which is a combination of MBE and VLS method [3]. However, twin boundaries formed in the NWs during growth. The occurrence of twin boundaries is controlled by various methods such as control of supersaturation, growth temperature, and diameter [4–8]. To control the twin boundaries, we focused on the pressure of arsenic because selfcatalyzed GaAs NW growth was nearly independent of Ga pressure, and arsenic flux plays an important role in the growth mechanism [3]. In addition, arsenic solubility in Ga solution is very low [9]. This means that the degree of supersaturation depends on arsenic pressure only.
Methods
Selfcatalyzed GaAs NWs were grown on a (111)Si substrate by MBEVLS method. The growth temperature was 580°C. When the arsenic flux varied from 5.0×10^{−6} to 1.9×10^{−5} Torr, the diameter and the length of the obtained GaAs NWs varied from 90 to 30 nm and 0.5 to 3.5 μm, respectively. We have previously discussed the tendency of the diameter and length to depend on the arsenic flux [3]. The obtained NWs were observed by transmission electron microscopy (TEM).
Results and discussion
TEM images of GaAs NWs are shown in Figure 1. In the case of low arsenic flux (5.0 × 10^{−6} Torr), wurtzite (WZ) structures are dominant as shown in Figure 1a. On the other hand, when the arsenic flux is high, Figure 1c shows that segments between the twin boundaries become large, and zinc blende (ZB) structures are dominant. The segment size follows the time between successive twincrystal nucleation events. Furthermore, distribution of the segment sizes is exponential from a stochastic point of view [10].
Figure 2 shows the distribution histogram of the segment sizes obtained from Figure 1. We assume a segment size of x and fitted the histogram with an exponential curve of the form exp (^{−}x/a) to estimate the expectation value of segment α. The estimated expectation values were 1.1 and 6.5 monolayers at arsenic fluxes of 7.0 × 10^{−6} and 1.9 × 10^{−5} Torr, respectively. The reciprocal of the expectation value α^{−1} is equivalent to the probability of occurrence of twincrystal nucleation. The probabilities were approximately 90% (7.0 × 10^{−6} Torr) and 15% (1.9 × 10^{−5} Torr). The dependence of the probability on the arsenic flux is in good agreement with [11] and [12].
To understand these phenomena, we calculate the degree of supersaturation and estimate the probability of twinboundary formation, following the procedure presented by Glas [13]. The degree of supersaturation Δμ is as follows:
where μ_{GaAs} is the half chemical potential of GaAs crystal nucleation; μ_{Ga}^{L} and μ_{As}^{L} are the chemical potentials of liquid gallium and liquid arsenic, respectively. We assume that the arsenic adatom was liquid arsenic existing on the gallium droplet. However, μ_{Ga}^{L} and μ_{As}^{L} include an interaction between gallium and arsenic in pure gallium and arsenic chemical potentials μ_{Ga}^{pL} and μ_{As}^{pL}, respectively. Therefore, μ_{ X }^{L} (X = Ga or As) is given by
where R is the gas constant, T is the growth temperature, and $a\frac{\text{L}}{X}$ is the activity of X in the liquid phase. At T_{0} = 298.15 K, when we adopt an enthalpy per mol in the X, the solid phase of h_{X,0}^{pS}, the degree of supersaturation Δμ is given by
All terms except the enthalpies depend on the growth temperature. The third, fourth, and fifth terms are the differences between the chemical potential and enthalpy of the pure gallium droplet, arsenic droplet, and GaAs crystal. Their values are tabulated in the paper of Ansara et al. [14]. The first and second terms are concerned with interactions in the alloy. They are denoted by the atomic concentration c_{ X } and the interaction parameter for gallium and arsenic ω_{Ga,As} as follows:
where V_{Ga(As)}, χ_{Ga(As)}, and δ_{Ga(As)} are the molar volume in the liquid phase, Pauling electronegativity, and Hildebrand solubility parameter, respectively. From these equations, we can obtain the relational expression for the supersaturation dependence on growth temperature and arsenic concentration in the gallium droplet.
Figure 3 shows the supersaturation as a function of the arsenic concentration in the gallium droplet at 580°C. In this calculation, the supersaturation per atom Δμ/N_{A}, where N_{A} is Avogadro's number, is adopted. In Figure 3, the equilibrium arsenic concentration is approximately 0.06% at Δμ/N_{A} = 0. This value is in fair agreement with the arsenic concentration of approximately 0.1% in the gallium droplet at 580°C [9]. Therefore, when the arsenic concentration is positive because of an oversupply of arsenic atoms greater than 0.06%, GaAs VLS growth may occur.
From supersaturation Δμ/N_{A}, we can obtain the probability of twinboundary formation. When arsenic adatoms on the gallium droplet surface diffuse to the threephase boundary of vapor, liquid, and solid phases, we assumed that the GaAs crystal nucleus would be formed into a rhombus shape with side length r at the vertex of the nanowire top surface. Figure 4 shows the growth model. When the ZB and WZ structures form, the amount of Gibbs free energy change in this growth system is given by
where Ω is the solid volume of the GaAs pair (4.51 × 10^{−29} m^{3}), h is the height of crystal nucleus, β is the contact angle of the droplet, γ_{LV} (γ_{SN}) is the interface energy between the droplet and the vapor phase (between the crystal nucleus and the NW top surface), and γ_{SL,ZB(WZ)} and γ_{SV,ZB(WZ)} are the interface energies of the GaAs ZB (WZ) crystal nucleus top and side surfaces. Obviously, h is the (111) GaAs lattice spacing, which is 0.32639 nm. γ_{SV,ZB} and γ_{SV,WZ} are the surface energies of (110)GaAs and (1120)GaAs, which are 0.62 and 0.54 J/m^{2}, respectively [15]. The interface energy γ_{LV} depends on temperature, and we use the following relation: γ_{LV} = 0.708 − 0.66 × 10^{− 4} × (T − 303) (J/m^{2}) [16]. If the crystal nucleus is ZB, γ_{SN} = 0. On the other hand, in the case of the WZ crystal nucleus, γ_{SN} is 0.023 (J/m^{2}), which is half of the GaAs stacking fault energy [17]. We assumed that the contact angle β is 45° [18]. Since the NW side surface has a certain asperity, we adopted the parameter ζ (0 < ζ < 1). Therefore, the maximum ΔG_{ZB(WZ)} * and the critical nucleus of Equation 4 for r are obtained as follows:
At steady state, it is known that the probability of crystal nucleation J_{ZB(WZ)} is proportional to the S_{ZB(WZ)} region, where a crystal nucleus can form, and Zeldovich factor Z_{ZB(WZ)}[19]. The relation is given by
where k is the Boltzmann constant, ${S}_{\text{ZB}\left(\text{WZ}\right)}=3\sqrt{3}{\left\{{r}_{\text{ZB}\left(\text{WZ}\right)}^{*}\right\}}^{2}$, and the Zeldovich factor is
Therefore, the probability of twincrystal nucleation P is
This equation indicates that the probability depends on the growth temperature and arsenic concentration. When we assume that ζ is 0.8 at 580°C, the relation between the probability P and arsenic concentration c_{As} is shown in Figure 5. As the droplet arsenic concentration increases with increasing arsenic flux, the probability of twincrystal nucleation decreases. This phenomenon agrees with the experimental results.
By using the probability of twincrystal nucleation in Figure 2, we can calculate the arsenic concentration and supersaturation per atom. When the arsenic fluxes are 7.0 × 10^{−6} and 1.9 × 10^{−5} Torr, the arsenic concentrations are 0.11% and 0.38%, and the supersaturations per atom are 53 and 143 meV, respectively. These supersaturation values are smaller than those of Aucatalyzed GaAs NWs (230 to 1,570 meV) [20]. This difference might be due to the difference in the side facet surface. Glas et al. adopted the {111} and {1100} facets in the ZB and WZ structures in their calculations, respectively. In addition, there might be a difference in the diffusion length between gold and gallium droplets. From the obtained arsenic concentration, we estimate the critical nucleus. When the arsenic fluxes are 7.0 × 10^{−6} and 1.9 × 10^{−5} Torr, the critical nuclei of ZB (WZ) are 1.1 (0.3) and 0.4 (0.1) nm, respectively. This means that the increase of arsenic flux decreases the critical nucleus and increases the growth rate. In the case of high arsenic flux, the size difference between critical ZB and WZ nuclei is small compared with the case of low arsenic flux. This means that the ZB structure appears easily as the arsenic flux increases. Therefore, we could improve the comprehension of the growth mechanism in the selfcatalyzed GaAs NWs. This comprehension might support a technological feasibility of a novel device like twinplane 1D superlattices [21].
Conclusions
Selfcatalyzed GaAs NWs were grown on a (111)Si substrate by MBEVLS method under various arsenic fluxes. From the TEM observations, we found that the segment size between the twin boundaries depends on the arsenic flux. In order to understand this phenomenon, we attempted to calculate the degree of supersaturation and estimate the probability of twinboundary formation. When the supersaturation increased with increasing arsenic flux, the size difference between the critical ZB and WZ nuclei decreased. As a result, the ZB structures were easier to obtain as the arsenic flux increased. This qualitatively explained the experimental results and the high probability of the incorporation of twin boundaries.
Authors’ information
MY is an associate professor, JP is a graduate school student, and HA is a professor at the Department of Electrical Engineering and Computer Science, Nagoya University.
Abbreviations
 GaAs:

gallium arsenide
 MBE:

molecular beam epitaxy
 NW:

nanowires
 Si:

silicon
 TEM:

transmission electron microscope
 VLS:

vaporliquidsolid
 WZ:

wurtzite
 ZB:

zinc blende.
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Acknowledgments
This study was partly supported by the GrantinAid for Scientific Research (KAKENHI) No. 23510148 of the Japan Society for the Promotion of Science (JSPS) and ‘Tatematsu Foundation.’ Nagoya University Venture Business Laboratory is gratefully acknowledged for allowing the use of facilities.
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Authors’ contributions
MY conceived, designed, and coordinated the study and drafted the manuscript. JP carried out the experiments and calculations and helped draft the manuscript. HA participated in the coordination of the study. All authors read and approved the final manuscript.
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Yamaguchi, M., Paek, J. & Amano, H. Probability of twin formation on selfcatalyzed GaAs nanowires on Si substrate. Nanoscale Res Lett 7, 558 (2012). https://doi.org/10.1186/1556276X7558
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Keywords
 GaAs nanowire
 Molecular beam epitaxy
 Vaporliquidsolid
 Twin boundary
 Supersaturation
 Wurtzite
 zinc blende