Growth mechanisms and crystallographic structure of InP nanowires ...
JOURNAL OF APPLIED PHYSICS 104, 044313 共2008兲
Growth mechanisms and crystallographic structure of InP nanowires on lattice-mismatched substrates Michael Moewe,1,a兲 Linus C. Chuang,1 Vladimir G. Dubrovskii,2,3 and Connie Chang-Hasnain1 1
Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, California 94720, USA 2 St. Petersburg Physical Technological Centre of the Russian Academy of Sciences for Research and Education, Khlopina 8/3, St. Petersburg 195220, Russia 3 Ioffe Physical Technical Institute of the Russian Academy of Sciences, Politekhnicheskaya 26, 194021, St. Petersburg, Russia
共Received 16 February 2008; accepted 12 June 2008; published online 27 August 2008兲 We present a growth model that predicts the growth phase and mechanism of InP nanowires 共NWs兲 and the experimental verifications of the model. The NWs were grown on lattice-mismatched GaAs substrates using metal-organic chemical vapor deposition via Au nanodrop-assisted vapor-liquid-solid growth. Nanodrops with larger diameters are shown to grow longer NWs because growth is governed mainly by direct precursor impingement on the nanodrop surface. The theoretical and experimental results also show that growth phase is dependent on NW diameter. We show that InP NWs with a diameter less than a certain value exhibit coherent growth of a single crystalline wurtzite 共WZ兲 phase, whereas larger diameter InP NWs often contain sequences of WZ and zincblende phases and stacking faults. These findings allow one to achieve coherent NW growth and WZ phases free from twinning if the NW diameter is below certain material-dependent critical diameters. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2968345兴 Nanowires 共NWs兲 grown by the vapor-liquid-solid 共VLS兲 mechanism are promising for monolithic integration of semiconductor materials with different lattice constants.1–4 It has been recently shown theoretically and experimentally that dislocation-free freestanding III–V NWs can be grown epitaxially on lattice-mismatched substrates if the wire diameter is below a certain critical value.2,3 The experimentally measured critical diameter De共0兲 equals 26 nm for InAs/Si 共lattice mismatch 0 = 11.6%兲, 36 nm for InP/ Si共0 = 8.1%兲, and 96 nm for InP/ GaAs共0 = 4.1%兲 systems.2 For the controlled production of such NWs, it is important to understand their growth properties and also the factors influencing the crystallographic structure of NWs. Many recent studies5–8 demonstrated that III–V NWs often adopt the hexagonal wurtzite 共WZ兲 phase in contrast to their bulk form with cubic zincblende 共ZB兲 phase. This phenomenon has been observed for most ZB compounds and epitaxial techniques, both for Au-assisted5–7 and selective area8 growth. Despite the surprising prevalence of WZ phase, NWs often contain sequences of different phases and stacking faults. This clearly affects material properties, so the control over the phase purity is now considered as one of the main challenges in III–V NW fabrication. This work addresses two issues of Auassisted low-pressure metal-organic chemical vapor deposition 共MOCVD兲 of InP NWs on the GaAs共111兲B substrates: 共1兲 study of growth mechanisms and 共2兲 investigation of crystallographic structure depending on the NW diameter. We show that the VLS growth of InP NWs is controlled by the direct impingement onto the drop surface and is strongly influenced by the Gibbs–Thomson effect. It will be demona兲
Electronic mail: [email protected].
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strated that the NW structure is diameter dependent so that 20 nm wide NWs are predominantly WZ and 60 nm NWs exhibit twinning between WZ and ZB structure. We will also present theoretical models qualitatively explaining the experimentally observed facts. InP NWs are grown using colloidal Au nanodrops as catalysts in a MOCVD reactor under the VLS growth mode. The mole fractions of the group V 共tertiarybutylphosphine兲 and group III 共trimethylindium兲 sources were 1.91⫻ 10−5 and 1.17⫻ 10−3, respectively, in a 12 l/min hydrogen carrier gas flow yielding a V/III ratio of 61. This ratio was found to be within a range that yields straight NWs with uniform diameter from base to top.4 The growth temperature was 470 ° C and the growth time was 3 min. The size of Au drops ranges from 10 to 160 nm. High-resolution scanning electron microscopy 共SEM兲 images were used to determine the critical diameter De as 96 nm for the epitaxial growth of InP NWs on the GaAs共111兲B substrate.3 In order to understand the growth mechanisms, we first study the length-diameter L共D兲 dependences of our NWs. The NW growth can be controlled either by the direct impingement of material onto the drop9,10 or by the adatom diffusion to the top.11,12 The first case is characterized by an increasing and the second by a decreasing dependence of the NW growth rate dL / dt on its diameter D. From the analysis of SEM images of different samples we construct experimental L共D兲 curves. The corresponding L共D兲 dependence is presented in Fig. 1 with a typical corresponding SEM image shown in the inset. We have found that all measured L共D兲 dependences are increasing. Such behavior is usually explained by the Gibbs–Thomson effect of elevation of chemical potential in a NW with a curved lateral surface.9,10 The
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FIG. 1. 共Color online兲 Experimental 共points兲 and theoretical 共line兲 lengthdiameter dependences of InP NWs. Inset is 20°-tilt SEM image of InP NWs grown by MOCVD on a GaAs共111兲B substrate. Growth temperature T = 470 ° C, group V mole fraction= 1.91⫻ 10−5, group III mole fraction = 1.17⫻ 10−3, and V/III ratio= 61 with the growth time t = 3 min.
adatom diffusion to the top is rather small, which is opposite to the case of molecular beam epitaxy 共MBE兲 共Ref. 11兲 and can be explained either by a slow rate of chemical reaction at the surface or by a low surface diffusivity by the growth species. According to the Givargizov–Chernov model,9 the addict to the bulk chemical potential in the NW of constant l / D. Here, ⍀ is the volume per diameter D equals 4⍀␥S−V l is the surface energy of the III–V pair in the crystal and ␥S−V lateral solid-vapor interface 共Fig. 3兲. The effective difference in chemical potentials in the vapor and the solid phase ⌬eff, driving the NW formation, is therefore lower than at the surface of bulk crystal 共D → ⬁兲 ⌬ = V − S: ⌬eff = ⌬ l − 4⍀␥S−V / D. The second assumption of the model is that the growth rate is proportional to 共⌬eff / kBT兲2, where T is the surface temperature and kB is the Boltzmann constant. Writing the growth rate in the form dL / dt = K共⌬ / kBT l / DkBT兲2, with K being the crystallization coeffi− 4⍀␥S−V cient and integrating, we arrive at
冉
L = L0 1 −
Dmin D
冊
2
.
共1兲
Here, L0 = Kt共⌬ / kBT兲2 is the maximum length of infinitely l / ⌬ is thick wires, t is the growth time, and Dmin = 4⍀␥S−V the minimum diameter below which the NWs would not grow. In Eq. 共1兲, we neglect the growth of a wetting layer, which is rather thin in all our samples. Comparison of theoretical and experimental L共D兲 dependences enables us to estimate several important characteristics of NW formation. From fitting the L共D兲 curve in Fig. 1 by Eq. 共1兲 we obtain L0 = 2070 nm and Dmin = 8 nm. For further estimates we use the parameters of InP from Refs. 13 and 14: the elementary volume per III–V pair ⍀ = 0.0506 nm3 and the 共110兲 surface energy of ZB phase l ␥ZB = 1.3 J / m2. According to the foregoing analysis, the formation of a WZ NW could lead up to 25% decrease in surl = 0.975 J / m2兲. Taking into account the face energy 共␥WZ possible polytypism between ZB and WZ structures, we obtain reasonable estimates for ⌬ / kBT = 2.45– 3.27 and K = 1.08– 1.92 nm/ s. The values of ⌬ = 157– 209 meV and
FIG. 2. TEM images of InP NWs show 共a兲 a NW with 12 nm diameter and a pure WZ phase as indicated by the FFT to the right and 共b兲 a larger 45 nm diameter NW. Twinning defects are clearly seen in this image. The electron diffraction pattern also shows the hybrid of typical ZB and WZ patterns.
the vapor supersaturation ⌽ = exp共⌬ / kBT兲 − 1 = 11.6– 26.4 are comparable to the case of MBE growth.11 Experimental evidence5–8 strongly suggests that the formation of WZ phase in ZB III–V NWs is somehow related to their small radius, resulting in a larger contribution of lateral surfaces to the total free energy of fully formed wires15–17 or monolayer islands mediating the growth of the wire top.18 We now present the results of transmission electron microscopy 共TEM兲 study of crystallographic structure in InP NWs of different diameters. Figure 2共a兲 shows a typical InP NW with a diameter of 12 nm imaged on the 具11– 20典 zone axis. For the smaller NWs, only the WZ phase is typically observed without twinning defects. This is shown in the fast Fourier transform 共FFT兲 on the right of the image, which shows a clear WZ pattern. For NWs larger than ⬃40 nm in diameter, the twinning between WZ and ZB phases becomes quite frequent, which can clearly be seen in Fig. 2共b兲. The diffraction pattern also shows a spread between diffraction peaks seen in both WZ and ZB phases, as indicated by the circled section of the pattern. Our model of NW structure is the following. Consider a single cylindrical NW growing perpendicular to the 共111兲 substrate. During the growth, N = 共D2L / 4⍀兲 InP pairs are deposited onto the substrate to form the NW of diameter D and length L 共Fig. 3兲. Assume that D is below the critical diameter De共0兲 for coherent NW growth.2,3 Neglecting a small variation in the drop size and shape which may occur
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FIG. 4. 共Color online兲 Side view of WZ and ZB structures. Star symbols indicate the dangling bonds on the lateral facets. The number of dangling bonds j equals 1 per layer in the WZ and 4/3 per layer in the ZB phase. FIG. 3. 共Color online兲 Schematics of NW growth: V is the vapor chemical l potential, S is the chemical potential in NW, and ␥S−V is the lateral surface l energy at the solid-vapor interface. The values of S and ␥S−V depend on the crystallographic phase in the NW.
during the growth, the difference in free energies in final 共the NW兲 and initial 共the drop on a bare substrate兲 states of the system is given by ⌬F = −
20 D 2L E D 2L l ⌬ + DL␥S−V . + 4⍀ 1 − 4 共1 + AL/D兲 共2兲
The first term in the right hand side stands for the change in volume free energy due to the vapor-solid phase transition. The second term gives the energy required to form the lateral surface. The last term describes the strain-induced elastic contribution to the free energy2 due to the lattice mismatch: E is the Young modulus and is the Poisson ratio of NW material. Coefficient A depends on . For further analysis it is convenient to rewrite Eq. 共2兲 in terms of the normalized formation energy per III–V pair ⌬f ⬅ ⌬F / N. Comparison of different terms in Eq. 共2兲 shows that the strain-induced contribution to ⌬f scales as D / L and cancels at L / D → ⬁. While the elastic energy determines the value of De共0兲,2,3 the stress relaxation involves a layer of thickness of a few D at most and has little effect on the formation of longer NWs. For the analysis of structural stability in the first approximation, it is therefore sufficient to consider the formation energies without the strain-induced term ⌬f k = − ⌬k +
4⍀␥lk . D
共3兲
Equation 共3兲 should be treated for ZB and WZ NWs 共k = ZB or WZ兲, with ⌬ZB = V − ZB ⬅ ⌬ being the difference in chemical potentials in the vapor and in the ZB phase, ⌬WZ = V − WZ the corresponding value for the WZ phase, 共110兲 l l ␥ZB = ␥ZB the lateral surface energy of ZB, and ␥WZ the lateral surface energy of WZ NWs. WZ structure should be predominant when ⌬f WZ ⬍ ⌬f ZB. On the other hand, the NW formation is possible only at ⌬f WZ ⬍ 0. At positive ⌬f WZ, the vapor supersaturation is insufficient to overcome the Gibbs– Thomson effect. These two conditions give the range of diameters Dmin ⬍ D ⬍ Dc, where the WZ phase should be prevalent. Characteristic diameters Dmin and Dc are given by
Dmin =
4 D 0, 共⌬/ − 1兲
Dc = 4共1 − 兲D0 ,
共4兲
共110兲 l l with D0 = ⍀␥ZB / , = WZ − ZB, and = ␥WZ / ␥ZB . The value of Dmin is equivalent to the Givargizov–Chernov expression in Eq. 共1兲. At D ⬍ Dmin the formation of NWs should be considered as thermodynamically forbidden. The parameter Dc therefore determines the critical diameter for the ZB to WZ structural transition. The parameter equals the difference in cohesive energies for the formation of bulk WZ and ZB crystals. For all ZB materials, ⬎ 0 since the ZB phase is stable in the bulk form. The surface energy coefficient equals the ratio of WZ to ZB lateral surface energy. According to Eq. 共4兲, the structural transition may occur only at ⬍ 1 when the surface energy of WZ structure is lower than that of ZB. The characteristic diameter D0 can 共110兲 13–18 be estimated from the available data on ⍀, , and ␥ZB . Surface dangling bonds on the NW lateral facets have been previously argued to have crucial effect to determine the structural stability.15–17 In order to estimate the coefficient , we simply count the number of dangling bonds j on the lateral facets of ZB and WZ NWs perpendicular to the 共111兲 surface.17 This procedure is illustrated in Fig. 4. It is seen that j equals 1 bond per III–V layer in the WZ and 4/3 bonds per III–V layer in the ZB phase. From these considerations we arrive at = 3 / 4, the formation of WZ phase thus leads to approximately 25% reduction in the surface energy. From the second Eq. 共4兲 we get the critical diameter for ZB to WZ transition Dc = D0. Our analysis shows that the diameter Dmin can be made arbitrarily small by increasing the vapor supersaturation ⌬. The critical diameter Dc does not depend on the lattice mismatch, whereas the critical diameter for the epitaxial growth of coherent NWs De is determined mainly by 0. In Table I we summarize the estimates for Dc obtained from Eq. 共4兲 at = 3 / 4 for different material systems. We use the data in Ref. 14 for the 共110兲 surface energies of ZB crystals. The values of are taken from Ref. 15. We also present the experimental values of D for the different material-substrate combinations.3 Comparison of Dc and De shows that InAs and InP NWs on Si substrates can be grown epitaxially only in the WZ phase 共De ⬍ Dc兲. The situation is different for the epitaxial InP NWs on the GaAs substrate 共De ⬎ Dc兲, which should form predominantly in the WZ phase below 60 nm and in the ZB phase for larger diameters between 60 and 96 nm. Because the difference in the formation energies ⌬f WZ − ⌬f ZB near Dc is much smaller than
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Material system InAs/Si InP/Si InP/GaAs
E0 共%兲
De 共nm兲
⍀ 共nm3兲
␥共110兲 ZB 共J / m2兲
共meV/pair兲
Dc 共nm兲
11.6 8.1 4.1
26 36 96
0.0567 0.0506 0.0506
1.0 1.3 1.3
10.6 6.8 6.8
34 60 60
kBT = 64 meV, thermal fluctuations will create rotational twins between WZ and ZB structures. The theoretical prediction is qualitatively in excellent agreement with the experimental data. The smaller diameter NWs exhibit single crystalline WZ phase without twinning, whereas the larger NWs have mixed WZ and ZB portions switching within a few to 20 nm range. The critical transition diameter experimentally observed was somewhat smaller than the theoretical value due to the possibility of inaccuracies and some simplifications of the analysis. To conclude, due to the interplay of strain-induced and surface energy effects on the growth performance, the technologically important fabrication of III–V NWs enables one to combine two major advantages: 共1兲 coherent growth and 共2兲 formation of pure WZ phase NWs without stacking faults. This requires two conditions: 共1兲 high enough supersaturation of the vapor phase and 共2兲 small enough diameter of NW. The proposed explanation of WZ structure of small NWs is not specific for III–V materials and would apply for other cubic materials, in particular for Si. This is important because WZ Si is the semiconductor with the direct band gap transition at the G point.19 Recently, WZ Si was observed experimentally in CVD grown NWs with the diameters as large as 100 nm.20 Moreover, ZB to WZ phase transformation can be observed in other nanostructures with low energy WZ facets when the surface energy gain outweighs the difference in bulk cohesive energies. This work was partially supported by the Russian Federal Agency for Science and Innovation, SANDIE Network of Excellence of European Commission, different grants of the Russian Foundation for Basic Research 共RFBR兲, Defense Advanced Research Projects Agency 共DARPA兲 Grant No. HR0011-04-1-0040 共CONSRT兲, and HP-CITRIS grants. The
authors also acknowledge the support of the National Center for Electron Microscopy, Lawrence Berkeley Laboratory, which is supported by the U.S. Department of Energy and fellowship support from the NSF-IGERT Program. R. S. Wagner and W. C. Ellis, Appl. Phys. Lett. 4, 89 共1964兲. L. C. Chuang, M. Moewe, S. Crankshaw, C. Chase, N. P. Kobayashi, and C. Chang-Hasnain, Appl. Phys. Lett. 90, 043115 共2007兲. 3 F. Glas, Phys. Rev. B 74, 121302共R兲 共2006兲. 4 L. C. Chuang, M. Moewe, S. Crankshaw, and C. Chang-Hasnain, Appl. Phys. Lett. 92, 013121 共2008兲. 5 J. C. Harmand, G. Patriarche, N. Péré-Laperne, M.-N. Mérat-Combes, L. Travers, and F. Glas, Appl. Phys. Lett. 87, 203101 共2005兲. 6 A. I. Persson, M. W. Larsson, S. Stenstrom, B. J. Ohlsson, L. Samuelson, and L. R. Wallenberg, Nat. Mater. 3, 678 共2004兲. 7 I. P. Soshnikov, G. E. Cirlin, A. A. Tonkikh, V. V. Nevedomskii, Yu. B. Samsonenko, and V. M. Ustinov, Phys. Solid State 48, 1440 共2007兲. 8 P. Mohan, J. Motohisa, and T. Fukui, Nanotechnology 16, 2903 共2005兲. 9 E. I. Givargizov, Highly Anisotropic Crystals 共Springer, Berlin, 1987兲. 10 V. G. Dubrovskii and N. V. Sibirev, Phys. Rev. E 70, 031604 共2004兲. 11 V. G. Dubrovskii, G. E. Cirlin, I. P. Soshnikov, A. A. Tonkikh, N. V. Sibirev, Yu. B. Samsonenko, and V. M. Ustinov, Phys. Rev. B 71, 205325 共2005兲. 12 W. Seifert, M. Borgstrom, K. Deppert, K. A. Dick, J. Johansson, M. W. Larsson, T. Martensson, N. Skold, C. P. T. Svensson, B. A. Wacaser, L. R. Wallenberg, and L. Samuelson, J. Cryst. Growth 272, 211 共2004兲. 13 Group IV elements, IV-IV and III-V compounds, Landolt-Börnstein, New Series, Group III, Pt. A 共Springer, Berlin, 2006兲. 14 J. W. Cahn and R. E. Hanneman, Surf. Sci. 1, 38 共1964兲. 15 T. Akiyama, K. Sano, K. Nakamura, and T. Ito, Jpn. J. Appl. Phys., Part 2 45, L275 共2006兲. 16 T. Akiyama, K. Nakamura, and T. Ito, Phys. Rev. B 73, 235308 共2006兲. 17 V. G. Dubrovskii and N. V. Sibirev, Phys. Rev. B 77, 035414 共2008兲. 18 F. Glas, J. C. Harmand, and J. Patriarche, Phys. Rev. Lett. 99, 146101 共2007兲. 19 J. M. Besson, E. H. Mokhtari, J. Gonzalez, and G. Weill, Phys. Rev. Lett. 59, 473 共1987兲. 20 A. Fontcuberta i Morral, J. Arbiol, J. D. Prades, A. Cirera, and J. R. Morante, Adv. Mater. 共Weinheim, Ger.兲 19, 1347 共2007兲. 1 2
Growth mechanisms and crystallographic structure of InP nanowires on lattice-mismatched substrates Michael Moewe,1,a兲 Linus C. Chuang,1 Vladimir G. Dubrovskii,2,3 and Connie Chang-Hasnain1 1
Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, California 94720, USA 2 St. Petersburg Physical Technological Centre of the Russian Academy of Sciences for Research and Education, Khlopina 8/3, St. Petersburg 195220, Russia 3 Ioffe Physical Technical Institute of the Russian Academy of Sciences, Politekhnicheskaya 26, 194021, St. Petersburg, Russia
共Received 16 February 2008; accepted 12 June 2008; published online 27 August 2008兲 We present a growth model that predicts the growth phase and mechanism of InP nanowires 共NWs兲 and the experimental verifications of the model. The NWs were grown on lattice-mismatched GaAs substrates using metal-organic chemical vapor deposition via Au nanodrop-assisted vapor-liquid-solid growth. Nanodrops with larger diameters are shown to grow longer NWs because growth is governed mainly by direct precursor impingement on the nanodrop surface. The theoretical and experimental results also show that growth phase is dependent on NW diameter. We show that InP NWs with a diameter less than a certain value exhibit coherent growth of a single crystalline wurtzite 共WZ兲 phase, whereas larger diameter InP NWs often contain sequences of WZ and zincblende phases and stacking faults. These findings allow one to achieve coherent NW growth and WZ phases free from twinning if the NW diameter is below certain material-dependent critical diameters. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2968345兴 Nanowires 共NWs兲 grown by the vapor-liquid-solid 共VLS兲 mechanism are promising for monolithic integration of semiconductor materials with different lattice constants.1–4 It has been recently shown theoretically and experimentally that dislocation-free freestanding III–V NWs can be grown epitaxially on lattice-mismatched substrates if the wire diameter is below a certain critical value.2,3 The experimentally measured critical diameter De共0兲 equals 26 nm for InAs/Si 共lattice mismatch 0 = 11.6%兲, 36 nm for InP/ Si共0 = 8.1%兲, and 96 nm for InP/ GaAs共0 = 4.1%兲 systems.2 For the controlled production of such NWs, it is important to understand their growth properties and also the factors influencing the crystallographic structure of NWs. Many recent studies5–8 demonstrated that III–V NWs often adopt the hexagonal wurtzite 共WZ兲 phase in contrast to their bulk form with cubic zincblende 共ZB兲 phase. This phenomenon has been observed for most ZB compounds and epitaxial techniques, both for Au-assisted5–7 and selective area8 growth. Despite the surprising prevalence of WZ phase, NWs often contain sequences of different phases and stacking faults. This clearly affects material properties, so the control over the phase purity is now considered as one of the main challenges in III–V NW fabrication. This work addresses two issues of Auassisted low-pressure metal-organic chemical vapor deposition 共MOCVD兲 of InP NWs on the GaAs共111兲B substrates: 共1兲 study of growth mechanisms and 共2兲 investigation of crystallographic structure depending on the NW diameter. We show that the VLS growth of InP NWs is controlled by the direct impingement onto the drop surface and is strongly influenced by the Gibbs–Thomson effect. It will be demona兲
Electronic mail: [email protected].
0021-8979/2008/104共4兲/044313/4/$23.00
strated that the NW structure is diameter dependent so that 20 nm wide NWs are predominantly WZ and 60 nm NWs exhibit twinning between WZ and ZB structure. We will also present theoretical models qualitatively explaining the experimentally observed facts. InP NWs are grown using colloidal Au nanodrops as catalysts in a MOCVD reactor under the VLS growth mode. The mole fractions of the group V 共tertiarybutylphosphine兲 and group III 共trimethylindium兲 sources were 1.91⫻ 10−5 and 1.17⫻ 10−3, respectively, in a 12 l/min hydrogen carrier gas flow yielding a V/III ratio of 61. This ratio was found to be within a range that yields straight NWs with uniform diameter from base to top.4 The growth temperature was 470 ° C and the growth time was 3 min. The size of Au drops ranges from 10 to 160 nm. High-resolution scanning electron microscopy 共SEM兲 images were used to determine the critical diameter De as 96 nm for the epitaxial growth of InP NWs on the GaAs共111兲B substrate.3 In order to understand the growth mechanisms, we first study the length-diameter L共D兲 dependences of our NWs. The NW growth can be controlled either by the direct impingement of material onto the drop9,10 or by the adatom diffusion to the top.11,12 The first case is characterized by an increasing and the second by a decreasing dependence of the NW growth rate dL / dt on its diameter D. From the analysis of SEM images of different samples we construct experimental L共D兲 curves. The corresponding L共D兲 dependence is presented in Fig. 1 with a typical corresponding SEM image shown in the inset. We have found that all measured L共D兲 dependences are increasing. Such behavior is usually explained by the Gibbs–Thomson effect of elevation of chemical potential in a NW with a curved lateral surface.9,10 The
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FIG. 1. 共Color online兲 Experimental 共points兲 and theoretical 共line兲 lengthdiameter dependences of InP NWs. Inset is 20°-tilt SEM image of InP NWs grown by MOCVD on a GaAs共111兲B substrate. Growth temperature T = 470 ° C, group V mole fraction= 1.91⫻ 10−5, group III mole fraction = 1.17⫻ 10−3, and V/III ratio= 61 with the growth time t = 3 min.
adatom diffusion to the top is rather small, which is opposite to the case of molecular beam epitaxy 共MBE兲 共Ref. 11兲 and can be explained either by a slow rate of chemical reaction at the surface or by a low surface diffusivity by the growth species. According to the Givargizov–Chernov model,9 the addict to the bulk chemical potential in the NW of constant l / D. Here, ⍀ is the volume per diameter D equals 4⍀␥S−V l is the surface energy of the III–V pair in the crystal and ␥S−V lateral solid-vapor interface 共Fig. 3兲. The effective difference in chemical potentials in the vapor and the solid phase ⌬eff, driving the NW formation, is therefore lower than at the surface of bulk crystal 共D → ⬁兲 ⌬ = V − S: ⌬eff = ⌬ l − 4⍀␥S−V / D. The second assumption of the model is that the growth rate is proportional to 共⌬eff / kBT兲2, where T is the surface temperature and kB is the Boltzmann constant. Writing the growth rate in the form dL / dt = K共⌬ / kBT l / DkBT兲2, with K being the crystallization coeffi− 4⍀␥S−V cient and integrating, we arrive at
冉
L = L0 1 −
Dmin D
冊
2
.
共1兲
Here, L0 = Kt共⌬ / kBT兲2 is the maximum length of infinitely l / ⌬ is thick wires, t is the growth time, and Dmin = 4⍀␥S−V the minimum diameter below which the NWs would not grow. In Eq. 共1兲, we neglect the growth of a wetting layer, which is rather thin in all our samples. Comparison of theoretical and experimental L共D兲 dependences enables us to estimate several important characteristics of NW formation. From fitting the L共D兲 curve in Fig. 1 by Eq. 共1兲 we obtain L0 = 2070 nm and Dmin = 8 nm. For further estimates we use the parameters of InP from Refs. 13 and 14: the elementary volume per III–V pair ⍀ = 0.0506 nm3 and the 共110兲 surface energy of ZB phase l ␥ZB = 1.3 J / m2. According to the foregoing analysis, the formation of a WZ NW could lead up to 25% decrease in surl = 0.975 J / m2兲. Taking into account the face energy 共␥WZ possible polytypism between ZB and WZ structures, we obtain reasonable estimates for ⌬ / kBT = 2.45– 3.27 and K = 1.08– 1.92 nm/ s. The values of ⌬ = 157– 209 meV and
FIG. 2. TEM images of InP NWs show 共a兲 a NW with 12 nm diameter and a pure WZ phase as indicated by the FFT to the right and 共b兲 a larger 45 nm diameter NW. Twinning defects are clearly seen in this image. The electron diffraction pattern also shows the hybrid of typical ZB and WZ patterns.
the vapor supersaturation ⌽ = exp共⌬ / kBT兲 − 1 = 11.6– 26.4 are comparable to the case of MBE growth.11 Experimental evidence5–8 strongly suggests that the formation of WZ phase in ZB III–V NWs is somehow related to their small radius, resulting in a larger contribution of lateral surfaces to the total free energy of fully formed wires15–17 or monolayer islands mediating the growth of the wire top.18 We now present the results of transmission electron microscopy 共TEM兲 study of crystallographic structure in InP NWs of different diameters. Figure 2共a兲 shows a typical InP NW with a diameter of 12 nm imaged on the 具11– 20典 zone axis. For the smaller NWs, only the WZ phase is typically observed without twinning defects. This is shown in the fast Fourier transform 共FFT兲 on the right of the image, which shows a clear WZ pattern. For NWs larger than ⬃40 nm in diameter, the twinning between WZ and ZB phases becomes quite frequent, which can clearly be seen in Fig. 2共b兲. The diffraction pattern also shows a spread between diffraction peaks seen in both WZ and ZB phases, as indicated by the circled section of the pattern. Our model of NW structure is the following. Consider a single cylindrical NW growing perpendicular to the 共111兲 substrate. During the growth, N = 共D2L / 4⍀兲 InP pairs are deposited onto the substrate to form the NW of diameter D and length L 共Fig. 3兲. Assume that D is below the critical diameter De共0兲 for coherent NW growth.2,3 Neglecting a small variation in the drop size and shape which may occur
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FIG. 4. 共Color online兲 Side view of WZ and ZB structures. Star symbols indicate the dangling bonds on the lateral facets. The number of dangling bonds j equals 1 per layer in the WZ and 4/3 per layer in the ZB phase. FIG. 3. 共Color online兲 Schematics of NW growth: V is the vapor chemical l potential, S is the chemical potential in NW, and ␥S−V is the lateral surface l energy at the solid-vapor interface. The values of S and ␥S−V depend on the crystallographic phase in the NW.
during the growth, the difference in free energies in final 共the NW兲 and initial 共the drop on a bare substrate兲 states of the system is given by ⌬F = −
20 D 2L E D 2L l ⌬ + DL␥S−V . + 4⍀ 1 − 4 共1 + AL/D兲 共2兲
The first term in the right hand side stands for the change in volume free energy due to the vapor-solid phase transition. The second term gives the energy required to form the lateral surface. The last term describes the strain-induced elastic contribution to the free energy2 due to the lattice mismatch: E is the Young modulus and is the Poisson ratio of NW material. Coefficient A depends on . For further analysis it is convenient to rewrite Eq. 共2兲 in terms of the normalized formation energy per III–V pair ⌬f ⬅ ⌬F / N. Comparison of different terms in Eq. 共2兲 shows that the strain-induced contribution to ⌬f scales as D / L and cancels at L / D → ⬁. While the elastic energy determines the value of De共0兲,2,3 the stress relaxation involves a layer of thickness of a few D at most and has little effect on the formation of longer NWs. For the analysis of structural stability in the first approximation, it is therefore sufficient to consider the formation energies without the strain-induced term ⌬f k = − ⌬k +
4⍀␥lk . D
共3兲
Equation 共3兲 should be treated for ZB and WZ NWs 共k = ZB or WZ兲, with ⌬ZB = V − ZB ⬅ ⌬ being the difference in chemical potentials in the vapor and in the ZB phase, ⌬WZ = V − WZ the corresponding value for the WZ phase, 共110兲 l l ␥ZB = ␥ZB the lateral surface energy of ZB, and ␥WZ the lateral surface energy of WZ NWs. WZ structure should be predominant when ⌬f WZ ⬍ ⌬f ZB. On the other hand, the NW formation is possible only at ⌬f WZ ⬍ 0. At positive ⌬f WZ, the vapor supersaturation is insufficient to overcome the Gibbs– Thomson effect. These two conditions give the range of diameters Dmin ⬍ D ⬍ Dc, where the WZ phase should be prevalent. Characteristic diameters Dmin and Dc are given by
Dmin =
4 D 0, 共⌬/ − 1兲
Dc = 4共1 − 兲D0 ,
共4兲
共110兲 l l with D0 = ⍀␥ZB / , = WZ − ZB, and = ␥WZ / ␥ZB . The value of Dmin is equivalent to the Givargizov–Chernov expression in Eq. 共1兲. At D ⬍ Dmin the formation of NWs should be considered as thermodynamically forbidden. The parameter Dc therefore determines the critical diameter for the ZB to WZ structural transition. The parameter equals the difference in cohesive energies for the formation of bulk WZ and ZB crystals. For all ZB materials, ⬎ 0 since the ZB phase is stable in the bulk form. The surface energy coefficient equals the ratio of WZ to ZB lateral surface energy. According to Eq. 共4兲, the structural transition may occur only at ⬍ 1 when the surface energy of WZ structure is lower than that of ZB. The characteristic diameter D0 can 共110兲 13–18 be estimated from the available data on ⍀, , and ␥ZB . Surface dangling bonds on the NW lateral facets have been previously argued to have crucial effect to determine the structural stability.15–17 In order to estimate the coefficient , we simply count the number of dangling bonds j on the lateral facets of ZB and WZ NWs perpendicular to the 共111兲 surface.17 This procedure is illustrated in Fig. 4. It is seen that j equals 1 bond per III–V layer in the WZ and 4/3 bonds per III–V layer in the ZB phase. From these considerations we arrive at = 3 / 4, the formation of WZ phase thus leads to approximately 25% reduction in the surface energy. From the second Eq. 共4兲 we get the critical diameter for ZB to WZ transition Dc = D0. Our analysis shows that the diameter Dmin can be made arbitrarily small by increasing the vapor supersaturation ⌬. The critical diameter Dc does not depend on the lattice mismatch, whereas the critical diameter for the epitaxial growth of coherent NWs De is determined mainly by 0. In Table I we summarize the estimates for Dc obtained from Eq. 共4兲 at = 3 / 4 for different material systems. We use the data in Ref. 14 for the 共110兲 surface energies of ZB crystals. The values of are taken from Ref. 15. We also present the experimental values of D for the different material-substrate combinations.3 Comparison of Dc and De shows that InAs and InP NWs on Si substrates can be grown epitaxially only in the WZ phase 共De ⬍ Dc兲. The situation is different for the epitaxial InP NWs on the GaAs substrate 共De ⬎ Dc兲, which should form predominantly in the WZ phase below 60 nm and in the ZB phase for larger diameters between 60 and 96 nm. Because the difference in the formation energies ⌬f WZ − ⌬f ZB near Dc is much smaller than
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Moewe et al. TABLE I. Critical diameters for different III–V materials on lattice-mismatched substrates.
Material system InAs/Si InP/Si InP/GaAs
E0 共%兲
De 共nm兲
⍀ 共nm3兲
␥共110兲 ZB 共J / m2兲
共meV/pair兲
Dc 共nm兲
11.6 8.1 4.1
26 36 96
0.0567 0.0506 0.0506
1.0 1.3 1.3
10.6 6.8 6.8
34 60 60
kBT = 64 meV, thermal fluctuations will create rotational twins between WZ and ZB structures. The theoretical prediction is qualitatively in excellent agreement with the experimental data. The smaller diameter NWs exhibit single crystalline WZ phase without twinning, whereas the larger NWs have mixed WZ and ZB portions switching within a few to 20 nm range. The critical transition diameter experimentally observed was somewhat smaller than the theoretical value due to the possibility of inaccuracies and some simplifications of the analysis. To conclude, due to the interplay of strain-induced and surface energy effects on the growth performance, the technologically important fabrication of III–V NWs enables one to combine two major advantages: 共1兲 coherent growth and 共2兲 formation of pure WZ phase NWs without stacking faults. This requires two conditions: 共1兲 high enough supersaturation of the vapor phase and 共2兲 small enough diameter of NW. The proposed explanation of WZ structure of small NWs is not specific for III–V materials and would apply for other cubic materials, in particular for Si. This is important because WZ Si is the semiconductor with the direct band gap transition at the G point.19 Recently, WZ Si was observed experimentally in CVD grown NWs with the diameters as large as 100 nm.20 Moreover, ZB to WZ phase transformation can be observed in other nanostructures with low energy WZ facets when the surface energy gain outweighs the difference in bulk cohesive energies. This work was partially supported by the Russian Federal Agency for Science and Innovation, SANDIE Network of Excellence of European Commission, different grants of the Russian Foundation for Basic Research 共RFBR兲, Defense Advanced Research Projects Agency 共DARPA兲 Grant No. HR0011-04-1-0040 共CONSRT兲, and HP-CITRIS grants. The
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