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Analysis of Morphological Evolution of Crystalline Domains in Nonequilibrium Shape by Using Minimization of Effective Surface Energy Yeonwoo Seo, Sanghwa Lee, Mi Yeon Ju, Donggyu Shin, Hyunkyu Park, and Chinkyo Kim* Department of Physics and Research Institute for Basic Sciences, Kyung Hee University, 1 Hoegi-dong Dongdaemun-gu, Seoul 130-701, Korea ABSTRACT:
Effective surface energy is proposed to analyze the morphological evolution of crystalline domains in nonequilibrium shape. This concept can be used to determine how favorable the formation of a certain crystallographic face is when a crystalline domain is grown even in kinetically limited conditions by incorporating various kinetic effects on the corresponding crystallographic face into the actual surface energy of that face. In this work, it is experimentally demonstrated that this concept, applied to the m-oriented GaN faceted domains grown under kinetically limited conditions on SiO2-patterned m-plane sapphire substrates, can successfully describe the morphological evolution of those GaN faceted domains in nonequilibrium shape. This method can give a new insight into the morphological analysis of crystalline domains in nonequilibrium shape.
’ INTRODUCTION During the growth of crystalline domains, they are either in nonequilibrium or equilibrium shape. The crystalline domain in nonequilibrium shape constantly changes its shape as growth time increases until it reaches an eventual equilibrium shape. Once the eventual equilibrium shape is reached, the domain will further grow only in size with no more change in shape. There are two different theoretical approaches to determine the equilibrium shape of a crystalline domain. If the surface energy of each crystallographic face of a certain domain is known, the Legendre transformation of the Wulff plot can determine the eventual equilibrium shape of that crystalline domain grown in near thermodynamic equilibrium. On the other hand, in kinetically limited conditions the Legendre transformation of the kinetic Wulff plot can determine the equilibrium shape of the crystalline domain if the growth velocity of each crystallographic face of that domain is known.14 Thus, the equilibrium shape of a faceted domain grown in near thermodynamic equilibrium can be predicted by the minimization of the total surface energy. When a faceted domain is grown under kinetically limited conditions, the growth velocity of each crystallographic face of the domain should be known to determine the equilibrium shape. On the r 2011 American Chemical Society
other hand, the nonequilibrium shape cannot be determined by the mere use of either of these theoretical approaches, but a numerical simulation technique such as level-set method is known to be necessary in combination with the Wulff plot (kinetic Wulff plot).3,5 With increasing demand for nonpolar-GaN-based optoelectronic devices, which are expected to have better radiative quantum efficiency in comparison with polar-GaN-based ones because of the invulnerability to the quantum-confined Stark effect,6 many attempts have been made to grow nonpolar (1010) (m-oriented) GaN films. Unlike c-oriented polar-GaN films grown on c-plane sapphire substrates, the nonpolar GaN films grown on m-plane sapphire substrates, however, tend to have domains with unintended crystallographic orientations and striated surface morphology.7,8 Since the surface morphology of a heteroepitaxial film is significantly affected by the shape of initially nucleated domains and their merging behavior, it is of importance to fully understand the surface evolution characteristics Received: April 26, 2011 Revised: July 13, 2011 Published: August 02, 2011 3930
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’ EXPERIMENTAL SECTION m-GaN faceted domains were laterally overgrown on SiO2-patterned m-plane sapphire substrates with 2-, 3-, and 4-μm holes by using hydride vapor phase epitaxy. Four samples were grown under the same growth condition except for the growth durations of 20, 30, 40, and 60 min, respectively. Synchrotron X-ray scattering measurements were carried out at the wiggler beamline 5A HFMS in the Pohang Accelerator Laboratory (PAL). The beamline was equipped with a MAR345 image plate. A double crystal Si(111) monochromator was used to select an
Figure 1. (a) SEI of m-GaN faceted domains with a schematic diagram showing height, width, and depth. m-Planes are shaded with lines. (b) A 1 2D projected map of reciprocal space at |q B| = 4.01 Å , corresponding to the Bragg peak of GaN (1120) plane. Sapphire {1126} Bragg peaks are also shown in this map because |q Å1 of the {1126} sapphire B| = 3.924 1 Bragg peak is so close to |q |=4.01 Å that the tails of the {1126} B sapphire Bragg peaks are seen in this map. X-ray wavelength of λ = 0.6199 Å at 20.00 keV. The details of 2-D projected mapping of reciprocal space can be found in our previous work.12
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’ RESULTS AND DISCUSSION Figure 1 illustrates a secondary electron image (SEI) of mGaN faceted domains with a schematic diagram of the domain and a two-dimensionally (2D) projected reciprocal space map at 1 |q B| = 4.01 Å , corresponding to the Bragg peak of GaN (1120) plane. Sapphire {1126} Bragg peaks are also seen in this map because their |q B| value is very close to that of the Bragg peak of GaN (1120) plane. The map reveals that the crystallographic relation between GaN and sapphire substrate is [1120]GaN [0002]sapphire and [0002]GaN [1120]sapphire, respectively. These X-ray diffraction data clearly confirm that these GaN domains are m-oriented. The dimensions of the m-oriented GaN domains grown on 2-, 3-, and 4-μm holes for 20, 30, 40, and 60 min were measured by using SEIs and are plotted as symbols in Figure 2. The morphological evolution of these domains can be theoretically described by using the minimization of effective surface energy in the following manner. Once the dimensions (width, depth, height) of the smallest domains grown on each hole are given, the dimensions of these domains in the very next moment can be calculated by the minimization of the total effective surface energy. In other words, a ratio between the partial derivatives of a total effective surface energy and a domain volume with respect to width, depth, height determines which direction the domain would grow.11 In this effective energy minimization scheme, the morphological evolution of m-oriented GaN domain depends on the effective surface energy ratio between m-plane )
of nucleated nonpolar-GaN faceted domains in nonequilibrium shape, but there has not been much systematic work on this issue. There have been already several reports on the nonpolar-GaNbased optoelectronic devices that were homoepitaxially grown on m-plane GaN substrates,9,10 but nonpolar-GaN-based optoelectronic devices on m-plane GaN substrates would not be costeffective for mass production, because they cost far more than those on m-oriented sapphire substrates. Thus, the investigation of the surface evolution characteristics of heteroepitaxially grown m-oriented GaN domains in nonequilibrium shape is of significant importance from the viewpoint of both scientific interest and industrial application. Recently, the surface morphological characteristics of moriented GaN faceted domains in nonequilibrium shape were qualitatively explained in terms of total surface energy minimization.11 The description of the surface morphological characteristics based on the minimization of the actual surface energy, however, would not be valid if the domains were grown under kinetically limited conditions. It can be regarded, from the estimated surface energy ratio between m-plane and c-plane being larger than the theoretically predicted one,3,11 that the surface energy described in the previous paper11 cannot be the actual surface energy and that the growth was done under the kinetically limited conditions. This implies that the use of surface energy in energy minimization needs to be modified in such a way that the surface morphology evolution of crystalline domains grown even under kinetically limited conditions can be described in terms of effective surface energy minimization. The effective surface energy incorporates the effect of crystallographic-plane-dependent surface processes, if any, into the actual surface energy in such a way that it can be used to determine, even for the kinetically limited growth, which crystallographic plane is more favored in terms of the minimization of total effective surface energy. For example, the effective surface energy of a crystallographic face can have higher value than that of the actual surface energy of the crystallographic plane if the transport of materials to that crystallographic face is limited in the kinetically limited growth regime. As a consequence, the formation of that crystallographic face becomes less favorable than before due to the increased effective surface energy. The main advantage of this approach over the kinetic Wulff plot, which can predict only the equilibrium shape when the growth is done under kinetically limited conditions, is that the morphological evolution of crystalline domains even in nonequilibrium shape can be described. In this work, it is demonstrated that this effective surface energy minimization scheme, which was found to describe in a qualitative manner the characteristic features of the crystalline domains in nonequilibrium shape, can quantitatively reproduce the morphological evolution of the crystalline domains in nonequilibrium shape, which is not expected to be done by the mere use of Legendre transformation of a kinetic Wulff plot.
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D, and H represent width, depth, and height, respectively. γ* = 3.0 is about the best fit which we can obtain with our model. The theoretical prediction of the morphological evolution by using this minimization scheme with different surface energy ratios is plotted as solid lines in Figure 2. The solid lines in red, blue, and green represent the domains in nonequilibrium shape grown on 2-, 3-, and 4-μm holes, respectively. The solid lines in black represent the equilibrium shape. As can be seen clearly, a larger (smaller) value of the effective surface energy ratio significantly suppresses (enhances) the growth along the depth direction because the growth along the depth results in larger fraction of mplane in the total surface. It should be also noted that the smallest domains are all away from the eventual equilibrium shape. As illustrated in Figure 2, the fit with γ* = 1.1 is not in good agreement with the experimental data, but the ones with larger γ* values appear to reproduce better the data. Both fits with γ* = 3.0 and 5.0 appear to reproduce the data equally well in this figure, but there is one big difference in the meaning of these two fits. The fit with γ* = 3.0 shows that some of the domains already reached the equilibrium shape, but the fit with γ* = 5.0 shows that none of the domains reached the equilibrium shape. It is not straightforward to determine in this figure which one between γ* = 3.0 and 5.0 is better in terms of the quality of the fitting. Thus, we calculate how much each domain is away from the eventual equilibrium shape in the following manner, so that we can determine which fit better describes our data. For an wurtzite m-oriented GaN domain in equilibrium shape, its shape does not change with growth time in such a way that (Wo,Do,Ho) can be expressed as (1,R1,R2)Ao, where R1 and R2 are constants solely dependent on γ* (actually it turns out that R2 does not depend on γ*, which will be discussed in the later section) and Ao is a proportionality factor. The volume of this equilibrium domain is given by pffiffiffi Vo ¼ Ho Wo Do þ Ho 2 Do = 3 pffiffiffi ¼ Ao 3 ð1 þ R2 = 3ÞR1 R2 ð2Þ Figure 2. Plots of W, D, and H for the faceted domains grown on patterned substrates. Solid symbols and thick curves represent the data points and fitting results, respectively. The curves are simulation results based on the minimization of the effective surface energy. Hollow symbols and thin curves, which are projection of data and fits on each plane, are drawn for clarity. Symbols in red, blue, and green represent the domains grown on the patterned substrates with a hole size of 2, 3, and 4 μm, respectively. Square, circle, up-triangle, and down-triangle represent the domains grown for 20, 30, 40, and 60 min, respectively. Note that the results are drawn on different scales.
and c-plane, γ* Em*/Ec*, which serves as a fitting parameter. The quality of each fit with a certain γ* value was determined by calculating the χ2 value, which is defined by 2 !2 !2 N 1 Wdata, i Wfit, i Ddata, i Dfit, i 2 4 þ χ ¼ 3N i ¼ 1 σ W, i σD, i
∑
þ
Hdata, i Hf it, i σH, i
!2 # ð1Þ
where N is a number of data points and σi is the error of each experimental data value, respectively. χ2 values are 145.4, 5.19, and 5.42 for those fits with γ* = 1.1, 3.0, and 5.0, respectively. W,
In turn, this implies that for any given domain with a volume, V, a corresponding proportionality factor, A, can be obtained in the following manner sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 3 pffiffiffi A¼ ð3Þ ð1 þ R2 = 3ÞR1 R2 So (1,R1,R2)A would be the dimension of the domain in equilibrium shape with the same volume, V, as the one under consideration. Note that (Wo,Do,Ho) = (1,R1,R2)Ao for a domain in equilibrium but that (W,D,H) 6¼ (1,R1,R2)A for a domain in nonequilibrium. Then, the normalized deviation of any domain from its eventual equilibrium shape is given by 2 2 2 W D H 1 R1 R2 þ þ ð4Þ δ¼ A A A This normalized deviation quantifies how much the shape of the domain under consideration deviates from the eventual equilibrium shape specified by R1 and R2, and it becomes zero if the domain under consideration has already reached its eventual equilibrium shape. Since R1 and R2 depend on the effective surface energy ratio, δ is a function of γ* as well. These normalized deviations of experimental data (plotted as symbols) and simulation results (plotted as solid lines) with different γ* 3932
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Figure 3. The deviation of each domain from the eventual equilibrium shape is plotted. The fit with γ* = 3.0 shows that some of the data points already reach the equilibrium shape, but the fit with γ* = 5.0 shows that no data points reach the eventual equilibrium shape. The domains grown on larger holes are further away from the equilibrium shape, and it takes longer growth time for them to reach the equilibrium shape. Error bars are shown for the fit with γ* = 5.0.
(=1.1, 3.0, and 5.0) values in Figure 3 illustrate how far the domains are away from the equilibrium shapes and how well the simulation results reproduce the experimental data. Consistent with Figure 2, the fit with γ* = 1.1 does not reproduce the experimental data at all. The fit with γ* = 5.0 is good, but the result with γ* = 3.0 is physically more meaningful, because once the largest domains grown on differently sized holes reach the equilibrium shape, their shape will not change back to the nonequilibrium shape unless there is an external perturbation. In other words, if there is any γ value that correctly predicts the reached equilibrium shape as an eventual equilibrium shape, that value (γ* = 3.0 in this case) is the one with which the experimental results can be properly described. It should be noted that fitting all three different data sets with 2-, 3-, and 4-μm-size hole rather than a single set places a stringent constraint on the fitting result in such a way that the effective surface energy ratio can be more accurately estimated. In Figure 3, we also need to note how each data point approaches the equilibrium shape (the line of δ = 0). It can be clearly seen that the domains starting from the smaller holes reach the equilibrium shape faster than those from the larger holes but that eventually all the domains starting from the 2-, 3-, and 4-μm-size hole are close to the equilibrium shape after the completion of 60-min growth. This is because the initial deviation
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from the equilibrium shape is largest for the domains starting from a 4-μm hole. γ* = 3.0, which is different from the theoretical surface energy ratio of 1.1 (Em = 136.7 meV/Å2 and Ec = 128.9 meV/Å2 based on broken bond model),3 implies that our GaN domains were grown in a kinetically limited regime. This effective surface energy ratio might be equal to the actual ratio of surface energy if the growth was done in thermodynamic equilibrium. The large value of the estimated effective surface energy ratio, 3, in our experiment may be related to the fact that the lattice mismatch along the GaN c-direction is 5 times larger than along the a-direction in such a way that the growth along the c-direction is highly suppressed, which is manifested by a large effective surface energy ratio. There is a clear advantage of using this effective surface energy concept over the kinetic Wulff’s plot, because our method can give a full description of the morphological evolution of domains in nonequilibrium shape, which cannot be done by using the kinetic Wulff’s plot. In Figure 2, it should be noted that the ratio (R1) of depth to width varies from 1.58 (when γ* = 1.1) to 0.58 (when γ* = 3.0) and 0.35 (when γ* = 5.0), but that the ratio (R2) of height to width is 0.87, independent of γ*. It is, however, found that R2 depends on how the total effective surface energy is calculated. In the previous paper,11 the bottom surface of the GaN domain was included in calculating the total surface energy, but the energy calculation with the bottom surface included could not properly reproduce the ratio between the width and height of m-oriented GaN domain in nonequilibrium shape. In this work, the bottom surface is excluded in calculating the total surface energy, which can be justified by the passivation effect of SiO2 on the bottom surface, in such a way that the theoretical prediction with γ* = 3.0 is in very good agreement with the experimental results.
’ CONCLUSIONS The concept of effective surface energy was proposed and it was experimentally demonstrated that the surface morphology evolution of GaN faceted domains in nonequilibrium shape can be successfully described with the effective surface energy ratio γ* = 3.0. Not only the equilibrium shape but also the nonequilibrium shape of the domain was successfully described by energy minimization schemed based on the effective surface energy. This concept of the effective surface energy may provide new insights into understanding the surface morphology evolution of the faceted domains in nonequilibrium shape. ’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT This work was financially supported by the National Research Foundation of Korea Grant funded by the Korean Government (2009-0067662). ’ REFERENCES (1) Wulff, G. Z. Krystallogr. Mineral. 1901, 34, 449. (2) Herring, C. Phys. Rev. 1951, 82, 87. (3) Jindal, V.; Shahedipour-Sandvik, F. J. Appl. Phys. 2009, 106, 083115. (4) Sekerka, R. F. Cryst. Res. Technol. 2005, 40, 291. 3933
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(5) Du, D.; Srolovitz, D. J.; Coltrin, M. E.; Mitchell, C. C. Phys. Rev. Lett. 2005, 95, 155503. (6) Chichibu, S.; et al. Nat. Mater. 2006, 5, 810. (7) Baker, T. J.; Haskell, B. A.; Wu, F.; Speck, J. S.; Nakamura, S. Jpn. J. Appl. Phys. 2006, 45, L154. (8) Armitage, R.; Hirayama, H. Appl. Phys. Lett. 2008, 92, 092121. (9) Masui, H.; Chakraborty, A.; Haskell, B. A.; Mishra, U. K.; Speck, J. S.; Nakamura, S.; DenBaars, S. P. Jpn. J. Appl. Phys. 2005, 44, L1329. (10) Koyama, T.; Onuma, T.; Masui, H.; Chakraborty, A.; Haskell, B. A.; Keller, S.; Mishra, U. K.; Speck, J. S.; Nakamura, S.; DenBaars, S. P. Appl. Phys. Lett. 2006, 89, 091906. (11) Seo, Y.; Kim, C. Appl. Phys. Lett. 2010, 97, 101902. (12) Lee, S.; Sohn, Y.; Kim, C.; Lee, D. R.; Lee, H. H. Nanotechnology 2009, 20, 215703.
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