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Stability diagrams for the surface patterns of GaN(0001¯) as a function of Schwoebel barrier height
Journal of Crystal Growth ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Stability diagrams for the surface patterns of GaN(0001̄ ) as a function of Schwoebel barrier height Filip Krzyżewski a, Magdalena A. Załuska-Kotur a,b,n a b
Institute of Physics Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland Faculty of Mathematics and Natural Sciences, Cardinal Stefan Wyszynski University, ul Dewajtis 5, 01-815 Warsaw, Poland
art ic l e i nf o
Keywords: Surface structure Computer simulations Growth model Surface processes Nitrides
a b s t r a c t Height and type of Schwoebel barriers (direct or inverse) decides about the character of the surface instability. Different surface morphologies are presented. Step bunches, double steps, meanders, mounds and irregular patterns emerge at the surface as a result of step (Schwoebel) barriers at some temperature or miscut values. The study was carried out on the two-component kinetic Monte Carlo (kMC) model of GaN(0001̄ ) surface grown in nitrogen rich conditions. Diffusion of gallium adatoms over N-polar surface is slow and nitrogen adatoms are almost immobile. We show that in such conditions surfaces remain smooth when gallium adatoms diffuse in the presence of low inverse Schwoebel barrier. It is illustrated by adequate stability diagrams for surface morphologies. & 2016 Elsevier B.V. All rights reserved.
1. Introduction Formation of various geometric step patterns during crystal growth process remains a subject of continuous interest of many researchers [1–4]. When steps stay straight and equally distanced interfaces between semiconductor layers are flat what gives a chance for high quality of grown nano-devices. From the other side ordered surface patterns (e.g. bunches or mounds) are used as templates for popular nano-objects like quantum dots or nanowires. Hence an identification of optimal growth conditions is important to maintain proper control under the nanostructures production process. Optimal growth conditions include proper temperature, miscut, external fluxes of components and microscopic crystal properties, like impurities or doping that have their influence onto the effective height of Schwoebel barrier. We show how these parameters affect formation of surface structures that appear during two component GaN(0001̄ ) crystal growth. Gallium nitride is a subject of interest because of its applications in many electronic devices. The most intense investigations of crystal growth focus on in its Ga-polar (0001) orientation because it is easier to obtain stable process with smooth surfaces of this polarization. However growth at N-polar GaN(0001̄ ) surface focuses recently more and more attention because of wide variety of applications as solar cells, sensors, high electron mobility transistors and light emitting diodes [5–7]. There are reasons to n Corresponding author at: Institute of Physics Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland.
believe that crystal grow in this polarization leads to structures of better quality. At the same time high barriers for diffusion of adatoms at this surface cause that the growth process is very unstable and leads to rough structures [8]. We show that even in so unfavorable conditions, assuming some values of internal parameters such as Schwoebel (step) barrier and for special choice of external parameters smooth step flow crystal grow is possible. Several phenomena happening in microscale have large impact on surface morphology of grown crystals [1,9–11]. Namely they are adatom diffusion and incorporation of surface atoms at steps. Slow diffusion, high direct step barrier, high external particle flux – all these factors cause that adatoms stick together creating islands at terraces before they approach steps. Those 2D (two-dimensional) structures evolve finally into 3D pattern of mounds. From the other side high temperature or low external flux of incoming particles can smoothen the surface. In this case important role plays imbalance of particle fluxes which are incorporated at steps [1,12–14]. When flux of atoms attaching step from the terrace below is lower than this from above step bunching is observed. Otherwise, when the amount of particles incorporated from the lower terrace is higher, meandered pattern emerges. Schwoebel effect, a possible source of such an imbalance, in real systems can be caused by the presence of impurities or doping [15,16]. Step barriers or diffusion barriers for different adatoms are internal model parameters which together with external parameters like temperature, incoming particle flux and miscut of the surface control the character of crystal growth. In the paper two component GaN(0001̄ ) model is used to study
http://dx.doi.org/10.1016/j.jcrysgro.2016.04.043 0022-0248/& 2016 Elsevier B.V. All rights reserved.
Please cite this article as: F. Krzyżewski, M.A. Załuska-Kotur, Journal of Crystal Growth (2016), http://dx.doi.org/10.1016/j. jcrysgro.2016.04.043i
F. Krzyżewski, M.A. Załuska-Kotur / Journal of Crystal Growth ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
Fig. 1. Potential profile as seen by the diffusing Ga adatom close to the step. At left side (A) inverse Schwoebel barrier is illustrated and at right side (B) direct one. Position of steps is shown below.
variety of surface morphologies. These observations are collected in stability diagrams which determine ranges of growth parameters needed to obtain particular patterns. Section 2 contains description of kinetic Monte Carlo (kMC) model. Next, in Section 3 we describe simulation results and present stability diagrams. Finally Section 4 contains conclusions.
2. Numerical model In order to investigate the influence of the internal and external growth parameters on the shapes of grown surfaces of GaN(0001̄ ) we adopted two component silicon carbide model described and studied in Ref. [17]. The model was used in GaN N-plane simulations [18] where experimental results were compared with simulated ones and several methods of surface smoothening were proposed. Here we investigate the dependence of the surface structure as it evolves during crystal growth process for broad range of investigated parameters. In particular we take into account the importance of the height of Schwoebel barrier at the step for the studied phenomenon. The height of the step barrier can be positive – which means direct Schwoebel barrier (SB) as well as negative – inverse Schwoebel barrier (ISB). Both types are discussed. Numerical systems consists of four layers of lattice sites which have their surface positions (x and y) and height (z) that denotes the number of crystallographic layers below. If, during any surface process, new particle appears at the site, its z value is increased by 2. If particle from the top disappears its z value is decreased by 2. Even and odd values of parameter z correspond to N and Ga atoms respectively. Ga and N atoms form GaN crystallographic lattice through appropriate geometry of interatomic bonds. They interact via nearest- and next nearest-neighbour (NN and NNN respectively) interactions. The energy of atom is
EX = EGaN
∑ ni + EXX ∑ NN
NNN
ni
(1)
where X = Ga , N and ni ¼ 1 if ith neighboring site is occupied and ni ¼0 otherwise. Sums go over all NNs and NNNs. Every atom is surrounded by up to 4 NNs, which are of different type and by up to 12 NNNs of the same type. Interactions between all atoms are chosen in such a way that the total energy of N atom in bulk crystal is 10.6 eV and this of Ga atom 10 eV. Such values
correspond to bond order calculations [19]. Namely, energies of interparticle bonds are: EGaGa = 0.3 eV for two Ga atoms, ENN = 0.35 eV for two N atoms and EGaN = 1.6 eV for bonds between Ga and N atom. Every simulation consists of several time steps during which all particles at top layer of the system execute jump procedure which runs as follows: In the first step at every site of the simulated crystal new atom can be adsorbed with probability dependent on fluxes of incoming Ga and N atoms.
P AX = FX
(2)
where FX is a flux of Ga or N. Both Ga and N atoms can be adsorbed all over the surface but each at its own lattice sites. They do not mix together. In the next step all surface particles can jump with the probability X ⎧ ⎪ νe β (Ef − E i − BD ) when Ei ≥ Ef PD = ⎨ ⎪ X − β B ⎩ νe D otherwise
T )−1
(3) 1011 s−1
In above formula β = (kB and ν = sets the time scale. BDX denotes diffusion barrier that is different for Ga and N adatoms. It was shown in [8] that barriers for diffusion at GaN(0001̄ ) surface are 1 eV for gallium and 1.8 eV for nitrogen. We used these values for parameter BDX in (3). For Ga adatoms an additional Schwoebel barrier (SB or ISB) SB is added and it modifies jump probability [1,10,12,17,20] as follows:
PDS = e−βSB PD
(4)
This barrier is activated only for sites just at the step and the landscape of potential energy is as shown in Fig. 1. Diffusion barriers for Ga adatoms do not change for all possible jumps except this between two sites at the upper side of the step for SB or between two sites at the lower side of the step for ISB. In such a way incorporation rate of adatoms from upper terrace is different than that from the lower one [10]. Setting an additional barrier at the lower terrace in the case of inverse Schwoebel effect is more effective method to control particle fluxes towards step than reduction of barrier at the higher terrace, as it was discussed in Ref. [20] and what we have also checked in our simulations. Ei and Ef are calculated and jumps of all particles are executed with probability PD. When jump procedure of all particles is finished we realize desorption process for each top atoms with the probability
Please cite this article as: F. Krzyżewski, M.A. Załuska-Kotur, Journal of Crystal Growth (2016), http://dx.doi.org/10.1016/j. jcrysgro.2016.04.043i
F. Krzyżewski, M.A. Załuska-Kotur / Journal of Crystal Growth ∎ (∎∎∎∎) ∎∎∎–∎∎∎ X
PdX = νe−β (Ei+ Bd )
3
(5)
X
where Bd are desorption barriers for Ga or N atoms. Their values are fixed in such a way that single N atoms residing at the surface desorb very easily and Ga atoms do not desorb at all. Namely BdN = − 0.7 eV and BdGa = 10 eV . Note that Ga adatoms adsorbed at Ga covered surface and N adatoms at N covered surface desorb easily, because they are not kept by the empty layer below and next neighbors within the lower layer are the only atoms they interact with. In all simulations gallium flux FGa = 4 nm/min is significantly lower then nitrogen flux FN = 16 nm/min . We simulate growth with an excess of nitrogen atoms. We use high flux of almost immobile N atoms and low flux of Ga atoms. Equilibrium in such conditions is controlled by easy N desorption process and almost impossible Ga desorption. Each simulation point of surface evolution starts from perfect ordering of equally distanced steps for N and Ga layers. Due to adsorption and desorption processes density of particles at top layer equilibrates during surface evolution reaching quantity dependent on growth conditions.
3. Stability diagrams In Figs. 2 and 3 we show stability diagrams illustrating how crystal growth process depends on the choice of parameters. Diagram in Fig. 2 illustrates pattern dependence on SB and ISB height and miscut at temperature T = 750 °C . It can be seen that ISB which are above 0.5 eV (plotted as negative values) for miscuts between 2° and 4° lead to the bunched surface (Fig. 4E). It is known that ISB usually induces step bunching during crystal growth processes [1,10,11,17]. ISB also slows down nucleation at terraces, because it hinders the jumping of particles to neighboring sites. For ISB height below 0.5 eV we see double step structure (Fig. 4D). It means that influence of ISB on the particle dynamics is too small to form bunches however it still causes imbalance of particle fluxes across steps. That difference induces formation of double steps which are easy to form due to the structure of GaN elementary cell consisting of two Ga-N layers. Direct Schwoebel barriers generate meandered patterns for high miscuts (Fig. 4C). For lower miscuts and relatively low SB rough structure arises (Fig. 4B) [18]. Finally higher SB lead to a formation of 3D mound structure at most of studied surface miscut. Generation of such structures is induced by high, direct SB and relatively high external
Fig. 2. Stability diagram in SB-miscut coordinates. Positive values of SB correspond to direct Schwoebel effect whereas negative SBs to the inversed one. All simulations were carried out at temperature 750 °C and fluxes FGa = 4 nm/min and FN = 16 nm/min . Letters from B to F correspond to surface patterns presented in Fig. 4. Dashed lines separate regions of different types of surface patterns. Horizontal dotted line represents place where the plot overlaps with diagram presented in Fig. 3.
Fig. 3. Stability diagram in SB-temperature coordinates. Positive values of SB correspond to direct Schwoebel effect whereas negative SBs to the inversed one. All simulations were carried out at fluxes FGa = 4 nm/min FN ¼16 nm/min and miscut 3°. Letters from A to F correspond to surface patterns presented in Fig. 4. Dashed lines separate regions of different types of surface patterns. Horizontal dotted line represents place where the plot overlaps with diagram presented in Fig. 2.
flux of particles. When 2D structures are formed next particles which are adsorbed on top of those islands remain there because high SB blocks downstep jumps. In such a way 3D mound structure is build. For very high miscuts there is not enough space to form 2D islands and steps bend instead. Closely spaced and bent steps can easily adjust their phases and form meandered pattern (Fig 4C). Fig. 3 shows stability diagram in other coordinations: temperature and height of Schwoebel barrier for fixed miscut 3°. Dotted, horizontal lines plotted in Figs. 2 and 3 represent area where these two diagrams overlap in 3D space. We can observe that at all studied temperatures steps bunch for ISB values above 0.5 eV (Fig. 4E). Interesting structures of bent steps form when ISB are lower and at the temperature range above 820 °C (Fig. 4A). Such structures are not observed in Fig. 2. In this case Schwoebel effect does not play crucial role in the system. Particles which are adsorbed on top of 2D-islands formed at terraces can easily diffuse to island edges, jump one terrace below and attach to the steps. In such a case islands which arise at terraces do not grow in z direction but they become wider and wider until they incorporate to steps. Such a phenomenon leads to emergence of wavy, double step pattern which is presented in Fig. 4A. Below 820 °C and for weak Schwoebel effect we note regular patterns of double steps. For temperatures below 760 °C and strict SB the structure changes again. It is due to the fact that diffusion along steps is affected by kinks. Particles can easily attach to the kink but detachment from it is connected with overcoming one or sum of two energy barriers of 0.3 eV for nitrogen and 0.35 eV for gallium. For two regions B and C where we have low SB and low temperature or high SB and high temperature ratio of jump rate across step to the jump rate along terrace DSB /D = 0.002. This can be compared with the ratio of jump rate along steps to jump rate along terrace Ds /D = 0.004 in the higher temperature region C and Ds /D = 0.0005 in the lower temperature region B. It appears that when we have dynamics with the same ratio of across step and along terrace rates, then rates along step matter in surface ordering. When jump probability along and across step are comparable particles have enough time to smoothen steps which create meanders. When wandering along steps is much slower than jumps down, steps are more rough and they do not arrange in a regular structure. In the region F, that is between B and C, rates for jumps down are smaller, they become again comparable with diffusion along steps and they lead to the formation of 3D mounds irrespectively of slope of the surface.
Please cite this article as: F. Krzyżewski, M.A. Załuska-Kotur, Journal of Crystal Growth (2016), http://dx.doi.org/10.1016/j. jcrysgro.2016.04.043i
F. Krzyżewski, M.A. Załuska-Kotur / Journal of Crystal Growth ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
Fig. 4. Surface patterns observed during simulations: bent steps (A), rough surface (B), smooth meanders (C), double steps (D), bunch (E) and mounds (F). Letters A–F correspond to those presented in stability diagrams in Figs. 2 and 3.
The multiplicity of microscopic events in the system under study is large and simple reasoning about the surface structure based on the comparison of jump rates across different paths is difficult. Note that there are two components in the system, one is immobile. Moreover SB and ISB are assigned only to Ga adatoms. Such properties of adatom dynamics also influence evolution of the surface structure. The general observation that can be drawn from the analysis of stability diagrams and the surface patterns in Fig. 4. is that the presence of ISB is associated with a more smooth surfaces than the presence of SB when other growth conditions are kept the same. ISB barriers usually lead to straight steps. In our study all patterns: meandering, bunching, mounding and roughening are described within one model, so regions where they happen cross and sometimes overlap. Detailed analysis of the transition lines between different regions is difficult. Dashed lines presented in Figs. 2 and 3 roughly denote transitions between observed surface morphologies. Regions F and C in Fig. 2 correspond to unstable surface morphologies. Following dashed lines which separate those areas from stable region D one can notice that for higher miscut stable patterns appear for broader range of SB height. The range of stable phase at miscut 4° is from 0.4 eV to 0.3 eV, whereas at 1° surface is stable between 0.5 eV and 0.25 eV which is narrower. It is consistent with the results obtained in [21] where miscut and SB dependent surface stability was investigated.
Schwoebel barrier values as a function of temperatures and miscuts. It has been shown that formation of concrete surface morphologies such as bunches, meandered patterns, double steps, bent steps, rough surface or mounds is strictly related to the character and the effective height of Schwoebel barrier. If ISB effect plays a crucial role in the system, steps are straight and have tendency to assembly in bunches. The presence of direct SB leads to the more rough steps that can order in regular patterns of meanders or mounds. Meanders are formed at high temperatures and miscuts whereas mounds at low temperatures and miscuts. Straight, double step pattern can be found in the region where SB or ISB do not play an important role in the surface evolution. We have also shown the sensitivity of surface structure on miscut and temperature parameters. In such a way we can conclude about the existing effective ISB and SB barriers in the system from the morphology of the surface. In particular a study of the impurity and doping influence on the SB height is very interesting question. Deeper knowledge about such relations is a step towards proper tuning of inversed or direct Schwoebel barrier height to obtain various surface morphologies.
Acknowledgment Research supported by the National Science Centre (NCN) of Poland (Grant NCN no. 2013/11/D/ST3/02700)
4. Conclusions References Two-component kinetic Monte Carlo model of wurtzite crystals was used to study surface evolution of GaN (0001¯ ). We investigated surface evolution in nitrogen rich conditions for various
[1] C. Misbah, O. Pierre-Louise, Y. Saito, Rev. Mod. Phys. 82 (2010) 981. [2] M. Valdmirova, A. Pimpinell, A. Videcoq, J. Cryst. Growth 220 (2000) 631.
Please cite this article as: F. Krzyżewski, M.A. Załuska-Kotur, Journal of Crystal Growth (2016), http://dx.doi.org/10.1016/j. jcrysgro.2016.04.043i
F. Krzyżewski, M.A. Załuska-Kotur / Journal of Crystal Growth ∎ (∎∎∎∎) ∎∎∎–∎∎∎ [3] A. Pimpinelli, V. Tonchev, A. Videcoq, M. Vladimirova, Phys. Rev. Lett. 88 (2002) 20. [4] X.Q. Shen, H. Okamura, J. Cryst. Growth 300 (2007) 75. [5] S. Keller, H. Li, M. Laurent, Y. Hu, N. Pfaff, J. Lu, D.F. Brown, N.A. Fichtenbaum, J. S. Speck, S.P. DenBaars, U.K. Mishra, Semicond. Sci. Technol. 29 (2014) 113001. [6] S. Rajan, A. Chini, M.H. Wong, J.S. Speck, U.M. Mishra, J. Appl. Phys. 102 (2007) 044501. [7] M.H. Wong, S. Keller, Nidhi, S. Dasgupta, D.J. Denninghoff, S. Kolluri, D. F. Brown, J. Lu, N.A. Fichtenbaum, E. Ahmadi, U. Singisetti, A. Chini, S. Rajan, S. P. DenBaars, J. Speck, U.K. Mishra, Semicond. Sci. Technol. 28 (2013) 074009. [8] T. Zywietz, J. Neugebauer, M. Scheffler, Appl. Phys. Lett. 73 (1998) 487. [9] M. Sato, M. Uwaha, Surf. Sci. 493 (2001) 494. [10] M. Sato, M. Uwaha, Y. Saito, Phys. Rev. Lett. 80 (1998) 4233. [11] K. Yagi, H. Minoda, M. Degawa, Surf. Sci. Rep. 43 (2001) 45.
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R.L. Schwoebel, E.J. Shipsley, J. Appl. Phys. 37 (1966) 3682. G.S. Bales, A. Zangwill, Phys. Rev. B 41 (1990) 5500. M.H. Xie, S.Y. Leung, S.Y. Tong, Surf. Sci. Lett. 515 (2002) L459. N. Ohtani, M. Katsumoto, J. Takahashi, H. Yashiro, M. Kanaya, Surf. Sci. 398 (1998) L303. N. Ohtani, M. Katsumoto, J. Takahashi, H. Yashiro, M. Kanaya, Phys. Rev. B 59 (1999) 4592. F. Krzyżewski, M.A. Załuska-Kotur, J. Appl. Phys. 115 (2014) 213517. F. Krzyżewski, M.A. Załuska-Kotur, H. Turski, M. Sawicka, C. Skierbiszewski, arXiv:1509.01035 [cond-mat.mtrl-sci] send to J Crystal Growth. K. Wang, J. Singh, D. Pavlidis, J. Appl. Phys. 76 (1994) 3502. M.H. Xie, S.Y. Leung, S.Y. Tong, Surf. Sci. 515 (2002) L459–L463. D. Vilone, C. Castellano, P. Politti, Phys. Rev. B. 72 (2005) 245414.
Please cite this article as: F. Krzyżewski, M.A. Załuska-Kotur, Journal of Crystal Growth (2016), http://dx.doi.org/10.1016/j. jcrysgro.2016.04.043i
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Stability diagrams for the surface patterns of GaN(0001̄ ) as a function of Schwoebel barrier height Filip Krzyżewski a, Magdalena A. Załuska-Kotur a,b,n a b
Institute of Physics Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland Faculty of Mathematics and Natural Sciences, Cardinal Stefan Wyszynski University, ul Dewajtis 5, 01-815 Warsaw, Poland
art ic l e i nf o
Keywords: Surface structure Computer simulations Growth model Surface processes Nitrides
a b s t r a c t Height and type of Schwoebel barriers (direct or inverse) decides about the character of the surface instability. Different surface morphologies are presented. Step bunches, double steps, meanders, mounds and irregular patterns emerge at the surface as a result of step (Schwoebel) barriers at some temperature or miscut values. The study was carried out on the two-component kinetic Monte Carlo (kMC) model of GaN(0001̄ ) surface grown in nitrogen rich conditions. Diffusion of gallium adatoms over N-polar surface is slow and nitrogen adatoms are almost immobile. We show that in such conditions surfaces remain smooth when gallium adatoms diffuse in the presence of low inverse Schwoebel barrier. It is illustrated by adequate stability diagrams for surface morphologies. & 2016 Elsevier B.V. All rights reserved.
1. Introduction Formation of various geometric step patterns during crystal growth process remains a subject of continuous interest of many researchers [1–4]. When steps stay straight and equally distanced interfaces between semiconductor layers are flat what gives a chance for high quality of grown nano-devices. From the other side ordered surface patterns (e.g. bunches or mounds) are used as templates for popular nano-objects like quantum dots or nanowires. Hence an identification of optimal growth conditions is important to maintain proper control under the nanostructures production process. Optimal growth conditions include proper temperature, miscut, external fluxes of components and microscopic crystal properties, like impurities or doping that have their influence onto the effective height of Schwoebel barrier. We show how these parameters affect formation of surface structures that appear during two component GaN(0001̄ ) crystal growth. Gallium nitride is a subject of interest because of its applications in many electronic devices. The most intense investigations of crystal growth focus on in its Ga-polar (0001) orientation because it is easier to obtain stable process with smooth surfaces of this polarization. However growth at N-polar GaN(0001̄ ) surface focuses recently more and more attention because of wide variety of applications as solar cells, sensors, high electron mobility transistors and light emitting diodes [5–7]. There are reasons to n Corresponding author at: Institute of Physics Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland.
believe that crystal grow in this polarization leads to structures of better quality. At the same time high barriers for diffusion of adatoms at this surface cause that the growth process is very unstable and leads to rough structures [8]. We show that even in so unfavorable conditions, assuming some values of internal parameters such as Schwoebel (step) barrier and for special choice of external parameters smooth step flow crystal grow is possible. Several phenomena happening in microscale have large impact on surface morphology of grown crystals [1,9–11]. Namely they are adatom diffusion and incorporation of surface atoms at steps. Slow diffusion, high direct step barrier, high external particle flux – all these factors cause that adatoms stick together creating islands at terraces before they approach steps. Those 2D (two-dimensional) structures evolve finally into 3D pattern of mounds. From the other side high temperature or low external flux of incoming particles can smoothen the surface. In this case important role plays imbalance of particle fluxes which are incorporated at steps [1,12–14]. When flux of atoms attaching step from the terrace below is lower than this from above step bunching is observed. Otherwise, when the amount of particles incorporated from the lower terrace is higher, meandered pattern emerges. Schwoebel effect, a possible source of such an imbalance, in real systems can be caused by the presence of impurities or doping [15,16]. Step barriers or diffusion barriers for different adatoms are internal model parameters which together with external parameters like temperature, incoming particle flux and miscut of the surface control the character of crystal growth. In the paper two component GaN(0001̄ ) model is used to study
http://dx.doi.org/10.1016/j.jcrysgro.2016.04.043 0022-0248/& 2016 Elsevier B.V. All rights reserved.
Please cite this article as: F. Krzyżewski, M.A. Załuska-Kotur, Journal of Crystal Growth (2016), http://dx.doi.org/10.1016/j. jcrysgro.2016.04.043i
F. Krzyżewski, M.A. Załuska-Kotur / Journal of Crystal Growth ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
Fig. 1. Potential profile as seen by the diffusing Ga adatom close to the step. At left side (A) inverse Schwoebel barrier is illustrated and at right side (B) direct one. Position of steps is shown below.
variety of surface morphologies. These observations are collected in stability diagrams which determine ranges of growth parameters needed to obtain particular patterns. Section 2 contains description of kinetic Monte Carlo (kMC) model. Next, in Section 3 we describe simulation results and present stability diagrams. Finally Section 4 contains conclusions.
2. Numerical model In order to investigate the influence of the internal and external growth parameters on the shapes of grown surfaces of GaN(0001̄ ) we adopted two component silicon carbide model described and studied in Ref. [17]. The model was used in GaN N-plane simulations [18] where experimental results were compared with simulated ones and several methods of surface smoothening were proposed. Here we investigate the dependence of the surface structure as it evolves during crystal growth process for broad range of investigated parameters. In particular we take into account the importance of the height of Schwoebel barrier at the step for the studied phenomenon. The height of the step barrier can be positive – which means direct Schwoebel barrier (SB) as well as negative – inverse Schwoebel barrier (ISB). Both types are discussed. Numerical systems consists of four layers of lattice sites which have their surface positions (x and y) and height (z) that denotes the number of crystallographic layers below. If, during any surface process, new particle appears at the site, its z value is increased by 2. If particle from the top disappears its z value is decreased by 2. Even and odd values of parameter z correspond to N and Ga atoms respectively. Ga and N atoms form GaN crystallographic lattice through appropriate geometry of interatomic bonds. They interact via nearest- and next nearest-neighbour (NN and NNN respectively) interactions. The energy of atom is
EX = EGaN
∑ ni + EXX ∑ NN
NNN
ni
(1)
where X = Ga , N and ni ¼ 1 if ith neighboring site is occupied and ni ¼0 otherwise. Sums go over all NNs and NNNs. Every atom is surrounded by up to 4 NNs, which are of different type and by up to 12 NNNs of the same type. Interactions between all atoms are chosen in such a way that the total energy of N atom in bulk crystal is 10.6 eV and this of Ga atom 10 eV. Such values
correspond to bond order calculations [19]. Namely, energies of interparticle bonds are: EGaGa = 0.3 eV for two Ga atoms, ENN = 0.35 eV for two N atoms and EGaN = 1.6 eV for bonds between Ga and N atom. Every simulation consists of several time steps during which all particles at top layer of the system execute jump procedure which runs as follows: In the first step at every site of the simulated crystal new atom can be adsorbed with probability dependent on fluxes of incoming Ga and N atoms.
P AX = FX
(2)
where FX is a flux of Ga or N. Both Ga and N atoms can be adsorbed all over the surface but each at its own lattice sites. They do not mix together. In the next step all surface particles can jump with the probability X ⎧ ⎪ νe β (Ef − E i − BD ) when Ei ≥ Ef PD = ⎨ ⎪ X − β B ⎩ νe D otherwise
T )−1
(3) 1011 s−1
In above formula β = (kB and ν = sets the time scale. BDX denotes diffusion barrier that is different for Ga and N adatoms. It was shown in [8] that barriers for diffusion at GaN(0001̄ ) surface are 1 eV for gallium and 1.8 eV for nitrogen. We used these values for parameter BDX in (3). For Ga adatoms an additional Schwoebel barrier (SB or ISB) SB is added and it modifies jump probability [1,10,12,17,20] as follows:
PDS = e−βSB PD
(4)
This barrier is activated only for sites just at the step and the landscape of potential energy is as shown in Fig. 1. Diffusion barriers for Ga adatoms do not change for all possible jumps except this between two sites at the upper side of the step for SB or between two sites at the lower side of the step for ISB. In such a way incorporation rate of adatoms from upper terrace is different than that from the lower one [10]. Setting an additional barrier at the lower terrace in the case of inverse Schwoebel effect is more effective method to control particle fluxes towards step than reduction of barrier at the higher terrace, as it was discussed in Ref. [20] and what we have also checked in our simulations. Ei and Ef are calculated and jumps of all particles are executed with probability PD. When jump procedure of all particles is finished we realize desorption process for each top atoms with the probability
Please cite this article as: F. Krzyżewski, M.A. Załuska-Kotur, Journal of Crystal Growth (2016), http://dx.doi.org/10.1016/j. jcrysgro.2016.04.043i
F. Krzyżewski, M.A. Załuska-Kotur / Journal of Crystal Growth ∎ (∎∎∎∎) ∎∎∎–∎∎∎ X
PdX = νe−β (Ei+ Bd )
3
(5)
X
where Bd are desorption barriers for Ga or N atoms. Their values are fixed in such a way that single N atoms residing at the surface desorb very easily and Ga atoms do not desorb at all. Namely BdN = − 0.7 eV and BdGa = 10 eV . Note that Ga adatoms adsorbed at Ga covered surface and N adatoms at N covered surface desorb easily, because they are not kept by the empty layer below and next neighbors within the lower layer are the only atoms they interact with. In all simulations gallium flux FGa = 4 nm/min is significantly lower then nitrogen flux FN = 16 nm/min . We simulate growth with an excess of nitrogen atoms. We use high flux of almost immobile N atoms and low flux of Ga atoms. Equilibrium in such conditions is controlled by easy N desorption process and almost impossible Ga desorption. Each simulation point of surface evolution starts from perfect ordering of equally distanced steps for N and Ga layers. Due to adsorption and desorption processes density of particles at top layer equilibrates during surface evolution reaching quantity dependent on growth conditions.
3. Stability diagrams In Figs. 2 and 3 we show stability diagrams illustrating how crystal growth process depends on the choice of parameters. Diagram in Fig. 2 illustrates pattern dependence on SB and ISB height and miscut at temperature T = 750 °C . It can be seen that ISB which are above 0.5 eV (plotted as negative values) for miscuts between 2° and 4° lead to the bunched surface (Fig. 4E). It is known that ISB usually induces step bunching during crystal growth processes [1,10,11,17]. ISB also slows down nucleation at terraces, because it hinders the jumping of particles to neighboring sites. For ISB height below 0.5 eV we see double step structure (Fig. 4D). It means that influence of ISB on the particle dynamics is too small to form bunches however it still causes imbalance of particle fluxes across steps. That difference induces formation of double steps which are easy to form due to the structure of GaN elementary cell consisting of two Ga-N layers. Direct Schwoebel barriers generate meandered patterns for high miscuts (Fig. 4C). For lower miscuts and relatively low SB rough structure arises (Fig. 4B) [18]. Finally higher SB lead to a formation of 3D mound structure at most of studied surface miscut. Generation of such structures is induced by high, direct SB and relatively high external
Fig. 2. Stability diagram in SB-miscut coordinates. Positive values of SB correspond to direct Schwoebel effect whereas negative SBs to the inversed one. All simulations were carried out at temperature 750 °C and fluxes FGa = 4 nm/min and FN = 16 nm/min . Letters from B to F correspond to surface patterns presented in Fig. 4. Dashed lines separate regions of different types of surface patterns. Horizontal dotted line represents place where the plot overlaps with diagram presented in Fig. 3.
Fig. 3. Stability diagram in SB-temperature coordinates. Positive values of SB correspond to direct Schwoebel effect whereas negative SBs to the inversed one. All simulations were carried out at fluxes FGa = 4 nm/min FN ¼16 nm/min and miscut 3°. Letters from A to F correspond to surface patterns presented in Fig. 4. Dashed lines separate regions of different types of surface patterns. Horizontal dotted line represents place where the plot overlaps with diagram presented in Fig. 2.
flux of particles. When 2D structures are formed next particles which are adsorbed on top of those islands remain there because high SB blocks downstep jumps. In such a way 3D mound structure is build. For very high miscuts there is not enough space to form 2D islands and steps bend instead. Closely spaced and bent steps can easily adjust their phases and form meandered pattern (Fig 4C). Fig. 3 shows stability diagram in other coordinations: temperature and height of Schwoebel barrier for fixed miscut 3°. Dotted, horizontal lines plotted in Figs. 2 and 3 represent area where these two diagrams overlap in 3D space. We can observe that at all studied temperatures steps bunch for ISB values above 0.5 eV (Fig. 4E). Interesting structures of bent steps form when ISB are lower and at the temperature range above 820 °C (Fig. 4A). Such structures are not observed in Fig. 2. In this case Schwoebel effect does not play crucial role in the system. Particles which are adsorbed on top of 2D-islands formed at terraces can easily diffuse to island edges, jump one terrace below and attach to the steps. In such a case islands which arise at terraces do not grow in z direction but they become wider and wider until they incorporate to steps. Such a phenomenon leads to emergence of wavy, double step pattern which is presented in Fig. 4A. Below 820 °C and for weak Schwoebel effect we note regular patterns of double steps. For temperatures below 760 °C and strict SB the structure changes again. It is due to the fact that diffusion along steps is affected by kinks. Particles can easily attach to the kink but detachment from it is connected with overcoming one or sum of two energy barriers of 0.3 eV for nitrogen and 0.35 eV for gallium. For two regions B and C where we have low SB and low temperature or high SB and high temperature ratio of jump rate across step to the jump rate along terrace DSB /D = 0.002. This can be compared with the ratio of jump rate along steps to jump rate along terrace Ds /D = 0.004 in the higher temperature region C and Ds /D = 0.0005 in the lower temperature region B. It appears that when we have dynamics with the same ratio of across step and along terrace rates, then rates along step matter in surface ordering. When jump probability along and across step are comparable particles have enough time to smoothen steps which create meanders. When wandering along steps is much slower than jumps down, steps are more rough and they do not arrange in a regular structure. In the region F, that is between B and C, rates for jumps down are smaller, they become again comparable with diffusion along steps and they lead to the formation of 3D mounds irrespectively of slope of the surface.
Please cite this article as: F. Krzyżewski, M.A. Załuska-Kotur, Journal of Crystal Growth (2016), http://dx.doi.org/10.1016/j. jcrysgro.2016.04.043i
F. Krzyżewski, M.A. Załuska-Kotur / Journal of Crystal Growth ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Fig. 4. Surface patterns observed during simulations: bent steps (A), rough surface (B), smooth meanders (C), double steps (D), bunch (E) and mounds (F). Letters A–F correspond to those presented in stability diagrams in Figs. 2 and 3.
The multiplicity of microscopic events in the system under study is large and simple reasoning about the surface structure based on the comparison of jump rates across different paths is difficult. Note that there are two components in the system, one is immobile. Moreover SB and ISB are assigned only to Ga adatoms. Such properties of adatom dynamics also influence evolution of the surface structure. The general observation that can be drawn from the analysis of stability diagrams and the surface patterns in Fig. 4. is that the presence of ISB is associated with a more smooth surfaces than the presence of SB when other growth conditions are kept the same. ISB barriers usually lead to straight steps. In our study all patterns: meandering, bunching, mounding and roughening are described within one model, so regions where they happen cross and sometimes overlap. Detailed analysis of the transition lines between different regions is difficult. Dashed lines presented in Figs. 2 and 3 roughly denote transitions between observed surface morphologies. Regions F and C in Fig. 2 correspond to unstable surface morphologies. Following dashed lines which separate those areas from stable region D one can notice that for higher miscut stable patterns appear for broader range of SB height. The range of stable phase at miscut 4° is from 0.4 eV to 0.3 eV, whereas at 1° surface is stable between 0.5 eV and 0.25 eV which is narrower. It is consistent with the results obtained in [21] where miscut and SB dependent surface stability was investigated.
Schwoebel barrier values as a function of temperatures and miscuts. It has been shown that formation of concrete surface morphologies such as bunches, meandered patterns, double steps, bent steps, rough surface or mounds is strictly related to the character and the effective height of Schwoebel barrier. If ISB effect plays a crucial role in the system, steps are straight and have tendency to assembly in bunches. The presence of direct SB leads to the more rough steps that can order in regular patterns of meanders or mounds. Meanders are formed at high temperatures and miscuts whereas mounds at low temperatures and miscuts. Straight, double step pattern can be found in the region where SB or ISB do not play an important role in the surface evolution. We have also shown the sensitivity of surface structure on miscut and temperature parameters. In such a way we can conclude about the existing effective ISB and SB barriers in the system from the morphology of the surface. In particular a study of the impurity and doping influence on the SB height is very interesting question. Deeper knowledge about such relations is a step towards proper tuning of inversed or direct Schwoebel barrier height to obtain various surface morphologies.
Acknowledgment Research supported by the National Science Centre (NCN) of Poland (Grant NCN no. 2013/11/D/ST3/02700)
4. Conclusions References Two-component kinetic Monte Carlo model of wurtzite crystals was used to study surface evolution of GaN (0001¯ ). We investigated surface evolution in nitrogen rich conditions for various
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