A challenge in performing crystal growth simulations is the large range of relevant length and time scales. While eventually being interested in a description on a mesoscopic/macroscopic scale (the size of typical defect or surface features is in the order of 10..100 nm and the growth time is in the order of seconds up to hours) the mechanisms leading to these structures (adatom adsorption, diffusion, desorption, island nucleation) require a resolution in the length scale of atomic bonds (10^-1nm) and in the time scale of atomic vibrations (10^-13s^-1). Therefore, common approaches to simulate growth have been restricted on specific properties (on the mesoscopic/macroscopic scale) and included microscopic information only indirectly by empirical/adjustable parameters. Examples are rate equations or continuum models. While these approaches give valuable insight into qualitative aspects of growth a quantitative analysis requires to include the microscopic mechanisms directly.

A rather new approach to describe microscopic growth mechanisms is the application of ab initio methods such as density-functional theory (DFT). The key idea of these methods is to describe nature on the most fundamental level: The growing crystal and its structural elements are decomposed into the most elementary building blocks such as atomic nuclei and electrons and the interaction between them is described by the fundamental laws of electrodynamics and quantum mechanics. In the present talk I will discuss how by combining density-functional theory with concepts of thermodynamics, continuum theory and/or statistical physics simulations can be performed which allow to bridge between microscopic and mesoscopic/macroscopic scales. Such multiscale simulations which combine methods developed for various length and time scale provide the unique opportunity to combine the advantages of ab initio methods (universality and predictive power) with the efficiency of mesoscopic/macroscopic models to describe system sizes relevant for crystal growth. To discuss the application but also the present limitations of ab initio based multiscale simulations the focus will be on various issues regarding crystal growth, defect formation and doping of wide bandgap semiconductors such as group-III nitrides (GaN, AlN, InN and their alloys).

In the first part of the talk the focus will be on equilibrium properties and structures. Specifically, it will be discussed how ab initio methods can be emplyed to predict bulk crystal properties such as thermal expansion coefficients, elastic constants, thermodynamic constants, formation energies of compounds, defects and impurities. Based on these results the presently achievable accuracy of these methods will be discussed [1-3].

A generalization of these concepts to surfaces allows to calculate the stability and electronic properties of surfaces/interfaces as function of the growth conditions (chemical potentials). Based on these results it became possible to identify conditions where e.g. surface reconstructions are stable or when the surface becomes unstable against step formation (surface roughening), faceting or the formation of nanostructures [4]. The same approach allowed also to identify the effect impurities/dopants have on these properties. An interesting result which emerged from these studies was that for certain growth conditions the dopant/impurity concentration on the surface can be orders of magnitude larger than in bulk. This has been shown to largely effect the growth morphology and can be even used to grow metastable phases which are otherwise not found in nature [5]. Also, this effect can be used to control the formation of nanostructures such as quantum dots, the formation of alloy fluctuations [6] or to achieve chemical ordering in semiconductors [4]. The effect dopants have on the surface morphology will be shown to also affect the doping efficiency. For example, the surface can enhance/reduce the formation of parasitic phases and thus the bulk solubility [7].

Finally, it will be discussed how the ab initio calculated barriers [8] can be used to perform growth simulations on a mesoscopic length and time scale. A direct approach would be to use the barriers to calculate the transition rates, construct a master equation and solve it by kinetic Monte Carlo (kMC). While this approach works well to study growth at low temperatures it becomes exceedingly expensive at high temperatures which are needed to achieve smooth surfaces. We have therefore developed a new method which is called adatom density Monte Carlo [9]. Using this method it will be shown how mechanisms controlling self-organization in V-grooves or lateral epitaxial overgrowth (which is used to reduce the dislocation density in group-III nitrides) can be identified.

In conclusion, the combination of density functional theory (giving an accurate description of the atomistic and electronic structure) with concepts of thermodynamics, statistical physics or continuum theory allows to address a wide range of crystal growth and doping problems which are not feasible by any of the methods alone due to the large range of relevant length and time scales. While this approach is still in its infancy first results are very promising and future improvements in the methods and in computers will allow to perform these types of studies routinely.

[1] C.G. Van de Walle, J. Neugebauer, J. Appl. Phys. 95, 3851-3879 (2004).

[2] J. Neugebauer, phys. stat. sol. (c) 6, 1651-1667 (2003).

[3] C.G. Van de Walle and J. Neugebauer, Nature 423, 626 (2003).

[4] J.E. Northrup, L.T. Romano, and J. Neugebauer, Appl. Phys. Lett. 74, 2319 (1999).

[5] J. Neugebauer, T. Zywietz, M. Scheffler, J.E. Northrup, and C. G. Van de Walle, Phys. Rev. Lett. 80, 3097 (1998).

[6] H. Chen, R.M. Feenstra, J.E. Northrup, T. Zywietz, J. Neugebauer, and D.W. Greve, Phys. Rev. Lett. 85, 1902 (2000).

[7] J. Neugebauer, phys. stat. sol. (b) 227, 93 (2001) .

[8] J. Neugebauer, T.K. Zywietz, M. Scheffler, J.E. Northrup, H. Chen, and R.M. Feenstra, Phys. Rev. Lett. 90, 056101 (2003).

[9] L. Mandreoli, J. Neugebauer, R. Kunert, and E. Schöll, Phys. Rev. B 68, 155429 (2003).

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1. First ab initio determination of the phase transformations in Ni₂MnGa
2. Ab initio based multiscale simulations of dislocations in GaN
3. Multiscale modelling of group-III nitride growth