Elsevier

Microelectronic Engineering

Volumes 43–44, 1 August 1998, Pages 497-505
Microelectronic Engineering

Crystalline lattice effects on tensionless surface dynamics

https://doi.org/10.1016/S0167-9317(98)00210-XGet rights and content

Abstract

A new model is introduced for two-dimensional crystalline interfaces with negligible surface tension. The model is given by a discrete version of the linear molecular beam epitaxy (MBE) equation plus an additional term periodic in the interface height variable. Langevin dynamics simulations and analytical arguments show that the model exhibits a roughening transition to the high temperature phase of the sine-Gordon model, whose initial stages are nevertheless described by the scaling of the linear MBE equation. Out of equilibrium, the model can have three different behaviors depending on temperature and deviation from equilibrium: A non-moving flat interface, a moving interface with oscillatory roughness, and a moving interface which shows scaling. Possible connections to experiments are discussed.

Introduction

Surface and interface roughness are exceedingly important problems in the fabrication of low dimensional semiconductor devices 1, 2. Rough surfaces develop as a consequence of competition among different effects, such as surface tension, surface diffusion, thermal fluctuations, lattice effects, applied forces, and so on. Thus, in equilibrium, a particular surface can be rough under certain conditions and macroscopically “flat” under some other ones, these two regimes being separated in many cases by a roughening transition 1, 3. Out of equilibrium, which is usually the case when growing micro and nanoelectronic devices, surfaces can grow in many different modes, ranging from layer by layer (LBL) to rough growth morphologies: Experimentally, ordered LBL growth is detected as oscillations in the reflection high-energy electron diffraction (RHEED) specular intensity [4], whereas for a growing rough interface the coherence leading to RHEED oscillations is lost.

Among the studies of surface dynamics, systems with negligible surface tension have received a lot of attention lately. Relevant instances of tensionless surfaces are thin films grown by, e.g. thermal evaporation, sputter deposition or Molecular Beam Epitaxy (MBE). Generically, we will refer to these as MBE growth 1, 4, 5. Villain [6]and Lai and Das Sarma [7], elaborating on classic works by Herring and Mullins 8, 9, proposed that surfaces grown in MBE-like conditions obey relaxation mechanisms (namely, surface diffusion) that locally minimize surface curvature. To linear order this is described by the so called linear MBE equation for surface height h(x,t):ht=−κ∇4h+I+ξ(x,t),where κ is a constant and ξ is a random fluctuation around the average flux I. This equation leads to scale invariant behavior of the growing surface both in time and space [1], which has been recently demostrated in real MBE systems 10, 11.

Besides the above relaxation mechanism, there are other processes which break this scale free picture of surface growth. One of them is the Schwoebel effect: a diffusion barrier at step edges which prevents an atom from jumping downward to a terrace, and therefore would generate a rough surface containing mounds, whose typical size provides a length scale [6]. On the other hand, lattice effects can also introduce a characteristic length scale equal to the distance between the surface coarse-grained units. There are several experimental situations in which the Schwoebel effect is absent, such as the dynamics of policrystalline interfaces, or surface growth at high enough temperatures. In this work we will neglect Schwoebel barriers and study the interplay between lattice effects and surface diffusion. In principle at high enough temperatures lattice effects are expected to become negligible so that the surface would evolve according to Eq. (1). However, we will see that lattice effects modify the scaling behavior at high temperatures so that Eq. (1)only holds up to a certain crossover time and therefore is not asymptotic.

The model we propose is hence given by the HamiltonianH=κin.n.hj−nhi2−V0icos(hi/a)−iIhi,where the functions hi(t) give the surface height above site i in a two-dimensional (2D) square (L×L) lattice, and the sum in the brackets runs over the nearest neighbors of site i, n=4 being the lattice coordination number; we emphasize that the values of hi are real numbers not restricted to being integers. The first term in Eq. (2)is a discrete version of the Laplacian squared of the height, which favors small surface curvatures and leads to the biharmonic term (4 h) in the evolution equations (see , below). The second term in Eq. (2)is a weighting function which favors hi to be 2nπa, with a the lattice spacing. Such a pinning potential was first introduced by Chui and Weeks [12]to study a continuous version of the discrete Gaussian model, which in this way becomes a sine-Gordon (sG) model [13]. Finally, the third term in Eq. (2)explicitly includes non-equilibrium effects through the interaction of the surface with applied driving fields I which represent, e.g., the chemical potential due to the flux of incoming particles in MBE [14].

We study model Eq. (2)by Langevin dynamics simulations, i.e., by integrating the corresponding overdamped equations of motion for hi [rescaled as to have κ=a=V0=1]:hit=−∇2d(∇2dhi)−sinhi+I+ξi(t).d2 stands for the discrete Laplacian in the square lattice and ξi(t) are independent Gaussian white noises of zero mean and 〈ξi(t)ξj(t′)〉=2ijδ(tt′), T being the temperature. Thus, Eq. (3)reproduces the long distance behavior of the continuum equationht=−∇4h−sinh+I(x)+ξ(x,t),Langevin dynamics has been very successful in the analysis of related models, such as the discrete 2D sG equation [15]: For instance, in [16], the equilibrium roughening temperature was determined, confirming the renormalization group (RG) results of Chui and Weeks [17], whereas in [18]it allowed to explain the system behavior near and far from equilibrium. From the MBE viewpoint, the study we carry out in this paper is interesting, as it is clear that if lattice (epitaxial) effects are important, an equation like Eq. (4)could belong in a general description of crystal growth by MBE 19, 20, 21.

The outline of this paper is as follows. In Section 2we present the simulation results of our model in equilibrium and discuss the existence for Eq. (3)of a scaling behavior different from that of Eq. (1)due to the lattice effects. Section 3is devoted to non-equilibrium results and the driving-temperature phase diagram. The final section is dedicated to a summary of our main conclusions and possible experimental implications.

Section snippets

Equilibrium behavior

Our study begins with the main question about the equilibrium (I=0) of our model: The existence or not of a roughening transition. As we know of no previous analytical or numerical results for , , we recall the main RG ideas as applied to the sG equation 14, 17, 22: There exists a roughening temperature TR above which the energy of a step on the surface becomes zero, because temperature “renormalizes” the sine term, effectively suppresing it at TR; along this process, the lattice potential

Non equilibrium behavior

For the the non equilibrium (I≠0) case, Fig. 3 summarizes our results, which lead to a separation of the (T,I) parameter space into three different regions. In region A, the lattice potential controls the dynamics, and the surface shows zero average velocity like in the low T phase at equilibrium. Increasing I further for fixed T within region A, a value of the driving is reached which is strong enough to pull the surface over the potential barriers, leading to a nonzero average velocity v for

Conclusion

We have proposed and studied a model for growth of crystalline tensionless surfaces [Eq. (2)] whose main features are an equilibrium roughening transition with TR≈10 (dimensionless units) to a high temperature phase similar to the sG model, and a non-equilibrium “phase diagram” composed of three regions where the surface is pinned, moves nonlinearly, or roughens kinetically. The nonlinear region, B, is in fact a crossover region from LBL growth (close to the pinned phase, A) to rough growth

Acknowledgements

We are indebted to L.A.N. Amaral, A.R. Bishop, J.A. Cuesta, E. Diez, and M. Kotrla for discussions. This work has been supported by CICyT (Spain) grant MAT95-0325.

References (27)

  • A.-L. Barabasi, H.E. Stanley, Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge UK,...
  • M. Scheffler, R. Zimmermann (Eds.), Proceedings of 23rd International Conference on the Physics of Semiconductors,...
  • S.A. Safran, Statistical Thermodynamics of Surfaces, Interfaces, and Membranes, Addison Wesley, Reading MA,...
  • M.G. Lagally (Ed.), Kinetics of Ordering and Growth at Surfaces, NATO ASI Series B, Vol. 239, Plenum, New York, 1990,...
  • J. Krug, Adv. Phys. (in...
  • J. Villain

    J. Physique I.

    (1991)
  • Z.-W. Lai et al.

    Phys. Rev. Lett.

    (1991)
  • C. Herring

    J. Appl. Phys.

    (1950)
  • W.W. Mullins

    J. Appl. Phys.

    (1957)
  • H.-N. Yang et al.

    Phys. Rev. Lett.

    (1996)
  • J.H. Jeffries, J.-K. Zuo, M.M. Craig, ibid. 76 (1996)...
  • S.T. Chui et al.

    Phys. Rev. B.

    (1976)
  • They also showed that any periodic function gives the same static behavior at the roughening point, so there is no loss...
  • Cited by (0)

    View full text