W. Dieterich and P. Maaß
Near an interface or a free surface of a solid the atomic structure normally changes relative to the bulk structure, as a consequence of the fact that interatomic interactions near the surface are modified or disrupted.
While this is a widely studied subject, only few investi-gations have been carried out with respect to surface ef-fects on two-point correlation functions determining the cross section in a surface scattering experiment.
Away from thermodynamic equilibrium, ordering and growth processes of adatoms on surfaces are of particu-lar interest, mostly in connection with molecuparticu-lar-beam- molecular-beam-epitaxy (MBE) experiments which have become impor-tant in the production and design of new materials.
Several investigations have been carried out during last year in order to improve our understanding of atomic structures and correlations near solid surfaces and of surface-induced ordering and growth processes.
Ordering kinetics near surfaces of metallic alloys.
The Cu-Au system represents an example of a fcc-alloy whose ordered phases have experimentally been studied in the past in detail and which also can be de-scribed successfully in terms of simplified lattice gas models. Several special ordering phenomena are known to occur at the (001)-surface of Cu3Au which in the bulk orders in the L12 structure. These include an oscillatory segregation profile which above the bulk ordering tem-perature T0 decreases from the surface towards the bulk, and a continuous vanishing of the order parameter at T0 in the first atomic layer, despite of the fact, that the bulk material undergoes a first order phase transition. Time-resolved small angle X-ray scattering experiments at Cu3Au (001) have revealed a two-step ordering process after a temperature-quench from the disordered phase to a final temperature Tf<T0. After a rapid penetration of a segregation wave from the surface into the bulk, slow evolution of the near-equilibrium domain pattern has been observed, which consists of 4 types of ordered do-mains separated by antiphase boundaries.
Previously (see the Annual Report 1998) the essential features of these experiments could be analyzed within time-dependent Ginzburg-Landau theory 1). More recent studies were based on Monte Carlo simulations, where elementary atomic migration steps were affected by a vacancy mechanism (“ABv-model”; A=Cu; B=Au;
v=vacancy). After verification of this model in compari-son with the static segregation amplitudes, the velocity of the penetrating segregation front as a function of tem-perature was extracted from the simulations.
The final penetration depths are a result of the com-petition of the ordering wave with the spontaneous growth of bulk fluctuations.
Fig. 1: Propagation of the segregation profiles y(z,t) after a temperature quench from the disordered phase (Ti>T0) to the ordered phase (Tf<T0) of Cu3Au. a) Pro-files y(z,t) for a fixed Tf = 0.97T0 b) Time-dependence of the penetration depth for different Tf. c) Penetration ve-locity at short times versus Tf.
The ordering kinetics in nonstoichiometric crystals turns out to be markedly different from the stoichiomet-ric composition, an effect which calls for experimental verification.
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
penetration depth in atomic layers
time in mcs
temperature Tfin units of To
a)
b)
c)
Density functional theory
The density functional formalism (DFT) for lattice gases was developed further mainly with respect to a de-scription of ordered phases and surface effects. The as-sumption of a nearest neighbor repulsion between parti-cles on a fcc-lattice is known to yield a phase diagram which reasonably describes the experimental phase dia-gram of the Cu1-xAux system. Using a particular version of the so-called “weighted density approximation”
(WDA) the phase diagram of such alloys in the regime x
< 0.3 could well be reproduced (see Fig. 2), establishing the DFT as a possible alternative to the well-known cluster variation method (CVM). In surface problems, however, the DFT appears to bear essential advantages over the CVM. This was demonstrated by calculating surface-induced two particle-correlation functions in a lattice gas with nearest-neighbor attraction in a strip ge-ometry. The results of that study compare favorably with computer simulations (Fig. 3). In such problems it is es-sential to use an interaction free energy functional that conforms with the particle-hole symmetry in lattice systems. A new type of functional, the “semilinear DFT”, turned out to be appropriate in reproducing es-sential qualitative aspects in lattice gases under dimen-sional reduction 2).
Fig. 2: Phase diagram of an AxB1-x alloy on a fcc lattice with effective nearst-neighbor repulsion V>0. The tran-sition temperature (in units of V/kB) as a function of the concentration cA of A-atoms was calculated using the hybrid weighted density approximation (HWDA). For comparison we also show the CVM in tetrahedron ap-proximation. The CVM correctly predicts ordered phases of AB3 and A2B2 type, whereas in the (non-sym-metric) HWDA only AB3 type ordering occurs.
Finally we generalized the DFT to the Potts-model.
Furthermore some steps were undertaken towards an extension of the DFT to a variational principle for multi-particle correlation functions. The latter aspect is of in-terest in particular in a comparison of the DFT with the general procedure in CVM theories.
Fig. 3: Lateral pair correlation function H||(x,y) in a lat-tice gas with strip geometry of width L=40 versus lateral coordinate y in the first row (x=0) and far from the sur-face (x=19) resulting from MC simulations and the se-milinear DFT. Temperatures in a) and b) are given in units of |V|/kB. Energetically neutral surfaces were as-sumed, with constant (x-independent) particle density px=0.5.
Wall induced correlations in Takahashi lattice gases
An exact formalism was developed to study the structural properties of one-dimensional lattice gases composed of particles with nearest neighbor interactions of arbitrary range (“Takahashi lattice gas”) in the pres-ence of an arbitrary confining wall potential 4). The pur-pose of these exact studies is twofold: On one hand they allow us to gain valuable insight into the behavior of higher-dimensional systems. On the other hand, it is possible to address the question for which physical quantities the canonical and grand canonical ensemble yield comparable results in such confined systems.
We derived linear recursion relations for generalized partition functions of the Takahashi lattice gas exposed to an external potential, from which thermodynamic quantities, as well as density distributions and correla-tion funccorrela-tions of arbitrary order can be determined. Ex-plicit results for density profiles and pair correlations near a wall were presented for various situations. In par-ticular we considered as a special case the hard rod lat-tice gas, for which a system of nonlinear coupled differ-ence equations for the occupation probabilities previ-ously derived by Robledo and Varea could be reduced to a much simpler system of independent linear equations.
It was further shown that for an external potential de-scribing two hard confining walls, various central
re-0.3
gions exist in the hard rod lattice gas, where the occupa-tion probabilities are exactly constant and the multi-point correlation functions are exactly translationally in-variant in the canonical ensemble, while in the grand ca-nonical ensemble such regions do not exist.
Exact density functionals in one dimension (d=1) Under geometrical restriction of a d-dimensional system in one direction one finally ends up with the cor-responding (d-1)-dimensional system. Exact knowledge of the statistical mechanics in reduced dimension there-fore yields an important reference for approximate theo-ries in confined systems of higher dimension. Under that aspect we further elaborated on the construction of exact density functionals for 1-d lattice gases. A general method was proposed for arbitrary nearest-neighbor in-teractions, including hard-core contributions of arbitrary size. Our procedure covers all previously known exact functionals for 1-dimensional lattice gas or fluid sys-tems, as developed predominantly in works by J. Percus et al. Our method is based on a generalized Markov-property for the probability of occupational configura-tions, where the “memory” in the conditional probabili-ties has a range given by the range of interactions 3).
The structure of the resulting grand free energy func-tional becomes most transparent in the continuum limit.
It contains both the 1-particle density r(x) and the 2-particle density r(x,y) which however is coupled to r(x) via a nonlinear integral equation. Again it is possible to generalize this theory to Potts-models.
It appears that with similar ideas approximate func-tionals for d=2 and d=3 can be constructed which reduce to the exact case d=1 (and of course also to d=0) under dimensional reduction.
Nucleation on top of islands in epitaxial growth An important process controlling the morphology of thin films grown by vapor deposition is the nucleation of stable atom clusters in the second layer on top of islands in the first layer. Smooth films growing by a layer-by-layer mode develop if stable clusters in the second layer-by-layer form after coalescence of islands in the first layer, while second layer nucleation preceding island coalescence leads to a rough film morphology. The onset of second layer nucleation occurs rather sharply, when the islands in the first layer have acquired a critical mean radius Rc. Accordingly, a simple criterion for the occurrence of rough multilayer growth is that Rc is smaller than the mean distance l of islands in the first layer (in the satu-ration regime of almost constant island density before coalescence).
We developed a stochastic theory for second layer nucleation based on scaling arguments 5), by which the overall nucleation rate on top of islands and the critical island radius Rc could be determined in dependence of the relevant experimental parameters.
These parameters are the incoming atom flux F, the jump rate D0 /a2 of adatoms (with a being the jump
dis-tance on the substrate surface), and the additional step edge barrier DES (Schwoebel barrier), which in addition to the bare surface diffusion barrier has to be sur-mounted by an adatom, when it crosses an island edge.
For small critical nuclei of size i£ 2 (a cluster composed of i+1 atoms is stable) the theory predicts
Rc~Ggam
where Gº D0 /Fa4, a º exp(-DES/kBT), and the expo-nents g and m have different values depending on how a and G relate to each other. Overall there exist 4 different regimes:
Fig. 4: The various scaling regions of second layer nu-cleation for a critical nucleus of size i=1 in an a-G dia-gram (see eq.(1)). The thick dashed line with slope (-6) marks the onset of smooth layer-by-layer growth and the circles refer to the onset of island coalescence found in the kinetic Monte-Carlo simulations.
The various regions I-IV for i=1 are pictured in the dynamical phase diagram shown in Fig. 4, which in ad-dition entails the transition line from rough multilayer to smooth layer-by-layer growth. Kinetic Monte Carlo simulations have been performed to test the validity of the theory, and the results from these simulations are in good agreement with the theoretical predictions (Fig. 5).
Since for large step edge barriers the result (1) deviates from the predictions of a mean-field type approach pro-posed earlier 5), we suggested to reexamine various
ex-10
4periments, which employed the mean-field approach to estimate the Schwoebel barrier in some materials. Re-cently, this reexamination has been carried out 7).
Fig. 5: Critical island size Rc obtained from kinetic Monte-Carlo simulations as a function of
) T k E exp(-D S/ B º
= for i=1 (upper figure) and
vari-ous ratios GºD0/Fa4 in a range 105 to 1012 , and for i=2 (lower figure) and various G in the range 105 to 109. The regimes I-IV (see text) are indicated together with their border lines.
Further elaboration on the stochastic theory shows that the mean field approach suggested in ref. 6 becomes essentially valid for critical nuclei of size i³3. For i=1 as well as for i=2 and large DES however, the nucleation process is dominated by fluctuations that are not ac-counted for by the mean field approach. In the language of critical phenomena, one may regard i=2 as the upper critical size of the critical nucleus above which mean field theory becomes applicable. Currently we are working on an extension of the stochastic approach in order to include the lifetimes of unstable clusters with comparatively high dissociation energies.
(1) H.P. Fischer, J. Reinhard, W. Dieterich and A.Majhofer, Europhys. Letters 46 (1999) 755
(2) J. Reinhard, W. Dieterich, P. Maass and H.L. Frisch, Phys. Rev. E 61 (2000) 422
(3) J. Buschle, P. Maass and W. Dieterich, J. Phys. A.:
Math. Gen. 33 (2000) L41
(4) J. Buschle, P. Maass and W. Dieterich, J. Stat. Phys. 99, No. 1/2 (April 2000), in press
(5) J. Rottler and P. Maass, Phys. Rev. Lett. 83 (1999) 3490 (6) J. Tersoff, A.W. Denier van der Gon and R.M. Tromp,
Phys. Rev. Lett. 72 (1994) 266
(7) J. Krug, P. Politi and Th. Michely, e-print cond/mat 9912410
10 100
10
−610
−510
−410
−310
−210
−110
0R
c/a
α =exp[− ∆ E
s/k
BT]
~α1/7
I
II
III IV
10 100
10
−610
−510
−410
−310
−210
−110
0R
c/a
α
=exp[−
∆E
s/k
BT]
~α1/5
~α3/7