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Surfactant-Mediated Growth of Nonequilibrium Interfaces

A number of recent experiments have showed that surfactants can modify the growth mode of an epitaxial film, suppressing islanding and promoting layer-by-layer growth. Here I introduce a set of coupled equations to describe the nonequilibrium roughening of an interface covered with a thin surfactant layer. The surfactant may drive the system into a novel phase, in which the surface roughness is negative, corresponding to a flat surface.

There is much theoretical interest in the statistical properties of nonequilibrium surfaces.
The growing interfaces naturally evolve into self-affine structures; the surface morphology and the dynamics of roughening exhibit simple scaling behavior despite the complicated nature of the growth process [1]. In particular, much attention has focused on different models to describe thin-film growth by molecular-beam epitaxy (MBE) [2][3][4][5][6].
Under ideal MBE conditions the primary relaxation mechanism is surface diffusion, which conserves the mass of the film. Experimentally both lattice strain and surface free energy determine whether the film undergoes layer-by-layer growth, islanding, or layer-by-layer growth followed by islanding. In experiments involving growth of Ge on Si(100) surface layer-bylayer growth is limited to 3-4 monolayers (ML) due to the lattice mismatch between Si and Ge and is followed by formation of unstrained Ge islands. It was shown recently that islanding in the Ge/Si system can be suppressed effectively by use of a surfactant monolayer, changing the growth mode from island growth to layer-by-layer growth [7]. Suitable surfactants such as As and Sb strongly reduce the surface free energy of both Si and Ge surfaces and segregate at the surface during growth. A number of subsequent experiments showed that surfactants can change the surface morphology in a wide variety of systems: Sb altered the growth behavior of Ag on Ag(111) [8], antimony was found to change the structure of islands in Ge/Si growth [9] and Te was used as surfactant to sustain layer-by-layer growth of InAs on GaAs(001) [10].
Microscopically, arsenic placed on the surface of Si or Ge lowers the surface tension by eliminating the dangling bond states. But for ordinary surfactants reduced surface tension enhances roughness. The reduced adatom diffusion might change the growth mode, but so far most continuum models with or without surface diffusion predict rough interfaces [2][3][4][5][6]11].
While the microscopic mechanism of As and Ge/Si interaction is much investigated [12], understanding in the framework of continuum growth models is still lacking.
In this paper I study the generic problem of nonequilibrium roughening of an interface covered by a thin surfactant layer (see Fig 1). Building on experimental results and general symmetry principles, a set of nonequilibrium equations are proposed to describe the growth 2 of an interface coupled to the fluctuations in the surfactant coverage. The analytic study of these equations indicates that the surfactant changes drastically the morphology of the interface in 2+1 dimensions. In particular, the coupled system supports the existence of a novel phase characterized by negative roughness exponent, which can be identified with a morphologically flat surface.
As mentioned above, under ideal MBE conditions, relaxation proceeds via surface diffusion. In contrast to the ideal MBE, there is experimental evidence that surfactant mediated growth of Ge on Si proceeds by highly local Ge incorporation with minimum surface diffusion [13]. Ge atoms that adhere to the As-capped surface rapidly exchange sites with the As atoms and incorporate into subsurface sites. In the absence of surface diffusion, the growth equation may contain terms which violate mass conservation [14]. The simplest nonlinear growth equation with nonconserved dynamics was introduced by Kardar, Parisi, and Zhang (KPZ) [15]: Here h(x, t) is the height of the interface in d = d ′ + 1 dimensions. The first term on the right hand side describes relaxation of the surface by a surface tension ν. The second term is the lowest order nonlinear term that can appear in the interface growth equation, and is related to lateral growth. η(x, t) is a stochastic noise driving the growth; it can describe thermal and beam intensity fluctuations. Additional terms in (1) will include the coupling to the surfactant fluctuations.
An efficient surfactant must fulfill two criteria: it must be sufficiently mobile to avoid incorporation, and it must surface segregate. Careful experimental studies showed for the Ge/Si system that the bulk As concentration is less than 1%; thus the effect of As on growth is a surface phenomena [7]. Neglecting the desorption of the surfactant atoms, the equation governing the surfactant kinetics obeys mass conservation. This leads to the is the width of the surfactant layer [16], η ′ is a conserved uncorrelated noise which incorporates the random local fluctuations of the surfactant, and j is the particle-number current density. The simplest linear equation with conserved dynamics correctly incorporating the effect of surface diffusion is [2] To account for the coupling between the growing surface and the surfactant it is necessary to introduce additional terms in Eq. (1) and (2). There are two main criterias which restrict our choice: The coupling terms must satisfy the symmetry conditions characteristic of the interface and the obtained set of equations should be self-consistent, i.e. the resulting dynamics should not generate further nonlinear terms. In addition the coupling terms included in Eq. (2) must obey the required mass conservation for the surfactant.
The simplest set of equations that satisfy the above conditions is where the noise terms η 0 and η 1 are assumed to be Gaussian distributed with zero mean and the following correlator: Here and The D 2 term is generated by D 0 and D 1 as will be shown below.
The generic nonlinear term (∇v) 2 in (3a) can be derived using symmetry principles. In (3b) the ∇ 2 [(∇h) · (∇v)] term results from a current j = −∇[(∇h) · (∇v)], and obeys mass conservation. Geometrical interpretation [5] of this term suggests that a positive γ drives the surfactant to cover uniformly the irregularities of the surface, i.e. enhances the wetting properties [17]. A negative γ has the opposite effect, assigning a non-wetting character to the surfactant. Since in experiments there is no evidence of surfactant agglomeration (nonwetting character), but it is energetically favorable to terminate the Ge layer with As atoms, we assume that the surfactant wets the surface, thus γ > 0.
The quantity of main interest is the dynamic scaling of the fluctuations characterized by  The diagrams contributing to λ cancel each other, resulting in the scaling relation z + χ 0 = 2.

(7)
This relation is known to be the property of the KPZ equation and it is a consequence of Galilean invariance (GI). Since the DRG conserves the GI, this scaling law is expected to remain valid to all orders of the perturbation theory.
A second scaling relation can be obtained from the non-renormalization of the diffusion coefficient D 1 : The diagrams that contribute to D 1 have a prefactor proportional to k 4 , thus they are irrelevant (k is the wave vector in the Fourier space). They in fact contribute to D 2 , justifying its introduction in (6).
These two scaling relations already indicate that the coupled interface/surfactant system is qualitatively different from the uncoupled one. For a planar interface (d ′ = 2) (7,8) give which means that at least one of the exponents has to be negative.
A third scaling relation unfortunately is not available, but insight can be obtained from numerical integration of the flow equations obtained from the DRG [20]. The integration showed the existence of two main regimes: (i) In the first regime one or both of the coupling terms (β, γ) scale to zero. In this case the two equations become completely (both coupling terms vanish) or partially (only one coupling term vanishes) decoupled, and the two equations might support different dynamic exponents z. The DRG scheme used is not reliable in this regime.
(ii) The presence of a strong coupling fixed point is expected when both of the nonlinear terms diverge. The integration shows that this coupled phase exists only for z ≥ 8/3 [21].
It is important to note that although there is no identifiable fixed point, in this phase the scaling relations (7,8) are exact. According to (7), for z ≥ 8/3 the roughness exponent of the interface χ 0 is negative. With a negative roughness exponent, every noise-created 6 irregularity is smoothed out by the growth dynamics and the resulting surface becomes flat.
Thus the coupling of the surfactant to the growing interface results in the suppression of the surface roughness. This corresponds exactly to the experimentally observed behavior, i.e.
the addition of the surfactant suppresses islanding, resulting in a morphological transition from rough (without surfactant) to flat (with surfactant) interface.
The roughness exponent of the surfactant from (8) is negative if z < 4, while for z > 4 it becomes positive. In the Ge/Si system, for example, the As has a saturation coverage of 1ML, which is independent of the system size and is governed only by the microscopic bonding of the As to the Ge dangling bonds. One expects no relevant fluctuations in the thickness of the coverage; this requires a negative roughness exponent for the surfactant and thus limits the dynamic exponent to values smaller than four.
The DRG analysis fails to provide the exact value of the dynamic exponent z. As in the case of many other growth phenomena, simple discrete models might be very helpful to obtain its value [22]. Summarizing the results of the direct integration of the DRG equations,