The microscopic dynamics of surface roughness and pattern formation induced by ion sputtering can be described by the noisy nonlinear Kuramoto-Sivashinsky (KS) equation [

12] which defines the surface height

*h*(

*x*,

*y*,

*t*) with

*x* and

*y* lying in the surface plane:

Here *v*
_{o}(*θ*) is the rate of erosion of the unperturbed planar surface; *υ*
_{
x
} and *υ*
_{
y
} represent the effective surface tensions generated by the erosion process [13]; *D*
_{
x
}
*D*
_{
y
} and *D*
_{
xy
} denote the surface relaxation kinetics; *τ*
_{
x
} and *τ*
_{
y
} describe the tilt-dependent erosion rates [14]; and *η*(*x,y,t*) represents an uncorrelated white noise component with zero mean, which incorporates the randomness resulting from the stochastic nature of ion arrival at the surface [15]. This expression recognizes the fact that surface relaxation is governed by two different diffusion processes. The terms with coefficients *D*
_{
x
} and *D*
_{
y
} are thermally activated. Their smoothing rates are based on mass transport on the surface. *D*
_{
xy
}∂^{2}
*/*∂*x*
^{2}(∂^{2}
*h/*∂*y*
^{2}), the sputtering induced diffusion, is regarded as a smoothing contribution in the morphology evolution without mass transport.

In general, ion bombardment provokes an anisotropic instability giving rise to characteristic ripple patterns [

13]. In a very special case where the ion beam impinges perpendicular to the target surface, coefficients in Eq. (

1) become isotropic and a regular matrix of dots is expected to be formed [

16]. The temporal surface height evolution can then be expressed as an isotropic KS equation [

5]:

where

*J* is the ion current density,

*p* the proportionality factor coupling the energy deposited to the erosion rate [

13],

*ε* the total energy deposited, and

*a* the average depth of energy deposition.

*α* and

*β* are the widths of the distribution parallel and perpendicular to the beam direction, respectively [

10] (generally

*a* *α*, and

*β* are comparable in magnitude [

13]). The diffusion coefficient

*D* in Eq. (

2), which is assumed isotropic, includes all diffusion coefficients, i.e., the thermal diffusion (

*D*
_{
t
}) and effective sputtering induced diffusion (

*D*
_{eff}).

Here

*E*
_{
a
} is the activation energy,

*k*
_{
B
} the Boltzmann’s constant, and

*T* the temperature. In Eq. (

2), the balance of the unstable erosion term (

*υ*∇

^{2}
*h*) and the surface diffusion term (−

*D*∇

^{4}
*h*) acting to smooth the surface, generates dots with characteristic wavelength that equals:

It is difficult to differentiate the two diffusion mechanisms when they simultaneously co-exist. However, at low temperature and comparably high ion energy,

*D*
_{
t
} is negligibly small compared to

*D*
_{eff}, and the effective ion-induced diffusion should dominate over thermal diffusion [

10]. Hence, based on Eqs. (

3), (5), and (6),

*l*
_{
c
} becomes:

Because

*β*, the lateral width of the energy deposited, is related to the sputtering energy (

*ε*) by

*β* *∝* *ε*
^{2m
} [

5], the characteristic wavelength is related to the sputtering energy by a power law [

10]:

The parameter *m* is between 0 and 1. *m* = 1 holds for Rutherford scattering. In the lower-keV and upper eV region, *m* = 1/3 should be adequate [17]. Equation (8) which implies the characteristic wavelength, is a strong function of the ion energy and independent of other parameters, for instance the ion beam flux and sample surface temperature, in the case of sputtering induced diffusion.