4  Determination of the Functions related to the Material Distribution

The material system refers to the set of species that are present in the original solid and the bombarding particles. The experimental conditions refer to the energy of the primary particles and their angle of incidence on the surface. In the IMPETUS codes it is the practice that the physical properties of general material distributions are derived from the known properties of the pure material species or from 50:50 mixtures (under the particular experimental conditions) which are communicated to the IMPETUS codes through a set of data files known as the material system database. The method of obtaining the values in the data files is beyond the scope of this paper and the reader is referred to reference [5] for further details. The databases used in this work are largely obtained from the TRIM program [9]. An example of a material system database is given in the IMPETUS II User Manual [7].

4.1  Properties of the Material System and Experimental conditions

The mathematical model of the previous section has introduced the diffusivity and range functions. The recession is a function of dose only and it is dependent on the sputter yield of the species which is in turn related to their surface concentrations. The diffusivity and range are functions of depth and vary with f only insofar as they are dependent on the material distribution and these functions can be modelled through reference to the properties of the material system. The diffusivities are related to the nuclear energy deposition function. The range and energy deposition functions are probability density functions and represent the proportion of bombarding particles or energy that is expected to be deposited per unit depth. In this section it is shown how these functions are determined for a general material distribution from the relevant properties of the material system.
The energy and range functions are probability density functions. The nuclear and inelastic energy functions FEi(x) and FIi(x) for each matrix of i-material represent the proportion of the bombarding energy that will be deposited as nuclear or inelastic energy in the unit of depth about x. The range function of a particle of the primary species in the solid of pure i-material is denoted fRi(x) for i=1,2,...,n.
It is assumed that the energy deposition functions for the pure i-species have the following form
FEi(x) = (aEi + bEi x) exp(-lEi x) ,
(4)

FIi(x) = (aIi + bIi x) exp(-lIi x) 
(5)
for i=1,2,..,n and the range functions for the pure species have the form
fRi(x) = (cRi + dRi x) exp(-
(x-
-
x
 
Ri 
 )2
2sRi2
 ) .
(6)
The values aEi, bEi, lEi, aIi, bIi, lIi,cRi, dRi, [`x]Ri, sRi (for i=1,2,..,n) are obtained from the material system database.

4.2  Energy Deposition Functions

The energy deposition functions for solids of the pure species alone consist of the nuclear energy FEi(x) and the inelastic energy FIi(x) deposited per unit depth in pure i material. Let Ei(x) for i=1,2,...n be the energy retained for deposition in a solid of the pure species at depths x¢ Î [x,¥), it follows that
Ei(x) = ó
õ 		¥

x 
 ( FEi(x¢) + FIi(x¢) ) dx¢ .
(7)
The energy retained at the surface of the pure species target must also be equal to the energy of bombardment less the backscattered energy
Ei(0) = E0 - Eiback 
(8)
where Eiback is the backscattered energy for the bombardment of the pure i-species with energy E0 and it is obtained from the material system database.
For an arbitrary material distribution the energy deposition functions can be determined by the following method. Let the conditions of bombardment state that the primary particles have energy E0. Let E(x, f) be the energy retained at depth x in the general solid at dose f, it follows that
E(x,f) = ó
õ 		¥

x 
 ( FE(x,f) + FI(x,f) ) dx .
(9)
although the functions E, FE and FI are still undetermined.
The energy retained at the surface is modelled by a linear approximation based on the surface concentrations of the species in the surface layer
E(0,f)=E0- n
å
i=1 
 qSi(f) Eiback
(10)
where qSi(f) (i=1,2,..,n) represents the surface concentrations at dose f, the modelling of these concentrations will be discussed later.
The energy retained functions are all decreasing functions of depth x. A method for determining the energy functions for a general material distribution is now given. Let it be assumed that E(x, f) is known up to the point x=x*( ³ 0). Let yi(x*) be the depth in the pure i-species for which Ei(yi(x*)) = E(x*,f). The energy functions FE and FI are obtained through assuming that they depend linearly on the local concentrations of the species:
FE(x*,f) = n
å
i=1 
 qi(x*,f) FEi(yi*),
(11)

FI(x*,f) = n
å
i=1 
 qi(x*,f) FIi(yi*).
(12)
The energy retained at x* + e is simply equal to the energy retained at x* less the energy deposited in [x*,x*+e]. Hence it can be approximated by
E(x* + e, f) » E(x*, f) - e( FE(x*,f) + FI(x*,f) )
or more accurately by the trapezium rule
E(x* + e,f) » E(x*,f) - e
2
 ( FE(x*,f) + FI(x*,f) + FE(x*+e,f) + FI(x*+e,f) )
for a given e, for example. In IMPETUS II, the energy is based on the latter approximation.
Since E(0,f) is given by equation (10), the energy functions FE, FI and E for any given material distribution at dose f can be (numerically) determined for all x from the pure species functions FEi, FIi and the backscattered energy Eiback for i=1,2,...n. See Badheka [5], section 3.2.2 for comparison of results from the above model with experimental results.
In figure 3 the nuclear energy functions for pure Silicon, pure Germanium and 50:50 mixture of the two are given. Figure 4 shows the inelastic energy function for the same target materials. The area beneath the curves represents the penetration energy, which is dependent on the surface concentration and is therefore different for the three cases.

Figure 3. Nuclear energy deposition functions for pure species and 50:50 mixture.

4.3  The Range Function

The depth reached in the solid by a bombarding particle is termed its range and it is governed by a probability density function. The functions for a particle of the primary species in the solid of pure i-material in [xA, xB] is denoted fRi(x) for i=1,2,...,n. Let Vi(x) be the volume of particles retained for deposition in a solid of the pure species at depths x¢ > x, it follows that
Vi(x) = ó
õ 		¥

x 
 fRi(x¢) dx¢ .
(13)


Figure 4. Inelastic energy deposition functions for the pure species and for a 50:50 mixture.

Let V(x,f) be the volume of particles retained at depth x in a general solid at dose f, it follows that
V(x,f) = ó
õ 		¥

x 
 fR(x¢,f) dx¢ .
(14)
although the functions V and fR are still undetermined.

Let Vipenet be the volume of particles penetrating the surface of a solid of the pure i-material, the values of which are stored in the material system datadase. For a general material distribution at dose f, the particles penetrating the surface and deposited in the solid is modelled by a linear approximation based on the concentrations in the surface layer, that is
V(0,f) = n
å
i=1 
 qSi(f) Vipenet ,
(15)
where the qSi(f) for i=1,2,..,n represent the surface concentrations at dose f.
The proportion of the particles that reach a point x and are deposited in that unit interval is taken to be a function of the energy retained only. Thus for each i-material the proportion pi(ei) can be written as
pi(ei) = fRi(x)
Vi(x)
(16)
where ei is the energy retained in the solid i-material at depth x.
A method is now outlined for determining the range function for a general material distribution. Let it be assumed that V(x,f) is known up to the point x=x* ³ 0 and that the function E(x,f), the energy retained at depth x in the composite material is known (see the previous sub-section). The probability that a particle that reaches x* will be deposited in the following unit interval is assumed to be linearly dependent on the concentrations, so that the volume of particles deposited is
fR(x*,f) = V(x*,f) n
å
i=1 
 qi(x*,f) pi(e*) ,
(17)
At the end of an interval [x*,x*+e], the volume retained can be approximated as follows:
V(x*+e,f) » V(x*,f) - efR(x*,f).
(18)
or by the trapezium rule
V(x* + e,f) » V(x*,f) - e
2
 ( fR(x*,f) + fR(x*+e,f) ).
(19)
for a given e. In IMPETUS II the latter approximation is employed.
Since V(0,f) is given by (15), the range functions fR and V for any given material distribution at dose f can be (numerically) obtained for all x from the energy retained function E, the pure species functions fRi and the penetration values for the pure species Vipenet The range functions for pure Silicon, pure Germanium and 50:50 mixture of the two are given in figure 5.

Figure 5. Range deposition functions for the pure species and for a 50:50 mixture.

4.4  The Diffusivity functions

The effective diffusivity of a particle of the i-species in a m-matrix is modelled by the following equation
dim(x) = gim R2im
Eim
 FE(x) .
(20)
The function FE(x) can be obtained by the method outlined in section 3.2. The other terms are properties of the material system and are obtained from the material system database: Rim denotes the displacement distance, Eim is the corresponding threshold energy, gim is a parameter which depends on the collision partners and the interatomic potential used to describe the collisions.
The diffusivity of the particle of the i-species in the general solid is modelled through the assumption of its linear dependence on the concentrations:
Di(x,f) = n
å
m=1 
 dim(x) qm(x,f) .
(21)

4.5  The Sputter Yield and Surface Recession Speed

One of the most important objectives of the work in this paper is to model the sputter yield from a given initial target material structure and bombardment conditions of the primary particles. The yield also determines the rate at which the solid erodes and hence the surface recession speed.
The yields yi(f) for i=1,2,..,.n at dose f are modelled in relation to the surface concentrations by a formula of the form
yi(f) = Ai qSi(f)
1+ n
å
m=1 
 Bim qSm(f)
(22)
where the qSi (f) (for i=1,2,..,n) represents the concentrations in the surface layer at dose f and Ai and Bim are constants based on known properties of the material system and are taken from the material system database. The surface recession speed can be computed from the yields by the following formula:
u(f) = n
å
i=1 
 yi(f) - t1 .
(23)
Hence, after dose f, the distance receded by the surface X0(f) is given by
X0(f) = ó
õ 		f

0 
 u(f¢) d f¢  .
(24)

4.6  Surface Concentrations

In this section the formulae (15), (10) and (22) show that the proportion of primary particles and energy pernetrating the surface and the sputter yields at dose f are related on the surface concentrations qS1(f), qS2(f), ..,qSn(f). For a general material distribution, it is understood that the material concentration at x=0 influence these quantities most with the influence of the qi(x,f) reducing as x increases. Unfortunately, the precise relationship between the representative surface concentration qSi(f) and the distribution of i-material qi(x,f) for i=1,2, ..,n is unknown.
The most straightforward way of determining the qSi(f) for i=1,2, ..,n is to put qSi(f) ¬ qi(0,f). Unfortunately, the qi(x,f) functions often have steep gradients at x=0 and hence this can easily be unrepresentative of the concentration in the surface layer. In IMPETUS II the qSi(f) are defined to be the average concentration in the first angstrom:
qSi(f) ¬ ó
õ 		1

0 
 qi(x,f) dx
(25)
for i=1,2, ..,n. The formula (25) is far from ideal but it expected to return a better measure of the concentrations in the surface region than the earlier formula.