Diffusion in molybdenum disilicide

Appendix

The concentration-depth profiles in the present investigation have been determined using a sputtering process for the serial sectioning of the diffusion sample. A detailed analysis of the sputtering process reveals that this sectioning technique can have an effect on the concentration profile in the sense that the measured profile deviates from the true profile in the sample. In the following some of these sputtering effects are briefly discussed and a technique is presented how the true profile can be determined even in the presence of sputtering effects. For a more detailed discussion on sputtering effects and their treatment the reader is referred to [11].

An ideal serial sectioning process would remove thin consecutive layers parallel to the surface. This would result in a perfect reproduction of the profile of the sample with the only difference that the measured profile is a ‘discretized’ representation of the original continuous profile. However, several different effects pertaining to the sputtering process can cause deviations of this ideal behaviour. These effects belong into one of the two following categories:

1. On a microscopic scale, the sputtering process does not strictly remove consecutive layers of atoms. Instead, the depth coordinate becomes a bit ‘uncertain’ because some mixing of atoms from different layers occurs. These effects cause a broadening of the profile because each layer may include tracer atoms from former and later layers. Examples of such ‘broadening’ effects are the roughening of the surface due to the statistical nature of the sputtering and the mixing of the atoms in the sample caused by knock-on effects.

   The measured profile f(x) after sputtering is given by a convolution (see, for example, Werner [31])

   (4) f(x)=(g∗c)(x)

   of the profile c(x) in the sample with a device function

   (5) g(x)=12σ2exp(−x22σ2)

   The width σ of the Gaussian device function g(x) determines the influence of the sputter broadening. This broadening influences the profile in a similar way as diffusion does (note the similarity between Eq. (5) and Eq. (2)). Thus, if c(x) obeys the thin-film solution according to Eq. (2), f(x) will also be a Gaussian solution but with an enhanced apparent diffusivity D′ which is given by

   (6) 4D′t=4Dt+2σ2

   However, sputter broadening effects are usually of short range. For low ion energies (only about 1 keV in our case) the broadening σ in Eq. (5) is of the order of a few nm only.

2. Due to the experimental setup described in Wenwer et al. [12], a tracer ‘reservoir’ can be present or formed during the sputtering process. Tracers removed from this ‘reservoir’ during the sputtering process will be added to the tracers removed from the diffusion zone resulting in a long-range ‘tail’ on the original diffusion profile. As the amount of tracers available in such a ‘reservoir’ is typically much lower than the amount of tracers within the diffusion zone, the ‘tail’ will have a low concentration and will thus not be detectable in the surface-near part of the profile. However, as the concentration due to volume diffusion decreases rapidly with increasing depth, the ‘tail’ will eventually determine the shape of the measured profile at larger depths. The resulting measured profile appears similar to what is known, for example, from type-B diffusion through the volume and along the grain boundaries in polycrystals. Our sputtering device uses a screen which shields parts of the sample from the sputter beam. Only a circular area in the middle with a diameter of about 8 mm is left open and defines the sputtered area on the sample. As the sputter beam is aligned at some angle against the surface normal (60° for our device) a sputter crater is formed during the sputtering process. The sputtered-off tracers do not only originate from the the crater bottom but, in addition, also from the edge regions. As a consequence, the tracer concentration, f(x), measured by sputtering deviates from the original tracer concentration, c(x), by the concentration of tracers, e(x), contributed from the edges:

   (7) f(x)=c(x)+e(x)

   A detailed inspection (Salamon [11]) shows that e(x) decreases approximately proportional to x^–2

   (8) e(x)∝1x2∫0xc(x′)x′dx′

   If the diffusion profile c(x) follows the thin-film solution the amount of tracers originating from the edges will at some depth dominate the total measured tracer concentration f (x) as this part decreases approximately with x^–2 and not exponentially. To minimize this effect we deposited the tracers only in a small region with a diameter of 7 mm which later is aligned within the center of the 8 mm sputter crater. However, evaporation and re-deposition of tracers during the annealing or the serial sectioning as well as surface diffusion can still cause some leaking of tracers into the outer areas.

   Eqs. (7) and (8) can also be written in the form of Eq. (4) with an approximative device function

   (9) g(x)=A⋅x−95

   where A is a fitting constant. The empirically derived exponent −95 in Eq. (9) was found to give a satisfactory approximation for practical purposes.

Eq. (6) permits an easy correction of measured profiles if only broadening effects are present and their broadening parameter σ is known. If tailing effects are present and/or σ is not known, it is still possible to determine the true diffusion profile c(x) from the measured profile f (x), but the technique is more complex as will be shown in what follows.

Let us consider diffusion for all x ∈ ℝ which allows us to avoid dealing with boundary conditions at the sample surface x = 0 for the sake of simplicity. Let us further denote the initial tracer concentration at time t = 0 with c₀(x) = c(x, t = 0). After diffusion for some time t the tracer concentration c_t(x) can be calculated by a convolution of the initial concentration c₀(x) with a kernel. The kernel K_t(x) is the thin-film solution according to Eq.(2) with N = ½ due to the boundary conditions chosen above. K_t(x) of the diffusion equation (see, for example, Glicksman [32]):

Both the evolution of a concentration profile by diffusion as well as the modification of such a profile by sputtering effects can be described by a convolution according to Eq. (10) and (4), respectively. As the convolution is both commutative and associative, one can mathematically switch the order of the two operations:

Eq. (11) shows that the kernel K_t produces f_t from f₀ in the same way as it produces c_t from c₀. Once the kernel is determined, the diffusion coefficient D is known (see Eq. (2)). Therefore, it is possible to simply use the profiles f_t and f₀ which are accessible by the sputtering process and not the true diffusion profiles c_t and c₀ to determine the Kernel K_t (and by doing this also to determine D).

The remaining task would be a deconvolution of f_t to f₀ in order to find the kernel K_t. However, such a deconvolution is prone to produce serious artifacts as the measured concentrations often scatter heavily around the profile (see, for example, Jansson [33]). An easier alternative is to start with an assumption for D as a zero-order approximation, use this value for a kernel K_t, convolute the kernel with the measured initial concentration profile f₀, and determine whether the result of this procedure is a good representation of the measured diffusion profile f_t. If it is not, a better value can be used as the next-order approximation for D, and the whole calculation is repeated until one ends with an acceptable representation of the measured profile. Such a straightforward calculation has been used in this investigation to evaluate the diffusion profiles of ⁹⁹Mo in MoSi₂. By doing this, both sputter broadening and ‘tailing’ effects have been taken fully into account. The advantage of this strategy is that one only has to measure a sputtered non-diffused sample under similar experimental conditions as for the diffusion samples to obtain f₀. Using the scheme described above allows to determine a ‘true’ diffusion coefficient which is not influenced by sputtering and which takes into account the real initial tracer concentration, i. e. does not depend on the assumption of, e. g., a thin-film source. In addition, it allows to both qualitatively and quantitatively explain the ‘tails’ observed in the diffusion profiles.