Interpreting Mössbauer spectra reflecting an infinite number of sites: An application to ${\mathrm{Fe}}_{1-x}\mathrm{Si}$ synthesized by pulsed laser annealing

We present a study on the interpretation of conversion electron Mössbauer spectra reflecting an infinite number of sites, in casu Mössbauer spectroscopy on ${\mathrm{Fe}}_{1-x}\mathrm{Si}$ layers on Si, synthesized by pulsed laser annealing. These spectra display a broad double-peaked resonance, reflecting the numerous different environments of the ${}^{57}\mathrm{Fe}$ probe due to a distribution of vacancies on the Fe sublattice. The spectra can be fitted in many different ways; hence finding a reliable physical interpretation is not straightforward. Therefore ab initio calculations have been performed in order to obtain a priori information about the hyperfine interaction parameter distributions. For this material, the electric-field gradient on the ${}^{57}\mathrm{Fe}$ atoms turns out to depend on details in the configuration of neighbors as far as the sixth neighbor shell. The isomer shift appears to be determined by the number of Fe atoms in the first and second Fe neighbor shells only. This leads to the construction of an ab initio based model predicting the mean isomer shift and its distribution for a given ${\mathrm{Fe}}_{1-x}\mathrm{Si}$ layer with a known Fe concentration profile. By applying this model new information from the experimental data can be extracted: we show that after applying one or two laser pulses, the Fe atoms are not completely randomized at an atomic scale. The relation of this model to other approaches of analyzing Mössbauer spectra with a distribution of sites is discussed, as well as the difference between the present results on $\left[\mathrm{Cs}\mathrm{Cl}\right]{\mathrm{Fe}}_{1-x}\mathrm{Si}$ and earlier interpretations in the literature. This work reveals how a combination of Mössbauer experiments and ab initio calculations leads to a more reliable interpretation of Mössbauer spectra reflecting an infinite number of sites.

Figure 1

Metastable cubic Fe silicide structures. For [CsCl]FeSi the different nearest-neighbor Fe shells around the central Fe atom are indicated by 1, 2, and 3. The high resemblance between the different phases is obvious: [CsCl]FeSi transforms into $\left[\mathrm{Cs}\mathrm{Cl}\right]{\mathrm{Fe}}_{1-x}\mathrm{Si}$ by randomly removing Fe atoms whereas $\gamma \text{−}\mathrm{Fe}{\mathrm{Si}}_2$ is obtained by an ordered removal of Fe atoms.

Figure 2

Fe distribution profiles as a function of the number of pulses applied for various laser energy densities. Note that the bottom value of the vertical axis is $x=1.0$ (pure Si) and the top value $x=0.0$ (FeSi). The horizontal axis shows the areal density, which can be unambiguously determined from RBS data. Dividing the areal density by the density gives the thickness. As the density is varying throughout our samples and hard to estimate due to the presence of vacancies, we do not make the explicit conversion. Still, the horizontal axis expresses “thickness” in a qualitative way. The arrow indicates the position of the surface. The dashed horizontal line indicates the average value of $x$ in the layer. The apparent increase in Si concentration at the left side of the arrow (below $50\times {10}^{15}\text{atoms}∕{\mathrm{cm}}^2$) originates from the detector resolution, which convolutes with the actual concentration profile. This does not indicate a very Si-rich layer on top of the surface.

Figure 3

(a) The RBS Fe distribution profile (see the caption of Fig. 2 for details on the units) and (b) the CEMS spectrum of a sample subjected to ten pulses of $330\mathrm{mJ}∕{\mathrm{cm}}^2$. This results in a homogeneous $\left[\mathrm{Cs}\mathrm{Cl}\right]{\mathrm{Fe}}_{1-x}\mathrm{Si}$ layer with $x=0.6$.

Figure 4

CEMS spectra for a sample subjected to one, two, and five pulses of $480\mathrm{mJ}∕{\mathrm{cm}}^2$, respectively. The dots represent the raw data. Note that although RBS reveals changes in the microstructure (Fig. 2, second column), the CEMS spectra are very similar.

Figure 5

Different configurations for the Fe vacancies in the first-nearest-neighbor Fe shell. The numbers indicate $V_{zz}$ (arbitrary units) obtained by a point-charge model calculation.

Figure 6

Calculated isomer shift as a function of the number of Fe atoms in the first Fe shell with 12 atoms in the second Fe shell. The different values shown for 0, 1, 2, 3, and 5 Fe atoms in the first shell are due to vacancies in the third Fe shell and to different angular positions in the first shell.

Figure 7

(Color online) Calculated isomer shift as a function of the number of Fe atoms in the first (0–6) and second (0–12) Fe shell.

Figure 8

(Color online) Binomial distribution for the number of Fe atoms in the first (0–6) and second (0–12) Fe shell for a layer with $x=0.6$.

Figure 9

Calculated isomer shift as a function of the lattice parameter for three different random environments, the $\gamma \text{−}\mathrm{Fe}{\mathrm{Si}}_2$ phase and the [CsCl]FeSi structure. The energy as a function of half of the lattice parameter for the $\gamma \text{−}\mathrm{Fe}{\mathrm{Si}}_2$ configuration is also added, the minimum occurs around $2.71\mathrm{\AA }$.

Figure 10

Plot of lattice parameters for various Fe-Si structures as a function of Si content, used for the calculation of the lattice parameter in $\left[\mathrm{Cs}\mathrm{Cl}\right]{\mathrm{Fe}}_{1-x}\mathrm{Si}$. The open circles represent the lattice parameter for Fe and one half of the lattice parameter for ${\mathrm{Fe}}_3\mathrm{Si}$ ($\mathrm{D}{\mathrm{O}}_3$ or $\mathrm{Al}{\mathrm{Fe}}_3$ type), two common reference structures. The filled circles are experimental values for cubic polycrystalline Fe-Si alloys (Ref. 39). The stars represent experimental values for various $\left[\mathrm{Cs}\mathrm{Cl}\right]{\mathrm{Fe}}_{1-x}\mathrm{Si}$ layers (Ref. 40). The vertical dashed line indicates the region that is of interest in this work.

Figure 11

The isomer shift dependence on $x$ in $\left[\mathrm{Cs}\mathrm{Cl}\right]{\mathrm{Fe}}_{1-x}\mathrm{Si}$, as determined from ab initio calculations. The dashed line describes the dependence for a constant lattice parameter $a$ whereas the full line takes into account the variation of the lattice parameter with $x$.

Figure 12

Calculated isomer shift as a function of the number of pulses for various laser energy densities (open squares). The line is drawn to guide the eye. The filled dots represent the experimental mean isomer shift values (error bars are small and fall within the black dots). The values for the calculated and experimental mean isomer shift are reported in Table .

Figure 13

Calculated sample-specific isomer shift distribution (dots) for a sample irradiated with one pulse of $730\mathrm{mJ}∕{\mathrm{cm}}^2$. A selected number of [${\mathrm{IS}}_{\min }$, ${\mathrm{IS}}_{\max }$] intervals are added as well ($\text{star}+\text{horizontal}$ bar, the interval is plotted at the same height as the corresponding $x$ value on the right-hand side axis). The line is a Gaussian fit through the distribution.