4.1. Total Sputtering Yield
The total sputtering yield depends on several parameters that are related to the impinging ion and the sputtered target material. An insight into the influence of these parameters can be obtained from the analytical equation for the sputtering yield. The analytical expressions that are most often used for the calculation of the total sputtering yield were derived by Sigmund in his seminal work on the theory of sputtering [
35]. For a perpendicular ion impact in the linear cascade regime, Sigmund derived the following expression for the total sputtering yield:
where
Esb is the surface binding energy and α is a dimensionless function which depends on the ion-target atomic mass ratio and the ion incidence angle, but it does not depend on the ion energy. In Equation (12), the nuclear stopping cross section
Sn is a complex function of the ion and the target atomic masses and the ion energy
with
Z and
M corresponding to the ion atomic number and mass (index
i) and the solid atomic number and mass (index
s), respectively. The
factor is the Lindhard screening length. Further complexity in Equation (13) arises from the reduced nuclear stopping power
sn. In practice, different forms of the reduced nuclear stopping power are used, which are based on the particular interatomic potential between the interacting atoms. Often a Thomas–Fermi or Krypton–Carbon potential is utilized [
57]. For the Thomas–Fermi potential, the reduced nuclear stopping power is:
while for the Krypton–Carbon potential, it is:
where the reduced energy
is defined as:
The nuclear stopping cross section Sn is the only term in Equation (12) which depends on the ion energy (Ei).
Over the years, the general equation for the total sputtering yield has been modified to achieve a better agreement with the experimental data (particularly at lower ion energies). An overview of various semi-empirical equations for the total sputtering yield can be found in
Table 2 of [
18]. Several authors have performed different modifications to Equation (12). Bohdansky [
38] and Yamamura [
23] introduced corrections for low-energy sputtering using very light or heavy ions (also at grazing incident angles). The correction factor that they introduced decreased the sputtering yield near the threshold energy. Eckstein et al. [
39] proposed a different empirically based correction factor for low-energy sputtering, whereas Wilhelm [
40] considered sputtering near the sputtering threshold by employing the quantum statistical approach of three-body sputtering. In the latest attempt, Shang et al. [
58] included a factor that is related to electronic energy loss for low-energy sputtering. They modified the equation of Zhang [
59], which is based on Yamamura’s semi-empirical formula.
The SRIM program uses the specific nuclear stopping power function to evaluate the propagation of the ions and the atoms through the solid. The authors of the program (Ziegler, Biersack, and Littmark) proposed the nuclear stopping power, which is derived from the so-called universal atomic potential, which is more often referred to as the ZBL potential. This interatomic potential was found to agree well with the experimentally determined interatomic potentials for numerous atomic pairs [
60]. The reduced nuclear stopping power in SRIM is calculated from the function:
where
a = 1.1383,
b = 0.01321,
c = 0.21226, and
d = 0.19593 are the best fit factors for the experimental data. The ZBL potential is an improvement of the Krypton–Carbon potential. Note that there are similarities between Equations (15) and (17). It should be emphasized that SRIM does not calculate the total sputtering yield from Sigmund’s equation, Equation (12). It uses the nuclear stopping power to determine the strength of the interactions (i.e., forces) between the particles that are involved in the binary collisions. The total sputtering yield is determined from the ratio between the number of sputtered atoms and the number of impinging ions. The SRIM program simulates the particle collisions with the stochastic Monte Carlo approach, which follows the path of all of the interacting atoms (and the impinging ions) and counts the number of atoms that leave the surface.
The general sputtering yield equation, Equation (12), should provide the most accurate data for the total sputtering yields; however, it is not very convenient for practical calculations due to the complexity of the
Sn function. Furthermore, it does not provide a clear relationship between the target and the ion-related parameters (e.g., mass and energy) and the sputtering yield. For this reason, Sigmund attempted to derive a simpler expression for the total sputtering yield. He considered the Thomas–Fermi potential for the nuclear stopping cross section and made several simplifications for the ion energies that are below 1 keV (i.e., energies typical for the sputter deposition). He derived a simple relation for the total sputtering yield [
35]:
This equation depends only on a few target and ion parameters: the energy transfer factor , the dimensionless factor (which is a function of the target-to-ion mass ratio), the surface binding energy Esb, and the incident ion energy Ei.
Sigmund analyzed the factor
for different sputtering conditions and proposed a general function in dependence of the target-to-ion mass ratio (i.e.,
Ms/
Mi). The
function is presented in Figure 13 of [
35]. By extracting the data from the figure and fitting a linear function (see graph in the
Supplementary Materials), we can find a very good fit to the following function:
where constant
k is approximately 0.15. By inserting Equation (19) into Equation (18), we obtain a very simple expression for the total sputtering yield:
Note that this more elegant equation depends only on two target parameters (the atomic mass and the surface binding energy) and two ion parameters (the atomic mass and the ion energy). The total sputtering yield, according to this equation, is linearly dependent on the ratio between the target atomic mass and the sum of target and ion masses. It also depends on the ratio between the ion energy and surface binding energy. In general, this equation should be useful for calculating sputtering yields of different ion-target combinations for ion energies that are up to 1200 eV. However, the accuracy of the calculated values in many instances differs significantly from the experimental sputtering yield data. Nevertheless, Equation (20) provides a very useful insight into the relations between the total sputtering yield, and the ion and the target properties.
In the following chapters, we examine the sputtering yields that were simulated by the SRIM program, which were optimized to be close to the experimental values. We compared them with the simplified expression, Equation (20). From this comparison, we attempted to draw some general trends and relations between the sputtering yield, and target- and ion-related properties as well with the order of elements in the periodic table.
4.1.1. Surface Binding Energy
The surface binding energy (SBE) is the energy that is needed for a particle in the solid to overcome the lattice bonding and leave the target material. This results in the ejection of the atoms from the surface (i.e., sputtering). The SBE is a crucial sputtering parameter. It appears in analytically derived sputtering yield equations and in the modeling of the sputtering process. Experimentally, the SBE values are not well established, therefore, in the majority of works (including the SRIM program), the heat of the sublimation is used as an approximation for the SBE. However, in the SRIM program, the surface binding energy is not defined as a simple chemical binding energy for the surface atoms; instead, the parameter also includes all of the surface non-linearities such as those which are produced by radiation damage, surface relaxation, surface roughness, and other effects as discussed by the authors of the program [
50]. The surface binding energy in SRIM is, therefore, a free input parameter, whereas the heat of sublimation is used as the default value for SBE.
The goal of this work was to obtain simulated total sputtering yields which are close to the experimental values. For this reason, we modified the SBE since it is not accurately defined in SRIM and because it has the strongest influence on the sputtering yield among the free parameters. We aimed to achieve a good agreement with the experimental sputtering yields in the 200–1200 eV range, which is the most relevant energy range for the sputter deposition techniques.
Figure 10a shows the modified SBE, and
Figure 10b shows the default SBE values that were used by SRIM (i.e., the heat of sublimation) in the dependence of the atomic number (Z) of the investigated materials. From the comparison it can be seen that for some materials, the SBE had to be modified significantly from the heat of sublimation values. The largest corrections had to be made for the lightest elements (i.e., B and C), where we had to decrease the value of SBE (e.g., for C, this was from 7.4 eV to 3.6 eV) and for the heaviest elements, where the values had to be increased (e.g., for W, this was from 8.7 eV to 15.0 eV). Smaller modifications were made for Cr, Cu, Zr, Mo, and Au, while the values for Ti, V, and Ag were close to the heat of the sublimation values.
In general, the increase in the atomic number is correlated with a larger SBE, whereas this is not the case for the heat of sublimation. A nearly linear relationship is observed for elements in the groups 4, 5, and 6, and for the lightest four elements (B, C, Al, and Si). For the heat of sublimation, the data are more scattered, and no clear trend was observed. It should be noted that if the heat of sublimation values were used in the simulations, then in some cases, the simulated values would differ significantly from the experimental values (sometimes even for a factor of two). The use of the modified SBE in the SRIM program was necessary and it was the only option to get a good agreement between the simulated and experimental data.
Before the development of the sputtering theory, it was already established experimentally that the total sputtering yield correlates with the reciprocal value of the heat of sublimation [
61]. Sigmund’s theoretically derived general equation for the total sputtering yield, Equation (12), and its simplified form, Equation (20), demonstrate such inverse proportionality. Hence, the sputtering yield is lower when the atoms are strongly bound near the surface (i.e., a larger SBE), and it is higher when they are weakly bound (i.e., a smaller SBE).
Figure 11a shows the sputtering yields that were calculated using SRIM for different target materials in dependence of the modified surface binding energies.
It can be seen that the sputtering yield is not simply an inverse function of the modified SBE. The belief that a smaller SBE results in a higher sputtering yield is not entirely correct. For example, the sputtering yield of the elements in group 11 (Ag, Au, and Cu) is much higher than the yield of the light elements (B, C, Al, and Si) and Ti, V, and Cr despite them having a very similar SBE. Many transition metals in the groups from 4 to 6 have similar sputtering yields (e.g., Ti, V, Zr, Nb, Mo, Hf, Ta, and W), although they have very different SBE values. The lowest sputtering yield was observed for the light elements (B, C, and Si), which also have low SBE values. The approximate inverse proportionality with the modified SBE can be seen for the elements within the same group of the periodic table. This is true for elements in the groups 5 and 6, but not for elements in group 4 where Y slightly increases with the SBE. In group 11, there is also a discrepancy in the trend. Hence, there is an inconsistency in the relationship between the sputtering yield and SBE.
This inconsistency in the inverse proportionality can be understood by analyzing Equation (20), which shows that the total sputtering yield also depends on the target-to-ion atomic mass ratio. If the sputtering yield is normalized by
and plotted as a function of the modified SBE (
Figure 11b), then the data decrease much closer to the 1/
Emsb trend. For some elements, the values deviate from this trend, therefore, it can be concluded that other factors which are contained in the complex general equation for total sputtering yield (Equation (12)) must contribute to the deviations from the fitting function.
In the literature, it was reported that the total sputtering yield also correlates with the filling of the
d-shells [
62]. Namely, the elements with filled
s and
d outer shells have the highest sputtering yield—these are the elements in group 12 (Zn, Cd, and Hg). The elements in group 11 (Cu, Ag, and Au) have a similar shell filling (with one missing electron in the d-shell). This effect, together with a low SBE, can explain the large sputtering yields that were observed among the elements of group 11.
4.1.2. Mass Ratio
The atomic masses of the impinging ion and the target material also influence the total sputtering yield. A simplified view suggests that the sputtering yield is higher when the atomic mass of the ion is close to the atomic mass of the target material. The difference between the masses of the two colliding particles is described by the energy transfer factor . For similar atomic masses, the Λ is close to one and for very different masses, it is significantly below one (typically closer to 0.5 when using argon ions). For example, in the case of sputtering a Ti target by Ar ions, the energy transfer factor Λ is 0.992 (i.e., MTi = 47.88 and MAr = 39.95, similar masses), while the sputtering W target it is 0.587 (i.e., MW = 183.84, different masses). In the case of sputtering with the same ions as the target material (i.e., self-sputtering Mi = Ms), the energy transfer factor equals one.
The total sputtering yield in the dependence of the energy transfer factor is shown in
Figure 12a. It can be seen that having similar masses for the impinging argon ion and the target material does not necessarily result in a higher total sputtering yield; for instance, the yields for Si, Ti, and V have a similar
Λ. Hence, the proximity of the atomic masses is not the best indicator for higher sputtering yields. Furthermore, there is no obvious trend between the total sputtering yield and the energy transfer factor. For the elements in group 6, the sputtering yield increases with
Λ, but not in the other groups. The opposite trend was observed in group 4 (i.e.,
Y slightly decreases when
Λ approaches 1), while in group 12, there is no clear trend. The reason for the absence of trends can be understood by examining Equation (18) where
Y also depends on the factor α, which is a function of the target-to-ion mass ratio as well as of SBE (as was discussed in
Section 4.1.1).
The product of the energy transfer factor
Λ and the factor α is approximately:
. According to Equation (20), the total sputtering yield linearly depends on the
. Hence, for a given ion, the heavier elements should in general have a higher sputtering yield than the lighter elements do.
Figure 12b shows the total sputtering yield for the investigated target materials in the dependence of the
ratio for the sputtering by argon ions with 600 eV. The trend of the sputtering yield increasing with the atomic mass is more obvious than in
Figure 12a, however, the relationship is not unambiguous. For example, the sputtering yield increases for the elements with Z ≤ 14 and the elements in group 11, but it decreases for the elements within groups 5 and 6. This inconsistency is a result of the surface binding energy, which strongly influences the sputtering yield. For this reason, in
Figure 12c, we present
YEmsb/
Ei in the dependence on the
as suggested by Equation (20). A relatively good linear trend can be observed in this case. It should be noted that the quantities on the
y-axis are related to the SRIM simulations (i.e.,
Y and
Emsb), while the values on the
x-axis correspond to the material properties. The plot shows that the simple sputtering yield equation that is presented (Equation (20)) can provide a reasonable estimate for the total sputtering yield, however, the influence of the ion energy should be analyzed as well (see
Section 4.1.3).
The total sputtering yield for sputtering under an oblique angle also significantly depends on the atomic masses of the ion and the target material. In
Figure 4a, the total sputtering yield for the target material is the highest for the oblique angles between 60° and 70°. The largest differences between the sputtering yield at the 70° ion incidence angle and the perpendicular ion sputtering are observed for the lightest elements B, C, Al, and Si (see
Figure 4d). Among the transition metals, the lighter elements (Ti, V, and Cr) show the largest relative sputtering yields when they are sputtered by argon ions under a 70° incidence angle (see
Figure 4c). On the other hand, in the case of the heaviest transition metals (i.e., Au, W, and Ta), we can observe that the total sputtering yield stays almost flat up to the incidence angle at around 60–70°—there is practically no increase in the sputtering yield under the larger incidence angles. At angles that are above 70°, the sputtering yield starts to drop sharply in all of the materials.
Hence, the atomic mass of the target material significantly influences the total sputtering yield for the oblique ion sputtering. To analyze this influence further, we show in
Figure 13 the relative sputtering yield
Y(70°)/
Y(0°) in the dependence of
. A clear inverse proportionality can be observed, i.e., the
Y(70°)/
Y(0°) ratio decreases with mass ratio.
Figure 4 and
Figure 13 demonstrate that the lighter elements sputter more substantially under the oblique ion angles than they do at the normal ion incidence. This difference becomes smaller with the increasing atomic mass of the target material, and it practically disappears for the heaviest elements. Hence, a viable approach to increase the sputtering yield of the lighter elements, which in general have the lowest sputtering yields at a normal ion incidence, is to sputter them under higher angles or to use targets with a large surface roughness.
The experimental sputtering yield data for ion bombardment under an oblique angle are rather scarce. In the collection by Behrisch and Eckstein [
22], the data are available for several elements that were analyzed in this work (Al, Ti, Cu, Zr, Ag, Ta, W, and Au) which were sputtered by 1.05 keV argon ions. In the
Supplementary Materials, we show a comparison between the simulated and empirical data for the selected elements. In general, the agreement is good for most of the materials except for Al and Ti, where larger deviations from the experimental data are observed. Overall, the simulated data show the same trends with regard to the ion incidence angle as the experimental data do. For example, the experimental data also demonstrate that the heavier elements have much smaller variations in the dependence of the ion incidence angle than the lighter elements do. Furthermore, the highest experimental values for the sputtering yield are also observed between the 60° and 70° incidence angles, as demonstrated by the SRIM simulations.
The sputtering yield at a normal incidence is usually considered to be the most reliable value, and it is generally used to obtain information on the sputtering yields at an oblique incidence using semi-empirical models. Rather complicated algebraic formulas are used to describe the angular dependence of the total sputtering. The reader that is interested in the evaluation of the sputtering yields under an oblique angle can find an overview of the semi-empirical fitting equations in
Table 3 of [
18].
4.1.3. Ion Energy
The number of sputtered atoms strongly depends on the energy of the impinging ions. Higher ion energies in general result in larger sputtering yields. The energy of the ions is of particular interest for the sputter deposition techniques since the ion energy can be adjusted to a large degree. In magnetron sputtering, the energy of the ions is controlled by the cathode potential. The typical cathode voltages in the DCMS regime are between 250 V and 400 V, in HiPIMS, these are between 450 V and 600 V, and in triode sputtering, these are around 1000 V or more. In the ion beam deposition, the energy of the ions is determined by the ion beam source, and it is also in the range of 1000 eV.
The simplified Sigmund Equation (20) suggests that
Y increases linearly with the ion energy. However, the results of the SRIM simulations that are presented in
Figure 3 show that
Y in the low-energy range (i.e., 200–1200 eV) deviates from the linear relationship. The same is observed in the experimental data [
22]. When calculating the sputtering yield values using Equation (20) and comparing them to experimental values, a large overestimate of the sputtering yield is found for most of the elements.
In order to account for the deviation from the linear relation with the ion energy, we fitted the simulated sputtering yields to the power-law Equation (10) using the parameters
a and
b. In
Section 3.1, we used both of the free parameters to obtain the best fit with the simulated data. Here, we use only the power parameter
b as a fitting parameter, while parameter
a was obtained from the analytical Equation (20). From the comparison of Equations (10) and (20) it follows that:
This approach reduces the number of fitting parameters and makes the allometric Equation (10) fall more in line with the sputtering theory that was developed by Sigmund. In
Table 4, we present the factor
a, the power coefficient
b, and the coefficient of determination
R2. The comparison of
R2 in
Table 2 and
Table 4 shows that the new fitting approach still provides a good agreement with the simulated data. The coefficient of determination is above 0.97 for most of the materials, and it is somewhat lower for C, Cu, and Ag. In [
63], a similar fitting approach was used as described in
Section 3.1. Both of the fitting parameters
a and
b were free parameters, however, their values differ from the ones in this work due to there being a few differences. In this work, we used the modified SBE values (instead of the heat of the sublimation) and considered the power equation
as opposed to
. Furthermore, we fitted the data to a lower energy range (up to 1200 eV). Hence, the coefficients
a and
b that are provided in
Table 2 should be more in line with the experimental sputtering yield values.
The fitting approach with the
a values which were calculated from Equation (21) and using
b as the fitting parameter is physically more reasonable. By combining the allometric Equation (10) and the simplified analytical equation, which was derived by Sigmund, we can modify the sputtering yield equation to account for the non-linear behavior with the ion energy:
This equation should provide more accurate results for the total sputtering yield than the linear Equation (20) can. It is valid for argon ion energies that are up to 1 keV. This semi-empirical relation depends on two empirical parameters: the modified surface binding energy which was derived from SRIM simulations (fitted to the experimental yield data) and the power coefficient b which was derived from fitting the energy-dependent sputtering yields (simulated by SRIM using modified SBE). The power coefficient b depends on the particular target material and the ion type. For a specific target material, the same value of b cannot be used for the other types of ions because the sputtering yield as a function of energy depends on the mass of the ions. The experimental data and SRIM simulations show a more linear-like relationship for the sputtering by heavier ions (e.g., Kr and Xe) and a more parabolic one for the sputtering by lighter elements (e.g., He and Ne). The power coefficient clearly depends on the ion mass. The usefulness of Equation (22) should be verified for other types of ions by making the same fitting approach which was performed for the argon ions in this work.
If we analyze the parameter
b, we can observe some interesting trends which demonstrate the soundness of the second fitting approach. In
Figure 14, we present the fitting parameter
b in dependence of the atomic number Z (a similar graph is obtained if it is plotted against the
ratio). Surprisingly, a sequential trend can be observed for the transition metal elements. The parameter
b increases practically linearly along the individual periods of the periodic table. For example, in period 4, the parameter
b changes from 0.68 (Ti) to 0.86 (Cu), while in period 5, it changes from 0.71 (Zr) to 0.82 (Ag). On the other hand, very similar values of the parameter
b are found within the individual groups of the periodic table. The elements in group 6 have values of
b that are close to 0.77. In group 5, a slightly increasing trend is observed, although the values of
b are close to 0.73. Thus, the elements in group 4 show the most parabolic behavior with the ion energy, while the elements in group 11 show the least parabolic behavior (see Equation (10)). Hence, for the transition metals, the trend shows a linear-like increase with the group number. However, for the lightest elements in groups 13 and 14 with Z ≤ 14, the opposite trend is observed—the value of
b for the elements in an individual group decreases with the atomic number (e.g., the values for B and C are different for those of Al and Si).
The sequential linear increase in the parameter
b for the transition metals gives us the confidence to estimate the fitting parameters for other elements of the periodic table that have not been studied in this work. From the linear increase of
b with respect to Z within the individual periods, we can calculate the values of
b for all of the elements of the individual period. For example, according to the trend in the period 4: Mn, Fe, Co, and Ni must have values that are between those of Cr and Cu (one should note the fitted linear curves in
Figure 14 for periods 4, 5, and 6). In the
Supplementary Materials, we provide the calculated values of the total sputtering yield for the transition metals and several other elements. The parameter
a was evaluated from Equation (21), and the fitting parameter
b was extrapolated from the linear functions in the individual periods that are shown in
Figure 14. By comparing the calculated sputtering yields with the experimental values, we can find a fairly good agreement for most of the elements. The transition metals which have not been investigated in this work (i.e., Mn, Fe, Co, Ni, and Zn; Ru, Rh, Pd, and Cd; Re, Os, Ir, and Pt) indeed show a relatively good agreement between the calculated data and experimental data, where they are available. The agreement is good for all three of the analyzed energies (i.e., 300 eV, 600 eV, and 1000 eV). This means that valuable sputtering yields could be also obtained for other ion energies.
The above results demonstrate that the simple semi-empirical Equation (22) is useful for a quick evaluation of the total sputtering yields using argon ions with energies that are up to 1200 eV. We should note that the simplified Sigmund’s equation for the total sputtering yield (Equation (20)) was derived from the Thomas–Fermi potential, which is too high at lower ion energies. However, with the use of the power parameter b for the ion energy, we can compensate for this discrepancy. In the derivation of Equation (18), it is assumed that sputtering occurs in the linear cascade regime. This is true for higher ion energies where the ion penetrates deeper into the solid and causes many collisions, which then result in the emission of atoms.
For the lower ion energies (e.g., those that are below 300 eV), the surface atoms are often sputtered through a single knock-on collision as demonstrated in the simulations by Biersack and Eckstein [
64]. For this reason, it can be expected that the sputtering yield values that were simulated using SRIM and calculated using Equation (20) could differ more significantly at lower ion energies than they might at higher ones. The single knock-on regime becomes even more important when the material is bombarded at higher incidence angles. Under an oblique ion sputtering, the
function also depends on the ion incidence angle [
59], which means that the Equation (18) cannot be simply reduced to Equation (20). For all of the above-mentioned reasons, noteworthy deviations between the calculated and experimental values can be expected, especially at lower ion energies. Further deviations can be attributed to the uncertainties that are related to the parameters in the SRIM simulations and the experimental methods that were used for measuring the sputtering yield.
The semi-empirical Equation (22) has many limitations, and it does not capture all of the complex relations that are contained in the general sputtering yield Equation (12). The most accurate approach for the evaluation of the total sputtering yields is therefore to use a general equation that was derived by Sigmund with corrections for the lower ion energy sputtering which are provided by the other authors.
4.2. Differential Sputtering Yield
The total sputtering yield is an important parameter in the theory of sputtering, however, for practical applications, the differential sputtering yield is often more relevant. The growth of thin films is determined by the deposition rate, which linearly depends on the differential sputtering yield [
65]. The atom flux from the sputtering sources, which is directed onto a specific region of the substrate, defines the film growth process. All of the target and ion properties that are discussed in the previous
Section 4.1 also influence the differential sputtering yield, but they are not discussed in detail in this chapter. Instead, we discuss the general trends in the differential sputtering yield of the investigated elements.
The angular distribution of the sputtered atoms is presented in
Figure 5 and
Figure 6, where the graphs are arranged in the same way as they can be found in the periodic table. The elements in the same column in the figure belong to the same group of the periodic table, meaning that they have the same valence shell electron configuration. In all of the investigated materials, the sputtered flux is the highest in the direction that is perpendicular to the target surface. According to simulations, the lightest elements seem to be sputtered predominantly near the normal direction (
Figure 6). This unusual angular distribution is discussed later in this paper.
In the theory of sputtering, Sigmund assumes that the angular distribution of the sputtered atoms is a cosine function [
35,
36]. Such a dependence is attributed to the isotropic flux inside the amorphous target material [
1]. A perfect cosine distribution is often not observed experimentally, especially when measuring at very low or very high ion energies. The angular distribution of the sputtered material is often fitted using the power fitting parameter
y [
66]:
Our simulations of angular distribution (
Section 3.2) show that the fitting parameter
y is close to one for the majority of the investigated materials. Therefore, the SRIM simulations show essentially a cosine angular distribution of the sputtered atoms. The exception is Si, where the
y was 2.172, while for elements B, C, and Al, we could not determine the power fitting factor because the simulations show a highly directed flux near the surface normal. Such an unrealistic atom distribution for the elements with Z < 14 was also observed by other authors.
Several possible reasons were suggested as the source of an error. Shulga [
67] suggested that directed atom emission in the light elements results from errors in the calculations of particle trajectories within the material. Wittmaack [
68] identified several problems in the earlier versions of the SRIM program (specifically, the 2000 and 2003 versions). The author attributed the sputtering yield artifacts to an incorrect approximation of ion-target scattering and too low electronic stopping power in the low-energy range. Furthermore, they identified the differences between the detailed and quick calculation mode, which are still present in the latest version of SRIM. Wittmaack also suggested that the problems in simulations might be caused by the assumed non-random atom spacing in the solid. As a result, the simulated sputtering yields that were achieved by earlier SRIM versions were compared to experimental values, and they were too large for
Zi/
Zs < 0.7 and too small for
Zi/
Zs > 2.
In our work, we compensated for any differences between the calculated and the experimental values by modifying the surface binding energy parameter. Other authors proposed additional sources of error in the SRIM, such as the algorithm for searching the collision partners when an ion approaches the surface [
69] or ignoring the extremely small scattering angles [
70]. We examined the collision cascades in the SRIM simulations for all of the investigated target materials, but we did not find any preferential direction for sputtering in the case of the elements with Z < 14. The nuclear stopping powers which are essential in the SRIM simulation are uncertain for lower ion energies, and this could be the source of the error in the simulation. We cannot identify the specific source of the error in SRIM, which results in highly directed atom emission in the case of the light elements.
In the literature, under-cosine, over-cosine, and cosine angular distributions of sputtered material are reported. In one of the earliest reports, Wehner and Rosenberg [
71] measured the angular distribution of Ni, Ge, Fe, Mo, and Pt for normal Hg
+ ion bombardment with energies that were between 100 eV and 1 keV. Their experiments demonstrate an under-cosine angular distribution with more material which was ejected to the sides than that which was demonstrated in the direction that was normal to the target. Tsuge et al. [
26] also measured an under-cosine distribution for the Au, Al, and NiFe targets, which were bombarded by 1 keV argon ions. They related the under-cosine spatial distribution to the crystallographic orientation. Oyarzabal et al. [
29,
72] performed measurements of an angular distribution for a very low ion energy range (75–225 eV) by analyzing the sputtering of C and Mo. Their measurements show a heart-like distribution. The differential sputtering yields which were measured for several metals using heavier ions (Kr and Xe) also showed a heart-like distribution [
73]. Turner et al. [
74] measured the angular distribution for Cu which was sputtered by argon ions with an energy of 510 eV. The distribution differed from the cosine distribution as more particles were sputtered near the target normal (between −10° and 10°) and at approximately 45°. Surla et al. [
75] measured an under-cosine distribution for Mo which was sputtered by argon ions with an energy of 750 eV. Chini et al. [
76] analyzed the sputtering of a Ge target with argon ions between 600 eV and 4000 eV. The authors found the distribution best fits with an over-cosine function, with the power parameters that were between 1.25 and 1.32.
Computer simulations that were performed by Yamamura et al. [
66] using the Monte Carlo simulation program ACAT showed that the degree of the over-cosine distribution depends on the energy of the incident ions. By analyzing an Fe target, they simulated over-cosine angular distributions for higher ion energies (above 2000 eV) and under-cosine angular distributions for lower ion energies (below 500 eV). They claim that the over-cosine angular distribution is due to the geometrical asymmetry near the surface. The authors explain the under-cosine distribution for low ion energy sputtering by an anisotropic velocity distribution of the cascade near the surface. Feder et al. [
77] measured and simulated the sputtering of Ag with argon ions. The shape of the angular distribution also qualitatively agreed with their simulations. The same authors sputtered Ge with argon and xenon ions and compared the measurements with the simulations that were performed using the TRIM.SP code [
31]. They measured a cosine-like angular distribution for 1 keV argon ions and an under-cosine distribution for the sputtering with 1 keV xenon ions. Hence, very different angular distributions were observed experimentally and by simulations. According to the SRIM simulations that were performed in this work, the angular distributions of the sputtered atoms are cosine for all of the elements (except for the elements with Z < 14), and they do not change with an increasing ion energy in the 300–1200 eV range.
It should be noted that the sputtering is also influenced by the roughness, composition, and crystallinity of the target material [
64], however, these parameters are not included in the SRIM program, but they often cannot be avoided in the experimental studies. Recent measurements and simulations by Cupak et al. [
15] show that when rather rough surfaces were bombarded by ions under large incidence angles, a more-or-less uniform distribution of the atoms resulted around the normal of the surface. On the other hand, the earlier simulations where the atomic-scale surface roughness had been implemented in the TRIM code show that the roughness significantly affected the total sputtering yield for the low ion energies and the non-normal angles of incidence [
78]. Similarly, Yamamura et al. [
66] stress the importance of roughness. Based on their sputtering simulations, they concluded that the roughness reduces the degree of the over-cosine distribution since the rougher surfaces have a larger effective surface area in comparison to the flat surfaces. Schlueter et al. [
79] found that polycrystalline metals with randomly oriented grains do not sputter with the same sputtering yield as an amorphous material does. They explained that the key reason for this is attributed to the linear collision sequences rather than the channeling effect. All of the above-discussed effects can influence the significant discrepancies between the experimentally determined angular distributions and the ones that are simulated by the SRIM program.
Several trends can also be recognized for the sputtering at oblique ion incidence angles.
Figure 8 shows the angular distribution for the sputtered elements in groups 4, 5, and 6 by argon ions impinging at a 60° angle with an energy of 600 eV. It can be seen that the symmetry of the atom distribution depends on the group of the periodic table. The elements in group 4 (Ti, Zr, and Hf) appear to have a more asymmetric distribution than the elements in group 5 (Cr, Mo, and W) do despite them having similar masses (i.e., for the elements in the same period). The most symmetric distribution is expressed by the elements in group 11 (Cu, Ag, and Au). However, the distribution also depends on the atomic mass of the target material. Within an individual group of elements, the lightest elements show the most asymmetric distribution (cf. columns in
Figure 8). For example, the lightest element in group 4 (i.e., Ti) experienced the largest asymmetry, while the most symmetric distribution can be observed in the heaviest element (Hf). Similar observations are valid for the other two analyzed groups of elements. The most symmetric distribution is observed in the sputtering of the heaviest analyzed element (Au). On the other hand, the asymmetry of the distribution does not appear to be associated with the surface binding energy. For example, Ti and Au have similar surface binding energies, but very different angular distributions of their sputtered atoms.
When analyzing the angular distribution for oblique sputtering with respect to the energy of the ions, we can see that distribution is more asymmetric at lower ion energies.
Figure 9 shows an example of the Cr distribution when it was sputtered by argon ions at a 60° angle at 300 eV, 600 eV, and 1200 eV. The atom distribution is the least symmetrical at 300 eV and the most symmetrical at 1200 eV. The sputtering by higher energy ions clearly causes a more symmetrical distribution of the sputtered atoms. This trend could be explained by the deeper penetration of the ions when a solid is bombarded by ions of a higher energy. Higher-energy ions cause more collisions in the solid, which results in a more uniform distribution of the collision cascades, while the lower-energy ions cause fewer cascades with a more asymmetric distribution, and consequently, an asymmetric emission of the atoms from the surface.
Lautenschläger et al. [
80] measured the angular distribution of Ti. They sputtered the target material perpendicularly and at 30° and 60° with 1 keV Ar. They also performed measurements and simulations for different Ar ion energies. They showed that sputtering is more isotropic for higher energies of incident ions and for lower incidence angles of the ions. Such sputtering enables more collisions along the ion and atom path inside the solid material. This leads to fully developed collision cascades which results in a more isotropic distribution of the sputtered particles.