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RBS Tutorial: Theory

Rutherford Backscattering (RBS) is based on collisions between atomic nuclei and derives its name from Lord Ernest Rutherford, who in 1911 was the first to present the concept of atoms having nuclei. It involves measuring the number and energy of ions in a beam which backscatter after colliding with atoms in the near-surface region of a sample at which the beam has been targeted.

With this information, it is possible to determine atomic mass and elemental concentrations versus depth below the surface. RBS is ideally suited for determining the concentration of trace elements heavier than the major constituents of the substrate. Its sensitivity for light masses, and for the makeup of samples well below the surface, is poor.

When a sample is bombarded with a beam of high energy particles, the vast majority of particles are implanted into the material and do not escape. This is because the diameter of an atomic nucleus is on the order of 1e-15 m while the spacing between nuclei is on the order of 2e-10 m. A small fraction of the incident particles do undergo a direct collision with a nucleus of one of the atoms in the upper few micrometers of the sample. This “collision” does not actually involve direct contact between the projectile ion and target atom. Energy exchange occurs because of Coulombic forces between nuclei in close proximity to each other. However, the interaction can be modeled accurately as an elastic collision using classical physics.

The energy measured for a particle backscattering at a given angle depends upon two processes. Particles lose energy while they pass through the sample, both before and after a collision. The amount of energy lost is dependent on that material’s stopping power. A particle will also lose energy as the result of the collision itself. The collisional lost depends on the masses of the projectile the target atoms. The ratio of the energy of the projectile before and after collision is called the kinematic factor.

The number of backscattering events that occur from a given element in a sample depend upon two factors: the concentration of the element and the effective size of its nucleus. The probability that a material will cause a collision is called its scattering cross section.

Kinematics

For scattering at the sample surface the only energy loss mechanism is momentum transfer to the target atom. The ratio of the projectile energy after a collision to the projectile energy before a collision is defined as the kinematic factor.

There is much greater separation between the energies of particles backscattered from light elements than from heavy elements, because a significant amount of momentum is transferred from the incident particle to a light target atom. As the mass of the target atom increases, less momentum is transferred to the target atom and the energy of the backscattered particle asymptotically approaches the incident particle energy. This means that RBS is more useful for distinguishing between two light elements than it is for distinguishing between two heavy elements. RBS has good mass resolution for light elements, but poor mass resolution for heavy elements.

For example, when He++ strikes light elements such as C, N, or O, a significant fraction of the projectile’s energy is transferred to the target atom and the energy recorded for that backscattering event is much lower than the energy of the beam. It is usually possible to resolve C from N or P from Si, even though these elements differ in mass by only about 1 amu.

However, as the mass of the atom being struck increases, a smaller and smaller portion of the projectile energy is transferred to the target during collision, and the energy of the backscattered atom asymptotically approaches the energy of the beam. It is not possible to resolve W from Ta, or Fe from Ni when these elements are present at the same depths in the sample, even though these heavier elements also differ in mass by only about 1 amu.

An important related issue is that He will not scatter backwards from H or He atoms in a sample. Elements as light as or lighter than the projectile element will instead scatter at forward trajectories with significant energy. Thus, these elements cannot be detected using classical RBS. However, by placing a detector so that these forward scattering events can be recorded, these elements can be quantitatively measured using the same principles as RBS.

Scattering Cross Sections

The relative number of particles backscattered from a target atom into a given solid angle for a given number of incident particles is related to the differential scattering cross section. The scattering cross section is basically proportional to the square of the atomic number of the target atom. The illustration shows relative yields for He backscattering from selected elements at an incident He energy of 2 MeV. The energies for He backscattering from these elements when present at the surface of a sample are also displayed. The graph indicates that RBS is over 100 times more sensitive for heavy elements than for light elements, due to the larger scattering cross sections of the heavier elements.

Stopping Power

Only a small fraction of the incident particles undergo a close encounter with an atomic nucleus and are backscattered out of the sample. The vast majority of the incident He atoms end up implanted in the sample. When probing particles penetrate to some depth in a dense medium, projectile energy dissipates due to interactions with electrons (electronic stopping) and to glancing collisions with the nuclei of target atoms (nuclear stopping). This means that a particle which backscatters from an element at some depth in a sample will have measurably less energy than a particle which backscatters from the same element on the sample surface. The amount of energy a projectile loses per distance traversed in a sample depends on the projectile, its velocity, the elements in the sample, and the density of the sample material. Typical energy losses for 2 MeV He range between 100 and 800 eV/nm. This energy loss dependence on sample composition and density enables RBS measurements of layer thicknesses, a process called depth profiling.

The majority of energy loss is caused by electronic stopping which behaves (roughly) like friction between the probing particles and the electron clouds of the target atoms. Nuclear stopping is caused by the large number of glancing collisions which occur along the path of the probing atom. Nuclear stopping contributes significant energy losses only at low particle energies. The ratio of energy loss to two-dimensional atom density for a given material is known as its stopping cross section (epsilon), commonly measured in units of eV-cm. Since the majority of energy loss is caused by interactions with electrons, the electronic structure of the target material has a significant affect upon its stopping power.

Theoretical predications of stopping power are both complicated and inaccurate. Therefore, empirical stopping powers are often used in RBS calculations. A polynomial equation and a table of coefficients provides calculations of stopping powers over a wide range of energies and elements. In order to calculate the energy loss per unit of depth in a sample one can multiply stopping cross section times the density of the sample material (atoms/cm2). Sample densities can vary significantly. It is necessary to know the density of the sample material in order to calculate the depth of a feature or the thickness of a layer by RBS.

Metal Silicide Example

An example of a sample well-suited for RBS analysis is a metal silicide film. They are commonly used as interconnections between semiconductor devices because they conduct better than aluminum or silicon. The conductivity depends on the ratio of the silicon to the metal and on the thickness of the film. Both parameters can be easily determined by RBS.

The figure illustrates the interaction between the kinematic factor and the scattering cross section. The two spectra come from two TaSi films of different Ta/Si compositions on Si substrates. In this example, one of the films is 230 nm thick, while the other film is 590 nm thick. The experiment uses an ion beam of He++ at 2.2 MeV.

In both spectra, the high energy peak arises by scattering from tantalum in the TaSi film layer. The peak at lower energy is from silicon, which appears in both the TaSi film on the surface and in the Si substrate. Silicon is much less likely to cause scattering events than tantalum due to its smaller scattering cross section. To make the features of the silicon signal in these two spectra easily discernible, the silicon peaks have been multiplied by five.

For scattering at the sample surface, the only energy loss is due to momentum transfer to the target atom. The high energy edge of the tantalum peaks near 2.1 MeV corresponds to backscattering from Ta at the surface. The high energy edge of the silicon peaks near 1.3 MeV corresponds to backscattering from Si at the surface.

Layer Thickness Measurements

By measuring the energy width of the Ta peak or the Si step and dividing by the energy loss of He per unit depth in a TaSi matrix, the thickness of the TaSi layer can be calculated.

For example, the low energy edge of the Ta peak corresponds to scattering from Ta at the TaSi/Si interface. The illustration shows that particles scattered from tantalum at the TaSi/Si interface of the 230 nm film have a final energy of about 1.9 MeV, while particles scattered from the same interface of the 590 nm film have less final energy (about 1.7 MeV) because they have passed through more TaSi. The entire Ta peak spans a greater energy range, because of the increased thickness of the layer it represents.

Elemental Ratios

By measuring the height of the Ta and Si peaks and normalizing by the scattering cross section for the respective element, the ratio of Ta to Si can be obtained at any given depth in the film. The stopping cross section for TaSi is significantly higher than for pure Si. This means that a backscattered particle will lose more energy per unit volume in TaSi than in pure Si. An implication of this fact is that, for a given energy loss (DE), there are fewer atoms contained in a volume of TaSi than for the same volume of pure Si. This results in fewer backscattering events, and that means the peak for silicon will be lower in the TaSi than in the pure Si layer. In the spectrum illustrated below, the silicon peak has a step at its high-energy end: the lower peak is the TaSi; the higher peak is the pure silicon.

The height of a backscattering peak for a given layer is inversely proportional to the stopping cross section for that layer. The stopping cross section of TaSi is known to be only 1.37 times that of Si. This explains why the height of the peak corresponding to Si in the TaSi layer is less than one-half the height of the peak corresponding to Si in the substrate, even for a film with a Si:Ta ratio of 2:3.

Elemental Concentrations

It is also possible to calculate the Si concentration in the TaSi film by comparing the height of the Si peak for the TaSi layer to that of the peak for the pure Si substrate, but only after correcting for the different stopping cross sections of the two materials.

Straggling

After passing through a target with a finite thickness the probing He atoms will not only lose energy, but they will also no longer be monoenergetic. Instead, they will have distribution about the energy which is predicted by energy loss calculations. The process through which a probing He atoms loses energy involves a large number of interactions with individual atoms along its trajectory through the sample. This causes statistical fluctuations in the energy loss process which, along with the inherent limits of energy resolution in the RBS detection system, limit the energy resolution which can be achieved for atoms backscattered from larger sample depths. Since the accurate determination of both depth and mass dependent on the energy of the backscattered particle, straggling limit depth and mass resolution for buried features.

As noted in the Stopping Power section, the majority of energy loss occurs through interactions with electrons. As a result, energy straggling increases with the atomic number of the target element since atomic number also reflects the number of electrons present. Energy straggling causes the low energy edge of peaks to slope. For thick, high Z materials this effect can be quite pronounced. If energy straggling is not taken into account, then sloping edges of peaks can be misinterpreted as intermixing between two layers. The accuracy of depth resolution depends on how accurately the contribution of straggling can be calculated.

Channeling

In addition to elemental compositional information, RBS can also be used to study the structure of single crystal samples. When a sample is channeled, the rows of atoms in the lattice are aligned parallel to the incident He ion beam. The bombarding He will backscatter from the first few monolayers of material at the same rate as a non-aligned sample, but backscattering from buried atoms in the lattice will be drastically reduced since these atoms are shielded from the incident He atoms by the atoms in the surface layers. For example, the back- scattering signal from a single crystal Si sample which is in channeling alignment along the <1-0-0> axis will be approximately 3% of the backscattering signal from a non-aligned crystal, or amorphous or poly-crystalline Si. By measuring the reduction in backscattering when a sample is channeled, it is possible to quantitatively measure and profile the crystal perfection of a sample, and to determine its crystal orientation.

Channeling can also be used to help improve the RBS sensitivity for light elements. For example, it is difficult to accurately measure N concentrations in TiN films deposited on Si substrates due to the interfering signal from the Si substrate. By channeling the substrate, its signal is reduced, thus improving the sensitivity for the N peak which is superimposed on the Si peak. Since TiN layers are typically polycrystalline, the channeling does not affect the backscattering signals from the Ti or N. Care must be taken, however, to avoid channeling in single crystal layers when performing compositional analysis, since the channeling effects can result in erroneously low concentrations for elements in these layers. In order to avoid channeling affects the orientation of the sample is continually manipulated in order to present a variety of crystal orientations during the analysis. Spectra acquired in this manner are frequently referred to as “Rotating Random” spectra, since the most common randomization routine involves tilting the sample 7 degrees off of the channeling axis, and then rotating the sample.

Density Effects

As noted in the discussion of Stopping Power, a probing He atom loses finite amounts of energy during encounters with atoms in a sample. As a result, the spacing (density) of atoms will have a direct effect on the amount of energy lost by a probe atom versus the distance (depth) it traverses.

For example, RBS analysis of a Ti film might produce a two-dimensional Ti concentration of 5.66 E 27 atoms/cm2. In order to calculate a thickness for this film, the density of the titanium must be assumed. If we use a value of 5.66 E 22 atoms/cm2 (density of bulk Ti), a thickness of 100 nm is obtained. Depending upon the method used to deposit this Ti film the actual film density may be significantly less than bulk density. In this case, the thickness obtained by measuring the film with a profilometer or using an SEM will be significantly higher than the value calculated from the RBS results.

It is easier to present RBS results in the form of concentration versus depth, so density assumptions are frequently used to convert the RBS results into this format. It should be pointed out whenever such an assumption has been made. If RBS thicknesses are presented without an appropriate warning about the assumptions made, however, questions should be raised about the accuracy of the technique. Whenever the actual density of a film is significantly different than the density assumed in an RBS calculation, the thickness obtained by RBS will diverge significantly from the thickness obtained by another technique.

It is useful to note that where TRBS is the thickness obtained by RBS, and DRBS is the density assumed when calculating the RBS thickness, TReal and DReal are the actual film thickness and density, and atoms/cm2 is the two-dimensional concentration of atoms in the film (the concentration which can be accurately calculated from the RBS results without making any assumptions). Another useful feature of RBS is that since RBS will provide an accurate concentration of the total atoms/cm which are present in a film, if the actual thickness of the film can be measured by another method, then the density of the film can be calculated.