9 Rutherford Back Scattering – III

Dr. Ajit K. Mahapatro

Learning Objectives

From this module students may get to know about the following

i. Depth scale for elemental analysis

ii. Surface energy and mean energy approximation

iii. Spectrum height for an element

1. Introduction

The quantification of atomic composition, the yield ratio of atoms scattered from the surface of the sample needs to be compared with energy and at a certain depth for the dependence of the energy width of the detector channel on backscattered energy. Ions when travel through a solid loses their energy by screened coulomb collision with the target nuclei and free target electrons. We already know that during scattering the nuclear energy loss is approximately three orders of magnitude smaller than the electronic energy loss. Energy loss of nuclei is mainly of interest in ion implantation and sputter-etching techniques like secondary ion mass spectrometry (referred as SIMS) whereas electronic interactions are the dominant mechanism for energy loss in the ion scattering analysis techniques. Thus one can selectively choose the incident beam species, energy and materials composition to get the desired result. As an energetic ion moves through matter, there is a loss of energy because of the Coulomb force acting on the electrons from the target atom. The passing of charged particle produces interaction which is columbic in nature. Since the electron has light mass, it does not affect the path of incident ions. This section describes the relation between the energy E1 of the detected particle and the depth x at which the backscattering event occurs in a mono isotopic elemental sample. Various approximations were taken in to account and are mathematically derived for the quantitative analysis. Depth scale analysis is important to get knowledge on the number atoms participating in the back scattering processes.

Depth Scale for an elemental sample:

Figure 1 shows the energy relation of an incident particle before and after scattering. Let the energy E₀ be the energy of incident particle, energy immediately before scattering at a depth x is E and E₁  be the energy of the particle emerging from the surface. It should be noted that the incident beam is smaller than the target. The incident particle, scattered particle and the normal to the surface of the sample lie in one plane. In the laboratory frame of reference it is given by θ = 180° – (θ1  + θ2), where θ1 and θ2 are the angles between the normal and the direction of the incident beam and of the scattered particle, respectively.

Fig 1:  Schematic  diagram  of  the  incident  and  back  scattered  beam.  θ₁   and  θ₂   are  positive; incident, back scattered beams and sample normal are coplanar.

θ₁, θ₂ are positive if they lie in the same side of the plane. We can relate the energy E with X by

Fig 2: Graphical representation of the energy loss in a sample in mean energy and surface energy approximations.

A graphical interpretation of these two equations is given in Fig. 2. Part (a) shows dE/dx as a function of energy as a light line. The heavy segments give the dE/dx values for the inward path from E0 to E and for the outward path path from KE to E1. The difference E0 — E is the energy loss along the inward path ΔEin; similarly, KE – E1 is the energy loss along the outward path ΔE_out.

Mathematical formulations of Energy Loss Factor [s] or stopping cross section [ε] If one assumes a constant value for dE/dx along the inward and outward paths, the two integrals can be written as

where the subscripts “in” and “out” refer to the (constant) values of dE/dx along the inward and outward paths. By eliminating E from these two equations, we have

Fig 3: If the energy loss is constant both outside and inside of the sample, energy ΔE is linearly related to the depth x which can be written as ΔE = [S]x represented in this schematic diagram.

Figure 4 depicts graphically the relation between the energy loss factor [S] and the actual depth x at which backscattering occurs for a given energy loss ΔE. It is clear from the graph that it is not exactly linear.

Fig. 4: Schematic of relation between the energy loss ΔE and the depth x at which backscattering occurs. The linear relation ΔE = AE = [S]x  is exact at one depth. The symbol [S₀] refers to the surface energy approximation.

2. Approximation to [S] and [ε]

2b. Mean energy Approximation

The surface energy approximation is no longer valid when the path length becomes appreciable. As a result, the surface approximation degrades. A better approximation can be obtained by selecting a constant value of dE/dx or ε at an energy E intermediate to that which the particle has at the end points of each track.

3. Energy Spectrum for a sample and height determination

We will see in this section the relation between the height of the energy and the number of scattering centers per unit area within the sample where backscattering occurs. In the remaining sections of this chapter, only stopping cross sections ε and stopping cross section factors [ε] will be used.

There is a functional dependence between the energy axis of a back scattering spectrum and the depth below the surface of a sample as shown in Fig. 4. Each energy width ε of a channel i in the multichannel analyzer is backscattering events recorded in channel i of thickness τi. The number of counts Hi in channel i is dependent on two factors including the thickness of the slab and the number of scattering centers in that cross section. The main goal is to find out the energy width S and the position El,i of channel ‘i’ in the energy spectrum, as indicated in Fig. 5 by correlating the number of counts Ht to the number of scattering centers per unit area Nτi in the slab of thickness τ, at depth xi which corresponds to the energy of a beam of normal incidence. The total number of particles detected in channel i is

?_?=  ?(??)Ω?N?_?/c??θ₁   

Fig. 5: Schematic diagram showing the relation between slab i at the depth xi (Left) and energy E1,i (Right).

Where, σ(Ei)  = Differential cross section at energy Ei.

Ω = solid angle.

Q = Total number of sample incident on the surface.

This is true if the incident particle is normal to the target surface which would give the total number of back scattering atoms of the sample. If there is a tilt in the incident particle from the normal to the surface with an angle θ1, then the equation can written as,

??= ?(?_?)Ω?N?_?/c??θ₁

4. Spectrum height for a scattering from the top layer

Fig. 6 gives a schematic diagram of the backscattering processes from the top surface of the sample and the spectrum resulting from it. The notation we have taken for the near-surface
region is H0 and ?0 instead of Hi and ??for regions within the sample.

Fig 6: Schematics of the backscattering process in the surface region of a sample consisting of a monoisotopic element (Left) and the resulting spectrum (Right).

?₀  =  ?(?₀ )Ω?­ξ/[?₀]c??θ₁

This equation states that the height of the energy spectrum at the surface is directly proportional to

(i) Q, the total number of incident particles on the sample;

(ii) ?(?₀ ), the average differential scattering cross section of the sample at the incident energy E₀;

(iii) Ω, the solid angle spanned by the detector aperture;

(iv) ?, the energy width of a channel

(v) [[?0]c??θ₁]-1  the inverse of the stopping cross section factor evaluated at the surface for a given scattering geometry multiplied by the cosine of the angle of incidence of the beam against the sample normal.

5. Spectrum height for a scatting at depth

The significance of depth profiling is to relate a spectrum height H_i  to a slab of material with thickness ?_? and number of atoms per unit area N?_? at depth xi. The height can be written as

??=  ?(??)Ω?N??/c??θ1

The cross section ?is evaluated at the energy Ei of the projectile immediately before scattering at depth xi. The amount of material N?? is defined by the energy width ? such that the particles backscattered from the slab will emerge from the sample with energies between E₁,i and E₁,i – ?. It would be wrong to conclude therefore that the energy width ?of these particles immediately after  scattering is  also  S.  The  reason  is  that  particles  with  slightly different  energies  after scattering at x undergo slightly different energy losses on their outward path.

As the incident beam enters the sample, the energy of the projectiles decreases. As a result the scattering cross section σ increases. This effect tends to increase the yield Hi  with decreasing energy E₁ of the detected particles. On the other hand, the stopping cross section also varies with E. In general this dependence is not as strong as that for σ (E) but ε can either increase or decrease with decreasing values of E. Consequently, the effect of the change in ε on the backscattering yield may either enhance or counteract the effect of the change in σ.

6. Depth scale for a homogeneous solid containing more than one element (compound sample)

In this section, we shall discuss the backscattering spectrum of a sample composed of a homogeneous mixture of several elements. For simplicity we denote the material as a compound sample although it could be either a mixture or a chemical compound. This case differs from that of the mono isotopic elemental sample considered thus in two significant ways. First, as the probing particles penetrate the film, they lose energy as a result of interactions with more than one element. Consequently, the stopping cross section depends on the composition of the sample. Second, when the probing particles with energy E are scattered at a specific depth within the sample, the value of the kinematic factor K and the scattering cross section σ will depend on the particular mass (atomic number) of the atom they strike. Since the stopping cross section varies with energy, the energy that the particles lose along identical outward tracks also depends on the atom struck in the scattering collision. For a compound sample, the yield of the backscattering spectrum and the energy-to-depth conversion thus depend on the element struck in the collision. All counts generated by backscattering from a given element constitute the signal of this element in the spectrum.

A particle penetrating the sample to a depth x under goes an energy loss given by

Summary

In this section we got the knowledge of depth scale analysis with the help of energy of the detected particle. The relation between the energy loss factor [S] and the actual depth x at which backscattering occurs for a given energy loss ΔE was calculated and was shown graphically which states that it is not linear. Two approximations were considered; one surface energy approximations and the other mean energy approximations. When the length scale is appreciable, surface energy approximation fails. The spectrum height was calculated which also depends on solid angle, scattering cross section and thickness of the material. By using the approximations described above we got the knowledge on the calculation of height of a spectrum of an elemental and compound film both on the surface and at a certain distance from the surface of the sample.

References:

1.Behrisch, R., and Scherzer, B. M. U. (1973).Thin Solid Films 19, 247.

2.Brice, D. K. (1973). Thin Solid Films 19, 121.

3.Chu, W. K., and Ziegler, J. F. (1975). J. Appl. Phys. 46, 2768.

4.Feng, J. S.-Y., Chu, W. K., Nicolet, M.-A., and Mayer, J. W. (1973). Thin Solid Films 19, 195.

5.Jack, H. E. Jr. (1973).Thin Solid Films 19, 267.6.Lever, R. F. (1976) in “Ion Beam Surface Layer Analysis” (O. Meyer, G. Linker, and F. Käppeler, eds.), Vol. 1, p. 111. Plenum Press New York.