Auger Electron Spectrum Interpretation

Auger electron spectroscopy is [81,82,83] an experimental technique for determining the chemical composition of a sample, which involves measurement of the rate of secondary electrons leaving a sample, on bombardment with a primary electron beam, as a function of energy. The results are usually presented as a graph of $\frac{\mathrm{d}N(E)}{\mathrm{d}E}$ against $E$, where $N(E)\mathrm{d}E$ is the rate of arrival at the detector of secondary electrons, of energies between $E$ and $E+\mathrm{d}E$. This graph contains a series of minima, each of which is generated by a particular chemical element or combination of chemical elements in the sample. It is possible to find look-up tables [83], which give the sensitivity $S_{E, X}$ of a minimum at energy $E$ to element $X$.

The author has been unable to find, in the literature, any pointers to the construction of a model of the Auger process detailed enough to allow Bayesian inference of the posterior probability distribution over the thickness of a cobalt film on a copper surface, given a measured Auger spectrum. Therefore, the author has devised the following, rough method of estimating the thickness:

1. The ``peak height'' of a minimum at energy $E$ can be estimated as
   

   $$h_E = \frac{n_L+n_R-2n_C}{2}\pm\left\vert\frac{n_L-n_R}{2}\right\vert\textrm{,}$$ (B.1)

   
   
   where $n_C$ is the $\frac{\mathrm{d}N(E)}{\mathrm{d}E}$ value at the minimum, $n_L$ that at the maximum immediately to the left of the minimum, and $n_R$ that at the maximum immediately to the right of the minimum. This method is used to estimate $h_{777.5\,\mathrm{eV}}$ and $h_{922\,\mathrm{eV}}$, since both of these minima have non-zero sensitivities to both copper and cobalt.
2. It is assumed that, in terms of an effective copper concentration in the sample $c_{Cu}$, and an effective cobalt concentration $c_{Co}$,
   

   $$h_{777.5\,\mathrm{eV}} = S_{777.5\,\mathrm{eV}, Cu}c_{Cu}+S_{777.5\,\mathrm{eV}, Co}c_{Co}\textrm{,}$$ (B.2)

   
   
   and
   

   $$h_{922\,\mathrm{eV}} = S_{922\,\mathrm{eV}, Cu}c_{Cu}+S_{922\,\mathrm{eV}, Co}c_{Co}\textrm{.}$$ (B.3)

   
   
   Therefore,
   

   $$c_{Cu} = \frac{S_{922\,\mathrm{eV}, Co}h_{777.5\,\mathrm{eV}... ...-S_{777.5\,\mathrm{eV}, Co}S_{922\,\mathrm{eV}, Cu}}\textrm{,}$$ (B.4)

   
   
   and
   

   $$c_{Co} = \frac{S_{922\,\mathrm{eV}, Cu}h_{777.5\,\mathrm{eV}... ...-S_{777.5\,\mathrm{eV}, Cu}S_{922\,\mathrm{eV}, Co}}\textrm{.}$$ (B.5)

   
   

3. The effective copper concentration due to copper between depths $z$ and $z+\mathrm{d}z$ is assumed to be
   

   $$\mathrm{d}c_{Cu} = A\exp{}(-\lambda{}z)\mathrm{d}z\textrm{,}$$ (B.6)

   
   
   where the exponential factor represents the loss of incident electrons to inelastic scattering, the mean free path $\frac{1}{\lambda}$ of which can be found in look-up tables [83]. Therefore, for a cobalt thickness $t$, the total effective copper concentration is
   

   $$c_{Cu} = \int_{z=t}^{\infty}A\exp{}(-\lambda{}z)\mathrm{d}z\protect$$

   $$= \frac{A}{\lambda}\exp{}(-\lambda{}t)\textrm{.}$$ (B.7)

   
   
   Similarly,
   

   $$c_{Co} = \frac{A}{\lambda}(1-\exp{}(-\lambda{}t))\protect$$

   $$= c_{Cu}(\exp{}(\lambda{}t)-1)$$ (B.8)

   
   
   
   

   $$\Rightarrow{} t = \frac{\ln{}\left(\frac{c_{Co}}{c_{Cu}}+1\right)}{\lambda}\textrm{.}$$ (B.9)

   
   

4. The estimate of cobalt thickness, in ordinary units of length, thus obtained can be compared with the thickness, in $\,\mathrm{\mu{}A}\,\mathrm{s}$, obtained from the ion flux monitor (section 4.3,) to give a conversion factor between the two unit systems.

Shortly before the experiments in section 5.4, two cobalt films were grown, and their thicknesses calibrated in this manner. The resulting conversion factors were $(104\pm{}20)\,\mathrm{\mu{}m}\,\mathrm{A}^{-1}\,\mathrm{s}^{-1}$ and $(67\pm{}13)\,\mathrm{\mu{}m}\,\mathrm{A}^{-1}\,\mathrm{s}^{-1}$. The average is $(78\pm{}11)\,\mathrm{\mu{}m}\,\mathrm{A}^{-1}\,\mathrm{s}^{-1}$.

Since this estimate has been produced in the absence of a genuinely quantitative understanding of the Auger process, it should be treated with some scepticism; the large random error estimate quoted above, resulting from the differing base signals on either side of an Auger peak, embodies just such a healthy scepticism. Hope [37], who devised a thickness estimation method in a similar spirit, suggested that it may introduce an extra calibration error of $\sim{}25\%$, of the same order as that quoted here.

It could be suggested that the errors in this estimate render Auger spectroscopy an unsuitable method of thickness measurement, compared with, say, the use of a quartz thickness monitor. It should be noted that Auger spectroscopy and the quartz thickness monitor play different roles in thickness measurement, and are not directly interchangeable. A quartz thickness monitor would reveal relative thicknesses, a task which is undertaken in this thesis, not with the Auger electron spectroscopy, but with the ion flux monitor in the cobalt evaporator. The Auger spectroscopy is used to calibrate the ion flux monitor, to produce absolute thicknesses; similar measures would be necessary for a quartz thickness monitor; one study [47] is an exception to this division of labour, having used Auger spectroscopy for relative thickness measurements, and a stylus profilometer, acting on thick films, for absolute calibration.

The literature on $Co/Cu$ structures seems to reveal Auger electron spectroscopy as the usual method for achieving this calibration [38,55,59,37]. The author is aware of one attempt [41] to calibrate the thickness monitor by observing the onset of ferro-magnetism with increasing thickness, then comparing with earlier measurements of the critical thickness for the onset of ferro-magnetism; however, these earlier measurements [55] were themselves made with a thickness monitor calibrated using Auger electron spectroscopy. Also, in one instance [38], the mono-layer-period oscillations in medium-energy electron diffraction and thermal-energy atom scattering intensity were used as back-ups to Auger electron spectroscopy for thickness calibration; the three methods produced results consistent with one another. Another alternative absolute thickness measurement method is scanning tunnelling microscopy [56], and one study [46] has used this as a means of calibrating an Auger electron spectroscopy apparatus, which was then used for relative thickness measurements. A third absolute thickness measurement method, which has been used [39] to calibrate a quartz thickness monitor, with a claimed calibration error of just $5\%$, is `ex situ X-ray interference.'