**1. Introduction**

124 Advances in Wavelet Theory and Their Applications in Engineering, Physics and Technology

Whitehouse, D. J., 2011. *Handbook of Surface and Nanometrology*, 2nd Edition, CRC Press,

pages.

Taylor and Francis Group, Boca Raton, FL, U.S.A, ISBN: 1978-1-4200-8201-2, 976

It has become more important to measure accurate depth profiles in developing more advanced devices. To this aim, Depth profiling in secondary ion mass spectrometry (SIMS) has been extensively used as an informative technique in the semiconductor and electronic devices fields due to its high sensitivity, quantification accuracy and depth resolution (Fujiyama et al, 2011; Seki et al, 2011). However, the depth resolution in SIMS analysis is still limited to provide reliable and precise information in very thin structures such as delta layers, abrupt interfaces, etc. By optimization of the experimental conditions, the depth resolution can be enhanced. In particular, lowering the primary energy seems to be a good solution, but this increases the measurement time and leads to other limitations, owing to the wrong focalization of primary ion beam, such as roughness in the crater bottom, not flat crater, etc. Therefore, the depth resolution remains so far to its perfect limit. It is only by numerical processing like deconvolution that the depth resolution can be improved beyond its experimental limits.

For the past several years, different approaches of deconvolution have been proposed taking into account the different physical phenomena that limit depth resolution, such as collisional mixing, roughness, and segregation ( Makarov, 1999; Gautier et al, 1998; Fares et al, 2006; Dowsett et al, 1994; Mancina et al, 2000; Shao et al, 2004; Collins et al, 1992; Allen et al, 1993; Fearn et al, 2005). However, most problems encountered in these deconvolution methods are due to the noise content in the measured profiles. This instrumental phenomenon, which cannot be eliminated by the improvement of operating conditions, strongly influences the depth resolution and therefore the quality of the deconvoluted profiles.

The deconvolution of depth profiling data in SIMS analysis amounts to the solution of an appropriate ill-posed problem in that any random noise in data leads to no unique and no stable solution (oscillatory signal with negative components, which are physically not acceptable in SIMS analysis). Thus, the results must be regularized (Tikhonov, 1963; Barakat et al, 1997; Prost et al, 1984; Burdeau et al, 2000; Herzel et al, 1995; Iqbal, 2003; Varah, 1983;

Multi-Scale Deconvolution of Mass Spectrometry Signals 127

This work is based on SIMS data, for which reason the results presented here are largely restricted to the conditions of SIMS analysis. The main objective of this work is to show that the MD gives much better deconvolution results than those obtained using monoresolution regularization methods. In particular, the results obtained are compared to those achieved using a regularized monoresolution deconvolution, which is Tikhonov-Miller regularization with a pre-deconvoluted signal as a model of the solution, denoted as TMMS (Mancina et al,

Depth profiling in SIMS analysis is mathematically described by the convolution integral which is governed by the depth resolution function (DRF), h(z). If the integral over h(z) is normalized to unity, then the measured (convolved) signal is given by the well-known

+ b(z), (1)

, k =0,…, 2N-2, (2)

y = Hx + b, (3)

*y*( ) ( ') ( ') ' *z h z z x z dz*

where x(z) is the compositional depth distribution function and b(z) is the additive noise.

This work deals with the deconvolution of depth profiling SIMS data. Therefore, it is important for further consideration to know the shape of the DRF that is typical of SIMS profiles. We have chosen to describe the DRF analytically in a form initially proposed by Dowsett et al (Dowsett et al, 1994), which is constituted by the convolution of double exponential functions with a Gaussian function. This DRF can be described by three parameters: λu (the rising exponential decay), σg (the standard deviation of the Gaussian function), and λd (the falling exponential decay). The latter characterizes the residual mixing effect, which is considered to be the main mechanism responsible for the degradation of the depth resolution (Boulakroune et al, 2007; Yang et al, 2006). For any possible values of these parameters, the DRF is normalized to unity. The consequences of the fact that the resolution function can be represented in the form of a convolution have been described elsewhere (Gautier et al, 1998; Dowsett et al, 1994; Collins et al, 1992; Allen

1

*k i ki k i y xh b* 

where N is the number of samples of vectors h, x. Equation (2) can be rewritten as

where H is a matrix built from h(z). In the case of a linear and shift-invariant system, H is a convolution operator (circular Toeplitz matrix). This means that the multiplication of H with the vector x leads numerically to the same operation as the analytical convolution of h(z)

*N*

0

2000).

**2.1 Background** 

convolution integral

et al, 1993).

with x(z).

For a discrete system, eq. (1) can be written as

**2. Deconvolution procedure** 

Essah, 1988; Brianzi, 1994; Stone, 1974; Connolly et al, 1998; Berger et al, 1999; Thompson et al, 1991; Fischer et al, 1998). To this end, the solution is superimposed with certain limitations by introducing some additional limitative operators, whose shape is chosen depending on the formalism used for the solution of the ill-posed problem, into a goal function; usually the goal function is the mismatch between the convoluted solution and the initial data (Makarov, 1999). Indeed, different forms of limitative operator have been used. For example, Collins and Dowssett (Collins et al, 1992) and Allen and Dowssett (Allen et al, 1993) have used the entropy function as a limitative operator. Based on the Tikhonov-Miller regularization, Gautier et al (Gautier et al, 1998) have used a limitative operator that was defined as smoothness of the solution. Mancina et al (Mancina et al, 2000) have introduced *a priori* a pre-deconvoluted signal as a model of solution in an iterative regularized method. Nevertheless, the results of most of these approaches contain artifacts with negative concentrations, which are not physically acceptable. The origin of these artifacts is related to the presence of strong local components of high frequencies in the signal which form part of the noise. To remove the negative components from the deconvoluted profile, some algorithms with non-negativity constraints have been proposed (Makarov, 1999; Gautier et al, 1998; Gautier et al; 1998; Prost et al, 1984). These methods, which constrain the data to be positive everywhere, are sensitive to noise, i.e., a little perturbation in the data can lead to a great difference in the deconvoluted solution. A truncation of the negative data is an arbitrary operation and it is not acceptable, since it results in an artificially steep slope and can lead to the adoption of subjective criteria for profile assessment (Herzel et al, 1995). Moreover, noise in the data increases the distance between the real signal and its estimate, therefore a priori constraint is not enough, and a free-oscillation deconvolution is necessary.

To overcome these limits, it is important to adopt a powerful deconvolution that leads to a smoothed and stable solution without application of any kind of constraints. In this context, multiscale deconvolution (MD), which is never used to recover SIMS profiles, may be the most appropriate technique.

The MD provides a local smoothness property with a high smoothness level in unstructured regions of the spectrum where only background occurs and a low smoothness level where structures arise (Fischer et al, 1998). Based on wavelet transform, the MD seems to be a good solution that can yield information about the location of certain frequencies in the profile on different frequency scales. Therefore, high frequencies, which are related to noise, can be localized and controlled at different levels of wavelet decomposition. The multiscale description of signals has facilitated the development of wavelet theory and its application to numerous fields (Averbuch & Zheludev, 2009; Charles et al, 2004; Fan & Koo, 2002; Neelamani et al, 2004; Zheludev, 1999; Rashed et al, 2007; Garcia-Talavera et al, 2003; Starck et al, 2003; Jammal et al, 2004; Rucka et al, 2006). This chapter is intended to explore capabilities of wavelets for the deconvolution framework. The proposed idea is to introduce a wavelet-based methodology in the Tikhonov-Miller regularization scheme and shrinking the wavelet coefficients of the blurred and the estimated solution at each resolution level allow a local adaptation of limitative operator in the quadratic Tikhonov-Miller regularization. This leads to compensation for high frequencies which are related to noise. As a result, the oscillations which appear in classical regularization methods can be removed. This leads to a smoothed and stable solution.

This work is based on SIMS data, for which reason the results presented here are largely restricted to the conditions of SIMS analysis. The main objective of this work is to show that the MD gives much better deconvolution results than those obtained using monoresolution regularization methods. In particular, the results obtained are compared to those achieved using a regularized monoresolution deconvolution, which is Tikhonov-Miller regularization with a pre-deconvoluted signal as a model of the solution, denoted as TMMS (Mancina et al, 2000).
