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ON THE APPLICATION OF CALIBRATION DATA TO SPREADING RESISTANCE ANALYSIS
REFERENCE: Berkowltz, H. L., "On the Application of Calibration Data to Spreading Resistance Analysis,"
Semiconductor Fabrication: Technology and Metrology, ASTM STP 990, Dlnesh C. Gupta, editor, American Society for Testing and Materials, 1989.
ABSTRACT: The requirements imposed by microelectronics structures on material characterization by spreading
resistance profiling (SRP) have made the development of sound calibration procedures for SRP essential. To this end, the Efficient Multilayer Analysis Program, also known as the Berkowltz-Lux technique, has been adapted to incorporate both variable probe radius and barrier resistance models. Since both models are heurlstically based, two versions of each method have been tried. Numerical stability and the ability to produce reasonable results are discussed for each version.
KEYWORDS: spreading resistance, profiling INTRODUCTION
The need for accuracy in material characterization of
microelectronics structures is increasing as device sizes and process tolerances shrink. As a result, it has become essential to develop sound methods of reconciling measured calibration data with procedures for spreading resistance profiling (SRP) analysis. To this end, the Efficient Multilayer Analysis Program (EMAP) [1-3] has been adapted to Incorporate both variable effective probe radius and barrier resistance models to analyze spreading resistance profiles.
In section 2, a brief review of the anatomy of an SRP analysis program is given. Xn particular, those features of the EMAP scheme not previously published are disclosed.
In section 3, the modification of an SRP program to accommodate profiles measured on semiconductor materials is discussed.
Dr. Berkowltz is Chief Scientist at Solid State Measurements Inc., 110 Technology Drive, Pittsburgh, PA 15275.
Implementations of the commonly invoked barrier resistance and variable effective probe contact radius models are described.
In section 4, representative spreading resistance profiles are presented, in which the resistivity and concentration profiles have been calculated using the barrier resistance and variable effective probe contact radius models.
Section 5 is devoted to conclusions and to a discussion of future trends for high resolution electrical profiling of semiconductor device structures.
REVIEW OF CURRENT PRACTICE
Most methods of analyzing spreading resistance profiles are based on the Schumann and Gardner (SG) multilayer solution to Laplace's equation [4] for a probe injecting current Into a conducting slab, where the resistivity of the slab is assumed to be a function of depth only.
Over the years much effort has been expended to reduce the time and computer resources needed to evaluate the SG integral, and several successful methods have been published [1,5-7]. This section describes the elements, in addition to an SG integrator, which are required to deconvolve a resistivity profile from a measured spreading resistance profile. As an example, the techniques used in the EMAP program are described.
The calculation of spreading resistance given a resistivity profile is straightforward and is described in Reference 1. To calculate resistivities given spreading resistances, however. Involves the inversion of a highly nonlinear system. Assuming that spreading resistances are measured atop sublayers of constant resistivity and thickness, the calculation is made as follows. Starting at the deepest point, the value of a trial sublayer resistivity is adjusted (and the SG correction factor recalculated) until the calculated sublayer spreading resistance agrees with the measured sublayer spreading resistance to within the desired tolerance. The resistivities thus obtained are then used in the calculation of the resistivities of higher sublayers until the surface sublayer is reached. The process used for ohmlc material is outlined in the flow chart In Figure 1.
Efficient methods for estimating the initial sublayer resistance and refining the estimate can significantly reduce the number of SG Integrals needed to achieve convergence- The method used to choose a trial sublayer resistivity depends on the position of the sublayer in the profile. On the substrate or junction, i.e. the lowest level, the resistivity p is calculated using the equation:
p = 2aRm/Csub
where Cg„i, Is the SG correction factor for an infinite substrate Rj, is the measured spreading resistance, and a Is the probe contact radius.
If the layer is within three layers above the substrate (or junction) then
Pi = Pl+1-
For other layers the initial guess is made by extrapolating Ln(p) for the three previously calculated resistivities with a parabola:
Pi = Pi+3(Pl+l/Pl+2)^
This appeal to continuity for the first guess of the sublayer resistivity frequently results In convergence In one or two iterations.
Commonly used methods to improve the sublayer resistivity estimate Include regula falsi, the secant method, or other zero finders which are readily available [8]. For the purposes of this paper, programs
prefixed with "TST" use the plodding but robust step and bisection method, while those prefixed with "EMAP" employ the method described be low.
To efficiently interpolate the SG correction factor as a function of p, a function with the following properties was sought: (1) it had to behave like the SG correction factor at large and small values of the resistivity; (ii) it had to lead to a numerically simple method of calculating the Improved sublayer resistivity estimate. At large and small values of the resistivity, the SG correction factor approaches constants Cj and Cg respectively, which depend only on geometry [1]. An approximate correction factor Cg Js defined by a simple Fade
approximation:
Ca = (Co+ACip/a)/(l+Ap/a)
The parameter A is adiusted so that the last calculated SG correction factor agrees with Cg when the last estimated p Is Inserted Into the above equation. By setting Rm equal to p Ca/(2 a) and finding the positive root of the quadratic equation,
:^l(p/a)2 + (Co-RjAXp/a) " Rm=0 the next estimate of p, is obtained.
As with most superllnear Interpolation schemes, oscillations can sometimes occur during which convergence Is very slow. Therefore EMAP... programs test for oscillatory behavior and use an averaging technique to damp the oscillations and restore rapid convergence.
Oscillations are detected by noting the trend of three consecutive estimates of p : rx, r2, and r3. If (r3-r2)(r2-r3) is negative then r3, the most recently calculated value of p, Is replaced with the value of the expression:
(r3ri-r22)/(r3+ri-2r2)
and the calculation continues. Both EMAP... and TST... programs require that j(l-Rj./R^)| be less than .0001 to terminate the calculation of a sublayer resistivity.