S.C. Choo

On the calculation of spreading resistance correction factors

Abstract It is shown that the calculations of spreading resistance correction factors for graded structures can be readily carried out by using a simple recurrence formula for the integration factor that occurs in Schumann and Gardners multilayer theory. The number of layers that can be used in the practical application of this theory has hitherto been limited by the computer core size requirement, because the earlier method of calculating the integration factor requires the inversion of a 2N × 2N matrix for an N-layer approximation. The use of the recurrence formula effectively removes this constraint. In terms of computation time, the recurrence-formula method is also very efficient. The economy thus achieved both in computation time and in core size requirement makes it possible now to make spreading resistance correction a routine matter, without having to resort to such measures as Hus interpolation and space partitioning scheme.

Theory of the photovoltage at semiconductor surfaces and its application to diffusion length measurements

Abstract This paper presents an analytical theory of the steady-state surface photovoltage, which takes into account recombination in the surface space charge region and at surface states, as well as bulk diffusion in the semiconductor. For a given wavelength of light used in the photo-excitation, the theory is able to predict the photon flux required to yield a specified surface photovoltage. The validity of the theory has been established by means of detailed comparison with exact numerical solutions. It is shown that for an Si specimen space charge recombination plays an important role in determining the surface photovoltage, particularly at photovoltages of the order 0.1 times the thermal voltage or less. The theory is applied to a rigorous examination of the validity of two standard test methods of the American Society for Testing and Materials for measuring the minority carrier diffusion length, which are based on the surface photovoltage. It is found that while the method due to Goodman works well in general and even in the presence of large surface recombination, the method due to Quilliet and Gosar does not always give the correct value of diffusion length, because of the importance of recombination in the space charge region and at the surface states—it does so only under certain restrictive conditions, i.e. the material must have a doping concentration greater than 1015 cm−3 and a long minority carrier lifetime (> 10 μs), with a surface in depletion (but not in inversion) at equilibrium and a very low surface recombination velocity.

The role of source boundary condition in spreading resistance calculations

Abstract Solutions are presented for the current density distribution at an equipotential disc electrode in contact with a slab backed by a perfect conductor. These exact solutions provide a basis for testing the validity of the two forms of source current density distribution assumed in approximate calculations of spreading resistance correction factors, viz. a uniform distribution and the distribution given by the classical solution for the infinitely thick slab. By using the latter distribution and the power loss definition for spreading resistance, a new correction factor integral has been obtained. Correction factors have been calculated by using this integral and those given by Schumann and Gardner, by Lee and by assuming a uniform current distribution. Except for Schumann and Gardners method, all the methods yield results consistent with those obtained for the current density distributions. In the case of Schumann and Gardners method, the correction factors obtained for thin slabs agree closely with those given by the exact method, despite the fact that the assumed source current distribution is in gross disagreement with the exact distribution. The close agreement in correction factors is fortuitous and is a consequence of the definition that Schumann and Gardner used for the spreading resistance. For a slab with a perfectly insulating substrate, exact solutions are not available. A comparative study has therefore been made in this case between the correction factors obtained by the four approximate methods themselves. The overall conclusion is that of the approximate methods, the uniform current density method is the most satisfactory from the point of view of self-consistency and overall accuracy.

Spreading resistance calculations by the use of Gauss-Laguerre quadrature

Abstract The Gauss-Laguerre quadrature is proposed as a numerical method for calculating the correction factor integrals that occur in spreading resistance calculations. The method is very efficient in terms of computation time and memory storage, requiring only 33 integrand values for each integral evaluation. The accuracy of the method has been investigated for a variety of graded structures, and found to be better than 5%. As a test of its practical utility, the method has been used in the correction of the spreading resistance profile of a practical buried layer structure, and it has been found that the CPU time taken to correct the entire profile of 57 data points is 1.0 min on an IBM 1130 System with a 16K word (16 bit) memory or 0.4 sec on a UNIVAC 1100/10 Multiprocessor System with a 393K word (36 bit) memory. These times are a factor of 6 to 8 less than those required by using the previously proposed adaptive Simpsons rule to compute the correction factor integrals.

An efficient numerical scheme for spreading resistance calculations based on the variational method

Abstract This paper presents a simple and efficient numerical scheme for evaluating the correction factor integrals that arise in the variational method. The scheme is a modification of one recently proposed by Berkowitz and Lux for the uniform flux method. The abscissae and the weights required for the integration are given in a form which allows the numerical scheme to be readily implemented. Using this scheme, it takes, on an average, 0.8 sec to compute one value of correction factor on an Apple II Microcomputer. For a slab of varying thickness, backed by either a perfectly conducting or a high resistivity substrate, the correction factors obtained agree with those derived from the exact constant-potential method to within 1%.

Spreading resistance calculations for graded structures based on the uniform flux source boundary condition

In the approximate calculations of spreading resistance correction factors, two different types of boundary conditions over the source region have hitherto been assumed, viz., a uniform flux distribution, and the specific flux distribution that obtains in the classical solution for the infinitely thick slab. This paper presents results of a theory which has been derived for multilayer structures by using the uniform source flux distribution. The results given include those obtained for a series of exponentially-graded structures of varying steepness in the resistivity profile, whose thickness h1 ranges from 0.1 to 10 times the circular source contact radius, a. The results show that, as a rule, for an insulating substrate and (h′1/a) < 0.5, the uniform flux assumption leads to correction factors which differ at most by about 3% from those derived from the infinitely-thick slab flux assumption, while for a conducting substrate and (h′1/a) = 0.1 to 10, there is a difference of about 8%. Whatever the nature of the resistivity profile studied, it has been found that the difference is never greater than about 8%.

A multilayer correction scheme for spreading resistance measurements

Abstract This paper presents a method for computing the correction factor integral that arises in Schumann and Gardners multilayer theory for spreading resistance measurements. From a detailed analysis of the integrand, a numerical scheme has been devised in which the integration is carried out with typically less than 150 integrand values and most of the functional values needed in the integration are pre-calculated and stored. The numerical scheme has been implemented on both an IBM 1130 System and a Univac 1106 Time Sharing System, and used to correct the spreading resistance profile of a buried layer structure. The results indicate that routine application of the multilayer theory is now practicable on either a minicomputer or a time sharing system.

The spreading resistance of an inhomogeneous slab with a disc electrode as a mixed boundary value problem

Abstract This paper presents a treatment of the mixed boundary value problem that arises in determining the spreading resistance of an inhomogeneous slab backed by a perfectly conducting substrate. It is shown that the inhomogeneity enters into the problem through the weight function of a pair of dual integral equations, and that this function is the same as the integration factor that occurs in previous approximate solutions based on assumed source current distributions. Except for the difference in the weight function, the dual integral equations are similar to those for the homogeneous slab. Calculations are performed for structures with exponential resistivity profiles, and the results used to determine the accuracy of three approximate methods currently available for spreading resistance calculations on semiconductor device structures.

The spreading resistance of a homogeneous slab on a high-resistivity substrate: Mixed boundary value solutions

Abstract The range of applicability of the mixed boundary value method for calculating spreading resistance is extended to a homogeneous slab with a disc contact source and backed by a substrate of arbitrary, but finite resistivity. Solutions are presented in terms of the spreading resistance correction factors and the source current density distributions for a slab of varying thickness and with various high resistivity substrates. In particular, the results for a thin slab indicate that, as the substrate resistivity increases, more and more of the source current is concentrated near the edge of the disc electrode. A comparison is made of the source current density and potential corresponding to the mixed boundary value method with those given by the uniform flux and the variable flux (power-loss) method. It is found that, except for large slab thicknesses, the source potential distributions for a slab with a high resistivity substrate are not strongly influenced by the particular form of the source current density distribution assumed in either the uniform flux or the variable flux method. In consequence, both these two methods yield correction factors which agree quite closely with those derived from the mixed boundary value method.

Extraction of semiconductor dopant profiles from spreading resistance data : an inverse problem

Abstract The traditional method of solving, on a layer-by-layer basis, the inverse problem of extracting resistivity values from spreading resistance measurements is found to produce wildly oscillatory, physically unacceptable resistivity profiles in the case of p-type silicon structures, where a resistivity-dependent probe contact radius is used in conjunction with the probe calibration data. These oscillations are manifestations of the fact that the inverse problem has non-unique solutions; they occur because the problem is inherently ill-posed. The well-known Tikhonov regularisation technique, which converts the present set of highly non-linear integral equations to an equivalent variational problem, is applied to stabilise the solution. Tests are performed on a variety of simulated profiles, and they reveal the existence of an optimum value for the regularisation parameter that is to be used with a second difference expression for the stabiliser of the cost function. When applied to measured spreading resistance data, the technique is found to produce results of reconstruction that are stable and physically reasonable.