REGULARIZATION OF BROADENED SPECTRA OF DEEP LEVEL TRANSIENT SPECTROSCOPY
A.V.Solomonov, V.I.Zubkov
State Electrotechnical University , St.Petersburg
Email: zubkov@fvleff.etu.spb.ru
Аннотация — Предлагается алгоритм анализа уширенных спектров DLTS полупроводниковых твердых растворов. Алгоритм заключается в построении регуляризованного решения уравнения Фре дгольма 1го рода путем минимизации сглаживающего функционала с учетом стабилизирующих параметров. Получающееся в результате уравнение Эйлера решается численными методами относительно искомой функции плотности состояний глубокого центра. В качестве крите рия сходимости используется принцип невязки.
Deep level transient spectroscopy (DLTS) is widely used in the last 20 years to characterise impurities and defects with deep levels in forbidden energy gap of semiconductors. Its specific features are extremely high sensiti vity (up to 10^{12} electrons/cm^{3}) and relative simplicity of hardware.
Unfortunately, there is no often one  to  one mathematical correspondence between the distribution of deep levels in forbidden energy gap of semiconductor and DLTS  spectrum; in a number of cases (semiconductor alloys, for example) it is possible to propose a lot of essentially different solutions, which match the measured spectrum within the bounds of experimental noise. Therefore methods of fitting by several parameters [1], used in modern measuring information DLTS  systems and based on subjective propositions, need some preliminary information about the object under treatment [2,3] and can lead to unstable results. Moreover, sometimes because of the quite broad apparatus function of DLTS  spectrometer one can not determine h ow many deep centers have formed the spectrum observed, and, thus, how many fitting parameters are needed.
We have proposed the algorithm of correct interpretation of complicated DLTS  spectra of semiconductor solid solutions. The algorithm is based on the method of Tikhonov regularization and allows to avoid the subjectivity in determinat ion of deep centers characteristics.
In general the temperaturedependent DLTS spectrum in semiconductor alloy may be described
, (1)
as where D(E) is the deep center density of states function (DSF)
to be found, e(T,E) is the deep center thermal emission rate, w(t)
is the weighting function of the DLTS  spectrometer, and (t_{1}, t
The solution of the Fredholm equation concerning the function D(E) on the obtained DLTS spectrum is unstable because of the experimental noise. We will solve it with the help of Tikhonov's regularization of illposed inverse problems [ 4]. According to it one should minimise the square of deviation of the calculated spectrum from the experimental one using the certain stabilising corrections
,
Obviously, among all existing solutions the most stable solution is the one that is most smooth. So we will find the approximate (regularized) solution D(E) as the most smooth of all solutions matching given error of the experimentally measured R(T). As a smoothness measure of the virtual approximate decisions it is possible to accept the value of a functional W :
where q and p are non  negative numbers (we have restricted the conditions with a demand of a continuity of the first order derivative).
To get the minimum of the functional Ma it is necessary to equate its first variation to zero:
Thus, the problem of obtaining of a stable solution of (1) is reduced to finding of the extremal of the Euler's integro  differential equation:
(2)
The parameter of regularization a , in accordance with the discrepancy principle [5], is determined by the equality of the discrepancies of the experimental and simulated spectra.
For practical computation we made a discretization of Euler equation (2) on an uniform mesh energytemperature in the region [a < E_{i} < b; c < T_{i} < d].
As a result we have got a set of linear algebraic equations:
,
where B_{a} is the quadratic and symmetrical matrix with a positive determinant; C is a three  diagonal matrix.
The resulting set of linear equations was solved by square root method [4] with the task dimension of 512*512 points.
The suggested algorithm has been used successfully for the resolution of two neighbouring deep centers, having the density of states function of an arbitrary form. Thе solving of this proble m is very important for the interpretation of DLTS data.
References
1. Istratov A.A., Hieslmair H., Flink C., and Weber E.K. Determination of parameters of deep level defects from numerical fit of deep level transient spectroscopy spectra: Analysis of accuracy and sensitivity to noise // Rev. Sci. Instr. 1998. v.69, N1. p.244250.
2. Omling P., Samuelson L., Grimmeiss H.G. Deep level transient spectroscopy evaluation of nonexponential transients in semiconductor alloys // J.Appl.Phys.  1983.  v.54, N9. p. 51175122.
3. Zubkov V.I., Pikhtin A.N., and Solomonov A.V. Deep level transient spectroscopy in solid solutions: Donor level in GaAs_{1x}P_{x} // // Sov. Phys. Semicond.  1989. v.23.  p. 39.
4. Tikhonov A.N., Arsenin V.Y. Solutions of illposed problems. Washington: Winston and Sons, 1977.
5. Hofmann B. Ill  posedness and regularization of inverse problem  A review of mathematical methods // The inverse problem: Symposium ad memoriam H. von Helmholtz. Berlin: Acad. Verl. 1995.  p. 45  66.
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