3. Theory of DLTFS 3. Theory of DLTFS

3.1  Fourier Transform

In the following we assume a digital system, which scans the analog signal f(t) with an analog-digital converter (ADC) in N discrete equidistant times kDt, k = 0,1,..,N-1. Dt is the sampling interval. For f(t) we postulate periodicity. The period width TW = NDt contains N intervals with the N+1 real values f0,...,fN.

For the subsequent comparison of measured and calculated values the following definitions are relevant:

a) Continuous (analytical) Fourier coefficents cn of the Fourier series (not the analytical Fourier transformation!):

cn = 1
TW
 TW
ó
õ
0 
 f(t) exp(-inw0t)  dt
(3.1)
with
w0 = 2p
TW
(3.2)
In case f(t) is real, then the cosine coefficients an and the sine coefficients bn are real, too, and represent real and imaginary parts of cn:
cn = 1
2
 (an - ibn)
(3.3)

b) Discrete (numerical) Fourier Transform (DFT):

Fn = N-1
å
 k = 0 
 fk exp(-2pink/N)
(3.4)
As the sampling values are real, only N/2 independent Fn exist. The following relation exists between DFT and discrete Fourier coefficients cnD:
Fn = N cnD
(3.5)

The excact reconstruction of a continuous time signal f(t), using discrete sampling values, is only possible if f(t) is spectrally limited, and if the sampling frequency 1/Dt is more than twice the highest frequency of f(t) (sampling theorem). The Nyquist frequency is half the sampling frequency. Spectral overlaps (aliasing effect) occur if the sampling theorem is not fulfilled.

Equation (3.4) represents the exact numerical integration according to the trapezium rule only if f0 = fN. Without this restriction the numerical integration rule takes the following form:

Fn = f0
2
 + N-1
å
 k = 1 
 fk exp(-2pink/N)+ fN
2
     ,     n = 0,1,...,N-1
(3.6)
If the function shows a discontinuity at the scan limit, i.e. f0 ¹ fN, then a correction is necessary if discrete and continuous coefficients are to be compared. For the analytical case the peripheral points f(0) and f(TW) are null sets. To avoid executing the correction for each Fn, f0 can be defined as follows for the input values of the DFT:
f0¢: = f0+fN
2
(3.7)
Numerical execution of the DFT can be done most efficiently with the FFT (Fast Fourier Transform) algorithm.

3.2  General idea of DLTFS

The general idea of the DLTFS (Deep Level Transient Fourier Spectroscopy) is as follows: N measuring values are sampled from a capacitance transient, and the discrete Fourier coefficients cnD are formed by numerical Fourier transformation.

Based on an adaquate theory for the charging of deep levels a certain time dependence of the transient is assumed. This function is developed into a Fourier series, and its continuous coefficients cn are calculated.

Assuming that the numerical coefficients originate from just this postulated function, the free parameters of the function can be determined by a comparison of the numerical and the analytical coefficients, usually in several ways.

Addititionally, it is immediately possible to check the basic theory by an appropriate selection of the number of Fourier coefficients. As each coefficient contains information about the entire transient, specifig ratios of some coefficients are characteristic for different signal forms.

Generally it would not be reasonable to use the entire frequency spectrum quantitatively for the evaluation. Starting with the assumption of a low-frequency active signal being overlapped by high-frequency noise signals, usually only the lowest orders will be evaluated.

In most cases it is favourable to adjust optimally the period width for each transient. In the software such a tempscan will be called tempscan with variable period width.

3.3  Direct evaluation

With the DLTFS method a direct ecaluation for each transient is possible. 'Direct' means that the evaluation values, for example the time constant, will be calculated direct from the transient and not 'indirect' by the maximum of a temperature curve.

3.3.1  Exponential law of time

The following discusses a real exponential law of time:

f(t) = A exp æ
ç
è 		- t+t0
t
 ö
÷
ø 		+ B
(3.8)
Where A is the amplitude , B the offset and t the time constant. For this real function following Fourier coeffients are obtained:
a0
=
2A
TW
 exp(-t0/t) (1-exp(-TW/t)) t+ 2B
(3.9)
an
=
2A
TW
 exp(-t0/t) (1-exp(-TW/t))
1
t
1
t2
 +n2w02
(3.10)
bn
=
2A
TW
 exp(-t0/t) (1-exp(-TW/t)) nw0
1
t2
 +n2w02
(3.11)
We get following relations to check whether the transient is exponential:
a)   
ak < an < k2
n2
 ak
for n < k
(3.12)
b)   
n
k
 bk < bn < k
n
 bk
for n < k
c)   
bn
an
 ak
bk
 = n
k
For the coefficients 1. and 2. order we get (in the software called 'exponential class'):
b2
2b1
 a1
a2
 = 1
(3.13)
The amplitude of the signal, and consequently the concentration of deep centers, can be calculated from each coeffient, for example from bn:
A = bn TW
2
 exp(t0/t)
(1-exp(-TW/t))
1
t2
 +n2w02
nw0
(3.14)

The time constant can be obtained from the ratio of two coefficents. There exist three principally different possibilities:

a)
t(an,ak) =
1
w0
   æ
 ú
Ö
an-ak
k2ak-n2an
 
(3.15)
b)
t(bn,bk) =
1
w0
   æ
 ú
Ö
kbn-nbk
k2nbk-n2kbn
 
c)
t(an,bn) =
1
nw0
 bn
an
It is a considerable advantage that here only ratios of coefficients and neither amplitude nor offset well be used.

3.4  Tempscan maximum analysis

With the Fourier coefficents is also a tempscan maximum analysis possible as at the conventional DLTS method. At this maximum analysis you define by the period width TW your time constant for the coefficient maximum. This value is only valid in the maximum and depends from TW, TW/t0 and the type of coeffcient. This value will be calculated numerical by simulation (variation of tau at fix TW and t0) and maximum search of the cofficient(tau) curve from eq. 3.10 resp. 3.11.

3.5  Isothermal evaluation

With the Fourier coefficents is also an isothermal maximum analysis possible as ICTS or frequency scan. For this method the period width will be variated at a fix temperature. From the maximum of this curve you get the time constant. With a numerical calculation by simulation of eq. 3.10 resp. 3.11 you get from your Tw-axis a tau-axis. The tau value is only valid in the coefficient maximum.