Laser technique for accurately determining the compensation density in N-type narrow gap semiconductor

A method for accurately determining the compensation density of n-type naw-gap semiconductors. A semiconductor sample is irradiated with laser pulses of a particular density and pulse width for a particular time length with the sample maintained at a low temperature to generate photo-excited carriers within the semiconductor sample. Photons of energy less than the energy gap, E.sub.g, but greater than, E.sub.g /2, generate carriers uniformly throughout the semiconductor via the nonlinear mechanism of two-photon absorption. Photo-Hall measurements are made on the semiconductor sample during and after the laser pulse to determine the mobility, .mu., and carrier density, n, as a function of time using suitable equipment such as a computer controlled digital processing oscilloscope to display the curves. The curves displayed by the oscilloscope are compared with previously calculated curves to obtain a match and thereby determine the quality of the sample. By combining measurements of the Hall effect and conductivity, one can deduce the carrier densities and mobilities as well as other various quantities by well-known formulas.

BACKGROUND OF THE INVENTION 
This invention relates to n-type narrow-gap semiconductors and more 
particularly to a method for more accurately determining the compensation 
density of such semiconductors. 
Narrow-gap semiconductors such as InSb, HgCdTe, etc., have important 
applications in electro-optics because of their use as infrared detectors 
and as laser materials. In order to provide the best quality materials for 
such devices, it is necessary to characterize the material as to its 
quality and suitability for use. In order to characterize such materials, 
it is very important to determine the donor concentration, N.sub.D, and 
the acceptor concentration, N.sub.A. The compensation density for n-type, 
material is equal to the acceptor concentration. 
For n-type material, the net impurity density, n, where n=N.sub.D -N.sub.A 
for monovalent defects, can be determined quite easily from Hall 
measurements. However, there are no known reliable techniques for 
determining the density of compensating acceptors, N.sub.A, in n-type 
narrow-gap materials. For wide-gap semiconductors, N.sub.A, can be 
determined from measurements of n as a function of temperature. As the 
temperature decreases, thermal excitation of carriers from the donors and 
acceptors becomes less probable. The decrease in n with decreasing 
temperature is known as carrier "freezeout". The change in n with 
temperature permits one to determine the compensation density. However, 
such carrier "freezeout" does not occur for n-type narrow gap materials 
where the donor levels are merged with the conduction band. 
A method for determining the compensation density in n-type InSb has been 
suggested previously. This method relies on a theoretical model to 
calculate the low temperature mobility for a given electron density, n, 
using the compensation density as a variable parameter. N.sub.A is 
obtained by comparing measured and calculated values of mobility. However 
two serious drawbacks make this approach unreliable. (1) The method is not 
applicable to the important class of ternary (e.g. Hg.sub.1-x Cd.sub.X Te) 
and quaternary (e.g. [PbSe].sub.1-x [SnTe].sub.x) compounds. For these 
mixed compounds, the mobility is extremely sensitive to the alloy 
composition x. Since this parameter is generally not well known, a simple 
comparison of the measured and calculated mobility is highly inaccurate. 
(2) Even for simple compounds like InSb the method is unreliable since it 
is based on a model which neglects important higher-order corrections to 
the theory. Comparison of the measured low-temperature mobility to the 
calculated value for a single value of n is unreliable. To be reliable, 
theory and experiment must be compared over a wide range of values of n. 
Consequently there is a need for a non-destructive technique for 
determining the compensation density for n-type narrow-gap semiconductors. 
A previous method has been set forth in an article: "Transport Properties 
of Photo-Excited Carriers in Slightly Compensated Hg.sub.0.785 
Cd.sub.0.215 Te." by F. J. Bartoli et al., Solid State Communications, 
Vol. 25, pp. 963-966, March 1978, and also has been set forth in patent 
application Ser. No. 011,821 filed Feb. 13, 1979. In this patent 
application, the method compares measured values of mobility as a function 
of photo-excited carrier density to theorectical curves which were 
calculated assuming a spatially uniform carrier distribution. However, it 
has been found that this method limits the accuracy to which N.sub.A can 
be determined because the carriers are generated in a thin layer on the 
surface of the sample, resulting in a non-uniform center distribution. The 
non-uniform carrier density results from the large linear absorption 
coefficient (.alpha.=10.sup.3 to 10.sup.5 cm.sup.-1) for photons with 
energies greater than the energy gap E.sub.g of the semiconductor. A 
single photon in this energy range has sufficient energy to excite a 
valence electron into the conduction band. Correction of the data for the 
non-uniform carrier distribution must be accomplished using approximate 
methods, hence introducing error in the technique. 
SUMMARY OF THE INVENTION 
The present invention comprises a non-destructive technique for determining 
the compensation density for n-type narrow-gap semiconductor materials 
such as InSb or Hg.sub.1-x Cd.sub.x Te which are fabricated into standard 
Van der Pauw photo-Hall samples such as commonly used for material 
characterization. Carriers are optically injected into the material by 
laser pulses having an energy of less than the energy gap, E.sub.g, but 
greater than, E.sub.g /2, and the photo-Hall mobility is measured as a 
function of photoexcited carrier density. For example a TEA or Q-switched 
laser may be used as an excitation source to generate sufficiently high 
density of carriers that the functional dependence of the photo-Hall 
mobility, .mu., on n may be determined for a wide range of n. It has been 
determined that the functional dependence of .mu. on n is very sensitive 
to the compensation density.

DETAILED DESCRIPTION 
In carrying out this invention, samples of a semi-conductor material to be 
tested are fabricated into Van der Pauw photo-Hall samples to obtain a 
mobility vs carrier density curve which is used as a basis for comparing 
curves of the tested material with compensation density curves previously 
calculated for materials with different compensation densities to 
determined the useful quality of the sample. The sample is cooled to a 
sufficiently low temperature, from 2.degree.-40.degree. K., that the 
mobility is limited by scattering of the electrons by ionized impurities. 
FIG. 1 illustrates a HgCdTe sample 10 prepared and arranged for determining 
a conductivity-voltage curve. Samples to be tested may be of any size and 
thickness except that best results are obtained when the thickness is no 
greater than 100 .mu.m with measurements of from 0.1 mm to 2.0 mm along 
the sides. It is important that the thickness be limited to a thickness 
such that the laser radiation will penetrate the sample; however, it 
should be thick enough so that the Hall measurements are not dominated by 
surface effects. The sample is secured to a larger substrate 12 so that 
the substrate can be used to support the electrical contacts 14, 16, 18 
and 20, as well as the samples. The electrical contacts are made at the 
corners of the sample. 
A TEA or Q-switch CO.sub.2 laser, for example, operating at a wavelength of 
10.6 .mu.m with a pulse duration of about 200 nsec may be used for InSb 
Hg.sub.1-x Cd.sub.x Te for some x values. The laser pulse is uniformly 
directed upon the total surface of the sample, as shown by the circle 26, 
to generate a uniform carrier density in the sample. The incident 
radiation is such that the photon energy, h.nu., is less than the energy 
gap E.sub.g of the sample but greater than E.sub.g /2. Such a photon 
energy generates carrier uniformly throughout the semiconductor via the 
nonlinear mechanism of two-photon absorption. The laser power density is 
kept sufficiently low so that sample heating is negligible. The laser 
power density should be in the range of about 10.sup.4 -10.sup.7 
w/cm.sup.2, depending on the material. During incidence of the laser 
radiation, photo-excited carriers are injected into the sample. Since the 
photon energy, h.nu., of the laser radiation is less than E.sub.g but 
greater than E.sub.g /2, two photons are absorbed simultaneously to create 
an electron-hole pair. The process is nonlinear in that the amount of 
energy absorbed by the material is proportional to the square of the laser 
intensity, rather than varying linearly as for single-photon absorption. 
The absorption depth chosen is large compared to the sampled thickness, in 
order to insure a uniform carrier density of photo-excited carriers. The 
conductivity of the sample changes as the carriers recombine to return to 
their original value. To eliminate transient effects, only data taken at 
least 200 nsec after the peak of the laser pulse is measured and recorded. 
This portion of the photo-excited electron decay yields steady-state 
values for mobility as a function of electron density as discussed below. 
In carrying out the measurements, a steady DC electrical source is 
connected across the contacts 14 and 20 (see FIG. 1) fixed to two adjacent 
corners of the sample to supply a steady current I of from 1-10 milliamps, 
and a voltage detector 24 is connected across the contacts 16 and 18 
affixed to the other two corners of the sample to detect the voltage 
output. The measured voltage output changes as the conductivity of sample 
changes due the incidence of the laser radiation. The voltage detector may 
be an oscilloscope which measures and displays a trace of the voltage 
output which can be recorded by taking a photograph of the displayed 
output voltage trace. The voltage output may also be stored in a computer 
28 for later retrieval and display on the oscilloscope. 
After photo-excited carriers have recombined and the voltage trace has been 
completed for the first test position, the polarity of the voltage source 
is changed and a second voltage trace is made for a second laser pulse 
identical to the first laser pulse. After the second voltage trace has 
been completed, the voltage sources is connected across contacts 18 and 20 
so that the applied voltage is perpendicular to the previously applied 
voltage and the voltage detector is connected across contacts 14 and 16. 
The same procedures are followed as described above to obtain voltage 
traces for each polarity in the latter arrangement. Voltage traces may 
also be obtained for the other contacts in the same manner. This gives 
four or eight voltage traces for the same parameters. The voltage values 
for each corresponding laser pulse may be averaged by hand, using ohm's 
law and the conductivity formula or averaged by the computer and displayed 
on the oscilloscope to present a single average conductivity-voltage 
curve. Alternatively the averaged voltage vs. time curve can be obtained 
from hand calculations on photographed oscilloscope traces of the voltage 
outputs by use of well known averaging formulas. 
After the average conductivity-voltage curve has been determined, the same 
sample is tested for Hall-voltages. As shown in FIG. 2, Hall voltages are 
obtained in the same manner as above except that the voltage source is 
applied across opposite corners such as 14 and 18, and the voltage 
detector is connected across opposite corners 16 and 20. Additionally, a 
magnetic field is applied perpendicularly to the surface plane of the 
sample and the sample is cooled to from 2.degree.-40.degree. K. as before. 
With the magnetic field in one direction, a Hall-voltage curve is made for 
each polarity of the applied voltage. Then the direction of the magnetic 
field is reversed and additional Hall-voltage curves are obtained for each 
polarity of the applied voltage. The voltage source and detector 
connections are then switched and more Hall-voltage curves are obtained 
for each polarity of the electrical source and for each direction of the 
magnetic field as set forth above. Hall-voltage curves may be obtained for 
each pair of opposite corner contacts and with the magnetic field in each 
direction to obtain either 8 or 16 curves as desired. The voltage values 
for corresponding times are averaged and an average Hall-voltage vs time 
curve is obtained. 
After the average conductivity-voltage and average Hall-voltage vs time 
curves have been obtained, these values are used to obtain carrier 
mobility, .mu., and carrier density, n. This can be accomplished using the 
appropriate formulas (described in Philips Research Reports, Vol. 13, 
pages 1-9, February 1958) either by hand or automatically by the use of a 
digital computer 28 connected to the voltage detectors and programmed to 
convert the average conductivity- and Hall-voltages directly to carrier 
density and mobility. For the sample configuration described in FIGS. 1 
and 2, the sample conductivity .sigma. as a function of time may be 
obtained from the conductivity-voltage curve V.sub.c (t) by means of the 
expression 
##EQU1## 
where I is the current, d is the sample thickness and ln(2) is the natural 
log of 2. The carrier density as a function of time may then be obtained 
from the Hall-voltage curve V.sub.H (t) by the expression 
##EQU2## 
where B is the magnetic field and e the charge of an electron. Finally the 
mobility .mu. is equal to .sigma./ne. The photo-Hall mobility, .mu., and 
electron density, n, may be monitored on a digital processing oscilloscope 
24 such as a Tektronix R7912 Waveform digitizer interfaced to a 
mini-computer 28 such as a PDP-11/40. FIG. 3 shows sample cathode ray tube 
traces of the carrier density, n, (curve A) and mobility, .mu., (curve B) 
as a function of time beginning with the peak of a 200 nsec laser pulse. 
After reaching their maximum values, both n and .mu. decay monotonically 
to their dark values. The mobility peaks approximately 700 nsec after n 
indicating that .mu. passes through a maximum as a function of n. As shown 
in FIG. 3, the horizontal scale is 200 nsec per division; the vertical 
scales are as follows: Curve A is 1.times.10.sup.15 cm.sup.-3 per division 
and Curve B is 1.times.10.sup.5 cm.sup.2 /V-sec per division. The zero 
level for .mu. and n is indicated on the vertical scale. If a computer is 
not used, then the curves as shown would have to be determined by hand by 
use of the conversion formulas. 
The solid line in FIG. 4 illustrates a theoretical curve for mobility .mu., 
as a function of carrier density, n, for the sample set forth above. The 
decrease in mobility for large carrier density is attributed to 
electron-hole scattering. The mobility enhancement at lower electron 
densities are explained in terms of neutralization of charged compensating 
acceptors. Photo-excited holes are captured by ionized acceptors because 
of the Coulomb attraction (each ionized monovalent acceptor can capture 1 
hole). Hole capture neutralizes the acceptors which then cannot scatter 
electrons as effectively. Consequently, this tends to increase the 
mobility. Furthermore, the photo-excited electrons increase the screening 
of the scattering potential and also tend to increase .mu.. Any 
photo-excited holes not captured by an ionized acceptor give rise to 
electron-hole scattering which tends to reduce the mobility. For 
sufficiently high photoexcited carrier densities (i.e., &gt;&gt;N.sub.A), most 
of the photoexcited holes are free and the mobility will decrease due to 
electron-hole scattering. 
The mobility depends strongly on the compensation density N.sub.A through 
the following relations .mu.=n/N ABM where n=N.sub.D -rN.sub.A +n.sub.e 
and 
##EQU3## 
where N.sub.A.sup.(-z) denotes the density of acceptors with a charge of 
-z, and N.sub.cc is the density of charged scattering centers. In these 
relations r is the valence of the acceptors and n.sub.e and p.sub.e are 
the excess electron and hole densities. The factor, A, represents the 
normal expression for scattering by single sites in the Born approximation 
and is described in Zawadski, W. and Szymanska, W., Phys. Status Solidi, 
(b) 45, 415 (1971). B is a phase-shift correction for the Born 
approximation described by J. B. Kriger and S. Strauss, Phys Rev., Vol. 
169, No. 3, p. 674-679, May 15, 1968); and M is a correction for 
multiple-ion-scattering (described in E. F. Moore and H. Ehrenreich, Solid 
State Communications, Vol. 4, p. 407, 1966). 
A family of .mu.vs n curves is calculated for different values of N.sub.cc 
by varying N.sub.CC and keeping .mu. and n equal to the measured values 
for the narrow gap semiconductor sample in the absence of optical 
excitation to obtain curves representing N.sub.A1, N.sub.A2, etc. By 
comparing the measured .mu.vs n curve to the calculated family of curves, 
the compensation density N.sub.A can be obtained directly by matching the 
measured curve with one of the .mu.vs n curves of the family of curves 
previously calculated. 
The laser power density and pulse width may be varied to optimize the above 
method for different narrow-gap semiconductors. Different optical sources 
such as CO, HF and DF lasers may be used. The technique may be used for 
lattice defects as well as substitutional impurities and can be used for 
defects of arbitrary valency. 
Obviously many modifications and variations of the present invention are 
possible in light of the above teachings. It is therefore to be understood 
that within the scope of the appended claims the invention may be 
practiced otherwise than as specifically described.