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Problems concerning the spatial distribution of deep impurities in semiconductors
PROBLEMS
CONCERNING
DEEP
IMPURITIES
the true bulk lifetime in A could be made on this assumption. A situation of direct practical interest is that of highlifetimematerialembedded withzones of low lifetime. As the zones of low lifetime are brought closer than the diffusion length within the high lifetime material, the observed photo-current decay will become shortened by recombination in these zones, and the recorded decay constant will become increasingly a function of the separation of the zones, and less characteristic of the lifetime in the good regions of the material. The decay constant will become insensitive to the concentration of defects within the low lifetime zones once this concentration is sufficiently high to ensure recombination of any carrier entering the zone. Also, under this condition, the surface area of the zone increases as the square of its linear dimensions, whilst the number of defects in the zone increases as the cube. Both of these effects show that, as the defects are allowed to group into clusters, the decay constant loses its inverse proportionality to the number of defects in the sample, as exists in the case of a uniform distribution. Thus photoconductive decay is no longer a good measure of the average concentration of defects if their spatial distribution varies rapidly. THERMALLY
GENERATED
CURRENT
The analysis of Sah, Noyce and Shockley attributes to mid-gap defects the ability to assist the thermal generation of electron-hole pairs, which in a p-i-n junction appear as reverse leakage current. Any given level has a certain probability per unit time of generating an electronhole pair, and therefore a specified defect produces a reverse current which is proportional to the total number of the defects within the depletion layer, and is independent of the actual spatial distribution of the defects. This needs qualification in those instances when the defects are sufficiently close to each other that the energy levels become perturbed. The maximum distance for this effect to be significant is 100 A i.e. defect concentrations that are of the order of 1 part in 105. In germanium, of a quality suitable for gamma-ray detectors, the average number of defects is more likely to be 1 part in lOlo or less, and therefore significant grouping of defects could occur without energy level perturbation. In such a situation, the leakage current due to carrier generation within the 5
IN
991
SEMICONDUCTORS
depletion layer is simply proportional to the number of defects within the depletion region (assuming that only one type of defect is predominant). SAH(~) has shown that the charge generation rate per cc of depletion layer is equal to (n,/27) where 7 is the photoconductive lifetime. This is on the understanding that the recombination centre concentration is uniform. As shown in section 2 the measurement of photoconductive lifetime is no longer a measure of the average concentration of recombination centres if their spatial distribution varies rapidly. To overcome this difficulty, a pragmatic definition of true carrier lifetime could be made on the basis of charge generation rate. For a depletion layer in zone A with a defect concentration N,, the charge generation rate = (~JZT,). The justification for calling 7, the true lifetime is that at any point it is inversely proportional to N,, and is independent of the gradient of N,. Further the estimation of its value is straight forward from measurement of the magnitude of the thermally generated current. Such a definition of 7, means that the recombination rate at any point is inversely proportional to 7,. However there are immediate difficulties in connecting this rate to any particular measurement of photoconductive lifetime as diffusion effects immediately spread the carriers into any adjacent volumes, and the observed decay is not related to the generation-recombination process in any simple manner. If the depletion layer contains two volumes, of cross-sectional areas S, and S,, and generation time constant, (or true lifetimes) 7, and TV, the total generation rate 1, for a p-i-n structure of unit intrinsic depth, ignoring diffusion of minority carriers from the p and n regions, ni
s, s,
=5.-+ibra
Tb
)*
The apparent generation time constant 7,. for the two zones in parallel is therefore given by
7i- =
(Sa+ SiJ
Sa7b
+
Sb7a.
(1)
P. E. GIBBONS
992
It is important to note that in this simple model that zone B does not have to be a single entity but could be the summation of small zones embedded within zone A. The details of the spatial distribution of zone B do not affect the value of rr, in marked contrast to the behaviour of the photoconductive decay as discussed in the previous section.
CHARGE STORAGE METHOD LIFETIME
r,+r,=
erf
I
t
G’
where 7 is the minority carrier lifetime. It is interesting to consider what happens if the value of 7 varies across the diode, for example a diode in which the base region is composed of two zones, each containing different concentrations of a deeplevel defect which controls carrier recombination and generation. For the sake of simplicity, allow the two zones to be big enough for the concept of minority carrier lifetime to be meaningful in the sense that the lifetime in a zone is inversely proportional to the deep-level defect concentration. The forward current of a diode is given as
The saturation current 1, is a function of lifetime, being inversely proportional to the square root of the lifetime for a p-n junction. For a p-i-n diode structure of the type constructed for radiation detectors, the saturation current is inversely proportional to lifetime, conditional on the displacement length (diffusion plus field assisted transport) being greater than the width of the i layer. If this
I .
Q,
OF MEASURING
LAX and NEUSTADTER(~)describe a measurement of carrier lifetime in a diode by means of observing the magnitude of the charge stored in the diode by a forward current, I,. This charge is measured by switching the diode from forward to reverse bias and observing the current as a function of time; the minority carriers diffuse across the junction to give an increased reverse current I,., which lasts for a time t. These parameters are related to the lifetime by the equation
If
latter condition is not met then 1, becomes inversely proportional to +, where n is 1 > n > 4. Thus the forward current flows predominantly into the low lifetime material, whilst, as will be shown, the effective charge storage occurs in the high lifetime zone. On a simple two zone model the total stored charge Q, in zone A is related to the forward current in A by (4) If the cross-sectional S, and S,.
areas of zones A and B are
(the saturation
Qa cc &r,(l-
current
in A)
n).
If n = 1, the stored charge is not a function of 7, but simply proportional to area, i.e. the stored charge density does not vary with lifetime. For n = 3, the stored charge density is proportional to the square root of lifetime. However, the problem of how this stored charge contributes to the reverse current pulse involves the value oft observed, which itself can be controlled by the test circuit parameters. If iR is allowed to be large, the stored charge flows away in a short time t, and if t is short compared with 7a and T*, no charge is lost by recombination in the sample. If i, is reduced, by simply increasing the resistance in the recovery circuit, then when 7a > t > 7b the observed stored charge is mainly attributable to that in the high lifetime zone. Of course, as t $ TV, the total observed storage charge falls further, as it is mainly lost by recombination in the diode. This is corrected for by the error function; it is important to realize that this correction does not apply in the intermediate case to the charge stored in the low lifetime zone since the time t is now controlled solely by charge in the high lifetime zone. To enable equation (2) to correctly describe this situation, it is necessary to adjust the observed value of I,. Since 1, consists of two components I fa and Ifb7 and it is only the former which is effective in producing charge storage, it is necessary to replace Ir by I,,.
Solid-State
Electronics
Pergamon
Press 1969. Vol. 12, pp. 989-995.
PROBLEMS CONCERNING
Printed
in Great Britain
THE SPATIAL
OF DEEP IMPURITIES
DISTRIBUTION
IN SEMICONDUCTORS
P. E. GIBBONS Atomic Energy Research
Establishment,
Harwell,
(Received 6 December 1968; in revisedform
Didcot,
Berks.,
England
10 April 1969)
Abstract-Variation in the spatial distribution of recombination centres within a semiconductor causes difficulties in the interpretation of lifetime measurements. These difficulties are assessed for three methods of lifetime assessment; photoconductive decay, charge storage and diode reverse leakage current. A quantitative comparison of these methods is made on the basis of the simple model of a semiconductor consisting of amatrix of low recombinationcentre concentration containing embedded volumes of a much higher concentration. Under these conditions it is shown that the diode reverse leakage current is the most meaningful method of estimating the average recombination centre concentration. The charge storage method is seriously affected by clustering of recombination centres and the analysis shows how this method can be used to estimate qualitatively that clustering is present. R&urn&La variation de la distribution spatiale des centres de recombinaison dans un semiconducteur cause des difficult& dans l’interpretation des mesures de longCvitt!. Ces difficult& sont estimCes pour trois methodes d’evaluation de longCvit6: dCclin photoconductif, emmagasinement de charge et courant de fuite inverse de diode. Une comparaison quantitative de ces mCthodes est faite sur la base d’un mod?le de semiconducteur simple consistant en une matrice de concentration centrale & faible recombinaison contenant des volumes encast& de plus grande concentration. Sous ces conditions, on demontre que le courant de fuite inverse de diode est la methode la plus significative pour Cvaluer la concentration moyenne des centres de recombinaison. La mCthode d’emmagasinement de charge est sCrieusement affectCe par l’amas de centres de recombinaison et l’analyse dCmontre comment cette mCthode peut &tre employ&e pour &aluer qualitativement la presence de cet amas. Zusammenfassung-Variationen in der rlumlichen Verteilung von Rekombinationszentren in einem Halbleiter verursachen Probleme bei der Interpretation von Lebensdauermessungen. Die Schwierigkeiten werden fiir drei Methoden der Lebensdauerbestimmung untersucht : Die Abklingzeit der Photoleitung, die Ladungsspeicherung und der Leckstrom von Dioden bei Betrieb in Sperrichtung. Ein quantitativer Vergleich dieser Methoden wird auf der Basis eines einfachen Modells fiir einen Halbleiter gegeben, der aus einer Matrix mit geringer Rekombinationszentrendichte besteht, die eingebettet kleine Volumina mit wesentlich hijherer Konzentration enthiilt. Unter diesen Bedingungen wird gezeigt, dass der Sperrleckstrom von Dioden die bedeutenste Methode fiir die Bestimmung der mittleren Rekombinationszentrendichte ist. Die Methode der Ladungsspeicherung dagegen wird ernstlich durch die Zusammenballung von Rekombinationszentren beeinflusst. Eine Analyse zeigt, wie diese Methode zur qualitativen Abschgtzung dafiir dienen kann, dass entsprechende Ausscheidungen vorhanden sind.
THE IMPORTANCE of minority carrier lifetime on semiconductor device performance has long been appreciated and has direct bearing on the performance of a diverse range of devices e.g. nuclear radiation detectors, solar cells, bi-polar transistors and fast switching diodes. SHOCKLEY and READ(~) formally showed how the lifetime is controlled by impurities which have
energy levels near the middle of the energy gap. A subsequent paper by SAH, NOYCE and SHOCKLEY’~’ related minority carrier lifetime to the thermally generated reverse bias currents of semiconductor junctions, showing how the generation process is assisted by the same impurity centres which control the recombination process. The value of the minority carrier lifetime is so 989
990
P. E. GIBBONS
important that it has become standard practice for suppliers of silicon or germanium crystals to specify the lifetime as measured by the photoconductive decay method, and device engineers generally find that such a measurement is useful for predicting those aspects of device performance which are dependent on minority carrier lifetime. It is important to note that it is usually assumed that the lifetime parameter only changes slowly with position throughout the crystal, and significant changes only occur over distances greater than several diffusion lengths. This paper will show that the concept of minority carrier lifetime runs into serious difficulties if the defect distribution which controls lifetime is changing rapidly within a diffusion length. Indeed the concept of a diffusion length as normally defined becomes meaningless. Large semiconductor radiation detectors, and in particular lithium drifted germanium p-i-n diodes for gamma-ray spectrometry which may have depletion volumes up to 100 cm3, do not show a simple correlation between device performance and photoconductive lifetime. This manifests itself in two ways. (i) The thermally generated current per unit volume of depletion region does not agree with the calculated values predicted by Sah, Noyce and Shockley, using values of photo-conductive lifetime and (ii) the quality of charge collection likewise does not appear to correlate with photoconductive lifetime. This second point is arguable in so far that charge collection could be affected by defects which have an energy level structure which allows the trapping of charge, but do not contribute to the true recombination mechanism. Quantitative measurements reported by GIBBONS et aLc3) of the charge collection process observed when a germanium diode is irradiated with gamma rays indicate that, whether the losses be purely temporary trapping, or permanent recombination, the spatial distribution of the defects responsible is far from uniform. The measurements indicate that some germanium crystals, of apparently good lifetime, have a distribution of electronically sensitive defects which approximates to clusters of dimensions 0.1-l mm across, spatially separated by several millimetres. Such crystals may appear almost indistinguishable in terms of photoconductive lifetime from those which are almost free from such cluster formation. Thus, although it might be argued that only a
spatial variation in trapping centres need be involved, it may be countered that if there is a mechanism causing clustering of defects giving rise to trapping levels, it is not unreasonable to suggest that a clustering of the true recombination centres also occurs. It is with these difficulties in the background that this paper discusses the effects of spatial variation in mid-gap defect concentration on photoconductive lifetime, thermal generation of current in the depletion zone of a diode, and minority carrier lifetime as measured by the charge storage method of LAX and NEUSTADTER.(~) PHOTOCONDUCTIVE
DECAY
The effect of variation in recombination centre concentration on photoconductive decay time can be analysed as follows. Consider two large zones of semiconductor A and B, in intimate contact with each other and of dimensions considerably greater than the minority carrier diffusion length. If the defect concentrations are respectively N, and Nb, and the recombination lifetimes 7a and 7b, then the recombination rate in A cc N,
i.e. :\‘, = -
and similarly Ta
If the defects
Kb
Nb = -.
Tb
in B are identical to those in A, then = Nbrb. The overall photoconductive decay would be composed of two exponential waveforms of time constant 7a and 7b. The intial magnitude of the two current components at t = 0 would be proportional to the number of carriers created in the respective zones by the light pulse. However as Nb becomes very much greater than N,, the observed photocurrent would become progressively that associated with zone A. Unless care was taken to measure the quantum efficiency of the photocurrent observed, the composite material could appear to have a lifetime of 7a if 7a 9 5-b. If the dimensions of the low lifetime zone B are contracted so that they are comparable or less than the diffusion length in zone A, the surface in contact with zone A will now appear to have a high surface recombination velocity, and an estimate of
K, = K,, and Nara
PROBLEMS
CONCERNING
I la -=
DEEP
IMPI.JRITIES
sa Tbn I,a -_ =p. ran S, I/ - I,a
‘lb
Ir
Ifa = 1+
(6)
2?+! 0 rb
The corrected relationship,
IN
993
SEMICONDUCTORS
the lifetime given by reverse current defined as To by equation (1). As an example consider a pi--n diode composed of two zones of true lifetime T= and 7b, cross-sectional area S, and Sb. A charge storage lifetime 7s is defined by the following equation.
a
-=
becomes
1
erf
l+‘r
J
If (7)
t -. 7s
(8)
Solving the equation simultaneously with equation (7), to eliminate t, leads to numerical relationships
IO Cross- sectional
0
2.0
b
I .o
c
0.5
d
02
e c”lCO.l-
f
oreo
I $-
Curve
0.1 for
comporison
_rL for +a T0
I O.,
FIG. 1. Dependence of charge storage lifetime on the ratio of true life-
times in the zones.
Unfortunately such data on ratios will not be available. However it is clear from this relation that if the correction is not applied, the calculated 7 will be less than the true 7,. THE RELATION CURRENT
BETWEEN REVERSE LEAKAGE AND CHARGE STORAGE
In the previous sections, it was shown how the photoconductive lifetime and a lifetime calculated from reverse bias currents could vary relative to each other depending on the spatial distribution of deep-lying impurities. It is of interest to see how the lifetime obtained by charge storage relates to
between all the remaining parameters. For a p-i-n structure, n = 1. The value of 78 can be enumerated for values of S, and 7b. 7s is also dependent on the ratio 1,/I, and this is shown in Fig. 1, where the relation between 7b and 78 is plotted for a series of values of 1,/I, for the case when S, = 0.10. The implication of this result is that the presence of low lifetime zones could be detected by the variation of 7s with, IT/If,, the latter being controlled by the test circuit parameters. The second point of interest is shown by the insertion of the curve relating 71. to 7b for the same value of S,/S,.
P. E.
994
FIG. 2. Dependence
GIBBONS
of charge storage lifetime and generation lifetime on ratio of true lifetime.
The physical significance of this is that the charge storage lifetime 75 is more sensitive to deep-level impurity clustering than is the generationrecombination lifetime or. This point is reinforced in Fig. 3, where the same data as Fig. 2 is replotted, now as a function of S, against 75 or 5-rfor values of
This demonstrates that the value of ~~ is lower than 7r for values of 1,./I, > 0.2. Taking I,./If = 0.5, Fig. 2 plots the variation of TV or or, for a range of values of S,. Under this condition, rs is consistently smaller than r. Further dTJdTb is greater than dr,/dTb (for constant S,).
u-
CWle
cn” cl? Ol- __
75
=o
cl
0 01
b c d
b’ c’ dl
Note
r,
\
002 0 05 0 !O calculated
FIG. 3. Dependence
\
u \
\
\
for +
\
\
>, \
L’>
\
\
\
=0 5
of lifetime parameters on ratio of areas of two-zone model.
PROBLEMS
CONCERNING
DEEP
IMPURITIES
7b. The value of dr,/dSb is greater than dr,/dS,, inferring that an increase in size of the clusters affects the charge storage lifetime 78 more rapidly than the value of or. As mentioned earlier, the model of a two-zone diode, with areas S, and S,, is not restricted to the case where S, is a single entity but rather the summation of small low life-time zones embedded within the high lifetime zone A. It should be noted that for values of 71, > 0.37, the basic assumption of the calculation becomes increasingly invalid, i.e. that the charge storage is due entirely to the high lifetime zone. This accounts for the failure of some of the 7b ---TV curves to extrapolate through the point of unit values. It should also be noted that the numerical analysis has been performed for a p--n structure. The application to a p-n junction would be straightforward. In equation (7), n would have a value of 4, whilst equation (l), giving the generation time constant 71.would become (‘% + s,)2
995
IN SEMICONDUCTORS CONCLUSIONS
By considering particular examples of lifetime variation within a crystal specimen, it has been shown that discrepancies may be observed between lifetime when measured by photoconductive decay, reverse leakage current and charge storage. In practice, very complex spatial variations of lifetime may occur. The simple examples however indicate a qualitative relationship between the three methods which probably holds in more complex situations. Taking the reverse current lifetime as the most meaningful averaging of deep impurities, then photoconductive decay underestimates the number of deep impurities, whilst charge storage can overestimate the number of impurities. This latter discrepancy has an interesting consequence for a sample in which the clustering of low lifetime zones varies across the specimen. This would show as a variation in the ratio of (7&r) across the specimen, decreasing in those areas where the deep-level impurities are increasingly segregated into clusters.
raTb
REFERENCES 1. W.
SHOCKLEY and
W.
T.
READ,
Phys. Rev. 87,
835 (1952).
Generally, for given values of the parameters 7b and S,, the values of 7s and or will be higher for the p-n structure than for the p--&z example, in other words, they are less sensitive to the presence of low lifetime zones.
2. C.
T.
SAH, R. N. NOYCE and W.
Instn R&o 3.
P.
E.
SHOCKLEY.
GIBBONS. 1. H.
HOWES and R.
PYOC.
.
Engrs. 45, 1228 (1957). B.
OWEN.
Nucleon. It&t& I.E.E. Conf. Publ. 47,152 (1968): 4. B. LAXand S. F. NEUSTADTER,J. appl. Phys. 31148 (1954).
CONCERNING
DEEP
IMPURITIES
the true bulk lifetime in A could be made on this assumption. A situation of direct practical interest is that of highlifetimematerialembedded withzones of low lifetime. As the zones of low lifetime are brought closer than the diffusion length within the high lifetime material, the observed photo-current decay will become shortened by recombination in these zones, and the recorded decay constant will become increasingly a function of the separation of the zones, and less characteristic of the lifetime in the good regions of the material. The decay constant will become insensitive to the concentration of defects within the low lifetime zones once this concentration is sufficiently high to ensure recombination of any carrier entering the zone. Also, under this condition, the surface area of the zone increases as the square of its linear dimensions, whilst the number of defects in the zone increases as the cube. Both of these effects show that, as the defects are allowed to group into clusters, the decay constant loses its inverse proportionality to the number of defects in the sample, as exists in the case of a uniform distribution. Thus photoconductive decay is no longer a good measure of the average concentration of defects if their spatial distribution varies rapidly. THERMALLY
GENERATED
CURRENT
The analysis of Sah, Noyce and Shockley attributes to mid-gap defects the ability to assist the thermal generation of electron-hole pairs, which in a p-i-n junction appear as reverse leakage current. Any given level has a certain probability per unit time of generating an electronhole pair, and therefore a specified defect produces a reverse current which is proportional to the total number of the defects within the depletion layer, and is independent of the actual spatial distribution of the defects. This needs qualification in those instances when the defects are sufficiently close to each other that the energy levels become perturbed. The maximum distance for this effect to be significant is 100 A i.e. defect concentrations that are of the order of 1 part in 105. In germanium, of a quality suitable for gamma-ray detectors, the average number of defects is more likely to be 1 part in lOlo or less, and therefore significant grouping of defects could occur without energy level perturbation. In such a situation, the leakage current due to carrier generation within the 5
IN
991
SEMICONDUCTORS
depletion layer is simply proportional to the number of defects within the depletion region (assuming that only one type of defect is predominant). SAH(~) has shown that the charge generation rate per cc of depletion layer is equal to (n,/27) where 7 is the photoconductive lifetime. This is on the understanding that the recombination centre concentration is uniform. As shown in section 2 the measurement of photoconductive lifetime is no longer a measure of the average concentration of recombination centres if their spatial distribution varies rapidly. To overcome this difficulty, a pragmatic definition of true carrier lifetime could be made on the basis of charge generation rate. For a depletion layer in zone A with a defect concentration N,, the charge generation rate = (~JZT,). The justification for calling 7, the true lifetime is that at any point it is inversely proportional to N,, and is independent of the gradient of N,. Further the estimation of its value is straight forward from measurement of the magnitude of the thermally generated current. Such a definition of 7, means that the recombination rate at any point is inversely proportional to 7,. However there are immediate difficulties in connecting this rate to any particular measurement of photoconductive lifetime as diffusion effects immediately spread the carriers into any adjacent volumes, and the observed decay is not related to the generation-recombination process in any simple manner. If the depletion layer contains two volumes, of cross-sectional areas S, and S,, and generation time constant, (or true lifetimes) 7, and TV, the total generation rate 1, for a p-i-n structure of unit intrinsic depth, ignoring diffusion of minority carriers from the p and n regions, ni
s, s,
=5.-+ibra
Tb
)*
The apparent generation time constant 7,. for the two zones in parallel is therefore given by
7i- =
(Sa+ SiJ
Sa7b
+
Sb7a.
(1)
P. E. GIBBONS
992
It is important to note that in this simple model that zone B does not have to be a single entity but could be the summation of small zones embedded within zone A. The details of the spatial distribution of zone B do not affect the value of rr, in marked contrast to the behaviour of the photoconductive decay as discussed in the previous section.
CHARGE STORAGE METHOD LIFETIME
r,+r,=
erf
I
t
G’
where 7 is the minority carrier lifetime. It is interesting to consider what happens if the value of 7 varies across the diode, for example a diode in which the base region is composed of two zones, each containing different concentrations of a deeplevel defect which controls carrier recombination and generation. For the sake of simplicity, allow the two zones to be big enough for the concept of minority carrier lifetime to be meaningful in the sense that the lifetime in a zone is inversely proportional to the deep-level defect concentration. The forward current of a diode is given as
The saturation current 1, is a function of lifetime, being inversely proportional to the square root of the lifetime for a p-n junction. For a p-i-n diode structure of the type constructed for radiation detectors, the saturation current is inversely proportional to lifetime, conditional on the displacement length (diffusion plus field assisted transport) being greater than the width of the i layer. If this
I .
Q,
OF MEASURING
LAX and NEUSTADTER(~)describe a measurement of carrier lifetime in a diode by means of observing the magnitude of the charge stored in the diode by a forward current, I,. This charge is measured by switching the diode from forward to reverse bias and observing the current as a function of time; the minority carriers diffuse across the junction to give an increased reverse current I,., which lasts for a time t. These parameters are related to the lifetime by the equation
If
latter condition is not met then 1, becomes inversely proportional to +, where n is 1 > n > 4. Thus the forward current flows predominantly into the low lifetime material, whilst, as will be shown, the effective charge storage occurs in the high lifetime zone. On a simple two zone model the total stored charge Q, in zone A is related to the forward current in A by (4) If the cross-sectional S, and S,.
areas of zones A and B are
(the saturation
Qa cc &r,(l-
current
in A)
n).
If n = 1, the stored charge is not a function of 7, but simply proportional to area, i.e. the stored charge density does not vary with lifetime. For n = 3, the stored charge density is proportional to the square root of lifetime. However, the problem of how this stored charge contributes to the reverse current pulse involves the value oft observed, which itself can be controlled by the test circuit parameters. If iR is allowed to be large, the stored charge flows away in a short time t, and if t is short compared with 7a and T*, no charge is lost by recombination in the sample. If i, is reduced, by simply increasing the resistance in the recovery circuit, then when 7a > t > 7b the observed stored charge is mainly attributable to that in the high lifetime zone. Of course, as t $ TV, the total observed storage charge falls further, as it is mainly lost by recombination in the diode. This is corrected for by the error function; it is important to realize that this correction does not apply in the intermediate case to the charge stored in the low lifetime zone since the time t is now controlled solely by charge in the high lifetime zone. To enable equation (2) to correctly describe this situation, it is necessary to adjust the observed value of I,. Since 1, consists of two components I fa and Ifb7 and it is only the former which is effective in producing charge storage, it is necessary to replace Ir by I,,.
Solid-State
Electronics
Pergamon
Press 1969. Vol. 12, pp. 989-995.
PROBLEMS CONCERNING
Printed
in Great Britain
THE SPATIAL
OF DEEP IMPURITIES
DISTRIBUTION
IN SEMICONDUCTORS
P. E. GIBBONS Atomic Energy Research
Establishment,
Harwell,
(Received 6 December 1968; in revisedform
Didcot,
Berks.,
England
10 April 1969)
Abstract-Variation in the spatial distribution of recombination centres within a semiconductor causes difficulties in the interpretation of lifetime measurements. These difficulties are assessed for three methods of lifetime assessment; photoconductive decay, charge storage and diode reverse leakage current. A quantitative comparison of these methods is made on the basis of the simple model of a semiconductor consisting of amatrix of low recombinationcentre concentration containing embedded volumes of a much higher concentration. Under these conditions it is shown that the diode reverse leakage current is the most meaningful method of estimating the average recombination centre concentration. The charge storage method is seriously affected by clustering of recombination centres and the analysis shows how this method can be used to estimate qualitatively that clustering is present. R&urn&La variation de la distribution spatiale des centres de recombinaison dans un semiconducteur cause des difficult& dans l’interpretation des mesures de longCvitt!. Ces difficult& sont estimCes pour trois methodes d’evaluation de longCvit6: dCclin photoconductif, emmagasinement de charge et courant de fuite inverse de diode. Une comparaison quantitative de ces mCthodes est faite sur la base d’un mod?le de semiconducteur simple consistant en une matrice de concentration centrale & faible recombinaison contenant des volumes encast& de plus grande concentration. Sous ces conditions, on demontre que le courant de fuite inverse de diode est la methode la plus significative pour Cvaluer la concentration moyenne des centres de recombinaison. La mCthode d’emmagasinement de charge est sCrieusement affectCe par l’amas de centres de recombinaison et l’analyse dCmontre comment cette mCthode peut &tre employ&e pour &aluer qualitativement la presence de cet amas. Zusammenfassung-Variationen in der rlumlichen Verteilung von Rekombinationszentren in einem Halbleiter verursachen Probleme bei der Interpretation von Lebensdauermessungen. Die Schwierigkeiten werden fiir drei Methoden der Lebensdauerbestimmung untersucht : Die Abklingzeit der Photoleitung, die Ladungsspeicherung und der Leckstrom von Dioden bei Betrieb in Sperrichtung. Ein quantitativer Vergleich dieser Methoden wird auf der Basis eines einfachen Modells fiir einen Halbleiter gegeben, der aus einer Matrix mit geringer Rekombinationszentrendichte besteht, die eingebettet kleine Volumina mit wesentlich hijherer Konzentration enthiilt. Unter diesen Bedingungen wird gezeigt, dass der Sperrleckstrom von Dioden die bedeutenste Methode fiir die Bestimmung der mittleren Rekombinationszentrendichte ist. Die Methode der Ladungsspeicherung dagegen wird ernstlich durch die Zusammenballung von Rekombinationszentren beeinflusst. Eine Analyse zeigt, wie diese Methode zur qualitativen Abschgtzung dafiir dienen kann, dass entsprechende Ausscheidungen vorhanden sind.
THE IMPORTANCE of minority carrier lifetime on semiconductor device performance has long been appreciated and has direct bearing on the performance of a diverse range of devices e.g. nuclear radiation detectors, solar cells, bi-polar transistors and fast switching diodes. SHOCKLEY and READ(~) formally showed how the lifetime is controlled by impurities which have
energy levels near the middle of the energy gap. A subsequent paper by SAH, NOYCE and SHOCKLEY’~’ related minority carrier lifetime to the thermally generated reverse bias currents of semiconductor junctions, showing how the generation process is assisted by the same impurity centres which control the recombination process. The value of the minority carrier lifetime is so 989
990
P. E. GIBBONS
important that it has become standard practice for suppliers of silicon or germanium crystals to specify the lifetime as measured by the photoconductive decay method, and device engineers generally find that such a measurement is useful for predicting those aspects of device performance which are dependent on minority carrier lifetime. It is important to note that it is usually assumed that the lifetime parameter only changes slowly with position throughout the crystal, and significant changes only occur over distances greater than several diffusion lengths. This paper will show that the concept of minority carrier lifetime runs into serious difficulties if the defect distribution which controls lifetime is changing rapidly within a diffusion length. Indeed the concept of a diffusion length as normally defined becomes meaningless. Large semiconductor radiation detectors, and in particular lithium drifted germanium p-i-n diodes for gamma-ray spectrometry which may have depletion volumes up to 100 cm3, do not show a simple correlation between device performance and photoconductive lifetime. This manifests itself in two ways. (i) The thermally generated current per unit volume of depletion region does not agree with the calculated values predicted by Sah, Noyce and Shockley, using values of photo-conductive lifetime and (ii) the quality of charge collection likewise does not appear to correlate with photoconductive lifetime. This second point is arguable in so far that charge collection could be affected by defects which have an energy level structure which allows the trapping of charge, but do not contribute to the true recombination mechanism. Quantitative measurements reported by GIBBONS et aLc3) of the charge collection process observed when a germanium diode is irradiated with gamma rays indicate that, whether the losses be purely temporary trapping, or permanent recombination, the spatial distribution of the defects responsible is far from uniform. The measurements indicate that some germanium crystals, of apparently good lifetime, have a distribution of electronically sensitive defects which approximates to clusters of dimensions 0.1-l mm across, spatially separated by several millimetres. Such crystals may appear almost indistinguishable in terms of photoconductive lifetime from those which are almost free from such cluster formation. Thus, although it might be argued that only a
spatial variation in trapping centres need be involved, it may be countered that if there is a mechanism causing clustering of defects giving rise to trapping levels, it is not unreasonable to suggest that a clustering of the true recombination centres also occurs. It is with these difficulties in the background that this paper discusses the effects of spatial variation in mid-gap defect concentration on photoconductive lifetime, thermal generation of current in the depletion zone of a diode, and minority carrier lifetime as measured by the charge storage method of LAX and NEUSTADTER.(~) PHOTOCONDUCTIVE
DECAY
The effect of variation in recombination centre concentration on photoconductive decay time can be analysed as follows. Consider two large zones of semiconductor A and B, in intimate contact with each other and of dimensions considerably greater than the minority carrier diffusion length. If the defect concentrations are respectively N, and Nb, and the recombination lifetimes 7a and 7b, then the recombination rate in A cc N,
i.e. :\‘, = -
and similarly Ta
If the defects
Kb
Nb = -.
Tb
in B are identical to those in A, then = Nbrb. The overall photoconductive decay would be composed of two exponential waveforms of time constant 7a and 7b. The intial magnitude of the two current components at t = 0 would be proportional to the number of carriers created in the respective zones by the light pulse. However as Nb becomes very much greater than N,, the observed photocurrent would become progressively that associated with zone A. Unless care was taken to measure the quantum efficiency of the photocurrent observed, the composite material could appear to have a lifetime of 7a if 7a 9 5-b. If the dimensions of the low lifetime zone B are contracted so that they are comparable or less than the diffusion length in zone A, the surface in contact with zone A will now appear to have a high surface recombination velocity, and an estimate of
K, = K,, and Nara
PROBLEMS
CONCERNING
I la -=
DEEP
IMPI.JRITIES
sa Tbn I,a -_ =p. ran S, I/ - I,a
‘lb
Ir
Ifa = 1+
(6)
2?+! 0 rb
The corrected relationship,
IN
993
SEMICONDUCTORS
the lifetime given by reverse current defined as To by equation (1). As an example consider a pi--n diode composed of two zones of true lifetime T= and 7b, cross-sectional area S, and Sb. A charge storage lifetime 7s is defined by the following equation.
a
-=
becomes
1
erf
l+‘r
J
If (7)
t -. 7s
(8)
Solving the equation simultaneously with equation (7), to eliminate t, leads to numerical relationships
IO Cross- sectional
0
2.0
b
I .o
c
0.5
d
02
e c”lCO.l-
f
oreo
I $-
Curve
0.1 for
comporison
_rL for +a T0
I O.,
FIG. 1. Dependence of charge storage lifetime on the ratio of true life-
times in the zones.
Unfortunately such data on ratios will not be available. However it is clear from this relation that if the correction is not applied, the calculated 7 will be less than the true 7,. THE RELATION CURRENT
BETWEEN REVERSE LEAKAGE AND CHARGE STORAGE
In the previous sections, it was shown how the photoconductive lifetime and a lifetime calculated from reverse bias currents could vary relative to each other depending on the spatial distribution of deep-lying impurities. It is of interest to see how the lifetime obtained by charge storage relates to
between all the remaining parameters. For a p-i-n structure, n = 1. The value of 78 can be enumerated for values of S, and 7b. 7s is also dependent on the ratio 1,/I, and this is shown in Fig. 1, where the relation between 7b and 78 is plotted for a series of values of 1,/I, for the case when S, = 0.10. The implication of this result is that the presence of low lifetime zones could be detected by the variation of 7s with, IT/If,, the latter being controlled by the test circuit parameters. The second point of interest is shown by the insertion of the curve relating 71. to 7b for the same value of S,/S,.
P. E.
994
FIG. 2. Dependence
GIBBONS
of charge storage lifetime and generation lifetime on ratio of true lifetime.
The physical significance of this is that the charge storage lifetime 75 is more sensitive to deep-level impurity clustering than is the generationrecombination lifetime or. This point is reinforced in Fig. 3, where the same data as Fig. 2 is replotted, now as a function of S, against 75 or 5-rfor values of
This demonstrates that the value of ~~ is lower than 7r for values of 1,./I, > 0.2. Taking I,./If = 0.5, Fig. 2 plots the variation of TV or or, for a range of values of S,. Under this condition, rs is consistently smaller than r. Further dTJdTb is greater than dr,/dTb (for constant S,).
u-
CWle
cn” cl? Ol- __
75
=o
cl
0 01
b c d
b’ c’ dl
Note
r,
\
002 0 05 0 !O calculated
FIG. 3. Dependence
\
u \
\
\
for +
\
\
>, \
L’>
\
\
\
=0 5
of lifetime parameters on ratio of areas of two-zone model.
PROBLEMS
CONCERNING
DEEP
IMPURITIES
7b. The value of dr,/dSb is greater than dr,/dS,, inferring that an increase in size of the clusters affects the charge storage lifetime 78 more rapidly than the value of or. As mentioned earlier, the model of a two-zone diode, with areas S, and S,, is not restricted to the case where S, is a single entity but rather the summation of small low life-time zones embedded within the high lifetime zone A. It should be noted that for values of 71, > 0.37, the basic assumption of the calculation becomes increasingly invalid, i.e. that the charge storage is due entirely to the high lifetime zone. This accounts for the failure of some of the 7b ---TV curves to extrapolate through the point of unit values. It should also be noted that the numerical analysis has been performed for a p--n structure. The application to a p-n junction would be straightforward. In equation (7), n would have a value of 4, whilst equation (l), giving the generation time constant 71.would become (‘% + s,)2
995
IN SEMICONDUCTORS CONCLUSIONS
By considering particular examples of lifetime variation within a crystal specimen, it has been shown that discrepancies may be observed between lifetime when measured by photoconductive decay, reverse leakage current and charge storage. In practice, very complex spatial variations of lifetime may occur. The simple examples however indicate a qualitative relationship between the three methods which probably holds in more complex situations. Taking the reverse current lifetime as the most meaningful averaging of deep impurities, then photoconductive decay underestimates the number of deep impurities, whilst charge storage can overestimate the number of impurities. This latter discrepancy has an interesting consequence for a sample in which the clustering of low lifetime zones varies across the specimen. This would show as a variation in the ratio of (7&r) across the specimen, decreasing in those areas where the deep-level impurities are increasingly segregated into clusters.
raTb
REFERENCES 1. W.
SHOCKLEY and
W.
T.
READ,
Phys. Rev. 87,
835 (1952).
Generally, for given values of the parameters 7b and S,, the values of 7s and or will be higher for the p-n structure than for the p--&z example, in other words, they are less sensitive to the presence of low lifetime zones.
2. C.
T.
SAH, R. N. NOYCE and W.
Instn R&o 3.
P.
E.
SHOCKLEY.
GIBBONS. 1. H.
HOWES and R.
PYOC.
.
Engrs. 45, 1228 (1957). B.
OWEN.
Nucleon. It&t& I.E.E. Conf. Publ. 47,152 (1968): 4. B. LAXand S. F. NEUSTADTER,J. appl. Phys. 31148 (1954).