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Transport in photo-conductors—I
Solid-State Electronics VoI. 25, No. 8, pp. 707-712, 1982 Printed in Great Britain.
0038-1101/82/080707-06503.00/0 Pergamon Press Ltd.
TRANSPORT IN PHOTO-CONDUCTORS--I FOCAL PLANE PROCESSING D. J. DAY and T. J. SHEPHERD Royal Signals & Radar Establishment, Malvern, Worcestershire, England
(Received 24 September 1981) Abstract--Transport of excess carriers in a photo-conductor allows photo-carriers to be sampled remote from their point of generation. In paper I the separation of a photo-conductor into a generation and sampling volume is described in terms of equivalent signal processing associated with the sampling volume. Device geometry for time-delayed integration, multiplexing and edge enhancement are discussed. The Green's function for excess carriers in an infinite solid is used to give a physical description of signal and noise and their modification by transport. Diffusion is identified as a cause of image degradation; noise correlation is shown to limit the processing advantage described. In paper II a more detailed analysis is outlined to include the effects due to finite length and give a discussion of coherent and incoherent sampling.
A D d E G(x - x',t - t')
~(x, ~o) d(k, oJ) H(t) g I k L LD I l+M MTF No no
NOTATION constant diffusion coefficient, mesec-t resolution length, m applied electric field, Vm-t point spread or Green's function for infinite solid, m-I Fourier Transform of Green's function m-~sec twice transformed Green's function, sec unit step function generation function, m-3sec-t constant bias current A wave number to/v, radian m-t device length, m diffusion length X/(Dr), m sample length, m transport length, m effective number of sample regions modulation transfer function equilibrium carrier number in given volume (electrons) equilibrium carrier concentration (electrons), m-3
OTF PTF
Po P R r
S t to Vn V X
Xo to
/z ot
0 8 f~ A
optical transfer function phase transfer function equilibrium carrier concentration (holes), m-3 noise power, joules sec -t [(1 +82)2 +tl2~q It2 resistance of sample region, ohms noise power spectrum, Hz-~ time, sec generation time, sec noise voltage, VHz -1/2 image scan velocity, msec -~ position coordinate, m generation point, m excess carrier lifetime, sec angular frequency, radian sec-~ mobility, mV-lsec -t phase angle of g(x, to), radian PTF for infinite solid, radian frequency dependent angle, radian 2V(Dr)/~Er normalised angular frequency or wavelength, m
INTRODUCTION Radiation absorbed from an image focussed on the surface of a photo-conductor will generate an excess carrier distribution representative of the image. An imaging system sampling that image with finite response time and finite spatial resolution invariably requires subsequent data processing to restore, relocate and enhance the sampled information. It has been demonstrated[l] that much of this post-detector processing is unnecessary if charge manipulation on the focal plane is considered. In this paper the processing advantages of charge manipulation in a photo-conductor are discussed and associated image degradation analysed. The technological impact and viability of the new detector concepts that emerge from this analysis are discussed elsewhere [2]. In section 1 the properties of excess carrier transport are discussed and the process of image degradation identified. A simplified analysis of the signal and noise in an infinite photo-conductor is developed to give a physical interpretation of the frequency response and power spectrum associated with these processes. In section 2 separation of charge accumulation and sampling in a three-terminal device is shown to be equivalent to time-delayed integration processing in a scanned system or multiplexing for unscanned systems. Correlation in sampling is also discussed in the context of image enhancement. In a companion paper[3] the physical models introduced here are given a more detailed analysis to provide a better appreciation of the effects of sampling and finite length.
I. TRANSPORTIN AMBIPOLARSEMICONDUCTORS Analysis in this section will be limited to simple photoconductive devices so that implicit in this discussion will be the assumption of recombination semiconductors and the 707
D. J. DAYand T. J. SHEPHERD
708
properties of excess carriers exhibiting ambipolar transport. Alternative classes of semiconductor exhibiting memory or charge displacement properties will not be included but could equally well be exploited to describe other processing functions. When excess carriers An are generated in a photodetector they must achieve a balance between their rate of generation in any volume and rate of loss by recombination or transport from that volume. These processes are described by the ambipolar continuity eqn [4] dAnot = D div grad An - _E.grad An - An¢+ g
Figure 2 demonstrates both these functions for different mismatch of velocities. This modification of MTF by velocity mismatch is only sigfiificant for large mismatch condition and small wave number k, as r ( / z E - v) > Drk > k '.
(7)
It is apparent that when the carrier drift and generation source velocities are matched, i.e./~E = v, then all image
(1) i I
where g is generation rate, r the recombination lifetime, E any applied electric field and D and g are ambipolar diffusion coefficient and mobility, given by (n + p ) D e D . ' ~ _ (n - p ) ~ . ~ p
D = riD, + pD~
- ntz, + pgp
I
t ~T -0.25
(2)
\
with n and p the equilibrium electron and hole concentrations or appropriate coefficient subscript. A solution for a one dimensional solid of unit crosssection in which carrier pairs are instantaneously generated at a point x' at time t' is
G \
t i "O'5
g(x', t') dx' dt' An(x - x', t - t') = 2V'[crD(t - t')l
\
t
-I
e x p - Ix - x'_4DU_i~#E(t - t'))2 + t -r t'] =- g(x', t')G(x - x', t - t') dx' dt'
(3)
which is the point spread or Green's function for an infinite solid, (see Fig. 1). Superposition of such instantaneous sources allows the excess carrier density to be calculated for any spatial and temporal source function. For a sinusoidal generation source of intensity g, wave number k and velocity v, the excess carrier distribution in an infinite solid becomes
0
6
8
I0
12
1,1
Fig. 1. Ambipolarpoint spread function with (~ErJX/(Dr)) = 6 for values of (tH) = 2, 1, 0.5, 0.25.
I~
J" G ( x - x ' , t - t ' ) e x p - i k ( x ' - v t ' )
4
x/o~7
An(x, t) = g J-
2
~
IO0
dx' dt'
exp - i~b = [(Drk 2 + 1)2 + ((/zE - v)~'k)2]112' g1"exp - ik(x - vt). (4)
C3
The first term is the optical transfer function (OTF) for an infinite solid, the modulus of which is the modulation transfer function (MTF) MTF = [(D'rk 2 + 1)2 + ((#E - v)~'k)2]-'/2
(5) 00l
I Ol
and phase the phase transfer function (PTF)
i©
Wave number
P T F = 4' = t a n - ' L(D,rk2 + l ) J '
(6)
R
Fig. 2. MTF and PTF for infinitephoto-conductor with mismatch ((#E- v)~Lo) = O, 1, 2, 4.
709
Transport in photo-conductors--I points are in phase and the OTF becomes real and equal to the MTF MTFI,.n= v = (Drk 2 + 1)-1. (8)
noise power spectrum S (shown in Fig. 4) at any point can be written
S:4(N_No)2 ~/ (D ) f)~ d Texp-itoT.
Diffusion then limits transport of any spatial information that has a wavelength less than a diffusion length LD = \/(D'r). It is the image size, not velocity, that then determines the MTF; velocity only determines the frequency at which the wavelength is observed. In a scanned imager the dwell time to of the image on the detector is usually quite short so that a more general solution of eqn (4) is for finite generation time limits 0 to t, to give (see Fig. 3)
ffdx'ff_dtG(x-x',t)'G(x-x',t+r, R
1/2 3
O
6 cos ~
20
Reds ~ - 1
(ll)
PN P+N
OTF[,o = OTF]®. { 1 e x p - [((k2Lo2 +
1)-i(#E-v)kr)~]}
(9)
where the variance (N-N~o) has been taken as NPI(N +P)[6] and we have defined 6 = 2 C ( D r ) , R - [(1 + 82)2+
with OTFI~ defined from eqn (4) for infinite dwell time. For matched velocities only diffusion terms are again left, but the wavelength for which diffusion broadening is effective becomes /
• \1/2
X< x / ( D r ) ~ l - e x p - ~ )
~
/zEz
(10) This increase in signal bandwidth can be understood simply as less broadening occurring in less time. The fluctuations in the excess carrier density due to generation and recombination at any point will constitute a noise process (G-R noise). The finite lifetime of the carriers will give a temporal correlation to this process [5]. Diffusion broadening will contribute a spatial correlation; the drift velocity of the carriers will transform this spatial correlation to a temporal one observed at any point. From the auto-correlation of the point spread function and the Wiener-Khintchine theorem the
(12)
and tan 0 = (1--i-T~
X/(Dto)(to
~4((/)7)2]1/2
0~<0~<
.
(13)
There are therefore uniform steady state signal and noise fluxes associated with and modified by transport of excess carriers in a semiconductor. Drift serves to replicate events generated at x' at the point of observation. Diffusion and recombination act to modify the contribution of any generation event to any point. This appreciation that the signal and noise fluxes are both manifestations of the same broadening processes limited by the carrier lifetime suggests that the spectra will possess the same essential features, i.e. S cc MTF[~,r=~
(14)
The noise voltage spectrum v, rather than the power spectrum S is usually of practical interest, giving v, x x/(MTFI,E=v).
oo
I
L~ k~
OOI 001
I Wave number
I0 # D~"
Fig. 3. MTF for finite dwell time with (t/r) = 0.03, 0.1, 0.3, 1, =.
This is equivalent to the observation that the autocorrelation function has its major contribution about the generation time t = 0 in eqn (l 1). It is important to realise that signal and noise, although of common origin, are observed in different frames of reference. The signal being driven by the generation function propagates at the velocity of the generation source v. The noise is associated only with the excess carriers and will propagate at their mean velocity #E. Optimum signal to noise bandwidth occurs about the matched condition. The analysis has been greatly simplified by assuming an infinite length photo-conductor and so avoids the consideration of boundary conditions. The finite generation response, given by eqn 9, will be in particular error for low applied drift fields. For this case diffusion to boundaries will determine the carrier distribution in a
710
D.J. DAYand T.J. SHEPHERD [0
Image line scan -- Velocity V
#Er/2L
• I
Corrier velocity /z E--~ q Bias current I
~ Ol
03
Voltage out
Fig. 5. Three-terminal device with serial scanning.
0 OPl
OI
i I
i IO
i
IO0
Frequency tar Fig. 4. Noise flux spectra for infinite photo-conductor with
(#Er/2L) = 1, 8, 32.
finite detector. At higher fields when drift dominates diffusion as the transport mechanism to the boundary the carrier distribution is determined by the dwell time, and the previous assumptions become valid.
excess carriers generated within the sample length will therefore be small compared with the carriers accumulated over the drift length. The number of sample lengths equivalent to this steady state accumulated signal flux can therefore be defined from eqn (9).
MTFIt/,,~ IzE~"for mid-band where k2Lo 2 < 1,/zE~- > 1. 1
(18) 2. THREE-TERMINALDEVICES The discussion of signal and noise in the previous section considered the flux at a point. Sampling the signal flux given by eqn 4 over a length 1 gives
l/2 An(x, t) dx = MTFI= sin (kl[2) l/2 k[----~ " gr exp + ikvt.
(15) The excess carrier accumulation can thus be considered separable from the sampling process, so that each contributes its own MTF, the usual sampling bandwidth being defined by
MFFtsample sin kl[2 =
kt/2
d ~ 2Lb,
(19)
(16)
Degradation of the spatial resolution due to sampling can therefore be avoided by making the sampling bandwidth greater than the diffusion bandwidth, which requires 1 ~<2L~
The three-terminal device is thus equivalent to the summed output of /zEdl multiple detectors of length /. It fulfils the function of time delay integration signal processing for a serial scanned linear array of such detectors. For an unscanned image (v = 0) the same separation of accumulation and sampling can be applied, (see Fig. 6). If a static carrier (#E = 0) distribution accumulates over a detector of length L then diffusion will limit spatial resolution d in that distribution to
Image line scan - Velocity
v~
C°rr"'.,oo,,, °u,. L.
(17)
A detector can therefore be configured as a three-terminal device long enough to allow excess carriers to achieve a near steady state signal flux which is then sampled in a small sample length 1, (see Fig. 5). For an image scanned at a velocity v = tzE along such a detector the carriers will travel approximately a drift length/zEr while accumulating to steady state and would then be sampled in a time t = lillE. The number of
P u l s e ~
Fig. 6. Three-terminal device with slow parallel scan or unscanned.
Transport in photo-conductors--I Over the length L there will be M such resolution elements L = Md = 2ML~
'1"
/
t a r -0.1
(20)
This distribution can then be sampled by being periodically swept by an applied electric field E into a sample region l = d in a time t where t-
711
L ttE
(21)
0
I0
\ I~1 ~
/
The number of resolution elements accessed by the sample region can be written in analogy to eqn (18) as 001
(22)
M = #Et
1'
The three-terminal device then acts as a multiplexing element giving a serial output from many detectors of length l sampling an image in parallel. The time t must be short compared with the carrier lifetime to minimise degradation due to generation and diffusion during sampling, as exp-[(kZLo2+
1-ip, Erk) t]
(k:Lo 2 + l)
OTFI°=° =
f~dT
e x p - ioJT f_~ G(x, t ) . G ( x , t + T) dt (25)
This describes the spatial influence, at any frequency o~, of a representative noise generation event (see Fig. 7). At high frequencies when diffusion dominates transport of carriers I(~12is symmetric about the point of generation x = 0 and is predominantly exponential in form with a characteristic attenuation length
a'V'(Dr) 2 ~ R cos O"
-I
0
I
3
I
I
4
5
I 6
Distance from generating paint, x/Dv/-D--~T(normalised)
Fig. 7.
[~j2 showing spatial correlation of a noise event with
At low frequencies when drift dominates the transport It~lz is asymmetric, being skewed in the direction of minority carrier drift so that it must be described by two characteristic lengths 1+ =
8V'(Dr)
(26)
(27)
0
(24)
Signal processing advantage can only be maintained while samples of the noise flux are uncorrelated. Noise correlation can be avoided by allowing re-accumulation of a new excess carrier distribution over a carrier lifetime between samples. This correlation can be demonstrated by considering the Fourier transform of the auto-correlation function for a point generation[3]
l=
-2
I 2
k
This short read-out cycle also allows z/t devices to be multiplexed to the same output, giving the number of resolution elements accessible at an output the same as for a scanned device:
I~(x, ,o)12 =
\ I
#E~-= 2L at tot= 0.1, 1, 10, 100.
sink~
(23)
M - #Er 1 '
,
Higher frequencies that are attenuated by diffusion contribute to the noise power over a distance less than a drift length #Er and can therefore be sampled more often than frequencies within the pass-band of the MTF. This allows the possibility of high frequency enhancement by intermediate sampling. An alternative description of this edge enhancement is to consider it as the bandwidth difference between short and long dwell time of the image. The important difference between these descriptions lies in the implied concept of sampling. The latter implies a simple-minded coherent or signal sampling concept as discussed in connection with eqn 16. The former requires an understanding of I(~12 and the implied relationship between coherent and incoherent sampling for any sample length. The properties of G and a more detailed discussion of sampling, coherence and finite device length is given in a companion paper [3].
CONCLUSIONS
It has been shown that charge manipulation in threeand multiple-terminal photo-conductors can describe signal processing functions previously considered exter-
D. J. DAYand T. J. SHEPHERD
712
nal to the detection process. Although this classification is arbitrary and dependent on previous prejudice towards two-terminal photo-conductors, it draws a useful analogy between the properties of charge (data) manipulation on or off the focal plane. REFERENCF.~
1. C. T. Elliott, Electron. Lett. 17 (8), 312 (1981).
2. C. T. E|liott, D. J. Day and D. J. Wilson, IR Phys. 22, 31--42(1982). 3. T. J. Shepherd and D. J. Day, Solid St. Electron. 25(8), 713-718. 4. See for example J. McKelvey, Solid State and Semiconductor Physics, p, 327. Harper & Row, New York (1966). 5. J. E, Hill and K. M. van Vliet, Physica XXIV, 709 (1958). 6. R. E. Burgess, Proc. Phys. Soc. B68, 661 (1955).
0038-1101/82/080707-06503.00/0 Pergamon Press Ltd.
TRANSPORT IN PHOTO-CONDUCTORS--I FOCAL PLANE PROCESSING D. J. DAY and T. J. SHEPHERD Royal Signals & Radar Establishment, Malvern, Worcestershire, England
(Received 24 September 1981) Abstract--Transport of excess carriers in a photo-conductor allows photo-carriers to be sampled remote from their point of generation. In paper I the separation of a photo-conductor into a generation and sampling volume is described in terms of equivalent signal processing associated with the sampling volume. Device geometry for time-delayed integration, multiplexing and edge enhancement are discussed. The Green's function for excess carriers in an infinite solid is used to give a physical description of signal and noise and their modification by transport. Diffusion is identified as a cause of image degradation; noise correlation is shown to limit the processing advantage described. In paper II a more detailed analysis is outlined to include the effects due to finite length and give a discussion of coherent and incoherent sampling.
A D d E G(x - x',t - t')
~(x, ~o) d(k, oJ) H(t) g I k L LD I l+M MTF No no
NOTATION constant diffusion coefficient, mesec-t resolution length, m applied electric field, Vm-t point spread or Green's function for infinite solid, m-I Fourier Transform of Green's function m-~sec twice transformed Green's function, sec unit step function generation function, m-3sec-t constant bias current A wave number to/v, radian m-t device length, m diffusion length X/(Dr), m sample length, m transport length, m effective number of sample regions modulation transfer function equilibrium carrier number in given volume (electrons) equilibrium carrier concentration (electrons), m-3
OTF PTF
Po P R r
S t to Vn V X
Xo to
/z ot
0 8 f~ A
optical transfer function phase transfer function equilibrium carrier concentration (holes), m-3 noise power, joules sec -t [(1 +82)2 +tl2~q It2 resistance of sample region, ohms noise power spectrum, Hz-~ time, sec generation time, sec noise voltage, VHz -1/2 image scan velocity, msec -~ position coordinate, m generation point, m excess carrier lifetime, sec angular frequency, radian sec-~ mobility, mV-lsec -t phase angle of g(x, to), radian PTF for infinite solid, radian frequency dependent angle, radian 2V(Dr)/~Er normalised angular frequency or wavelength, m
INTRODUCTION Radiation absorbed from an image focussed on the surface of a photo-conductor will generate an excess carrier distribution representative of the image. An imaging system sampling that image with finite response time and finite spatial resolution invariably requires subsequent data processing to restore, relocate and enhance the sampled information. It has been demonstrated[l] that much of this post-detector processing is unnecessary if charge manipulation on the focal plane is considered. In this paper the processing advantages of charge manipulation in a photo-conductor are discussed and associated image degradation analysed. The technological impact and viability of the new detector concepts that emerge from this analysis are discussed elsewhere [2]. In section 1 the properties of excess carrier transport are discussed and the process of image degradation identified. A simplified analysis of the signal and noise in an infinite photo-conductor is developed to give a physical interpretation of the frequency response and power spectrum associated with these processes. In section 2 separation of charge accumulation and sampling in a three-terminal device is shown to be equivalent to time-delayed integration processing in a scanned system or multiplexing for unscanned systems. Correlation in sampling is also discussed in the context of image enhancement. In a companion paper[3] the physical models introduced here are given a more detailed analysis to provide a better appreciation of the effects of sampling and finite length.
I. TRANSPORTIN AMBIPOLARSEMICONDUCTORS Analysis in this section will be limited to simple photoconductive devices so that implicit in this discussion will be the assumption of recombination semiconductors and the 707
D. J. DAYand T. J. SHEPHERD
708
properties of excess carriers exhibiting ambipolar transport. Alternative classes of semiconductor exhibiting memory or charge displacement properties will not be included but could equally well be exploited to describe other processing functions. When excess carriers An are generated in a photodetector they must achieve a balance between their rate of generation in any volume and rate of loss by recombination or transport from that volume. These processes are described by the ambipolar continuity eqn [4] dAnot = D div grad An - _E.grad An - An¢+ g
Figure 2 demonstrates both these functions for different mismatch of velocities. This modification of MTF by velocity mismatch is only sigfiificant for large mismatch condition and small wave number k, as r ( / z E - v) > Drk > k '.
(7)
It is apparent that when the carrier drift and generation source velocities are matched, i.e./~E = v, then all image
(1) i I
where g is generation rate, r the recombination lifetime, E any applied electric field and D and g are ambipolar diffusion coefficient and mobility, given by (n + p ) D e D . ' ~ _ (n - p ) ~ . ~ p
D = riD, + pD~
- ntz, + pgp
I
t ~T -0.25
(2)
\
with n and p the equilibrium electron and hole concentrations or appropriate coefficient subscript. A solution for a one dimensional solid of unit crosssection in which carrier pairs are instantaneously generated at a point x' at time t' is
G \
t i "O'5
g(x', t') dx' dt' An(x - x', t - t') = 2V'[crD(t - t')l
\
t
-I
e x p - Ix - x'_4DU_i~#E(t - t'))2 + t -r t'] =- g(x', t')G(x - x', t - t') dx' dt'
(3)
which is the point spread or Green's function for an infinite solid, (see Fig. 1). Superposition of such instantaneous sources allows the excess carrier density to be calculated for any spatial and temporal source function. For a sinusoidal generation source of intensity g, wave number k and velocity v, the excess carrier distribution in an infinite solid becomes
0
6
8
I0
12
1,1
Fig. 1. Ambipolarpoint spread function with (~ErJX/(Dr)) = 6 for values of (tH) = 2, 1, 0.5, 0.25.
I~
J" G ( x - x ' , t - t ' ) e x p - i k ( x ' - v t ' )
4
x/o~7
An(x, t) = g J-
2
~
IO0
dx' dt'
exp - i~b = [(Drk 2 + 1)2 + ((/zE - v)~'k)2]112' g1"exp - ik(x - vt). (4)
C3
The first term is the optical transfer function (OTF) for an infinite solid, the modulus of which is the modulation transfer function (MTF) MTF = [(D'rk 2 + 1)2 + ((#E - v)~'k)2]-'/2
(5) 00l
I Ol
and phase the phase transfer function (PTF)
i©
Wave number
P T F = 4' = t a n - ' L(D,rk2 + l ) J '
(6)
R
Fig. 2. MTF and PTF for infinitephoto-conductor with mismatch ((#E- v)~Lo) = O, 1, 2, 4.
709
Transport in photo-conductors--I points are in phase and the OTF becomes real and equal to the MTF MTFI,.n= v = (Drk 2 + 1)-1. (8)
noise power spectrum S (shown in Fig. 4) at any point can be written
S:4(N_No)2 ~/ (D ) f)~ d Texp-itoT.
Diffusion then limits transport of any spatial information that has a wavelength less than a diffusion length LD = \/(D'r). It is the image size, not velocity, that then determines the MTF; velocity only determines the frequency at which the wavelength is observed. In a scanned imager the dwell time to of the image on the detector is usually quite short so that a more general solution of eqn (4) is for finite generation time limits 0 to t, to give (see Fig. 3)
ffdx'ff_dtG(x-x',t)'G(x-x',t+r, R
1/2 3
O
6 cos ~
20
Reds ~ - 1
(ll)
PN P+N
OTF[,o = OTF]®. { 1 e x p - [((k2Lo2 +
1)-i(#E-v)kr)~]}
(9)
where the variance (N-N~o) has been taken as NPI(N +P)[6] and we have defined 6 = 2 C ( D r ) , R - [(1 + 82)2+
with OTFI~ defined from eqn (4) for infinite dwell time. For matched velocities only diffusion terms are again left, but the wavelength for which diffusion broadening is effective becomes /
• \1/2
X< x / ( D r ) ~ l - e x p - ~ )
~
/zEz
(10) This increase in signal bandwidth can be understood simply as less broadening occurring in less time. The fluctuations in the excess carrier density due to generation and recombination at any point will constitute a noise process (G-R noise). The finite lifetime of the carriers will give a temporal correlation to this process [5]. Diffusion broadening will contribute a spatial correlation; the drift velocity of the carriers will transform this spatial correlation to a temporal one observed at any point. From the auto-correlation of the point spread function and the Wiener-Khintchine theorem the
(12)
and tan 0 = (1--i-T~
X/(Dto)(to
~4((/)7)2]1/2
0~<0~<
.
(13)
There are therefore uniform steady state signal and noise fluxes associated with and modified by transport of excess carriers in a semiconductor. Drift serves to replicate events generated at x' at the point of observation. Diffusion and recombination act to modify the contribution of any generation event to any point. This appreciation that the signal and noise fluxes are both manifestations of the same broadening processes limited by the carrier lifetime suggests that the spectra will possess the same essential features, i.e. S cc MTF[~,r=~
(14)
The noise voltage spectrum v, rather than the power spectrum S is usually of practical interest, giving v, x x/(MTFI,E=v).
oo
I
L~ k~
OOI 001
I Wave number
I0 # D~"
Fig. 3. MTF for finite dwell time with (t/r) = 0.03, 0.1, 0.3, 1, =.
This is equivalent to the observation that the autocorrelation function has its major contribution about the generation time t = 0 in eqn (l 1). It is important to realise that signal and noise, although of common origin, are observed in different frames of reference. The signal being driven by the generation function propagates at the velocity of the generation source v. The noise is associated only with the excess carriers and will propagate at their mean velocity #E. Optimum signal to noise bandwidth occurs about the matched condition. The analysis has been greatly simplified by assuming an infinite length photo-conductor and so avoids the consideration of boundary conditions. The finite generation response, given by eqn 9, will be in particular error for low applied drift fields. For this case diffusion to boundaries will determine the carrier distribution in a
710
D.J. DAYand T.J. SHEPHERD [0
Image line scan -- Velocity V
#Er/2L
• I
Corrier velocity /z E--~ q Bias current I
~ Ol
03
Voltage out
Fig. 5. Three-terminal device with serial scanning.
0 OPl
OI
i I
i IO
i
IO0
Frequency tar Fig. 4. Noise flux spectra for infinite photo-conductor with
(#Er/2L) = 1, 8, 32.
finite detector. At higher fields when drift dominates diffusion as the transport mechanism to the boundary the carrier distribution is determined by the dwell time, and the previous assumptions become valid.
excess carriers generated within the sample length will therefore be small compared with the carriers accumulated over the drift length. The number of sample lengths equivalent to this steady state accumulated signal flux can therefore be defined from eqn (9).
MTFIt/,,~ IzE~"for mid-band where k2Lo 2 < 1,/zE~- > 1. 1
(18) 2. THREE-TERMINALDEVICES The discussion of signal and noise in the previous section considered the flux at a point. Sampling the signal flux given by eqn 4 over a length 1 gives
l/2 An(x, t) dx = MTFI= sin (kl[2) l/2 k[----~ " gr exp + ikvt.
(15) The excess carrier accumulation can thus be considered separable from the sampling process, so that each contributes its own MTF, the usual sampling bandwidth being defined by
MFFtsample sin kl[2 =
kt/2
d ~ 2Lb,
(19)
(16)
Degradation of the spatial resolution due to sampling can therefore be avoided by making the sampling bandwidth greater than the diffusion bandwidth, which requires 1 ~<2L~
The three-terminal device is thus equivalent to the summed output of /zEdl multiple detectors of length /. It fulfils the function of time delay integration signal processing for a serial scanned linear array of such detectors. For an unscanned image (v = 0) the same separation of accumulation and sampling can be applied, (see Fig. 6). If a static carrier (#E = 0) distribution accumulates over a detector of length L then diffusion will limit spatial resolution d in that distribution to
Image line scan - Velocity
v~
C°rr"'.,oo,,, °u,. L.
(17)
A detector can therefore be configured as a three-terminal device long enough to allow excess carriers to achieve a near steady state signal flux which is then sampled in a small sample length 1, (see Fig. 5). For an image scanned at a velocity v = tzE along such a detector the carriers will travel approximately a drift length/zEr while accumulating to steady state and would then be sampled in a time t = lillE. The number of
P u l s e ~
Fig. 6. Three-terminal device with slow parallel scan or unscanned.
Transport in photo-conductors--I Over the length L there will be M such resolution elements L = Md = 2ML~
'1"
/
t a r -0.1
(20)
This distribution can then be sampled by being periodically swept by an applied electric field E into a sample region l = d in a time t where t-
711
L ttE
(21)
0
I0
\ I~1 ~
/
The number of resolution elements accessed by the sample region can be written in analogy to eqn (18) as 001
(22)
M = #Et
1'
The three-terminal device then acts as a multiplexing element giving a serial output from many detectors of length l sampling an image in parallel. The time t must be short compared with the carrier lifetime to minimise degradation due to generation and diffusion during sampling, as exp-[(kZLo2+
1-ip, Erk) t]
(k:Lo 2 + l)
OTFI°=° =
f~dT
e x p - ioJT f_~ G(x, t ) . G ( x , t + T) dt (25)
This describes the spatial influence, at any frequency o~, of a representative noise generation event (see Fig. 7). At high frequencies when diffusion dominates transport of carriers I(~12is symmetric about the point of generation x = 0 and is predominantly exponential in form with a characteristic attenuation length
a'V'(Dr) 2 ~ R cos O"
-I
0
I
3
I
I
4
5
I 6
Distance from generating paint, x/Dv/-D--~T(normalised)
Fig. 7.
[~j2 showing spatial correlation of a noise event with
At low frequencies when drift dominates the transport It~lz is asymmetric, being skewed in the direction of minority carrier drift so that it must be described by two characteristic lengths 1+ =
8V'(Dr)
(26)
(27)
0
(24)
Signal processing advantage can only be maintained while samples of the noise flux are uncorrelated. Noise correlation can be avoided by allowing re-accumulation of a new excess carrier distribution over a carrier lifetime between samples. This correlation can be demonstrated by considering the Fourier transform of the auto-correlation function for a point generation[3]
l=
-2
I 2
k
This short read-out cycle also allows z/t devices to be multiplexed to the same output, giving the number of resolution elements accessible at an output the same as for a scanned device:
I~(x, ,o)12 =
\ I
#E~-= 2L at tot= 0.1, 1, 10, 100.
sink~
(23)
M - #Er 1 '
,
Higher frequencies that are attenuated by diffusion contribute to the noise power over a distance less than a drift length #Er and can therefore be sampled more often than frequencies within the pass-band of the MTF. This allows the possibility of high frequency enhancement by intermediate sampling. An alternative description of this edge enhancement is to consider it as the bandwidth difference between short and long dwell time of the image. The important difference between these descriptions lies in the implied concept of sampling. The latter implies a simple-minded coherent or signal sampling concept as discussed in connection with eqn 16. The former requires an understanding of I(~12 and the implied relationship between coherent and incoherent sampling for any sample length. The properties of G and a more detailed discussion of sampling, coherence and finite device length is given in a companion paper [3].
CONCLUSIONS
It has been shown that charge manipulation in threeand multiple-terminal photo-conductors can describe signal processing functions previously considered exter-
D. J. DAYand T. J. SHEPHERD
712
nal to the detection process. Although this classification is arbitrary and dependent on previous prejudice towards two-terminal photo-conductors, it draws a useful analogy between the properties of charge (data) manipulation on or off the focal plane. REFERENCF.~
1. C. T. Elliott, Electron. Lett. 17 (8), 312 (1981).
2. C. T. E|liott, D. J. Day and D. J. Wilson, IR Phys. 22, 31--42(1982). 3. T. J. Shepherd and D. J. Day, Solid St. Electron. 25(8), 713-718. 4. See for example J. McKelvey, Solid State and Semiconductor Physics, p, 327. Harper & Row, New York (1966). 5. J. E, Hill and K. M. van Vliet, Physica XXIV, 709 (1958). 6. R. E. Burgess, Proc. Phys. Soc. B68, 661 (1955).