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Theory of transient space-charge perturbed currents in insulators
J. Phys. C’hem. Solids
THEORY
Pergamon Press 1967. VoL 28, pp. 641-647.
Printed in Great Britain.
OF TRANSIENT SPACE-CHARGE CURRENTS IN INSULATORS* A.
c.
PERTURBED
PAPADAEIS
Post Of&e Research Station, Dollis Hill, London, N.W.2. (Received 20 July 1966) Abetract-The problem of space charge perturbed (SCP) currents resulting from the drift of excess carriers in insulators is considered. Assuming the excess carriers to be instantaneously produced in a narrow sheet close to one of the electrodes expressions for the resulting SCP currents and the carrier transit times are derived. Secondaryinjectioncurrents are excluded by mesns of blocking contacts and carrier capture at both shallow and deep trapping centres during transit is allowed. It is shown that the field perturbations due to the space-charge of the excess carriers cause the initially narrow carrier sheet to broaden, even in the absence of diffusion. The time dependence of the resulting current pulse is found to be strongly dependent on the ratio of the applied field to the field due to the excess carriers. The results of the present theory are compared to those for the transient space-charge limited currents obtained by MANYand R.+x~vv.(~)
1. INTRODUCTION THIS PAPER forms part of a general study of carrier drift in high resistivity materials where the
dielectric relaxation time is long compared with the carrier transit time. Excess carriers may be introduced into these materials either by injection through the electrodes or by irradiation with light or energetic charged particles. Irradiation produces electron-hole pairs which may be separated with the aid of external electric field. Space-charge neutrality, which prevents this separation in more conducting materials, is not a requirement here. Injection of carriers into an insulator was first proposed by MOTTand Gun~x~(l) and inevitably leads to a space-charge build-up in the insulator. The study of carrier injection and the resulting space-charge limited (SCL) currents has since been extensively pursued.(12) Comparatively little attention, however, has been given in the literature to the study of transient currents in insulators. Such currents might arise, for example, as a result of the absorption of an energetic charged particle in a solid state radiation detector or in the measurement of carrier properties in high resistivity * A preliminary account of this work was presented at the Conference on Solid State Physics at Bristol, January 1965.
materials by the method of SPEAR.(~)This method is based on the measurement of transient currents arising from the drift of excess carriers induced in the insulator by a short duration ionization process -e.g. a light flash or an electron-beam pulse. Due to the lack of a detailed theory, for interpretational purposes it was necessary to use expressions obtained by HECHT(~) from simple considerations. More recent experiments, however, have indicated that appreciable deviations from the simple theory can occur both in the carrier transit times and in the time dependence of the current. These deviations, which are due to space-charge perturbations, become more pronounced at high excitation intensities and at low applied fields. This is well illustrated in the results of MANY et uk with iodine(4) and anthracenec5) and of GIBBONSand PAPADAK@) with sulphur. MANY and RAK&w(‘) and HFXFRICH and MARIP) have independently considered the theoretical problem where the excess carriers are introduced in a narrow region close to one of the (injecting) electrodes. Hence a single carrier system through the insulator is considered. It is assumed that the excess carrier density is sufliciently large to form an infinite reservoir close to the 641
642
A. C. PAPADAKIS
crystal surface. Space charge effects allow only a relatively small portion of the reservoir to be injected into the crystal bulk at any given time. Under these conditions a time dependent SCL current flows. An essential condition of both these treatments is that the carrier density in the reservoir remains sufficiently large so as not to limit the injection of the full number of carriers required to sustain the SCL current, i.e. there must be a virtual “ohmic” contact to the insulator. In Many’s experiments with iodine,(*) the reservoir was maintained by steady intense illumination with strongly absorbed light through a transparent electrode. Carriers were injected into the crystal bulk by means of a it is doubtful steep voltage pulse, However, whether the full SCL current can be maintained at high current densities and/or in the presence of fast initial recombination, particularly when pulsed irradiation is used with a constant drift field.(e) This uncertainty does not arise in Spear’s experiments where secondary injection is excluded by the use of “blocking” contacts and a finite number of carriers are generated close to one electrode by a short duration ionizing pulse. Spear’s arrangement also has the advantage of giving a better signal-to-noise ratio due to the absence of injection currents. The displacement current observed when a steady drift field is applied is due solely to one type of carrier generated by the ionization and will be referred to as the space-charge perturbed (SCF) current to distinguish it from Many’s SCL current. The time dependence of SCP currents that result from the drift of carriers from a narrow region of ionization in an insulator has been considered by KEATSNG and PAPADAKIS.(~*)In their treatment secondary injection was excluded by generating the excess carriers in a region remote from the electrodes. Both electrons and holes were considered mobile and their analysis is valid up to a time when the iirst carriers reach one of the electrodes, this time being left undefined. In the present treatment, by assuming that the region of ionization is close to a blocking electrode, we obtain expresaions for the transit time of the first carriers to reach the opposite electrode. For insulators free from deep trapping centres we extend the analysis to times beyond this and hence
obtain expressions for the complete time dependence of the SCP current and for the transit time of the last carriers to reach the opposite electrode.
2. THEORY We consider a homogeneous and initially neutral insulator crystal of thickness L with an infinite cross-section and plane parallel electrodes at either end. As a result of an instantaneous ionization process a number of electron-hole pairs is produced in the form of an infinite sheet of small (but finite) thickness close to one of the electrodes. We shall assume, without loss of generality, that the sheet is close to the anode. With an applied field that is sutBciently large to overcome the attractive forces between them, the electrons and holes separate, the holes being drawn into the crystal bulk while the electrons are collected at the neighbouring electrode. Since the initial sheet thickness is small, complete charge separation takes place within a short time after ionization, and this causes negligible field perturbations. Hence, we take the time immediately after separation as our origin and consider the drift of holes alone in the bulk. The holes are drifting towards the cathode in the form of a space charge sheet in a field which is perturbed by this space charge. As a result, the sheet broadens as it drifts until the time when its leading edge reaches the cathode and carriers begin to leave the crystal. Assuming that at t = 0 the carrier density is uniform within the sheet, in the absence of diffusion it will remain uniform throughout the transit, although the carrier density will be continuously decreased due both to the broadening of the packet and the loss of carriers due to capture. Two types of carrier will be considered: (a) capture at deep trapping centres with release times long compared to the carrier transit time, and(b) shallow trapping at centres with both capture and release times short compared to the carrier transit time. We shall assume that deep trapping can be described by a bulk lifetime T which is independent of the carrier density. Shallow trapping will be taken into account by using an effective trap-controlled drift mobility p in place of the lattice controlled mobility.‘a) Carrier spread due to shallow trapping will be neglected.
THEORY
OF TRANSIENT
SPACE-CHARGE
The theoretical formulation will be divided into two parts: (i) from t = 0 when carrier separation is completed, until t =: Tl when the leading edge of the carrier sheet reaches the anode; and (ii) from 2’s when the trailing edge of the t= T,untilt= sheet reaches the cathode and hence all the excess carriers have been collected. MKS units are used throughout. 2.1 0 < t Q Figure 1 distribution distributions
T, shows the space charge and field at some time t (0 < t < TX). The at t+dt are also shown in this figure.
PERTURBED
CURRENTS
IN INSULATORS
643
trailing edges of the sheet which by integration of the above equations is given by enw 8
c---z
c
F
exp(,- t/T) = fife exp( - t/7)
where w is the sheet width and the subscript 0 denotes quantities at t = 0. Since the crystal is neutral initially, the field at the leading edge of the sheet El is equal to the field at the cathode, and the leading edge drift velocity is given by
The trailing edge drift velocity is smalfer and is given by
An expression for the time dependence of EL can be readily derived by equating the mean potential change dV during dt due to the drift of holes, and the change dEl in the general field level: dV = 04J(dx,
+dxz) = L dE,
i.e.
drs, ~8 0
I X2
I
- -(a51 -Z--L
- tp/2).
(3)
*I x
(b) FIG. 1. Space-charge and field distributions at times t (continuous lines) and tf dt (dotted lines) 0 < t < T.
Within the region occupied by the sheet there is a constant field gradient given by dE
We now change into the dimensionless variables @ = E,/&,, A = rjTo and rl = EI/& tr = exp( - t/r). To = L~~~~ is the carrier transit time in the absence of space charge perturbations to the field. With the boundary condition q = j3 at a = 1 equation (3) has the solution
en
Z==T where E is the field strength, x the coordinate normal to the electrodes, en the charge per unit volume in the sheet immediately after separation and Bthe permittivity of the insulator. Thus there is a field difference between the leading and
We note that equation (4) becomes identical with the expression for 7 given in Ref. 10 when the contribution of one type of carrier is negligible. The charge Q displaced at the electrodes and the displacement current i can be obtained from equation (4) using the relations 4 = @, -Es) and
644
A. C. PAPADAKIS
i = a$/dt or, in dimensionless
In general Tl is then obtained by numerical solutions of this equation. Two special cases of interest lead to useful further simplification.
form
Q = de = 7-B and I=;/&=
---
(a) Small signal case. When the field perturbation
adrl
Ada = -&Q
-(U%%9--
1 +B/X)exp[(h/B)(l-)~l).
(5) Here q,, = en,,wo is the total charge per unit area of the sheet immediately after separation, and * = qo/To is the current density due to charge q0 *a in the absence of space charge perturbations to the field. The time dependence of the position of the leading and trailmg edges of the sheet can be readily obtained from equations (1) and (2) respectively :
Xl+=
W,,--.
s
ho? -da 8, =
(6)
due to the mobile holes is negligible (i.e. A/#l + 1) broadening of the sheet is negligible (W 2: W,) and Tl N T2 N T,,. This case arises when very few carriers are present initially or when a strong field is applied (16 9 1). It may also arise when the perturbation is confined to the initial part of the transit because of rapid trapping (7 4 To). From equation (5) the displacement current up to t = To is now given by I = exp( - t/T)[l -(l/2/?)
(b) No deep trappi% case. We now consider
the case where carrier capture at deep trapping centres during the transit time is negligible i.e. r > To, X B 1. In this approximation equations (5), (7) and (8) reduce to
(1-
l/2/3) exp(#TO)
(10)
W = W, + @To
1
where W, = coo/L is the dimensionless initial sheet width. Hence the time dependent sheet width w is given by
= W,+(@)(l-a).
(9)
and drops sharply to zero when t = T,,.
I=
w = w/L = x,-x,
exp( - t/T)]
(7)
and TlPo
=
2-[2~-ll[exp(T~/ST~)-11.
(11)
SCHWARTZ and HORNIG(~~) have shown that equations (10) and (11) can be obtained from Many’s formulation provided that the boundary conditions are altered to allow for the absence of the infinite reservoir at the anode. i.e. by replacing Many’s equation (7) which is E(0, t) = 0 by the equation E(L, t)-E(0, t) = bo.
Broadening of the sheet may well be sufficient to spread the carriers that are generated in a sheet of negligible width, over almost the entire interelectrode distance and so give quite a wide range of transit times. The transit time of the leading edge Tl can be 2.2 T, < t < Ta obtained from equation (6) by setting X1 = 1 We now consider the collection of carriers at the for 77 cathode from Tl until T, when all excess charge (when 0 = ~1 = exp( - T$-)). Substituting from equation (4) and neglecting W, we obtain has been removed. For simplicity, however, we shall confine ourselves to the case of an insulator 2g bl free from deep trapping centres, i.e. where the --= (l-j3/xa+(2/3-1+/3/A) space-charge perturbations are most pronounced. s A 1 Figure 2 shows the space-charge and field x expKWXl41/4 dc. (8) distributions at times t and t+dt (Tl < t < T,).
THEORY
OF TRANSIENT
SPACE-CHARGE
PERTURBED
The field difference between the trailing edge of the sheet and the cathode is given by g2 =
eny,o
where y is the distance of the trailing edge from the cathode. Equating the number of carriers per
CURRENTS
IN INSULATORS
645
change dE, we obtain 0.5(yd8s+8,dy)
= LdE,.
(13)
With the boundary condition Es = E. when y = 0, t = Ta (since the crystal is then free of any space-charges), equation (13) has the solution
E, = E,-y&,/2L where dy/dt = -pE,,
or in dimensionless form
PO Y2
E2/go = j3 - -
71~ =
2t
(14)
and t
y=yjL=isq,dt. BTO=1
(15)
The transit time of the trailing edge is obtained from numerical solutions of equation (15) at t = T2 when Y = 0. The displacement current I can be obtained from equations (14) and (15) using the relation I
=
T
3. RESULTS
d&
Odt
=
AND
T
dr12
0-g
(16)
DISCUSSION
Figure 3 shows the time dependence of the SCP current in the absence of deep trapping for various FIG. 2. Space-charge and field t (TV < t < T.J when the trailing a distance y from the cathode. t-l-& (dotted lines) are
3
distributions at time edge of the sheet is at The distributions at also shown.
2
unit area which leave the crystal, in time dt, to the cbrresponding decrease in the remaining number of carriers, we obtain - np(E2 + 82) = -npE,+ydn/dt
< I
(12)
where Es is the field at the trailing edge. With the boundary condition n = nI = nowo/wl at t = TI equation (12) has the solution ?+
s
TI)]-”
Equating the changes in potential, the time dt, due to the drift of holes and to the overall field
0
05
I.5
2.0
t/To
FIG. 3. The time dependence of the transient current in the absence of deep trapping and for different values of the parameter fi = I&,/B,,. The solid curves are for the SCP current computed from equations (10) and (16). The dashed curve shows the SCL current derived from MANY and RAKA~~.(‘)
646
A. C. PAPADAKIS
values of the parameter /3,which is a measure of the field perturbation @ = condo = ~~o/~}. The region 0 < t < T has been obtained from equation (9) and the region Tl < t < T, from numerical solutions of equations (14), (15) and (16). The results of Many and Rakavy for the SCL current [equations (27) and (37) of Ref. 73 are also shown in this figure. In the region 0 < t -c Tl the SCP current increases with time except in the small signal limit (/I g l), when it is constant. This increase is a consequence of the perturbation of the applied field by the excess carrier spacecharge, As the sheet moves away from the anode, it moves in an increasing field due to the perturbations. Since no carriers are permanently lost by capture or through the electrodes during this period, the current increases. The rate of increase is strongly dependent of the parameter 8. During the period Tl < t < Ta the current decreases since carriers are leaving the crystal. The current I I drops to zero at t = Ta when all the excess I I I 0 85 5.0 FS IO carriers have been collected. As a result of the P field perturbations the rate of decrease is nonFIG. 4. The initial value of the SCP current and the linear but once again dependent on ,8. ratio I(Tl)/I(O) as a function of B and for different In contrast to the SCP current, the time dependvalues of the parameter h = T/T,,. ence of the SCL current is independent of the applied strength or the excess carrier density 1.25 provided that the latter is suffi~en~y l~ge.Therate I of increase of the SCL current during the initial period is appreciably higher than that of the SCP current, even for /3= 1. This additional increase is due to the continuous injection of carriers into the crystal when “non-blocking” contacts are used. Similarly the rate of decrease of the SCL current is smaller and the current now decays to the steady state SCL value of 1.125 i,,.(l) From equation (5), the initial value of the SCP current i(0) is independent of the extent of bulk trapping and is given by
1
i(0) = ie(l -l/2/3). At ,3 = 1 we have ~(O)/i~= O-5 which is identical with the initial value of the SCL current. The SCP, however, allows a continuous variation of i(O)/&,in the range 0.5-l depending on the density of excess carriers and the applied field strength. Figure 4 shows the variation of i(0)/io with 8. The ratio of the SGP current at t = Tl I(T,) to the initial current, for various values of the parameter X - -r/T, is also shown in this figure.
t/To
FIG. 5. Variation of the initilplpart of the SCP current with time at fi = 1 h = T/T~as parameter. l
THEORY OF TRANSIENT SPACE-CHARGE PERTURBED CURRENTS IN INSULATORS
The effect of deep trapping on the time dependence of the SCP current is illustrated in Fig. 5. Here i/is [given by equation(S)] at ,8 = 1 is shown for various values of the parameter j3.As X decreases both the number of carriers and the resulting field perturbations are decreased. However, the increase of the current with time persists even for X = 1
647
Figure 6 shows the variation of the transit time
Tl obtained from numerical solutions of equation (8) as a function of ,9 for various values of the parameter A. The transit time of the trailing edge Tl for X g 1 is also shown in this figure.
Acknowledgements-The author wishes to thank Prof. J. C. hDExSON of Imperial College, London, for his encouragement and support during the early part of r;rk. Acknowledgement is made to the Engineer-in-Chief of the British Post Office for permission to publish this
(r ‘;;[
REFERENCES
2. I.0
3. 4.
5. 6. 7. 8.
FIG. 6. Variation of the transit time TX as a function of j for different values of the parameter h. The transit time Ta for I\> 1 is also shown (dotted line).
11. 12.
MOTT N. F. and GURNEY R. W., Electronic Processes in Ionic Crystds (Oxford: Clarendon Press 1940). SPEAR W. E. and MORT J., Proc.Phys. Sot. 81,130 (1963). HECHT K., 2. Phys. 77,235 (1932). SILVER M., SWICORDM., JARNAGINR. C., MANY A., WEISS S. Z. and SIMHONY M., J. Phys. Chews. Solids 23,419 (1962). MANY A., WEISS S. Z. and SIMHONY M., Phys. Rev. 126,1989 (1962). GIBBONSD. J. and PAPAIXUCIS A. C., To be published. MANY A. and RAKAVY G., Phys. Rev. 126, 1980 (1962). HELPRICH W. and MARK P., Z. Phvs. 166. 370 (1962). SIMHONY A. and GORELIK J., J. Phys. Chem. Solids 26,1133 (1965). KEA%NG P. N. and PAPAD,+XISA. C., PYOC. Int. Conf. on the Phys. of Semicond. Paris 1964 (Paris: Dunod) p. 519. SCHWARTZL. M. and HORNIG J. F., J. Phys. Chem. Solids 26,182l (1965). LAMPERT M. A., Rep. Prog. Phys. 27, 329 (1964).
THEORY
Pergamon Press 1967. VoL 28, pp. 641-647.
Printed in Great Britain.
OF TRANSIENT SPACE-CHARGE CURRENTS IN INSULATORS* A.
c.
PERTURBED
PAPADAEIS
Post Of&e Research Station, Dollis Hill, London, N.W.2. (Received 20 July 1966) Abetract-The problem of space charge perturbed (SCP) currents resulting from the drift of excess carriers in insulators is considered. Assuming the excess carriers to be instantaneously produced in a narrow sheet close to one of the electrodes expressions for the resulting SCP currents and the carrier transit times are derived. Secondaryinjectioncurrents are excluded by mesns of blocking contacts and carrier capture at both shallow and deep trapping centres during transit is allowed. It is shown that the field perturbations due to the space-charge of the excess carriers cause the initially narrow carrier sheet to broaden, even in the absence of diffusion. The time dependence of the resulting current pulse is found to be strongly dependent on the ratio of the applied field to the field due to the excess carriers. The results of the present theory are compared to those for the transient space-charge limited currents obtained by MANYand R.+x~vv.(~)
1. INTRODUCTION THIS PAPER forms part of a general study of carrier drift in high resistivity materials where the
dielectric relaxation time is long compared with the carrier transit time. Excess carriers may be introduced into these materials either by injection through the electrodes or by irradiation with light or energetic charged particles. Irradiation produces electron-hole pairs which may be separated with the aid of external electric field. Space-charge neutrality, which prevents this separation in more conducting materials, is not a requirement here. Injection of carriers into an insulator was first proposed by MOTTand Gun~x~(l) and inevitably leads to a space-charge build-up in the insulator. The study of carrier injection and the resulting space-charge limited (SCL) currents has since been extensively pursued.(12) Comparatively little attention, however, has been given in the literature to the study of transient currents in insulators. Such currents might arise, for example, as a result of the absorption of an energetic charged particle in a solid state radiation detector or in the measurement of carrier properties in high resistivity * A preliminary account of this work was presented at the Conference on Solid State Physics at Bristol, January 1965.
materials by the method of SPEAR.(~)This method is based on the measurement of transient currents arising from the drift of excess carriers induced in the insulator by a short duration ionization process -e.g. a light flash or an electron-beam pulse. Due to the lack of a detailed theory, for interpretational purposes it was necessary to use expressions obtained by HECHT(~) from simple considerations. More recent experiments, however, have indicated that appreciable deviations from the simple theory can occur both in the carrier transit times and in the time dependence of the current. These deviations, which are due to space-charge perturbations, become more pronounced at high excitation intensities and at low applied fields. This is well illustrated in the results of MANY et uk with iodine(4) and anthracenec5) and of GIBBONSand PAPADAK@) with sulphur. MANY and RAK&w(‘) and HFXFRICH and MARIP) have independently considered the theoretical problem where the excess carriers are introduced in a narrow region close to one of the (injecting) electrodes. Hence a single carrier system through the insulator is considered. It is assumed that the excess carrier density is sufliciently large to form an infinite reservoir close to the 641
642
A. C. PAPADAKIS
crystal surface. Space charge effects allow only a relatively small portion of the reservoir to be injected into the crystal bulk at any given time. Under these conditions a time dependent SCL current flows. An essential condition of both these treatments is that the carrier density in the reservoir remains sufficiently large so as not to limit the injection of the full number of carriers required to sustain the SCL current, i.e. there must be a virtual “ohmic” contact to the insulator. In Many’s experiments with iodine,(*) the reservoir was maintained by steady intense illumination with strongly absorbed light through a transparent electrode. Carriers were injected into the crystal bulk by means of a it is doubtful steep voltage pulse, However, whether the full SCL current can be maintained at high current densities and/or in the presence of fast initial recombination, particularly when pulsed irradiation is used with a constant drift field.(e) This uncertainty does not arise in Spear’s experiments where secondary injection is excluded by the use of “blocking” contacts and a finite number of carriers are generated close to one electrode by a short duration ionizing pulse. Spear’s arrangement also has the advantage of giving a better signal-to-noise ratio due to the absence of injection currents. The displacement current observed when a steady drift field is applied is due solely to one type of carrier generated by the ionization and will be referred to as the space-charge perturbed (SCF) current to distinguish it from Many’s SCL current. The time dependence of SCP currents that result from the drift of carriers from a narrow region of ionization in an insulator has been considered by KEATSNG and PAPADAKIS.(~*)In their treatment secondary injection was excluded by generating the excess carriers in a region remote from the electrodes. Both electrons and holes were considered mobile and their analysis is valid up to a time when the iirst carriers reach one of the electrodes, this time being left undefined. In the present treatment, by assuming that the region of ionization is close to a blocking electrode, we obtain expresaions for the transit time of the first carriers to reach the opposite electrode. For insulators free from deep trapping centres we extend the analysis to times beyond this and hence
obtain expressions for the complete time dependence of the SCP current and for the transit time of the last carriers to reach the opposite electrode.
2. THEORY We consider a homogeneous and initially neutral insulator crystal of thickness L with an infinite cross-section and plane parallel electrodes at either end. As a result of an instantaneous ionization process a number of electron-hole pairs is produced in the form of an infinite sheet of small (but finite) thickness close to one of the electrodes. We shall assume, without loss of generality, that the sheet is close to the anode. With an applied field that is sutBciently large to overcome the attractive forces between them, the electrons and holes separate, the holes being drawn into the crystal bulk while the electrons are collected at the neighbouring electrode. Since the initial sheet thickness is small, complete charge separation takes place within a short time after ionization, and this causes negligible field perturbations. Hence, we take the time immediately after separation as our origin and consider the drift of holes alone in the bulk. The holes are drifting towards the cathode in the form of a space charge sheet in a field which is perturbed by this space charge. As a result, the sheet broadens as it drifts until the time when its leading edge reaches the cathode and carriers begin to leave the crystal. Assuming that at t = 0 the carrier density is uniform within the sheet, in the absence of diffusion it will remain uniform throughout the transit, although the carrier density will be continuously decreased due both to the broadening of the packet and the loss of carriers due to capture. Two types of carrier will be considered: (a) capture at deep trapping centres with release times long compared to the carrier transit time, and(b) shallow trapping at centres with both capture and release times short compared to the carrier transit time. We shall assume that deep trapping can be described by a bulk lifetime T which is independent of the carrier density. Shallow trapping will be taken into account by using an effective trap-controlled drift mobility p in place of the lattice controlled mobility.‘a) Carrier spread due to shallow trapping will be neglected.
THEORY
OF TRANSIENT
SPACE-CHARGE
The theoretical formulation will be divided into two parts: (i) from t = 0 when carrier separation is completed, until t =: Tl when the leading edge of the carrier sheet reaches the anode; and (ii) from 2’s when the trailing edge of the t= T,untilt= sheet reaches the cathode and hence all the excess carriers have been collected. MKS units are used throughout. 2.1 0 < t Q Figure 1 distribution distributions
T, shows the space charge and field at some time t (0 < t < TX). The at t+dt are also shown in this figure.
PERTURBED
CURRENTS
IN INSULATORS
643
trailing edges of the sheet which by integration of the above equations is given by enw 8
c---z
c
F
exp(,- t/T) = fife exp( - t/7)
where w is the sheet width and the subscript 0 denotes quantities at t = 0. Since the crystal is neutral initially, the field at the leading edge of the sheet El is equal to the field at the cathode, and the leading edge drift velocity is given by
The trailing edge drift velocity is smalfer and is given by
An expression for the time dependence of EL can be readily derived by equating the mean potential change dV during dt due to the drift of holes, and the change dEl in the general field level: dV = 04J(dx,
+dxz) = L dE,
i.e.
drs, ~8 0
I X2
I
- -(a51 -Z--L
- tp/2).
(3)
*I x
(b) FIG. 1. Space-charge and field distributions at times t (continuous lines) and tf dt (dotted lines) 0 < t < T.
Within the region occupied by the sheet there is a constant field gradient given by dE
We now change into the dimensionless variables @ = E,/&,, A = rjTo and rl = EI/& tr = exp( - t/r). To = L~~~~ is the carrier transit time in the absence of space charge perturbations to the field. With the boundary condition q = j3 at a = 1 equation (3) has the solution
en
Z==T where E is the field strength, x the coordinate normal to the electrodes, en the charge per unit volume in the sheet immediately after separation and Bthe permittivity of the insulator. Thus there is a field difference between the leading and
We note that equation (4) becomes identical with the expression for 7 given in Ref. 10 when the contribution of one type of carrier is negligible. The charge Q displaced at the electrodes and the displacement current i can be obtained from equation (4) using the relations 4 = @, -Es) and
644
A. C. PAPADAKIS
i = a$/dt or, in dimensionless
In general Tl is then obtained by numerical solutions of this equation. Two special cases of interest lead to useful further simplification.
form
Q = de = 7-B and I=;/&=
---
(a) Small signal case. When the field perturbation
adrl
Ada = -&Q
-(U%%9--
1 +B/X)exp[(h/B)(l-)~l).
(5) Here q,, = en,,wo is the total charge per unit area of the sheet immediately after separation, and * = qo/To is the current density due to charge q0 *a in the absence of space charge perturbations to the field. The time dependence of the position of the leading and trailmg edges of the sheet can be readily obtained from equations (1) and (2) respectively :
Xl+=
W,,--.
s
ho? -da 8, =
(6)
due to the mobile holes is negligible (i.e. A/#l + 1) broadening of the sheet is negligible (W 2: W,) and Tl N T2 N T,,. This case arises when very few carriers are present initially or when a strong field is applied (16 9 1). It may also arise when the perturbation is confined to the initial part of the transit because of rapid trapping (7 4 To). From equation (5) the displacement current up to t = To is now given by I = exp( - t/T)[l -(l/2/?)
(b) No deep trappi% case. We now consider
the case where carrier capture at deep trapping centres during the transit time is negligible i.e. r > To, X B 1. In this approximation equations (5), (7) and (8) reduce to
(1-
l/2/3) exp(#TO)
(10)
W = W, + @To
1
where W, = coo/L is the dimensionless initial sheet width. Hence the time dependent sheet width w is given by
= W,+(@)(l-a).
(9)
and drops sharply to zero when t = T,,.
I=
w = w/L = x,-x,
exp( - t/T)]
(7)
and TlPo
=
2-[2~-ll[exp(T~/ST~)-11.
(11)
SCHWARTZ and HORNIG(~~) have shown that equations (10) and (11) can be obtained from Many’s formulation provided that the boundary conditions are altered to allow for the absence of the infinite reservoir at the anode. i.e. by replacing Many’s equation (7) which is E(0, t) = 0 by the equation E(L, t)-E(0, t) = bo.
Broadening of the sheet may well be sufficient to spread the carriers that are generated in a sheet of negligible width, over almost the entire interelectrode distance and so give quite a wide range of transit times. The transit time of the leading edge Tl can be 2.2 T, < t < Ta obtained from equation (6) by setting X1 = 1 We now consider the collection of carriers at the for 77 cathode from Tl until T, when all excess charge (when 0 = ~1 = exp( - T$-)). Substituting from equation (4) and neglecting W, we obtain has been removed. For simplicity, however, we shall confine ourselves to the case of an insulator 2g bl free from deep trapping centres, i.e. where the --= (l-j3/xa+(2/3-1+/3/A) space-charge perturbations are most pronounced. s A 1 Figure 2 shows the space-charge and field x expKWXl41/4 dc. (8) distributions at times t and t+dt (Tl < t < T,).
THEORY
OF TRANSIENT
SPACE-CHARGE
PERTURBED
The field difference between the trailing edge of the sheet and the cathode is given by g2 =
eny,o
where y is the distance of the trailing edge from the cathode. Equating the number of carriers per
CURRENTS
IN INSULATORS
645
change dE, we obtain 0.5(yd8s+8,dy)
= LdE,.
(13)
With the boundary condition Es = E. when y = 0, t = Ta (since the crystal is then free of any space-charges), equation (13) has the solution
E, = E,-y&,/2L where dy/dt = -pE,,
or in dimensionless form
PO Y2
E2/go = j3 - -
71~ =
2t
(14)
and t
y=yjL=isq,dt. BTO=1
(15)
The transit time of the trailing edge is obtained from numerical solutions of equation (15) at t = T2 when Y = 0. The displacement current I can be obtained from equations (14) and (15) using the relation I
=
T
3. RESULTS
d&
Odt
=
AND
T
dr12
0-g
(16)
DISCUSSION
Figure 3 shows the time dependence of the SCP current in the absence of deep trapping for various FIG. 2. Space-charge and field t (TV < t < T.J when the trailing a distance y from the cathode. t-l-& (dotted lines) are
3
distributions at time edge of the sheet is at The distributions at also shown.
2
unit area which leave the crystal, in time dt, to the cbrresponding decrease in the remaining number of carriers, we obtain - np(E2 + 82) = -npE,+ydn/dt
< I
(12)
where Es is the field at the trailing edge. With the boundary condition n = nI = nowo/wl at t = TI equation (12) has the solution ?+
s
TI)]-”
Equating the changes in potential, the time dt, due to the drift of holes and to the overall field
0
05
I.5
2.0
t/To
FIG. 3. The time dependence of the transient current in the absence of deep trapping and for different values of the parameter fi = I&,/B,,. The solid curves are for the SCP current computed from equations (10) and (16). The dashed curve shows the SCL current derived from MANY and RAKA~~.(‘)
646
A. C. PAPADAKIS
values of the parameter /3,which is a measure of the field perturbation @ = condo = ~~o/~}. The region 0 < t < T has been obtained from equation (9) and the region Tl < t < T, from numerical solutions of equations (14), (15) and (16). The results of Many and Rakavy for the SCL current [equations (27) and (37) of Ref. 73 are also shown in this figure. In the region 0 < t -c Tl the SCP current increases with time except in the small signal limit (/I g l), when it is constant. This increase is a consequence of the perturbation of the applied field by the excess carrier spacecharge, As the sheet moves away from the anode, it moves in an increasing field due to the perturbations. Since no carriers are permanently lost by capture or through the electrodes during this period, the current increases. The rate of increase is strongly dependent of the parameter 8. During the period Tl < t < Ta the current decreases since carriers are leaving the crystal. The current I I drops to zero at t = Ta when all the excess I I I 0 85 5.0 FS IO carriers have been collected. As a result of the P field perturbations the rate of decrease is nonFIG. 4. The initial value of the SCP current and the linear but once again dependent on ,8. ratio I(Tl)/I(O) as a function of B and for different In contrast to the SCP current, the time dependvalues of the parameter h = T/T,,. ence of the SCL current is independent of the applied strength or the excess carrier density 1.25 provided that the latter is suffi~en~y l~ge.Therate I of increase of the SCL current during the initial period is appreciably higher than that of the SCP current, even for /3= 1. This additional increase is due to the continuous injection of carriers into the crystal when “non-blocking” contacts are used. Similarly the rate of decrease of the SCL current is smaller and the current now decays to the steady state SCL value of 1.125 i,,.(l) From equation (5), the initial value of the SCP current i(0) is independent of the extent of bulk trapping and is given by
1
i(0) = ie(l -l/2/3). At ,3 = 1 we have ~(O)/i~= O-5 which is identical with the initial value of the SCL current. The SCP, however, allows a continuous variation of i(O)/&,in the range 0.5-l depending on the density of excess carriers and the applied field strength. Figure 4 shows the variation of i(0)/io with 8. The ratio of the SGP current at t = Tl I(T,) to the initial current, for various values of the parameter X - -r/T, is also shown in this figure.
t/To
FIG. 5. Variation of the initilplpart of the SCP current with time at fi = 1 h = T/T~as parameter. l
THEORY OF TRANSIENT SPACE-CHARGE PERTURBED CURRENTS IN INSULATORS
The effect of deep trapping on the time dependence of the SCP current is illustrated in Fig. 5. Here i/is [given by equation(S)] at ,8 = 1 is shown for various values of the parameter j3.As X decreases both the number of carriers and the resulting field perturbations are decreased. However, the increase of the current with time persists even for X = 1
647
Figure 6 shows the variation of the transit time
Tl obtained from numerical solutions of equation (8) as a function of ,9 for various values of the parameter A. The transit time of the trailing edge Tl for X g 1 is also shown in this figure.
Acknowledgements-The author wishes to thank Prof. J. C. hDExSON of Imperial College, London, for his encouragement and support during the early part of r;rk. Acknowledgement is made to the Engineer-in-Chief of the British Post Office for permission to publish this
(r ‘;;[
REFERENCES
2. I.0
3. 4.
5. 6. 7. 8.
FIG. 6. Variation of the transit time TX as a function of j for different values of the parameter h. The transit time Ta for I\> 1 is also shown (dotted line).
11. 12.
MOTT N. F. and GURNEY R. W., Electronic Processes in Ionic Crystds (Oxford: Clarendon Press 1940). SPEAR W. E. and MORT J., Proc.Phys. Sot. 81,130 (1963). HECHT K., 2. Phys. 77,235 (1932). SILVER M., SWICORDM., JARNAGINR. C., MANY A., WEISS S. Z. and SIMHONY M., J. Phys. Chews. Solids 23,419 (1962). MANY A., WEISS S. Z. and SIMHONY M., Phys. Rev. 126,1989 (1962). GIBBONSD. J. and PAPAIXUCIS A. C., To be published. MANY A. and RAKAVY G., Phys. Rev. 126, 1980 (1962). HELPRICH W. and MARK P., Z. Phvs. 166. 370 (1962). SIMHONY A. and GORELIK J., J. Phys. Chem. Solids 26,1133 (1965). KEA%NG P. N. and PAPAD,+XISA. C., PYOC. Int. Conf. on the Phys. of Semicond. Paris 1964 (Paris: Dunod) p. 519. SCHWARTZL. M. and HORNIG J. F., J. Phys. Chem. Solids 26,182l (1965). LAMPERT M. A., Rep. Prog. Phys. 27, 329 (1964).