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Investigations of transient space-charge-limited currents at saw-tooth voltage pulses
Solid-State
Electronics
Pergamon
Press 1968. Vol. 11, pp. 577-582.
INVESTIGATIONS LIMITED
CURRENTS
A. G. ZHDAN,
Printed in Great Britain
OF TRANSIENT AT SAW-TOOTH
T. U. MUSABEKOV.
V. B. SANDOMIRSKY,
SPACE-CHARGEVOLTAGE M. I. ELINSON
PULSES and
M. E. CHUGUNOVA Institute
of Radiotechnics (Received
and Electronics, 20 November
Moscow,
U.S.S.R.
1967)
Abstract-A theory of transient SCLC for saw-tooth voltage pulses has been developed on the basis of MANY and RACAVY’S method. (I) Expressions have been obtained which allow the determination of some parameters of the material, particularly the effective mobility of carriers using special points of the transient characteristics. Experimental investigation of the transient characteristics of thin-film In-Cd%SiO,-Al heterojunction diodes have been carried out to verify the theory developed for the saw-tooth voltage pulse. The I-V characteristics observed with the help of a curve-tracer at sufficiently fast linear voltage increase exhibit a region of n-type negative differential resistance. The theory is found to be in satisfactory agreement with the experiment in the regions where the solutions are derived in analytical form. R&mm&-On a developpe une theorie de courant B charge d’espace limitee, SCLC, transitoire relative aux impulsions de tension a dents de scie sur la base de la methode de Many et Recavy”). On a obtenu des expressions qui ont permis de determiner certains parametres du materiau, particulierement la mobilit effective des porteurs en employant des points speciaux dans les caracteristiques transitoires. L’examen experimental des caracteristiques transitoires des diodes d’heterojonctions In-SCd-O,Si-Al a pellicule fine a 6td fait pour verifier la theorie developpee pour les impulsions de tension a dents de scie. Les caracteristiques courant-tension observees a l’aide d’un traceur de courbes a un taux d’augmentation de tension linbire relativement dleve pr&sente une region a resistance diff&entielle negative de type n. On a trouve que la theorie est en bon accord avec I’experience dans les regions oh les solutions sont derivees en forme analytique. Zusammenfassung-Eine Theorie des zeitabhangigen raumladungsbegrenzten Stroms fiir Siigezahn-Spannungsimpulse wurde nach der Methode von Many und Racavy”’ entwickelt. Die erhaltenen Ausdriicke erlauben die Bestimmung einiger Materialparameter, insbesondere die Tragerbeweglichkeit, wobei spezielle Punkte der Zeitcharakteristik verwendet werden. Das Zeitverhalten von In-CdS-SiOa-Al-Diinnschicht-Heteroilbergangen wurde experimentell untersucht urn die fiir Sagezahnimpulse entwickelte Theorie zu verifizieren. Die mit Hilfe eines SBgezahn-Generators bei geniigend schnellem linearem Spannungsanstieg beobachteten StromSpannungs-Kennlinien zeigen ein Gebiet negativen differentielen Widerstands. Die Theorie befindet sich in befriedigender Ubereinstimmung mit dem Experiment in den Gebieten, wo die Losungen in analytischer Form abgeleitet wurden.
INTRODUCTION A. MANY and G. RACAVY’~’ have developed a simplified theory of transient space-charge-limitedcurrents (SCLC) in insulators for the case of rectangular voltage pulses. It would be interesting to carry out an analogous analysis for saw-tooth pulses of applied voltages. The use of such a form of voltage pulse permits us to observe not only the 577
behaviour of current corresponding to the fixed voltage value as it is in the case of rectangular voltage pulses, but the dynamics of full I-V characteristics of SCLC-diodes, as well. Such a method of measurement seems promising in yielding additional information on the properties of solids. Besides, some simplification of the interpretation of the experimental data can be
578 A. C;. ZHDAN,
T. U. MUSABEKOV,
1’. B. SANDOMIRSKY,
expected as a result of using a more simple time dependence of the applied voltage, compared for example, with the sine-shaped law. The purpose of this paper is to give a theoretical analysis of transient SCLC for the case of saw-tooth voltage pulses and to verify the agreement of the theory and experimental data obtained when using thin film metal-semiconductor-insulator--metal hcterojunction structures.
THEORETICAL
2!f+ iT
j(T)= n(s, T) . C(S. 7) +
and initial conditions
I1,
for system
1
E(O,7) = ‘(S, 0) = 0;
03
1E(.S,T) ds
at
4s, (1)= I, ,) at
Here the following
and M. E. CHUGUNOV.4
notation
has been used,
ANALYSIS
The analysis for voltage pulses U(t) = IL/ (U = voltage, 1 = time, a = constant) will bc carried out usingtheschemeof calculation suggested by h,fANY and K.~cAvY.(~) h electron flow through a plane-parallel insulator layer arranged between an injecting cathode (X = 0) and a blocking anode (X = I,) is considered. All the traps in the insulator are supposed to he placed at one discrete energy level and to bc uniformI! distributed in the layer. The problem will be solved within the framework of the approximation of a virtual cathode. The equilibrium bctwecn the conduction band and the traps is assumed to hc established in a short time compared to the transit time. The carrier flow is governed by the equation for the total current, Poisson’s equation, and the continuity equation. These equations, in term of dimensionless variables, may be cspressed as follows,
The boundary are given by,
$1. 1. ELlNSON
5
;o
(I
=
_I K
=
s ::
1.
-T
where y = the absolute electron charge, iV and p = the free carrier concentration and mobilit! respectively, E = the electrical field, I = the total current density (the conductionand displacement-current density), 8-l = the fraction of trapped electrons, K = permittivity. As it is seen from equations (la), the influence of fast trapping is reduced to the normalization of the electron mobility and charge, I+0
@ = $-. 1 +B’
Q” = “a-’
System (1) can bc solved by the method OI characteristics.‘2’ Rlanv_I and Kacavv have S~OWII the flow lines to be the characteristics in this ~1s~‘. The flow lines can be defined by,
There arc two differential characteristics,
relationships
(1)
(1)
cl7
- .i(d
for the-t,
INVESTIGATIONS
OF TRANSIENT
SPACE-CHARGE-LIMITED
of zone I as they all originate from the cathode at the same field ~(0,0) = 0, and hence they do not differ from one another. Zone III inludes all the flow lines originating from the cathode at moment 7 > 0. Zone IV includes the lines originating from the cathode at large values of 7 with the displacement current being neglected in comcurrent. Physically parison to the conduction zone III includes lines of zone IV that are distinguished to make the calculation more simple. The solution of the system (1) yields, j(-r) = -cc~s-~
1 i-r = 2/2 arc cos de j(7,)
(zone I)
(2)
where e = the natural logarithms base. Zone III may be described by means of a system of three usual differential equations; but their analytical solutions cannot be obtained. The solution for zone IV is, j(7)
at T2 > 1.
= - i72,
dr1t,-0 dj,
/
/
“2;
e
----i
= -ed[2(e-l)]
and the current tively,
= - 2/[2(e - l)] e_
I -J ’
r,
1
c
I-
density at the break are, respec(4)
I(Tp) = -eFL. It may be seen from these equations that in the case of fast trapping the curve deformation is reduced only to the break shifting to the larger values of t, while value of I(Tp) does not change. I(Tcp) doesn’t depend on Tp, and hence the carrier mobility because of the fact that I(Ttp) represents in essence the maximum displacement current,
= -5.03;
df,,-o
[
‘/
Ts, = 1,29J(y9
I(Tp)
x
-0)
7r
(3)
The examination has shown that the curve j(T) exhibits a break at point 7i, that may be characterized with different derivative values to the left and to the right, djr
bj B
FIG. 1. Qualitative theoretical curve j(r) in terms of dimensionless variables; curve A: 7 < TV; curve B: 7 2 TV; curve E’ represents the possible kind of time dependence of j at 7 3 TV.
at 0 < 7 < 7r I
= -e
579
-j(T)
ov
5
CURRENTS
z/(e-1) arc.cos e-112 I
= -2.41. The time dependence of SCLC density j(T) is represented in Fig. 1; solid lines A and B correspond to the computed curves j(~) for zones I and IV, the dashed line C is a qualitative dependence j(T) for zone III. In terms of dimensional variables the time of the leading front arrival Tp,
El TV) - -.UCTV> - __ TP, LTP,
It is seen that the value I(T) doesn’t depend on Tp, in the case of the chosen form of the signal U = at. Equations (4) and (5) can be used to verify the accordance between the theory and experiment using the dependence of Tp and I(Ttp) on the rate of voltage pulse amplitude increase. If the theory agrees with the experiment, equations (4) and (5) can be used for determining the value of the effective mobility II* as well. JXPERIMENTAL VERIFXCATION OF THEORY For an experimental investigation a In-CdsSiOx-Al diode structure was used. Monocrystalline
580 A. G. ZHDAN, T. U. MUSABEKOV, V. B. SANDOMIRSKY, M. I. ELINSON and M. E. CHUGUNOVA
CdS films with specific resistivity of lo4 !&cm were deposited in vacuum upon cleavage faces of mica to a thickness of 0.5 p.(3) Two electrodes of indium (of 0.8 mm width) which makes an ohmic contact to CdS with 9 p spacing between them were evaporated upon a CdS film. SiO, films (0.05 - 0.08 p thick)(4) and an aluminium electrode (21 30 p width) were deposited over the top of the structure. The geometry of the system did not differ from a planar one because the SiO, resistance was much larger than that of cadmium sulphide. Such geometry was desirable from the point of view of production convenience as the absence of the lower metallic contact allowed the deposition of epitaxial semiconductor films. In carrying out measurements both contacts of indium were connected together, forming one of the diode electrodes, the upper aluminium film being the second electrode. As has been shown in MULLER and ZULEEG'Spaperc5’ such a structure is a heterojunction in which an excellent rectification has been observed (Fig. 2). In this structure the forward direction corresponds to a positive bias applied to the aluminium electrode. The present measurements of transient SCLC are different from those carried out in.(5) We have employed both a curve tracer with a sinusoidal voltage scanning and a pulse curve tracer as well. It should be noticed that the former introduces a quantitative uncertainty of results on account of unknown time factors. The pulse curve tracer supplied saw-tooth voltage pulses with regulated amplitude (O-50 V), pulse duration (15 psec-10 msec) and pulse repetition frequency (100 c/s-20 kc/s). As a result of pulse investigations I-V characteristics of the In-CdS-SiO,-Al system are found to exhibit systematically an n-type negative differential resistance. Curves represented in Fig. 3 give four distinct regions: (a) an initial jump of current due to paralleled parasitic input capacitance C, z 3 pf of the measuring scheme and geometrical diode capacitance C, N 12 pF ; (b) a region of rapid current rise; (c) a region of current decrease and (d) a region of the second current rise. When comparing Figs. 3 and 1 it is seen that the theoretical curve differs from the experimental one by the lack of minimum. However if the maximum on the experimental curves of Fig. 3 is interpreted as a break at t = Trp it may be seen from-Fig. 4 that the experimental relations I(Tcp-a and T~J-u-~‘~
FIG. 4. (a) Experimental current corresponding to the curve j(r) maximum vs. the rate of voltage increase a. (b) Ty corresponding to the current maximum vs. n. Pulse duration-15 psec, pulse repetition frequency5 kc/s.
are linear according to equations (4) and (5). Moreover, the values K = 14 and CL*= 1.4.10-s cm21 V-set, found from Fig. 4 and equations (4) and (5) are quite reasonable: similar values of K Y 12- 15 for SiO, are referred toliterature;‘4*1) the possibility of low 0 N 10-V and hence small CL*is beyond doubt.(7’ Besides, the experimental and theoretical curves I(t) are analogous in regions t < TV and t > Tqx Therefore the observed curves T(t) may be considered as transient SCLC. Quantitative discrepancies between experimental and theoretical curves, the absence of current minimum on the latter in particular, the difference between the theoretical (e) and err:erimental [5*6 for Fig. 3(b)] values of relations i(Tlp)/l(O) can be accounted ft,r 1;~ the fact that the theoretical model doesn’t take into consideration some peculiarities of the real case: the geometry of the experimental diodes, the regime of periodic pulse train, the existence of 1: :ri’” .i: eributed in energy, the slov, trapping and
INVESTIGATIONS
OF
TRANSIENT
SPACE-CHARGE-LIMITED
the nonuniform space distribution of traps as well. For example, the probable localization of traps near the anode may result in an intensified trapping at t 5 TV and in a decay of the current. Qualitatively this case is illustrated by the dashed curve C’ Fig. 1. On the other hand, an experimental confirmation of the theoretical relations (4) and (5) may mean that more precise theoretical relations taking into consideration the factors mentioned above differ from relations indicated above by small numerical coefficients. In fact a satisfactory quantitative agreement has been obtained between the theoretical curves corresponding to equations (2) and (3) at 0 < t < TCJI and t > Tp, and experimental ones when parameters found from equations (4) and (5) and used in the calculation are corrected according to the experiment. Then equations (4) and (5) can be expressed as, I 12 Ttp=y ” (4a)
J#a
I(Tp) = Z-r”--. 47rL Using equations less form,
(5a)
(2) and (3) we have in dimension-
L’(t) = - -
1
-
(2a)
(0 < t 6 Tp,) and I’(t)
=
-
__9Yp St - Tp2
(t > TV).
(3a)
The value of y is defined from relation cos2 (y/2/2) = I( T)jI(O) = Z; Ts, and Z being found from the experiment. The change of pulse repetition frequency f,, modifies the I-V characteristics [Figs. 3(a)-(d)]. At high f, and long time pulse curves I(t) exhibit a second and sometimes a third maximum [Fig. 3(d)], the nature of which is not yet clear. It should be noted that curves I(t) and the values of Y and Z give a better agreement with the theory when f, decreases and this is quite natural because the theory has been developed for the case of single pulses. Although the current maximum in the region f,, = 1a15 - 5 kc/s is
CURRENTS
1,
581
r3ec
FIG. 5. Theoretical (solid curves) and experimental (dots) curves Z(t)/Z(O). The experimental value of Z(0) has been found from extrapolation of an initial part of the characteristic [Fig. 3(b)] up to the point of intersection with the ordinates axis. The experimental value of 2=5,6 (y = 1.4) was used for the calculation of the theoretical curve.
always observed, we can believe that in the case of very short pulses and small f, when the trapping is poor the curve I(t) with a maximum will be transformed into a curve with a break according to the theory. Then it is a real p but not an effective value of mobility that will appear in equation (4). The value of 0 for traps ‘working’ at a given frequency can be found when comparing the real TV and CL* according to ‘long’ time pulses. An information about characteristical parameters of dominating traps can be received from the temperature and frequency dependencies of 0. The transition from the curve with a maximum into one with a break is a suitable criterion for ‘switching off’ the traps. It should be stressed that the I-V characteristics with a current maximum may be obtained when a curve tracer with a sinusoidal voltage scanning is used provided the rectified sinusoidal signal frequency is sufficiently high. (> 100 c/s). These characteristics are very much alike to the characteristics that have been observed for the Meoxide-Me structures by KREYNINA, HICKMOTT et aZ.@) This apparently means that in some cases the behaviour of such systems can be interpreted in terms of the mechanism discussed in this paper. It should be taken into consideration that the capacitance of the structure can be wholly conditioned
582 A. G. ZHDAN,
T. U. MUSABEKOV,
V. B. SANDOMIRSKY,
by the thickness of a very thin region of the insulator near one of the metal contacts. However, a full understanding of the phenomenon requires detailed investigation of the frequency-response characteristics of such structures. Acknowledgement-The authors would like to thank V. Philippov for his help in performing the experiment.
REFERENCES 1. A. MANY and G. RACAVY, Phys. Rev. 126, (6), 1980 (1962).
M. I. ELINSON
and M. E. CHUGUN0V.X
2. 1. L. BEREZINand N. P. ZHIDKOV, i\fetody z:\*chislenii, II, p. 461, GIFML, Moscow (1960). 3. A. G. ZHD.AN. R. N. SIIEFTAL, M. 15 CHL!GUNOV.~ and M. I. ELINSON, Radiotekh. Elektron. 11, (8\, 1536 (1960). 4. A. G. ZIIDAN, M. E. CHLGUNOVX and M. I. ELINSON, Radiotekh. Hektron. 18, 305 (1968). 5. R. S. ~TULLERand R. ZULEEC, J. appl. Phys. 35, (5). 1550 (1964). 6. H. HIROSE and J. WADA, Jap. Jnl appl. Phys. 3, (3), 179 (1964): Jap. Jnl appl. Pkys. 4, (9), 639 (1965). 7. K. S. MULLER, Solid-St. Electron. 6, (I), 25 (1963). 8. RI. I. ELINSON, Collected articles. Voprosyplenochnoi elektroniki, p. 174, Soviet Radio, Moscow (1966).
Electronics
Pergamon
Press 1968. Vol. 11, pp. 577-582.
INVESTIGATIONS LIMITED
CURRENTS
A. G. ZHDAN,
Printed in Great Britain
OF TRANSIENT AT SAW-TOOTH
T. U. MUSABEKOV.
V. B. SANDOMIRSKY,
SPACE-CHARGEVOLTAGE M. I. ELINSON
PULSES and
M. E. CHUGUNOVA Institute
of Radiotechnics (Received
and Electronics, 20 November
Moscow,
U.S.S.R.
1967)
Abstract-A theory of transient SCLC for saw-tooth voltage pulses has been developed on the basis of MANY and RACAVY’S method. (I) Expressions have been obtained which allow the determination of some parameters of the material, particularly the effective mobility of carriers using special points of the transient characteristics. Experimental investigation of the transient characteristics of thin-film In-Cd%SiO,-Al heterojunction diodes have been carried out to verify the theory developed for the saw-tooth voltage pulse. The I-V characteristics observed with the help of a curve-tracer at sufficiently fast linear voltage increase exhibit a region of n-type negative differential resistance. The theory is found to be in satisfactory agreement with the experiment in the regions where the solutions are derived in analytical form. R&mm&-On a developpe une theorie de courant B charge d’espace limitee, SCLC, transitoire relative aux impulsions de tension a dents de scie sur la base de la methode de Many et Recavy”). On a obtenu des expressions qui ont permis de determiner certains parametres du materiau, particulierement la mobilit effective des porteurs en employant des points speciaux dans les caracteristiques transitoires. L’examen experimental des caracteristiques transitoires des diodes d’heterojonctions In-SCd-O,Si-Al a pellicule fine a 6td fait pour verifier la theorie developpee pour les impulsions de tension a dents de scie. Les caracteristiques courant-tension observees a l’aide d’un traceur de courbes a un taux d’augmentation de tension linbire relativement dleve pr&sente une region a resistance diff&entielle negative de type n. On a trouve que la theorie est en bon accord avec I’experience dans les regions oh les solutions sont derivees en forme analytique. Zusammenfassung-Eine Theorie des zeitabhangigen raumladungsbegrenzten Stroms fiir Siigezahn-Spannungsimpulse wurde nach der Methode von Many und Racavy”’ entwickelt. Die erhaltenen Ausdriicke erlauben die Bestimmung einiger Materialparameter, insbesondere die Tragerbeweglichkeit, wobei spezielle Punkte der Zeitcharakteristik verwendet werden. Das Zeitverhalten von In-CdS-SiOa-Al-Diinnschicht-Heteroilbergangen wurde experimentell untersucht urn die fiir Sagezahnimpulse entwickelte Theorie zu verifizieren. Die mit Hilfe eines SBgezahn-Generators bei geniigend schnellem linearem Spannungsanstieg beobachteten StromSpannungs-Kennlinien zeigen ein Gebiet negativen differentielen Widerstands. Die Theorie befindet sich in befriedigender Ubereinstimmung mit dem Experiment in den Gebieten, wo die Losungen in analytischer Form abgeleitet wurden.
INTRODUCTION A. MANY and G. RACAVY’~’ have developed a simplified theory of transient space-charge-limitedcurrents (SCLC) in insulators for the case of rectangular voltage pulses. It would be interesting to carry out an analogous analysis for saw-tooth pulses of applied voltages. The use of such a form of voltage pulse permits us to observe not only the 577
behaviour of current corresponding to the fixed voltage value as it is in the case of rectangular voltage pulses, but the dynamics of full I-V characteristics of SCLC-diodes, as well. Such a method of measurement seems promising in yielding additional information on the properties of solids. Besides, some simplification of the interpretation of the experimental data can be
578 A. C;. ZHDAN,
T. U. MUSABEKOV,
1’. B. SANDOMIRSKY,
expected as a result of using a more simple time dependence of the applied voltage, compared for example, with the sine-shaped law. The purpose of this paper is to give a theoretical analysis of transient SCLC for the case of saw-tooth voltage pulses and to verify the agreement of the theory and experimental data obtained when using thin film metal-semiconductor-insulator--metal hcterojunction structures.
THEORETICAL
2!f+ iT
j(T)= n(s, T) . C(S. 7) +
and initial conditions
I1,
for system
1
E(O,7) = ‘(S, 0) = 0;
03
1E(.S,T) ds
at
4s, (1)= I, ,) at
Here the following
and M. E. CHUGUNOV.4
notation
has been used,
ANALYSIS
The analysis for voltage pulses U(t) = IL/ (U = voltage, 1 = time, a = constant) will bc carried out usingtheschemeof calculation suggested by h,fANY and K.~cAvY.(~) h electron flow through a plane-parallel insulator layer arranged between an injecting cathode (X = 0) and a blocking anode (X = I,) is considered. All the traps in the insulator are supposed to he placed at one discrete energy level and to bc uniformI! distributed in the layer. The problem will be solved within the framework of the approximation of a virtual cathode. The equilibrium bctwecn the conduction band and the traps is assumed to hc established in a short time compared to the transit time. The carrier flow is governed by the equation for the total current, Poisson’s equation, and the continuity equation. These equations, in term of dimensionless variables, may be cspressed as follows,
The boundary are given by,
$1. 1. ELlNSON
5
;o
(I
=
_I K
=
s ::
1.
-T
where y = the absolute electron charge, iV and p = the free carrier concentration and mobilit! respectively, E = the electrical field, I = the total current density (the conductionand displacement-current density), 8-l = the fraction of trapped electrons, K = permittivity. As it is seen from equations (la), the influence of fast trapping is reduced to the normalization of the electron mobility and charge, I+0
@ = $-. 1 +B’
Q” = “a-’
System (1) can bc solved by the method OI characteristics.‘2’ Rlanv_I and Kacavv have S~OWII the flow lines to be the characteristics in this ~1s~‘. The flow lines can be defined by,
There arc two differential characteristics,
relationships
(1)
(1)
cl7
- .i(d
for the-t,
INVESTIGATIONS
OF TRANSIENT
SPACE-CHARGE-LIMITED
of zone I as they all originate from the cathode at the same field ~(0,0) = 0, and hence they do not differ from one another. Zone III inludes all the flow lines originating from the cathode at moment 7 > 0. Zone IV includes the lines originating from the cathode at large values of 7 with the displacement current being neglected in comcurrent. Physically parison to the conduction zone III includes lines of zone IV that are distinguished to make the calculation more simple. The solution of the system (1) yields, j(-r) = -cc~s-~
1 i-r = 2/2 arc cos de j(7,)
(zone I)
(2)
where e = the natural logarithms base. Zone III may be described by means of a system of three usual differential equations; but their analytical solutions cannot be obtained. The solution for zone IV is, j(7)
at T2 > 1.
= - i72,
dr1t,-0 dj,
/
/
“2;
e
----i
= -ed[2(e-l)]
and the current tively,
= - 2/[2(e - l)] e_
I -J ’
r,
1
c
I-
density at the break are, respec(4)
I(Tp) = -eFL. It may be seen from these equations that in the case of fast trapping the curve deformation is reduced only to the break shifting to the larger values of t, while value of I(Tp) does not change. I(Tcp) doesn’t depend on Tp, and hence the carrier mobility because of the fact that I(Ttp) represents in essence the maximum displacement current,
= -5.03;
df,,-o
[
‘/
Ts, = 1,29J(y9
I(Tp)
x
-0)
7r
(3)
The examination has shown that the curve j(T) exhibits a break at point 7i, that may be characterized with different derivative values to the left and to the right, djr
bj B
FIG. 1. Qualitative theoretical curve j(r) in terms of dimensionless variables; curve A: 7 < TV; curve B: 7 2 TV; curve E’ represents the possible kind of time dependence of j at 7 3 TV.
at 0 < 7 < 7r I
= -e
579
-j(T)
ov
5
CURRENTS
z/(e-1) arc.cos e-112 I
= -2.41. The time dependence of SCLC density j(T) is represented in Fig. 1; solid lines A and B correspond to the computed curves j(~) for zones I and IV, the dashed line C is a qualitative dependence j(T) for zone III. In terms of dimensional variables the time of the leading front arrival Tp,
El TV) - -.UCTV> - __ TP, LTP,
It is seen that the value I(T) doesn’t depend on Tp, in the case of the chosen form of the signal U = at. Equations (4) and (5) can be used to verify the accordance between the theory and experiment using the dependence of Tp and I(Ttp) on the rate of voltage pulse amplitude increase. If the theory agrees with the experiment, equations (4) and (5) can be used for determining the value of the effective mobility II* as well. JXPERIMENTAL VERIFXCATION OF THEORY For an experimental investigation a In-CdsSiOx-Al diode structure was used. Monocrystalline
580 A. G. ZHDAN, T. U. MUSABEKOV, V. B. SANDOMIRSKY, M. I. ELINSON and M. E. CHUGUNOVA
CdS films with specific resistivity of lo4 !&cm were deposited in vacuum upon cleavage faces of mica to a thickness of 0.5 p.(3) Two electrodes of indium (of 0.8 mm width) which makes an ohmic contact to CdS with 9 p spacing between them were evaporated upon a CdS film. SiO, films (0.05 - 0.08 p thick)(4) and an aluminium electrode (21 30 p width) were deposited over the top of the structure. The geometry of the system did not differ from a planar one because the SiO, resistance was much larger than that of cadmium sulphide. Such geometry was desirable from the point of view of production convenience as the absence of the lower metallic contact allowed the deposition of epitaxial semiconductor films. In carrying out measurements both contacts of indium were connected together, forming one of the diode electrodes, the upper aluminium film being the second electrode. As has been shown in MULLER and ZULEEG'Spaperc5’ such a structure is a heterojunction in which an excellent rectification has been observed (Fig. 2). In this structure the forward direction corresponds to a positive bias applied to the aluminium electrode. The present measurements of transient SCLC are different from those carried out in.(5) We have employed both a curve tracer with a sinusoidal voltage scanning and a pulse curve tracer as well. It should be noticed that the former introduces a quantitative uncertainty of results on account of unknown time factors. The pulse curve tracer supplied saw-tooth voltage pulses with regulated amplitude (O-50 V), pulse duration (15 psec-10 msec) and pulse repetition frequency (100 c/s-20 kc/s). As a result of pulse investigations I-V characteristics of the In-CdS-SiO,-Al system are found to exhibit systematically an n-type negative differential resistance. Curves represented in Fig. 3 give four distinct regions: (a) an initial jump of current due to paralleled parasitic input capacitance C, z 3 pf of the measuring scheme and geometrical diode capacitance C, N 12 pF ; (b) a region of rapid current rise; (c) a region of current decrease and (d) a region of the second current rise. When comparing Figs. 3 and 1 it is seen that the theoretical curve differs from the experimental one by the lack of minimum. However if the maximum on the experimental curves of Fig. 3 is interpreted as a break at t = Trp it may be seen from-Fig. 4 that the experimental relations I(Tcp-a and T~J-u-~‘~
FIG. 4. (a) Experimental current corresponding to the curve j(r) maximum vs. the rate of voltage increase a. (b) Ty corresponding to the current maximum vs. n. Pulse duration-15 psec, pulse repetition frequency5 kc/s.
are linear according to equations (4) and (5). Moreover, the values K = 14 and CL*= 1.4.10-s cm21 V-set, found from Fig. 4 and equations (4) and (5) are quite reasonable: similar values of K Y 12- 15 for SiO, are referred toliterature;‘4*1) the possibility of low 0 N 10-V and hence small CL*is beyond doubt.(7’ Besides, the experimental and theoretical curves I(t) are analogous in regions t < TV and t > Tqx Therefore the observed curves T(t) may be considered as transient SCLC. Quantitative discrepancies between experimental and theoretical curves, the absence of current minimum on the latter in particular, the difference between the theoretical (e) and err:erimental [5*6 for Fig. 3(b)] values of relations i(Tlp)/l(O) can be accounted ft,r 1;~ the fact that the theoretical model doesn’t take into consideration some peculiarities of the real case: the geometry of the experimental diodes, the regime of periodic pulse train, the existence of 1: :ri’” .i: eributed in energy, the slov, trapping and
INVESTIGATIONS
OF
TRANSIENT
SPACE-CHARGE-LIMITED
the nonuniform space distribution of traps as well. For example, the probable localization of traps near the anode may result in an intensified trapping at t 5 TV and in a decay of the current. Qualitatively this case is illustrated by the dashed curve C’ Fig. 1. On the other hand, an experimental confirmation of the theoretical relations (4) and (5) may mean that more precise theoretical relations taking into consideration the factors mentioned above differ from relations indicated above by small numerical coefficients. In fact a satisfactory quantitative agreement has been obtained between the theoretical curves corresponding to equations (2) and (3) at 0 < t < TCJI and t > Tp, and experimental ones when parameters found from equations (4) and (5) and used in the calculation are corrected according to the experiment. Then equations (4) and (5) can be expressed as, I 12 Ttp=y ” (4a)
J#a
I(Tp) = Z-r”--. 47rL Using equations less form,
(5a)
(2) and (3) we have in dimension-
L’(t) = - -
1
-
(2a)
(0 < t 6 Tp,) and I’(t)
=
-
__9Yp St - Tp2
(t > TV).
(3a)
The value of y is defined from relation cos2 (y/2/2) = I( T)jI(O) = Z; Ts, and Z being found from the experiment. The change of pulse repetition frequency f,, modifies the I-V characteristics [Figs. 3(a)-(d)]. At high f, and long time pulse curves I(t) exhibit a second and sometimes a third maximum [Fig. 3(d)], the nature of which is not yet clear. It should be noted that curves I(t) and the values of Y and Z give a better agreement with the theory when f, decreases and this is quite natural because the theory has been developed for the case of single pulses. Although the current maximum in the region f,, = 1a15 - 5 kc/s is
CURRENTS
1,
581
r3ec
FIG. 5. Theoretical (solid curves) and experimental (dots) curves Z(t)/Z(O). The experimental value of Z(0) has been found from extrapolation of an initial part of the characteristic [Fig. 3(b)] up to the point of intersection with the ordinates axis. The experimental value of 2=5,6 (y = 1.4) was used for the calculation of the theoretical curve.
always observed, we can believe that in the case of very short pulses and small f, when the trapping is poor the curve I(t) with a maximum will be transformed into a curve with a break according to the theory. Then it is a real p but not an effective value of mobility that will appear in equation (4). The value of 0 for traps ‘working’ at a given frequency can be found when comparing the real TV and CL* according to ‘long’ time pulses. An information about characteristical parameters of dominating traps can be received from the temperature and frequency dependencies of 0. The transition from the curve with a maximum into one with a break is a suitable criterion for ‘switching off’ the traps. It should be stressed that the I-V characteristics with a current maximum may be obtained when a curve tracer with a sinusoidal voltage scanning is used provided the rectified sinusoidal signal frequency is sufficiently high. (> 100 c/s). These characteristics are very much alike to the characteristics that have been observed for the Meoxide-Me structures by KREYNINA, HICKMOTT et aZ.@) This apparently means that in some cases the behaviour of such systems can be interpreted in terms of the mechanism discussed in this paper. It should be taken into consideration that the capacitance of the structure can be wholly conditioned
582 A. G. ZHDAN,
T. U. MUSABEKOV,
V. B. SANDOMIRSKY,
by the thickness of a very thin region of the insulator near one of the metal contacts. However, a full understanding of the phenomenon requires detailed investigation of the frequency-response characteristics of such structures. Acknowledgement-The authors would like to thank V. Philippov for his help in performing the experiment.
REFERENCES 1. A. MANY and G. RACAVY, Phys. Rev. 126, (6), 1980 (1962).
M. I. ELINSON
and M. E. CHUGUN0V.X
2. 1. L. BEREZINand N. P. ZHIDKOV, i\fetody z:\*chislenii, II, p. 461, GIFML, Moscow (1960). 3. A. G. ZHD.AN. R. N. SIIEFTAL, M. 15 CHL!GUNOV.~ and M. I. ELINSON, Radiotekh. Elektron. 11, (8\, 1536 (1960). 4. A. G. ZIIDAN, M. E. CHLGUNOVX and M. I. ELINSON, Radiotekh. Hektron. 18, 305 (1968). 5. R. S. ~TULLERand R. ZULEEC, J. appl. Phys. 35, (5). 1550 (1964). 6. H. HIROSE and J. WADA, Jap. Jnl appl. Phys. 3, (3), 179 (1964): Jap. Jnl appl. Pkys. 4, (9), 639 (1965). 7. K. S. MULLER, Solid-St. Electron. 6, (I), 25 (1963). 8. RI. I. ELINSON, Collected articles. Voprosyplenochnoi elektroniki, p. 174, Soviet Radio, Moscow (1966).