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Journal of Electrostatics, 2 (1976) 59--78 59 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands EFFECTS OF EXCITATION RISE-TIME AND CHARGE INJECTION CONDITIONS ON THE TRANSIENT FIELD AND CHARGE BEHAVIOR FOR UNIPOLAR CONDUCTION MARKUS ZAHN, SHING-CHONG PAO and CHEUNG FUNG TSANG Department of Electrical Engineering, University of Florida, Gainesville, Florida 32611 (USA) (Received January 1, 1975; in revised form August 26, 1975) Summary Recent analysis using a unipolar ion mobility conduction model is continued and extended by taking into account the experimental realities of excitation rise-time and charge injection threshold conditions. The transient behavior of the bulk electric field and space charge distributions as a function of time and position as well as the resulting time dependence of the terminal voltage and current are found for an imposed voltage or current excitation with non-zero rise-time, contrasting the behavior between field or charge specification at the emitter electrode with charge injection onset once the emitter electric field exceeds a threshold value. Representative plots of field and charge distributions as well as voltage and current time dependence for various excitations, rise-times, and boundary conditions are presented. Important time constants which can be correlated to experiments under these various conditions are tabulated and where possible, results are presented in closed form. 1. I n t r o d u c t i o n Recent analysis has used a unipolar ion mobility conduction law to model electrical conduction mechanisms in electrical insulators [1,2]. The analysis, also appropriate for solid state conduction [ 3], neglects diffusion which is allowed if applied voltages are much greater than the thermal voltage (v >> kT/q) and if the charge density gradients are not very large [4]. Under these assumptions, this recent work solved for the transient time and spatial responses of the electric field and space charge distributions as well as the terminal voltage--current characteristics for any type of applied excitation and any physically allowed initial and b o u n d a r y conditions. Examples presented in those works [1,2] include step changes in excitation of applied current or voltage to an initially unexcited or pre-stressed system with specification of the electric field at the emitter electrode as a b o u n d a r y condition. Space charge limited conditions are treated when the emitter electric field is zero, and as a model for a system where charge injection occurs at some threshold field strength, the emitter electric field was also allowed to be 60 clamped at some non-zero value. Specific applications include correlation to Kerr electro-optic measurements where precision electric field measurements are made optically for those dielectrics (typically nitrobenzene) which become significantly birefringent when stressed b y high voltages [5,6]. We disagree with other related work which has argued that specification of the electric field at the injecting electrode as a b o u n d a r y condition is n o t valid, b u t rather that the charge density must be specified [7,8]. The mathematics requires kn.owing the emitter electric field either because it is specified or calculated from a model relating the emitter charge density to the electric field. In earlier work, the specific case where the emitter charge density is proportional to the emitter electric field was treated, which would be appropriate for conduction b y particulate impurities charged u p o n contact with the electrodes. We do agree that specification of the emitter charge density is often more convenient as solutions can be obtained for all values of charge density (0 ~< q < o~) which however only corresponds to a finite range for the emitter electric field (0 <~Eo <~ V/d). For specification of E0 outside these bounds no physically realizable solution is possible. Thus, the b o u n d a r y condition to be used depends on what is known about the physical nature of the emitting electrode or, for analytical purposes, on convenience. As we will show here, for many cases specification of the emitter electric field allows for closed form solutions over wide intervals, while with the emitter charge density specified, numerical techniques must often be used. The work presented here is an extension and continuation of this recent analysis. We continue our practice of considering voltage or current excitations where the analytically simpler case of current specification is usually appropriate for semiconductor applications, while for dielectric liquids, the voltage is usually imposed. For completeness, we repeat the cases of step voltage and current excitations, now specifying the charge density at the emitter electrode rather than the electric field. We also examine the effects of excitations which have non-zero rise-times and tabulate important time constants for various rise-times and b o u n d a r y conditions. These time constants are important to related experimental measurements of ionic mobility by time-of-flight techniques [3]. Where possible, analysis will be presented in closed form. In our earlier examples [1,2] with a step voltage excitation and specification of the emitter electric field t w o characteristic curves emanate from the origin. These t w o curves were due to the field discontinuity at x = 0, t = 0, where approaching the origin along the line t = 0, the field was given b y the voltage divided b y the electrode spacing, while approaching the origin along x = 0, the electric field was clamped at some lower value. Here the electric field is continuous at the origin for all cases, so only one characteristic curve emanates from there. 61 2. Review of past analysis We consider again a parallel plate geometry, where the electrode at x = 0 is a source of positive ions with mobility p in the dielectric. All physical parameters are assumed constant, and with the neglect of fringing so that the electric field and current are in the x-direction and only depend on the coordinate x, the field equations reduce to: ~E e 0x ~Jc (1) =q aq 4---=0 ~x ~t (2) Jc = q u E (3) z f Edx = v (4) v 0 where e is the fluid permittivity, E is the electric field in the x-direction, q is the charge density, Jc is the c o n d u c t i o n current, and v is the voltage. Substituting eqns. (1) and (3) into eqn. (2), we obtain: aE -- at + pE ~E 8x = J(t)/e (5) where J ( t ) is the current per unit electrode area flowing in the wires to the electrodes due to b o t h conduction current in the fluid bulk given by eqn. (3) as well as displacement current due to the time rate of change of the surface charge on the electrode. By integrating eqn. (5) and using eqn. (4) we obtain the voltage--current relation dv - - dt + (p/2)[E:(l,t) - - E2(0,t)] = J ( t ) l / e (6) Note that the first term on the left yields the familiar capacitive current, while the second term is the modification due to the presence of bulk charge with mobility p. Equation (5) falls into the class of quasi-linear partial differential equations of first order which is solved in the standard way of Lagrange and the m e t h o d of characteristics using the subsidiary equations [ 9] dt -- 1 dx = ttE edE - J(t) (7) U p o n integration, we obtain the solutions: E = Q(t)/e + e, (8) 62 q = qo/[1 + gqo(t -- t0)/e] (9) t x = gtc, + (g/e) ; Q(t')dt' + c2 (10) o where Q(t) is the total charge per unit electrode area flowing through the electrodes t t Q(t) = f J(t')dt' = (e/l){v + (g/2) ; [E2(l,t ') -- E2(O,t')]dt ' } o (11) o Equations (8) and (9) give the electric field and charge density on the family of curves in the x--t planes with the trajectories given by eqn. (10). The parameter q0 is the charge density on a particular trajectory at any time t = to. Usually to is picked as the time where a particular characteristic trajectory begins at a boundary, usually t = 0 or x = 0. The parameters cl and c2 are constants of integration to be determined from initial and boundary conditions. These constants are invariant on any characteristic curve but generally differ between curves. As discussed in our earlier work, if the parameter Q(t) is known, it is a simple straightforward procedure to solve for E(x, t) and q(x,t) from eqns. (8)--(10). From eqn. (11), Q(t) is determined by a simple integration if J(t) is known. However, if the voltage is imposed, then the last equality of eqn. (11) must be used. Generally, Q(t) cannot be found in closed form this way because the electric field at the electrodes is n o t k n o w n until the total solution is obtained, so finite difference schemes must be used. In the following section, we will use the results of eqns. (8)--(11) to consider excitations of step voltages or currents with the boundary condition at x = 0 being the specification of the charge density. Then we will generalize past analysis by including voltage and current excitations with non-zero rise times and specification of the electric field at the emitter electrode. 3. Specification of the charge density at the emitting electrode 3.A Step current excitation We assume an input J(t) = t0 J0 t< 0 (12) t>0 Then from eqn. (11) Q(t) = Jot t> 0 We introduce the normalized quantities (13) 63 = x/l, ~ = [eMJoll'~E, ~" = [ g J o / e l ] v ' t , Y = (eu/Jol3)'av, = [pl/eJoll~q (14) The initial and b o u n d a r y conditions of interest are E(.~,~" = 0) = 0, ~(E = 0,~') = qo (15) All t h a t remains to be done is to find the parameters cl and c2 in eqns. (8) and (10). Those characteristic curves which emanate from the ~ = 0 boundary, as in Fig. 1, have zero values for E(E,~ = 0) and Q(t = 0), so that cl = 0. Similarly I.O 0.8 / 0.6 0.4 0.2 0 0 0.5 1.0 [5 20 i Fig. 1. T r a j e c t o r i e s o f c h a r a c t e r i s t i c curves in n o r m a l i z e d x - - t space for a s t e p c u r r e n t e x c i t a t i o n w i t h t h e n o r m a l i z e d charge d e n s i t y specified at t h e e m i t t i n g e l e c t r o d e as q'0 -- 1.O q0 = 0, so from eqn. (9) the charge density remains zero along those curves. Using eqn. (13) in eqns. (8) and (10) yields the solutions in this region as: E - - E 0 =t'2/2, E=~, ~ =0 (16) where c2 = x0 is the position where a characteristic curve begins at ~ = 0. This region extends over the time interval of 0 < ~ < 2 ~ which is obtained by solving for the time the trajectory in eqn. (16) which starts at x0 = 0 and reaches the other electrode at E = 1. The solutions of eqn. (16) are valid to the left of this demarcation curve, exemplified by the heavy curve in Fig. 1. To the right of this demarcation curve, characteristic curves begin at the = 0 b o u n d a r y with the charge density specified as ~(E = 0,~) = q0. From eqn. (9), we immediately know the charge density along these curves as = 5o/[1 + qo(}'-- }~o)] (17) where To is the time where the characteristic curve begins at E = 0. However, we cannot obtain the equations of the characteristic curves given by eqn. (10) until we find the parameter c,. This parameter can be found from eqn. (8) 64 when the solution for the emitter electric field is known. FindingE(~ = 0,~) is easy when the current is constant, as eqn. (5) at ~ = 0 in normalized form: ct~(~ = 0,~) d{ + ~oE(~ = 0,~) = 1 (18) has solution E(~ = 0,~) = [1 -- exp(--~ot)]/qo (19) Then using eqn. (19) we find cl from eqn. (8), which when substituted into eqn. (10) yields the trajectories and electric field as = (t'-- t'o)[(~-- to)/2 + (1 -- exp(--qot'o))/qo] (20) = t - - t'o + (1 -- exp(--~oto))/~o (21) Note that as }'o > > 1/~o, the steady state solutions are tim /~(~) = [2~ + 1 1 ~ ] '/2 ~-~ ~(~) = iI[2~ * II~] '~ (22) Typical results for q0 = 1 are plotted in Fig. 1 for the characteristic trajectories, and for the time and spatial dependences of the electric field and charge density in Fig. 2. The heavy line emanating from the origin separates those curves emanating from the ~= 0 boundary and from the ~ = 0 boundary. The curves to the left of this demarcation curve obey eqn. (16) while to the right they obey eqn. (20). Note that the demarcation curve reaches the boundary ~ = 1 at T = t~ = 2 '/2,independent of the boundary condition at = 0. Figures 2(a) and (b) show the electric field and space charge distributions as a function of ~ for various times. For ~ < t'l, at a fixed time, the electric field distribution has curvature due to the space charge injected over the limited spatial region below the demarcation curve. The charge is zero above the demarcation curve, resulting in the zero slopes shown in this region in Fig. 2(a). At }'= ~,, space charge fills the volume, and as time increases further, the electric field increases as in eqn. (21), finally reaching the steady state of eqn. (22). Figures 2(c) and (d) show the electric field and space charge distributions at various positions as a function of time. Here to the left of the demarcation curve, the electric field increases linearly with time with no space charge as given by eqn. (16). To the right of the demarcation curve the field and charge distributions are given by eqns. (17) and (21). Once the electric field distribution is known, the terminal voltage can be found from eqn. (4). Figure 3 plots this voltage versus time for various values of q0- Note that for ~ < ~,, we can calculate the voltage, in closed form using eqn. (6), to be ~(~) = ~-- t3/6 + ( 1 / 4 ~ ) [ 2 ~ o ~ - - exp(--2~ot) + 4exp(--~o}') -- 3] 0 ~< ~ For ~ > ~,, numerical techniques are necessary. t~ (23) 65 1.5 1.5 io I0 05 o | o 025 05 075 ID (c) (o) 1.0 '~'00 '~ J fQ'~ 075 075 !" "'-2o ~ BUr2 : o , , ~ Oglo 0.5 O2'5 on 0 0.2,5 1" 05 0 075 0 1.0 1.0 2.0 3.0 40 (d) (b) Fig. 2. Normalized electric field and space charge distributions as functions of normalized t i m e and position for a step current e x c i t a t i o n with q'0 = 1.0. 1.5 ~°,a5~7 ~ 1.0 io. 5 1.0 ~ 0.5 0 0 I I I IO 20 3.0 4D Fig. 3. Normalized terminal voltage for a step current e x c i t a t i o n as a f u n c t i o n o f normalized t i m e for various values of normalized e m i t t e r charge densities, q0. 66 3 . B S t e p voltage e x c i t a t i o n For a step current excitation, the analysis was straightforward because the parameter Q ( t ) was known. For a step voltage excitation, the parameter Q ( t ) is not easily obtainable because it depends on the electric fields at the electrodes as given by eqn. (11). Despite this difficulty, with the b o u n d a r y condition of a constant emitter electric field, large parts of the solution were obtained in closed form. Here, with specification of the charge density at the emitter at x = 0, we have not been able to obtain any part of the solution in closed form so numerical techniques are used throughout. With a step voltage applied: v(t) = i Vo t>0 0 t<O (24) initially the field is uniform with no space charge E ( x , t = O) = Vo/l, (25) q ( x , t = O) = 0 We also maintain the constant charge density boundary condition at the emitter electrode (26) q ( x = O,t) = qo We now normalize all variables to the voltage as = x/l, E = El/Vo, t = u V o t / 2 l 2, J = 2Jl3/epV~, 5 = ql:/eVo (27) All characteristic curves emanating from the ~= 0 boundary have 5 = 0, while from eqns. (9) and (26) those emanating from the ~ = 0 boundary at a time t0 have 5 = 50/[1 + 2~0(F-- F0)] (28) However, we cannot calculate the characteristic trajectories directly until the parameter cl is evaluated, which as previously stated cannot be obtained until E(~ = 0,F) and Q ( t ) are known. Unlike the development for a step current in eqn. (18) the equation for J~(~ = 0,T) obtained from eqns. (5) and (6): dE(~ = 0,~) dF + 250E(£ = 0,~) = [E~(~ = 1,t) --/~2(~ = 0,~)] (29) cannot be directly solved until the electric field at ~ = I is known, although finite difference methods can be easily applied to obtain solutions. Analogously to Fig. 1 a demarcation curve emanates from the origin. To the left of this curve the charge density is zero while to the right the charge density is given by eqn. (28). Figure 4 shows typical plots of the time and spatial dependences of the electric field and space charge distributions for q0 = 10. An important time constant of the system is the time T1 for the demarcation curve to reach = 1. For space charge limited conditions (q0 -* 0% ~(~ = 0,F) -* 0), past work has shown that T1 = 1 - - exp(--1A) ~ 0.3935. As q0 -+ 0 so that E(~ = 0,~) -~ 1, 67 ] qo ~ I 0 . 0 .,~ 7, - j 15 1.5 £,i .o 0.9 08 I .25 1.25 O? 06 ID 7_- -- - I0 - 05 0 4 - 0.3 075 ).75 0 . 2 - - ).5 0.5 Ol - - I .05 0.25 - - L25 O0 , OJ [ll 0 0.2 0'.4 OJ6 o's fo o:,, 0 I 0.'8 20 1.6 i" (o) (c) ro ;'=0,0 j e1 D04- 6 .01 0 2 ~ 4 J 2¸ ............. OI 0 I, 0.25 I , Q5 I , 0.75 T'..._ I.O . .3- 0 0 04 Q8 [2 16 T (b) (d) Fig. 4. N o r m a l i z e d electric field a n d s p a c e charge d i s t r i b u t i o n s as f u n c t i o n s o f t i m e a n d p o s i t i o n f o r a s t e p voltage e x c i t a t i o n w i t h q'0 = 10. we have that ~1 -~ 0.5. For charge densities in between these values, ~, is bounded b y these t w o values as listed in Table 1 for various values o f q0. Figure 5 shows the current per unit electrode area as calculated from eqn. (6) with the time derivative of the voltage being zero for ~r> 0, b u t an impulse at ~-- 0 due to the instantaneous capacitive charging. Note that for finite emitter charge densities, besides the impulse current, the current density approaches zero at T~ 0, while for space charge limited conditions, q0 = oo, the normalized current density jumps to unity. 68 TABLE 1 Characteristic time constant ~ for a step voltage excitation for various values of emitter charge densities r, 0.0 0.01 0.1 0.25 0.5 0.7 1.0 1.5 2.5 10.0 oo 0.5000 0.4992 0.4923 0.4826 0.4700 0.4621 0.4527 0.4415 0.4277 0.4034 0.3935 2.5 2o 25 15 ~5 J lOI.O o 7 - - 0,5 o25 0 0 Q25 0.5 0:75 IO 125 Fig. 5. Normalized terminal current for a step voltage excitation as a function of normalized time for various emitter charge density values. 4. C u r r e n t e x c i t a t i o n w i t h n o n - z e r o r i s e - t i m e We w i s h t o c o n s i d e r h e r e a c u r r e n t e x c i t a t i o n o f t h e f o r m J ( t ) = J0(1 - - e x p ( - - t / r ) ) t > 0 (30) N o t e t h a t i n t h e l i m i t as r -~ 0, o u r r e s u l t s s h o u l d r e d u c e t o t h o s e p r e v i o u s l y reported for a step current [1]. From eqn. (11): Q(t) = Jot + Jor(exp(--t/~) - - 1) (31) 69 and since E(x, t = 0) = 0, t h o s e curves e m a n a t i n g f r o m t h e t = 0 b o u n d a r y have t h e p a r a m e t e r c, d e f i n e d in eqn. (8) as zero. Thus, t h e solution in this region f r o m eqns. (8)--(11) is: (32) -- Xo = t2/2 -- ~r" + F2( 1 -- exp(--~/~')) F, = ~+ ~'(exp(--~/r') -- 1) (33) ~=0 (34) where we use the normalizations d e f i n e d in eqn. (14). Again, we have the d e m a r c a t i o n curve which starts at the origin with E = 0. To the right of this curve, we m u s t specify a b o u n d a r y c o n d i t i o n at E = 0. We t a k e t h e p o i n t of view here t h a t for t h e e m i t t e r electric field less t h a n some critical value Ec, no charge emission occurs, b u t once this value is reached t h e e m i t t e r electric field is c l a m p e d at this value. Thus for ~" < Ec the s y s t e m acts as a capacitor w i t h no charge injection, so t h e p e r t i n e n t e q u a t i o n in normalized f o r m is, f r o m eqn. (5), [~/~x] [E(x = 0, T)] = 0: dE(E = 0,~) = 1 -- exp(--~/F), d~ w i t h solution: = = l E(E = 0,£) ~< Ec ~+ F(exp(--~/F) -- 1) 0 ~< ~ (35) ~/ (36) where T/is d e f i n e d as t h e t i m e t h a t E(E = 0,Y) just reaches Ec. Thus, as s h o w n in Fig. 6 for F = 0.1 and Ec = 0.5, we have t w o d e m a r c a t i o n curves, one f 0.8 06 04 0.2, .? ti ?t TI T2 2 3 4 Fig. 6. C h a r a c t e r i s t i c t r a j e c t o r i e s f o r a n imposed current w i t h n o r m a l i z e d r i s e - t i m e ~" = 0 . 1 and critical emitter electric field E c = 0.5. 70 emanating from the origin and one emanating from ~j. To the left o f the latter curve, the space charge density remains zero so the electric field in this region is still given by eqn. (33) but the trajectory o f the curves is n o w given by the relation = (}-2 _ }-o2)/2 _ ~(~ _ t0) + ~2(exp(--~0/~) -- exp(--~/~')) (37) For those curves starting at times ~ > ~/the electric field, space charge and trajectories are given by the relations = (~'-- t'0)(/~c -- ~'0 -- 7exp(--~'o/~) + (~2 _ T~)/2 + 72(exp(--~0/~') -- - exp(--F/~)) = Ec + ( ~ - - ~o) + ~'(exp(--}'/~') -- exp(--to/7)) "~ = ll[.Ec + (1"-- (38) (39) ~'o)] (40) Figure 7 shows the time and spatial dependences for the electric field and space charge distributions for ~'c = 0.5 and F = 0.1. The pertinent time constants are ~',, where the curve emanating from the origin reaches the other electrode at ~ = 1, ~., the time when charge injection begins, and }'2, the time w h e n the 15 ~c:05 :01 ~ c =05 T :01 15 T,t / QQ j I0 ~ T, 05 9~ E"°/ ~2 ~o 9~ Q9 0.5 o2 0 & oi~ oi. o18 i I0 }i r 0 t i i 3 ~l 2 T (c) (o) ~'9 2.0 ~c=o5 "f" =Ol I.fi 15 ~i t.o O.4 0.5 ,T, 02 06 O4 (b) Q8 I0 ~2 (d) Fig. 7. Time and space distributions of the electric field and space charge density for current excitation with ?'= 0.1 and/~c = 0.5. 71 TABLE2 Characteristic t i m e constants ~ , ~ and ~ for c u r r e n t e x c i t a t i o n w i t h non-zero rise-time ~" for various values of steady state e m i t t e r electric field E ¢ 0.0 0.05 0.5 1.0 2.5 5.0 t2 0 1.4142 0 1.4633 0 1.8277 0 2.1142 0 2.6784 0 3.2730 t2 0.1000 1.4177 0.1474 1.4675 0.3534 1.8570 0.4832 2.1277 0.7421 2.7027 1.0345 3.3128 t2 0.2000 1.4283 0.2497 1.4782 0.5250 1.8557 0.7068 2.1539 1.0714 2.7482 1.4841 3.3858 t2 0.3000 1.4457 0.3500 1.4957 0.6687 1.8816 0.8889 2.1892 1.3335 2.8074 1.8381 3.4784 t2 0.4000 1.4697 0.4500 1.5197 0.7988 1.9139 1.0501 2.2318 1.5611 2.8769 2.1427 3.5849 t~ t2 0.5000 1.5000 0.5500 1.5500 0.9207 1.9518 1.1983 2.2807 1.7669 2.9543 2.4159 3.7011 t2 0.6000 1.5362 0.6500 1.5862 1.0372 1.9950 1.3375 2.3351 1.9573 3.0380 2.6669 3.8243 t~ t= 0.7000 1.5780 0.7500 1.6280 1.1499 2.0429 1.4701 2.3942 2.1363 3.1266 2.9012 3.9526 0.8000 1.6248 0.8500 1.6748 1.2598 2.0952 1.5976 2.4573 2.3062 3.2192 3.1222 4.0844 0.9000 1.6763 0.9500 1.7263 1.3676 2.1514 1.7211 2.5240 2.4687 3.3149 3.3325 4.2187 o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t~ 1.0 0.8 0.6 0.4 0.2 I 2 3 4 5 Fig. 8. Terminal voltage v e r s u s t i m e with current e x c i t a t i o n for various values of rise-time with Ec ffi 0 and E c = 0 . 5 . 72 curve beginning at ~ reaches ~ = 1. For space charge limited conditions ~ = 0, and ~1 = ~2. The time ~1 can be found from eqn. (32) with x0 = 0 and ~ = 1 while }) can be found from eqn. (36) with E = Ec, and Y2 can be found from eqn. (37) with ~ = 1 and T0 --- ~/. The transcendental nature of these equations requires numerical techniques for solution. Table 2 lists these various time constants for different values of Ec and ~. Figure 8 plots the terminal voltage v e r s u s time for various rise-times ~ with Ec = 0 and Ec = 0.5. 5. V o l t a g e e x c i t a t i o n w i t h n o n - z e r o rise-time In practice, the voltage imposed across a dielectric or insulating liquid has a non-zero rise-time because the capacitive charging current is limited to a finite value. If this rise-time is small compared to the transit time for an injected charge to reach the other electrode, the step voltage solutions obtained in previous work and here in Section 3.B are approximately correct. Normalizing all variables to the voltage as in eqn. (27), the rise-time ~ must obey the inequality ~"< < 0.5 as seen from Table 1. For ~ comparable or greater than 0.5, the problem must be reconsidered using eqns. (6) and (8)--(11) where we choose the voltage to be given as v(t) = V0(1 -- exp(--t/r)) t/> 0 (41) As found in earlier examples, a characteristic trajectory emanates from the origin in the x - - t plane which separates those characteristics arising from initial conditions at the Y= 0 b o u n d a r y from those due to the charge emitter boundary condition at ~ = 0. We again assume that charge injection begins only after the electric field at the emitter electrode reaches a threshold value, Ec and that for higher voltages the emitter electric field remains clamped at this value. For lesser field values, no charge emission occurs. Thus for the excitation of eqn. (41), the electric field remains uniform given by the instantaneous voltage divided by the electrode spacing with no charge injection until the normalized field strength reaches ~'c. The normalized time this threshold field strength is reached is defined as ~/. Two other pertinent time constants and their relative values to the rise-time ~ describe the system. We define tl as the time when the curve emanating from the origin, (~ = 0, ~= 0), reaches the other electrode at ~ = 1. The time the characteristic trajectory which begins at time ~j reaches the other electrode at ~ = 1 is defined as t2. These system time constants, shown in Fig. 9 and very important in correlation between analysis and experiments, are tabulated for various rise-times and charge emission threshold conditions in Table 3. At time t2, space charge is distributed over the entire region between the electrodes. For ~ < < ~2, the electric field and space charge distributions are very close to the steady state distributions at Y2 as can be seen in Figs. 10 and 12 for Y = 0.1 with Ec = 0 and Ec = 0.5. If ~ is comparable or greater than t2, the field and charge distributions gradually approach the steady state distributions for times greater than t2 as demonstrated in Figs. 11 and 13 for 73 ,.o,o,- o,.-[./f, o' //// No charge injection until Tj : / 2/ / /// // t Onset of charge injection Electric field cloml~d at I~.c for T I "fj Fig. 9. Typical characteristic trajectories for an applied voltage excitation with normalized rise-time ~. Charge injection begins at time ~ when the emitter electric field reaches value /~c- Note that ~ can be greater or less than t~. TABLE 3 Characteristic time constants ~, ~/, and ~ for voltage excitation with non-zero rise-time ~" for various values of steady state emitter electric field Ec /•c• 0 ~ =~ 0.0 0.1 0.5 1.0 1.5 2.5 0 0.3935 0 0.4886 0 0.8115 0 1.0722 0 1.2753 0 1.5996 0.25 ~/ 0.0 0.3976 0.0288 0.5072 0.5096 0.1438 0.8384 0.8546 0.2877 1.1169 1.1575 0.4315 1.3388 1.4095 0.7192 1.7003 1.8455 0.50 ~ 0.0 0.4118 0.0693 0.5307 0.5443 0.3466 0.8845 0.9755 0.6932 1.1591 1.3978 1.0397 1.4115 1.8082 1.7329 1.7669 2.5610 0.75 ~ tL ~2 0.0 0.4396 0.1386 0.5662 0.6117 0.6931 0.9190 1.2535 1.3863 1.2000 1.9831 2.0794 1.4200 2.6937 3.4657 1.7700 4.0976 0.90 t~ t~ 0.0 0.4701 0.2303 0.5904 0.7190 1.1513 0.9300 1.6747 2.3026 1.2000 2.8388 3.4539 1.4200 3.9919 5.7565 1.7700 6.3073 = 1.0 with Ec = 0 and Ec = 0.5. Note that for space charge limited conditions = o), = Figure 14 plots the time dependence of the resulting terminal current for Ec = 0 and Ec = 0.5. Note that for ~ small, the current is very large near -- 0 due to the large capacitive charging current necessary. If ~ -- 0, this charging current is infinite (an impulse) at ~= 0, but immediately drops to finite values for ~ > 0. For non-zero but small ~, the initial current is large but still finite, quickly dropping down with some small oscillations as it approaches the steady state. For large ~, the initial charging current is small and 74 1-751 I¢o.o j ,./.-/'" ,.~5~ 75 I o,_1 ~ 0.75 0.75 025 025 / 1 i 0.25 05. ,;'°1 0.75 r~.o o., oz ~ ' ~ : i 1.0 0.5 o . o 5 i -- i 1.0 1.5 2.0 T (al (c) 6.0 4.0 2.0 l "? :o.i 50 I~¢=0 "C=OI 40 005 \L 5.0 20 ~a3 "-.. ~ - 0.2 O.4 o.~Qt,.o - 10 I0 0.25 0.5 (b) 075 I0 05 I0 15 20 (d) Fig. 10. Electric field and space charge distributions for an applied voltage with Ec = 0 and ~'= 0 . I . there are no oscillations as the current approaches the steady state. These curves are important in correlation to time-of-flight measurements, which determine the ionic mobility # through measurement of the time for the current to reach its peak (~= ~1) when a step voltage is applied. In an experi- 75 15 125 ~¢= 0 1.5 //J] "~o,o // ~',=°0 ;~J/I // 1.0 075 0.75 05 0.5 0 25 025 015 0 8 "~---- i 0.~5 ,0 I I I I 2 3 /~;" ~o.o I 2.5 I~,-o 15 1.5 I0 I0 °s I 025 I 'I 08 05 y (b) 4 r {c) (Q) z5 ~''°~ 2o 1.0 0.15 / I 125 075 IO 005/ 0z~ ~ o f - - o5 I 2 3 4 T (d) Fig, 11. Electric field and space charge distributions for an applied voltage with Ec = 0 and F'= 1.0. ment, if the voltage rise-time is small, t h e current peak at ~ = tl is so m u c h smaller than t h e initial charging current that it can be easily o v e r s h a d o w e d and difficult t o measure. If F is large there is n o current peak at ~ = tl so t h a t the ionic m o b i l i t y c a n n o t be measured. These problems make terminal measurements o f this t y p e difficult. Further r e f i n e m e n t o f bulk optical measurem e n t s using Schlieren p h o t o g r a p h y and the Kerr electro-optic effect will allow greater precision in measurements w h e n used in c o n j u n c t i o n w i t h terminal measurements. 76 1.5 Ec=0'5 "~=01 T,- 1.25 '~~o~ °4- 1.5 c=05 I. 25 = //~ " f I0 ~ 08 O6 ,o /// o. 1.0 /// ~ O4 0.2 02 075 i'=0.1 0 1 - - 0.5 ~~ t=lj=00693 T:005 0.5 0.25 i:oo3 025 ' ~ = 0 - - ~{:ooi I 0.25 I ~l'=0 05 I 1.0 075 ~j I i i O4 O l8 I 2 (c) (o) 20 - 1.6 f ~c :0.5 20 00 E¢:0.5 'T :01 010 1.5 1.5 025 - "70I 7=01 -.. ID "~ 05 0.5 025 i 05 (b) 05-- T2. ~:iO- 0.5 Q5 i;" ~il - i i I 0.75 ID Q4 08 12 16 T (d) Fig. 12. Electric field and space charge distributions for an applied voltage w i t h Ec = 0.5 and ~'= 0.1. 6. Concluding remarks The analysis presented here is of great value in correlation to experimental measurements of the conduction and breakdown mechanisms in dielectric liquids as well as semiconductors for single carrier charge injection. By fitting 77 I .25 ~c=O..~ .~-'" f" : ~.o J ./ / I0 0 75 /.'J" 12 5 Ec =0.5 ;: ~o / " i", i.o 2.0 oo.B.6~ 1.0 /~ 04~ 02~ 0.75 1,0- / . / ~ T=OB O0 05 0.5 --05 - - 0 25 - - 0.25 0 3 OI I i 0 5 i 0.5 0.75 iI 1,0 i i 2 :5 4 R (o) (c) 20 ~ ~c=0.5 2.0 l I~c=O.5 ~:l.O I ~,oo ~ ~ ~I 02 ~ 1.0 ~ O. 5 ° 1 ~ iO o 4 ~ ' 0.5 Tz I_ 0.25 05 0,75 I0 ijt I $[, T~ 2 3 4 T (b) (d) Fig. 13. Electric field and space charge distributions for an applied voltage with Ee ffi 0.5 and 7 = 1.0. experimental measurements with the analysis, values of ionic mobilities and other physical parameters can be calculated. However, as emphasized in this work, the pertinent time constants measured and derived from analysis are strongly related to the emitter boundary conditions as well as the rise-time of the applied excitation. The results presented here, both for the bulk electric field and space charge distributions as well as the time dependence of the terminal voltage and current, allow for greater precision in the calculation of physical parameters by taking into account these experimental realities of excitation rise-time and charge injection thresholds. 78 [ ~, Vo, i ,~ 1 2 / I0 ] ~ -- i .¢ / ,," / "..... f" " . t5 05~.25 "~ ~o j ./ 7 2 3 Fig. 14. Terminal current for an applied voltage excitation for various values of ~ w i t h = 0 (solid lines) and E c = 0.5 (dotted lines). Ec Acknowledgements Ms. Marjorie A. Niblack and the photography staff of the Office of Instructional Resources at the University of Florida are gratefully acknowledged for their help in the preparation of the plots presented in this work. This work was supported by the National Science Foundation under Grant Nos. GK--37594 and ENG72--04214 A01. References 1 2 3 4 5 6 M. Zahn, C.F. Tsang and S.C. Pao, J. Appl. Phys., 45 (1974) 2432. M. Zahn and S.C. Pao, J. Electrostatics, 1 (1975) 235. M.A. Lampert and P. Mark, Current Injection in Solids, Academic Press, New York, 1970 R.B. Schilling and H. Schachter, J. Appl. Phys., 38 (1967) 841. E.C. Ca~idy and H.N. Cones, J. Res. Natl. Bur. Stand. (U.S.), 73C (1969) 5. E.C. Ca~idy, R.E. Hebner, Jr., M. Zahn and R.J. Sojka, IEEE Trans. Electr. Insul., EI--9 (1974) 43. 7 P. Atten and J.P. Gosse, J. Chem. Phys., 51 (1969) 2804. 8 P. Atten, C. R. Acad. Sci., 266 B (1969) 1188. 9 I.N. Sneddon, Elements of Partial Differential Equations, McGraw-Hill, New York, 1957, Ch. 2.