.

De 6. 7.

I. V. Bodnar', A. G. Karoza, A. I. Lukomskii, and G. F. Smirnova, "A study of the optical properties of the ternary compound AglnS2," Zh. Prikl. Spektrosk., 31, No. 5, 880-882 (1979). J. L. Shay and J. H. Wernick, Ternary Chalcopyrite Crystals: Growth, Electronic Structure, and Applications, Pergamon Press, New York (1975), p. 244. Yu. V. Rud' and K. Ovezov, "Photoelectric properties of diodes based on n-ZnSiAs2," Fiz. Tekh. Poluprovodn., i0, No. 5, 951-957 (1976). T. F. Moss, G. Barell, and B. Ellis, Semiconductor Opto-electronics, Halsted Press (1973).

THEORY OF PHOTOINDUCED ELECTRICAL CONDUCTIVITY Yu. T. Mikhailov

UDC 537.312.5:541.14/15

A convenient method of researching photochemical reactions is that of photoinduced electrical conductivity PEC, which involves observing the current pulse in a cell irradiated by a short light pulse between electrodes inserted in the solution [1-3]. The current arises in the circuit composed of the cell with the electrodes, a voltage source, and a load resistor because of the change in resistance of the solution, which arises in general from the production of new charges and change in the mobility of the existing ones. The lack of a theory relating the photocurrent to the excitation conditions hinders the use of PEC in research on processes in solutions (theoretical analysis for solid polymers can be found in [4]). The principles given in the literature for recording current induced by radiation in a cell containing electrodes relate either to the conditions of pulse radiolysis [5, 6] or else are excessively simplified [I]. In all cases it is assumed [i, 5, 6] that the photolysis or radiolysis products are uniformly distributed in the gap between the electrodes, and also that the solution before excitation is nonconducting, i.e., one can only examine solutions of neutral substances, while the voltage on the electrodes does not alter when the induced current occurs. In photoexcitation, it is much commoner for one to have a nonuniform distribution of the products. Also, the compounds may dissociate in solution into ions, and the solvent may be a conductor, while the voltage on the electrodes may alter when the photocurrent appears. Here we consider the theoretical principles of the PEC method for the general case of a conducting solution: the finite resistance in the cell containing the solution and an inhomogeneous bulk distribution for the products. We consider a cell consisting of two identical plane-parallel inert metal electrodes immersed in a conducting solution that on the whole is electrically neutral. The leakage field is small if the minimum transverse dimension of the electrodes a is related to the distance Lo between them by a/Lo > i. The thickness of the double electrical layer is less than 10 -3 cm in a pure conducting solvent [7, 8], and it is much less in the presence of a dissolved substance that dissodiates into ions, and it is much less than the distance Lo, which is usually 0.i-i cm. The sum of the polarizing voltage and the electrode potential (due to the double electrical layer) does not exceed a few volts at all reasonable current densities [7, 8]. Therefore, when a voltage of more than 20-30 V is applied briefly [9] to the electrodes, one can assume that there is a uniform distribution of the electric field E(x) between the electrodes (where x is the coordinate along the direction from one electrode to the other), along with a uniform concentration for the initial ions (equal to the ion concentration before the voltage is applied). The pulse voltage also eliminates the distortion of the photocurrent kinetics arising from the loss of the photoproducts at the electrodes [9]. Before the excitation, a dark current with density Jo flows between the electrodes:

Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 38, No. 5, pp. 818-825, May, 1983. Original article submitted April 5, 1982.

600

0021-9037/83/3805-0600507.50

9 1983 Plenum Publishing Corporation

/N

s

Jo = e X Zic%t~ Eo -,'=1 i J

0

/N

"~ Zic ~PP, Lo ~ i I

(I)

where JN is the number of kinds of charged particles that produce the dark current, Zj is the valency of charged particles of type j (Zj = 0 for neutral particles), and the concentration of dark-current charges in the gap is uniformly distributed, so c~(S, x, t) = c73 = const, where S is a two-dimensional coordinate in a plane parallel to the electrodes and t is time, while Eo = Uo/Lo is the initial electric field strength in the gap, with Uo the voltage between the electrodes before the excitation, ~jo = eD?Z 3 J /kT the mobility of the dark-current charges before excitation, and D? the molecular diffusion coefficient before excitation. J The excitation by a short light pulse of length tu at time t = 0 produces in general an inhomogeneous distribution of the photoproducts Ck(S , x, t) between the electrodes. Also, some fraction of the exciting radiation goes to heat the volume as a result of radiationless deactivation, and this increases the mobility and diffusion coefficient of the charge carriers. Therefore, the total current density J after excitation is equal to the sum of the current densities due to displacement and conduction, and it is written as

J (S, x, 0 = ~'o

0 [~ (S, x, t)]

Ot

iN

~~

i=l

a=l

+ e ~ Z~P~(S, x, t ) + ~ Z~P~(S,x, t), i--/:k,

(2)

where eo is the dielectric constant of vacuum, r x, t) is the dielectric constant of the solution, ko is the number of kinds of particle produced by the photoexcitation, Zk is the valency of the particles produced by the excitation of type k, and the fluxes Pj and Pk are found from the transport equations

Pj(S, x, t ) = ~ j ( S , x, t) E(S, x, t) cj(S, x, t ) - - D : ( S ,

x, t) Oci(S, x, t)/Ox,

(3)

Pk (S, x, t)= Fh (S, x, t)E (S, x, t)ch (S, x, t ) - Dh (S, x, t)Ock (S, x, t)/O:~, ] = 1, . . . , ]N; k :

1. . . . .

ko

with boundary conditions that reflect the absence of photoproducts around the electrodes on excitation between them:

c~(S, o, t)=cj(s, L o , / ) = c ~ Dj(S, O, t ) = V j ( S ,

~ ( S , O, t)=,~j(S, Lo, t ) = ~ o ,

Lo, t ) = V ~ ] = 1 , . . . , IN, (4)

ch(S, O, t ) = c h ( S , Lo, t ) = O

~th(S, O, t)=~t h(S, Lo, t ) = ~ t ~

Dh(S, 0, t ) = D k ( S , Lo, t ) = D o, k = l ,

. . . , ko.

Here E(S, x, t) is the strength of the electric field between the electrodes after excitation. We also assume that the photoproduct concentration is much less than the concentrations of the dark-current charges:

c ~x~ (s, x, 0 ~ c~ .

(5)

The initial conditions for cj and ck are put as follows, where cj undergoes photoconversion:

cj (s, x, o ) :

c~ - c ~

(s, x) ] = 1

i~ m = 1,

ch (S, x, 0 ) : ch (S, x) k : 1. . . . .

h {m} E {1}, (6)

ko c., (S, x)<< e~,

where jx is the number of kinds of dark-current charges that undergo photoconversion, and cm(S , x) is the decrease in the concentration of the charges of type m = j involved in the photoreaction. For the cj that do not participate in the photoreaction,

cj(S, x, O)= c~

(7)

1

601

In (5)-(7) allowance has been made for the fact that thermal expansion occurs only during the time t T > Lo/va, where v a is the speed of sound in the solvent. For example, v a = i000 m/sec [i0] in ethanol, and correspondingly for Lo = 1 cm we have t T > I0 ~sec. Therefore, from the instant of excitation by a pulse of length t u < t T up to time t T one needs to incorporate only the changes in mobility and diffusion coefficient due to the temperature step in the excited volume, since the charge concentration will not be able to change up to times of the order of t T. The charge mobility is dependent on the solution temperature,

and as ~j m I/H, where

is the viscosity of the medium, we have as follows for temperature changes AT(S, x) small relative to the initial value:

~(S, x ) = 11o[1--~AT(S, x)],

(8)

where ~o is the viscosity before the excitation and heat production, and a = (dn/dT)/no is the linear term in the variation of viscosity with temperature (at constant volume for the solution). From (8) we get

py (S, x) = / x ~ [1 @ r

(S, x)],

~*k(S, x) = po [1 q- r

(S, x)],

(9)

where

AT(S, x ) -

1

pCv

dQ(S, x) dV

(i0)

Here CV is the specific heat of the medium at constant volume, 0 is the density, and dQ/dV is the volume density of the heat production on excitation. We neglect the change in the radius of an ion with temperature on the basis that the electrical conductivity of an electrolyte solution is linearly dependent on temperature over a wide range [ii]. For a small temperature change, we have from (5)-(7) that

Dj(S, x) = D o q- ADj(S, x), Hi(S, x) = V~ -~ A~xj(S, x), ] = 1, . . . ,

Dh(S, x)=D~

x), [*h(S, x)=-~x~

x), k = l . . . . .

IN,

(ii)

ko,

E (S, x) = Eo -}- AE (S, x), where ADj(S, x), ADk(S, x), A~j(S, x), A~k(S, x), and AE(S, x) are quantities of the first order of smallness relative to D j,

D~, ~ ,

~,

and Eo.

We consider (2) at the instant tm corresponding subject to the condition that

to the maximum in the photocurrent pulse

t~ ~ tm <
(12) Then the initial concentrations

de-

We substitute (3), (6), (7), (9)-(11) into (2) and integrate (2) with respect to x with limits from 0 to Lo, and with respect to S from 0 to So (So is the electrode area) on the basis of the boundary conditions of (4) to find the amplitude of the current pulse up to terms of the second order as IN

L~

"

602

IN

'I ~ /'==- 1

~

LSpCv Vo

h~

Vf /'=1

,

Lg

~ h~l

l~

h h~h--

m m~l

~17~ 7 9

(13)

Here Qo is the total amount of heat deposited in the excited volume, Vo is the volume of the gap between the electrodes, and N4, o NI., o and N~LL~ o are the total numbers of particles of types j, j ~ k, and m in the space between the electrodes. Further,

in deriving

(13) we have used the equation L0.

c)U. . . . Ot fm

c) Ot

_

E (S, x, t) dx

J" 0

= O,

tm

which implies that

OE(S, x, t) I at

= O.

(14)

It,,,

The dielectric constant of the medium alters abruptly only during the exciting pulse, so (14) means that the displacement current is zero at tm. From (i) we have the current before excitation as

(15)

Io = JoSo = Uo/R~

where fN

1/R~--

eSo

E

Zjc? ~?;

(16)

and R~ is the resistance of the cell containing the solution as measured between the electrodes before excitation. We subtract the Io of (15) from (13) to get the photocurrent pulse If = I -- Io as measured. Joint solution of this equation with the equation expressing constancy of the source voltage,

Rt(I--4)=Uo--U, where R l is the load resistance,

(17)

gives the photocurrent amplitude as

If =

eUoF L ~', ( I + R z /

U0

o~Q.o

R z § R~

~Cv Vo

' R k) o -~

(18)

where ho

~TNO, h=l

O

/1

o

(19)

m=l

We see from (18) that we can process the data by introducing the quantity

q : IfL~ (1 -t- Rz/R~)[Uo : eF -t- L~aQo/R~pCvVo characterizing

(20)

the photoconductivity.

Note that the greater the resistance of the cell and the less the amount of heat produced the greater the proportion of the photoc~rrent due to photochemical reactions. Clearly, the most favorable case is that of a solution of a substance that does not dissociate into ions and where the solute does not reduce the resistance. The amount of heat deposited is directly proportional to the radiationless deactivation constant and can be small in solutions of highly luminescent substances. The expansion of the volume in which the heat is deposited during a time of the order of t T reduces the charge density, and some of the charges at the electrodes are displaced from the electrode gap. After the end of this process, the current density Ja is

603

/N

J~ (S) : e X

Z:c~ (S, x) pj (S, x) E~ (S, x),

(21)

i=1

where Ea(S , x) is the steady--state electric field. Here we assume that the current due to the photochemical reactions has vanished by this instant. The charge density after the thermal expansion is described by 0

c~(S, x):= cj/[1 + ~ A T ( S , x)]. Here B is the thermal-expansion

(22)

coefficient of the solution and

AT(S, x ) - - 1 pCs

dQ(S, x) dV

,

(23)

where C s = CV + R/M is the specific heat of the solution at constant pressure, with R the universal gas constant and M the molecular weight of the solvent. We substitute (9), (22), and (23) into (21) and integrate (20) with respect to x with limits 0 and Lo and with respect to S with limits 0 and So to get the current I a in terms of first-order terms:

ia :: Uo/R ko ~_ , (~ __ ~), QoUo/gCp Rk~ o,

(24)

where U a is the steady-state voltage on the electrodes. We subtract (24) from (13) and solve the resulting equation together with the equation for the constancy of the source voltage

(25)

R z ( / ~ - - 0 -- U - - U ~ , which gives us the change in the current from the maximal value AI T

AIT=

loQo pCsV o

~ + eR/MCv 1 + Rz/R~ + ~RvQo/PCsR~Vo

(26)

From (18) and (26) we have that the current change due to thermal expansion related to the photocurrent due to the thermal mechanism is R/MC V + 8/a and can be 10-20%. Therefore, the PEC method can be applied to electrolyte solutions if one allows for the effects of the heat deposition on the amplitude of the photocurrent and the steady--state value after the excitation. Here it appears that it is difficult to examine photochemical reactions when the current due to the photoreaction in unit primary act is less than or equal to the current due to the thermal effect. For a one-quantum reaction (quantum yield Y*) and a twoquantum reaction (quantum yield 72 on absorption of the second quantum) the equality of these currents is written as

e?iAp = ~t,vL~o (1

__

~,f~f/v--

~i)/pCvRk0 V o,

e~,2hp = czh~L~ (1 -- ?2)/pCv R~

(27) (28)

Here A~ is the mobility change in the primary unit act of the photochemical process as referred to the elementary charge e, v is the frequency of the exciting radiation, and yf, '~ are the quantum yield and mean frequency of the luminescence. The quantum yield in luminescence from highly excited states is much less than one, so there is no corresponding term in (28). From (27) and (28) we get the minimal quantum yield in one-quantum and two-quantum photoreactions, which can without difficulty be examined by the PEC method:

2~i, = (1 - - ? f ~-f/~)/(1 -b ehppCv VoR~ ~ i . = 1/(1 + eAppCv VoR~

(29) (30)

We substitute into these expressions the values Vo = 1 cm s, Lo = i cm; for water and ethanol ~/QC V ~ 10 -2 cm3/J at 20~ [12], and hg(X = 530 mn) = 3.75.10 -I' J; for reduction of the rhodamines by ethanol, Ap = 3.10 -~ cm2/V.sec [9], and usually with Lo = 1 cm So = 1 cm=, R~ ~

604

i06 ~ for ethanol solutions, while the maximum quantum yield in luminescence is 0.9-0.96, so ~f = u and we have ymin % I0-~, y~in % i0-5. Therefore, the thermal effect restricts the limiting sensitivity of the PEC in the examination of photochemical reactions in conducting solutions. The noise restricts the sensitivity of the recording system. In fact, in our case, t h e main part is played by the thermal noise in the load, which has the mean--square voltage U~n = 4kTRzAf, where Af is the recording passband, and the shot noise in the dark current, which has the following mean--square voltage at the load U~--n = 2eloR~hf = 2eUoR~hf/R~. The meansquare noise voltage is

V Un

sn .

Reliable recording requires a signal-to-noise voltage 91.

eU~

L~ (1 + RF~ ~

Then

9~ V U--~n

We substitute F = Nk&~ to get L~(1

N~>~

/ o

-5 R ~.Rk) V-(4kT/R~ § 2eUo/R~) Af.

eUoA~

We s u b s t i t u t e i n t o (31) t h e v a l u e s R~ = 10 n ~, and Af = 0 . 3 5 / t r [ 1 3 ] , from t h e 0 . 1 l e v e l t o t h e 0 . 9 o n e , h a v e h f = 3 . 5 . 1 0 7 Hz, t h e n w i t h R l 50 kfi we h a v e N k ) 2 . 1 0 -13 m o l e ) .

Lo = 1 cm; Uo = 500 V, 5~ = 3 . 1 0 -~ c m a / V . s e c , T = where t r i s the r i s e time of the edge of a square and f o r t h e n a n o s e c o n d r e c o r d i n g r a n g e ( t r = 10 -8 = 50 ~ we f i n d Nk ) 10 - ~ mole ( f o r t r = 10 -5 s e c

(31) 300~ pulse s e c ) we and R~ =

I f t h e r e i s no s a t u r a t i o n i n t h e l e v e l s o f t h e e x c i t e d s u b s t a n c e s , w h i c h i s so i f t h e f o l l o w i n g c o n d i t i o n i s o b e y e d [14] f o r d y e s w i t h low quantum y i e l d s f o r t h e t r i p l e t s t a t e on e x c i t a t i o n by n a n o s e c o n d p u l s e s :

Go=Wom/cS:ctu ~ pjB3~,

(32)

t h e n t h e y i e l d s o f t h e o n e - q u a n t u m and t w o - q u a n t u m s t e p p h o t o r e a c t i o n s (with the second quantum a b s o r b e d b y a s i n g l e t - e x c i t e d m o l e c u l e ) a r e d e s c r i b e d by t h e f o l l o w i n g e x p r e s s i o n s , w h e r e Go i s t h e v o l u m e d e n s i t y o f t h e e x c i t a t i o n e n e r g y a t t h e i n p u t t o t h e c e l l , Wo i s t h e e n e r g y o f t h e e x c i t i n g p u l s e , c i s t h e v e l o c i t y o f l i g h t i n vacuum, m i s t h e r e f r a c t i v e index of the s o l v e n t , Sc i s t h e c r o s s s e c t i o n a l a r e a o f t h e e x c i t e d o b j e c t , t u i s t h e d u r a t i o n o f t h e e x citing pulse, Pc: is the probability of spontaneous transition from the f i r s t s i n g l e t - e x c i t e d s t a t e o f t h e dye m o l e c u l e t o t h e g r o u n d s t a t e , and Ba4 i s t h e E i n s t e i n c o e f f i c i e n t for the induced transition from the f i r s t s i n g l e t - e x c i t e d state to higher-lying ones:

N(kl): yIWo/hv N~~')-

y~rnB3~tW~ 2ct~S c o3ihv

(33)

From (32) we h a v e a c o n s t r a i n t on t h e l i m i t i n g e x c i t i n g p u l s e e n e r g y ( t o p r o v i d e e x p l i c i t d e p e n d e n c e o f t h e l i n e a r and q u a d r a t i c r e l a t i o n s h i p s on t h e e x c i t i n g e n e r g y f o r t h e o n e - q u a n t u m and t w o - q u a n t u m r e a c t i o n s ) :

Wo ~ cS c!~p31/mB3~.

(34)

On the basis of (34) and (31) we get constraints on the quantum yields in the photoreactions: y~, y j 2 ~

hvL~ (1 §

Rz/B ~ eUohl~Wo

.V(4kT/R l § 2eUo/R~ A/. '

(35)

The quantity S c is restricted by the distance between the electrodes and the size of these and usually S c < i cm 2, Pa~ = 2.10 8 s e c - I, and Ba4 = 2.10 1 2 cm 3 /J.sec for rhodamines [15, 16]; m = 1.33 for ethanol [i0], and then for t u = 30-40 nsec we get Wo < 0.I J, and on substituting this value of Wo into (35) we get for the nanosecond range (tr = i0 -8 see) that y~, y2/2 2.10 -5 (4.10 -7 for t r = 10 -5 sec). 605

This discussion shows that the PEC method can be used to examine photoreactions. The expressions enable one to select the working conditions and to determine the characteristics of the photochemical reactions from the results. LITERATURE CITED i. 2.

3. 4. 5. 6.

7.

8. 9. i0. ii. 12. 13. 14. 15. 16.

606

H. S. Piloff and A. C. Albrecht, "Biphotonic ionization: a flash photoconductivity study of TMPD in 3-methylpentane solutions," J. Chem. Phys., 49, No. ii, 4891-4901 (1968). A. A. Mak, Yu. T. Mikhailov, and V. V. Ryl'kov, "Studies on the laser photochemistry of dye solutions by photoinduced electrical conduction," Opt. Mekh. Prom., No. 3, 20-22 (1981). Yu. T. Mikhailov and V. V. Ryl'kov, "Photocurrent observation in solutions of xanthene dyes on laser excitation," Khim. Vys. Energ., 15, No. 3, 263-266 (1981). D. Fox, M. M. Labes, and A. Weissberger, Physics and Chemistry of the Organic Solid State, Wiley--Interscience, New York, Vol. 2 (1965); Vol. 3 (1967). A. A. Chernenko and E. G. Vinkler, "Theory of pulse radiolysis in an electric field," Dokl. Akad. Nauk SSSR, 195, No. 3, 650-653 (1970). A. V. Vannikov, E. I. Mal'tsev, V. I. Zolotarevskii, and A. V. Rudnev, "A study of the short-lived charged radiolysis products in organic liquids by electrical and optical methods," Int. J. Radiat. Phys. Chem., ~, No. 2, 135-148 (1972). A. L. Rotinyan, K. I. Tikhonov, and I. A. Shoshina, Theoretical Electrochemistry [in Russian], Khimiya, Leningrad (1981). K. J. Vetter, Electrochemical Kinetics: Theoretical and Experimental Aspects, Academic Press (1967). Yu. T. Mikhailov and V. V. Ryl'kov, "Cooperative and stepwise photoprocesses in rhodamine solutions," Zh. Prikl. Spektrosk., 36, No. 3, 445-450 (1982). N. I. Koshkin and M. G. Shirkevich, Handbook on Elementary Physics [in Russian], GIFML, Moscow (1962). A. I. Levin, Theoretical Principles of Electrochemistry [in Russian], Metallurgiya, Moscow (1972). Chemist's Handbook, 2nd ed. [in Russian], Goskhimizdat, Moscow--Leningrad, Vol. 1 (1962); Vol. 3 (1965). A. M. Bonch-Bruevich, Electronics in Experimental Physics [in Russian], Nauka, Moscow (1966). Yu. T. Mikhailov and V. V. Ryl'kov, "Kinetics of nonlinear photoreactions in dye solutions," Zh. Prikl. Spektrosk., 35, No. 6, 979-983 (1981). E. N. Viktorova and I. A. Gofman, "A study of the fluorescence characteristics of a series of rhodamine dyes," Zh. Fiz. Khim., 39, No. Ii, 2643-2649 (1965). A. V. Aristov and V. S. Shevandin, "Induced singlet-singlet absorption spectra of rhodamine dyes in the range 15,000-25,000 cm-1,'! Opt. Spektrosk., 43, No. 2, 228-232 (1977).