September 1963 BRIGHTNESS CONTRAST

observed in Experiment II had longer exposure dura-tions been used. (It should be recognized that, at thereceptor level, the effective exposure time is deter-mined by eye movements, as well as by when the experi-menter turns the stimulus on and off.) However, thecritical consideration would seem to be the exposureduration necessary to elicit the phenomenon to be ex-plained. In the case of Mach bands, if a stimulus ex-posure of 80 msec is sufficient to elicit the phenomenonin well-developed form, and if, as Experiment II indi-cates, linear summation does not occur within this time,then it can be concluded that linear summation is nota necessary feature of the mechanisms involved.

Data bearing on the Mach Band phenomenon arepresented in Table III. The stimulus was a verticallydivided bipartite field, approximately 30' on a side,superimposed on a continuously illuminated adaptationfield. The luminances, in ft-L, were as follows: brightpart of stimulus, 90; dark part of stimulus, 4; surround,

0.15. Mach bands were elicited by throwing out offocus the central dividing contour, only, by variousamounts. Variously blurred stimuli were presented incounterbalanced random sequences and at exposuredurations of 4 sec and 80 msec. On each presentationthe subject estimated the width of the bright line. Thedata are from a single subject who did not participatein the previous studies. For each exposure condition,the data were gathered on two days.

The actual values of the estimates are not importantto the present discussion. What is important is that thesubject perceived Mach bands under both 4 sec and 80msec exposure conditions, as evidenced by the consist-ency of the estimates and the systematic variation inestimated width as the stimulus was defocused. Sincelinear summation does not occur under the shorter ex-posure condition it can be concluded that the mecha-nisms which produce Mach bands do not depend onlinear summation.

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA VOLUME 53, NUMBER 9 SEPTEMBER 1963

Nonuniformly Excited Photoconductivity in Sensitive PhotoconductorsPAUL H. WENDLAND*

Molectro, Inc., Los Angeles 64, California(Received 7 January 1963)

An analysis of the photocurrents produced by nonuniform illumination of a sensitive photoconductor isgiven. It is shown that for sensitive photoconductors in the steady state, the current density is a constantthroughout the bulk of the photoconductor, under all conditions of excitation. The expression for the photo-current density as a function of illumination and applied field is then obtained through an integration ofPoisson's equation, with a model of the photoconductor as a thin, flat slab and illumination incident perpen-dicular to one face. The boundary conditions take into account the existence of secondary photocurrentsdrawn in from the electrodes by space charge resulting from the primary excitation. It is found that largesecondary photocurrents can be obtained under nonuniform excitation if the shallow electron trap densityis low and electron capture cross section of hole-containing centers is low. Experimental results are reportedfor photocurrents in sensitive evaporated films of CdS:Ag, which satisfy the geometry requirements of theanalysis.

PURE insulators (or semiconductors at low tempera-ture) generally exhibit a low photoconductive

sensitivity. A high sensitivity, however, can be inducedin these materials through the introduction of com-pensated acceptors." 2 A simplified energy level schemedepicting such acceptor centers is given in Fig. 1. Theacceptor levels are initially filled with electrons fromdonor levels and obtain a net negative charge, over-allneutrality being maintained by the positive charge onthe donors. The compensated acceptor centers then havea high Coulombic probability for capturing photo-excited holes and a low probability for subsequent cap-ture and recombination of electrons. Electrons that are

*Now at Hughes Research Laboratories, Malibu, California.lW. W. Tyler, R. Newman, and H. H. Woodbury., Phys. Rev.

97, 669 (1955).2R. H. Bube, Photoconductivity of Solids (John Wiley & Sons,

Inc., New York, 1960), p. 171.

optically excited out of the valence band thereby obtainrelatively long free lifetimes which result in large ob-served photocurrents.

When a geometry such as that shown in Fig. 2 is usedwith the photoconductor, carriers are produced non-

photo -electronv _

ionized donors

ompens ated acceptors

photo-hole

FIG. 1. Energy diagram illustrating compensated acceptors andtheir effect on photoexcited carriers.

AND STIMULUS OUTPUT 1037

PAUL H. WENDLAND

FIG. 2. Schematic dia-gram of physical arrange-ment of photoconductor.

thickness

uniformly through the volume, and diffusion and spacecharge effects enter the problem. Such a geometry oftenfinds use in practice, and can be used to gain informa-

tion on the relationship between spectral response andabsorption coefficient. For both applications, it isnecessary to establish the theoretical relationships be-tween the photoconductor material and geometricalparameters, and the photocurrent. DeVore3 has analyzedthis problem assuming diffusion effects alone, andGoodman4 has found a solution assuming field-drivenphotocurrents only. In both papers, trapping effectswere neglected, space charge effects were taken as small,

and the lifetime was assumed independent of carrierconcentration.

Here, we consider the effects of nonuniform excitationin sensitive photoconductors, which command special

considerations, and develop an analysis including space-charge effects and resultant secondary photocurrents.A sensitive photoconductor is defined for our purposesas one containing a concentration of compensatedacceptors, with a capture cross section for holes largeenough that the free electron lifetime is much greaterthan the free hole lifetime. Recombination occursthrough the compensated acceptor after these centershave trapped holes. The density of compensated accep-tors is assumed always greater than the density ofphotoexcited holes, so that there will always be unfilledcenters ready to trap holes. Shallow electron traps areassumed present with a common energy level. Theenergy diagram of our sensitive photoconductor is thatof Fig. 1, with one dominant compensated acceptorlevel, and one dominant shallow electron trapping level.

The geometry of our model is that of Fig. 2, in whichthe thickness of photoconductor d is smaller than any

other physical dimension. Ohmic electrodes are applied

to both faces, with the electrode facing the incidentillumination being transparent. The Ohmic electrodes

3 H. B. DeVore, Phys. Rev. 102, 86 (1956).4 A. M. Goodman, J. Appl. Phys. 30, 144 (1959).

allow charge to freely enter and leave the photocon-ductor, and the free electron lifetime is only terminatedby recombination. Trapped hole charge in the body ofthe photoconductor can create an additional spacecharge field at the cathode which causes secondaryphotocurrents to enter the material, and our analysisis particularly concerned with these effects.

LIST OF SYMBOLS

it, free electron concentrationut, trapped electron concentrationPt, trapped hole concentrationTe', free electron lifetimeTpt, trapped hole lifetimeJa, electron current densityf(), creation rate of electron-hole pairsa, absorption constantE, electric fieldx, distance from front illuminated surfaced, sample thickness,, mobilitye, charge of an electronq, total charge densityk, dielectric constantD,, electron diffusion constantV, applied voltagev, electron thermal velocityS,,, capture cross section for electrons of hole-containing

centerE, energy distance from conduction band of shallow

electron trapsN, density of states in conduction bandNt, density of shallow electron traps

ANALYSIS

The general equations that a photoconductive systemmust satisfy have been written down before,5 and inthe one-dimensional case with trapping effects includedonly via the recombination term, they become forelectrons:

continuity equation for electrons

dn/dt = - n/ur+ f (x) + e- (dJ,Ydx); (1)

transport equation for electrons

J,=,4enE+eD,(dn/dx); (2)

Poisson's equation

dE/dx= 47rq/k. (3)

The equations for holes, similar to Eqs. (1) through(4) for electrons, are simplified in the case of sensitivephotoconductors.

Since photoexcited holes are rapidly captured at, ornear, the site of creation, there is negligible hole trans-

' E. S. Ritner, Paper in Photoconductivity Conference (JohnWiley & Sons, Inc., New York, 1956), p. 215.

applied field -

ohmic electrodes

incident radiation P

F 'Vol. 53103X

September1963 NONITNIFORMLY FXCITEFD PHOTOCONDUCTIVITY

port and negligible free hole density. The steady-statecontinuity equation for trapped holes is thus

Pt= f (X),rt, (4)

where rpt is the trapped hole lifetime, and Pt is the trap-ped hole density. The length of time that a hole remainstrapped is determined by the interaction of the hole-containing center with the free electrons. If the free-electron density moves with velocity v past a hole-containing center, the number of interactions per unittime will be given by vS,,n, where S. is defined as thecapture cross section of the hole-containing center. Thetime per interaction Tt, is then

Tpt= (vS~n)-1. (5)

The free electron lifetime is determined by the interac-tion of an electron with the density of hole-containingcenters. The relationship between the capture crosssection of these centers and the free electron lifetimehas been given before as

combination of Eqs. (4) and (5) gives the relationshipbetween trapped hole density and free electron densityPt =f(x)/vSn. The total charge density is thus relatedto the free electron density by the relation:

(10)

Equations (3), (9), and (10) determine the differentialequation that the photoconductive system must satisfy,and in terms of the field this equation becomes

wheredE/dx+C2f(x)E/2- C3/E= 0, (11)

C2= E (8ire/k) (vSC 1)-l, C3 = [ (47reC /k) (1+)].

This nonlinear differential equation of the first ordercan be transformed into a linear one by multiplyingthrough by E, and letting E0/2=U. Equation (11)becomes

dU/dx+C 2f(x) U= C3. (12)

This equation has the well-known solution

Substituting (4) and (5) into (6), we find

T = n/f (X). (7)Substitution of Eq. (7) in the continuity equation (1)demands that the current density be stationary understeady-state conditions, i.e., when n/at=0. Thus inthe steady state

aJnlax=0. (8)

A general condition that must be obeyed by all photo-conductive systems for which Eqs. (4) and (5) can bewritten is that the current density must be stationary(J. is a constant over x), under steady-state conditions.Equation (4) is only true for systems in which photo-excited holes are created in trapping sites or are trappedin such sites immediately, so that the free hole density iseffectively zero. Equation (5) is only true when the holecapturing center is deep enough so that thermal freeingof trapped holes has a much lower probability thanrecombination.

With the assumption that the applied field be alwayslarger than the "photodiffusion" or Dember field, whichwill always be equal to or less than (kr/e) (1/n) (dn/dx),we can neglect the diffusion term in Eq. (2) and sub-stituting Eq. (8) into (2), it is found that

C1= nE, (9)

where C = J/eA, with C constant.The total space charge density is made up of posi-

tive contributions from trapped holes and negativecontributions from free electrons and trapped electrons,q= e(n+nt-pt). Considering a material with one type ofshallow trap, the ratio of trapped to free electron chargeis given by a, where = [(N/Nt) exp(- /kT)]-'. The

6 Reference 2, p. 70.7A. Rose, Phys. Rev. 97, 1538 (1955).

U= C exp -c 2 fdx)+C3 exp(- C2ffdx)

Xf eP(C2 ffdx)dx, (13)

where the boundary conditions will determine theconstant C. We assume Ohmic contacts to the photo-conductor, and by this we mean a sheath of electrons atthe cathode ready to enter the condition band and onlyrestrained by the self-created space charge field and thefield of the photoexcited charge. Without any illumina-tion on the photoconductor, Eq. (12) reduces to the dif-ferential equation for one carrier space charge limited-current flow, as it must, and the boundary conditionwould be that the field be zero at the cathode. Whenillumination creates electron-hole pairs, the electronsmove toward the anode and the trapped hole chargeproduces an additional field at the cathode whichcreates typical secondary photocurrents.8 Electrons aredrawn in from the cathode by this field until the com-bination of primary and secondary carriers reduces thefield at the cathode to zero. With the assumption ofsecondary injected photocurrents, then, the boundarycondition is that the field be zero at the cathode. Apply-ing the condition that E=O at x=O to Eq. (13), andusing the definition E2/2= U, the expression for E is

E= [2C3 exp(-C 2 ffdx)

rZ 1 /2

Xj exp C2 f fdx dx I8 P. J. Van Heerden, Phys. Rev. 106, 468 (1957).

(14)

(6)

1039

q= eEn(1+6)-f W/vs-nl-

,r.= (VS.wPt)-1.

PAUL H. WENDLAND

d/2 d

Distance from front surface x

FIG. 3. General form of the electric field distribution andthe electron density distribution.

For the geometry of Fig. 2, f (.) is given by,'

f W = Ia Eexp (- o) ]. (15)

The result of the integrations involved in Eq. () isnot expressible in closed form when Eq. (15) is useddirectly as the generating function. We consider, then,the two important limiting cases of very strong absorp-tion and very weak absorption of the incident energy,where reasonable approximations can be given for Eq.(15).

For weak absorption, ad<<1, and with no reflection atthe back electrode surface of the photoconductor, Eq.(15) becomes

f(x) :Ika. (16)

For weak absorption, using Eq. (16) and (14), the ex-pression for the electric field becomes

E= {2C31 - exp(-C 2 Iax)/C2Ia} 1 2 . (17)

The electron density distribution is readily obtainedfrom Eqs. (17) and (9), and the general form of both Eand n are plotted in Fig. 3. We note, from Eqs. (5) and

(9), that pt will have the same form as E, except for a

constant factor.For strong absorption, the rapid exponential decay of

Eq. (15) will be replaced by a step function, where

f=Ic 0<X<1/a, (18)

f= /a<x<d.The expression for the electric field, for strong absorp-

tion, is then given by Eq. (17) when x has the range,

0<x < 1/a. When 1/a <x<d, we find that

E= {2CEx-1/a+ (1C-e-C2T)/Ca]}ls2. (19)

The constants Ci and C2 contain the current densityJn and since this is the quantity that is actually meas-ured, its value is of prime interest. The current densityis determined by the boundary condition that the poten-tial at the anode be the applied potential V. If the effectof the space charge density on the electric field is takenas small in comparison with the applied field at theanode, we can readily determine the current density bysetting E= V/d at x=d in Eqs. (19) and (17). In sup-port of such a procedure, we note that correct resultsare obtained when this procedure is used in the solutionof one carrier space charge limited current flow with onetype of shallow trap.7 Taking the fn term out of theconstants C2 and C3, C 2 =C 2

1 /J., C3 =C 3lJt, and settingE= V/d at x=d in Eqs. (17) and (19), the expressionsfor the current density are obtained for weak and strongabsorption. Two limiting cases result, depending on theincident intensity I. When C2Ictd<<1 in the weak absorp-tion case, and C2I<<1 in the strong absorption case, thecurrent density, from Eqs. (17) and (19), is given by

J= (2C31 d)-(V/d) 2, (20)

which is recognized as the equation for pure spacecharge limited current flow with one type of shallowtrap.7 For C2I<<1, then, space charge currents dominatephotocurrents in this model.

When C2Jad>>1, we find for the weak absorptioncase, from Eq. (17)

J = (Ia)"/2 (C21/2C3s)"2 (Vld). (21)

For strong absorption, ad>>A, and when C21>>t, wefind from Eq. (19)

(V/d) 2= (2C3'd)Jn+ (2C3'/C 21IOa)J 2. (22)

This quadratic equation has the solution, when V/d> (C 3

1 C21Id 2 /4)1 2

(23)

We find then that large photocurrents can be observedin sensitive photoconductors with nonuniform excita-tion and, in order to obtain the highest values, Eqs.(21) and (23) require large C2

1 and small C31. This in

turn presents the reasonable requirement that the crosssection of the compensated acceptor center be as smallas possible and the density of electron trapping centersbe as small as possible, in order to observe the largestphotocurrents.

MEASUREMENTS

Experimental data for sensitive photoconductors,using the volume electrode configuration of Fig. 2, hasbeen taken in order to investigate the validity of Eqs.(21) and (23).

Vol. 531040

'j.;Z::; (.[U)112 (C2'12C-,')"'(Vld).

September1963 NONUNIFORMLY EXCITED

Thin films of CdS were evaporated under controlledconditions 9 onto SnO2 coated glass slides. The films werecoevaporated with Ag, which served as the activatorafter a vacuum heating to establish the Ag in the lattice.In was evaporated on the film surface and served as theback electrode, with the transparent and conductingSnO2 serving as the front electrode.

Film thicknesses were measured using the interferencefringes of the optical transmission curves and rangedfrom 4 to 6 . The experimental equipment used tomeasure photocurrents has been described by us be-fore,9 and it was used here to take data on photocurrentas a function of applied field and as a function of illumi-nation intensity. The illumination source was a tungstenfilament operated at 2500'K, and all measurementswere made at 23C.

X-ray data showed that, after processing, the filmswere microcrystalline with a crystal size below 300 A.These microcrystals showed both the cubic and hexago-nal modifications, with hexagonal predominating. Elec-tron micrographs showed a uniform and smooth surfacewith a very small number of defects.

A primary concern in these measurements was to de-termine whether or not large sican be obtained in nonuniformlyconductors. If secondary photo

I 03

I02 I

>la

r)

0

101 I

l 00 _

102

Applied fie

FIG. 4. Spectral response of

I P. H. Wendland, J. Opt. Soc. Am.

0

./

a;

.4)0

a,

:I

.45 .50 .55 .60.!avelength (microns)

I .65 .70

FIG. 5. Photocurrent as a function of illumination intensity.

econdary photocurrents cathode does not occur, the large space charge fromexcited sensitive photo- trapped holes will remain unneutralized and only smallcurrent injection at the space charge limited currents will flow. With nonuniform

excitation, the magnitude of the photocurrent and theform of the current voltage curve can indicate the rela-tive importance of secondary injection. The spectral

Breakdown - d response of photoconductivity is particularly valuablein evaluating Eqs. (21) and (23), since we can movecontinuously from the wavelength region of ad <1 anduniform excitation to the wavelength region of ad <1and nonuniform excitation. For CdS, the absorptionedge, and thus the beginning of the transition betweenweak and strong absorption, occurs at approximately5200 A. The spectral response of photoconductivity forour samples is given in Fig. 4. While there is a drop inresponse in the strongly absorbed region below 5000 A,this drop is no greater than is observed for sampleswith both electrodes on the same crystal face.

Photocurrent as a function of illumination is plottedin Fig. 5 for the "volume electroded" sample, with 10V/cm applied. Since the source was a tungsten filament,the spectral output included wavelengths that are bothweakly and strongly absorbed. The observed photocur-rent density of 5.5 A/cm2 at 102 ft-c and at 104 V/cm,serves, however, to index the spectral response of Fig.4 in familiar terms, and represents the integrated out-put from all components of the spectral response.Since the drop in photoresponse between 0.60 and 0.42 is no greater than a factor of 10, the large observed

103 104 photocurrent density for both weak and strong absorp-tion demonstrates that large secondary photocurrents

Id (volts/cm) can be driven through nonuniformly excited sensitivephotoconductivity. photoconductors. The shift at 1 ft-c from a linear to a

square root dependence of photocurrent on illumination52, 51 (1962). intensity is indicative of the fact- that the simple energy

PHOTOCONDUCTIVITY 1041

traps as well as additional recombination centers wouldbe expected in the most general case. The distribution ofholes among the recombination centers is determinedby the position of the hole demarcation level, whichthereby influences the photosensitivity. 10 Changes in

temperature and illumination level, leading to a motionof the demarcation level, may therefore be expected toproduce changes in the functional relationship betweenphotocurrent and illumination intensity. However, inthe range of illumination levels in which holes princi-pally occupy the compensated acceptor centers, the fore-going analysis would be expected to best correlate re-sults. For CdS, this situation occurs at the higher lightlevels.6 A distribution in energy of electron traps, ratherthan a single level, would also be expected to affect a

3 12 1 l change in the relationship between photocurrent andI 102 0~ 102 i03 applied voltage, if one makes direct comparisons with

llulnination Intensity (Yont-Candks) the results of one-carrier space charge limited current

1G. 6. Photocurrent as a function of applied field. flow.With due consideration for the necessarily simple

cture of our theoretical model is not completely energy level scheme of the theoretical model, the experi-

it. However, the high light intensities bound the mental data gives qualitative support to the expressions

where the model should best describe the real derived for the dependence of photocurrent density on

on, and this is where correlation is found. applied field and illumination intensity. Particularly,

dependence of photocurrent on applied field is the evidence is in favor of secondary injection as a means

i Fig. 6, with a constant illumination level of 10 of neutralizing the trapped hole space charge, and pro-

reakdown occurs at 104 V/cm. The photocurrent viding large observed photocurrents that do not exhibit

the 1.4 power of the applied field throughout the the high-power dependence on voltage that is char-Asanin. bnth weak and strong absorption occur acteristic of space-charge limited currents.' -0 -A - -w--n _ o_ - -

simultaneously. With monochromatic light of 0.4 5-Awavelength, the photocurrent varies as the 1.5 power ofthe applied field.

DISCUSSION

The basic model used in the analysis, with one com-pensated acceptorcenter and one shallow electron trap-ping center, is certainly simplified over the actual situa-tion. In particular, a distribution in energy of electron

ACKNOWLEDGMENTS

The author gratefully acknowledges the assistanceof Mr. S. C. Requa in obtaining the experimentaldata, and Mr. J. M. Hanlet in obtaining the electronmicrographs.

'0 A. Rose, RCA Review 12, 362 (1951).

105

104

103

lo1

level pisufficieiregionsituatic

Thegiven iift-c. Brgoes asran ae

Vol. 531042 PAUL H. WENDLAND

.t�

I

I

r:IIII