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Current-voltage dependencies of heterogeneous semiconductor systems
J. Phys. Chem. Solids, 1974, Vol. 35, pp. 865-869.
Pergamon Press.
Printed in Great Britain
C U R R E N T - V O L T A G E D E P E N D E N C I E S OF HETEROGENEOUS SEMICONDUCTOR SYSTEMS* P. MARK and B. SANG LEE Department of Electrical Engineering, Princeton University, Princeton. N.J. 08540, U.S.A. (Received 17 September 1973)
Abstract--Computed current-voltage (J-V) dependencies of heterogeneous (powder) semiconductor systems reveal an anomalous dependence between the constant-voltage current J and the uncompensated donor (acceptor) concentration N. Over a range of N ( N , < n < N2) of approximately one decade, J decreases by as much as four decades with increasing N. For N > N.., the grain Schottky barrier thickness d is less than the grain half-width 1/2, the grain surface potential V, is almost independent of N and the J - V dependence is superlinear. For N, < N < N2, d > I/2, V, decreases linearly with N, J increases strongly with decreasing N and the J - V dependence is superlinear. For N < N,, d > I/2. V, ~ V,h ( = k T / q ) and J ~ NV. The phenomenon is used to account for some observed J - V dependencies with column II-chalcogenide and ZnO powder semiconductor systems (electro-optic displays, electrophotographic receptors and heterogeneous catalysts).
I. INTRODUCTION Wide bandgap semiconductor materials are frequently used as heterogeneous systems in the form of powders whose grain diameters range from less than one micron to several tens of microns. Their principal applications lie in the fields of optoelectronics and catalysis. In the former, the powder grains are usually embedded in a binder for mechanical rigidity. Application is found chiefly as electrophotographic receptors [ 1-11 ] and as components in electro-optic displays [12-24]. Other applications, e.g. bi-stable switches [15, 25, 26], have also been explored. As catalysts[27-29], the powder grains are held in either fixed or fluid bed reactors to maximize access of the reactants to the grain surfaces. In both applications, the electronic properties of the powder grains are important as they give insight into the operating mechanism. Experience with semiconductor powders has revealed two general classes of behavior. The column II-chalcogenide powders generally display superlinear current-voltage ( J - V ) characteristics[9-16]. The superlinear characteristics are usually attributed to intergrain barriers and are explained in terms of Schottky emission or tunneling[15, 16]. Occasionally certain specially treated powders can also be made to exhibit a switching effect [15, 25, 26] which has been attributed to a *Supported by the Office of Naval Research. 865
regenerative internal photoconductivity triggered by double injection[26]. On the other hand, the ZnO-based electrophotographic receptors [5-8] and catalysts[30] generally display linear J - V dependencies as though they were devoid of intergrain barriers. We attempt here to reconcile this apparent conflict of behavior with the aid of a model based on unipolar, low-field conduction through symmetric intergrain barrier junctions that originate from surface states capable of localizing electronic charge from the grain interiors. We investigate the J - V b e havior for different semiconductor parameters and grain sizes in the diffusion-controlled, tow-field limit [31 ]. Our major finding is an anomalous dependence between the constant-voltage current and the uncompensated donor (or acceptor) concentration N. This is shown in Fig. 1 which presents the computed J - N dependence at 5 V applied to a 0.05 cm thick powder-binder sample with powder grain diameter l = 2 x 10-' cm (2/x). The parameters next to the points on the curve are the corresponding Schottky barrier heights (surface potential) V,. The remaining semiconductor parameters used in this computation are given in the figure caption. There is a range of N of about one decade, bounded b y N~ and N2, over which the current decreases with increasing N. Selected computed J - V characteristics spanning this anomalous region are plotted in Fig. 2. F o r N > N2, the Schottky barrier thickness d <
P. MARK and B. SANG LEE
866 16 IO
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1/2 and V, d e p e n d s only w e a k l y on N [32]. H e r e the J - V d e p e n d e n c e is similar to than of a f o r w a r d biased junction. In the a n o m a l o u s range N, < N < N2, d >1/2 and the relation b e t w e e n V, and N b e c o m e s [33, 34]
i
'
14 IO
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V, = qNI2/8e 0271
i O i3
0 520
/
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e 0469
12 10
ao 459 0362e
II IO
~lbaDO 439 0 454 N I
10
, I O I O 12
N 2
I,
I,
I O 13
I O 14 N
,
I O 15
I O 16
I O 17
( ¢m-5l
Fig. 1. Computed dependence of the constant-voltage current (units: J/q~) on the uncompensated donor concentration N. The following material parameters were used. Bandgap: 1.5 eV; dielectric constant: 10; acceptorlike surface state of concentration 10"cm -~ located 0.55 eV below the conduction band; I = 2 x 10-" cm; L = 0.05 cm; bulk donor level 0.05 eV below the conduction band. The curve was computed assuming 5 V applied. The numerical parameters next to each point is the corresponding surface potential in eV. 17 IO
I
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i O Is
I
/
/oo.o.
I°I
0.161
/ / 14
,gs
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O. 69
6.6,,,o,,/
~
/
in practical units, where q is the electronic charge and e is the d.c. permittivity of the s e m i c o n d u c t o r . H e n c e V, d e c r e a s e s linearly as N d e c r e a s e s , which is the source of the strong increase of J with decreasing N. The lower end of the a n o m a l o u s region N = N, c o r r e s p o n d s a p p r o x i m a t e l y to V, = V,h = k T / q . H e n c e for N < N,, the J - V and J - N d e p e n dencies are linear; the current is Ohmic. The a n o m a l o u s region is shifted to higher values of N as l is d e c r e a s e d , as anticipated by e q u a t i o n (1) with V, essentially constant. F o r a one d e c a d e reduction in grain size, the a n o m a l o u s region is shifted u p w a r d on the N - s c a l e by about two decades which makes N, ~ 5 x 10'Scm -3. H e n c e we e x p e c t a strong t e n d e n c y for fine-grain p o w d e r s to display an O h m i c J - V d e p e n d e n c e . 2. MODEL
FOR
d <
o. . . .
V ( x ) = (V,/d2)(2xd - x 2)
0.362
Vs
4, , I0
I / 2 ( N > N2)
We adopt the c o n v e n t i o n a l o n e - d i m e n s i o n a l model for the s c h e m a t i c band diagram of the h e t e r o g e n e o u s system[15]. As s h o w n in Fig. 3. it consists of a series of grains e a c h of which is c h a r a c t e r i z e d by a band structure having a surface barier. W e a s s u m e each grain to be n - t y p e and of d i a m e t e r I. To derive the J - V b e h a v i o r we single out a r e p r e s e n t a t i v e intergrain j u n c t i o n defined on the figure by x = 0 . A c c o r d i n g to the diffusionlimited a p p r o x i m a t i o n , the barrier potential profile V ( x ) , m e a s u r e d positive d o w n w a r d , is given by [31]
id 2
I0
(1)
o
6*"I~'%J" I
,
4
/
,
~
Ni
/
a
|
I0
I00
v (volts)
Fig. 2. Computed current-voltage curves for the Nvalues listed on the left side of the figure. The underlined N-values signify N ~ N2. The numerical parameters on the right side of the figure are the corresponding surface potentials in eV. The same materials parameters as in Fig. 1 were used.
V(x
i
I i
t
J
(2)
~
Ec
E v IL x
d
Fig. 3. Schematic energy bands of the semiconductor powder grains for the case d < 1/2 with no applied voltage.
Current-voltage dependencies of heterogeneous semiconductor systems so long as the barrier thickness d is less than 1[2. The surface potential V, = V ( d ) and the barrier thickness d are related by (3)
V~ = qNd2/2~.
The relation between V, and N depends on the semiconductor bulk and surface parameters and must be computed numerically from the over-all charge neutrality condition[32]. If a bias VA smaller than Vs is applied to the junction, the voltage will drop almost completely across the depletion region with VR on the right and VL on the left (VA = VL + VR) as shown in Fig. 4. The respective barrier thicknesses are dL2= 2E(V, + V,.)/qN and dR2 = 2 e ( V , - VR)/qN with VR < V~. If we assume that the intergrain charge in V$ ~ V R
3_
867
ing with respect to x. We obtain[31] J/qlx = V, h N [ e x p - V J V , h] x / [(exp V./V,h)-- 1_] [ d . r [ e x p - V ( x ) / V , h ] dx Jo
(7)
with VR related to the applied voltage V through equations (4) and (5). The denominator can be integrated in closed form only if the quadratic term in equation (2) can be neglected. This restricts VR to values substantially less than Vs. With this approximation, we obtain[31] J/qp. = (2q/e) It2Nal2(Vs - VI~)at2[exp - V, / Vth]
x t~[I [(exp V . [ V , h ) - 1]
-
-Up-
-
"~
37v,.lr
(8)
The curves of Figs. 1 and 2 in the range N > N2 were computed using equations (4), (5) and (8) with VR/V, <~0.2 and using numerically computed V s - N dependencies [32].
t ~-f
I- . . . . - " 3. M O D E L
I I
J
I
I I
dL
da
Fig. 4. Schematic energy band diagram for one junction of the structure of Fig. 3 with voltage VA applied to the junction.
F O R d > 1 / 2 ( N < Nz)
When d > 1/2, the grain is depleted and the semiinfinite assumption embodied in equation (2) is invalid. This can happen either by considering a smaller grain size, by decreasing N below N2 or by increasing the charge in the intergrain surface states[32]. Then the pertinent energy band diagrams are those shown in Figs. (5a) and (5b) for the Y S
the surface states is not affected by Va, then by charge neutrality d L + dR = 2d and VR is calculated to be
•
•
•
• •
°
•
•
•
Vim Ef V(
VR = Va + 8 % --[(VA +8V.)2-8VaV~] ''z. (4) The junction voltage VA and the externally applied voltage V related by "CA = (1/ L ) V
0
~/z
(o)
(5)
where L is the sample thickness. The unipolar current density flowing across the junction is given by J/ql~ = V , h ( d n / d x ) - n ( d V / d x )
(6)
where /z is the mobility and n is the conduction band electron concentration. As J ~ J(x), we proceed by multiplying both sides of equation (6) by the integrating factor [exp - V(x)[ V,h ] and integrat-
(b)
Fig. 5. (a) Schematic energy bands of the semiconductor powder grains for the case d > 112 with no voltage applied. (b) Same as part (a) with an applied voltage.
868
P. MARK a n d B. SANG LEE
cases of no applied voltage and an applied voltage, respectively. The current is again given by equation (7) but the barrier potential profile now becomes [33]
ZnO powder grains are always less than 1 tz in dia. [5-8]. Presumably this consistent difference arises from the ways the powders are made or from the requirements of the systems for which they are V (x ) = (4 V, [12)(xl - x"). (9) intended. The parameters of the computations illustrated in Figs. 1 and 2 correspond to the chalcogenide case and the superlinear curves of F o r small applied voltage ValVe-<0"2, we may again neglect the quadratic term in equation (9) Fig. 2 can certainly be used to account for several characteristics[14-16]. Because of the condition from which we obtain d > o r < 1/2 and equation (1), a smaller grain diameter shifts the anomalous region to larger N Y/qlz = (4N/I)(V~ - V a ) [ e x p - V~IVth] values. For example, assume all the parameters the × [(exp VA/Vfh)-- 1]][1 - - e x p - 2 ( V , - Va)/~,Sh] (lo) same as in Figs. 1 and 2 except l and replace that with 1 = 0-1/z = 10-~ cm. The critical value Nj is then increased approximately by the square of the where V, is now given by equation (1) rather than ratio of the new to the old thickness, i.e. (2× by equation (3) and where VA = Vl/2L. The curves 10-'/10-5)-'= 400 or from about 1014 c m -3 to about of Figs. 1 and 2 in the anomalous range N~ < N < 4 × 10 ~6cm -3. With such a system, N would have to N2 were computed using equations (10) and (1). Finally, for N < N~, we assume the Ohmic de- exceed this value in order to observe a superlinear J - V dependence and all smaller N - v a l u e s would pendence lead to a linear J - V dependence. An uncompenJ/qt~ = N V / L (11) sated donor concentration of 4 × 10 '6 cm -3 is rather large for wide bandgap materials especially in because V5 < V,h so that the surface potential no powder form and it is unlikely, in the opinion of the longer limts the current. authors, that the ZnO electrophotographic receptors[8] or catalysts are doped that heavily. 4. DISCUSSION Hence ZnO-binder electrophotographic receptors The principal feature of the analysis, the anomal- would tend to display an Ohmic J - V dependence ous J - N dependence over a restricted range of N, as has often been reported in the literature. Linear has already been mentioned in Section I. The phys- J - V dependencies have also been observed with ical reason for this anomalous behavior is that once ZnO catalyst powders having the same grain size as depletion of each grain has been achieved for N < the ZnO grains in a photoreceptor sheet[30]. As N2, V~ decreases linearly with N[33,34] rather conductivity changes can be induced in a catalyst than very weakly with N [32], as shown by the V~- by the adsorption of the reactants and during a values next to the points in Fig. l, Thus, in the catalytic reaction [27-29, 35], the fact that d.c. conanomalous region a slight reduction in N will have ductivity measurements can be made with a two simultaneous and opposing consequences: a catalyst powder with minimal fear of influence linear reduction in N which will work toward a from surface barriers may prove quite useful in slight decrease in J through the pre-exponential helping to establish chemisorption and catalysis term in equation (10) and a linear reduction in V~ mechanisms. through equation (1) which will work towards a The column II-chalcogenide powders, on the very large increase in J through the e x p - V ~ / V , , other hand, most always are coarser grain than the term in equation (I0). In practice, N need drop ZnO powders. Grain sizes of I g or larger are genbelow N2 by only a factor of 3-5 before the surface erally reported [l l, 15]. Here N, is of the order of potential becomes of the order of k T and hence in- 10 ~3 to 10~5cm-3, as in the above sample computaoperative in limiting the current. tion. Consequently, steep J - V dependencies are The various effects observed with powder sys- usually reported with these materials as their nortems now become plausible. Consider first the mal doping can easily exceed these values. difference between the column II-chalcogenides The switching effect observed with certain speand the ZnO powder-binder systems. One principal cially doped CdSe and CdS pokder-binder distinguishing feature between them as they have systems [25, 26] might also derive from the anomalbeen reported in the literature is their respective ous J - N dependence reported here. Suppose the grain sizes. The chalcogenide powder grains are "special doping" happens to set N just above N,. so usually larger than 1 ~ [ 9 - 1 6 ] (10-'cm) while the that d is slightly less than 1/2. One would then ex-
Current-voltage dependencies of heterogeneous semiconductor systems pect to o b s e r v e a J - V d e p e n d e n c e s o m e w h a t like the l o w e s t c u r v e of Fig. 2. If at s o m e voltage, say, s o m e w h e r e b e t w e e n 20 and 6 0 V w h e r e it is o b s e r v e d in practice, the applied voltage w e r e to induce a change in the sample so that it would c o m ply with a J - V characteristic c o r r e s p o n d i n g to an N - v a l u e slightly less than N:, a large s p o n t a n e o u s increase in the c o n s t a n t - v o l t a g e current could be e x p e c t e d . Such a d e c r e a s e in N with increasing V could o c c u r if there w e r e s o m e (one-carrier) injection present near the switching voltage and if s o m e of the injected charge w e r e trapped by the ionized donors ( N ) in interior of the grain. Acknowledgements--We wish to thank Mr. John T. Wallmark, Jr. for supplying the V~-N data in the computation of Figs. 1 and 2. One of us (P.M.) also thanks Dr. Fred Jeffers of the Bell and Howell Research Laboratories, Pasadena, California for many fruitful discussions.
REFERENCES
1. Schaffert R. M.. Electrophotography. Focal Press, London (1965). 2. Comizzoli R. B., Lozier G. S. and Ross D. A., Proc. I E E E 60, 348 (1972). 3. Mitsui M. J., I E E E Trans. Electron. Dev. ED-19, 396 (1972). 4. Nakamura K., t E E E Trans. Electron. Dev. ED-19, 405 (1972). 5. Young C. J. and Grieg H. G., R C A Rev. 15, 469 (1954). 6. Ruppel W., Gerritsen H. J. and Rose A.. Heir. Phys. Acta 30, 495 (1957). 7. Gerritsen H. J., Ruppel W. and Rose A., Heir. Phys. Acta 30, 235, 504 (I957). 8. Amick J. A., R C A Rev. 20, 753, 770 (1959). 9. Wainer E., Phot. Engng 3, 12 (I952). 10. Kallmann H. and Rosenberg B., Phys. Roy. 97, 1595 (1955).
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11. Smith M. and Behringer A. J., J. appl. Phys. 36, 3475 (1965). 12. Thomsen S. M. and Bube R. H., Rev. scient. Instrum. 26, 664 (1955). 13. Nicoll F. H. and Kazan B., J. Opt. Soc. Am. 45, 647 (1955). 14. DeVore H. B., R C A Rev. 20, 79 (1959). 15. Bube R. H., J. appl. Phys. 31, 2239 (1960). See also Bube R. H. In: Physics and Chemistry of H - V I Compounds (Edited by M. Avert and J. S. Prener) pp. 695-700. Wiley, New York (1967). 16. Miyake T., Japan J. appl. Phys. 10, 427 (1971). 17. Orthuber R. K. and Ullery L. R., J. Opt. Soc. Am. 44, 297 (1954). 18. Kazan B. and Nicoll F. H., Proc. IRE 43, 1888 (1955). 19. Kazan B., RCA Roy. 19, 19 (1958). 20. Nicoll F. H., RCA Rev. ZO, 658 (1959). 21. Hadley C. P. and Christensen R. W., R C A Rev. 20, 670 (1959). 22. Loebner E. E., RCA Rev. 20, 715 (1959). 23. Hook H. O., RCA Rev. 20, 744 (1959). 24. Vecht A. and Werring N. J., J. Phys. D: AppL Phys. 3, 105 (1970). 25. Nicoll F. H.. RCA Roy. 19, 77 (1958). 26. Nicastro L. J. and Offenbacher E. L., RCA Rev. 33~ 357 (1972). 27. Garner W. E., Ado. Catal. 9, 169 (1957). 28. Amigues P. and Teichner S. J., Disc. Faraday Soc. 41, 362 (1966). 29. Chan H. and Prater D. C., Disc. Faraday Soc. 41,380 (1966). 30. Connoy J. V., Ph.D. Dissertation, Department of Chemical Engineering, Princeton University, 1968 (unpublished result). 31. Henisch H. K., Rectifying Semiconductor Contacts, Chapter 7, pp. 179-182, pp. 203-209. Oxford University Press. London (1957). 32. Mark P., Surface Sei. 25, 192 (1971). 33. Goodwin T. A. and Mark P., Prog. Surface Sct 1, 1 (1971). 34. Baidyaroy S. and Mark P., Surface Sci. 30, 53 (1972). 35. Chong V. M., Connoy J. V. and Mark P., Phys. Status Solidi (a) 9, K133 (1972).
Pergamon Press.
Printed in Great Britain
C U R R E N T - V O L T A G E D E P E N D E N C I E S OF HETEROGENEOUS SEMICONDUCTOR SYSTEMS* P. MARK and B. SANG LEE Department of Electrical Engineering, Princeton University, Princeton. N.J. 08540, U.S.A. (Received 17 September 1973)
Abstract--Computed current-voltage (J-V) dependencies of heterogeneous (powder) semiconductor systems reveal an anomalous dependence between the constant-voltage current J and the uncompensated donor (acceptor) concentration N. Over a range of N ( N , < n < N2) of approximately one decade, J decreases by as much as four decades with increasing N. For N > N.., the grain Schottky barrier thickness d is less than the grain half-width 1/2, the grain surface potential V, is almost independent of N and the J - V dependence is superlinear. For N, < N < N2, d > I/2, V, decreases linearly with N, J increases strongly with decreasing N and the J - V dependence is superlinear. For N < N,, d > I/2. V, ~ V,h ( = k T / q ) and J ~ NV. The phenomenon is used to account for some observed J - V dependencies with column II-chalcogenide and ZnO powder semiconductor systems (electro-optic displays, electrophotographic receptors and heterogeneous catalysts).
I. INTRODUCTION Wide bandgap semiconductor materials are frequently used as heterogeneous systems in the form of powders whose grain diameters range from less than one micron to several tens of microns. Their principal applications lie in the fields of optoelectronics and catalysis. In the former, the powder grains are usually embedded in a binder for mechanical rigidity. Application is found chiefly as electrophotographic receptors [ 1-11 ] and as components in electro-optic displays [12-24]. Other applications, e.g. bi-stable switches [15, 25, 26], have also been explored. As catalysts[27-29], the powder grains are held in either fixed or fluid bed reactors to maximize access of the reactants to the grain surfaces. In both applications, the electronic properties of the powder grains are important as they give insight into the operating mechanism. Experience with semiconductor powders has revealed two general classes of behavior. The column II-chalcogenide powders generally display superlinear current-voltage ( J - V ) characteristics[9-16]. The superlinear characteristics are usually attributed to intergrain barriers and are explained in terms of Schottky emission or tunneling[15, 16]. Occasionally certain specially treated powders can also be made to exhibit a switching effect [15, 25, 26] which has been attributed to a *Supported by the Office of Naval Research. 865
regenerative internal photoconductivity triggered by double injection[26]. On the other hand, the ZnO-based electrophotographic receptors [5-8] and catalysts[30] generally display linear J - V dependencies as though they were devoid of intergrain barriers. We attempt here to reconcile this apparent conflict of behavior with the aid of a model based on unipolar, low-field conduction through symmetric intergrain barrier junctions that originate from surface states capable of localizing electronic charge from the grain interiors. We investigate the J - V b e havior for different semiconductor parameters and grain sizes in the diffusion-controlled, tow-field limit [31 ]. Our major finding is an anomalous dependence between the constant-voltage current and the uncompensated donor (or acceptor) concentration N. This is shown in Fig. 1 which presents the computed J - N dependence at 5 V applied to a 0.05 cm thick powder-binder sample with powder grain diameter l = 2 x 10-' cm (2/x). The parameters next to the points on the curve are the corresponding Schottky barrier heights (surface potential) V,. The remaining semiconductor parameters used in this computation are given in the figure caption. There is a range of N of about one decade, bounded b y N~ and N2, over which the current decreases with increasing N. Selected computed J - V characteristics spanning this anomalous region are plotted in Fig. 2. F o r N > N2, the Schottky barrier thickness d <
P. MARK and B. SANG LEE
866 16 IO
i
,°-y"
1
00452 0 OI61D/~
i
00633 0 0904
1/2 and V, d e p e n d s only w e a k l y on N [32]. H e r e the J - V d e p e n d e n c e is similar to than of a f o r w a r d biased junction. In the a n o m a l o u s range N, < N < N2, d >1/2 and the relation b e t w e e n V, and N b e c o m e s [33, 34]
i
'
14 IO
~r
E u
V, = qNI2/8e 0271
i O i3
0 520
/
:L -)
e 0469
12 10
ao 459 0362e
II IO
~lbaDO 439 0 454 N I
10
, I O I O 12
N 2
I,
I,
I O 13
I O 14 N
,
I O 15
I O 16
I O 17
( ¢m-5l
Fig. 1. Computed dependence of the constant-voltage current (units: J/q~) on the uncompensated donor concentration N. The following material parameters were used. Bandgap: 1.5 eV; dielectric constant: 10; acceptorlike surface state of concentration 10"cm -~ located 0.55 eV below the conduction band; I = 2 x 10-" cm; L = 0.05 cm; bulk donor level 0.05 eV below the conduction band. The curve was computed assuming 5 V applied. The numerical parameters next to each point is the corresponding surface potential in eV. 17 IO
I
i O 16
i O Is
I
/
/oo.o.
I°I
0.161
/ / 14
,gs
2x1040
O. 69
6.6,,,o,,/
~
/
in practical units, where q is the electronic charge and e is the d.c. permittivity of the s e m i c o n d u c t o r . H e n c e V, d e c r e a s e s linearly as N d e c r e a s e s , which is the source of the strong increase of J with decreasing N. The lower end of the a n o m a l o u s region N = N, c o r r e s p o n d s a p p r o x i m a t e l y to V, = V,h = k T / q . H e n c e for N < N,, the J - V and J - N d e p e n dencies are linear; the current is Ohmic. The a n o m a l o u s region is shifted to higher values of N as l is d e c r e a s e d , as anticipated by e q u a t i o n (1) with V, essentially constant. F o r a one d e c a d e reduction in grain size, the a n o m a l o u s region is shifted u p w a r d on the N - s c a l e by about two decades which makes N, ~ 5 x 10'Scm -3. H e n c e we e x p e c t a strong t e n d e n c y for fine-grain p o w d e r s to display an O h m i c J - V d e p e n d e n c e . 2. MODEL
FOR
d <
o. . . .
V ( x ) = (V,/d2)(2xd - x 2)
0.362
Vs
4, , I0
I / 2 ( N > N2)
We adopt the c o n v e n t i o n a l o n e - d i m e n s i o n a l model for the s c h e m a t i c band diagram of the h e t e r o g e n e o u s system[15]. As s h o w n in Fig. 3. it consists of a series of grains e a c h of which is c h a r a c t e r i z e d by a band structure having a surface barier. W e a s s u m e each grain to be n - t y p e and of d i a m e t e r I. To derive the J - V b e h a v i o r we single out a r e p r e s e n t a t i v e intergrain j u n c t i o n defined on the figure by x = 0 . A c c o r d i n g to the diffusionlimited a p p r o x i m a t i o n , the barrier potential profile V ( x ) , m e a s u r e d positive d o w n w a r d , is given by [31]
id 2
I0
(1)
o
6*"I~'%J" I
,
4
/
,
~
Ni
/
a
|
I0
I00
v (volts)
Fig. 2. Computed current-voltage curves for the Nvalues listed on the left side of the figure. The underlined N-values signify N ~ N2. The numerical parameters on the right side of the figure are the corresponding surface potentials in eV. The same materials parameters as in Fig. 1 were used.
V(x
i
I i
t
J
(2)
~
Ec
E v IL x
d
Fig. 3. Schematic energy bands of the semiconductor powder grains for the case d < 1/2 with no applied voltage.
Current-voltage dependencies of heterogeneous semiconductor systems so long as the barrier thickness d is less than 1[2. The surface potential V, = V ( d ) and the barrier thickness d are related by (3)
V~ = qNd2/2~.
The relation between V, and N depends on the semiconductor bulk and surface parameters and must be computed numerically from the over-all charge neutrality condition[32]. If a bias VA smaller than Vs is applied to the junction, the voltage will drop almost completely across the depletion region with VR on the right and VL on the left (VA = VL + VR) as shown in Fig. 4. The respective barrier thicknesses are dL2= 2E(V, + V,.)/qN and dR2 = 2 e ( V , - VR)/qN with VR < V~. If we assume that the intergrain charge in V$ ~ V R
3_
867
ing with respect to x. We obtain[31] J/qlx = V, h N [ e x p - V J V , h] x / [(exp V./V,h)-- 1_] [ d . r [ e x p - V ( x ) / V , h ] dx Jo
(7)
with VR related to the applied voltage V through equations (4) and (5). The denominator can be integrated in closed form only if the quadratic term in equation (2) can be neglected. This restricts VR to values substantially less than Vs. With this approximation, we obtain[31] J/qp. = (2q/e) It2Nal2(Vs - VI~)at2[exp - V, / Vth]
x t~[I [(exp V . [ V , h ) - 1]
-
-Up-
-
"~
37v,.lr
(8)
The curves of Figs. 1 and 2 in the range N > N2 were computed using equations (4), (5) and (8) with VR/V, <~0.2 and using numerically computed V s - N dependencies [32].
t ~-f
I- . . . . - " 3. M O D E L
I I
J
I
I I
dL
da
Fig. 4. Schematic energy band diagram for one junction of the structure of Fig. 3 with voltage VA applied to the junction.
F O R d > 1 / 2 ( N < Nz)
When d > 1/2, the grain is depleted and the semiinfinite assumption embodied in equation (2) is invalid. This can happen either by considering a smaller grain size, by decreasing N below N2 or by increasing the charge in the intergrain surface states[32]. Then the pertinent energy band diagrams are those shown in Figs. (5a) and (5b) for the Y S
the surface states is not affected by Va, then by charge neutrality d L + dR = 2d and VR is calculated to be
•
•
•
• •
°
•
•
•
Vim Ef V(
VR = Va + 8 % --[(VA +8V.)2-8VaV~] ''z. (4) The junction voltage VA and the externally applied voltage V related by "CA = (1/ L ) V
0
~/z
(o)
(5)
where L is the sample thickness. The unipolar current density flowing across the junction is given by J/ql~ = V , h ( d n / d x ) - n ( d V / d x )
(6)
where /z is the mobility and n is the conduction band electron concentration. As J ~ J(x), we proceed by multiplying both sides of equation (6) by the integrating factor [exp - V(x)[ V,h ] and integrat-
(b)
Fig. 5. (a) Schematic energy bands of the semiconductor powder grains for the case d > 112 with no voltage applied. (b) Same as part (a) with an applied voltage.
868
P. MARK a n d B. SANG LEE
cases of no applied voltage and an applied voltage, respectively. The current is again given by equation (7) but the barrier potential profile now becomes [33]
ZnO powder grains are always less than 1 tz in dia. [5-8]. Presumably this consistent difference arises from the ways the powders are made or from the requirements of the systems for which they are V (x ) = (4 V, [12)(xl - x"). (9) intended. The parameters of the computations illustrated in Figs. 1 and 2 correspond to the chalcogenide case and the superlinear curves of F o r small applied voltage ValVe-<0"2, we may again neglect the quadratic term in equation (9) Fig. 2 can certainly be used to account for several characteristics[14-16]. Because of the condition from which we obtain d > o r < 1/2 and equation (1), a smaller grain diameter shifts the anomalous region to larger N Y/qlz = (4N/I)(V~ - V a ) [ e x p - V~IVth] values. For example, assume all the parameters the × [(exp VA/Vfh)-- 1]][1 - - e x p - 2 ( V , - Va)/~,Sh] (lo) same as in Figs. 1 and 2 except l and replace that with 1 = 0-1/z = 10-~ cm. The critical value Nj is then increased approximately by the square of the where V, is now given by equation (1) rather than ratio of the new to the old thickness, i.e. (2× by equation (3) and where VA = Vl/2L. The curves 10-'/10-5)-'= 400 or from about 1014 c m -3 to about of Figs. 1 and 2 in the anomalous range N~ < N < 4 × 10 ~6cm -3. With such a system, N would have to N2 were computed using equations (10) and (1). Finally, for N < N~, we assume the Ohmic de- exceed this value in order to observe a superlinear J - V dependence and all smaller N - v a l u e s would pendence lead to a linear J - V dependence. An uncompenJ/qt~ = N V / L (11) sated donor concentration of 4 × 10 '6 cm -3 is rather large for wide bandgap materials especially in because V5 < V,h so that the surface potential no powder form and it is unlikely, in the opinion of the longer limts the current. authors, that the ZnO electrophotographic receptors[8] or catalysts are doped that heavily. 4. DISCUSSION Hence ZnO-binder electrophotographic receptors The principal feature of the analysis, the anomal- would tend to display an Ohmic J - V dependence ous J - N dependence over a restricted range of N, as has often been reported in the literature. Linear has already been mentioned in Section I. The phys- J - V dependencies have also been observed with ical reason for this anomalous behavior is that once ZnO catalyst powders having the same grain size as depletion of each grain has been achieved for N < the ZnO grains in a photoreceptor sheet[30]. As N2, V~ decreases linearly with N[33,34] rather conductivity changes can be induced in a catalyst than very weakly with N [32], as shown by the V~- by the adsorption of the reactants and during a values next to the points in Fig. l, Thus, in the catalytic reaction [27-29, 35], the fact that d.c. conanomalous region a slight reduction in N will have ductivity measurements can be made with a two simultaneous and opposing consequences: a catalyst powder with minimal fear of influence linear reduction in N which will work toward a from surface barriers may prove quite useful in slight decrease in J through the pre-exponential helping to establish chemisorption and catalysis term in equation (10) and a linear reduction in V~ mechanisms. through equation (1) which will work towards a The column II-chalcogenide powders, on the very large increase in J through the e x p - V ~ / V , , other hand, most always are coarser grain than the term in equation (I0). In practice, N need drop ZnO powders. Grain sizes of I g or larger are genbelow N2 by only a factor of 3-5 before the surface erally reported [l l, 15]. Here N, is of the order of potential becomes of the order of k T and hence in- 10 ~3 to 10~5cm-3, as in the above sample computaoperative in limiting the current. tion. Consequently, steep J - V dependencies are The various effects observed with powder sys- usually reported with these materials as their nortems now become plausible. Consider first the mal doping can easily exceed these values. difference between the column II-chalcogenides The switching effect observed with certain speand the ZnO powder-binder systems. One principal cially doped CdSe and CdS pokder-binder distinguishing feature between them as they have systems [25, 26] might also derive from the anomalbeen reported in the literature is their respective ous J - N dependence reported here. Suppose the grain sizes. The chalcogenide powder grains are "special doping" happens to set N just above N,. so usually larger than 1 ~ [ 9 - 1 6 ] (10-'cm) while the that d is slightly less than 1/2. One would then ex-
Current-voltage dependencies of heterogeneous semiconductor systems pect to o b s e r v e a J - V d e p e n d e n c e s o m e w h a t like the l o w e s t c u r v e of Fig. 2. If at s o m e voltage, say, s o m e w h e r e b e t w e e n 20 and 6 0 V w h e r e it is o b s e r v e d in practice, the applied voltage w e r e to induce a change in the sample so that it would c o m ply with a J - V characteristic c o r r e s p o n d i n g to an N - v a l u e slightly less than N:, a large s p o n t a n e o u s increase in the c o n s t a n t - v o l t a g e current could be e x p e c t e d . Such a d e c r e a s e in N with increasing V could o c c u r if there w e r e s o m e (one-carrier) injection present near the switching voltage and if s o m e of the injected charge w e r e trapped by the ionized donors ( N ) in interior of the grain. Acknowledgements--We wish to thank Mr. John T. Wallmark, Jr. for supplying the V~-N data in the computation of Figs. 1 and 2. One of us (P.M.) also thanks Dr. Fred Jeffers of the Bell and Howell Research Laboratories, Pasadena, California for many fruitful discussions.
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