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J. Phys. Chem. Solids Pergamon Press 1961. Vol. 22, pp. 189-197. Printed in Great Britain.
THE ROLE OF INJECTING CONTACTS IN
PHOTOCONDUCTORS
MURRAY A. LAMPERT Radio Corporation of America, RCA Laboratories, Princeton, N. J.
Abstract-It is usually assumed that the only role of the “ohmic,” i.e. injecting, contacts is that of supporting Ohm’s law currents through the photoconductor. In actual fact the injecting contact plays a key role in limiting the available gain (G)-bandwidth (l/70) product of the photoconductor, through the agency of space-charge which is either injected and volume-distributed or spatially localized near the potential minimum at the contact. G/TO has a limiting value M/n where w is the dielectric relaxation time under operating conditions and M is a dimensionless quantity determined by either the volume-distributed or contact-localized space charge. Where the negative contact is injecting for electrons and the positive contact injecting for holes, two-carrier current flow in the insulator is supplied by the contacts. Double injection and photo- conductivity are closely related in that the characteristics of both phenomena are governed by the same recombination centers. Double injection has been analyzed for an idealized insulator with a single set of recombination centers completely filled in thermal equilibrium. Outstanding results are: (i) a threshold voltaee for the double iniection, and (ii) a negative resistance associated with a hole lifetime which increases with injection ievel.
I. INTRODUCTION A.N OHMIC contact is by definition and tradition one that is experimentally invisible, i.e. one that plays no role in electrical measurements other than to permit the flow of Ohm’s law currents
corresponding to the bulk value of conductivity.
On this account one might well wonder how there can be enough content in the subject to merit a published article. The point is that, in actuality, a so-called ohmic contact is generally an injecting contact, i.e. a contact which can inject excess carriers into the solid, thereby promoting non- ohmic current flow. Further, it is a remarkable fact that, even in the regime of (steady-state) Ohm’s law currents, the ohmic contact can make itself felt experimentally, in transient phenomena, e.g. in the response time for photoconductivity. In this paper we shall touch on two areas of problems. First we shall review earlier work by Dr. ROSE and the author, concerned with one- carrier or majority-carrier photocurrent flow. Here the significant physical parameter through which the injecting contact may exert an influence on photoconductivity is space charge. In steady- state photoconductivity the space charge in
question is injected and volume distributed; in transient photoconductivity it may alternatively be spatially localized near the potential minimum at the contact. Then we shall discuss more recent work concerned with two-carrier, or double- injection, current flow in insulators. Here the important physical parameter is the free carrier lifetime-a parameter which is, of course, also central to photoconductivity. The different kinds of contacts a metal can make to the conduction band of an insulator are illustrated by the schematic energy-band diagrams of Fig. 1. The left-hand diagrams correspond to
thermal equilibrium. EC denotes the bottom con-
duction band level, p the thermodynamic Fermi level. (There will also, of course, be numerous localized states in the forbidden gap. These are not shown on the diagrams.) The right-hand figures show the energy diagrams for the corresponding contacts with a voltage applied across the insulator, the polarity being such that the contact shown is negative. The top pair of diagrams represent the ohmic or injecting contact. At the contact, in the downward-bending region, is a reservoir of free electrons available for injection into the bulk 189
M. A. LAMPERT
BLOCKING (RECTIFYING)
FIG. 1. Schematic energy-band diagrams illustrating the different types of contacts to the conduction band. The left-hand diagrams correspond to thermal equili- brium, the right-hand ones to an applied voltage.
of the insulator. At sufficient voltage this reservoir delivers a space-charge-limited, excess electron current into the conduction band-a fact of con- siderable importance for the later discussion. Under an applied voltage there is a potential mini- mum or energy maximum in the insulator near the
contact interface, shown at position Pin the upper-
right figure. This is quite analogous to the famous potential minimum near the cathode of thermionic vacuum tubes operated under space-charge- limited conditions. For future reference we note
that all excess electrons to the right of the energy
maximum P, i.e. in the bulk of the insulator, are
compensated by an equal number of positive charges on the anode contact, not shown in the figures. Also, when an electron leaves the insulator at the anode contact an electron will automatically enter the insulator at the cathode contact, a fact which makes possible high photoconductive gain. This last is also true of the neutral or flat-band contact, middle figures. With this kind of contact an applied voltage is ohmically distributed across the insulator, and only Ohm’s law currents will flow-at least up to the point where temperature- limited, or saturated, current is drawn from the metal. This type of contact is not easy to come by experimentally. Finally, bottom figures, we have the blocking, or rectifying, contact. Here most of the applied voltage is localized at the contact. An electron leaving at the anode can no longer be
replaced by injection at the cathode, and so, in the absence of contact breakdown, the photocon- ductive gain is limited to unity at most. It is im- portant to recognize that under photoexcitation of the insulator the injecting contact may be con- verted into a neutral contact (straight dashed line in upper-left figure) or even a blocking contact
(lower dashed line in upper-left figure) ; likewise
the neutral contact may be converted into a blocking contact. Such effects have been observed experimentally.(l) Everything we have said about contacts to the conduction band applies equally well to contacts to the valence band, using the appropriate, complementary energy-band dia- grams. For the remainder of this paper we restrict our discussion to injecting contacts.
II. MAJORITY-CARRIEFt PHOTOCONDUCTMTY. THE GAIN-BANDWIDTH PRODUCT In discussing one-carrier or majority-carrier photoconductivity we assume, for the sake of definiteness, that the current carriers are electrons.
Photoconductive gain G is defined by: G = AI/eF
where AI is the induced photocurrent and F is
the absorbed light flux producing it. Since an electron leaving at the anode is replaced by an equivalent one entering at the cathode, G is also the number of transits a photo-excited electron makes in its lifetime, i.e.
G+
d-
5-l = -
F
with 71 the electron lifetime and T’ its time of
transit from cathode to anode. The second relation in (1) is the defining relation for -rl,Jtr being the total number of photo-excited, free electrons. There will generally be a much larger number of trapped electrons in quasi-thermal equilibrium with the free electrons. During the photocon- ductive rise the light must supply these trapped electrons as well as the free ones. Therefore the rise time 7s may be written as:
where JV, is the increment inthe total number of photo-excited electrons, free plus trapped, ac- companying an increase in the absorbed light flux
(^192) M. A. L
The time constant rf is the dielectric relaxation time for the photoconductor at the operating point. Equation (3) represents a universal form for the gain-bandwidth product. It is a rather convenient form in which to express this product because of
a very interesting property of M discovered
several years ago by ~T~XMANN, (2) ROSE(~) and REDINGTONW under a wide variety of circum-
stances M, as a function of the applied voltage V,
has a maximum possible value of unity. The
(AMPERT
many injected, excess electrons as are excited by the light. This marks the onset of space-charge- limited current injection, which occurs then at
M = 1. Beyond this voltage X(&T) increases
together with JYA and M = 1 at all higher voltages.
The origin of this maximum M-value of unity lies in the fact that the onset of space-charge-limited current flow occurs where MA = NT. This will be true for other, more complicated, models of photoconductors as well. A key to obtaining M-values exceeding unity is the invocation of
-FC -E
--------- F -----------__
1()() - ___--- ER 100 -
10-ATEV IO-
I - I-
.I - .I -
.Ol
v-, V
SC V+^ Vsc
FIG. 3. Variation of M = JVA/JVT with voltage V for two simple photo-
conductors : left, a trap-free photoconductor with deep-lying recombina-
tion centers, (F-&)/kT > 1 and right a trap-free photoconductor with recombination centers near the Fermi level, IF--ERl/kT < 1.
reason why this is true can be seen by considering localized states in the forbidden gap which a simple photoconductor, namely the trap-free influence^ space-charge-limited^ current^ flow^ but photoconductor with deep-lying recombination
centers (~-FR > kT) illustrated in Fig. 3(a).
are not electron traps, i.e. are not in quasi-thermal equilibrium with the free electrons. Recombina-
In order to compute the variation of M with tion centers are such states since they are in kinetic,
voltage it is necessary to know how NA and NT rather than quasi-thermal, equilibrium with the separately vary with voltage. The anode charge free electrons. Consider, for example, a trap-free 4”~ simply varies linearly with voltage. On the photoconductor where the recombination centers
other hand, so long as excess charge has not been are located within KT of the Fermi level, illustrated
injected into the insulator, J$~T is a constant in Fig. 3(b). In this case space-charge-limited
independent of voltage and dependent only on the current flow will set in at that voltage V,,
electronic structure of the insulator and the light where _,@-A = Jr,,& the number of empty re- excitation level, as shown above. Thus, in the combination^ centers.^ Since^ MT^ =^ J^ as^ pre-
Ohm’s law region of current flow M, hence G/TO viously, M will increase linearly with voltage up to
increases linearly with voltage. This result clearly its maximum value of_Hr,R/N, attained at voltage
holds for any photoconductor. For the particular V,,. This ratio can, of course, be much greater
case of Fig. 3(a) there are no trapped electrons, than^ unity.^ Between^ V,,^ and^ 2V,,,^ Jf^ increases
and so NT = &‘” at all voltages. Now, at that up to NA^ so that^ M^ drops^ to unity,^ and remains
voltage V,, where JYA = JY = JYT there are as so at higher voltages. Detailed analysis of other
THE ROLE OF INJECTING CONTACTS IN PHOTOCONDUCTORS 193
models of photo~onducto~ including electron sweep-out of the ad~tion~ p~t~lectrons excited
traps is given in an article by ROSE(~)and the by the increased light, the light-excited holes
author. being^ trapped^ and^ left^ behind^ in^ the^ photo-
We next consider the influence of the injecting conductor.^ The^ time^ required^ to^ build^ up^ the
contact on the photoconductive transient, i.e., necessary positive charges is T,,,E^ and is given by:
on the response time. Referring to Fig, 4, let there
be a steady photocurrent Jo flowing in response to ,yi.,
some steady level of illumination. The corre- ro,c
Z-----+
F
Mlp& = ~~~TNT~‘/’ (4)
sponding potential minimum or energy maximum
occurs at some position PO near the cathode with^ F^ the absorbed light flux. Correspondingly,
FIG. 4. Transient response of the ohmic (injecting) contact.
interface. Let the level of incident light flux
suddenly be doubled, the applied voltage being
held fixed. The new, steady photocurrent will
also be approximately doubled. In order to deliver
the increased current the energy barrier must
shift downwards, by an amount eAV and to the
left from Po to 5, a distance A. In a simplified
diffusionmodel for the energy barrier,(%AV N kT
and A is a Debye length corresponding to the
total, excess charge density NT at the potential
minimum (instead of just the free, excess charge
density), A = (~kT~ez~T)1/2, where E is the static
dielectric constant. The time T~,~it takes for the
energy barrier to shift from PO to PI is the contact-
controlled response time. It remains to compute
T~,~.Between POand PI tl-rereare NT, c = AN&i negative clrarges, where A is the cross-sectional
area of the photoconductor. At the lower current
JO these negative charges are all compensated
by positive charges to the left, at the catkode
interface. At the higher current ZJo they must
be compensated by positive charges to the right
of 5. Since the applied voltage is unchanged,
these positive charges can only be obtained through
0
the contact-controlled gain-bandwidth product is:
G-=^ Li%,
70,C rr
with 7r the same dielectric
operating point as in (3).
derivation is independent
M
MA
o=- J-T&
relaxation time at the
Note that the above
of the nature of the
current JO, i.e. whether it is an ohmic or a space-
charge-limited current.
In any given photoconductor the controlling
response time is obviously whichever is the longer
time, 7s or 70,~.
The corresponding response time also
determines the gain-bandwidth product.
III. DOUBLE JNJJXTION IN INSULATORS
We turn now to tke subject of two-carrier
current flow in insulators, or double injection.
The schematic energy-band diagram for the
insulator is shown in Fig. 5. At the negative
electrode, or cathode, there is an electron-injecting
contact, as before. Now, at the positive electrode,
or anode, there is a hole-injecting contact which,
THE ROLE OF INJECTING CONTACTS IN PHOTOCONDUCTORS 195
the low-level hole lifetime ~~~~~~is just NR. At
high injection levels, where n >> NR andp + NE,
n and p being the injected free electrons and hole
densities respectively, charge neutrality requires
that n m p. With op > a, the electron and hole
recombination rates can only be balanced if the
recombination centers are largely empty. The
electrons formerly in these centers have been
transferred to the conduction band. Therefore,
at high injection levels the density of electron-
capturing centers to be inserted into the lifetime
formula to obtain the high-level electron lifetime
rta,l,igh is again Nn. With p1M p it follows neces-
sarily that I@&$, w Tn,h$s&the high-level hole
lifetime. These results are summarized as follows:
From (6) ii follows, taking f$$$ RS 1, that
This drastic increase of hole lifetime with in-
jection level between low and high levels means
that, in this region of currents, the more holes
that are injected the “easier” it is for them to
get across the insulator (in the face of recombina-
tion). In fact, it is so much easier that the voltage
required actually decreases as the current increases.
The double-injection current-voltage character-
istic for the model of Fig. 6 contains a regime of
negative resistance.
Another extremely interesting feature of double
injection in the model of Fig. 6 is the existence of a
voltage threshold for two-carrier current flow.
This result, which is contrary to previous
speculation(7) on this question, can be obtained
through a rather simple argument. At low injection
IeveIs the hole Iifetime throughout the insulator
has its low-lifetime value rP,row. Define the drift
length d as the distance a hole moves in one
lifetime: d = vp,drift7p,low = ppc%~p,~ow, With d
the applied field, which varies with position. As
d increases, d increases proportionately. Let
successive drift lengths be measured off, starting
from the anode, as in Fig. 7. At the end of each
drift length the free hole density p decreases by a
factor of approximately two, Because of neutrality
requirements the free electron density n does like-
wise. In order tkat the total current remain
constant, the field d must increase by the same
factor of approximately two over each drift length.
(Variations of p, n and d with position are indicated
in Fig. 7.) Thus each drift length is roughly twice
n<< N_
FIG. 7. Drift-length diagram illustrating the voltage threshold for double injection. The G$ denote successive drift lengths for holes.
as long as the preceding drift length: dz CT2d
d3 -N 2dg N 4d1, etc. Hence the final drift length
df(ds in Fig. 7) 2: L/2, where L is the cathode-anode
spacing. Correspondingly, by definition of drift
length, tke hole transit-time across half of the
insulator = rP,low, so that we have finally the
result:
LZ
kth 1: - 2 qqow tcpvth
where subscript “th” denotes threshold and tP is
the hole transit time. Rigorous matkematical
~alysis(*) gives L’/~~vth = 27, I,,,+ The complete
current-voftage ckaracteristic for the model of
Fig. 6 is shown on the log-log plot of Fig. 8.
Three regimes are apparent: a steeply rising
current near the voltage threshold, a negative
resistance down to some minimum voltage VM, and
finally a square-law dependence of current on
voltage at high currents. Tkis last regime corre-
sponds to injection levels sufficiently high that
n M p % NR so that Q-nM Ql M Tn,high M ‘+,hie;h
as given in (6). The transfer of electrons initially in
the recombination centers Nn to the conduction
band through double injection has been completed.
196 M.^ A.^ LAMPERT
VM (^) LOG V Vth
FIG. 8. Current (J-voltage (V) characteristic, on log-log plot, for the model of Fig. 6.
As a result, at high injection levels the insulator
behaves exactly like a semiconductor with an
equivalent thermal free carrier density no = NR.
The insulator has, in a manner of speaking, been
electronically converted, through double injection,
into a semiconductor. (The same effect could be
produced by shining bandgap light on the
insulator.) The square law shown in Fig. 8 is
precisely that derived by ROSE and the author(s)
for the equivalent semiconductor. At still higher
injection levels, when injected space charge takes
on importance, the square law goes over into a
cube law, not shown in Fig. 8. The low current
end of the square law in Fig. 8 is determined by
the condition:(s) tp,M = L2/ppV’ M TpTp,h&h, a
result essentially identical to that for the equivalent
semiconductor problem. (9) Comparing this result
with (7) We See that v&vM zw ~p,h&&p,~ow %
+,/v~ $. 1. A further feature to be noted in
Fig. 8 is the relatively small variation of current
witk voltage over most of the negative-resistance
regime. J increases by only a factor of five between
Vth/2 and VM.
Upon reflection it will be seen that the double-
injection characteristic exhibited in Fig. 8, which
was derived neglecting space charge, cannot
possibly be valid down to arbitrarily low currents.
For, corresponding to the threshold voltage there
must be in the insulator a space charge Qth = CVth,
where C is the capacitance of the insulator. The
neutr~ity-bred, double-injection solution cannot
be valid until the injected electron charge exceeds
Qm. At lower injection levels the current in the
idealized insulator of Fig. 6 is actually the one-
carrier, space-charge-limited electron current for
a trap-free insulator shown as the dashed line.
On the experimental side, a current-voltage
characteristic such as exhibited in Fig. 8 might
be expected to produce one of two striking effects:
first, because of the negative resistance, spontan-
eous oscillations under d.c. applied voltage, or, if
the osciIlations are quenched by external circuit
resistance, a marked hysteresis in the d.c. current-
versus-voltage curve. Both effects have been
reported for a number of insulators and high-
resistance semiconductors : spontaneous oscilla-
tions in Au-doped Ge,(lO) high-resistivity GaAs,(ll)
CdS(12) and ZnSe(ls); hysteresis in Fe-doped
Ge,(lJ) CdSe(15) and CdS(ls? Almost certainly a
lifetime increasing with injection level is the under-
lying cause in some, and perhaps all, of these
observations.
It remains to establish quantitatively, both by
experiment and theory, the relationships between
photoconductivity and double injection. The basis
for a close relationship has already been suggested
-both depend profoundly on the behavior of the
free carrier lifetimes. The correlations between
infrared quenching excitation and double-
injection “breakdown” observed earlier by
TYLER in his studies of Fe-doped Ge augur
well for future studies of such relationships.
Acknowledgements-The work reported in Section II of this paper was initiated by Dr. A. ROSE and carried out jointly by Dr. ROSE and the author. The author is further pleased to acknowledge many conversations with Dr. ROSE which helped to clarify certain aspects of the double-ejection studies reported in Section III of this paper.
REFERJ3NczREis
**1. RUPPEL W., paper in these Proceedings.
- ST&KMANN F., z. Phys. 147, 544 (1957).
- ROSEA., HeZv.** Phys. Actu **30, 242 (1957).
- REDINGTONR. W.,** J Appl. Phys. 24, 189 (1958). 5._ ROLEA. and LAMPERTM. A., Phys. Reo. 113, 1227 (1959).
