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Role of injecting contacts, Schemes and Mind Maps of Physics

Role of injecting contacts in semiconductor physics

Typology: Schemes and Mind Maps

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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
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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.

  1. ST&KMANN F., z. Phys. 147, 544 (1957).
  2. ROSEA., HeZv.** Phys. Actu **30, 242 (1957).
  3. REDINGTONR. W.,** J Appl. Phys. 24, 189 (1958). 5._ ROLEA. and LAMPERTM. A., Phys. Reo. 113, 1227 (1959).