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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.
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
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
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
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
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
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.
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.
5-l = -
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
several years ago by ~T~XMANN, (2) ROSE(~) and REDINGTONW under a wide variety of circum-
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
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
SC V+^ Vsc
FIG. 3. Variation of M = JVA/JVT with voltage V for two simple photo-
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
are not electron traps, i.e. are not in quasi-thermal equilibrium with the free electrons. Recombina-
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
injected into the insulator, J$~T is a constant in Fig. 3(b). In this case space-charge-limited
electronic structure of the insulator and the light where _,@-A = Jr,,& the number of empty re- excitation level, as shown above. Thus, in the combination^ centers.^ Since^ MT^ =^ J^ as^ pre-
increases linearly with voltage. This result clearly its maximum value of_Hr,R/N, attained at voltage
be a steady photocurrent Jo flowing in response to ,yi.,
F
Mlp& = ~~~TNT~‘/’ (4)
occurs at some position PO near the cathode with^ F^ the absorbed light flux. Correspondingly,
diffusionmodel for the energy barrier,(%AV N kT
energy barrier to shift from PO to PI is the contact-
T~,~.Between POand PI tl-rereare NT, c = AN&i negative clrarges, where A is the cross-sectional
JO these negative charges are all compensated
interface. At the higher current ZJo they must
of 5. Since the applied voltage is unchanged,
0
70,C rr
o=- J-T&
current JO, i.e. whether it is an ohmic or a space-
THE ROLE OF INJECTING CONTACTS IN PHOTOCONDUCTORS 195
FIG. 7. Drift-length diagram illustrating the voltage threshold for double injection. The G$ denote successive drift lengths for holes.
kth 1: - 2 qqow tcpvth
VM (^) LOG V Vth
FIG. 8. Current (J-voltage (V) characteristic, on log-log plot, for the model of Fig. 6.
the condition:(s) tp,M = L2/ppV’ M TpTp,h&h, a
Vth/2 and VM.
must be in the insulator a space charge Qth = CVth,
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.