The tunneling behavior of Schottky barriers has been investigated by several authors. The I-V characteristics exhibit an exponential form in the forward direction which can be used to determine the energy vs complex momentum dispersion relation for charge carriers in the forbidden

gap. In this paper we show that under proper conditions the presence of traps can increase the tunneling probability and result in a reduction in the slope of the log I vs V characteristic by a factor of 2

Mead, C. A.

Parker, G. H.

Caltech Authors - Main

THE EFFECT OF TRAPPING STATES ON TUNNELING IN METAL
SEMICONDUCTOR JUNCTIONS
G. H. Parker and C. A. Mead 
 
Citation: Applied Physics Letters 14, 21 (1969); doi: 10.1063/1.1652641 
View online: http://dx.doi.org/10.1063/1.1652641 
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Published by the AIP Publishing 
 
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Volume 14, Number 1 APPLIED PHYSICS LETTERS 1 January 1969 
3G. Quentin and J. M. Thuillier, Proc. Intern. 
Con/. Phys. Semiconductors, Kyoto, p. 493 (1966). 
4K. Hess, G. Nimtz, and K. Seeger, Solid State 
Electron. (to be published). 
9 R. B. Parfen'ev, A. M. Pogarskij, I. I. Farb-
shtejn, and S. S. Shalyt, Sov. Phys.-Solid State 4, 
2630 (1963). 
lOR. Wagner, Z. Physik 193, 218 (1966). 
5 J. D. Zook, Phys. Rev. 136, A869 (1964). 
6 R. Stratton, Proc. Roy. Soc. (London) 246 A, 406 
(1958); after eliminating typographical errors 
Conwell,12 p. 197, reproduced the treatment. 
11 J. H. Mendum and R. N. Dexter, Bull. Am. Phy s. 
Soc. 9 (1964). 
7 G. Lucovski, R. C. Keezer, and E. Burstein, 
Solid State Commun. 5, 439 (1967). 
BR. Geick, P. Grosse, and W. Richter, First In-
ternational Symposium "Physics of Selenium and 
Tellurium," Montreal, October 1967. 
12 E. M. Conwell, "High Field Transport in Semi-
conductors," in Solid State Physics, Suppl, F. Seitz, 
D. Turnbull, and H. Ehrenreich, Eds. (Academic 
Press Inc., New York, 1967), Vol. 9, p. 82. 
13 R. S. Caldwell and H. Y. Fan, Phys. Rev. 114, 
664 (1959). 
THE EFFECT OF TRAPPING STATES ON TUNNELING IN METAL-SEMICONDUCTOR 
JUNCTIONS 
C. H. Parker and C. A. Mead 
California Institute of Technology 
Pasadena, California 91109 
(Received 28 October 1968; in final form 27 November 1968) 
The tunneling behavior of Schottky barriers has been investigated by several authors. The I-V 
characteristics exhibit an exponential form in the forward direction which can be used to deter-
mine the energy vs complex momentum dispersion relation for charge carriers in the forbidden 
gap. In this paper we show that under proper conditions the presence of traps can increase the 
tunneling probability and result in a reduction in the slope of the log I vs V characteristic by a 
factor of 2. 
Schottky barriers were fabricated by evaporating 
palladium onto n-type CdTe, prepared by cleaving 
on the (110) crystal plane and immediately placing 
the crystal in an ion-pumped vacuum system. The 
palladium was evaporated through a metal mask in 
a vacuum of 10- 7 Torr. The carrier concentration 
was determined in each case by (capacitancet 2 VB 
voltage plots. These plots showed no significant 
change from room temperature to 77°K. 
The I - V characteristics of diodes having differ-
ent carrier concentrations are shown in Fig. 1. The 
upper portion of the curves agrees (± 10%) with 
that expected from tunneling calculations,',2 includ-
ing a slight temperature dependence, presented by 
Padovani and Stratton l where 
I ex: exp{-s(CfiB - V - ~)} 
and 
s = ~ e~:t) V2 tanh[fT ~ (:~JV2J . 
-I 
-2 
-3 
" 
J 
~
- -4 
o 
<5 
o 
--' 
-5 
-6 
-7 
T= 77 OK 
Pd - '"n'" CdT. 
.2 .3 .5 .6 
v 
Fig. 1. I-V characteristics of Pd ''n'' CdTe 
barriers for several carrier concentrations 
(Jo = 1 A/cm2). 
fers by a factor of 2 (± 10%) from that predicted 
Here I is the current, CfiB the barrier energy, V the 
applied voltage, ~ the Fermi energy, Ii Planck's 
constant, m* the effective mass of the electron, E 
the dielectric constant of the semiconductor, ND 
the carrier concentration, q the charge on the elec-
tron, k Boltzman's constant, and T the absolute 
temperature. 
by Eq. (1). We attribute this change in slope to tun-
neling via an intermediate state as illustrated in 
Fig. 2. 
The lower portion of the log I vs V curves dif-
The tunneling rate R (and hence the current) 
arising from electron tunneling through the forbid-
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Volume 14, Number 1 APPLIED PHYSICS LETTERS 1 January 1969 
(0) (b) 
(0) 
LOG J 
v 
Fig. 2. Tunneling through a Schottky barrier 
with and without trapping center. 
den gap3-5 of a semiconductor is of the form 6 
R ($; exp {- 2 iX:2 kdX} , (3) 
where k is the complex momentum wavevector of 
the electron. The integral is taken from the con-
duction band at x 0 to the metal at x 2. 
Padovani and Stratton l have completed the de-
tailed calculation for tunneling through a Schottky 
barrier by assuming a parabolic energy momen-
tum dispersion relation with the result given in 
Eq. (1). However, if traps are present we must also 
concern ourselves with a two-step tunneling proc-
ess with the trap as an intermediate state. In the 
forward direction, (1) the electron tunnels from 
the semiconductor conduction band to the state and 
then (2) from the state to the metal. If the concen-
tration of states is given by Nt and F is the proba-
bility of occupation of the state by an electron, then 
the number of occupied states is NtF. Rewriting 
Eq. (3) in terms of the new limits we find that the 
rates for the two steps are 
Rl =C1Nt (1- !)exP{-2J
X1 kdX} (4) 
Xo 
and 
where the factors C l and C2 are slowly varying 
functions of the voltage, etc. 
(5) 
Under steady-state conditions, F will adjust it-
self until two rates are equal. Thus 
ClC2Nt exp(-2 {Jx:1kdX + i~~2kdX}) 
R 1 = R2 = {~} {- x } 
C1exp -2)x:1kdX +C 2 exp -2 JX1
2kdx 
(6) 
Equation (6) represents a peaked function which, 
22 
provided C l and C2 are not too different, attains its 
maximum value when 
exp{-2Jx:2kdx}=exp{-2J~lkdX}. (7) 
The rate for the completed transition will be pro-
portional to the peak value of Eq. (6), which is just 
the square root of the expression for single-step 
tunneling. 
R ex: exp{ - iX:2 kdX} . (8) 
The number of traps and their distribution in ener-
gy will determine the magnitude and range over 
which Eq. (8) is valid. If states are available in the 
context described above, we can repeat the pro-
cedure used by Padovani and Stratton 1 and show 
that the current is now of the form 
I 0: exp { - i S (qJ B - V - ~)} • (9) 
The slope of the log I vs V curves has been re-
duced to 1/2 5 . An analogous situation is known to 
occur in p -n junctions 7-9 at low biases for the 
case of recombination via intermediate states in 
the space-charge region. There the states change 
the slope of the log I vs V curve from q/kT to 
q/2kT. 
The wide range of energy over which tunneling 
occurs via intermediate states can be attributed to 
thermal broadening of the peak of Eq. (6) at 77°K. 
The diode I-V characteristics began to exhibit 
structure in the low applied voltage range when the 
samples were cooled to 4.2°K, thus confirming that 
trapping states are responsible for the observed 
behavior. The method described here is a potential-
ly powerful technique for studying the distribution 
in energy of traps in these materials. The current 
should show a maximum (or at least a minimum in 
d logI/dV), where qJ - V - ~ is equal to twice the 
trap energy [Eq. (7)]. 
In samples fabricated from a different ingot of 
CdTe the two-step tunneling process was found to 
dominate the entire measurable forward I-V char-
acteristic. Guard-ring experiments and area-depen-
dent I-V characteristics indicated that anomalous 
edge effects were not responsible for the observed 
behavior. Thus we have shown that the presence of 
traps can under the proper conditions significantly 
alter the tunneling characteristics of a Schottky 
barrier diode, and can be unambiguously identified 
by the slope of the log I vs V plot. 
We wish to thank T. C. McGill for many helpful 
discussions, and L. van Atta, G. Picus, and N. Kyle 
of the Hughes Research Laboratories for supplying 
the CdTe, and an anonymous reviewer for valuable 
suggestions. The work was supported in part by the 
U.S. Office of Naval Research. 
1 F. A. Padovani and R. Stratton, Solid State Elec-
tron. 9, 695 (1966). 
2 J. W. Conley and G. D. Mahan, Phys. Rev. 161, 
681 (1967). 
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Volume 14, Number I APPLIED PHYSICS LETTERS I January 1969 
3 F. A. Padovani and R. Stratton, Phys. Rev. Let-
ters 16, 1202 (1966). 
4G. H. Parker and C. A. Mead, Phys. Rev. Letters 
21, 605 (1968). 
5M. F. Millea, M. McColl, and C. A. Mead (to be 
published in Phys. Rev.). 
sR. Stratton, J. Phys. Chem. Solids 23,1177 
TIME AVERAGE HOLOGRAPHY EXTENDED* 
(1962) . 
7W. Schockley and W. T. Read, Phys. Rev. 87, 
825 (1952). 
8R. N. Hall, Phys. Rev. 83,228 (1951); 87, 387 
(1952). 
9 C . T. Sah, R. N. Noyce, and W. Schockley, Proc. 
IRE 45, 1228 (1957). 
C. C. Aleksojj 
Radar and Optics Laboratory, 
Willow Run Laboratories, 
Institute of Science and Technology, 
The University of Michigan, 
Ann Arbor, Michigan 48107 
(Received 8 November 1968) 
It is demonstrated that the usefulness of time-average holography in analyzing vibration can be ex-
tended by modulating the reference beam. 
Powell and Stetson have shown that time-average 
holography can be used to analyze vibrations. 1 This 
letter will describe how time-average holography 
can be extended in its range of usefulness by modu-
lating the reference beam. 
A basic concept in time-average holography is 
that a sinusoidally vibrating object will phase 
modulate light reflected from it with a modula-
tion depth proportional to the amplitude of vibra-
tion. Consider a point, with coordinate vector x, on 
the vibrating object. Light scattered from this point 
to a point on the photographic plate will be phase 
modulated by the function a(x)cos[wf + Q! (x)], where 
a(x) is the modulation depth, w the angular fre-
quency of vibration, and Q! (x) the phase of modula-
tion. Powell and Stetson demonstrated that the 
brightness of the reconstructed point is propor-
tional to J~(a), where Jo(a) is the zero-order Bessel 
function. Thus fringes are formed on the recon-
structed object from which the vibration ampli-
tudes of the object can be analyzed. However, note 
that as the amplitude of vibration increases, the 
fringe peaks decrease, and it becomes difficult 
to analyze the vibrations. Further note that the 
phase information Q! (x) is lost. 
Both of these deficiencies can be corrected by 
phase modulating the reference beam with the func-
tion bcos(wt + (3). It can then be shown2 that the ob-
ject point will reconstruct with a brightness pro-
portional to J~(c), where c is given by the cosine 
law 
c 2 = a 2 + b 2 - 2 ab cos(QI - {3). 
Thus the fringes are no longer formed about zero 
* This work was sponsored in part by the USAF 
Avionics Laboratory, Wright-Patterson Air Force 
Base, Ohio. 
vibration amplitude, but about a vibration ampli-
tude determined by the modulation depth and phase 
of the reference beam. 
These results have been experimentally demon-
strated by observing the mode structure of a reso-
nant crystal. A l-in.-cube ADP crystal (0° cut) 
was electrically excited into standing shear-wave 
modes. Such shear modes have been studied by 
crossed polaroid techniques.3 Figure l(a) shows 
the fifth-order shear mode in a crossed polaroid 
system. 
For the hologram exposure collimated light is 
directed along the z axis of the crystal and is plane 
polarized at 450 to the crystal's x and y axes. Thus 
the output light remains plane polarized, and is 
phase modulated by the changing index of refrac-
tion. The output light is then incident onto a diffus-
ing plate, and a hologram is made of the diffusing 
plate. Figure l(b) shows the virtual image recon-
structed from a normal time-average hologram of 
the fifth-order shear mode. Note that the nulls re-
construct brightest. Figure 1(c) is the reconstruc-
tion when the reference beam is modulated with a 
modulation depth and phase approximately equal to 
the peak of the central lobe of the fifth-order shear 
mode. Note that the central lobe is brightest, while 
the adjacent lobes, vibrating 1800 out of phase with 
the central lobe, are less bright. A 1800 phase 
shift in the reference beam modulation inverts the 
role of the lobes as is shown in Fig. 1(d). 
The reference beam modulation for Figs. 1(c) 
and 1(d) was obtained by sending a narrow refer-
ence beam through the same crystal used as the 
object. Thus any modulation depth and phase that 
the object can produce can also be used as the 
reference bias. In fact the entire light distribu-
tion of the end face of the crystal has been imaged 
23 
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The Effect of Trapping States on Tunneling in Metal Semiconductor Junctions

http://authors.library.caltech.edu/52898/1/1.1652641.pdf

The Effect of Trapping States on Tunneling in Metal Semiconductor Junctions

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