Solid-State Electronics Pergamon
THE
Press 1965. Vol. 8, pp. 395-399.
RICHARDSON
EMISSION
CONSTANT
IN SCHOTTKY
Printed in Great Britain
FOR THERMIONIC
BARRIER
DIODES
C. R. CROWBLL
Bell Telephone
Laboratories,
Incorporated,
Murray Hill, New Jersey
(Received 9 September 1964; in revisedform 28 October 1964)
Abstract-The
Richardson equation appropriate to thermionic emission in Schottky barrier diodes
is derived. For a semiconductor having an energy band with ellipsoidal constant-energy surfaces
in momentum space, the Richardson constant A: associated with a single energy minimum is
where 1, m and n are the direction cosines of the normal to the emitting plane relative to the principal
axes of the ellipsoid and m,, rnU and trzzare the components of the effective mass tensor. In the Ge
conduction band, summation of emission from all the energy minima gives maximum and minimum
ratios of A* to the free electron value A (= 120 A/ cms/“Ks) of 1.19 and l-07 for the <lOO> and
<ill > directions respectively. In the silicon conduction band, maximum and minimum ratios of
2.15 and 2.05 occur for the <ill > and <lOO> directions respectively. The theoretical predictions
are in good agreement with experimental results from W-Si and Au-GaAs diodes.
RCsumC-L’equation
de Richardson appropriee & l’emission thermionique dans les diodes g
barritre Schottky est d&iv&e. Pour un semiconducteur ayant une bande d’energie avec des surfaces
ellipsoi’dales a Cnergie constante dans l’espace de quantite de mouvement, la constante de Richardson
Af associee a un minimum d’energie simple est:
oh I, m et n sont les cosinus directeurs de la normale au plan Cmetteur relatif aux axes principales
de l’ellipso’ide et nz~,my et m, sont les composantes du tenseur de masse effectif. Dans la bande de
conduction de Ge, la somme des emissions de tous les minima d’energie donne des rapports maxima
et minima d’A* B la valeur d’electrons libres A (= 120 A/ cms/‘Ks) d’1,19 et 1,07 pour les directions
<lOO) et <ill ) respectivement. Dans la bande de conduction de silicium, des rapports maxima
et minima 2,15 et 2,OS se produisent pour les directions <ill > et <lOO> respectivement. Les
predictions theoriques sont en bon accord avec les resultats experimentaux obtenus des diodes de
W-Si et Au-AsGa.
Zusammenfaasung-Richardsons
Gleichung fur die thermionische Emission in Schottky Sperrschichtdioden wird abgeleitet. Fiir einen Halbleiter, der ein Energieband mit ellipsenfiirmigen
Oberflachen konstanter Energie im Impulsraum besitzt, ist die mit einem einzelnen Energieminimum verkniipfte Richardsonsche Konstante
qk2
A* = ~~(12m,m~+m~m,m,:+nzm,m,)~~2
wo I,m und rz die Richtungskosinusse
der Normalen zur Emissionsebene in Bezug auf die Hauptachsen des Ellipsoids sind, und mz, my und ms die Komponenten des effektiven Massentensors.
Im Leitungsband von Ge ergibt die Addition der Emission von allen Energieminima maximale
und minimale Verhaltnisse von A* zur freien Elektronenmasse A (= 120 A/cm2rK2)
von 1,19
395
396
C.
R.
CROWELL
und 1,07 fiir die <lOO )- und (111 )-Richtungen.
Im Leitungsband van Silizium findet man
maximale und minimale Verhaltnisse van 2,15 und 2,OS fiir die <ill )- und <lOO )-Richtungen.
Die theoretischen Voraussagen stehen in guter iibereinstimmung
mit Versuchsergebnissen
an
W-Si- und Au-GaAs-Dioden.
CURRENT interest in the application
of the thermionic emission
model to the explanation
of the
current-voltage
characteristics
of surface barrier
diodes is exemplified
by the work of KAHNG on
Au-SitI)
and Au-GaAst2)
diodes, that of GOODMAN@) on metal CdS diodes, that of CROWELL,
SARACE and SZE(~) on W-semiconductor
diodes as
well as the Schottky
effect measurements
of SZE,
CROWELL and KAHNG@) on Au-Si diodes. Use of
the thermionic
emission model originally proposed
by BETHE@) raises the question of the appropriate
effective
mass
m* to use in evaluating
the
Richardson
constant
A* = 4nm*qk”/h3,
(1)
where q is the electronic
charge, k Boltzmann’s
constant,
and h Planck’s constant.
The use of the free electron
mass for conventional
thermionic
emitters
(yielding
A = 120
A/cms/“Ks)
is attractive
since for saturation
or
space charge limited
emission
into vacuum,
an
electron is only emitted if it surmounts
a potential
energy maximum in the vacuum outside the emitter.
The emitter is considered
to maintain
an equilibrium distribution
of charge in the vacuum region
between
the emitter
and potential
energy maximum.
This
is emphasised
in HERRING and
?31CHOLSt7) derivation
of the Richardson
equation
and is implicit
in the Epstein-Fry-Langmuir
analysis@) which
has been further
clarified
by
FERRIScg) and slightly modified by CROWELL@) to
include
the effect of the emitter
work function
more explicitly.
Thus the universal use of the free
electron mass to predict the emission into vacuum
is entirely plausible.
In a Schottky
barrier diode,
an analogous
case exists except that the potential
energy maximum
occurs in the semiconductor
and
a corresponding
modification
of the derivation
of
the Richardson
equation is needed to account for
the effective mass in the semiconductor.
This is
developed
here for a semiconductor
with ellipsoidal constant
energy
surfaces
in momentum
space.
The current density Ji incident on one side of a
plane surface under equilibrium
conditions
can be
related to the
space :
Ji =
energy
2P
__
cw3
distribution
s
vz>o
of states
vi
d& d&, d&
I+
exp(E/kT)
in R
(2)
where pi is the velocity normal to the plane and E
is the energy (relative to the Fermi energy) of a
carrier
of charge
q with wave number
vectors
kZz, 6, and R,.
If the above plane is located at the maximum
of
the potential
energy barrier in a Schottky
diode,
Ji is then the current
density
which would be
collected
on one side of the barrier if the flow of
carriers from that side were suppressed
without
otherwise
altering the equilibrium.
If Ji has components
of current
which arise from carriers in
several energy minima in 4 space, the current from
each minimum
can be calculated
separately.
By
choice of an origin in R space at one minimum
and
definition of directions with respect to the principal
axes of the minimum,
l&l
vi =
L
h -+
mz
ky?n
-++
m_Y
$n
m,
1
(3)
where I, m and n are the direction
cosines of the
normal to the plane and m,, my and m, are the
components
of the effective mass tensor. If q& is
the maximum
potential
energy
(measured
with
respect to the Fermi level), the energy of the carrier
is
+qc$m.
(4)
The expression
for vi in equation
(3) requires
that the integration
in equation (2) be carried out
over all R values on one side of a plane in R space.
If this integration
is done first, for example, with
respect to iZ, the lower limit on ,4Z is a function of
&, and R, as determined
by equation
(3) with
vi = 0. By substitution
of the expressions
for vi
and E in equation
(2) and the assumption
of a
Maxwellian
distribution
(i.e.
u&
is
greater
than
a
1I .._ v
\
THE
RICHARDSON
CONSTANT
FOR
few KT), the integration for a single energy minimum becomes (except for a common multiplicative constant) the sum of three integrals of the
form
sss
00
03
exp [ - (As+ ~2 + us)] dh dp dv
-m --oo -(/T/L+yv)/a
?I
as
(5)
= Z (a2+p2+y2)1/2
and
where
c( = Z~(m,m,),
p = m 2/(mzmz)
y = n~(m,my),
when the contribution of & to vi
is considered. The current density J1 from a single
minimum is then
51 = b+(12m,mz
+ m2m,m,+ n2m,m,)W
x exp( -q&P).
(6)
The relationship between this current density and
the actual current density flowing in a Schottky
diode needs further elaboration. Consider first the
distribution
of carriers reaching the potential
energy maximum moving away from the metal
contact. If this distribution
remains in equilibrium with the semiconductor up to the potential
energy maximum, and the semiconductor in turn
remains in equilibrium with the metal, the current
density caused by this flow of carriers is given by
equation (6) with & = 4~ where qq5B is the
potential energy barrier height measured with
respect to the Fermi level in the metal. If carriers
reaching the top of the barrier from the body of
the semiconductor are characterised by a difference
qY in Fermi energy from that of the previous
distribution, the net current is then
J = Ji[exp(qV/kT)-
11
= A*T2 exp[-(q+B/kT)][exp(qV/kT)-
11.
(7)
V is very close to the applied voltage if the field
near the potential energy maximum is large enough
that the carrier density throughout most of the
space charge depletion region is large in comparison with the density at the potential energy
maximum.
In practice a net flow of current will disturb the
equilibrium conditions that have been assumed
here. BETHE’S@) criterion for neglecting imref drops
due to the transport of carriers to within a mean
4
THERMIONIC
EMISSION
397
free path, L, of the potential energy maximum can
be stated as follows : the applied field at the barrier
must be greater than kT/qL, i.e. the barrier must
be abrupt enough that the carrier density in the
rest of the depletion layer will not be seriously
reduced by the net current flow. The value of L
determines the extent to which the distribution of
carriers incident on the metal side of the barrier is
characteristic solely of the semiconductor.
If the
distance of the potential energy maximum from
the metal-semiconductor
interface, xm, is comparable to or greater than L, or if the exchange of
carriers is mainly from the semiconductor to the
metal, the properties of the metal will not influence
the current ilow. When xm < L, the reverse
characteristic of the diode should be affected by the
nature of the metal, since carrier flow from the
metal to the semiconductor without intermediate
scattering will be predominant. Quantum mechanical reflections at the potential maximum and the
metal-semiconductor
interface may be expected
to decrease the current below the above prediction.
This limitation is found also in thermionic emission
into vacuum,(7) but may be expected to affect the
current in either case by a few per cent at most.
Back-scattering of carriers over the potential energy
maximum due to lattice and impurity scattering
has no vacuum analogue. This effect depends on
the shape of the potential maximum and the mean
free path for scattering
events.
Such backscattering may well be of the order of 25 per
cent.(ll) These effects, however, will not be considered here.
Using the results of equation (6), for current
from a single minimum in k space, the ratio of the
effective Richardson constant AT [cf. equation (I)]
to the free electron value A (= 120 A/cm2.‘Ks)
is
AT/A = (Pmym, + m2mzm, + n2mymz)l:2
(8)
where the mass components are now expressed in
units of the free electron mass. This shows that the
value of AT is independent of the mass component
in the direction of the current for current normal to
a principal axis.
The following results can be deduced by application of equation (8). For electrons in GaAs the
effective mass in the lowest minimum of the conduction band is isotropic.(ls) Thus
(A*/A)n-~a~s
= m* = 0.072
(9)
398
For
R =
flow
mass
due
C.
R.
CROWELL
holes in Ge and Si, the two energy minima at
0 give rise to approximately
isotropic current
from both the light and heavy hole&s)
(of
ml and mh respectively).
Adding the currents
to these carriers,
(Ll*/&-oe
= ml+mh
= 0.04+0.32
= 0.36
(10)
(A*/A),-si
= ml+mh
= 0*16+0.50
= 0.66
(11)
Emission in the conduction
band in Ge arises from
minima at the edges of the Brillouin zone in the
(111) directions.
These minima are equivalent
to
four ellipsoids
with longitudinal
mass m,, = 1.59
and transverse
mass rnl = 0.082.(13) The sum of
all the AT values has a minimum
in the (111)
directions :
(A*/A)n-~e~~ll)
= m,+
(mt+
8mi,ml)li2
= 1.07
(12)
The maximum
A* occurs
m;-+2m,,m,
(A*/&-Ge(lOO>
= 4
:
for the (100 ) directions
3
1/Z
= 1.19 (13)
-i
For electrons
in Si, minima occur in the (100)
directions
and m,, = 0.9163, m, = 0*1905.(14) All
minima contribute
equally to the current
in the
(111 j direction yielding the maximum
A* :
(A*lA)n-si(lll>
= 6
The minimum
directions:
value
(-4*/A),-si<loo>
= 2.15
of A* occurs
= 2m,+4(m,,m,)1’x
for the
(14)
(100 )
= 2.05
(15)
The increased
values of A* for electrons
in Si in
comparison
with the value for free electrons
are
due to the number of minima in the E-R relationship for Si in contrast to the single minimum
for
free electrons.
The value of A* for electrons
in
GaAs is small in comparison
with the free electron
value in spite of the high velocity
of electrons
normal to the emitting surface in GaAs. The small
A* is due to the low effective density of states in
the conduction
band.
Thus in general the Richardson
constant
for
thermionic
emission in Schottky diodes reflects the
nature of the effective mass and number of valleys
for carriers
in the semiconductor.
Where anisotropy exists in the effective mass, the predicted
Richardson
constant
also shows a slight variation
with orientation
of the semiconductor
crystal
structure
relative to the Schottky barrier.
An experimental
check of the predicted
A*
values requires a very precise knowledge
of T and
40. For a diode with C#JB= 0.8 V at room temperature, an error of 1 % in either $s or T will affect
the value of A* deduced from a saturation
current
density measurement
by -30 %. There must also
be some assurance that the measured J-I’ characteristic is dominated
by majority
carrier current
over the barrier. In addition c$B is dependent
on
both the applied field and the temperature
at the
metal-semiconductor
interface. If (kTlq)(d In J/dY)
from the forward characteristic
of a Schottky diode
is within 1 per cent of the value expected
taking
into account the image force lowering of 4,,(s) and
an independent
measurement
of 4~ is available, it
is reasonable
to attempt a comparison
with theory.
A study of recently published work(l-5) reveals that
these criteria are satisfied only by measurements
on Au-GaAs(s)
and W-Si(4) diodes. A comparison
of xm and L for the W-Si and Au-GaAs
diodes
shows that the condition xm > L is satisfied at low
forward bias. When the applied field at the metalsemiconductor
interface is 4 x 104 V/cm, xm M 25-k
for both GaAs and Si.(s) The electron mean free
paths for optical phonon generation
are z 15 A in
Table 1. Comparison of measured and predicted barrier heights
c PhotO
W-Si
Au-GaAs
0.65 + 0.02
[l 1 I]
[III1
11001
4 CaD.
Volts
4A*
+A
0.65 f 0.02
0.67 f 0.01
0.65 + 0.01
1.01 T!z0.01
0.95 + o-02
0.89 z!I 0.02
0.99 f 0.01
0.95 + 0.01
0.89 + 0.01
1.06 +_ 0.01
1.02 rf: 0.01
0.95 f 0.01
THE
RICHARDSON
CONSTANT
GaAs(l5) and ~60 A in Si.(ls) Table 1 lists the
measured values of +B from photothreshold
and
capacitive methods?, and the values of 4~ from the
measured saturation emission first calculated with
the theoretical A* values, yielding $A*, and then
with the free electron value, yielding 4~. The data
from W-Si diodes indicate that either A* or A
gives results consistent with experimental accuracy.
For Au-GaAs diodes, however, the value of A* is
much smaller than A and its use is necessary to
reconcile theory and experiment.
REFERENCES
1. D. KAHNG, Solid-State
Electron. 6,281 (1963).
2. D. KAHNG, Bell. Syst. Tech. J. 43, 215 (1964).
3. A. M. GOODMAN,J. Appl. Phys. 35,573 (1964).
t The capacitively measured barrier heights for the
W-Si and Au-GaAs diodes have been corrected by 7
and 19 mV, respectively for the image force lowering
expected in the current-voltage measurements.
FOR
THERMIONIC
EMISSION
399
4. C. R. CROWELL, J. C. SARACEand S. M. SZE (to be
published) Tram. Met. Sot. Amer. Inst. Mech.
Engrs., March (1965).
5. S. M. SZE, C. R. CROWELL and D. KAHNG, J. Appl.
Phys. 35, 2534 (1964). Also C. R. CROWiLL,
S. M. SZE and D. KAHNG. Bull. Amer. Phvs. Sot.
Ser. 11,9,281
(1964).
’
6. H. A. BETHE, MIT Radiation Lab. Rept. 43,12
(1942).
7. C. HERRING and M. H. NICHOLS, Rev. Mod. Phys.
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8. P. S. EPSTEIN, Verhandb.
Dtsch. Plzys. Ges. 21
(1919). Also T. C. FRY, Phys. Rea. 17,441 (1921)
and I. LANGMUIRPhys. Rev. 21,419 (1923).
9. W. R. FERRIS, RCA Rev. 10, 134 (1949).
10. C. R. CROWELL, J. Appl. Phys. 17, 93 (1956).
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Sot. (Lond). 74, 131 (1959).
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15. R. A. LOGAN, A. G. CHYNOWETHand B. G. COHEN,
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