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Current transport in monolithic hot-electron structures
Thin Solid Films, 89 (1982) 21-26
21
ELECTRONICS AND OPTICS
C U R R E N T T R A N S P O R T IN M O N O L I T H I C H O T - E L E C T R O N STRUCTURES* J. M. SHANNON AND B. J. GOLDSMITH
Philips Research Laboratories, Redhill, Surrey ( Gt. Britain) (Received August 13, 1981; accepted September 24, 1981)
Current transport is discussed in the context of shallow structures about 750 thick in silicon containing two potential barriers, one for the emission and one for the collection of hot electrons. Measurements indicate that the current gains are higher than those predicted from simplified models using field-free diffusion of hot electrons.
1. INTRODUCTION Current gain has been reported for monolithic hot-electron structures 1. These structures are composed of a single semiconductor material in which potential barriers are formed using narrow highly doped layers which emit and collect hot electrons. Being of a single material with continuity in electronic structure, the structures offer the possibility of efficient collection of hot electrons and therefore high transport factors. In this paper the properties of barriers and current transport across them are discussed in the context of simplified structures in which current transport is assumed to be one dimensional. 2. SAMPLEPREPARATION The monolithic hot-electron structure investigated in this work is shown schematically in Fig. 1. The structure is made up of narrow highly doped layers grown on a silicon substrate. Two barriers are formed in n-type material using narrow regions highly doped with acceptor atoms. These barrier-forming regions are fully depleted of holes by the natural band bending which occurs within the semiconductor. To obtain barriers of high quality the thin layers have to be about 150/~ thick. During the fabrication of these structures in silicon the temperature should not exceed about 750 °C; otherwise diffusion of impurities will prevent the formation of narrow layers. Techniques such as molecular beam epitaxy might be used to form a * Paper presented at the Fifth International Thin Films Congress, Herzlia-on-Sea,Israel, September 21-25, 1981. 0040-6090/82/0000-0000/$02.75
© ElsevierSequoia/Printedin The Netherlands
22
J . M . SHANNON, B. J. GOLDSMITH
Emitter
Bose
',
Me
Collector
N;
~i
II
I I
- ~ w ~Fig. 1. A monolithic hot-electron structure with the corresponding band diagram. A bias Vc~ is applied between the front and the back.
single-crystal film on silicon at a low temperature particularly if used in conjunction with ion implantation and transient annealing 2, or alternatively recrystallization of evaporated silicon films on single-crystal silicon offers the possibility of producing thin n- or p-type layers on single-crystal substrates during an annealing treatment at about 600 °C 3. The work reported here, however, is concerned with structures fabricated in thin recrystallized ion-implanted layers on single-crystal substrates. The advantage of using ion implantation at low energies 4 is that, as it is a nonequilibrium process, the common dopant impurity atoms can be forced into the silicon lattice to form highly doped metastable layers after annealing at temperatures of about 550 °C or more when the bombarded layers recrystallize epitaxially on the undamaged substrate. A disadvantage, however, is that if the layers are to be narrow then they must also be shallow and therefore some of the most desirable structures from a performance point of view s cannot be formed without involving additional techniques such as those mentioned above. The measurements reported below were made on Si(100) implanted with arsenic and BF 2 at energies of less than 20 keV. The arsenic dose was always greater than that required for amorphization (about 2 × 1013 cm-2) and after implantation the layers were recrystallized by annealing at 750 °C in an evacuated ampoule to give structures about 750 A thick. 3.
HOT-ELECTRON COLLECTION
Efficient collection of hot electrons requires the formation of what we shall call "bulk unipolar diodes". As the name implies, these barriers occur within the bulk of the material, and under bias conditions current flows via the transport of majority
23
C U R R E N T T R A N S P O R T IN H O T - E L E C T R O N STRUCTURES
carriers. There are a number of possible structures, including the camel diode 6' 7 and the planar barrier and variations 8, but where we need to maximize the quality of the diode the camel diode seems to be the most useful as it attempts to maximize the number of impurities in the barrier region and thereby to minimize the ideality factor. For thermionic emission to be the dominant transport mechanism we need to satisfy Bethe's criterion 9' 10. >
kT
(I)
- -
q
where 2 is the mean free path for electrons around the top of the barrier and E, is the average field. This criterion is not difficult to satisfy in silicon because the high solubility of impurities enables narrow layers to be formed with significant band bending over interatomic distances 4. Assuming that thermionic emission occurs the diode characteristic is similar to that ofa Schottky diode and J=
SAB*T2exp(-q~C~exP(Kq-.~V~Iy-exp(-\qn~--K-T.)}/\l
(2,
where S is the area of the diode and n is an ideality factor. However, as can be seen from Fig. 2 carrier scattering around the top of the barrier due to quantum mechanical reflections is substantially reduced compared with that for a metali0 -I Ref I~ ~ I
X
,14~x~
//~x ~X
i0 -2
×
10
t~ I0"3
,~
=~ 0.8
~
~
6c (ev)
v
Xo 4 2 x , , ~
b
~ " ~ ~×
16 4 /
u i..:0.4
I A u - Si
I
10-5
00.2
0
0.5 Electron normal
I ! 10 15 energy (eV)
21.0
10 -6
0
~0 54 ~ I J I i I l I i I , l 2 4 6 8 10 12 Reverse volts(V)
to barrier
Fig. 2. Q u a n t u m m e c h a n i c a l t r a n s m i s s i o n across a b u l k u n i p o l a r d i o d e with an a b r u p t p o t e n t i a l barrier a n d zero field. A m e t a l - s e m i c o n d u c t o r barrier is s h o w n for c o m p a r i s o n . (m*l = m* 2) Fig. 3. Reverse characteristics of c a m e l collectors for various 20 keV BF 2 ÷ i m p l a n t a t i o n doses s h o w i n g changes in the b a r r i e r h e i g h t w i t h c h a n g e s in the p-layer c o n c e n t r a t i o n (10 keV As ÷ i m p l a n t a t i o n dose, 5 x 1014 c m - 2 ; N D - = 5 x 10 l~ c m - 3 ) : curve a, 2.5 x 1013 c m -2 (NA ÷ = 2.6 X 1017 c m - 3 ) ; curve b, 5.0 x 1013 c m - 2 (NA + = 3.0 x 1017 cm - 3); curve c, 7.5 x 1013 cm - 2 (N A÷ = 3.4 x l 0 t 7 c m - 3). The r a n g e s w i t h i n w h i c h the m e a s u r e d characteristics lie for 65% of the diodes studied are indicated. The c a l c u l a t e d curves ( ) are fitted at Vce = 1 V a n d d = 480 .~.
24
J . M . SHANNON, B. J. GOLDSMITH
semiconductor barrier, even assuming the most pessimistic case of an abrupt barrier, and the value of Richardson's constant A is greater. Assuming uniformly doped layers the effective barrier height ~bRc' of a camel d i o d e is 6 qd2NA
q~c'-
2ee,°
N D d 2 2 ~AA(V+qSs)--~{d q ND(ND+NA)--2EaoqND(V+Os)} 1/2 (3)
and this together with eqn. (2) can be used to calculate the reverse characteristic. An example is shown in Fig. 3 where the only adjustable parameter was the effective thickness d of the p+ layer. 4.
HOT-ELECTRON EMISSION
Band diagrams corresponding to the hot-electron emitters used in this work are shown schematically in Fig. 4(a). Above a threshold concentration t t the surface field reverses and there is an increase in the effective barrier height. The Bethe criterion (eqn. (1)) for thermionic emission will be met if the acceptor layer is highly doped and since 2 is shortened as a result of impurity scattering the potential m a x i m u m can be arranged to be at a distance greater than 2 from the metal semiconductor interface in which case emission is entirely characterized by the electronic structure of the semiconductor.
09{
~ 0 e, qba---
N£
I
b
i
V R ~ o
-
V
d
;° 0
(a)
De )th
(b)
I - --
2
[
1
l
4 6 8 5 KeV BF2 (cm-2 )
X~
10 10lj
Fig. 4. The influence of acceptor concentration for emitters with aluminium contacts: (a) Schematic representation of h o w the band shape changes with acceptor concentration (V, = 0); (b) measured barrier heights vs. the dose of implanted acceptors.
Under a reverse bias, the barrier is pulled down and current transport changes to thermionic-field emission as the bias is increased. Assuming this model, the barrier height would be expected to vary around the height qSBE of the metalsemiconductor barrier as indeed is found (Fig. 4(b)). Above the threshold dose of about 5 × 1013 cm 2 (t ~ 75/~) the barrier height increases while below this dose the barrier decreases as a result of thermionic-field emission. 5. TRANSISTOR ACTION
An important feature of these structures is the possibility of injecting electrons
CURRENT
TRANSPORT
IN HOT-ELECTRON
25
STRUCTURES
with energies well above the collector barrier. The electrons can therefore make m a n y collisions and still be collected. The main energy loss mechanism is the emission of optical p h o n o n s with effective energy hco = 0.054 eV. An electron with an energy of say 1 eV above the collector barrier can make a b o u t 20 collisions and still be collected. Using the Boltzmann transport equation, Ridley 12 has considered the diffusive transport of hot electrons across a semiconductor base with the assumption that electrons make m a n y collisions during the energy loss process. The analysis (parabolic bands) shows that transport can be described in the form of a diffusion equation with time replaced by energy, i.e.
,~¢,(e,x)
L~
~-t
~(e,x) - 0
ho~ ~ 8x
(4)
where ¢ is the electron energy density per unit energy interval and L is a hotelectron mean free path. The solution for a semi-infinite medium (ff -~ 0, x --, ~ ) gives the base transport factor ~ as ct=l-
erf(~**)
(5)
where L* = 2L{(E o - ~ac')/hog} 1/2 is a characteristic length for energy loss. As an example the results of two-terminal measurements on a transistor structure with a floating base are shown in Fig. 5. Since current transport can be considered to be one dimensional and complications due to forward-biased junctions are absent it is instructive to c o m p a r e the results of such a measurement with those from eqn. (5) above. 5 X102 4 'E v
~) -~
i//t
10
"~ ~~ . . . . . .
~
08
~
3
0 6 ~~
2
04 L o [3.
1 0
g
~
f
~ 5
I
10
02 ~ c
A
15
20
I
25
0
30
VcE(V)
Fig. 5. Current-voltage characteristics ( ) of a camel collector with no emitter barrier (10 keV As ÷ implantation dose, 5 x 1014cm- 2; 20 keV BF2÷ implantation dose, 5 x 1013cm - 2)(curve A) and with an emitter barrier (5 keV BF2 + implantation dose, 5 x 1013cm- 2)(curve B) showing current gain with a base transport factor e (- - -). In this case the leakage Ic due to thermionic emission over the collector barrier is amplified since we have
where • is the base transport factor. C o m p a r i n g the camel collector current (curve A)
26
J . M . SHANNON, B. J. GOLDSMITH
with the collector current when an emitter barrier is also present (curve B) we obtain an ~ of about 0.9 which is similar to values determined from three-terminal common emitter characteristics la. Both chemical and electrical profile data gave w ~ 250 A for this transistor with Eo/q ~ 2.5 V and ~bBc' ~ 0.4 V at VcE = 8 V. The value of obtained from eqn. (5), however, is only about 0.65 even with the unrealistic case of negligible non-parabolicity. It seems therefore that eqn. (5) underestimates the value of~. One reason for this is the neglect of the electric field in the collector which, since it helps to sweep the electrons away, increases the collector current. ACKNOWLEDGMENT
It is a pleasure to acknowledge technical help from A. Gill and secondary ion mass spectrometry measurements by J. B. Clegg. REFERENCES 1 J . M . Shannon, IEEJ. SolidState Electron Devices, 3 (t979) 142. 2 F . W . Saris, Appl. Phys. Lett., 40 (1982) 64. 3 L.S. Hung, S. S. Lau, M. von Allmen, J. W. Mayer, B. M. Ullrich, J. E. Baker, P. Williams and W. F. Tseng, Appl. Phys. Lett., 37 (1980) 909. 4 J . M . Shannon, Nucl. Instrum. Methods, 18~183 (1981) 545. 5 J . M . Shannon, Proc. Inst. Electr. Eng., Part 1, 128 (1981) 134. 6 J . M . Shannon, Jpn. J. Appl. Phys., Suppl. 19-1, 19 (1980) 301. 7 J . M . Shannon, Appl. Phys. Lett., 35 (1979) 63. 8 R.J. Malik, T. R. Aucoin, R. L. Ross, C. E. C. Wood, L. F. Eastman and K. Board, Electron. Lett., 16 (1980) 836. 9 H . A . Bethe, M1TRadiat. Lab. Rep. 43-12, 1942 (Massachusetts Institute of Technology). 10 H . K . Henisch, Rectifying Semiconductor Contacts, Clarendon, Oxford, 1957, p. 196. 11 J . M . Shannon, Solid-State Electron., 19 (1976) 537. 12 B . K . Ridley, Solid-State Electron., 24 (1981) 147. 13 J . M . Shannon and A. Gill, Electron. Lett., 17 ( 1981 ) 620.
21
ELECTRONICS AND OPTICS
C U R R E N T T R A N S P O R T IN M O N O L I T H I C H O T - E L E C T R O N STRUCTURES* J. M. SHANNON AND B. J. GOLDSMITH
Philips Research Laboratories, Redhill, Surrey ( Gt. Britain) (Received August 13, 1981; accepted September 24, 1981)
Current transport is discussed in the context of shallow structures about 750 thick in silicon containing two potential barriers, one for the emission and one for the collection of hot electrons. Measurements indicate that the current gains are higher than those predicted from simplified models using field-free diffusion of hot electrons.
1. INTRODUCTION Current gain has been reported for monolithic hot-electron structures 1. These structures are composed of a single semiconductor material in which potential barriers are formed using narrow highly doped layers which emit and collect hot electrons. Being of a single material with continuity in electronic structure, the structures offer the possibility of efficient collection of hot electrons and therefore high transport factors. In this paper the properties of barriers and current transport across them are discussed in the context of simplified structures in which current transport is assumed to be one dimensional. 2. SAMPLEPREPARATION The monolithic hot-electron structure investigated in this work is shown schematically in Fig. 1. The structure is made up of narrow highly doped layers grown on a silicon substrate. Two barriers are formed in n-type material using narrow regions highly doped with acceptor atoms. These barrier-forming regions are fully depleted of holes by the natural band bending which occurs within the semiconductor. To obtain barriers of high quality the thin layers have to be about 150/~ thick. During the fabrication of these structures in silicon the temperature should not exceed about 750 °C; otherwise diffusion of impurities will prevent the formation of narrow layers. Techniques such as molecular beam epitaxy might be used to form a * Paper presented at the Fifth International Thin Films Congress, Herzlia-on-Sea,Israel, September 21-25, 1981. 0040-6090/82/0000-0000/$02.75
© ElsevierSequoia/Printedin The Netherlands
22
J . M . SHANNON, B. J. GOLDSMITH
Emitter
Bose
',
Me
Collector
N;
~i
II
I I
- ~ w ~Fig. 1. A monolithic hot-electron structure with the corresponding band diagram. A bias Vc~ is applied between the front and the back.
single-crystal film on silicon at a low temperature particularly if used in conjunction with ion implantation and transient annealing 2, or alternatively recrystallization of evaporated silicon films on single-crystal silicon offers the possibility of producing thin n- or p-type layers on single-crystal substrates during an annealing treatment at about 600 °C 3. The work reported here, however, is concerned with structures fabricated in thin recrystallized ion-implanted layers on single-crystal substrates. The advantage of using ion implantation at low energies 4 is that, as it is a nonequilibrium process, the common dopant impurity atoms can be forced into the silicon lattice to form highly doped metastable layers after annealing at temperatures of about 550 °C or more when the bombarded layers recrystallize epitaxially on the undamaged substrate. A disadvantage, however, is that if the layers are to be narrow then they must also be shallow and therefore some of the most desirable structures from a performance point of view s cannot be formed without involving additional techniques such as those mentioned above. The measurements reported below were made on Si(100) implanted with arsenic and BF 2 at energies of less than 20 keV. The arsenic dose was always greater than that required for amorphization (about 2 × 1013 cm-2) and after implantation the layers were recrystallized by annealing at 750 °C in an evacuated ampoule to give structures about 750 A thick. 3.
HOT-ELECTRON COLLECTION
Efficient collection of hot electrons requires the formation of what we shall call "bulk unipolar diodes". As the name implies, these barriers occur within the bulk of the material, and under bias conditions current flows via the transport of majority
23
C U R R E N T T R A N S P O R T IN H O T - E L E C T R O N STRUCTURES
carriers. There are a number of possible structures, including the camel diode 6' 7 and the planar barrier and variations 8, but where we need to maximize the quality of the diode the camel diode seems to be the most useful as it attempts to maximize the number of impurities in the barrier region and thereby to minimize the ideality factor. For thermionic emission to be the dominant transport mechanism we need to satisfy Bethe's criterion 9' 10. >
kT
(I)
- -
q
where 2 is the mean free path for electrons around the top of the barrier and E, is the average field. This criterion is not difficult to satisfy in silicon because the high solubility of impurities enables narrow layers to be formed with significant band bending over interatomic distances 4. Assuming that thermionic emission occurs the diode characteristic is similar to that ofa Schottky diode and J=
SAB*T2exp(-q~C~exP(Kq-.~V~Iy-exp(-\qn~--K-T.)}/\l
(2,
where S is the area of the diode and n is an ideality factor. However, as can be seen from Fig. 2 carrier scattering around the top of the barrier due to quantum mechanical reflections is substantially reduced compared with that for a metali0 -I Ref I~ ~ I
X
,14~x~
//~x ~X
i0 -2
×
10
t~ I0"3
,~
=~ 0.8
~
~
6c (ev)
v
Xo 4 2 x , , ~
b
~ " ~ ~×
16 4 /
u i..:0.4
I A u - Si
I
10-5
00.2
0
0.5 Electron normal
I ! 10 15 energy (eV)
21.0
10 -6
0
~0 54 ~ I J I i I l I i I , l 2 4 6 8 10 12 Reverse volts(V)
to barrier
Fig. 2. Q u a n t u m m e c h a n i c a l t r a n s m i s s i o n across a b u l k u n i p o l a r d i o d e with an a b r u p t p o t e n t i a l barrier a n d zero field. A m e t a l - s e m i c o n d u c t o r barrier is s h o w n for c o m p a r i s o n . (m*l = m* 2) Fig. 3. Reverse characteristics of c a m e l collectors for various 20 keV BF 2 ÷ i m p l a n t a t i o n doses s h o w i n g changes in the b a r r i e r h e i g h t w i t h c h a n g e s in the p-layer c o n c e n t r a t i o n (10 keV As ÷ i m p l a n t a t i o n dose, 5 x 1014 c m - 2 ; N D - = 5 x 10 l~ c m - 3 ) : curve a, 2.5 x 1013 c m -2 (NA ÷ = 2.6 X 1017 c m - 3 ) ; curve b, 5.0 x 1013 c m - 2 (NA + = 3.0 x 1017 cm - 3); curve c, 7.5 x 1013 cm - 2 (N A÷ = 3.4 x l 0 t 7 c m - 3). The r a n g e s w i t h i n w h i c h the m e a s u r e d characteristics lie for 65% of the diodes studied are indicated. The c a l c u l a t e d curves ( ) are fitted at Vce = 1 V a n d d = 480 .~.
24
J . M . SHANNON, B. J. GOLDSMITH
semiconductor barrier, even assuming the most pessimistic case of an abrupt barrier, and the value of Richardson's constant A is greater. Assuming uniformly doped layers the effective barrier height ~bRc' of a camel d i o d e is 6 qd2NA
q~c'-
2ee,°
N D d 2 2 ~AA(V+qSs)--~{d q ND(ND+NA)--2EaoqND(V+Os)} 1/2 (3)
and this together with eqn. (2) can be used to calculate the reverse characteristic. An example is shown in Fig. 3 where the only adjustable parameter was the effective thickness d of the p+ layer. 4.
HOT-ELECTRON EMISSION
Band diagrams corresponding to the hot-electron emitters used in this work are shown schematically in Fig. 4(a). Above a threshold concentration t t the surface field reverses and there is an increase in the effective barrier height. The Bethe criterion (eqn. (1)) for thermionic emission will be met if the acceptor layer is highly doped and since 2 is shortened as a result of impurity scattering the potential m a x i m u m can be arranged to be at a distance greater than 2 from the metal semiconductor interface in which case emission is entirely characterized by the electronic structure of the semiconductor.
09{
~ 0 e, qba---
N£
I
b
i
V R ~ o
-
V
d
;° 0
(a)
De )th
(b)
I - --
2
[
1
l
4 6 8 5 KeV BF2 (cm-2 )
X~
10 10lj
Fig. 4. The influence of acceptor concentration for emitters with aluminium contacts: (a) Schematic representation of h o w the band shape changes with acceptor concentration (V, = 0); (b) measured barrier heights vs. the dose of implanted acceptors.
Under a reverse bias, the barrier is pulled down and current transport changes to thermionic-field emission as the bias is increased. Assuming this model, the barrier height would be expected to vary around the height qSBE of the metalsemiconductor barrier as indeed is found (Fig. 4(b)). Above the threshold dose of about 5 × 1013 cm 2 (t ~ 75/~) the barrier height increases while below this dose the barrier decreases as a result of thermionic-field emission. 5. TRANSISTOR ACTION
An important feature of these structures is the possibility of injecting electrons
CURRENT
TRANSPORT
IN HOT-ELECTRON
25
STRUCTURES
with energies well above the collector barrier. The electrons can therefore make m a n y collisions and still be collected. The main energy loss mechanism is the emission of optical p h o n o n s with effective energy hco = 0.054 eV. An electron with an energy of say 1 eV above the collector barrier can make a b o u t 20 collisions and still be collected. Using the Boltzmann transport equation, Ridley 12 has considered the diffusive transport of hot electrons across a semiconductor base with the assumption that electrons make m a n y collisions during the energy loss process. The analysis (parabolic bands) shows that transport can be described in the form of a diffusion equation with time replaced by energy, i.e.
,~¢,(e,x)
L~
~-t
~(e,x) - 0
ho~ ~ 8x
(4)
where ¢ is the electron energy density per unit energy interval and L is a hotelectron mean free path. The solution for a semi-infinite medium (ff -~ 0, x --, ~ ) gives the base transport factor ~ as ct=l-
erf(~**)
(5)
where L* = 2L{(E o - ~ac')/hog} 1/2 is a characteristic length for energy loss. As an example the results of two-terminal measurements on a transistor structure with a floating base are shown in Fig. 5. Since current transport can be considered to be one dimensional and complications due to forward-biased junctions are absent it is instructive to c o m p a r e the results of such a measurement with those from eqn. (5) above. 5 X102 4 'E v
~) -~
i//t
10
"~ ~~ . . . . . .
~
08
~
3
0 6 ~~
2
04 L o [3.
1 0
g
~
f
~ 5
I
10
02 ~ c
A
15
20
I
25
0
30
VcE(V)
Fig. 5. Current-voltage characteristics ( ) of a camel collector with no emitter barrier (10 keV As ÷ implantation dose, 5 x 1014cm- 2; 20 keV BF2÷ implantation dose, 5 x 1013cm - 2)(curve A) and with an emitter barrier (5 keV BF2 + implantation dose, 5 x 1013cm- 2)(curve B) showing current gain with a base transport factor e (- - -). In this case the leakage Ic due to thermionic emission over the collector barrier is amplified since we have
where • is the base transport factor. C o m p a r i n g the camel collector current (curve A)
26
J . M . SHANNON, B. J. GOLDSMITH
with the collector current when an emitter barrier is also present (curve B) we obtain an ~ of about 0.9 which is similar to values determined from three-terminal common emitter characteristics la. Both chemical and electrical profile data gave w ~ 250 A for this transistor with Eo/q ~ 2.5 V and ~bBc' ~ 0.4 V at VcE = 8 V. The value of obtained from eqn. (5), however, is only about 0.65 even with the unrealistic case of negligible non-parabolicity. It seems therefore that eqn. (5) underestimates the value of~. One reason for this is the neglect of the electric field in the collector which, since it helps to sweep the electrons away, increases the collector current. ACKNOWLEDGMENT
It is a pleasure to acknowledge technical help from A. Gill and secondary ion mass spectrometry measurements by J. B. Clegg. REFERENCES 1 J . M . Shannon, IEEJ. SolidState Electron Devices, 3 (t979) 142. 2 F . W . Saris, Appl. Phys. Lett., 40 (1982) 64. 3 L.S. Hung, S. S. Lau, M. von Allmen, J. W. Mayer, B. M. Ullrich, J. E. Baker, P. Williams and W. F. Tseng, Appl. Phys. Lett., 37 (1980) 909. 4 J . M . Shannon, Nucl. Instrum. Methods, 18~183 (1981) 545. 5 J . M . Shannon, Proc. Inst. Electr. Eng., Part 1, 128 (1981) 134. 6 J . M . Shannon, Jpn. J. Appl. Phys., Suppl. 19-1, 19 (1980) 301. 7 J . M . Shannon, Appl. Phys. Lett., 35 (1979) 63. 8 R.J. Malik, T. R. Aucoin, R. L. Ross, C. E. C. Wood, L. F. Eastman and K. Board, Electron. Lett., 16 (1980) 836. 9 H . A . Bethe, M1TRadiat. Lab. Rep. 43-12, 1942 (Massachusetts Institute of Technology). 10 H . K . Henisch, Rectifying Semiconductor Contacts, Clarendon, Oxford, 1957, p. 196. 11 J . M . Shannon, Solid-State Electron., 19 (1976) 537. 12 B . K . Ridley, Solid-State Electron., 24 (1981) 147. 13 J . M . Shannon and A. Gill, Electron. Lett., 17 ( 1981 ) 620.