- Email: [email protected]
Temperature dependence of resistivity and hole conductivity mobility in p-type silicon
Solid-St&
E.khvdc&
1976, Vol. 19.99.949-953.
Perlpmon Rcss.
Printed in Great Britain
TEMPERATURE DEPENDENCE OF RESISTIVITY AND HOLE CONDUCTIVITY MOBILITY IN p-TYPE SILICON? K. Y. TSAOand C. T. SAH Department of Electrical Engineering and Materials Research Laboratory, University of Illinois, Urbana, IL 61801, U.S.A. (Received 10March 1976;in revisedfon
19April 1976)
Ah&act-A new method is employed to determine the temperature dependence of the resistivity and hole conductivity mobility of p-type silicon. This method involves the use of an aluminum-on-p-Si ohmic diode and a magnesium on p-Si Schottky barrier diode on the same silicon chip. The resistivity is determined from the Al/p-Si ohmic diode. The hole concentration is evaluated from the C-V data of the Mg/p-Si Schottky barrier diode. The conductivity mobility is then computed from the resistivity and hole concentration data. The following ranges are covered: 77-300K, 0.4400 ohm-cm and 5 x 1016-2x10” holes/cm’ at room temperature.
1. INTRODUCTION
of resistivity, [l-6] drift mobility [11 and Hall mobility&51 of n - and p-type silicon has been investigated extensively. Information on the temperature dependence of conductivity mobility of heavily doped silicon has been published[2,7). The main diiculty in all of these earlier works is that the carrier concentration cannot be determined to a factor of better than about 1.5. In these works, the carrier concentration was usually calculated from P = AIqR, using the measured Hall coefficient, RH. The factor A is of the order of 1. Its value is a function of the temperature and energy dependences of the scattering mechanisms. This uncertainty in A makes it impossible to obtain carrier mobility values more accurate than the uncertainty in A which can vary from as low as 0.5 to as high as 2 or more. The purpose of this paper is to present accurate mobility data of holes in p-type silicon using a new method which combines two well known methods in semi-conductor device studies. The well-known and accurate capacitance+voltage method is used to determine the carrier concentration. This C-V measurement is made on a Mg/p-Si Schottky barrier diode and it is combined with the resistivity measurement, p, made on a Al/p -Si ohmic diode adjacent to the SB diode to give the hole mobility using the relationship p,, = (qPp)-‘. The experimental procedures and results are given in Section 2 and a brief and qualitative discussion of the results in terms of the scattering mechanisms is given in Section 3. The
temperature
dependence
The cross sectional view of the devices is shown in Pii. 1 which are fabricated as follows. Seven slices of 10 mil thick boron-doped p-type silicon of (111) orientation and resistivity range from 0.5 to 100&cm were used in this experiment. After degreasing, chemical polishing and removing oxide, a 0.2~ aluminum layer was evaporated
Whis work is supported by the Air Force O&e of Scientific Research and the Air Force Cambridge Research Laboratory.
onto one surface and 0.3~ aluminum dots were evaporated onto the other surface through a metal mask with 12 mils diameter holes. The slice was then annealed in an open tube furnace at 575°C for 6 min in an argon ambient. Magnesium dots of 0.5~ thickness were evaporated immediately onto the silicon between the 12 mil aluminum dots through 30 mils diameter holes of a metal mask. Then, 0.4~ thick aluminum dots of 12 mils diameter were evaporated onto the 30 mil magnesium dots for wire bonding. A very thin layer of epo-tek (resistivity = 0.006 n-cm) was used to bond the chip to the g-pin TO-5 header. Each chip is a 100 mils square and 4-8 mil thick. It contains one aluminum dot and four aluminum-onmagnesium dots (hereafter referred to as Mg dot) with the aluminum dot in the center. The distance between the edges of the Al dot and the Mg dots is about 15 mils which is ample since the Mg Schottky barrier can be considered insulating. The Al dot gives an ohmic diode for resistivity measurement and the Mg dots give good Schottky barrier diodes on p-Si for C-V and carrier concentration measurements. Table 1 gives the device numbers, their resistivities and carrier (hole) concentrations at room temperature, the latter, determined from C-V measurements described below.
2.2 Measurements The I-V characteristics of the Al ohmic diodes were displayed on a calibrated Tektroix type 575 transistor curve tracer to ascertain that it is ohmic. The resistances of the Al ohmic diodes were measured with a Data Precision 3500 digital meter. The proximity of the Mg Schottky diodes to the Al ohmic diode introduces little error in the resistance measurements since the Schottky barrier has an insulating depletion layer boundary of only a few microns from the Mg/Si interface. The capacitanc*voltage measurements were made on the Mg/p-Si Schottky diodes using a Boonton 71 AB capacitance meter with a 1 MHz 15mv measuring signal. The analog output of the Boonton 71 AB was point-bypoint recorded. Temperature used in these measurements was from
949
K. Y. TSAOand C. T.
950
SAH
Al (0.4@
I
30 mils 12 mils-l
45
I
mils
Al (0.2~) 100 mils Fig. LThecross-sectionalviewof theAl-on-p-SiohmicdiodeandtheMg-on-p-SiSchottkydiode.
has been used in calculating curve (b) in Fig. 2. Here a is the radius of the aluminum dot, H is the thickness of the silicon wafer and h = H/a.
Table 1. Properties of devices at 300 K
68C24
9.67x1014
7412
5.74xd5
14.5
445
2.96
367 344
7645
1.08XlP
1.69
7735
2.36xlcP
0.836
317
7914
2.83~10~~
0.710
311
8053
4.85x1o16
0.452
285
77°K to room temperature,
and the emf of the thermocouple located near the diode header was read from a Data Precision 3500 digital meter. The system is calibrated against a secondary standard traceable to NBS to give better than +0.2”K accuracy.
2.3 Resistivity calculation The resistance through the aluminum ohmic diode structure is composed of three resistances in series: the contact resistances of aluminum to the two silicon surfaces, and the resistance of the silicon body. After heat treatment, the aluminum-silicon contacts become very good and their resistances can be neglected. The measured resistance can be regarded as due to silicon only. Since the diameter of the aluminum dot on the front silicon surface is about the same as wafer thickness, lateral current flow is important and spreading resistance correction must be made. The one-dimensional resistance of a cylinder is shown as case (a) in Fig. 2. The true resistance, including spreading effect, is given as case (b) in Fig. 2. The resistance of a thin infinite slab with a disk electrode has been investigated by Foxhall and Lewis[8] and their asymptotic solution,
2.4 Experimental results The typical Z-V characteristics for the magnesium Schottky barrier diode and the aluminum ohmic diode are shown in Fig. 3a at 300K and Fig. 3b at 77 K. The magnesium Schottky barrier diodes give high breakdown voltages. Thus, reliable C-V measurements can be made. The aluminum diode shows a linear Z-V behavior at both 77 and 300 K in Fig. 3. The 77 K curve of the Al diode shows that it is a very good ohmic contact at this low temperature and this permits the measurement of silicon bulk resistance with very small error. Hole concentrations were calculated from the data points of C-V measurements. Figure 4 shows that the hole concentration is almost independent of temperature in the entire temperature range from 77 to 300 K. The resistivity was calculated from the measured resistance and device dimensions using eqn (1). The results are shown in Fig. 5. All curves are shaped concave upward and the minimum point shifts to higher temperature as carrier concentration increases. The relation of the resistivity and the carrier concentration is plotted in Fig. 6. From the values of hole concentration and resistivity, the hole conductivity mobility can be determined. The result of the temperature dependences of the hole mobility is given in Fig. 7. 3. DJSCUS!SlON The
fabrication of aluminum ohmic diodes which retain good ohmic behavior at 77 K as well as the magnesium in p-Si diodes with good rectification characteristics at 300K are the keys of this work. The use of Mg was arrived at after extensive trials with a number of low work function metals. However, it is generally thought that gold on p-type silicon eutectic junction is ohmic. Our tests
Resistivity and mobility in p-type silicon
951
RFSISTANCF
,.6 IO-'
2
4
6
6
I
2
4
6
e
IO
h (-H/a) Fig. 2. Curve (a) is the ideal one-dimensional resistance of a cylindrical conductor. Curve (b) is the exact resistance of a diskcontactonasemi-infiniteslab(seelowfigureintheinset)whoseasymtoticvalue,0.25,is(c). Al
OHMIC
DIODE Mg SCHOTTKY BARRIER DIODE
(a)
I
OHMIC
DIODE Mg SCHOTTKY BARRIER DIODE
(b) Fii. 3. The oscilloscope traces of the current-voltage characteristics of the Al-p-S ohmic diode and the Mep-Si Schottkydiode.(a)3OOKand(b)77K.
952
K. Y. TSAO and C. T. SAH
g
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
7645
. . . .
7412 . . .
.
. .
.
.
s B
5 B
Y 8
zlol/f . . . .. . . .
.6?C?4. . .
t
lol.~
1
10-j
10 RESISTIVITY
Fii.
IO2
(ohm-cm)
6. The hole concentration from Fig. 4 vs the resistivity from Fig. 5 at 300 K. 5
,
,
,
,
,
1
I
I
,
,
,
Fig. 4. The hole concentrations as a function of sample temperature for six Schottky diodes from capacitance-voltage measurements.
/ozo2 50
100
150
TEMPERATURE
200
250
300
i
(K)
Fig. 7. The conductivity mobility of holes computed from the data in Figs. 4 and 5 as a function of temperatures for the six ohmic and six Schottky diodes.
TEMPERATURE
(K)
Fig. 5. The resistivity as a function of sample temperature ohmic diodes from current-voltage measurements.
for six
show that the aluminum on p-Eli junction gives a better ohmic contact than gold on p-Si eutectic junction after heat treatment at 400°C for 1 minute. At 77 K the aluminum on p-Si junction shows a linear Z-V characteristic whereas the gold in psi eutectic junction shows rectification. This is probably due to a thin layer of aluminum acceptors on silicon [9] making a p +/p junction as well as the reduction of the oxide film by Al. The measured resistances among the aluminum ohmic
diodes made from one silicon slice differ only by a few percent. This variation appears to be due to the nonuniform heat treatment as well as true resistivity variation. Diodes from the lower resistance group were measured to give a resistance variation of about 1% and C-V variation of about 2% after repeated heating and cooling cycles between 77 and 300 K. It is well known that there are two predominant scattering mechanisms which control the carrier mobilities in silicon, namely scattering due to longitudinal and intervalley acoustic phonons and to ionized impurities [ lo]. The mobility from the combined scattering process can be roughly calculated by the Mathiessen rule,
Resistivity and mobility in p-type silicon
The mobility due to phonon scattering, pL, will decrease as the temperature increases, and the mobility due to ionized impurities, p,, will increase as the temperature increases. The trend of mobility variation (or resistivity variation) in Fii. 7 show that the higher temperature side of the mobility peak is dominated by lattice scattering mechanism, and the lower temperature side is dominated by impurity scattering mechanism. As the carrier concentration increases, the magnitude of the peak decreases and broadens. This indicates that the role of ionized impurity scattering center becomes increasingly important in low resistivity material. The resistivity versus carrier concentration at room temperature as shown in Fii. 6 is in good agreement with a best fit of published data[ll]. The method described here is limited by the breakdown voltage of the magnesium on p -Si Schottky diode. So far we have achieved a 28 V breakdown voltage for a hole concentration of 4.8 x 10’6cm-’ (0.5 n-cm) at room temperature. It appears that this method is useful up to
953
10” acceptors/cm3. At higher impurity concentrations, deionization of the impurity at low temperatures must be taken into account in the analysis of the capacitancevoltage data.
REBERENCES
1. M. Prince, Phys. Rev. 93, 1204(1954). 2. G. W. Ludwig and R. L. Watters, Phys. Rev. 101,1699(1956). 3. F. J. Morin and J. P. Maita, Phys. Reu. 96, 28 (1954). 4. R. 0. Carlson, Phys. Rev. 100, 1075(1975). 5. D. Long and J. Myers, Phys. Rev. 115, 1107(1959). 6. P. W. Chapman, 0. N. Tufte, J. D. Zook and D. Long, J. Appl. Phys. 34, 3291 (1963). 7. G. L. Pearson and J. Bardeen, Phys. Rev. 75,865 (1949). 8. G. F. Foxhall and J. A. Lewis, Bell Svsfem Tech. J. 1609(Julv , I 1964). 9. T. M. Reith and J. D. Schick, Appl. Phys. Letts. 25,524 (1974). 10. See for example, C. T. Sah, T. I-I. Nina and L. L. Tschooo. Surface Science 32, 561 (1972);W. Sh&kley, Electrons a& Holes in Semiconductors,Chap. 11.Van Nostrand, New York (1950). 11. C. T. Sah (unpublished).
E.khvdc&
1976, Vol. 19.99.949-953.
Perlpmon Rcss.
Printed in Great Britain
TEMPERATURE DEPENDENCE OF RESISTIVITY AND HOLE CONDUCTIVITY MOBILITY IN p-TYPE SILICON? K. Y. TSAOand C. T. SAH Department of Electrical Engineering and Materials Research Laboratory, University of Illinois, Urbana, IL 61801, U.S.A. (Received 10March 1976;in revisedfon
19April 1976)
Ah&act-A new method is employed to determine the temperature dependence of the resistivity and hole conductivity mobility of p-type silicon. This method involves the use of an aluminum-on-p-Si ohmic diode and a magnesium on p-Si Schottky barrier diode on the same silicon chip. The resistivity is determined from the Al/p-Si ohmic diode. The hole concentration is evaluated from the C-V data of the Mg/p-Si Schottky barrier diode. The conductivity mobility is then computed from the resistivity and hole concentration data. The following ranges are covered: 77-300K, 0.4400 ohm-cm and 5 x 1016-2x10” holes/cm’ at room temperature.
1. INTRODUCTION
of resistivity, [l-6] drift mobility [11 and Hall mobility&51 of n - and p-type silicon has been investigated extensively. Information on the temperature dependence of conductivity mobility of heavily doped silicon has been published[2,7). The main diiculty in all of these earlier works is that the carrier concentration cannot be determined to a factor of better than about 1.5. In these works, the carrier concentration was usually calculated from P = AIqR, using the measured Hall coefficient, RH. The factor A is of the order of 1. Its value is a function of the temperature and energy dependences of the scattering mechanisms. This uncertainty in A makes it impossible to obtain carrier mobility values more accurate than the uncertainty in A which can vary from as low as 0.5 to as high as 2 or more. The purpose of this paper is to present accurate mobility data of holes in p-type silicon using a new method which combines two well known methods in semi-conductor device studies. The well-known and accurate capacitance+voltage method is used to determine the carrier concentration. This C-V measurement is made on a Mg/p-Si Schottky barrier diode and it is combined with the resistivity measurement, p, made on a Al/p -Si ohmic diode adjacent to the SB diode to give the hole mobility using the relationship p,, = (qPp)-‘. The experimental procedures and results are given in Section 2 and a brief and qualitative discussion of the results in terms of the scattering mechanisms is given in Section 3. The
temperature
dependence
The cross sectional view of the devices is shown in Pii. 1 which are fabricated as follows. Seven slices of 10 mil thick boron-doped p-type silicon of (111) orientation and resistivity range from 0.5 to 100&cm were used in this experiment. After degreasing, chemical polishing and removing oxide, a 0.2~ aluminum layer was evaporated
Whis work is supported by the Air Force O&e of Scientific Research and the Air Force Cambridge Research Laboratory.
onto one surface and 0.3~ aluminum dots were evaporated onto the other surface through a metal mask with 12 mils diameter holes. The slice was then annealed in an open tube furnace at 575°C for 6 min in an argon ambient. Magnesium dots of 0.5~ thickness were evaporated immediately onto the silicon between the 12 mil aluminum dots through 30 mils diameter holes of a metal mask. Then, 0.4~ thick aluminum dots of 12 mils diameter were evaporated onto the 30 mil magnesium dots for wire bonding. A very thin layer of epo-tek (resistivity = 0.006 n-cm) was used to bond the chip to the g-pin TO-5 header. Each chip is a 100 mils square and 4-8 mil thick. It contains one aluminum dot and four aluminum-onmagnesium dots (hereafter referred to as Mg dot) with the aluminum dot in the center. The distance between the edges of the Al dot and the Mg dots is about 15 mils which is ample since the Mg Schottky barrier can be considered insulating. The Al dot gives an ohmic diode for resistivity measurement and the Mg dots give good Schottky barrier diodes on p-Si for C-V and carrier concentration measurements. Table 1 gives the device numbers, their resistivities and carrier (hole) concentrations at room temperature, the latter, determined from C-V measurements described below.
2.2 Measurements The I-V characteristics of the Al ohmic diodes were displayed on a calibrated Tektroix type 575 transistor curve tracer to ascertain that it is ohmic. The resistances of the Al ohmic diodes were measured with a Data Precision 3500 digital meter. The proximity of the Mg Schottky diodes to the Al ohmic diode introduces little error in the resistance measurements since the Schottky barrier has an insulating depletion layer boundary of only a few microns from the Mg/Si interface. The capacitanc*voltage measurements were made on the Mg/p-Si Schottky diodes using a Boonton 71 AB capacitance meter with a 1 MHz 15mv measuring signal. The analog output of the Boonton 71 AB was point-bypoint recorded. Temperature used in these measurements was from
949
K. Y. TSAOand C. T.
950
SAH
Al (0.4@
I
30 mils 12 mils-l
45
I
mils
Al (0.2~) 100 mils Fig. LThecross-sectionalviewof theAl-on-p-SiohmicdiodeandtheMg-on-p-SiSchottkydiode.
has been used in calculating curve (b) in Fig. 2. Here a is the radius of the aluminum dot, H is the thickness of the silicon wafer and h = H/a.
Table 1. Properties of devices at 300 K
68C24
9.67x1014
7412
5.74xd5
14.5
445
2.96
367 344
7645
1.08XlP
1.69
7735
2.36xlcP
0.836
317
7914
2.83~10~~
0.710
311
8053
4.85x1o16
0.452
285
77°K to room temperature,
and the emf of the thermocouple located near the diode header was read from a Data Precision 3500 digital meter. The system is calibrated against a secondary standard traceable to NBS to give better than +0.2”K accuracy.
2.3 Resistivity calculation The resistance through the aluminum ohmic diode structure is composed of three resistances in series: the contact resistances of aluminum to the two silicon surfaces, and the resistance of the silicon body. After heat treatment, the aluminum-silicon contacts become very good and their resistances can be neglected. The measured resistance can be regarded as due to silicon only. Since the diameter of the aluminum dot on the front silicon surface is about the same as wafer thickness, lateral current flow is important and spreading resistance correction must be made. The one-dimensional resistance of a cylinder is shown as case (a) in Fig. 2. The true resistance, including spreading effect, is given as case (b) in Fig. 2. The resistance of a thin infinite slab with a disk electrode has been investigated by Foxhall and Lewis[8] and their asymptotic solution,
2.4 Experimental results The typical Z-V characteristics for the magnesium Schottky barrier diode and the aluminum ohmic diode are shown in Fig. 3a at 300K and Fig. 3b at 77 K. The magnesium Schottky barrier diodes give high breakdown voltages. Thus, reliable C-V measurements can be made. The aluminum diode shows a linear Z-V behavior at both 77 and 300 K in Fig. 3. The 77 K curve of the Al diode shows that it is a very good ohmic contact at this low temperature and this permits the measurement of silicon bulk resistance with very small error. Hole concentrations were calculated from the data points of C-V measurements. Figure 4 shows that the hole concentration is almost independent of temperature in the entire temperature range from 77 to 300 K. The resistivity was calculated from the measured resistance and device dimensions using eqn (1). The results are shown in Fig. 5. All curves are shaped concave upward and the minimum point shifts to higher temperature as carrier concentration increases. The relation of the resistivity and the carrier concentration is plotted in Fig. 6. From the values of hole concentration and resistivity, the hole conductivity mobility can be determined. The result of the temperature dependences of the hole mobility is given in Fig. 7. 3. DJSCUS!SlON The
fabrication of aluminum ohmic diodes which retain good ohmic behavior at 77 K as well as the magnesium in p-Si diodes with good rectification characteristics at 300K are the keys of this work. The use of Mg was arrived at after extensive trials with a number of low work function metals. However, it is generally thought that gold on p-type silicon eutectic junction is ohmic. Our tests
Resistivity and mobility in p-type silicon
951
RFSISTANCF
,.6 IO-'
2
4
6
6
I
2
4
6
e
IO
h (-H/a) Fig. 2. Curve (a) is the ideal one-dimensional resistance of a cylindrical conductor. Curve (b) is the exact resistance of a diskcontactonasemi-infiniteslab(seelowfigureintheinset)whoseasymtoticvalue,0.25,is(c). Al
OHMIC
DIODE Mg SCHOTTKY BARRIER DIODE
(a)
I
OHMIC
DIODE Mg SCHOTTKY BARRIER DIODE
(b) Fii. 3. The oscilloscope traces of the current-voltage characteristics of the Al-p-S ohmic diode and the Mep-Si Schottkydiode.(a)3OOKand(b)77K.
952
K. Y. TSAO and C. T. SAH
g
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
7645
. . . .
7412 . . .
.
. .
.
.
s B
5 B
Y 8
zlol/f . . . .. . . .
.6?C?4. . .
t
lol.~
1
10-j
10 RESISTIVITY
Fii.
IO2
(ohm-cm)
6. The hole concentration from Fig. 4 vs the resistivity from Fig. 5 at 300 K. 5
,
,
,
,
,
1
I
I
,
,
,
Fig. 4. The hole concentrations as a function of sample temperature for six Schottky diodes from capacitance-voltage measurements.
/ozo2 50
100
150
TEMPERATURE
200
250
300
i
(K)
Fig. 7. The conductivity mobility of holes computed from the data in Figs. 4 and 5 as a function of temperatures for the six ohmic and six Schottky diodes.
TEMPERATURE
(K)
Fig. 5. The resistivity as a function of sample temperature ohmic diodes from current-voltage measurements.
for six
show that the aluminum on p-Eli junction gives a better ohmic contact than gold on p-Si eutectic junction after heat treatment at 400°C for 1 minute. At 77 K the aluminum on p-Si junction shows a linear Z-V characteristic whereas the gold in psi eutectic junction shows rectification. This is probably due to a thin layer of aluminum acceptors on silicon [9] making a p +/p junction as well as the reduction of the oxide film by Al. The measured resistances among the aluminum ohmic
diodes made from one silicon slice differ only by a few percent. This variation appears to be due to the nonuniform heat treatment as well as true resistivity variation. Diodes from the lower resistance group were measured to give a resistance variation of about 1% and C-V variation of about 2% after repeated heating and cooling cycles between 77 and 300 K. It is well known that there are two predominant scattering mechanisms which control the carrier mobilities in silicon, namely scattering due to longitudinal and intervalley acoustic phonons and to ionized impurities [ lo]. The mobility from the combined scattering process can be roughly calculated by the Mathiessen rule,
Resistivity and mobility in p-type silicon
The mobility due to phonon scattering, pL, will decrease as the temperature increases, and the mobility due to ionized impurities, p,, will increase as the temperature increases. The trend of mobility variation (or resistivity variation) in Fii. 7 show that the higher temperature side of the mobility peak is dominated by lattice scattering mechanism, and the lower temperature side is dominated by impurity scattering mechanism. As the carrier concentration increases, the magnitude of the peak decreases and broadens. This indicates that the role of ionized impurity scattering center becomes increasingly important in low resistivity material. The resistivity versus carrier concentration at room temperature as shown in Fii. 6 is in good agreement with a best fit of published data[ll]. The method described here is limited by the breakdown voltage of the magnesium on p -Si Schottky diode. So far we have achieved a 28 V breakdown voltage for a hole concentration of 4.8 x 10’6cm-’ (0.5 n-cm) at room temperature. It appears that this method is useful up to
953
10” acceptors/cm3. At higher impurity concentrations, deionization of the impurity at low temperatures must be taken into account in the analysis of the capacitancevoltage data.
REBERENCES
1. M. Prince, Phys. Rev. 93, 1204(1954). 2. G. W. Ludwig and R. L. Watters, Phys. Rev. 101,1699(1956). 3. F. J. Morin and J. P. Maita, Phys. Reu. 96, 28 (1954). 4. R. 0. Carlson, Phys. Rev. 100, 1075(1975). 5. D. Long and J. Myers, Phys. Rev. 115, 1107(1959). 6. P. W. Chapman, 0. N. Tufte, J. D. Zook and D. Long, J. Appl. Phys. 34, 3291 (1963). 7. G. L. Pearson and J. Bardeen, Phys. Rev. 75,865 (1949). 8. G. F. Foxhall and J. A. Lewis, Bell Svsfem Tech. J. 1609(Julv , I 1964). 9. T. M. Reith and J. D. Schick, Appl. Phys. Letts. 25,524 (1974). 10. See for example, C. T. Sah, T. I-I. Nina and L. L. Tschooo. Surface Science 32, 561 (1972);W. Sh&kley, Electrons a& Holes in Semiconductors,Chap. 11.Van Nostrand, New York (1950). 11. C. T. Sah (unpublished).