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Negative resistance in silicon doped with gold
Solid-Stare
Ekclronics
Pergamon
Press
197 1. Vol. 14, pp. I I 19- I122.
NEGATIVE
Printed in Great Britain
RESISTANCE IN SILICON DOPED WITH GOLD*
P. ANTOGNETTI,
Istituto di Elettrotecnica,
A. CHIABRERA
Universitadi
and S. RIDELLA
Genova, Viale Cambiaso 6, 16145 Genova, Italy
(Received 18 January 197 1; in revisedform 22 February 197 1) this paper, we have derived a simple analytical expression of the J-V characteristic for gold-doped silicon; such expression applies to the low injection and high injection regimes and to the negative resistance region when the quasineutrality approximation holds. The J-V characteristic. evaluated for a gold-doped silicon p+-i-n + diode, shows the existence of the negative resistance region.
Abstract-In
Resume - Dans cet article, nous avons derive une simple expression analytique de la caracteristique J-V du silicium dope a l’or. Une telle expression s’applique aux bas et hauts regimes d’injection et a la zone de resistance negative lorsque se maintient I’approximation de quasi-neutralite. La caracteristique J-V, evaluee pour une diode $-i-n+ au silicium dope a l’or, fait apparaitre I’existence d’une zone de resistance negative. ZusammenfassungFur golddotiertes Silizium werden einfache analytische Ausdrilcke fur die I-I’Charakteristik abgeleitet. Sie sind anwendbar bei geringer und hoher Injektion sowie im Bereich negativen Widerstandes, wenn naherungsweise eine QuasineutralM vorliegt. Die I-l/-Charakteristik einer golddotierten p+-i-n+-Diode aus Silizium weist einen solchen Bereich negativen Widerstandes auf.
BULK semiconductors doped with deep two-level traps exhibit interesting properties. Their J-V characteristic is of the S-type showing a differential negative resistance [ 1,2] furthermore sinusoidal oscillations have been observed in the positive resistance region of the characteristic [3,4]. These oscillations have been interpreted as recombination wave instabilities [ 51. The case of gold-doped silicon p+-i-n+ diodes has been studied extensively[2,3,5]; computer solutions for the evaluation of the J-V characteristic have been outlined in [2,6]. Simple power laws for the various regimes of the characteristic have been found to agree well with the numerical solution[2]; yet such analytical solutions apply only to the positive resistance regions of the J-V curve; as far as the negative resistance region is concerned, they only predict its lower terminal voltage Vt. The purpose of this paper is to formulate an analytical expression of the J-V characteristic of *Work supported (CNR) of Italy.
by the National Research
Council
silicon doped with gold, which includes the negative resistance region; this is done by applying the Regional Approximation Method[7] to the treatment given in [2]. We assume that the reader is familiar with [2], as we will use the same notation. Let us now recall the various regimes of the J-P’ characteristic [2] : (1) ohmic regime (2) low injection regime (3) high injection regime (4) space-charge limited regime The quasineutrality approximations, i.e. E + 0, holds in regimes 1, 2, 3; the negative resistance (NR) region is comprised between regimes 2 and 3 and is separated from regime 2 by a constant voltage curve at voltage V,: such breakover voltage, characterizing the onset of NR, is Edependent. This seems to suggest that the NR region is function of 6, but such assumption is in contrast with the quasineutrality approximation in regimes 2 and 3. Furthermore, even if one assumes to approximate the NR region by connecting
1119
P. ANTOGNETTI.
I120
A. CHIABRERA
V(, and V, with a line. no information is available on the x-distribution of II, p and E. Here we deal with regimes 2. 3 and with the NR region assuming the quasineutrality approximation to hold. This assumption will be further discussed at the end of the paper. In regime 3. the characteristic is given by [ 71:
and S. RIDELLA
given to L, it stems out that p(f.,) and tz(L, ) do not depend on the value of the abscissa L, if: C’, = L/,x-: Equation
?‘ = L,/l..
(5) can now be rewritten:
t IO)
.I = .I,> J =
&
V’/L:’
(1)
For region II we can write [2]: being L the device
length. and: .I = K,C’,‘l(f.-
K,, = (9i8)e~,,~,,((;(.,,
(7) Equation (1) has been derived approximations: t1.p > N. P:
Let V, be the voltage region we have [ 21:
being I/, = VIl.and:
V, the
vnoltage drop
I1 = p
(3)
< N. P. across
(4) region
I; for this
p(L,) From
= 3V,/2L,
-- ML,) = Jlr(p,,+pn!E(L,).
equation
(6) (7)
of (5) and (6) into (7) gives:
=s ML,) = 2K,(V,/L,‘)/3e(~,,+~,,j. (8) and from the physical
( 13)
V, = V,,,Y”‘( I _ ,p2
?
113)
where: v,,, = I’,( K,,lK, )’ is the voltage
obtained
( 14)
from ( I I ) by formally
setting
J = J, and L, = 0.
The complete device can now be considered as the series connection of two diodes corresponding to the two regions shown in Fig. 1: the diode of region I works in the high injection regime. while the diode of region II works in the low injection regime. The resulting .I-V characteristic of the complete device is obtained from equations (9). (IO)and(l3): J/J,,‘,‘(
1 ~ J/J, ):“I
(IS)
(5)
and:
The substitution
region
In this region the electric field is zero at the cathode and reaches its maximum at .Y= I_, The voltage V, can be computed from ( I I ):
V/V, = (J/J, )’ i- (V,,,/V,)(
/l(L,)
across
K, = (9/S)f’~,r~L,,((~l,(.,r’+ c,, (‘,,)?,,/2(C,,C~,, )
.I = K,, v,211_1:I
E(L,)
(111
using the following
where IV. Pare the densities of ionized trap centers. The values of J and V at which (3) is no longer satisfied are labelled J, and Vt. In regime 3, the electric field is zero near the anode. increases along the device and reaches its maximum value near the cathode. The carrier densities decrease along the device reaching their minimum value near the cathode. Thus, the first point of the device, where condition (3) is not satisfied lies near the cathode, when J is decreased below J,. This means that. for J < Jt, the device can be divided into two regions (Fig. I); in region I (0 < .r i L,). condition (3) holds. whereas in region 11 it is: n.p
L, i::
- C,l(‘,,’ J/c,, c,,Jc,, + C’,,).
Equation (15) is valid for J < .I,, i.e. it represents the J-V characteristic in regime 2 and in the NR region. Regime 3. i.e. for J > J,. is described by equation (5) where L, = L and V, = V. In order to find the upper and lower terminal voltages and the corresponding currents of the N R region, one has to set:
(8) meaning
dV/dJ=O
Let
V’ be the
lower
terminal
(16)
voltage
to which
NEGATIVE
RESISTANCE
IN SILICON
DOPED
1121
i n+
“i”
P+I
WITH GOLD
I I
-I--b 0 o+
Fig.
L
Ll
1. Schematic
v
-P
cross-section
corresponds J’, and let V” be the upper terminal voltage to which corresponds J”. The condition for the negative resistance.to exist is that V’ < V” and J’ > J”. It is worth pointing out that the above results apply only to a diode which is not space-chargelimited at breakover, i.e. in which V,(J”) is not greater than G’*.The breakover voltage however, is essentially temperature independent, while V, decreases with increasing temperature: thus equation (I 5) holds at a high enough temperature so that V,( J”) < V,“. The profiles of p(x). n(x) and E(x) can also be determined in regions I and I1 according to the procedure outlined in [2] and the results are in agreement with the computer solutions reported in [2]. One might observe that p(x), n(x) and E(x) are not continuous at x = 15,; this is due to our rough approach, as for x close to L1 neither condition (3). nor (4) is satisfied, so that a more complete theory would be necessary. It is worth noting, however, that the analytical expressions reported in [2] show a discontinuity at x = 0 in regime 2, and at n = L in regime 3. In our work the discontinuity region, x = L,, moves from cathode to anode as the current is decreased along the NR region. Note also that equation (I 5) does not apply to the ohmic regime. As an example we consider the case of a p+-i-n+ silicon diode in which the intrinsic region, doped with NA, = 10lfi atoms of gold per cubic centimeter, is 100 pm long. Figure 2 shows the J-V
X
of the device.
‘5 Id_ a
T, 7
IO'-
10
-
11
1
I
I
/
IO2
10
I
IO3
v (volts)
Fig. 2. J-V characteristic of the gold-doped p+-i-n+ diode. characteristic of the device, as computed from equation (15). Recalling that at 300°K V,,, * VI, the approximate solutions of equation (16) are: V’ = Vt
J’ = Jt
(17)
at the low terminal voltage and: *This remark has been suggested-by the Referee’s comments.
V” = (3/16)31’Vt,;
J” = (l/4)5,
(18)
P. ANTOGNETTI,
1122
A. CHlABRERA
at the upper terminal voltage. The fact that V’ < V” and J’ > .I” shows the existance of the NR region. The value of the average negative resistance per unit area is given by: R = (V”-
V’)/(./“-./‘)
= -(3/16)‘!‘V,,//,
and S. RIDELLA
mation holds. The J-V characteristic. evaluated for a gold-doped silicon p--i-n’ diode. shows the existance of the negative resistance region. This world represents an extension of the results obtained by Lampert[7] in the case of semiconductors doped with one level traps.
(19) No attempt has been made to compare the numerically evaluated data reported in [2] with the characteristic of Fig. 2, as in our case we have I’,(/“) = I/” > V,,. Nevertheless the curve of Fig. 2 is representative of equation (15) and by sufficiently increasing the temperature one is able to reach the condition V,(/“) < Vb. In summary, we have derived a simple analytical expression of the J-V characteristic for gold-doped silicon; such expression applies to the low injection and high injection regimes and to the negative resistance region when the quasineutrality approxi-
REFERENCES
1. M. A. Lampert, Ph~s. Kev. 125. I26 ( 1962).
2, W. H. Weber and G. W. Ford, Solid-St. Elrctron. 13. 1333 (1970). 3. J. S. Moore. N. Holonyak Jr. and M. D. Sirkis. Solid-St. Electron. 10, 823 (I 967). 4. M. M. Blouke, N. Holonyak Jr., B. G. Streetman and H. R. Zwicker. Solid-St. Electron. 13, 337 (I 970). 5. P. Antognetti. A. Chiabrera and S. Ridella. Eurooean Semiconductor Devices Research ConferLnce. Munich. March I97 I. 6. H. R. Zwicker. B. G. Streetman. A. M. Andrews and H. J. Deuling. Appl. Phys. Lrtt. 16.63 ( 1970). 7. M. A. Lampert and P. Mark. Current frljec.tiorz ;/I Solids. Chapter 11,Academic Press. New York (1970).
Ekclronics
Pergamon
Press
197 1. Vol. 14, pp. I I 19- I122.
NEGATIVE
Printed in Great Britain
RESISTANCE IN SILICON DOPED WITH GOLD*
P. ANTOGNETTI,
Istituto di Elettrotecnica,
A. CHIABRERA
Universitadi
and S. RIDELLA
Genova, Viale Cambiaso 6, 16145 Genova, Italy
(Received 18 January 197 1; in revisedform 22 February 197 1) this paper, we have derived a simple analytical expression of the J-V characteristic for gold-doped silicon; such expression applies to the low injection and high injection regimes and to the negative resistance region when the quasineutrality approximation holds. The J-V characteristic. evaluated for a gold-doped silicon p+-i-n + diode, shows the existence of the negative resistance region.
Abstract-In
Resume - Dans cet article, nous avons derive une simple expression analytique de la caracteristique J-V du silicium dope a l’or. Une telle expression s’applique aux bas et hauts regimes d’injection et a la zone de resistance negative lorsque se maintient I’approximation de quasi-neutralite. La caracteristique J-V, evaluee pour une diode $-i-n+ au silicium dope a l’or, fait apparaitre I’existence d’une zone de resistance negative. ZusammenfassungFur golddotiertes Silizium werden einfache analytische Ausdrilcke fur die I-I’Charakteristik abgeleitet. Sie sind anwendbar bei geringer und hoher Injektion sowie im Bereich negativen Widerstandes, wenn naherungsweise eine QuasineutralM vorliegt. Die I-l/-Charakteristik einer golddotierten p+-i-n+-Diode aus Silizium weist einen solchen Bereich negativen Widerstandes auf.
BULK semiconductors doped with deep two-level traps exhibit interesting properties. Their J-V characteristic is of the S-type showing a differential negative resistance [ 1,2] furthermore sinusoidal oscillations have been observed in the positive resistance region of the characteristic [3,4]. These oscillations have been interpreted as recombination wave instabilities [ 51. The case of gold-doped silicon p+-i-n+ diodes has been studied extensively[2,3,5]; computer solutions for the evaluation of the J-V characteristic have been outlined in [2,6]. Simple power laws for the various regimes of the characteristic have been found to agree well with the numerical solution[2]; yet such analytical solutions apply only to the positive resistance regions of the J-V curve; as far as the negative resistance region is concerned, they only predict its lower terminal voltage Vt. The purpose of this paper is to formulate an analytical expression of the J-V characteristic of *Work supported (CNR) of Italy.
by the National Research
Council
silicon doped with gold, which includes the negative resistance region; this is done by applying the Regional Approximation Method[7] to the treatment given in [2]. We assume that the reader is familiar with [2], as we will use the same notation. Let us now recall the various regimes of the J-P’ characteristic [2] : (1) ohmic regime (2) low injection regime (3) high injection regime (4) space-charge limited regime The quasineutrality approximations, i.e. E + 0, holds in regimes 1, 2, 3; the negative resistance (NR) region is comprised between regimes 2 and 3 and is separated from regime 2 by a constant voltage curve at voltage V,: such breakover voltage, characterizing the onset of NR, is Edependent. This seems to suggest that the NR region is function of 6, but such assumption is in contrast with the quasineutrality approximation in regimes 2 and 3. Furthermore, even if one assumes to approximate the NR region by connecting
1119
P. ANTOGNETTI.
I120
A. CHIABRERA
V(, and V, with a line. no information is available on the x-distribution of II, p and E. Here we deal with regimes 2. 3 and with the NR region assuming the quasineutrality approximation to hold. This assumption will be further discussed at the end of the paper. In regime 3. the characteristic is given by [ 71:
and S. RIDELLA
given to L, it stems out that p(f.,) and tz(L, ) do not depend on the value of the abscissa L, if: C’, = L/,x-: Equation
?‘ = L,/l..
(5) can now be rewritten:
t IO)
.I = .I,> J =
&
V’/L:’
(1)
For region II we can write [2]: being L the device
length. and: .I = K,C’,‘l(f.-
K,, = (9i8)e~,,~,,((;(.,,
(7) Equation (1) has been derived approximations: t1.p > N. P:
Let V, be the voltage region we have [ 21:
being I/, = VIl.and:
V, the
vnoltage drop
I1 = p
(3)
< N. P. across
(4) region
I; for this
p(L,) From
= 3V,/2L,
-- ML,) = Jlr(p,,+pn!E(L,).
equation
(6) (7)
of (5) and (6) into (7) gives:
=s ML,) = 2K,(V,/L,‘)/3e(~,,+~,,j. (8) and from the physical
( 13)
V, = V,,,Y”‘( I _ ,p2
?
113)
where: v,,, = I’,( K,,lK, )’ is the voltage
obtained
( 14)
from ( I I ) by formally
setting
J = J, and L, = 0.
The complete device can now be considered as the series connection of two diodes corresponding to the two regions shown in Fig. 1: the diode of region I works in the high injection regime. while the diode of region II works in the low injection regime. The resulting .I-V characteristic of the complete device is obtained from equations (9). (IO)and(l3): J/J,,‘,‘(
1 ~ J/J, ):“I
(IS)
(5)
and:
The substitution
region
In this region the electric field is zero at the cathode and reaches its maximum at .Y= I_, The voltage V, can be computed from ( I I ):
V/V, = (J/J, )’ i- (V,,,/V,)(
/l(L,)
across
K, = (9/S)f’~,r~L,,((~l,(.,r’+ c,, (‘,,)?,,/2(C,,C~,, )
.I = K,, v,211_1:I
E(L,)
(111
using the following
where IV. Pare the densities of ionized trap centers. The values of J and V at which (3) is no longer satisfied are labelled J, and Vt. In regime 3, the electric field is zero near the anode. increases along the device and reaches its maximum value near the cathode. The carrier densities decrease along the device reaching their minimum value near the cathode. Thus, the first point of the device, where condition (3) is not satisfied lies near the cathode, when J is decreased below J,. This means that. for J < Jt, the device can be divided into two regions (Fig. I); in region I (0 < .r i L,). condition (3) holds. whereas in region 11 it is: n.p
L, i::
- C,l(‘,,’ J/c,, c,,Jc,, + C’,,).
Equation (15) is valid for J < .I,, i.e. it represents the J-V characteristic in regime 2 and in the NR region. Regime 3. i.e. for J > J,. is described by equation (5) where L, = L and V, = V. In order to find the upper and lower terminal voltages and the corresponding currents of the N R region, one has to set:
(8) meaning
dV/dJ=O
Let
V’ be the
lower
terminal
(16)
voltage
to which
NEGATIVE
RESISTANCE
IN SILICON
DOPED
1121
i n+
“i”
P+I
WITH GOLD
I I
-I--b 0 o+
Fig.
L
Ll
1. Schematic
v
-P
cross-section
corresponds J’, and let V” be the upper terminal voltage to which corresponds J”. The condition for the negative resistance.to exist is that V’ < V” and J’ > J”. It is worth pointing out that the above results apply only to a diode which is not space-chargelimited at breakover, i.e. in which V,(J”) is not greater than G’*.The breakover voltage however, is essentially temperature independent, while V, decreases with increasing temperature: thus equation (I 5) holds at a high enough temperature so that V,( J”) < V,“. The profiles of p(x). n(x) and E(x) can also be determined in regions I and I1 according to the procedure outlined in [2] and the results are in agreement with the computer solutions reported in [2]. One might observe that p(x), n(x) and E(x) are not continuous at x = 15,; this is due to our rough approach, as for x close to L1 neither condition (3). nor (4) is satisfied, so that a more complete theory would be necessary. It is worth noting, however, that the analytical expressions reported in [2] show a discontinuity at x = 0 in regime 2, and at n = L in regime 3. In our work the discontinuity region, x = L,, moves from cathode to anode as the current is decreased along the NR region. Note also that equation (I 5) does not apply to the ohmic regime. As an example we consider the case of a p+-i-n+ silicon diode in which the intrinsic region, doped with NA, = 10lfi atoms of gold per cubic centimeter, is 100 pm long. Figure 2 shows the J-V
X
of the device.
‘5 Id_ a
T, 7
IO'-
10
-
11
1
I
I
/
IO2
10
I
IO3
v (volts)
Fig. 2. J-V characteristic of the gold-doped p+-i-n+ diode. characteristic of the device, as computed from equation (15). Recalling that at 300°K V,,, * VI, the approximate solutions of equation (16) are: V’ = Vt
J’ = Jt
(17)
at the low terminal voltage and: *This remark has been suggested-by the Referee’s comments.
V” = (3/16)31’Vt,;
J” = (l/4)5,
(18)
P. ANTOGNETTI,
1122
A. CHlABRERA
at the upper terminal voltage. The fact that V’ < V” and J’ > .I” shows the existance of the NR region. The value of the average negative resistance per unit area is given by: R = (V”-
V’)/(./“-./‘)
= -(3/16)‘!‘V,,//,
and S. RIDELLA
mation holds. The J-V characteristic. evaluated for a gold-doped silicon p--i-n’ diode. shows the existance of the negative resistance region. This world represents an extension of the results obtained by Lampert[7] in the case of semiconductors doped with one level traps.
(19) No attempt has been made to compare the numerically evaluated data reported in [2] with the characteristic of Fig. 2, as in our case we have I’,(/“) = I/” > V,,. Nevertheless the curve of Fig. 2 is representative of equation (15) and by sufficiently increasing the temperature one is able to reach the condition V,(/“) < Vb. In summary, we have derived a simple analytical expression of the J-V characteristic for gold-doped silicon; such expression applies to the low injection and high injection regimes and to the negative resistance region when the quasineutrality approxi-
REFERENCES
1. M. A. Lampert, Ph~s. Kev. 125. I26 ( 1962).
2, W. H. Weber and G. W. Ford, Solid-St. Elrctron. 13. 1333 (1970). 3. J. S. Moore. N. Holonyak Jr. and M. D. Sirkis. Solid-St. Electron. 10, 823 (I 967). 4. M. M. Blouke, N. Holonyak Jr., B. G. Streetman and H. R. Zwicker. Solid-St. Electron. 13, 337 (I 970). 5. P. Antognetti. A. Chiabrera and S. Ridella. Eurooean Semiconductor Devices Research ConferLnce. Munich. March I97 I. 6. H. R. Zwicker. B. G. Streetman. A. M. Andrews and H. J. Deuling. Appl. Phys. Lrtt. 16.63 ( 1970). 7. M. A. Lampert and P. Mark. Current frljec.tiorz ;/I Solids. Chapter 11,Academic Press. New York (1970).