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Excess base noise in the bipolar junction transistor
Solrd-Sure Elecrronrc.~ Vol. 28. No. 9, pp. 875-876. Printed in Great Britain.
1985
003x-1101/x5 $3.00 + .lo ‘J 1985 Pergamon Press Ltd.
EXCESS BASE NOISE IN THE BIPOLAR JUNCTION TRANSISTOR L. M. RUCKERand T. D. DAVIS Electrical
Engineering
(Received
Department,
University of Texas at Arlington, U.S.A.
10 September 1984; in revisedform
Arlington,
TX 75019,
10 December 1984)
Abstract-It has been known for some time that the bipolar transistor base region exhibits a noise in excess of that predicted by thermal noise associated with the base spreading resistance measured by largeor small-signal methods. This paper presents a relatively simple mechanism and model involving a transconductance gradient that accounts for the excess noise.
The bipolar junction transistor (BJT) base region has been studied for over two decades with varying degrees of success. One difficulty is that measurements of the base resistance by dc., a.c. (large- and smallsignal) and noise methods do not agree. The a.c. and to some extent the less reliable d.c. techniques are fairly well understood[l-31. The measured noise resistance is typically a factor of 2-4 greater than expected. This note will present a mechanism that accounts for the excess base noise with a model that is relatively easy to use. The base resistance is generally broken into the extrinsic part between the base contact and the edge of the emitter and the intrinsic part under the active emitter. At low frequency the extrinsic part is a simple resistor. The intrinsic resistance is a thin region under the emitter with a resistance typically two orders of magnitude larger than the extrinsic part. Both regions exhibit thermal noise voltage spectral density as given by the Nyquist relation S,(f) = 4kTR, where k is Boltzman’s constant, T is temperature in Kelvins and R is resistance. At a low level the emitter current is evenly distributed and there is no gradient except at the edge of the emitter. However, at moderate levels there is a crowding of the emitter current toward the base contact due to the base current flowing through the intrinsic resistance. For much practical work this does not become important until higher current levels, but any current gradient in the emitter causes a transconductance gradient. This gradient gives rise to an excess noise that is associated with the base region. A differential element of the intrinsic base has a resistance d R with an associated thermal noise voltage e,. The instantaneous noise voltages at the ends of the element will have equal magnitude but opposite signs. Each voltage will induce a current in the collector given by the relation g,,,e,, where g,,, is the transconductance. For a constant g,,,, e, induces a positive collector current increment at one end and a negative current at the other end and they cancel. However, if g,,, is not constant the two currents will 875
not be equal and the excess noise not accounted for in earlier models results. If the thermal noise voltage density associated with the resistance element dr is 7de, = 4kTdR = 4kTR dx, and the transconductance at one end of the element is g, and at the other end is g, + dg,,,, then the collector current noise induced by this mechanism is given by di2=(4kTRdx)[(
gm +dgm)2
-(g,)‘]
(la)
=(4kTRdx)[g;+2g,dg,+dg;-g;] (lb) = (4kTR dx)(2g,
dg,).
(lc)
This assumes that dg: is small compared with d g,,,, which presents no difficulties. To apply eqn (1) it is necessary to find a relation for the current crowding and write the transconductance as a function of distance perpendicular to the base contact. Hauser’s model[4] has been used for this purpose and gives the result R ITL = R,,{ 1 + 2[ln(cos Z)/(
Z tan Z)]}.
(2)
R, is the maximum (low-current) intrinsic base resis-
tance, and Z is defined in eqn (4). Equation (2) represents the excess base noise resistance with low injection into the base. However, the conductivity modulation effects that occur at higher injection levels will reduce the noise as the base resistivity drops, When this effect is included the high injection excess base noise resistance is R
@a L nH = u,W(l - a)
X
sinhL&? i
L&
tan’
sinhL&---2[
Lfi Xln(cosh
L&)
, (3)
L. M. RUCKER and T. D. DAVIS
876
where
Z is defined
by the relation
zmz=( -gi+j,
(4)
diffusion l,(g<22, D, and u, are the ambipolar constant and mobility, a is the common base current gain, L is the base width perpendicular to the base contact, h is the thickness of the base (emitter to collector), W is the base width perpendicular to L and h, K = (1 - a)/W2. le is emitter current, and Ix = 2Vr W/(1 - a)phr where Vr is kT/q and ph is the base resistivity. A more direct method of determining Z is used in SPICE models[5]. The combination of eqns (2) and (3) is difficult to obtain in one derivation. However, they are easily combined as two uncorrelated noise sources to give the total equivalent excess base noise resistance R,,T.: 1 (5) R’zr=
i (l/R,,,)*
+(1/R,,,,)’
This noise resistance is in addition to the well-established base noise resistance 4kTr,. However, the excess term is added to the model in series with the internal base node usually identified as B’ and is not altered by the input impedance as is 4kTr,. Significant data exists for four devices of varying geometries to verify this part of the model and will be presented at a later data when other aspects can be included. Because each elemental area of the base generates noise, the associated resistance and capacitance
would be diminishingly small resulting in a very broadband noise. Transit time effects do not effect this noise so the high frequency limit will be set by the same quantum mechanical considerations applicable to any thermal noise. This excess noise will not appear in the usual SPICE analysis where a distributed transistor is simulated with five or ten elements. The reason is that the noise produced by the finite base resistance element is divided by the input resistance of each transistor element. In a real transistor the division between the infinitesimal d R and input resistance is negligible. The relation given in eqn (5) can be included in SPICE 2 models [3] by altering the program’s coding. The usual base resistance noise, 4kTr,, remains. This model can use the adaptation of Hauser’s current crowding model already in recent versions of SPICE which includes the more complex parts of the relations in eqns (2)-(5). An equivalent circuit model to inject the noise can be used but is cumbersome and not recommended. It is not possible to simply add a resistor since this would alter other important circuit characteristics.
REFERENCES 1. Ian Getreu, Modeling the B~polur Trunsisfor, pp. 151-164. Tektronix, City (1976). 2. G. Blasquez and .I. Caminade, Phvs. Status Solidi (a), 713 (1975). 3. G. Blasquez, J. Caminade and K. M. van Vliet, So/Id-St. Electron. 423 (1980). 4. J. R. Hauser, IEEE Truns. Electron Devices, 238 (1964). 5. Base region modeling defined in SPICE 2 manuals.
1985
003x-1101/x5 $3.00 + .lo ‘J 1985 Pergamon Press Ltd.
EXCESS BASE NOISE IN THE BIPOLAR JUNCTION TRANSISTOR L. M. RUCKERand T. D. DAVIS Electrical
Engineering
(Received
Department,
University of Texas at Arlington, U.S.A.
10 September 1984; in revisedform
Arlington,
TX 75019,
10 December 1984)
Abstract-It has been known for some time that the bipolar transistor base region exhibits a noise in excess of that predicted by thermal noise associated with the base spreading resistance measured by largeor small-signal methods. This paper presents a relatively simple mechanism and model involving a transconductance gradient that accounts for the excess noise.
The bipolar junction transistor (BJT) base region has been studied for over two decades with varying degrees of success. One difficulty is that measurements of the base resistance by dc., a.c. (large- and smallsignal) and noise methods do not agree. The a.c. and to some extent the less reliable d.c. techniques are fairly well understood[l-31. The measured noise resistance is typically a factor of 2-4 greater than expected. This note will present a mechanism that accounts for the excess base noise with a model that is relatively easy to use. The base resistance is generally broken into the extrinsic part between the base contact and the edge of the emitter and the intrinsic part under the active emitter. At low frequency the extrinsic part is a simple resistor. The intrinsic resistance is a thin region under the emitter with a resistance typically two orders of magnitude larger than the extrinsic part. Both regions exhibit thermal noise voltage spectral density as given by the Nyquist relation S,(f) = 4kTR, where k is Boltzman’s constant, T is temperature in Kelvins and R is resistance. At a low level the emitter current is evenly distributed and there is no gradient except at the edge of the emitter. However, at moderate levels there is a crowding of the emitter current toward the base contact due to the base current flowing through the intrinsic resistance. For much practical work this does not become important until higher current levels, but any current gradient in the emitter causes a transconductance gradient. This gradient gives rise to an excess noise that is associated with the base region. A differential element of the intrinsic base has a resistance d R with an associated thermal noise voltage e,. The instantaneous noise voltages at the ends of the element will have equal magnitude but opposite signs. Each voltage will induce a current in the collector given by the relation g,,,e,, where g,,, is the transconductance. For a constant g,,,, e, induces a positive collector current increment at one end and a negative current at the other end and they cancel. However, if g,,, is not constant the two currents will 875
not be equal and the excess noise not accounted for in earlier models results. If the thermal noise voltage density associated with the resistance element dr is 7de, = 4kTdR = 4kTR dx, and the transconductance at one end of the element is g, and at the other end is g, + dg,,,, then the collector current noise induced by this mechanism is given by di2=(4kTRdx)[(
gm +dgm)2
-(g,)‘]
(la)
=(4kTRdx)[g;+2g,dg,+dg;-g;] (lb) = (4kTR dx)(2g,
dg,).
(lc)
This assumes that dg: is small compared with d g,,,, which presents no difficulties. To apply eqn (1) it is necessary to find a relation for the current crowding and write the transconductance as a function of distance perpendicular to the base contact. Hauser’s model[4] has been used for this purpose and gives the result R ITL = R,,{ 1 + 2[ln(cos Z)/(
Z tan Z)]}.
(2)
R, is the maximum (low-current) intrinsic base resis-
tance, and Z is defined in eqn (4). Equation (2) represents the excess base noise resistance with low injection into the base. However, the conductivity modulation effects that occur at higher injection levels will reduce the noise as the base resistivity drops, When this effect is included the high injection excess base noise resistance is R
@a L nH = u,W(l - a)
X
sinhL&? i
L&
tan’
sinhL&---2[
Lfi Xln(cosh
L&)
, (3)
L. M. RUCKER and T. D. DAVIS
876
where
Z is defined
by the relation
zmz=( -gi+j,
(4)
diffusion l,(g<22, D, and u, are the ambipolar constant and mobility, a is the common base current gain, L is the base width perpendicular to the base contact, h is the thickness of the base (emitter to collector), W is the base width perpendicular to L and h, K = (1 - a)/W2. le is emitter current, and Ix = 2Vr W/(1 - a)phr where Vr is kT/q and ph is the base resistivity. A more direct method of determining Z is used in SPICE models[5]. The combination of eqns (2) and (3) is difficult to obtain in one derivation. However, they are easily combined as two uncorrelated noise sources to give the total equivalent excess base noise resistance R,,T.: 1 (5) R’zr=
i (l/R,,,)*
+(1/R,,,,)’
This noise resistance is in addition to the well-established base noise resistance 4kTr,. However, the excess term is added to the model in series with the internal base node usually identified as B’ and is not altered by the input impedance as is 4kTr,. Significant data exists for four devices of varying geometries to verify this part of the model and will be presented at a later data when other aspects can be included. Because each elemental area of the base generates noise, the associated resistance and capacitance
would be diminishingly small resulting in a very broadband noise. Transit time effects do not effect this noise so the high frequency limit will be set by the same quantum mechanical considerations applicable to any thermal noise. This excess noise will not appear in the usual SPICE analysis where a distributed transistor is simulated with five or ten elements. The reason is that the noise produced by the finite base resistance element is divided by the input resistance of each transistor element. In a real transistor the division between the infinitesimal d R and input resistance is negligible. The relation given in eqn (5) can be included in SPICE 2 models [3] by altering the program’s coding. The usual base resistance noise, 4kTr,, remains. This model can use the adaptation of Hauser’s current crowding model already in recent versions of SPICE which includes the more complex parts of the relations in eqns (2)-(5). An equivalent circuit model to inject the noise can be used but is cumbersome and not recommended. It is not possible to simply add a resistor since this would alter other important circuit characteristics.
REFERENCES 1. Ian Getreu, Modeling the B~polur Trunsisfor, pp. 151-164. Tektronix, City (1976). 2. G. Blasquez and .I. Caminade, Phvs. Status Solidi (a), 713 (1975). 3. G. Blasquez, J. Caminade and K. M. van Vliet, So/Id-St. Electron. 423 (1980). 4. J. R. Hauser, IEEE Truns. Electron Devices, 238 (1964). 5. Base region modeling defined in SPICE 2 manuals.