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Integral aspect of transistor differential conductances
Solid~SIalr Ekcrmnics Pemmnon Press Ltd
Vol. 24. pp. 13Fl34
, 1981. Printed inGreatBnuin
INTEGRAL ASPECT OF TRANSISTOR DIFFERENTIAL CONDUCTANCES A. MIHNEA, T. DUNCA,S. GEORGESCU and A. A. VILPMAIOR CCSITS/IPRS BANEASA, Str.ErouIancu Nicolae 32/32B, 72996_Bucharest, Romania (Received 25 March
1980; in revised form 21 June 1980)
Abstract-There is experimental evidence that several transistor types in specified regimes have I-V characteristics with identical features, namely: constant differential conductances; translation symmetry. Therefore, they might admit similar approaches. We intend to describe them analytically using integral expressions of the current, in order to connect the differential conductances with the boundary values rather than with the detailed charge and field distributions.
tance of the lightly doped collector region in equilibrium.
BIPOLAR TRANSISTOR IN QUASI-SATUIWT’ION
Many bipolar transistors (e.g. planar-epitaxialtransistors) Varying eqn (2), we get: feature I-V characteristics as shown in Fig. l[l]. The quasi-saturation is usually explained by the gradual extension of a high injection domain from the collectorbase junction through the lightly doped collector region[2]. When this whole region is at high injection, hard saturation is reached. We shall assume that the current in the collector region is uniformly distributed across an effective area A(x), dependent upon the depth in the collector, X, and independent from the operating point I=-VCE: SA(x) = 0:
1, = qA(x)p.n(d,
-4p~)d4./dx,
-h/N),=owm
rc = JO, (qA(x)p.N)-‘dx
is the electrical
=
. W.(w) SVC,
exp ( VB&~TI~))SVBC,
+ (k/N)
(3)
where VBc = c#J,,~ - &(O). With this definition for Vet, Gummel’s charge control relation holds in the quasisaturation too. Consequently, the difference between the collector current in quasi-saturation and in active regime is:
(1)
where &(x) is the quasi-Fermi level of the majority carriers in the collector; & is the quasi-Fermi level of the minority carriers, constant in the high injection domain; n(& -4PB) is the majority carrier concentration, assuming that quasi-neutrality condition holds. Separating the variables in eqn (I ) and integrating over the lightly doped collector region (from x = 0 to x = tv), we obtain:
where
= rc(&)+Ppr( = (n/N),=,
rc(&)v,,
L -
Iactive
=
-
10 exp
(4)
( V*d(kTlq)),
where I0 is considered constant. Varying eliminating VBc from eqn (3), we obtain:
eqn (4) and
(5)
this equation explains both the translation symmetry of the characteristics at small reverse current (Zactive- Ic) and their constant slope, equal to I/rc, at large reverse current. It is interesting to note that the above considerations apply as well for the “conductivity controlled transistor”[3]. This device has no metallurgical base; hence, its active regime collector current, as given by Gummel’s charge control relation, goes to infinity. Consequently, its I-V characteristics follow the broken lines in Fig. I.
resis-
JUNCTION-GATE FIELD EFFECT TRANSISTOR
The gate-grounded source I-V characteristics of a FET are shown in Fig. 2. The channel is supposed to present a neutral
region
region
extended
near the source toward
electrode
the drain
and a depleted
electrode
(see insert
in
Fig. 2). First, conduction is neglected by taking the majority carrier mobility IL. = 0. In the Shockley’s “gradual
Fig. 1. Characteristics of bipolar transistors with low collector doping. In quasi-saturation. near the boundary with the active regime, they re-iterate each other by translation along the l/rc-slope broken lines.
channel any 133
approximation”,
section
of the
neutral
the
integral channel
mobile
charge
depends
upon
in the
134
A. MIHNEA
et a/.
tion x is:
Integrating along the neutral channel (from 0 to x~):
Fig. 2. Characteristicsof a FET. They re-iterate each other by translation along the
V,
axis. The considered FET structure is shown in the insert.
This expression suggests that the FET current is controlled by the electrical charge of carriers concentrated in the neutral channel. Equation (11) explains the features of the FET I-V characteristics at intermediate and high current (at low current, I-V characteristics are exponential [4]). Assuming 6,~ = 0:
electrostatic potential I/I in the channel axis: n(x, y)dy = NW,
(12) (SIsdSVs).,..
0)).
(13)
WsdSV&.,
We will take into account the two-dimensional effect in the channel, considering this function dependent also on the drain voltage VD: N(x) = A”($, V,).
(7)
Specifically, we suppose that a simultaneous variation SV, SV, with the condition &M(x)=O, for any x, is possible. Then, S$ is the difference of two Poisson equation solutions with identical charge distribution; i.e. 8+(x, y) is a solution of the Laplace equation for the domain given in Fig. 2. For low longitudinal fields in the neutral channel, this may be regarded as a “Laplace voltage divider”: (6$(x, O)/SVD),,,, = l/p,
(8)
where CL= (32/~‘)(I/w)’ is the Laplace “voltage gain”[41. Accordingly: N(x) = J(t), with < = IL- Vd( V,)
= - (d&f) 4”Jv”( vs- VA Vd = (VCL)(P~,/X~,) 4.v^( vs - VAVD)).
(14)
In the low current limit, &*((s’) = -.X[, with X>O. The characteristics Iso vs V. at constant VD observe the space-charge limited current law [S]: ISD= (p”Ixd)(qx/m vs- VA vLd*,
(15)
and present translation symmetry. In the high current limit, the mobile charge approaches the fixed charge in the channel sections near the source: K = NW. It cannot exceed this value because of the ohmic nature of the source contact. Consequently, the characteristics become linear. The usual transconductance (13) approaches the open channel conductance, as predicted by the classical theory. Regardless of the current value, the voltage gain p is constant and the drain conductance equals the transconductance divided by the Laplace voltage gain [4].
(9)
REFERENCES
L. A. Hahn, ZEEETrans. Electron Deu. ED-M, 654 (1%9). 2. L. E. Clark, IEEE Trans. Electron Deu. ED-16, 113(1%9). I.
where V,( V,) observes the conditions: d VJd V, = l/p and K = 0 for 4 = Vd(V,). So, V,( V,) is the voltage in the channel axis at complete depletion boundary xd (see Fig. 2). Now, let the carrier mobility take its actual, non-zero value. The conduction current in a neutral channel sec-
3. A. Caruso, P. Spirito, G. Vitale, Electron. Left. IO, 267 (1979). 4. J. Nishizawa. T. Terasaki and J. Shibata. IEEE Trans. Elecfron Deu. ED-22, I85 (1975). 5. D. Dascrilu, Electronic Processes in Unioo/or Solid-State Devices. Editura Academiei, Bucuresti and Abacus Press, Tunbridge Wells (1977).
Vol. 24. pp. 13Fl34
, 1981. Printed inGreatBnuin
INTEGRAL ASPECT OF TRANSISTOR DIFFERENTIAL CONDUCTANCES A. MIHNEA, T. DUNCA,S. GEORGESCU and A. A. VILPMAIOR CCSITS/IPRS BANEASA, Str.ErouIancu Nicolae 32/32B, 72996_Bucharest, Romania (Received 25 March
1980; in revised form 21 June 1980)
Abstract-There is experimental evidence that several transistor types in specified regimes have I-V characteristics with identical features, namely: constant differential conductances; translation symmetry. Therefore, they might admit similar approaches. We intend to describe them analytically using integral expressions of the current, in order to connect the differential conductances with the boundary values rather than with the detailed charge and field distributions.
tance of the lightly doped collector region in equilibrium.
BIPOLAR TRANSISTOR IN QUASI-SATUIWT’ION
Many bipolar transistors (e.g. planar-epitaxialtransistors) Varying eqn (2), we get: feature I-V characteristics as shown in Fig. l[l]. The quasi-saturation is usually explained by the gradual extension of a high injection domain from the collectorbase junction through the lightly doped collector region[2]. When this whole region is at high injection, hard saturation is reached. We shall assume that the current in the collector region is uniformly distributed across an effective area A(x), dependent upon the depth in the collector, X, and independent from the operating point I=-VCE: SA(x) = 0:
1, = qA(x)p.n(d,
-4p~)d4./dx,
-h/N),=owm
rc = JO, (qA(x)p.N)-‘dx
is the electrical
=
. W.(w) SVC,
exp ( VB&~TI~))SVBC,
+ (k/N)
(3)
where VBc = c#J,,~ - &(O). With this definition for Vet, Gummel’s charge control relation holds in the quasisaturation too. Consequently, the difference between the collector current in quasi-saturation and in active regime is:
(1)
where &(x) is the quasi-Fermi level of the majority carriers in the collector; & is the quasi-Fermi level of the minority carriers, constant in the high injection domain; n(& -4PB) is the majority carrier concentration, assuming that quasi-neutrality condition holds. Separating the variables in eqn (I ) and integrating over the lightly doped collector region (from x = 0 to x = tv), we obtain:
where
= rc(&)+Ppr( = (n/N),=,
rc(&)v,,
L -
Iactive
=
-
10 exp
(4)
( V*d(kTlq)),
where I0 is considered constant. Varying eliminating VBc from eqn (3), we obtain:
eqn (4) and
(5)
this equation explains both the translation symmetry of the characteristics at small reverse current (Zactive- Ic) and their constant slope, equal to I/rc, at large reverse current. It is interesting to note that the above considerations apply as well for the “conductivity controlled transistor”[3]. This device has no metallurgical base; hence, its active regime collector current, as given by Gummel’s charge control relation, goes to infinity. Consequently, its I-V characteristics follow the broken lines in Fig. I.
resis-
JUNCTION-GATE FIELD EFFECT TRANSISTOR
The gate-grounded source I-V characteristics of a FET are shown in Fig. 2. The channel is supposed to present a neutral
region
region
extended
near the source toward
electrode
the drain
and a depleted
electrode
(see insert
in
Fig. 2). First, conduction is neglected by taking the majority carrier mobility IL. = 0. In the Shockley’s “gradual
Fig. 1. Characteristics of bipolar transistors with low collector doping. In quasi-saturation. near the boundary with the active regime, they re-iterate each other by translation along the l/rc-slope broken lines.
channel any 133
approximation”,
section
of the
neutral
the
integral channel
mobile
charge
depends
upon
in the
134
A. MIHNEA
et a/.
tion x is:
Integrating along the neutral channel (from 0 to x~):
Fig. 2. Characteristicsof a FET. They re-iterate each other by translation along the
V,
axis. The considered FET structure is shown in the insert.
This expression suggests that the FET current is controlled by the electrical charge of carriers concentrated in the neutral channel. Equation (11) explains the features of the FET I-V characteristics at intermediate and high current (at low current, I-V characteristics are exponential [4]). Assuming 6,~ = 0:
electrostatic potential I/I in the channel axis: n(x, y)dy = NW,
(12) (SIsdSVs).,..
0)).
(13)
WsdSV&.,
We will take into account the two-dimensional effect in the channel, considering this function dependent also on the drain voltage VD: N(x) = A”($, V,).
(7)
Specifically, we suppose that a simultaneous variation SV, SV, with the condition &M(x)=O, for any x, is possible. Then, S$ is the difference of two Poisson equation solutions with identical charge distribution; i.e. 8+(x, y) is a solution of the Laplace equation for the domain given in Fig. 2. For low longitudinal fields in the neutral channel, this may be regarded as a “Laplace voltage divider”: (6$(x, O)/SVD),,,, = l/p,
(8)
where CL= (32/~‘)(I/w)’ is the Laplace “voltage gain”[41. Accordingly: N(x) = J(t), with < = IL- Vd( V,)
= - (d&f) 4”Jv”( vs- VA Vd = (VCL)(P~,/X~,) 4.v^( vs - VAVD)).
(14)
In the low current limit, &*((s’) = -.X[, with X>O. The characteristics Iso vs V. at constant VD observe the space-charge limited current law [S]: ISD= (p”Ixd)(qx/m vs- VA vLd*,
(15)
and present translation symmetry. In the high current limit, the mobile charge approaches the fixed charge in the channel sections near the source: K = NW. It cannot exceed this value because of the ohmic nature of the source contact. Consequently, the characteristics become linear. The usual transconductance (13) approaches the open channel conductance, as predicted by the classical theory. Regardless of the current value, the voltage gain p is constant and the drain conductance equals the transconductance divided by the Laplace voltage gain [4].
(9)
REFERENCES
L. A. Hahn, ZEEETrans. Electron Deu. ED-M, 654 (1%9). 2. L. E. Clark, IEEE Trans. Electron Deu. ED-16, 113(1%9). I.
where V,( V,) observes the conditions: d VJd V, = l/p and K = 0 for 4 = Vd(V,). So, V,( V,) is the voltage in the channel axis at complete depletion boundary xd (see Fig. 2). Now, let the carrier mobility take its actual, non-zero value. The conduction current in a neutral channel sec-
3. A. Caruso, P. Spirito, G. Vitale, Electron. Left. IO, 267 (1979). 4. J. Nishizawa. T. Terasaki and J. Shibata. IEEE Trans. Elecfron Deu. ED-22, I85 (1975). 5. D. Dascrilu, Electronic Processes in Unioo/or Solid-State Devices. Editura Academiei, Bucuresti and Abacus Press, Tunbridge Wells (1977).