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Bipolar transistor: Two-dimensional effects on current gain and base transit time
Solid-State ElectronicsVol. 31, No. 12, pp. 1715-1724,1988 Printed in Great Britain. All rights reserved
l-1038-1 101/88 S3.00+ 0.00 Copyright 0 1988Pergamon Press pk
BIPOLAR TRANSISTOR: TWO-DIMENSIONAL EFFECTS ON CURRENT GAIN AND BASE TRANSIT TIME H. F. F. Jos Development Department of Discrete Semiconductors, Philips, 6534 AE Nijmegen, The Netherlands (Receiued 29 March 1988; in revised form 18 July 1988)
Abstract-Two-dimensional &e&s in bipolar devices with small dimensions are calculated analytically if simplifying assumptions are made e.g. uniformly doped regions and zero recombination. In this paper we treat effects of the emitter sidewalls on the current gain and the base transit time. The current flows through the base and emitter are calculated for rectangular as well as curved emitter-base junctions. Current gain and base transit time are analysed as local variables, i.e. functions that vary along the 2-D emitter-base junction. In this way it can be seen where and how several parameters influence the overall performance of the device. The total current gain is modified by the sidewalls; it increases if the distance between the emitter-base junction and the emitter metallisation is increased. However, this distance cannot be increased indefinitely because of finite recombination lifetimes. The diffusion capacitance is strongly affected by the sidewalls. Because the sidewalls carry a considerable part of the current the transit time is less affected. However, the influence of the sidewalls is still noticeable. It is expected that in a transistor with a 1 pm wide and 0.15 pm deep emitter the transition frequency is already negatively influenced by the emitter sidewalls.
1. INTRODUCTlON
2. RECI’ANGULAREMITTER
The dimensions of bipolar transistors have been reduced since their invention in an ongoing effort to improve the high frequency performance of discrete as well as integrated devices[ 11.A reduction of lateral dimensions of, for instance, discrete bipolar transistors results in a lowering of the base resistance and smaller parasitic capacitances. This leads to a device with a higher cut-off frequency and lower noise. The reduction of vertical dimensions can be achieved by techniques as low energy implantation, rapid thermal annealing, co-implantation[2] and pre-amorphisation[3]. The lateral dimensions can be diminished by technologies as direct slice writing with an electron beam pattern generator (EBPG), X-ray lithography, plasma etching, etc. Submicron lateral dimensions can be realised. However, as the emitterwidth is reduced to the same order of magnitude as the emitterdepth 2-D effects are likely to occur. Up till now most analytical descriptions of bipolar devices have been 1-D. Recently Hurkx published a 2-D analytical model of a simplified bipolar transistor with a rectangular shaped emitter[rl]. In that model the consequences of 2-D effects for the current gain have been treated. In high frequency devices the current paths of the minority charge carriers in the base and emitter and the transit times belonging to these paths are of interest. In this paper we present an extension of the model of Hurkx by which the current paths and transit times can be calculated analytically for a simplified bipolar device. Our model not only treats rectangular shaped emitters but also curved emitters. We shall treat them separately.
In Fig. 1 a 2-D cross section of a simplified bipolar transistor with rectangular emitter is shown. We shall treat two important current densities: J&x, y) of minority carriers in the emitter and J,,(x, y) of minority carriers in the base. Analytical expression for J&-C,y) and J&y) can be obtained by means of conformal mapping. To do this we make some simplifying assumptions: -The state is stationary: all derivatives with respect to time are zero. -There is no generation or recombination. Especially for the emitter this is a difficult condition that only holds for very shallow emitters (b Q 0.25 pm for an npn device). -The bandgap is constant in the emitter as well as in the base. -There is no electric field in the emitter and base, hence the emitter and base are uniformly doped. -The surface recombination velocity at the emitter contact is infinite. It is possible to give an analytical expression for J,,(x, y) if there is finite recombination in the emitter and at the emitter contact. Hurkx[4] has treated this case. However, the solution is rather complex and consists of infinite series. At least a finite number of them must be numerically processed to gain insight. In the case of a curved emitter the solution will be even more complex. Since our model is developed for devices with very small dimensions it is justified to neglect recombination. We shall come back to this at the end of this paper. Using conformal mapping we
1715
H. F. F. Jos
1716
Integration onto z:
Emitter
of eqn
(2) yields
the
mapping
1 _ u>’
Collector
(4)
section of simplified emitter. The collector
bipolar
Fig. I. Cross rectangular
of w
transistor
with
is contacted at the back-
side.
obtain a solution with a small number of terms and extension to the curved emitter case is straightforward. Several effects, however, are not taken into account by this model, e.g. high injection, ohmic losses and the existence of space charge regions. 2.1. The base For an npn transistor the minority base are electrons. The concentration by the Laplace equation:
(1)
We model the base region as a region in the complex plane z (Fig. 2a). The boundary conditions and meaning of symbols are defined in Fig. 2(a). The electron concentration at the emitter-base junction is given by n,. In this model the base region extends to x = - co. Now the effect of one sidewall can be examined. Of course it should be checked afterwards whether this approach is justified in the case of two sidewalls. We shall show that it is because under the emitter the 1-D solution is rapidly reached for small negative values of x. We map the complex z plane onto the upper half plane w (Fig. 2b) by the Schwarz-Christoffel transformation[5]: dz
dl nw+l
s=F
i=J-l,
a=E.
(5)
N = in, In(u) + nj. The current curves are given by 101= constant and the equiconcentration curves by arg(v) = constant. The current curves are concentric circles centred at the origin and the equiconcentration curves are straight lines through the origin. Now v=w+Jm and hence plane is:
the
complex
concentration
in the
Differentiation
of eqn (6) with respect dN
i
dw
n
-c-n
current
to w yields:
1 ‘,,/m’
in the z plane can be found by:
(2)
J- w - 1’
(7)
where r =2{(d+b)/d}2-
,
(O,b)
(0) b
1’
n = nj
,f (40) /I
1.
3
(3)
an JY
n=n,
;O
Here D is the diffusivity
of electrons
in the base.
2”
(b)
4
d 1”
an
-0
1" n=O
v
lY
I-
2’
n=O
2 I
3
v-
-1
1
n=n,
4 I
I
4 IY’ x
w
(6)
The complex
w--t
and
Note that all functions in eqns (4) and (5) are complex and that their arguments can be negative. The point (t, 0) in the w plane is mapped onto the origin in the z plane. The boundary conditions in the w plane are denoted in Fig. 2(b). To find the complex concentration in the w plane we map the w plane onto the upper half plane u with the semicircle with radius 1 removed. For reasons of symmetry the complex concentration in the v plane is:
carriers in the is determined
V&(x, y) = 0.
dw=---
where
L- x,
Fig. 2. (a) Complex plane z = x + iy. Emitter depth is b, base thickness under emitter is d. (b) Upper half plane w = x’ + iy’. The points 14 correspond with those in Fig. 2(a).
1’
Bipolar transistor
Unfortunately we cannot invert eqn (4) and substitute w(z) in eqn (7). We must start at a point in the w plane and find the corresponding point z and the current at z. A simple computer programme has been written which performs this task. The points taken in the w plane are the nodes between the equiconcentration and current curves. The current curves were chosen in such a way that they divide the base in segments through which equal currents flow. The equiconcentration curves correspond to lines through the origin in the v plane with angles ilr/n with the horizontal axis, where i = 0, 1,. . . , n and n is the total number of curves. A vectorplot of the current densities in the base is shown in Fig. 3 (arbitrary units). The values for b and d are: 6 = 0.12 pm and d = 0.30 pm. The number of points taken on each current curve is 15. Note that the current density diverges near the sharp corner of the emitter. Furthermore it shows that the current density is already almost 1-D for x = -O.lOpm, which means that if the emitter is wider than 0.20 pm both sidewalls will not influence each other. At distances x > 0.5 pm the current densities are almost negligible. Since the basecontact is far away (at about x = 1.5 pm in the finest structured devices) its influence on the current flow can be ignored. 2.2. The transit time Now that we know the current flow we can calculate the transit time for each current curve. Look at a infinitesimal narrow current segment in the base. In Fig. 4 the concentration of electrons in the segment is plotted. We take the equiconcentration and current curves as orthogonal coordinate system. The coordinate along the current curve is 1, where 0 < I< I,,; the width of the curve is r(l). The current density in the segment is assumed to be dependent only on I:
0.12
---_ --_
1
-0.3
n
Fig. 4. Concentration of electrons in a current segment with length I,, and width r(l).
Furthermore: Jr(l) = constant = Jg(O),
(9)
where Jo is the current density at the emitter edge. From eqns (8) and (9) it follows: dn - JoW dl=qDro and hence: -J,,r(O) ’ dl’ n(l) = ~ + nj . qD s oW The stored charge in the segment is:
dQ=q
L.X
n(040 dl,
(10)
(11)
s0 Substitution of eqns (10) in (11) yields the transit time ~9in the segment:
e=+j 0
=-
1
D
ss c..
o
k..
r(l)
1
dl’
qqdl.
(12)
Note that if r(l) is constant 0 = 1:,,/2D as is expected. Since we know the shape of the current segments we can calculate the nested integral of
dn J = -qDdr.
-0.10
1717
_
- - _ - _ - _ -:.
-
,,,,1,,,,,,.,,,.... -0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
Fig. 3. Vector plot of minority carrier current density inside base. The vectors are of arbitrary length. Near the emitter comer the density diverges. All segments carry the same total current. Under the emitter the current flow rapidly becomes 1-D.
0.6
H. F. F. Jos
1718
eqn (12) numerically. However, eqn (12) consists of a nested integral and therefore it is not very sensitive to the shape of the current segments provided r(l) is a smooth function. For instance, if r(l,,,,,)/r(O) < 10, which is true almost everywhere except near points 3 and 4 in Fig. 2(a), the nested integrals of a linearly shaped and an exponentially shaped segment differ only by less than 10%. It is possible to give an analytical expression for eqn (12) if an exponentially shaped current segment is assumed, i.e. if:
-
b/d ‘2
-.-
b/d’1
-m-
b/d ‘2/3
6-
r(l) = r(O)exp(Z/a)
where 0 = I,,,/ln(r(O)/r(l,,)). Hence: 2
B=%{f-;(1--$.$)},
(13)
where I
c = ln{r(L,,)/r(O)). With the help of eqn (13) 0 can be calculated for a current segment if only ZmBX and r(l,,,)/r(O) are known. The values for lm, are given by the integral of Idz 1along the current flow line:
OL
1.0 p/b
Fig. 5. Transit time vs distance along the emitter-base junction for some values of b/d.
depends on b/d. Note that 8 has a sharp minimum at p = b where 0 is zero. This is the point where the current density diverges. (u2 exp(2ei$) - 2tu exp(id) + 1) do,
v exp(id,) + 1 where v is the point on the real axis in the v plane corresponding to the point on the base-emitter junction for which l,,, is calculated. However, it is much simpler to determine l,,,,, numerically from the shapes of the current curve. From eqn (9) it follows that r(I,,,,,)/r(O) = J,/J(I = I,,,). Since the point I = 0 corresponds to a point w = w, in the w plane, where w. is real, and the point I = l,,,,, to the point w = -w. eqn (7) can be used to calculate rU,,,)lr(O): r(L) -= r(0)
(w. + l)(wo + t) (Wo-t)(Wo1) .
We introduce p as the parametrisation of the emitter-base junction: p = 0 at the point (0, b), p = b at the point (0,O) and p = b -x at all points (x, 0) with negative x. In Fig. 5 0 is plotted as a function of p/b for several values of b/d. Note that 0 has been calibrated in such a way that 0 = 1 in the 1-D case under the emitter. There is an asymptote for p = 0 where 6 goes to infinity. In practice this will not happen because of finite recombination lifetimes. However, as we shall demonstrate in Section 2.3 there is no contradiction between our model and practice in this point. The behaviour of 0 for p < 1 strongly
2.3. The dajiiion capacitance Multiplying 0 by J yields the charge stored in a current segment. The total charge stored in the base is obtained by adding the charges stored in all segments. The total charge of a base with two sidewalls can be described by: Q = W,( bf’ + W), where 0, and J, are the 1-D base transit time and current density respectively and W is the width of the emitter. The contribution of one sidewall, which is not only determined by the geometrical sidewall but also by the region under the emitter where the current density cannot be described by the 1-D solution, is given by the function f: It is obtained by considering a region that extends so far under the emitter that at its left boundary x = -x,, the current flow can be accurately approximated by 1-D flow. From the total charge stored in this region a virtual charge Q = t9,J,xo is subtracted to calculate $ In Fig. 6 the function f is plotted vs d/b. The function f reaches a minimum of about 3.4 for d = 1.7b. The equation above yields the diffusion capacitance: C = C,(l + 2bf/ W).
(14)
Bipolar transistor
-f -.-
1719
b = 0.20pm and a basedepth of d = O.lOpm. The transit time belonging to the point at about an atomic radius (4A) below the surface at the base-emitter junction is still less than 50 times the 1-D transit time under the emitter. Since the 1-D transit time (0 = d*/2D) is typically 5 to 10 ps this transit time is less than 500~s. This is still much smaller than Shockley-Read-Hall recombination times of about lo-’ s[6].
g
2.4. The current gain
./’ ./ ./ / I
I
I
1
2
3
I 4
d/b
Fig. 6. Functionsfand
The hole current in the emitter is the most important contribution to the base current especially when the emitter is shallow. In our model we assume that it is the only contribution. Again we disregard recombination (see Section 5) and the hole concentration is governed by the Laplace equation. We solve the problem by conformal mapping. In Fig. 7 the emitter is represented in the complex z plane together with the boundary conditions for the hole current. The distance between the emitter contact and the emitter sidewall is a. Note that Fig. 7 is a mirror image to the structure in Fig. 1. We map the z plane onto the w plane of Fig. 2(b) and in this case we can invert the transformation: w _ 2 cosh(xz/b) + cosh(na/b) + 1 cosh(na/b) - 1 *
g vs d/b (see text).
The 1-D diffusion capacitance C, N e,J, IV. Since the minimum value of f is 3.4 the sidewalls contribute considerably to the total diffusion capacitance. For instance, if b = 0.15 pm and W = 1.0 pm the sidewalls are responsible for at least half of the diffusion capacitance. The asymptote for 8 at p = 0 suggests that an infinite charge might be stored at the surface of the structure. This is, however, not true. To find the total charge stored due to sidewall effects J@)B(p)dp has to be integrated along the base-emitter junction. We can prove that the part of this integral at the surface is finite, by integrating from p = 0 to p = 6, where 6 is arbitrarily small but finite. The point p = 0 corresponds to w = 1, while p = 6 corresponds to w = 1 + 7. From eqn (7) it follows that J(p) is independent of p if p goes to zero; from eqn (4) that p is proportional to fi, where 0 Q z’ d 7. From eqn (13) it can be seen that 0 is proportional to &,,/ln(l/r’). If we assume that I,,,, is of the same order as the real part of z, where z corresponds to w = - 1 - 7 (z is the point on the base-collector junction where I = I,,,) we find that 0 N ln(l/t’). The charge stored at the surface is therefore proportional to:
(15)
Note that the boundary conditions in the w plane remain the same as in Fig. 2(b). The hole concentration at the emitter-base junction is pj. Now the complex current can be calculated in the same way as in the base but now explicitly as a function of z: J=
iqDPj b
2 sinh(xz/b) ’ {2 cosh(xz/b) + cosh(aa/b) + 1}2 ’ - {cosh(sa/b)I}’
(16)
Now D is the diffusivity of holes in the emitter. The points z are merely the mirror images to the points z in Fig. 2(a). By taking the negative complex conjugates we can transform them into each other.
-----iY
n.
J
’ mln(l/r’)
dt’,
j
I-
x
0
which is a finite integral. Furthermore, recombination in the base can be neglected even for the longest transit times. As an example consider a structure with an emitterdepth of
Fig. 7. Emitter in the complex z plane. The emitter depth is b; the distance between metallisation and sidewall is a.
1720
H. F. F. Jos
lE2_
3. CURVED EMITTER
1-
3.1. The base
z54-
The case of a curved emitter is analogous to that of a rectangular emitter. We assume that the curvature of the emitter starts at the surface at point (b, b) and ends in the origin of the z plane (Fig. 9). The Schwarz-Christoffel transformation is given by[6]:
32-
lEl_ %3!544
dz C Clw-t dw=w+l+w+1 w-l’ J
3-
(18)
where 2
lE0 % 3 5 4
/
-d
-
cc---w(l + r)
a/b=025
-.-
tVb'O.7
-.-
a/b'l.O
Integration
and
r =
b
d(l/a -1)-b’
of eqn (18) yields:
:
3
+d - l-
2
rC1+r
: E-1
0
1
I 1
I 2
3
gain as function of distance along the for some values of the metallisation to base distance.
To calculate the current gain along the emitter-base junction we have to start in the w plane and calculate z by eqn (4) and the collector current density by eqn (7). Now the base current density is given by eqn (16) and hence Hfe is known as a function of parameter p. In the rest of this paper we shall use Hfe as the normalised current gain, i.e. the current gain calibrated in such a way that it equals one in the 1-D case under the emitter. It turns out that Hfe does not depend very much on the value of b/d but it does on the value of u/b. In Fig. 8 Hfe is plotted as a function of p/b for several values of a. The total current gain of the sidewall increases when a is larger because the sidewall base current decreases. For small emitters this effect can be noticeable. We shall come back to this in Section 3 in the curved emitter case. The total collector current Z, is related to the 1-D current density by: I,= J,W+2I,, where Z, is the total sidewall current that can be obtained by adding the currents in all segments or by integration of eqn (7). Note that here we also have to take into account the contribution of the region under the emitter where the current flow is not 1-D. The effect of the sidewall current can be described by a function g as: I, = J,( W + 2bg).
>
I
p/b
Fig. 8. Local current emitter-base junction
S-l In {( s+l
(17)
In Fig. 6 g has been plotted as a function of d/b.
us - 1 - l/a In 3 + irr(l/u - 1) , (19) ( > 1 where s and a are defined in eqn (5). Note that t has not yet been defined and is determined by the condition that z = b + ib for w = 1. Hence t is determined by two equations: tI
1+r=
=exp{-$1
+r)}, d
d-bcr/(l
(20)
-a)’
The current density is calculated in the same way as eqn (7):
We can proceed in the same way as in Section 2 and make a vector plot of the current density in the base (arbitrary units). In Fig. 10 the current density is plotted not only in the base but also in the emitter. We shall come back to this in Section 3.2. Note that
-J-y+ d 1"
2' iY
t
Fig. 9. Base in the complex
x
z plane in the curved case.
emitter
Bipolar transistor
1721
i
t
-0.1
-0.2
Fig. 10. Vector plot of minority carrier current densities inside emitter and base. The vectors are of arbitrary length. All segments carry the same total current in the base (not in the emitter). Clearly can be seen that the current gain in the sidewall is relatively large. Under the emitter the current flow rapidly
becomes 1-D. the curvature of the emitter is in general not circular and that the current flow becomes approximately I-D for small negative values of x. A small remainder of the current divergence in the curved emitter case can still be seen at the origin. The transit time has been calculated and 0 has been plotted in Fig. 11 as a function of p/b for the same values of b/d as in Fig. 5. Again 0 has been calibrated IO-
-
b/d*2
-.-
b/d.1
-m-
b/d’P/J
I
e .
to equal 1 under the emitter. Now the parameter p is
the distance along the curved emitter-base junction. Since the curvature is not quite circular in all cases the values for p at the origin are slightly different for the various values of b/d. However, p/b always equals about n/2 at the origin. Qualitatively Fig. 5 and Fig. 11 do not differ very much apart from the minimum in Fig. 5. The diffusion capacitance can be described by eqn (14). In Fig. 12 the functionfhas been plotted vs d/b. Note that f is larger than in the rectangular emitter case for d/b > 0.4. This is due to the fact that the sidewall now carries more current. The function f has a minimum value of 3.74 at d = 1Sb. In a curved emitter the sidewall accounts for an even larger part of the diffusion capacitance than in a rectangular emitter, although the transit times of the curved emitter are smaller. 3.2. The current gain
I 0
I 1
I 2
I 3
p/b
Fig. II. Transit time vs distance along the emitter-base junction for some values of b/d.
In the case of the rectangular emitter we found a conformal transformation of the emitter’s interior onto the w plane that can be inverted. This inversion is necessary for the calculation of the current gain because the points in the z plane where the hole current in the emitter has to be calculated are already determined by the electron current in the base. In the case of the curved emitter we also need an invertable transformation. Furthermore, the non-circular curvature of the base demands a transformation of the emitter of which the curvature can be adapted to fit the base curvature. We start the transformation of eqn (15). We rewrite eqn (15) as a transformation of the z plane to the u plane. Consider the equiconcentration line in the o plane that has an angle 4 with the horizontal axis
1722
H. F. F. Jos
Y-
the origin. In order to obtain an emitter of the right size and the current densities within it eqns (15) and (16) can be used by replacing 6 by 6’ and a by a’, where b’ and a’ are determined by: b’=ba/(n
-4)
6-
2 cosh{rc(a’ - a)/b’} = cosh(lta’/b’) + 1 - {cosh(na’/b’) - l}cos($). In Fig. 10 an example is shown of a current vector plot in the curved emitter case. The points at the junction in which the electron current density is calculated do not exactly coincide with the points in which the hole current density is calculated. The parameter Cphas been adapted to find the best overlap of the curvatures. Note that for negative x both junctions are exactly the same. In Fig. 14 the current gain is plotted as a function of the parameter p/b for several values of a/b. Again Hfe has been set to 1 under the emitter. As with the rectangular emitter the parameter a has a strong influence on Hfe. The total current gain is calculated with the help of eqn (17): d/b
Fig. 12. Functions f and g vs d/b (see text). (Fig. 13a). This equiconcentration line is mapped onto the .z plane as a curve in the emitter’s interior (Fig. 13b). In fact this curve can be used as a new emitter boundary. However, our new emitter is smaller than the old one and shifted with respect to
I, = J]( w + 2bg).
In Fig. 12 g has been plotted as a function of d/b. It is possible to define an analogue of eqn (17) for the base current: (22)
Ib = Jb,(W +2&J. 3.5
-
a/b*0.5
-.-
a/b’0.7
-.-
oIb’l.0
2.5
i
.
(b)
1.5
7 . . \ fi . \ .
--------
0.5
Fig. 13. (a) Complex v plane consisting of the upper half plane without the semicircle with radius 1. The equiconcentration line that makes an angle r$ with the horizontal axis is mapped to the z plane as the emitter-base junction, (see (b). (b) Emitter in z plane. The equiconcentration curve corresponding to 4 is shown.
I
I
1
I
2
3
p/b
Fig. 14. Local current gain as function of distance along the emitter-base junction for some values of the metallisation to base distance.
Bipolar transistor
1723
2.5r
0.2 ,um. As an example we take b = d = 0.15 pm and W = 1 pm. Then we find: e = 1.448,. This shows that the sidewalls already contribute a large amount to the effective base transit time. For completeness we also give an empirical formula for g,, by which eqn (23) can be evaluated. g,
-0.4 a0
I
I
0.5
1.0
I 1.5
a/b
2:
,/(0.26/a)
- 0.9a/b + 1.
The empirical formula given in this section are only valid for the ranges shown in the Figs 12 and 15. They are not meant to be used as design rules since they are based on a simplified model.
Fig. 15. Function g, vs a/b (see text). 5. RECOMBINATION
By adding the base currents in the emitter in all segments g,, can be calculated. In Fig. 15 g, is shown as a function of a/b. The total current gain is given by: (23) As mentioned before this is only valid for W > 0.20pm. Equation (23) only contains the modification of Hfe due to 2-D effects. It must be kept in mind that Hfe, is the 1-D current gain that still depends on b and d. The current gain can be increased independently from b and d by increasing a. However, in real devices recombination can be important for large values of a[4]: the increase in current gain might therefore be limited. 4. THE EFFECTIVE
BASE TRANSIT TIME
We have disregarded recombination of holes in the emitter. This is allowed for shallow emitters, For instance, if the emitter doping concentration is about 10Mcrn3 the diffusion length Lp is about 0.25 pm. If infinite recombination is assumed at the contact the current density in the I-D case is given by: (25) If b/L, < 0.6 this current density equals the density without recombination within 10%. Thus in the 1-D case recombination can be neglected for b Q 0.15 pm. In the 2-D case we estimate the influence of recombination in a first approximation by replacing b/L, by &%?@, where 0, is the recombination lifetime in the emitter and 0 is the transit time in a given segment. We write: 28 28 8, -=__=_ % e1 0,
In a charge model an effective base transit time is defined by:
where Q is the stored minority charge in the base. In Section 2.3 an expression has been given for Q. Using that expression together with eqn (17) yields: _
-
e
W + 2bf
‘VT?&+
b 2 _ . 8, 0 L,
Hence:
6 = Q/4
e
28
(24)
Ignoring parasitic and depletion capacitances the transition frequency can be calculated by using Ft = l/(2&) and e.g. Fig. 12. Some insight in the influence of the geometry on the transit time can be obtained by fitting the functions f and g to empirical formulae. It appears that f and g can be described with a reasonable accuracy by:
f z (b/d)0.*5+ 0.37d/b + 2.55, g z 1 + 0.6d/b. Substitution in eqn (24) yields the change in 8 due to the sidewalls. Note that W needs to be larger than
J
qPjD
1 ’
Lo rgh (b I&/@@% This function can be approximated totes: 1. Il.
b/&/‘@XGl b/LpJm
by two asymp-
J = qPjD/bJ&P& > 1 J = qpjD/Lp.
(26) (27)
In fact case I is assumed to be valid in our paper. The effects of recombination can be demonstrated in a case in which b = 0.12 pm and a varies from 0. I to 0.2 pm. For each value of a the current density can be integrated over the part of the sidewall for which case II should apply. This current can be compared to the current obtained by integrating eqn (27) over the same region, which is easy since the density of eqn (27) is a constant. Now an effective g, can be defined that describes the sidewall base current crudely corrected for recombination. It shows that for u < 0.12 pm recombination can be neglected, but
H. F. F. Jos
1724
for higher values of a the base current can hardly be decreased by increase of a. The effective g, reaches the value 0.37 at a = 0.2 pm. Since usually a x 0.76 it seems to be justified to neglect recombination if b ~0.15~~ and a -~O.lOpm. If recombination is important for less shallow emitters conformal mapping cannot be performed and the Helmholtz equation[4] has to be solved for the emitter. 6. CONCLUSIONS
We presented a model by which the current flow in of bipolar devices can be calculated. This proved to be possible for rectangular as well as curved emitters. The device has been divided in segments by the current flow lines and each segment can be treated as a transistor with its own current gain and transit time. We showed that recombination in the base can be ignored even for the large transit times of the current leaving the top of the emitter. Furthermore, the current density drops sharply to zero in the lateral direction, hence the current flow will not be affected by the base contacts. The current gain of the device as a whole depends on the position of the emitter metallisation. Increasing the distance between the emitter metallisation and the base increases the current gain. However, at large distances recombination is expected to become the base and emitter
important and the increase in current gain will be limited. The diffusion capacitance is strongly modified by the presence of the sidewalls. In the case of a curved emitter the relative contribution of the sidewalls to the capacitance is at least a factor 7.5 times larger than that of the 1-D emitter. The influence of the sidewalls on the effective base transit time is smaller because they carry a relatively large part of the current. However, in the case of an emitter with a width of 1 pm (emitterdepth 0.15 pm and basedepth 0.15 pm) the sidewalls increase the transit time by a factor 1.44. Therefore it seems likely that in real transistors with submicron lateral dimensions the influence of the sidewalls on the transition frequency cannot be ignored. REFERENCES
1. S. M. Sze, Physics of Semiconductor p. 157. Wiley, New York (1981).
Devices, 2nd edn,
2. S. Aronowitz, J. appl. Phys. 61, 2495 (1987). 3. S. D. Brotherton, J. P. Gowers, N. D. Young, J. B. Clegg and J. R. Ayres, J. appl. Phys. 60, 3567 (1986).
4. G. A. M. Hurkx, IEEE Trans. Electron Dev. ED-34, 1939 (1987).
5. M. R. Spiegel, Complex Variables, Schaum’s Outline Series, p. 204. McGraw-Hill, New York (1972).
6. D. C. D’Avanzo, M. Vanzi and R. W. Dutton, Report G-201-5, Stanford University (1979). Fielak, p. 301. Wiley, New York (1950).
7. E. Weber, Electromagnetic
l-1038-1 101/88 S3.00+ 0.00 Copyright 0 1988Pergamon Press pk
BIPOLAR TRANSISTOR: TWO-DIMENSIONAL EFFECTS ON CURRENT GAIN AND BASE TRANSIT TIME H. F. F. Jos Development Department of Discrete Semiconductors, Philips, 6534 AE Nijmegen, The Netherlands (Receiued 29 March 1988; in revised form 18 July 1988)
Abstract-Two-dimensional &e&s in bipolar devices with small dimensions are calculated analytically if simplifying assumptions are made e.g. uniformly doped regions and zero recombination. In this paper we treat effects of the emitter sidewalls on the current gain and the base transit time. The current flows through the base and emitter are calculated for rectangular as well as curved emitter-base junctions. Current gain and base transit time are analysed as local variables, i.e. functions that vary along the 2-D emitter-base junction. In this way it can be seen where and how several parameters influence the overall performance of the device. The total current gain is modified by the sidewalls; it increases if the distance between the emitter-base junction and the emitter metallisation is increased. However, this distance cannot be increased indefinitely because of finite recombination lifetimes. The diffusion capacitance is strongly affected by the sidewalls. Because the sidewalls carry a considerable part of the current the transit time is less affected. However, the influence of the sidewalls is still noticeable. It is expected that in a transistor with a 1 pm wide and 0.15 pm deep emitter the transition frequency is already negatively influenced by the emitter sidewalls.
1. INTRODUCTlON
2. RECI’ANGULAREMITTER
The dimensions of bipolar transistors have been reduced since their invention in an ongoing effort to improve the high frequency performance of discrete as well as integrated devices[ 11.A reduction of lateral dimensions of, for instance, discrete bipolar transistors results in a lowering of the base resistance and smaller parasitic capacitances. This leads to a device with a higher cut-off frequency and lower noise. The reduction of vertical dimensions can be achieved by techniques as low energy implantation, rapid thermal annealing, co-implantation[2] and pre-amorphisation[3]. The lateral dimensions can be diminished by technologies as direct slice writing with an electron beam pattern generator (EBPG), X-ray lithography, plasma etching, etc. Submicron lateral dimensions can be realised. However, as the emitterwidth is reduced to the same order of magnitude as the emitterdepth 2-D effects are likely to occur. Up till now most analytical descriptions of bipolar devices have been 1-D. Recently Hurkx published a 2-D analytical model of a simplified bipolar transistor with a rectangular shaped emitter[rl]. In that model the consequences of 2-D effects for the current gain have been treated. In high frequency devices the current paths of the minority charge carriers in the base and emitter and the transit times belonging to these paths are of interest. In this paper we present an extension of the model of Hurkx by which the current paths and transit times can be calculated analytically for a simplified bipolar device. Our model not only treats rectangular shaped emitters but also curved emitters. We shall treat them separately.
In Fig. 1 a 2-D cross section of a simplified bipolar transistor with rectangular emitter is shown. We shall treat two important current densities: J&x, y) of minority carriers in the emitter and J,,(x, y) of minority carriers in the base. Analytical expression for J&-C,y) and J&y) can be obtained by means of conformal mapping. To do this we make some simplifying assumptions: -The state is stationary: all derivatives with respect to time are zero. -There is no generation or recombination. Especially for the emitter this is a difficult condition that only holds for very shallow emitters (b Q 0.25 pm for an npn device). -The bandgap is constant in the emitter as well as in the base. -There is no electric field in the emitter and base, hence the emitter and base are uniformly doped. -The surface recombination velocity at the emitter contact is infinite. It is possible to give an analytical expression for J,,(x, y) if there is finite recombination in the emitter and at the emitter contact. Hurkx[4] has treated this case. However, the solution is rather complex and consists of infinite series. At least a finite number of them must be numerically processed to gain insight. In the case of a curved emitter the solution will be even more complex. Since our model is developed for devices with very small dimensions it is justified to neglect recombination. We shall come back to this at the end of this paper. Using conformal mapping we
1715
H. F. F. Jos
1716
Integration onto z:
Emitter
of eqn
(2) yields
the
mapping
1 _ u>’
Collector
(4)
section of simplified emitter. The collector
bipolar
Fig. I. Cross rectangular
of w
transistor
with
is contacted at the back-
side.
obtain a solution with a small number of terms and extension to the curved emitter case is straightforward. Several effects, however, are not taken into account by this model, e.g. high injection, ohmic losses and the existence of space charge regions. 2.1. The base For an npn transistor the minority base are electrons. The concentration by the Laplace equation:
(1)
We model the base region as a region in the complex plane z (Fig. 2a). The boundary conditions and meaning of symbols are defined in Fig. 2(a). The electron concentration at the emitter-base junction is given by n,. In this model the base region extends to x = - co. Now the effect of one sidewall can be examined. Of course it should be checked afterwards whether this approach is justified in the case of two sidewalls. We shall show that it is because under the emitter the 1-D solution is rapidly reached for small negative values of x. We map the complex z plane onto the upper half plane w (Fig. 2b) by the Schwarz-Christoffel transformation[5]: dz
dl nw+l
s=F
i=J-l,
a=E.
(5)
N = in, In(u) + nj. The current curves are given by 101= constant and the equiconcentration curves by arg(v) = constant. The current curves are concentric circles centred at the origin and the equiconcentration curves are straight lines through the origin. Now v=w+Jm and hence plane is:
the
complex
concentration
in the
Differentiation
of eqn (6) with respect dN
i
dw
n
-c-n
current
to w yields:
1 ‘,,/m’
in the z plane can be found by:
(2)
J- w - 1’
(7)
where r =2{(d+b)/d}2-
,
(O,b)
(0) b
1’
n = nj
,f (40) /I
1.
3
(3)
an JY
n=n,
;O
Here D is the diffusivity
of electrons
in the base.
2”
(b)
4
d 1”
an
-0
1" n=O
v
lY
I-
2’
n=O
2 I
3
v-
-1
1
n=n,
4 I
I
4 IY’ x
w
(6)
The complex
w--t
and
Note that all functions in eqns (4) and (5) are complex and that their arguments can be negative. The point (t, 0) in the w plane is mapped onto the origin in the z plane. The boundary conditions in the w plane are denoted in Fig. 2(b). To find the complex concentration in the w plane we map the w plane onto the upper half plane u with the semicircle with radius 1 removed. For reasons of symmetry the complex concentration in the v plane is:
carriers in the is determined
V&(x, y) = 0.
dw=---
where
L- x,
Fig. 2. (a) Complex plane z = x + iy. Emitter depth is b, base thickness under emitter is d. (b) Upper half plane w = x’ + iy’. The points 14 correspond with those in Fig. 2(a).
1’
Bipolar transistor
Unfortunately we cannot invert eqn (4) and substitute w(z) in eqn (7). We must start at a point in the w plane and find the corresponding point z and the current at z. A simple computer programme has been written which performs this task. The points taken in the w plane are the nodes between the equiconcentration and current curves. The current curves were chosen in such a way that they divide the base in segments through which equal currents flow. The equiconcentration curves correspond to lines through the origin in the v plane with angles ilr/n with the horizontal axis, where i = 0, 1,. . . , n and n is the total number of curves. A vectorplot of the current densities in the base is shown in Fig. 3 (arbitrary units). The values for b and d are: 6 = 0.12 pm and d = 0.30 pm. The number of points taken on each current curve is 15. Note that the current density diverges near the sharp corner of the emitter. Furthermore it shows that the current density is already almost 1-D for x = -O.lOpm, which means that if the emitter is wider than 0.20 pm both sidewalls will not influence each other. At distances x > 0.5 pm the current densities are almost negligible. Since the basecontact is far away (at about x = 1.5 pm in the finest structured devices) its influence on the current flow can be ignored. 2.2. The transit time Now that we know the current flow we can calculate the transit time for each current curve. Look at a infinitesimal narrow current segment in the base. In Fig. 4 the concentration of electrons in the segment is plotted. We take the equiconcentration and current curves as orthogonal coordinate system. The coordinate along the current curve is 1, where 0 < I< I,,; the width of the curve is r(l). The current density in the segment is assumed to be dependent only on I:
0.12
---_ --_
1
-0.3
n
Fig. 4. Concentration of electrons in a current segment with length I,, and width r(l).
Furthermore: Jr(l) = constant = Jg(O),
(9)
where Jo is the current density at the emitter edge. From eqns (8) and (9) it follows: dn - JoW dl=qDro and hence: -J,,r(O) ’ dl’ n(l) = ~ + nj . qD s oW The stored charge in the segment is:
dQ=q
L.X
n(040 dl,
(10)
(11)
s0 Substitution of eqns (10) in (11) yields the transit time ~9in the segment:
e=+j 0
=-
1
D
ss c..
o
k..
r(l)
1
dl’
qqdl.
(12)
Note that if r(l) is constant 0 = 1:,,/2D as is expected. Since we know the shape of the current segments we can calculate the nested integral of
dn J = -qDdr.
-0.10
1717
_
- - _ - _ - _ -:.
-
,,,,1,,,,,,.,,,.... -0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
Fig. 3. Vector plot of minority carrier current density inside base. The vectors are of arbitrary length. Near the emitter comer the density diverges. All segments carry the same total current. Under the emitter the current flow rapidly becomes 1-D.
0.6
H. F. F. Jos
1718
eqn (12) numerically. However, eqn (12) consists of a nested integral and therefore it is not very sensitive to the shape of the current segments provided r(l) is a smooth function. For instance, if r(l,,,,,)/r(O) < 10, which is true almost everywhere except near points 3 and 4 in Fig. 2(a), the nested integrals of a linearly shaped and an exponentially shaped segment differ only by less than 10%. It is possible to give an analytical expression for eqn (12) if an exponentially shaped current segment is assumed, i.e. if:
-
b/d ‘2
-.-
b/d’1
-m-
b/d ‘2/3
6-
r(l) = r(O)exp(Z/a)
where 0 = I,,,/ln(r(O)/r(l,,)). Hence: 2
B=%{f-;(1--$.$)},
(13)
where I
c = ln{r(L,,)/r(O)). With the help of eqn (13) 0 can be calculated for a current segment if only ZmBX and r(l,,,)/r(O) are known. The values for lm, are given by the integral of Idz 1along the current flow line:
OL
1.0 p/b
Fig. 5. Transit time vs distance along the emitter-base junction for some values of b/d.
depends on b/d. Note that 8 has a sharp minimum at p = b where 0 is zero. This is the point where the current density diverges. (u2 exp(2ei$) - 2tu exp(id) + 1) do,
v exp(id,) + 1 where v is the point on the real axis in the v plane corresponding to the point on the base-emitter junction for which l,,, is calculated. However, it is much simpler to determine l,,,,, numerically from the shapes of the current curve. From eqn (9) it follows that r(I,,,,,)/r(O) = J,/J(I = I,,,). Since the point I = 0 corresponds to a point w = w, in the w plane, where w. is real, and the point I = l,,,,, to the point w = -w. eqn (7) can be used to calculate rU,,,)lr(O): r(L) -= r(0)
(w. + l)(wo + t) (Wo-t)(Wo1) .
We introduce p as the parametrisation of the emitter-base junction: p = 0 at the point (0, b), p = b at the point (0,O) and p = b -x at all points (x, 0) with negative x. In Fig. 5 0 is plotted as a function of p/b for several values of b/d. Note that 0 has been calibrated in such a way that 0 = 1 in the 1-D case under the emitter. There is an asymptote for p = 0 where 6 goes to infinity. In practice this will not happen because of finite recombination lifetimes. However, as we shall demonstrate in Section 2.3 there is no contradiction between our model and practice in this point. The behaviour of 0 for p < 1 strongly
2.3. The dajiiion capacitance Multiplying 0 by J yields the charge stored in a current segment. The total charge stored in the base is obtained by adding the charges stored in all segments. The total charge of a base with two sidewalls can be described by: Q = W,( bf’ + W), where 0, and J, are the 1-D base transit time and current density respectively and W is the width of the emitter. The contribution of one sidewall, which is not only determined by the geometrical sidewall but also by the region under the emitter where the current density cannot be described by the 1-D solution, is given by the function f: It is obtained by considering a region that extends so far under the emitter that at its left boundary x = -x,, the current flow can be accurately approximated by 1-D flow. From the total charge stored in this region a virtual charge Q = t9,J,xo is subtracted to calculate $ In Fig. 6 the function f is plotted vs d/b. The function f reaches a minimum of about 3.4 for d = 1.7b. The equation above yields the diffusion capacitance: C = C,(l + 2bf/ W).
(14)
Bipolar transistor
-f -.-
1719
b = 0.20pm and a basedepth of d = O.lOpm. The transit time belonging to the point at about an atomic radius (4A) below the surface at the base-emitter junction is still less than 50 times the 1-D transit time under the emitter. Since the 1-D transit time (0 = d*/2D) is typically 5 to 10 ps this transit time is less than 500~s. This is still much smaller than Shockley-Read-Hall recombination times of about lo-’ s[6].
g
2.4. The current gain
./’ ./ ./ / I
I
I
1
2
3
I 4
d/b
Fig. 6. Functionsfand
The hole current in the emitter is the most important contribution to the base current especially when the emitter is shallow. In our model we assume that it is the only contribution. Again we disregard recombination (see Section 5) and the hole concentration is governed by the Laplace equation. We solve the problem by conformal mapping. In Fig. 7 the emitter is represented in the complex z plane together with the boundary conditions for the hole current. The distance between the emitter contact and the emitter sidewall is a. Note that Fig. 7 is a mirror image to the structure in Fig. 1. We map the z plane onto the w plane of Fig. 2(b) and in this case we can invert the transformation: w _ 2 cosh(xz/b) + cosh(na/b) + 1 cosh(na/b) - 1 *
g vs d/b (see text).
The 1-D diffusion capacitance C, N e,J, IV. Since the minimum value of f is 3.4 the sidewalls contribute considerably to the total diffusion capacitance. For instance, if b = 0.15 pm and W = 1.0 pm the sidewalls are responsible for at least half of the diffusion capacitance. The asymptote for 8 at p = 0 suggests that an infinite charge might be stored at the surface of the structure. This is, however, not true. To find the total charge stored due to sidewall effects J@)B(p)dp has to be integrated along the base-emitter junction. We can prove that the part of this integral at the surface is finite, by integrating from p = 0 to p = 6, where 6 is arbitrarily small but finite. The point p = 0 corresponds to w = 1, while p = 6 corresponds to w = 1 + 7. From eqn (7) it follows that J(p) is independent of p if p goes to zero; from eqn (4) that p is proportional to fi, where 0 Q z’ d 7. From eqn (13) it can be seen that 0 is proportional to &,,/ln(l/r’). If we assume that I,,,, is of the same order as the real part of z, where z corresponds to w = - 1 - 7 (z is the point on the base-collector junction where I = I,,,) we find that 0 N ln(l/t’). The charge stored at the surface is therefore proportional to:
(15)
Note that the boundary conditions in the w plane remain the same as in Fig. 2(b). The hole concentration at the emitter-base junction is pj. Now the complex current can be calculated in the same way as in the base but now explicitly as a function of z: J=
iqDPj b
2 sinh(xz/b) ’ {2 cosh(xz/b) + cosh(aa/b) + 1}2 ’ - {cosh(sa/b)I}’
(16)
Now D is the diffusivity of holes in the emitter. The points z are merely the mirror images to the points z in Fig. 2(a). By taking the negative complex conjugates we can transform them into each other.
-----iY
n.
J
’ mln(l/r’)
dt’,
j
I-
x
0
which is a finite integral. Furthermore, recombination in the base can be neglected even for the longest transit times. As an example consider a structure with an emitterdepth of
Fig. 7. Emitter in the complex z plane. The emitter depth is b; the distance between metallisation and sidewall is a.
1720
H. F. F. Jos
lE2_
3. CURVED EMITTER
1-
3.1. The base
z54-
The case of a curved emitter is analogous to that of a rectangular emitter. We assume that the curvature of the emitter starts at the surface at point (b, b) and ends in the origin of the z plane (Fig. 9). The Schwarz-Christoffel transformation is given by[6]:
32-
lEl_ %3!544
dz C Clw-t dw=w+l+w+1 w-l’ J
3-
(18)
where 2
lE0 % 3 5 4
/
-d
-
cc---w(l + r)
a/b=025
-.-
tVb'O.7
-.-
a/b'l.O
Integration
and
r =
b
d(l/a -1)-b’
of eqn (18) yields:
:
3
+d - l-
2
rC1+r
: E-1
0
1
I 1
I 2
3
gain as function of distance along the for some values of the metallisation to base distance.
To calculate the current gain along the emitter-base junction we have to start in the w plane and calculate z by eqn (4) and the collector current density by eqn (7). Now the base current density is given by eqn (16) and hence Hfe is known as a function of parameter p. In the rest of this paper we shall use Hfe as the normalised current gain, i.e. the current gain calibrated in such a way that it equals one in the 1-D case under the emitter. It turns out that Hfe does not depend very much on the value of b/d but it does on the value of u/b. In Fig. 8 Hfe is plotted as a function of p/b for several values of a. The total current gain of the sidewall increases when a is larger because the sidewall base current decreases. For small emitters this effect can be noticeable. We shall come back to this in Section 3 in the curved emitter case. The total collector current Z, is related to the 1-D current density by: I,= J,W+2I,, where Z, is the total sidewall current that can be obtained by adding the currents in all segments or by integration of eqn (7). Note that here we also have to take into account the contribution of the region under the emitter where the current flow is not 1-D. The effect of the sidewall current can be described by a function g as: I, = J,( W + 2bg).
>
I
p/b
Fig. 8. Local current emitter-base junction
S-l In {( s+l
(17)
In Fig. 6 g has been plotted as a function of d/b.
us - 1 - l/a In 3 + irr(l/u - 1) , (19) ( > 1 where s and a are defined in eqn (5). Note that t has not yet been defined and is determined by the condition that z = b + ib for w = 1. Hence t is determined by two equations: tI
1+r=
=exp{-$1
+r)}, d
d-bcr/(l
(20)
-a)’
The current density is calculated in the same way as eqn (7):
We can proceed in the same way as in Section 2 and make a vector plot of the current density in the base (arbitrary units). In Fig. 10 the current density is plotted not only in the base but also in the emitter. We shall come back to this in Section 3.2. Note that
-J-y+ d 1"
2' iY
t
Fig. 9. Base in the complex
x
z plane in the curved case.
emitter
Bipolar transistor
1721
i
t
-0.1
-0.2
Fig. 10. Vector plot of minority carrier current densities inside emitter and base. The vectors are of arbitrary length. All segments carry the same total current in the base (not in the emitter). Clearly can be seen that the current gain in the sidewall is relatively large. Under the emitter the current flow rapidly
becomes 1-D. the curvature of the emitter is in general not circular and that the current flow becomes approximately I-D for small negative values of x. A small remainder of the current divergence in the curved emitter case can still be seen at the origin. The transit time has been calculated and 0 has been plotted in Fig. 11 as a function of p/b for the same values of b/d as in Fig. 5. Again 0 has been calibrated IO-
-
b/d*2
-.-
b/d.1
-m-
b/d’P/J
I
e .
to equal 1 under the emitter. Now the parameter p is
the distance along the curved emitter-base junction. Since the curvature is not quite circular in all cases the values for p at the origin are slightly different for the various values of b/d. However, p/b always equals about n/2 at the origin. Qualitatively Fig. 5 and Fig. 11 do not differ very much apart from the minimum in Fig. 5. The diffusion capacitance can be described by eqn (14). In Fig. 12 the functionfhas been plotted vs d/b. Note that f is larger than in the rectangular emitter case for d/b > 0.4. This is due to the fact that the sidewall now carries more current. The function f has a minimum value of 3.74 at d = 1Sb. In a curved emitter the sidewall accounts for an even larger part of the diffusion capacitance than in a rectangular emitter, although the transit times of the curved emitter are smaller. 3.2. The current gain
I 0
I 1
I 2
I 3
p/b
Fig. II. Transit time vs distance along the emitter-base junction for some values of b/d.
In the case of the rectangular emitter we found a conformal transformation of the emitter’s interior onto the w plane that can be inverted. This inversion is necessary for the calculation of the current gain because the points in the z plane where the hole current in the emitter has to be calculated are already determined by the electron current in the base. In the case of the curved emitter we also need an invertable transformation. Furthermore, the non-circular curvature of the base demands a transformation of the emitter of which the curvature can be adapted to fit the base curvature. We start the transformation of eqn (15). We rewrite eqn (15) as a transformation of the z plane to the u plane. Consider the equiconcentration line in the o plane that has an angle 4 with the horizontal axis
1722
H. F. F. Jos
Y-
the origin. In order to obtain an emitter of the right size and the current densities within it eqns (15) and (16) can be used by replacing 6 by 6’ and a by a’, where b’ and a’ are determined by: b’=ba/(n
-4)
6-
2 cosh{rc(a’ - a)/b’} = cosh(lta’/b’) + 1 - {cosh(na’/b’) - l}cos($). In Fig. 10 an example is shown of a current vector plot in the curved emitter case. The points at the junction in which the electron current density is calculated do not exactly coincide with the points in which the hole current density is calculated. The parameter Cphas been adapted to find the best overlap of the curvatures. Note that for negative x both junctions are exactly the same. In Fig. 14 the current gain is plotted as a function of the parameter p/b for several values of a/b. Again Hfe has been set to 1 under the emitter. As with the rectangular emitter the parameter a has a strong influence on Hfe. The total current gain is calculated with the help of eqn (17): d/b
Fig. 12. Functions f and g vs d/b (see text). (Fig. 13a). This equiconcentration line is mapped onto the .z plane as a curve in the emitter’s interior (Fig. 13b). In fact this curve can be used as a new emitter boundary. However, our new emitter is smaller than the old one and shifted with respect to
I, = J]( w + 2bg).
In Fig. 12 g has been plotted as a function of d/b. It is possible to define an analogue of eqn (17) for the base current: (22)
Ib = Jb,(W +2&J. 3.5
-
a/b*0.5
-.-
a/b’0.7
-.-
oIb’l.0
2.5
i
.
(b)
1.5
7 . . \ fi . \ .
--------
0.5
Fig. 13. (a) Complex v plane consisting of the upper half plane without the semicircle with radius 1. The equiconcentration line that makes an angle r$ with the horizontal axis is mapped to the z plane as the emitter-base junction, (see (b). (b) Emitter in z plane. The equiconcentration curve corresponding to 4 is shown.
I
I
1
I
2
3
p/b
Fig. 14. Local current gain as function of distance along the emitter-base junction for some values of the metallisation to base distance.
Bipolar transistor
1723
2.5r
0.2 ,um. As an example we take b = d = 0.15 pm and W = 1 pm. Then we find: e = 1.448,. This shows that the sidewalls already contribute a large amount to the effective base transit time. For completeness we also give an empirical formula for g,, by which eqn (23) can be evaluated. g,
-0.4 a0
I
I
0.5
1.0
I 1.5
a/b
2:
,/(0.26/a)
- 0.9a/b + 1.
The empirical formula given in this section are only valid for the ranges shown in the Figs 12 and 15. They are not meant to be used as design rules since they are based on a simplified model.
Fig. 15. Function g, vs a/b (see text). 5. RECOMBINATION
By adding the base currents in the emitter in all segments g,, can be calculated. In Fig. 15 g, is shown as a function of a/b. The total current gain is given by: (23) As mentioned before this is only valid for W > 0.20pm. Equation (23) only contains the modification of Hfe due to 2-D effects. It must be kept in mind that Hfe, is the 1-D current gain that still depends on b and d. The current gain can be increased independently from b and d by increasing a. However, in real devices recombination can be important for large values of a[4]: the increase in current gain might therefore be limited. 4. THE EFFECTIVE
BASE TRANSIT TIME
We have disregarded recombination of holes in the emitter. This is allowed for shallow emitters, For instance, if the emitter doping concentration is about 10Mcrn3 the diffusion length Lp is about 0.25 pm. If infinite recombination is assumed at the contact the current density in the I-D case is given by: (25) If b/L, < 0.6 this current density equals the density without recombination within 10%. Thus in the 1-D case recombination can be neglected for b Q 0.15 pm. In the 2-D case we estimate the influence of recombination in a first approximation by replacing b/L, by &%?@, where 0, is the recombination lifetime in the emitter and 0 is the transit time in a given segment. We write: 28 28 8, -=__=_ % e1 0,
In a charge model an effective base transit time is defined by:
where Q is the stored minority charge in the base. In Section 2.3 an expression has been given for Q. Using that expression together with eqn (17) yields: _
-
e
W + 2bf
‘VT?&+
b 2 _ . 8, 0 L,
Hence:
6 = Q/4
e
28
(24)
Ignoring parasitic and depletion capacitances the transition frequency can be calculated by using Ft = l/(2&) and e.g. Fig. 12. Some insight in the influence of the geometry on the transit time can be obtained by fitting the functions f and g to empirical formulae. It appears that f and g can be described with a reasonable accuracy by:
f z (b/d)0.*5+ 0.37d/b + 2.55, g z 1 + 0.6d/b. Substitution in eqn (24) yields the change in 8 due to the sidewalls. Note that W needs to be larger than
J
qPjD
1 ’
Lo rgh (b I&/@@% This function can be approximated totes: 1. Il.
b/&/‘@XGl b/LpJm
by two asymp-
J = qPjD/bJ&P& > 1 J = qpjD/Lp.
(26) (27)
In fact case I is assumed to be valid in our paper. The effects of recombination can be demonstrated in a case in which b = 0.12 pm and a varies from 0. I to 0.2 pm. For each value of a the current density can be integrated over the part of the sidewall for which case II should apply. This current can be compared to the current obtained by integrating eqn (27) over the same region, which is easy since the density of eqn (27) is a constant. Now an effective g, can be defined that describes the sidewall base current crudely corrected for recombination. It shows that for u < 0.12 pm recombination can be neglected, but
H. F. F. Jos
1724
for higher values of a the base current can hardly be decreased by increase of a. The effective g, reaches the value 0.37 at a = 0.2 pm. Since usually a x 0.76 it seems to be justified to neglect recombination if b ~0.15~~ and a -~O.lOpm. If recombination is important for less shallow emitters conformal mapping cannot be performed and the Helmholtz equation[4] has to be solved for the emitter. 6. CONCLUSIONS
We presented a model by which the current flow in of bipolar devices can be calculated. This proved to be possible for rectangular as well as curved emitters. The device has been divided in segments by the current flow lines and each segment can be treated as a transistor with its own current gain and transit time. We showed that recombination in the base can be ignored even for the large transit times of the current leaving the top of the emitter. Furthermore, the current density drops sharply to zero in the lateral direction, hence the current flow will not be affected by the base contacts. The current gain of the device as a whole depends on the position of the emitter metallisation. Increasing the distance between the emitter metallisation and the base increases the current gain. However, at large distances recombination is expected to become the base and emitter
important and the increase in current gain will be limited. The diffusion capacitance is strongly modified by the presence of the sidewalls. In the case of a curved emitter the relative contribution of the sidewalls to the capacitance is at least a factor 7.5 times larger than that of the 1-D emitter. The influence of the sidewalls on the effective base transit time is smaller because they carry a relatively large part of the current. However, in the case of an emitter with a width of 1 pm (emitterdepth 0.15 pm and basedepth 0.15 pm) the sidewalls increase the transit time by a factor 1.44. Therefore it seems likely that in real transistors with submicron lateral dimensions the influence of the sidewalls on the transition frequency cannot be ignored. REFERENCES
1. S. M. Sze, Physics of Semiconductor p. 157. Wiley, New York (1981).
Devices, 2nd edn,
2. S. Aronowitz, J. appl. Phys. 61, 2495 (1987). 3. S. D. Brotherton, J. P. Gowers, N. D. Young, J. B. Clegg and J. R. Ayres, J. appl. Phys. 60, 3567 (1986).
4. G. A. M. Hurkx, IEEE Trans. Electron Dev. ED-34, 1939 (1987).
5. M. R. Spiegel, Complex Variables, Schaum’s Outline Series, p. 204. McGraw-Hill, New York (1972).
6. D. C. D’Avanzo, M. Vanzi and R. W. Dutton, Report G-201-5, Stanford University (1979). Fielak, p. 301. Wiley, New York (1950).
7. E. Weber, Electromagnetic