Bipolar semiconductor devices
Bipolar semiconductor devices Erik V Thomsen March 6, 2005
1
Introduction
This short lecture note describes some important aspects of bipolar devices. It is a supplement to the textbook ”Semiconductor Physics & Devices” by Neamen used in the Solid State Electronics and Microtechnology course (33253) at the Technical University of Denmark. Carrier transport in semiconductors is described for the special case of no recombination (τ → ∞). This leads to mathematically simple solutions of the transport equations and allows for a clear and simple, yet accurate, description of both the diode and the bipolar transistor in terms of • bias • doping levels • thickness of the different device regions • other material parameters such as intrinsic concentration and diffusivity
2
Carrier transport in semiconductors
Carrier transport in semiconductors is described mathematically by the continuity and current density equations. The continuity equations for electron and holes are given by (Neamen p. 6.18 and 6.19) ∂n 1 ∂Jn = + (Gn − Rn ) ∂t e ∂x 1 ∂Jp ∂p =− + (Gp − Rp ) ∂t e ∂x
(1) (2)
and they state, that matter does not disappear. These equations are easily derived by a simple book keeping of the particles that enter and leave a small volume (see Neamen p. 195). The current density equations (Neamen, p. 173, Eqn. (5.33)) describe the flux 1
of carriers due to drift (in an electric field E) and diffusion (with diffusion coefficients Dn and Dp for electrons and holes respectivily) dn Jn = eµn nE + eDn | {z } | {zdx} drift
(3)
diffusion
dp Jp = eµp pE − eDp | {z } | {zdx} drift
(4)
diffusion
These equations are valid in general, and can be applied to many different transport problems. To describe the carrier transport, for example in the diode or transistor, these differential equations must be solved with appropriate boundary conditions. We will look for a steady state solution and neglect recombination and generation. In steady state, the carrier concentrations do not depend on time, ∂p ∂n =0 =0 ∂t ∂t
(5)
and if there is no net recombination or generation G n − Rn = 0 G p − Rp = 0
(6)
In this case, the continuity equations become very simple dJn =0 dx dJp =0 dx
(7) (8)
This simply means, that the current density of both electrons and holes is constant. If we assume, that there is no electric field (or only a very small field), E = 0, transport is only due to diffusion. This is the case in the neutral regions outside the depletion layers. In these field free areas, the current density equations reduce to dn dx dp Jp = −eDp dx
Jn = eDn
(9) (10)
Thus, the current density is simply proportional to the gradient of the carrier concentration. Combining the continuity equations, Eqns. (7) and (8), and the current density equations, Eqns. (9) and (10), yields d2 n =0 dx2 d2 p =0 dx2 2
(11) (12)
Figure 1: In the abscence of recombination, the carrier concentration in a neutral region is simply linear with position, and the current density is proportional to the slope. The solutions to these equations are very simple Jn x eDn Jp x p(x) = ap + bp x = ap − eDp
n(x) = an + bn x = an +
(13) (14)
where we have used Eqns. (9) and (10). The coefficients an , bn , ap and bp are constants given by the boundary conditions. This tell us, that the electron and hole distributions are straight lines, and that the current density is proportional to the slope. Fig. 1 illustrates this concept.In the following, these simple results will be used to describe first the diode and then the bipolar transistor. These results can also be derived from the ambipolar transport equation (Neamen p. 5.18 and 5.19) 2 ∂(δn) ′ ∂ (δn) ′ ∂(δn) =D + µ E +g−R ∂t ∂x2 ∂x Dn Dp [(n0 + δn) + (p0 + δp)] D′ = Dn (n0 + δn) + Dp (p0 + δp)
using the simplifications (in this case for p-type) p-type p0 ≫ n0 Low injection p0 ≫ δn No excess generation/recombination g − R = 0 Field free region E=0 Same order of magnitude Dn ∼ Dp ∂(δn) =0 Steady state ∂t 3
(15) (16)
we arrive to the result that
∂ 2n =0 ∂x2
(17)
Likewise (for n-type) ∂ 2p =0 (18) ∂x2 This means that the n and p are straight lines - the same result as obtained above.
3
The diode
In this section, we will treat the transport of carriers through a pn junction. The starting point is the equations derived above. Consider the diode shown on Fig. 2. The thickness of the depletion layer is W = xn + xp . At each end of the diode there is an ohmic contact, located at −xcp and xcn for the p-type and n-type regions, respectively. The regions outside the depletion layers are neutral. Clearly, the thickness of the neutral region in the n-type material is Wn = xcn − xn and in the p-type material we have Wp = xcp − xp . In the analysis, we will neglect recombination, and then the electron and hole profiles are straight lines as shown above.
3.1
Boundary conditions
In order to proceed, we must find the proper boundary conditions. This means, that we have to specify the minority carrier concentrations at the contacts and at the edge of the depletion layer. 3.1.1
At the contacts
We assume, that the contacts are perfect ohmic contacts. This means, that there is thermal equilibrium and the carrier densities are pn (xcn ) = pno np (xcp ) = npo 3.1.2
(19) (20)
Law of mass action
The minority carrier density in thermal equilibrium is related to the majority carrier density by the law of mass action: np = n2i . This means, nno pno = n2i ppo npo = n2i
4
(21) (22)
Wp
Wn
W
p
n
x ¡ xcp
¡ xp
pn (x) np (x)
0
xn qV kT
¶
qV npo exp kT
¶
pno exp
µ µ
xcn
pno
npo
x ¡ xcp
¡ xp
0
xn
xcn
Figure 2: The carrier profiles in the diode are straight lines when recombination is ignored. Boundary conditions at the depletion layer edges and at the contacts are shown.
5
Assuming complete ionization of the impurity atoms, nno ≈ ND and ppo ≈ NA , we can express the minority carrier density in thermal equilibrium as function of the doping level n2i ND n2 ≈ i NA
pno ≈
(23)
npo
(24)
This result is very useful, and will be used many times in the following sections. 3.1.3
Law of the junction
The minority carrier density at the junction can be found with help of the current density equations, Eqns. (3) and (4). Let us first consider the current density equation for holes dp (25) Jp = eµp pE − eDp dx We know, that in thermal equilibrium Jp = 0. Therefore, we might expect, that even with a small current through the device the current density is so small, that Jp ≈ 0. With these assumptions the current density equation reduces to µp E − Dp
1 dp =0 p dx
(26)
µ , and solving for the electric Using the Einstein relation (Neamen p. 154), Dp = kT e p field, we have kT 1 dp (27) E= e p dx This equation is easily integrated over the depletion layer, and we know, that the integral is the difference between the built-in voltage and the applied voltage, Vbi − V . Therefore, Z xn Z Z kT xn 1 dp kT pn (xn ) 1 Vbi − V = − E dx = − dx = − dp (28) e −xp p dx e ppo (−xp ) p −xp µ ¶ ppo (−xp ) kT ln (29) = e pn (xn ) This result might be written as pn (xn ) = ppo (−xp ) exp
µ
eV kT
¶
µ ¶ eVbi exp − kT
(30)
From the solution of the Poisson equation, we know that the built-in potential is given by (Neamen, p. 241) ¶ µ kT ppo Vbi = (31) ln e pno 6
Inserting this in the above expression for pn yields µ ¶ eV pn (xn ) = pno exp kT In the same manner, the electron concentration at −xp is µ ¶ eV np (−xp ) = npo exp kT
(32)
(33)
These equations are referred to as the ”law of the junction”. The important thing is, that the minority carrier densities at the depletion layer edge depends exponentially on the applied voltage. This result is crucial for the understanding of both the diode and the bipolar transistor. We note, that ratio of the minority carrier densities at both sides of the depletion is independent of the applied voltage, NA pno n2i NA pn (xn ) = = = np (−xp ) npo ND n2i ND
(34)
Thus, the ratio of the minority carrier densities is simply the inverse ratio of the doping levels in the two regions.
3.2
Current-voltage characteristics
In the following, we will determine the current-voltage characteristics of the ideal diode. Because we neglect recombination, the current density of electron and holes is constant. In the n-type region, the holes are minority carriers, and we denote the hole current density as Jpn . The hole current density is obtained with the help of Fig. 2 and Eq. (10) ¶ µ µ ¶ dpn eV eDp pno Jpn = −eDp −1 (35) exp = dx Wn kT In the same way, the electron current density in the p-type region, Jnp , is ¶ µ µ ¶ dnp eV eDn npo Jnp = eDn −1 exp = dx Wp kT
(36)
The total current density through the device is the sum of the electron and hole current densities. We obtain (using the law of mass action), J = Jpn + Jnp ¶ ¶µ µ ¶ µ eV Dp pno Dn npo + −1 exp =e Wn Wp kT µ ¶ ¶µ µ ¶ Dn n2ip Dp n2in eV ≈e −1 exp + ND Wn NA Wp kT ¶ µ µ ¶ eV ≡ Js exp −1 kT 7
(37) (38) (39) (40)
8 10
8
10
6
10
4
10
2
10
0
6
J/Js
|J/Js|
4
2
0
-2
-6
-4
-2
0
2
4
10
6
-2
-20
-10
qV/(kT)
0
10
20
qV/(kT)
Figure 3: In the forward direction, the current through the diode depends exponentially on the applied bias. With reverse bias, there is only a small current, the leakage currrent. where Js is called the saturation current density. This important equation is the current-voltage characteristics for an ideal diode, shown in Fig. 3. We note, that the current depends exponentially on the applied bias. For a positive bias, the current flow is very large. For negative bias, the current through the device is very small. This means, that the pn-junction has rectifying properties. Note, that in the forward direction, a change in bias of ∼ 60mV (= kT ln(10)/e) increases the current with a decade. For a one sided abrupt junction the current density depends only on the properties of low doped material µ µ ¶ ¶ DL n2iL eV J ≈e exp −1 (41) NL WL kT The subscript L refers to the low doped side of the junction. 3.2.1
Electron and hole profiles
The hole profile in the n-type region is given by Eq. (14). From Fig. 2 we obtain µ µ ¶ ¶ eV xcn − x exp −1 (42) pn (x) = pno + pno xcn − xn kT The first term on the left is simply the thermal equilibrium hole concentration, and the second term describes the change in carrier concentration when we apply a voltage. Likewise, the electron profile in the p-type region is ¶ µ µ ¶ x + xcp eV np (x) = npo + npo −1 (43) exp xcp − xp kT 8
Figure 4: The stored excess minority carrier charge in the diode is shown as the shaded areas.
3.3
Minority carrier storage
In the diode, minority and majority carriers are stored in the neutral regions. It is important to calculate the stored excess minority charge because these carriers can only be removed through diffusion, and can therefore limit the switching speed of the device. The stored charge is simply the area below the carrier profile as shown on Fig. 4. The excess hole charge per unit area stored in the n-type region is Z xcn Qp = e (pn (x) − pno ) dx (44) =
xn 1 e (pn (xn ) 2
− pn (xcn )) (xcn − xn ) µ µ ¶ ¶ epno Wn eV exp −1 = 2 kT epno Wn J = 2 Js
(45) (46) (47)
In the last equation we have used the diode characteristics, Eq. (40). Note, that the stored charge is directly proportional to the current through the device. In the same way, the excess electron charge per unit area stored in the p-type region is Z −xp Qn = e (np (x) − npo ) dx (48) −xcp µ µ ¶ ¶ eV enpo Wp exp −1 (49) = 2 kT enpo Wp J = (50) 2 Js
9
The total stored charge is the sum of the charge stored on the two sides of the junction. Therefore, (using ND pno = n2i , and NA npo = n2i ) Q = Qp + Qn
(51)
¶ µ µ ¶ eV e = 2 (pno Wn + npo Wp ) exp −1 kT µ ¶ ¶µ µ ¶ Wn n2in Wp n2ip eV e ≈2 −1 exp + ND NA kT µ ¶ Wn n2in Wp n2ip J e =2 + ND NA Js
(52) (53) (54)
where nin and nip are the intrinsic concentration in the n and p-type material, respectively. For a low doped silicon device nin = nip but for highly doped material or for heterojunction devices they are different. If the thickness of the two neutral regions is comparable, the stored excess charge has its largest contribution from the low doped side of the junction. For a one sided abrupt junction, the stored excess charge per unit area is µ ¶µ µ ¶ ¶ WL n2iL eV e Q≈ 2 exp −1 (55) NL kT WL n2iL J (56) = 2e N L Js Note, the in this case the stored excess charge depends only on the properties of the low doped region.
3.4
Transit time
In steady state, there is a constant flux of minority carriers through the neutral regions. The transit time, τ , is defined as ¯ ¯ ¯Q¯ τ = ¯¯ ¯¯ (57) J
For a one sided abrupt junction we find (using Eqns. (41) and (55)) ¢ ¡ eV ¢ 2 ¡ e WL niL − 1 exp WL2 2 N kT = τ = D nL2 ¡ ¢ ¡ ¢ 2DL e NLLWiLL exp eV −1 kT
(58)
This is the diffusion transit time, and it depends on the minority carrier diffusivity and the square of the thickness of the neutral region.
3.5
Small signal equivalent circuit of a diode
The small signal equivalent circuit of a diode is shown on Fig. 5. It consists of the depletion layer capacitance, diffusion capacitance and the diode conductance. 10
Figure 5: The small signal equivalent circuit of a diode consist of the diode conductance, Gd , the junction capacitance, Cj , and the diffusion capacitance Cd . The series resistance, rs , is also shown. 3.5.1
Depletion layer capacitance
The depletion layer capacitance is given by ǫA ǫA =r ³ Cj = ´ W 2ǫ NA +ND e
NA ND
(59) (Vbi − V )
where A is the area of the device. To see how this depends on the current through the device we can rearrange Eq. (40) µ µ ¶ ¶ eV J = Js exp −1 (60) kT µ ¶ J eV (61) + 1 = exp Js kT µ ¶ J kT ln +1 (62) V (J) = e Js and substitute the expression for V (J) into Eq. (59) Cj = r
ǫA 2ǫ e
³
NA +ND NA ND
´³
Vbi −
kT e
ln
³
J Js
´´ +1
(63)
A plot of this equation is shown in Fig. 6. At low current densities, the junction capacitance is almost constant. 3.5.2
Diffusion capacitance
The diffusion capacitance is obtained from Eq. (53) ¶ µ ¶ µ dQ eV e2 A Wn n2in Wp n2ip Cd = A exp ≈ + dV 2kT ND NA kT µ ¶ µ ¶ J e2 A Wn n2in Wp n2ip + +1 = 2kT ND NA Js Note, that the diffusion layer capacitance depends linearly on the current. 11
(64) (65)
5
10
5
10
4
10
3
10
2
10
1
10
0
Cj/Cj(0) Cd/Cj(0)
3
Ct/Cj(0)
Ct/Cj(0)
4
2
1
0 10
0
10
2
10
4
10
6
10
8
10
10
10
12
10
0
10
2
10
J/Js
4
10
6
10
8
10
10
10
12
J/Js
Figure 6: The total capacitance for a diode depends critically on the current. The left figure shows how Cj and Cd depend on the normalized current density. The figure at the right shows how the total capacitance, Ct = Cj + Cd , depends on the normalized current density. 3.5.3
Total capacitance
The total capacitance per unit area is the sum of the depletion layer and the diffusion capacitance Cj + Cd Ct = A Aµ ¶µ ¶ 2 J e Wn n2in Wp n2ip = + +1 2kT ND NA Js · µ ¶½ µ ¶¾¸− 21 2 NA + ND kT J + Vbi − ln +1 eǫ NA ND e Js
(66) (67) (68)
This equation is shown on Fig. 6. At low current densities the capacitance is determined by the depletion layer capacitance, but at high current densities the diffusion capacitance dominates. 3.5.4
Conductance
The magnitude of the current through the diode depends on the applied voltage, Eq. (40). Therefore, the diode has a small signal conductance, gd , given by dI dJ eA gd = =A = Js exp dV dV kT eA J ≈ kT 12
µ
eV kT
¶
(69) (70)
where the last equation is valid for V ≫ linearly on the current.
kT . e
13
Note, that the conductance depends
WE
WB
WC
n
p
n
x XEC
XEB XBE
XBC
npoB exp
pn (x) np (x) pnoE exp
³
qVBE kT
³
qVBE kT
XCC
´
npoB exp
´
XCB
³
qVBC kT
´
pnoC exp
³
qVBC kT
´
pnoC
pnoE x XEC
XEB XBE
XBC
XCB
XCC
Figure 7: When recombination is ignored, all minority carrier profiles in the transistor are straight lines. In this example, the transistor is working in the forward active mode where the emitter-base junction is forward biased whereas the base-collector junction is reverse biased. The metallurgical junctions are shown with broken lines.
4
The bipolar transistor
The transistor is a semiconductor device composed of a pnp or npn layer sequence. In this section we will describe the npn transistor, shown on Fig. 7. To describe the carrier flow through the device, we must solve the current density and continuity equations as was done for the diode. The transistor is a little more difficult to describe because it has one more pn junction than the diode, but the procedure is basically the same. We will assume that recombination can be ignored. For the base, this is an excellent approximation because the base is very thin in the transistors used today. When recombination is ignored, we know that all minority carrier profiles in the field-free areas are straight lines. To determine the minority carrier current densities we just have to determine the boundary conditions at the contacts and the depletion layer edges.
14
4.1
Boundary conditions
For the ohmic contacts we assume that the minority carrier concentration is equal to the thermal equilibrium value. Thus pnE (XEC ) = pnoE pnC (XCC ) = pnoC
(71)
At the emitter-base and the base-collector junctions we use the law of the junction. Therefore, ¢ ¡ ¢ ¡ (72) pnE (XEB ) = pnoE exp eVkTBE npB (XBE ) = npoB exp eVkTBE In the same way, the minority carrier concentration at the base-collector junction is ¡ ¢ ¡ ¢ (73) npB (XBC ) = npoB exp eVkTBC pnC (XCB ) = pnoC exp eVkTBC
These boundary conditions are shown on Fig. 7.
4.2
Minority carrier current densities
The minority carrier current densities in the different regions of the transistor are obtained as the slope of the minority carrier profiles. The most important current in the transistor consists of the minority carriers injected from the emitter to the base. These carriers travel over the base, and because we neglect recombination, they all arrive at the collector where they become majority carriers. The current density is1 µ µ ¶ µ ¶¶ eVBC eDn eVBE dnpB npoB exp = − npoB exp JnB = eDn dx WB kT kT ¶ µ ¶¶ µ µ eDn npoB eVBC eVBE =− − exp exp WB kT kT ¶ µ ¶¶ µ µ 2 eDn niB eVBC eVBE =− − exp (74) exp NB WB kT kT For the holes injected from the base to the emitter we have ¶ µ µ ¶ dpnE eVBE eDp JpE = −eDp − pnoE pnoE exp =− dx WE kT ¶ ¶ µ µ eDp pnoE eVBE =− −1 exp WE kT ¶ ¶ µ µ eDp n2iE eVBE =− −1 exp NE WE kT 1
(75)
Note, that in this context, the width of the neutral base is denoted WB . In the textbook by Neamen, the corresponding symbol is xB .
15
Finally, the minority carrier current density in the collector is ¶¶ µ µ dpnC eVBC eDp JpC = −eDp pnoC − pnoC exp =− dx WC kT ¶¶ µ µ eVBC eDp pnoC 1 − exp =− WC kT ¶¶ µ µ 2 eDp niC eVBC =− 1 − exp NC WC kT
(76)
These equations describe the minority carrier current densities in the transistor for any bias condition. In the following we will use these equations to describe the different modes of operation.
4.3
Forward active mode, VBE > 0, VBC ≪ 0
In the forward active mode, the emitter-base junction is forward biased whereas the base-collector junction is reverse biased. This mode is used widely for analog and high speed digital circuits. In this mode, the emitter emits electrons into the base and the collector collects them. Using that VBE > 0 and VBC ≪ 0 the equations for the current densities (Eqns. (74,75, 76)) reduce to ¶ µ eDn n2iB IC eVBE JnB = − ≈− (77) exp NB WB kT A ¶ µ eDp n2iE IB eVBE JpE = − ≈− (78) exp NE WE kT A eDp n2iC JpC = − (79) NC WC ¢ ¡ Because the emitter-base junction is forward biased, exp eVkTBE is large, and JpC can be ignored compared to JnB . Therefore, the collector current is the same as the injected current in the base, IC = −AJnB , and the base current is simply the injected current in the emitter, IB = −AJpE . Thus, by applying a voltage between the emitter and the base, we can control the collector current. 4.3.1
Current gain
In the forward active mode, the common emitter current gain, βF , is given by IC βF = = IB =
eDn n2iB NB W B eDp n2iE NE W E
exp exp
¡ eV
BE
kT
¡ eVBE ¢
kT 2 Dn NE WE niB Dp NB WB n2iE
16
¢ (80)
This equation is very important as it relates the current gain to the properties of the semiconductor material. Let us now examine the three different parts of Eq. (80) in some detail. In silicon, the ratio of diffusivities, Dn /Dp , is around 2. This explains why the most widely used transistor is the npn structure, because for the pnp transistor the relevant ratio is Dp /Dn which is around 0.5. Therefore, the current gain is increased at no cost. The next part of Eq. (80) is the ratio NE WE /(NB WB ). First, we note that NE WE is simply the total fixed charge per unit area in the neutral region of the emitter. Similarly, NB WB is the total fixed charge per unit area in the base. To obtain a high current gain, the emitter must be wide (thick) and highly doped whereas the base should be narrow (thin) and low doped. Assuming WE = WB and substituting typical values NE = 1019 cm−3 , NB = 1017 cm−3 and Dn = 2Dp yields β = 50. It is important to realize, that WE and WB are the thicknesses of the neutral regions. As the basecollector junction is reverse biased, part of the depletion layer will extend into the base, and this will become wider as the reverse voltage is increased. Thus, the neutral base width is decreased and, from Eq. (77), the collector current, and therefore β, increases. This is known as the Early effect. The ratio of the intrinsic concentration in the base to the intrinsic concentration in the emitter, n2iB /n2iE , is also an important factor. For low doped material the intrinsic concentration is simply ni and the ratio equals unity. However, it turns out, that the intrinsic concentration depends on the doping level. Remember, that the intrinsic concentration in a semiconductor depends exponentially on the bandgap, n2i ∝ exp(−Eg /kT ). When a semiconductor is highly doped, the bandgap decreases and the intrinsic concentration goes up. As the emitter is normally highly doped, the intrinsic concentration in the emitter is larger than in the base, and the ratio n2iB /n2iE is less than unity. This effect reduces the current gain. Clearly, it would be very advantageous if it was possible to have a high intrinsic concentration in the base. This is possible in the Heterojunction Bipolar Transistor (HBT) to be discussed later on. The transistor can also be operated in the inverse mode where the collector is used as emitter and vice versa. The current gain in this mode is simply (use Eq. (80) and exchange E with C) Dn NC WC n2iB βR = (81) Dp NB WB n2iC As the collector is lower doped than the base, the reverse current gain is typically much smaller than the forward current gain.
17
10
-2
10
-4
10
-6
10
-8
10
-10
10
-12
10
-14
200
IB IC CURRENT GAIN
IC IB [A]
Gummel Plot
150
100
50
0.0
0.2
0.4
0.6
0
0.8
VBE [V]
0.0
0.2
0.4
0.6
0.8
VBE [V]
Figure 8: The figure at the left shows a Gummel Plot of a standard BC547 transistor. The figure at the right shows the common emitter current gain IC /IB . The peak current gain equals the value obtained from Eq. 80. The loss in current gain at low currents is due to an additional base current, which is caused by recombination in the base-emitter depletion layer.¡ This¢contribution is also exponentially dependent on the BE base-emitter voltage, ∝ exp qV2kT 4.3.2
Gummel Plot
The above expressions for the collector and base currents, Eqns. (77) and (78), can be written as ¶ µ eVBE (82) IC = IS exp kT µ ¶ IS eVBE IB = exp (83) βF kT where the saturation current, IS is given by IS =
eDn An2iB NB WB
(84)
Fig. 8 shows a plot of IC and IB as a function of VBE with VBC = 0V . This is known as a Gummel plot. Notice, that simple theory, as expressed by Eqns. (82) and (83), is in excellent agreement with the measurements over many decades of current. 4.3.3
Base transit time
The base transit time, τb , is obtained from the stored excess minority carrier charge, QB , as ¯ ¯ ¯ QB ¯ ¯ ¯ (85) τb = ¯ JnB ¯ 18
npoB exp
pn (x) np (x) pnoE exp
³
qVBE kT
³
qVBE kT
´
npoB exp
´
³
qVBC kT
´
pnoC exp
QB
³
qVBC kT
´
pnoC
pnoE x XEC
XEB XBE
XBC
XCB
XCC
Figure 9: The stored excess hole charge in the base is shown as the shaded area. The stored charge is simply the area below the carrier profile as shown on Fig. 9. Therefore, ¶ µ ¶¶ µ µ eVBC eVBE 1 − exp (86) QB = 2 enpoB WB exp kT kT ¶ µ ¶¶ µ µ 2 eVBC eVBE 1 niB = 2e WB exp − exp (87) NB kT kT Using Eq. (74) we then obtain ¢ ¡ ¢¢ ¡ ¡ 2 ¯ ¯ 1 niB ¯ QB ¯ e NA WB exp eVkTBE − exp eVkTBC 2 ¯= τb = ¯¯ ¢ ¡ ¢¢ ¡ ¡ JnB ¯ eDn n2iB exp eVBE − exp eVBC NB W B kT kT =
WB2 2Dn
(88) (89)
This is the base transit time, and it depends on the square of the base width. To have a low base transit time we must make the base thin. Note, that the base transit time depends on the inverse of the minority carrier diffusivity. Therefore, a high diffusivity gives a low base transit time. This is again a reason for using the npn transistor because of the larger carrier diffusivity in the base.
4.4
Modes of operation
In the above discussion we considered the forward active mode. Clearly, the terminal currents for any bias condition can be expressed using the equations for the minority carrier current densities, Eqns. (75, 74, 76). In general, the terminal currents are given
19
Figure 10: The nonlinear hybrid-π model. by IE = (JnB + JpE ) A IC = − (JnB + JpC ) A IB = −IE − IC = (JpC − JpE ) A For the emitter current, we use Eqns. (74) and (75) and obtain µ µ ¶ µ ¶¶ eVBE eDn n2iB A eVBC exp − exp IE = − NB WB kT kT ¶ ¶ µ µ 2 eDp niE A eVBE − −1 exp NE WE kT µ µ ¶ µ ¶¶ ¶ ¶ µ µ eVBE IS eVBC eVBE = −IS exp − exp − −1 exp kT kT βF kT Similarly, the collector current is obtained using Eqns. (74) and (76) µ µ ¶ µ ¶¶ eVBE eVBC eDn n2iB A exp − exp IC = NB WB kT kT ¶¶ µ µ 2 eDp niC A eVBC + 1 − exp NC WC kT µ µ ¶ µ ¶¶ ¶ ¶ µ µ eVBE IS eVBC eVBC = IS exp − exp − −1 exp kT kT βR kT
(90) (91) (92)
(93) (94)
(95) (96)
These equations are the basis for the nonlinear hybrid-π model of the bipolar transistor, shown in Fig. 10. This model uses one current parameter, IS , and the common emitter current gains βF and βR . The transport currents are described by ¢ ¢ ¡ ¡ ¢ ¢ ¡ ¡ (97) ICC = IS exp eVkTBE − 1 IEC = IS exp eVkTBC − 1 and the base current is described by
IBF =
ICC βF
IBR = 20
IEC βR
(98)
Therefore, the collector, emitter and base currents are IC = ICC − IEC − IBR IE = IEC − ICC − IBF IB = IBF + IBR
(99) (100) (101)
The nonlinear hybrid-π model of the bipolar transistor is widely used in circuit simulation programs.
21
Figure 11: The common emitter small signal equivalent circuit
4.5
Small signal equivalent circuit
The small signal equivalent circuit of the transistor in the common emitter configuration is shown in Fig. 11. The transconductance, gm , is given by gm =
dIC = dVBE
e I kT C
(102)
and it depends linearly on the collector current. The input conductance is gπ =
1 dIB dIB dIC gm = = = rπ dVBE dIC dVBE hf e
(103)
where the small signal current gain is given by hf e =
dIC dIB
(104)
In general, this current gain depends on the frequency, and at low frequency, hf e ≈ βF . Finally, the output conductance is often modeled as ¯ dIC ¯¯ IC go = (105) ≈ ¯ dVCE VBE VCE + VA
where VA is the Early voltage. The capacitance Cπ is the capacitance of the forward biased emitter-base junction and consists of the depletion layer capacitance, CjEB , in parallel with the diffusion capacitance, CdEB . The capacitance Cµ is the depletion capacitance of the reverse biased base-collector junction. Thus, Cπ = CjEB + CdEB Cµ = CjBC
(106) (107)
The diffusion capacitance of the forward biased emitter-base junction, is defined as (ignoring the charge stored in the emitter) CdEB = A 22
dQB dVBE
(108)
Figure 12: Circuit for determining fT . Using Eqns. (86) and (77) the diffusion capacitance is µ ¶ e2 n2iB eVBE 1 exp CdEB = 2 AWB kT NB kT 2 W e ≈ B IC 2Dn kT e = τ b IC kT = τ b gm
(109) (110) (111) (112)
and it depends on the base transit time and the transconductance. 4.5.1
Transit frequency
One of the most important parameters to describe the high frequency performance of bipolar transistors is the common emitter transit frequency, fT . This is the frequency where the magnitude of the short circuit common emitter current gain extrapolates to unity. Fig. 12 shows the circuit for determining fT . Using Kirchoffs current law we obtain ic = gm veb − jωCµ veb ib = gπ veb + jω (Cπ + Cµ ) veb
(113) (114)
Therefore, the small signal current gain is hf e (ω) =
gm − jωCµ ic = ib gπ + jω (Cπ + Cµ )
(115)
Because the admittance of Cµ is small compared to gm we can ignore the term jωCµ veb in the above equation, and hf e (ω) ≈
gm gπ + jω (Cπ + Cµ )
23
(116)
Using βF = gm /gπ we can write hf e (ω) =
1+
βF (Cπ + Cµ )
jω gβmF
(117)
The frequency at which the magnitude of this gain extrapolates to unity is fT . To find fT , we first calculate v u βF2 gm (118) |hf e | = u ³ ´2 ≈ t ω (Cπ + Cµ ) βF 1 + ω gm (Cπ + Cµ ) because the second term in the denominator is much larger than unity. Therefore, the frequency where |hf e | = 1 is fT =
1 gm 1 ωT = 2π 2π Cπ + Cµ
(119)
To find the dependance of fT on collector current we calculate 1 1 gm ωT = 2π 2π Cπ + Cµ gm 1 = 2π CjEB + Cd + CjBC 1 1 = CjEB +CjBC 2π + Cd
(121)
1 = 2π
(123)
fT =
gm
(120)
(122)
gm
1 kT CjEB +CjBC e IC
+ τb
This result might be rewritten as kT CjEB + CjBC 1 = + τb 2πfT e IC Evidently, fT is ultimately limited by the base transit time.
24
(124)
10
CURRENT GAIN
βF
3
10
2
10
1
10
0
10
-1
10
-2
fβ 10
0
10
2
10
4
fT 10
6
10
8
10
10
FREQUENCY (Hz) Figure 13: The small signal current gain, hf e , depends on the frequency. At low frequencies, hf e equals the DC current gain, βF . At high frequencies hf e decreases. The frequency where the magnitude of the small signal current gain equals unity is the transit frequency,√ fT . The β cutoff frequency, fβ , is the frequency where the magnitude of hf e equals βF / 2.
25
1
Introduction
This short lecture note describes some important aspects of bipolar devices. It is a supplement to the textbook ”Semiconductor Physics & Devices” by Neamen used in the Solid State Electronics and Microtechnology course (33253) at the Technical University of Denmark. Carrier transport in semiconductors is described for the special case of no recombination (τ → ∞). This leads to mathematically simple solutions of the transport equations and allows for a clear and simple, yet accurate, description of both the diode and the bipolar transistor in terms of • bias • doping levels • thickness of the different device regions • other material parameters such as intrinsic concentration and diffusivity
2
Carrier transport in semiconductors
Carrier transport in semiconductors is described mathematically by the continuity and current density equations. The continuity equations for electron and holes are given by (Neamen p. 6.18 and 6.19) ∂n 1 ∂Jn = + (Gn − Rn ) ∂t e ∂x 1 ∂Jp ∂p =− + (Gp − Rp ) ∂t e ∂x
(1) (2)
and they state, that matter does not disappear. These equations are easily derived by a simple book keeping of the particles that enter and leave a small volume (see Neamen p. 195). The current density equations (Neamen, p. 173, Eqn. (5.33)) describe the flux 1
of carriers due to drift (in an electric field E) and diffusion (with diffusion coefficients Dn and Dp for electrons and holes respectivily) dn Jn = eµn nE + eDn | {z } | {zdx} drift
(3)
diffusion
dp Jp = eµp pE − eDp | {z } | {zdx} drift
(4)
diffusion
These equations are valid in general, and can be applied to many different transport problems. To describe the carrier transport, for example in the diode or transistor, these differential equations must be solved with appropriate boundary conditions. We will look for a steady state solution and neglect recombination and generation. In steady state, the carrier concentrations do not depend on time, ∂p ∂n =0 =0 ∂t ∂t
(5)
and if there is no net recombination or generation G n − Rn = 0 G p − Rp = 0
(6)
In this case, the continuity equations become very simple dJn =0 dx dJp =0 dx
(7) (8)
This simply means, that the current density of both electrons and holes is constant. If we assume, that there is no electric field (or only a very small field), E = 0, transport is only due to diffusion. This is the case in the neutral regions outside the depletion layers. In these field free areas, the current density equations reduce to dn dx dp Jp = −eDp dx
Jn = eDn
(9) (10)
Thus, the current density is simply proportional to the gradient of the carrier concentration. Combining the continuity equations, Eqns. (7) and (8), and the current density equations, Eqns. (9) and (10), yields d2 n =0 dx2 d2 p =0 dx2 2
(11) (12)
Figure 1: In the abscence of recombination, the carrier concentration in a neutral region is simply linear with position, and the current density is proportional to the slope. The solutions to these equations are very simple Jn x eDn Jp x p(x) = ap + bp x = ap − eDp
n(x) = an + bn x = an +
(13) (14)
where we have used Eqns. (9) and (10). The coefficients an , bn , ap and bp are constants given by the boundary conditions. This tell us, that the electron and hole distributions are straight lines, and that the current density is proportional to the slope. Fig. 1 illustrates this concept.In the following, these simple results will be used to describe first the diode and then the bipolar transistor. These results can also be derived from the ambipolar transport equation (Neamen p. 5.18 and 5.19) 2 ∂(δn) ′ ∂ (δn) ′ ∂(δn) =D + µ E +g−R ∂t ∂x2 ∂x Dn Dp [(n0 + δn) + (p0 + δp)] D′ = Dn (n0 + δn) + Dp (p0 + δp)
using the simplifications (in this case for p-type) p-type p0 ≫ n0 Low injection p0 ≫ δn No excess generation/recombination g − R = 0 Field free region E=0 Same order of magnitude Dn ∼ Dp ∂(δn) =0 Steady state ∂t 3
(15) (16)
we arrive to the result that
∂ 2n =0 ∂x2
(17)
Likewise (for n-type) ∂ 2p =0 (18) ∂x2 This means that the n and p are straight lines - the same result as obtained above.
3
The diode
In this section, we will treat the transport of carriers through a pn junction. The starting point is the equations derived above. Consider the diode shown on Fig. 2. The thickness of the depletion layer is W = xn + xp . At each end of the diode there is an ohmic contact, located at −xcp and xcn for the p-type and n-type regions, respectively. The regions outside the depletion layers are neutral. Clearly, the thickness of the neutral region in the n-type material is Wn = xcn − xn and in the p-type material we have Wp = xcp − xp . In the analysis, we will neglect recombination, and then the electron and hole profiles are straight lines as shown above.
3.1
Boundary conditions
In order to proceed, we must find the proper boundary conditions. This means, that we have to specify the minority carrier concentrations at the contacts and at the edge of the depletion layer. 3.1.1
At the contacts
We assume, that the contacts are perfect ohmic contacts. This means, that there is thermal equilibrium and the carrier densities are pn (xcn ) = pno np (xcp ) = npo 3.1.2
(19) (20)
Law of mass action
The minority carrier density in thermal equilibrium is related to the majority carrier density by the law of mass action: np = n2i . This means, nno pno = n2i ppo npo = n2i
4
(21) (22)
Wp
Wn
W
p
n
x ¡ xcp
¡ xp
pn (x) np (x)
0
xn qV kT
¶
qV npo exp kT
¶
pno exp
µ µ
xcn
pno
npo
x ¡ xcp
¡ xp
0
xn
xcn
Figure 2: The carrier profiles in the diode are straight lines when recombination is ignored. Boundary conditions at the depletion layer edges and at the contacts are shown.
5
Assuming complete ionization of the impurity atoms, nno ≈ ND and ppo ≈ NA , we can express the minority carrier density in thermal equilibrium as function of the doping level n2i ND n2 ≈ i NA
pno ≈
(23)
npo
(24)
This result is very useful, and will be used many times in the following sections. 3.1.3
Law of the junction
The minority carrier density at the junction can be found with help of the current density equations, Eqns. (3) and (4). Let us first consider the current density equation for holes dp (25) Jp = eµp pE − eDp dx We know, that in thermal equilibrium Jp = 0. Therefore, we might expect, that even with a small current through the device the current density is so small, that Jp ≈ 0. With these assumptions the current density equation reduces to µp E − Dp
1 dp =0 p dx
(26)
µ , and solving for the electric Using the Einstein relation (Neamen p. 154), Dp = kT e p field, we have kT 1 dp (27) E= e p dx This equation is easily integrated over the depletion layer, and we know, that the integral is the difference between the built-in voltage and the applied voltage, Vbi − V . Therefore, Z xn Z Z kT xn 1 dp kT pn (xn ) 1 Vbi − V = − E dx = − dx = − dp (28) e −xp p dx e ppo (−xp ) p −xp µ ¶ ppo (−xp ) kT ln (29) = e pn (xn ) This result might be written as pn (xn ) = ppo (−xp ) exp
µ
eV kT
¶
µ ¶ eVbi exp − kT
(30)
From the solution of the Poisson equation, we know that the built-in potential is given by (Neamen, p. 241) ¶ µ kT ppo Vbi = (31) ln e pno 6
Inserting this in the above expression for pn yields µ ¶ eV pn (xn ) = pno exp kT In the same manner, the electron concentration at −xp is µ ¶ eV np (−xp ) = npo exp kT
(32)
(33)
These equations are referred to as the ”law of the junction”. The important thing is, that the minority carrier densities at the depletion layer edge depends exponentially on the applied voltage. This result is crucial for the understanding of both the diode and the bipolar transistor. We note, that ratio of the minority carrier densities at both sides of the depletion is independent of the applied voltage, NA pno n2i NA pn (xn ) = = = np (−xp ) npo ND n2i ND
(34)
Thus, the ratio of the minority carrier densities is simply the inverse ratio of the doping levels in the two regions.
3.2
Current-voltage characteristics
In the following, we will determine the current-voltage characteristics of the ideal diode. Because we neglect recombination, the current density of electron and holes is constant. In the n-type region, the holes are minority carriers, and we denote the hole current density as Jpn . The hole current density is obtained with the help of Fig. 2 and Eq. (10) ¶ µ µ ¶ dpn eV eDp pno Jpn = −eDp −1 (35) exp = dx Wn kT In the same way, the electron current density in the p-type region, Jnp , is ¶ µ µ ¶ dnp eV eDn npo Jnp = eDn −1 exp = dx Wp kT
(36)
The total current density through the device is the sum of the electron and hole current densities. We obtain (using the law of mass action), J = Jpn + Jnp ¶ ¶µ µ ¶ µ eV Dp pno Dn npo + −1 exp =e Wn Wp kT µ ¶ ¶µ µ ¶ Dn n2ip Dp n2in eV ≈e −1 exp + ND Wn NA Wp kT ¶ µ µ ¶ eV ≡ Js exp −1 kT 7
(37) (38) (39) (40)
8 10
8
10
6
10
4
10
2
10
0
6
J/Js
|J/Js|
4
2
0
-2
-6
-4
-2
0
2
4
10
6
-2
-20
-10
qV/(kT)
0
10
20
qV/(kT)
Figure 3: In the forward direction, the current through the diode depends exponentially on the applied bias. With reverse bias, there is only a small current, the leakage currrent. where Js is called the saturation current density. This important equation is the current-voltage characteristics for an ideal diode, shown in Fig. 3. We note, that the current depends exponentially on the applied bias. For a positive bias, the current flow is very large. For negative bias, the current through the device is very small. This means, that the pn-junction has rectifying properties. Note, that in the forward direction, a change in bias of ∼ 60mV (= kT ln(10)/e) increases the current with a decade. For a one sided abrupt junction the current density depends only on the properties of low doped material µ µ ¶ ¶ DL n2iL eV J ≈e exp −1 (41) NL WL kT The subscript L refers to the low doped side of the junction. 3.2.1
Electron and hole profiles
The hole profile in the n-type region is given by Eq. (14). From Fig. 2 we obtain µ µ ¶ ¶ eV xcn − x exp −1 (42) pn (x) = pno + pno xcn − xn kT The first term on the left is simply the thermal equilibrium hole concentration, and the second term describes the change in carrier concentration when we apply a voltage. Likewise, the electron profile in the p-type region is ¶ µ µ ¶ x + xcp eV np (x) = npo + npo −1 (43) exp xcp − xp kT 8
Figure 4: The stored excess minority carrier charge in the diode is shown as the shaded areas.
3.3
Minority carrier storage
In the diode, minority and majority carriers are stored in the neutral regions. It is important to calculate the stored excess minority charge because these carriers can only be removed through diffusion, and can therefore limit the switching speed of the device. The stored charge is simply the area below the carrier profile as shown on Fig. 4. The excess hole charge per unit area stored in the n-type region is Z xcn Qp = e (pn (x) − pno ) dx (44) =
xn 1 e (pn (xn ) 2
− pn (xcn )) (xcn − xn ) µ µ ¶ ¶ epno Wn eV exp −1 = 2 kT epno Wn J = 2 Js
(45) (46) (47)
In the last equation we have used the diode characteristics, Eq. (40). Note, that the stored charge is directly proportional to the current through the device. In the same way, the excess electron charge per unit area stored in the p-type region is Z −xp Qn = e (np (x) − npo ) dx (48) −xcp µ µ ¶ ¶ eV enpo Wp exp −1 (49) = 2 kT enpo Wp J = (50) 2 Js
9
The total stored charge is the sum of the charge stored on the two sides of the junction. Therefore, (using ND pno = n2i , and NA npo = n2i ) Q = Qp + Qn
(51)
¶ µ µ ¶ eV e = 2 (pno Wn + npo Wp ) exp −1 kT µ ¶ ¶µ µ ¶ Wn n2in Wp n2ip eV e ≈2 −1 exp + ND NA kT µ ¶ Wn n2in Wp n2ip J e =2 + ND NA Js
(52) (53) (54)
where nin and nip are the intrinsic concentration in the n and p-type material, respectively. For a low doped silicon device nin = nip but for highly doped material or for heterojunction devices they are different. If the thickness of the two neutral regions is comparable, the stored excess charge has its largest contribution from the low doped side of the junction. For a one sided abrupt junction, the stored excess charge per unit area is µ ¶µ µ ¶ ¶ WL n2iL eV e Q≈ 2 exp −1 (55) NL kT WL n2iL J (56) = 2e N L Js Note, the in this case the stored excess charge depends only on the properties of the low doped region.
3.4
Transit time
In steady state, there is a constant flux of minority carriers through the neutral regions. The transit time, τ , is defined as ¯ ¯ ¯Q¯ τ = ¯¯ ¯¯ (57) J
For a one sided abrupt junction we find (using Eqns. (41) and (55)) ¢ ¡ eV ¢ 2 ¡ e WL niL − 1 exp WL2 2 N kT = τ = D nL2 ¡ ¢ ¡ ¢ 2DL e NLLWiLL exp eV −1 kT
(58)
This is the diffusion transit time, and it depends on the minority carrier diffusivity and the square of the thickness of the neutral region.
3.5
Small signal equivalent circuit of a diode
The small signal equivalent circuit of a diode is shown on Fig. 5. It consists of the depletion layer capacitance, diffusion capacitance and the diode conductance. 10
Figure 5: The small signal equivalent circuit of a diode consist of the diode conductance, Gd , the junction capacitance, Cj , and the diffusion capacitance Cd . The series resistance, rs , is also shown. 3.5.1
Depletion layer capacitance
The depletion layer capacitance is given by ǫA ǫA =r ³ Cj = ´ W 2ǫ NA +ND e
NA ND
(59) (Vbi − V )
where A is the area of the device. To see how this depends on the current through the device we can rearrange Eq. (40) µ µ ¶ ¶ eV J = Js exp −1 (60) kT µ ¶ J eV (61) + 1 = exp Js kT µ ¶ J kT ln +1 (62) V (J) = e Js and substitute the expression for V (J) into Eq. (59) Cj = r
ǫA 2ǫ e
³
NA +ND NA ND
´³
Vbi −
kT e
ln
³
J Js
´´ +1
(63)
A plot of this equation is shown in Fig. 6. At low current densities, the junction capacitance is almost constant. 3.5.2
Diffusion capacitance
The diffusion capacitance is obtained from Eq. (53) ¶ µ ¶ µ dQ eV e2 A Wn n2in Wp n2ip Cd = A exp ≈ + dV 2kT ND NA kT µ ¶ µ ¶ J e2 A Wn n2in Wp n2ip + +1 = 2kT ND NA Js Note, that the diffusion layer capacitance depends linearly on the current. 11
(64) (65)
5
10
5
10
4
10
3
10
2
10
1
10
0
Cj/Cj(0) Cd/Cj(0)
3
Ct/Cj(0)
Ct/Cj(0)
4
2
1
0 10
0
10
2
10
4
10
6
10
8
10
10
10
12
10
0
10
2
10
J/Js
4
10
6
10
8
10
10
10
12
J/Js
Figure 6: The total capacitance for a diode depends critically on the current. The left figure shows how Cj and Cd depend on the normalized current density. The figure at the right shows how the total capacitance, Ct = Cj + Cd , depends on the normalized current density. 3.5.3
Total capacitance
The total capacitance per unit area is the sum of the depletion layer and the diffusion capacitance Cj + Cd Ct = A Aµ ¶µ ¶ 2 J e Wn n2in Wp n2ip = + +1 2kT ND NA Js · µ ¶½ µ ¶¾¸− 21 2 NA + ND kT J + Vbi − ln +1 eǫ NA ND e Js
(66) (67) (68)
This equation is shown on Fig. 6. At low current densities the capacitance is determined by the depletion layer capacitance, but at high current densities the diffusion capacitance dominates. 3.5.4
Conductance
The magnitude of the current through the diode depends on the applied voltage, Eq. (40). Therefore, the diode has a small signal conductance, gd , given by dI dJ eA gd = =A = Js exp dV dV kT eA J ≈ kT 12
µ
eV kT
¶
(69) (70)
where the last equation is valid for V ≫ linearly on the current.
kT . e
13
Note, that the conductance depends
WE
WB
WC
n
p
n
x XEC
XEB XBE
XBC
npoB exp
pn (x) np (x) pnoE exp
³
qVBE kT
³
qVBE kT
XCC
´
npoB exp
´
XCB
³
qVBC kT
´
pnoC exp
³
qVBC kT
´
pnoC
pnoE x XEC
XEB XBE
XBC
XCB
XCC
Figure 7: When recombination is ignored, all minority carrier profiles in the transistor are straight lines. In this example, the transistor is working in the forward active mode where the emitter-base junction is forward biased whereas the base-collector junction is reverse biased. The metallurgical junctions are shown with broken lines.
4
The bipolar transistor
The transistor is a semiconductor device composed of a pnp or npn layer sequence. In this section we will describe the npn transistor, shown on Fig. 7. To describe the carrier flow through the device, we must solve the current density and continuity equations as was done for the diode. The transistor is a little more difficult to describe because it has one more pn junction than the diode, but the procedure is basically the same. We will assume that recombination can be ignored. For the base, this is an excellent approximation because the base is very thin in the transistors used today. When recombination is ignored, we know that all minority carrier profiles in the field-free areas are straight lines. To determine the minority carrier current densities we just have to determine the boundary conditions at the contacts and the depletion layer edges.
14
4.1
Boundary conditions
For the ohmic contacts we assume that the minority carrier concentration is equal to the thermal equilibrium value. Thus pnE (XEC ) = pnoE pnC (XCC ) = pnoC
(71)
At the emitter-base and the base-collector junctions we use the law of the junction. Therefore, ¢ ¡ ¢ ¡ (72) pnE (XEB ) = pnoE exp eVkTBE npB (XBE ) = npoB exp eVkTBE In the same way, the minority carrier concentration at the base-collector junction is ¡ ¢ ¡ ¢ (73) npB (XBC ) = npoB exp eVkTBC pnC (XCB ) = pnoC exp eVkTBC
These boundary conditions are shown on Fig. 7.
4.2
Minority carrier current densities
The minority carrier current densities in the different regions of the transistor are obtained as the slope of the minority carrier profiles. The most important current in the transistor consists of the minority carriers injected from the emitter to the base. These carriers travel over the base, and because we neglect recombination, they all arrive at the collector where they become majority carriers. The current density is1 µ µ ¶ µ ¶¶ eVBC eDn eVBE dnpB npoB exp = − npoB exp JnB = eDn dx WB kT kT ¶ µ ¶¶ µ µ eDn npoB eVBC eVBE =− − exp exp WB kT kT ¶ µ ¶¶ µ µ 2 eDn niB eVBC eVBE =− − exp (74) exp NB WB kT kT For the holes injected from the base to the emitter we have ¶ µ µ ¶ dpnE eVBE eDp JpE = −eDp − pnoE pnoE exp =− dx WE kT ¶ ¶ µ µ eDp pnoE eVBE =− −1 exp WE kT ¶ ¶ µ µ eDp n2iE eVBE =− −1 exp NE WE kT 1
(75)
Note, that in this context, the width of the neutral base is denoted WB . In the textbook by Neamen, the corresponding symbol is xB .
15
Finally, the minority carrier current density in the collector is ¶¶ µ µ dpnC eVBC eDp JpC = −eDp pnoC − pnoC exp =− dx WC kT ¶¶ µ µ eVBC eDp pnoC 1 − exp =− WC kT ¶¶ µ µ 2 eDp niC eVBC =− 1 − exp NC WC kT
(76)
These equations describe the minority carrier current densities in the transistor for any bias condition. In the following we will use these equations to describe the different modes of operation.
4.3
Forward active mode, VBE > 0, VBC ≪ 0
In the forward active mode, the emitter-base junction is forward biased whereas the base-collector junction is reverse biased. This mode is used widely for analog and high speed digital circuits. In this mode, the emitter emits electrons into the base and the collector collects them. Using that VBE > 0 and VBC ≪ 0 the equations for the current densities (Eqns. (74,75, 76)) reduce to ¶ µ eDn n2iB IC eVBE JnB = − ≈− (77) exp NB WB kT A ¶ µ eDp n2iE IB eVBE JpE = − ≈− (78) exp NE WE kT A eDp n2iC JpC = − (79) NC WC ¢ ¡ Because the emitter-base junction is forward biased, exp eVkTBE is large, and JpC can be ignored compared to JnB . Therefore, the collector current is the same as the injected current in the base, IC = −AJnB , and the base current is simply the injected current in the emitter, IB = −AJpE . Thus, by applying a voltage between the emitter and the base, we can control the collector current. 4.3.1
Current gain
In the forward active mode, the common emitter current gain, βF , is given by IC βF = = IB =
eDn n2iB NB W B eDp n2iE NE W E
exp exp
¡ eV
BE
kT
¡ eVBE ¢
kT 2 Dn NE WE niB Dp NB WB n2iE
16
¢ (80)
This equation is very important as it relates the current gain to the properties of the semiconductor material. Let us now examine the three different parts of Eq. (80) in some detail. In silicon, the ratio of diffusivities, Dn /Dp , is around 2. This explains why the most widely used transistor is the npn structure, because for the pnp transistor the relevant ratio is Dp /Dn which is around 0.5. Therefore, the current gain is increased at no cost. The next part of Eq. (80) is the ratio NE WE /(NB WB ). First, we note that NE WE is simply the total fixed charge per unit area in the neutral region of the emitter. Similarly, NB WB is the total fixed charge per unit area in the base. To obtain a high current gain, the emitter must be wide (thick) and highly doped whereas the base should be narrow (thin) and low doped. Assuming WE = WB and substituting typical values NE = 1019 cm−3 , NB = 1017 cm−3 and Dn = 2Dp yields β = 50. It is important to realize, that WE and WB are the thicknesses of the neutral regions. As the basecollector junction is reverse biased, part of the depletion layer will extend into the base, and this will become wider as the reverse voltage is increased. Thus, the neutral base width is decreased and, from Eq. (77), the collector current, and therefore β, increases. This is known as the Early effect. The ratio of the intrinsic concentration in the base to the intrinsic concentration in the emitter, n2iB /n2iE , is also an important factor. For low doped material the intrinsic concentration is simply ni and the ratio equals unity. However, it turns out, that the intrinsic concentration depends on the doping level. Remember, that the intrinsic concentration in a semiconductor depends exponentially on the bandgap, n2i ∝ exp(−Eg /kT ). When a semiconductor is highly doped, the bandgap decreases and the intrinsic concentration goes up. As the emitter is normally highly doped, the intrinsic concentration in the emitter is larger than in the base, and the ratio n2iB /n2iE is less than unity. This effect reduces the current gain. Clearly, it would be very advantageous if it was possible to have a high intrinsic concentration in the base. This is possible in the Heterojunction Bipolar Transistor (HBT) to be discussed later on. The transistor can also be operated in the inverse mode where the collector is used as emitter and vice versa. The current gain in this mode is simply (use Eq. (80) and exchange E with C) Dn NC WC n2iB βR = (81) Dp NB WB n2iC As the collector is lower doped than the base, the reverse current gain is typically much smaller than the forward current gain.
17
10
-2
10
-4
10
-6
10
-8
10
-10
10
-12
10
-14
200
IB IC CURRENT GAIN
IC IB [A]
Gummel Plot
150
100
50
0.0
0.2
0.4
0.6
0
0.8
VBE [V]
0.0
0.2
0.4
0.6
0.8
VBE [V]
Figure 8: The figure at the left shows a Gummel Plot of a standard BC547 transistor. The figure at the right shows the common emitter current gain IC /IB . The peak current gain equals the value obtained from Eq. 80. The loss in current gain at low currents is due to an additional base current, which is caused by recombination in the base-emitter depletion layer.¡ This¢contribution is also exponentially dependent on the BE base-emitter voltage, ∝ exp qV2kT 4.3.2
Gummel Plot
The above expressions for the collector and base currents, Eqns. (77) and (78), can be written as ¶ µ eVBE (82) IC = IS exp kT µ ¶ IS eVBE IB = exp (83) βF kT where the saturation current, IS is given by IS =
eDn An2iB NB WB
(84)
Fig. 8 shows a plot of IC and IB as a function of VBE with VBC = 0V . This is known as a Gummel plot. Notice, that simple theory, as expressed by Eqns. (82) and (83), is in excellent agreement with the measurements over many decades of current. 4.3.3
Base transit time
The base transit time, τb , is obtained from the stored excess minority carrier charge, QB , as ¯ ¯ ¯ QB ¯ ¯ ¯ (85) τb = ¯ JnB ¯ 18
npoB exp
pn (x) np (x) pnoE exp
³
qVBE kT
³
qVBE kT
´
npoB exp
´
³
qVBC kT
´
pnoC exp
QB
³
qVBC kT
´
pnoC
pnoE x XEC
XEB XBE
XBC
XCB
XCC
Figure 9: The stored excess hole charge in the base is shown as the shaded area. The stored charge is simply the area below the carrier profile as shown on Fig. 9. Therefore, ¶ µ ¶¶ µ µ eVBC eVBE 1 − exp (86) QB = 2 enpoB WB exp kT kT ¶ µ ¶¶ µ µ 2 eVBC eVBE 1 niB = 2e WB exp − exp (87) NB kT kT Using Eq. (74) we then obtain ¢ ¡ ¢¢ ¡ ¡ 2 ¯ ¯ 1 niB ¯ QB ¯ e NA WB exp eVkTBE − exp eVkTBC 2 ¯= τb = ¯¯ ¢ ¡ ¢¢ ¡ ¡ JnB ¯ eDn n2iB exp eVBE − exp eVBC NB W B kT kT =
WB2 2Dn
(88) (89)
This is the base transit time, and it depends on the square of the base width. To have a low base transit time we must make the base thin. Note, that the base transit time depends on the inverse of the minority carrier diffusivity. Therefore, a high diffusivity gives a low base transit time. This is again a reason for using the npn transistor because of the larger carrier diffusivity in the base.
4.4
Modes of operation
In the above discussion we considered the forward active mode. Clearly, the terminal currents for any bias condition can be expressed using the equations for the minority carrier current densities, Eqns. (75, 74, 76). In general, the terminal currents are given
19
Figure 10: The nonlinear hybrid-π model. by IE = (JnB + JpE ) A IC = − (JnB + JpC ) A IB = −IE − IC = (JpC − JpE ) A For the emitter current, we use Eqns. (74) and (75) and obtain µ µ ¶ µ ¶¶ eVBE eDn n2iB A eVBC exp − exp IE = − NB WB kT kT ¶ ¶ µ µ 2 eDp niE A eVBE − −1 exp NE WE kT µ µ ¶ µ ¶¶ ¶ ¶ µ µ eVBE IS eVBC eVBE = −IS exp − exp − −1 exp kT kT βF kT Similarly, the collector current is obtained using Eqns. (74) and (76) µ µ ¶ µ ¶¶ eVBE eVBC eDn n2iB A exp − exp IC = NB WB kT kT ¶¶ µ µ 2 eDp niC A eVBC + 1 − exp NC WC kT µ µ ¶ µ ¶¶ ¶ ¶ µ µ eVBE IS eVBC eVBC = IS exp − exp − −1 exp kT kT βR kT
(90) (91) (92)
(93) (94)
(95) (96)
These equations are the basis for the nonlinear hybrid-π model of the bipolar transistor, shown in Fig. 10. This model uses one current parameter, IS , and the common emitter current gains βF and βR . The transport currents are described by ¢ ¢ ¡ ¡ ¢ ¢ ¡ ¡ (97) ICC = IS exp eVkTBE − 1 IEC = IS exp eVkTBC − 1 and the base current is described by
IBF =
ICC βF
IBR = 20
IEC βR
(98)
Therefore, the collector, emitter and base currents are IC = ICC − IEC − IBR IE = IEC − ICC − IBF IB = IBF + IBR
(99) (100) (101)
The nonlinear hybrid-π model of the bipolar transistor is widely used in circuit simulation programs.
21
Figure 11: The common emitter small signal equivalent circuit
4.5
Small signal equivalent circuit
The small signal equivalent circuit of the transistor in the common emitter configuration is shown in Fig. 11. The transconductance, gm , is given by gm =
dIC = dVBE
e I kT C
(102)
and it depends linearly on the collector current. The input conductance is gπ =
1 dIB dIB dIC gm = = = rπ dVBE dIC dVBE hf e
(103)
where the small signal current gain is given by hf e =
dIC dIB
(104)
In general, this current gain depends on the frequency, and at low frequency, hf e ≈ βF . Finally, the output conductance is often modeled as ¯ dIC ¯¯ IC go = (105) ≈ ¯ dVCE VBE VCE + VA
where VA is the Early voltage. The capacitance Cπ is the capacitance of the forward biased emitter-base junction and consists of the depletion layer capacitance, CjEB , in parallel with the diffusion capacitance, CdEB . The capacitance Cµ is the depletion capacitance of the reverse biased base-collector junction. Thus, Cπ = CjEB + CdEB Cµ = CjBC
(106) (107)
The diffusion capacitance of the forward biased emitter-base junction, is defined as (ignoring the charge stored in the emitter) CdEB = A 22
dQB dVBE
(108)
Figure 12: Circuit for determining fT . Using Eqns. (86) and (77) the diffusion capacitance is µ ¶ e2 n2iB eVBE 1 exp CdEB = 2 AWB kT NB kT 2 W e ≈ B IC 2Dn kT e = τ b IC kT = τ b gm
(109) (110) (111) (112)
and it depends on the base transit time and the transconductance. 4.5.1
Transit frequency
One of the most important parameters to describe the high frequency performance of bipolar transistors is the common emitter transit frequency, fT . This is the frequency where the magnitude of the short circuit common emitter current gain extrapolates to unity. Fig. 12 shows the circuit for determining fT . Using Kirchoffs current law we obtain ic = gm veb − jωCµ veb ib = gπ veb + jω (Cπ + Cµ ) veb
(113) (114)
Therefore, the small signal current gain is hf e (ω) =
gm − jωCµ ic = ib gπ + jω (Cπ + Cµ )
(115)
Because the admittance of Cµ is small compared to gm we can ignore the term jωCµ veb in the above equation, and hf e (ω) ≈
gm gπ + jω (Cπ + Cµ )
23
(116)
Using βF = gm /gπ we can write hf e (ω) =
1+
βF (Cπ + Cµ )
jω gβmF
(117)
The frequency at which the magnitude of this gain extrapolates to unity is fT . To find fT , we first calculate v u βF2 gm (118) |hf e | = u ³ ´2 ≈ t ω (Cπ + Cµ ) βF 1 + ω gm (Cπ + Cµ ) because the second term in the denominator is much larger than unity. Therefore, the frequency where |hf e | = 1 is fT =
1 gm 1 ωT = 2π 2π Cπ + Cµ
(119)
To find the dependance of fT on collector current we calculate 1 1 gm ωT = 2π 2π Cπ + Cµ gm 1 = 2π CjEB + Cd + CjBC 1 1 = CjEB +CjBC 2π + Cd
(121)
1 = 2π
(123)
fT =
gm
(120)
(122)
gm
1 kT CjEB +CjBC e IC
+ τb
This result might be rewritten as kT CjEB + CjBC 1 = + τb 2πfT e IC Evidently, fT is ultimately limited by the base transit time.
24
(124)
10
CURRENT GAIN
βF
3
10
2
10
1
10
0
10
-1
10
-2
fβ 10
0
10
2
10
4
fT 10
6
10
8
10
10
FREQUENCY (Hz) Figure 13: The small signal current gain, hf e , depends on the frequency. At low frequencies, hf e equals the DC current gain, βF . At high frequencies hf e decreases. The frequency where the magnitude of the small signal current gain equals unity is the transit frequency,√ fT . The β cutoff frequency, fβ , is the frequency where the magnitude of hf e equals βF / 2.
25
