2

Models for Semiconductor Devices Used in Analog Electronic Systems

2.1Introduction

This chapter will describe the properties and characteristics of the semiconductor components of analog integrated circuits (AICs). These include diodes, bipolar junction transistors (BJTs), and various types of field-effect transistors (FETs). Biomedical engineers probably will never be called upon to design circuits with or use discrete transistors, with the possible exception of power transistors. Instead, modern analog electronic systems use integrated circuits. To appreciate the behavior and limitations of ICs, however, it is necessary to understand the fundamental behavior of their semiconductor components. (The actual design of ICs is beyond the scope of this text.) One of the most ubiquitous analog ICs is the operational amplifier (op amp); op amps are used in many signal conditioning applications, including differential amplifiers; instrumentation amplifiers; active filters; true RMS converters; precision rectifiers; track-and-hold circuits; etc. Other commonly encountered ICs in biomedical engineering are dc voltage regulators, temperature sensors, phase-lock loops, synchronous rectifiers, analog multipliers, dc-to-dc converters, medical isolation amplifiers, analog-to-digital converters (ADCs), digital-to-analog converters (DACs), etc.

Specifically, the following sections will examine the salient circuit characteristics of pn junction diodes, light-emitting diodes (LEDs), laser diodes (LADs), npn and pnp small-signal BJTs, junction field-effect transistors (JFETs), and n- and p-MOSFETs. Large-signal and midand high-frequency small-signal models will be considered. The midand high-frequency behavior of IC “building blocks” that use two transistors will also be analyzed.

23

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24	Analysis and Application of Analog Electronic Circuits

2.2pn Junction Diodes

2.2.1Introduction

There are several uses for pn junction diodes as discrete components and in ICs. As stand-alone components, power diodes are used to rectify ac to produce dc in power supplies. They are also used with inductive components such as relay coils, motor coils, and loudspeaker windings to clamp inductive voltage switching transients to prevent them from destroying the switching transistors. Small, discrete pn diodes are also used in op amp precision rectifier circuits, peak detectors, sample-and-hold circuits, logarithmic amplifiers, and exponential amplifiers. Diodes are used in IC designs for temperature compensation, to make current mirrors, and for dc level shifting (in the avalanche mode) (Millman, 1979).

2.2.2The pn Diode’s Volt–Ampere Curve

Figure 2.1 illustrates the static volt–ampere curve of a typical silicon pn signal diode (as opposed to power rectifier diode). Note the three major regions in the curve: (1) the forward conduction region in the first quadrant, the (2) blocking region, and the (3) avalanche breakdown (or zener) region in the third quadrant. The volt–ampere behavior of a diode’s forward and blocking regions can be approximated with the well-known approximate mathematical model:

iD = Irs [exp(vD/ηVT) − 1]

(2.1)

vD is the dc voltage across the diode, VT ∫ kT/q, where T is the Kelvin temperature of the junction; q is the electron charge magnitude (1.6 ∞ 10–19 Cb); k is Boltzmann’s constant (1.38 ∞ 10−23 J/K); and η is a bugger factor between 1 and 2 used to fit the diode curve to experimental data. η 2 for silicon diodes; Irs is the reverse saturation current (on the order of μA for

signal diodes). Irs is a strong increasing function of temperature and is taken to be non-negative. At 25∞C, VT = 0.0257 V. The crudeness of the model can

be appreciated in the blocking region where the exponential argument is large and negative, so iD −Irs. In reality, there is a strong ohmic (leakage) component to the reverse iD, so the approximation, iD −Irs + vD/ρ, is more realistic for vD < 0. ρ has the dimensions of resistance (ohms).

Avalanche occurs under reverse-biased conditions when minority carriers conducting the reverse iD gain enough kinetic energy by accelerating in the strong electric field and, through collisions with the substrate atoms’ electrons, create new electron–hole pairs. These new carriers in turn are accelerated in the high E-field and cause still more carriers to be formed. This avalanche process can be modeled by the empirical equation:

© 2004 by CRC Press LLC

Models for Semiconductor Devices Used in Analog Electronic Systems

25

iD

iD

+

+

p

vD

vD

n

−

−

iD

8

A

(mA)

B

6

4

VZ

vD < 0

. . .

Irs

−0.5 V

2

vD

. . .

0

0.5 V 1.0 V

. . .

(µA)

.

IZ

.

.

(mA)

C

FIGURE 2.1

(A) “Layer cake” cross section of a silicon pn junction diode. (B) Diode symbol. (C) Typical I–V curve for a small-signal, Si diode, showing avalanche (zener) breakdown at reverse bias, vD =

−Vz.

iD	−Irs	(2.2)
	1− (vD Vz )n

The exponent, n, can range from 3 to 6 and vD lies between 0 and Vz . Depending on how the pn junction is doped (the density of donor atoms in the n material and acceptor atoms in the p material), Vz can range from approximately 3.2 to over 100 V.

Diodes operated in their avalanche regions can be used as voltage sources or references, and for dc voltage level shifting. Figure 2.2 illustrates an

© 2004 by CRC Press LLC

26	Analysis and Application of Analog Electronic Circuits
IS	RS		IL
		VZ

IZ

VS

RL

FIGURE 2.2

Use of an avalanche (zener) diode as a DC voltage source for RL. The diode’s zener resistance is neglected.

avalanche dc voltage source circuit. The tantalum filter capacitor is used to lower the source impedance of the avalanche supply at high frequencies. From Ohm’s law:

			IL = VZ /RL		(2.3)
			IS = IL + IZ		(2.4)
	RS	= (VS	− VZ ) IS =	(VS − VZ )	(2.5)
				IZ + VZ RL

Every avalanche (zener) diode has a preferred operating point, (VZ, IZ), given by the manufacturer, at which its dynamic impedance is low and power dissipation is safe. Thus in Equation 2.5 VZ and IZ are known and RL is known (or IL at VZ), enabling one to find the required series RS. The required

power dissipated in RS is simply PRs = (VS − VZ)2/RS = IS2 RS watts. Excluding the avalanche region, the dc characteristics of a pn junction

diode can be modeled by an ideal diode (ID) model, shown in Figure 2.3(A). For the ideal diode, iD = 0 for vD < 0, and vD = 0 for iD > 0. Figure 2.3(B) shows that, by adding a reverse-biasing voltage source, VF, model diode volt–ampere curve can be shifted to the right by VF . The model is made yet more real by adding the series resistance, RF, in the conduction path, as shown in Figure 2.3(C). In Figure 2.3(D), a leakage conductance is added in parallel with the model of C to model reverse diode leakage.

The ideal diode and its variations are useful for pencil-and-paper circuit analysis. Detailed simulation of electronic circuits containing diodes is done with electronic circuit analysis programs (ECAPs) such as PSPICE and MicroCap, which handle the nonlinearity algebraically as well as compute the diode’s voltage-dependent shunt capacitance.

© 2004 by CRC Press LLC

Models for Semiconductor Devices Used in Analog Electronic Systems

27

iD	iD				iD	+
					ID	vD
					+
ID
					VF
					−
vD			vD			−
0	0	VF			B
A
iD				iD	+

	ID
	+	vD
GF
	VF
	−
	RF
vD		−

0 VF

C

iD			iD	+
			+
			VF	vD
			−	RL
			GF + GL
			RF	−
			vD
	0	VF	D
GL

FIGURE 2.3

Various I–V models for junction diodes, excluding avalanche behavior. (A) Ideal diode. (B) Ideal diode in series with fixed forward voltage drop. (C) Ideal diode in series with fixed forward voltage drop and forward resistance. (D) Ideal diode in series with fixed forward voltage drop and forward resistance, and reverse leakage resistance.

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28	Analysis and Application of Analog Electronic Circuits

2.2.3High-Frequency Behavior of Diodes

A fundamental limitation to the high-frequency performance of pn semiconductor diodes is their voltage-dependent, small-signal junction capacitance. The junction capacitance mechanism is different for forward conduction (vD > 0) and reverse bias conditions (vD < 0).

As demonstrated earlier, when a diode is dc reverse-biased, a very small (nA) reverse leakage current flows, part of which is due to the thermal (random) generation of hole–electron pairs in the depletion region of the device. The depletion region is a volume around the junction in the p- and n-doped semiconductor sides that is free of mobile carriers (holes and electrons, respectively). The electric field from the negative vD causes the carriers to move away from the junction, creating what is essentially a layer of chargefree intrinsic semiconductor around the junction. The more negative vD is, the stronger the electric field and the larger the depletion region. The chargefree depletion region behaves like a leaky dielectric in a parallel-plate capacitor. The capacitor’s “plates” are the dense conductive semicon region where majority carriers still exist and conductivity is high. Recall that the capacitance of a simple parallel-plate capacitor is given by:

C =	Aκεo	farads	(2.6)
	d

where κ is the dielectric constant of silicon; A is the area of the diode junction in m2; and d is the effective “plate” separation in m. Note that the effective d increases as vD goes more negative and the electric field increases at the junction. Thus Cd decreases as vD goes more negative and is maximum at vD = 0. Diode depletion capacitance can be modeled by the function:

Cd (vD ) =		Cd (0)		farads, for vD < 0	(2.7)
	(1	− vD ψo )	n

where Cd(0) is the value of the depletion capacitance at vD = 0 and ψo is the contact potential of the pn junction determined by the carrier doping densities at the junction and their gradients. ψo ranges from 0.2 to 0.9 V, and is in the neighborhood of 0.75 V at room temperature for a typical silicon smallsignal diode. The exponent, n, ranges from 0.33 to 3, depending on the doping profile near the pn junction; n = 0.5 for an abrupt (symmetrical step) junction. For depletion, vD < 0 in Equation 2.7. Note that, as vD goes positive and approaches ψo, Cd approaches •. Clearly, Equation 2.7 is intended to model diode capacitance for −VZ < vD < 0. Typical values for Cd(0) are in the range of tens of picofarads. A typical reverse-biased, pn diode’s junction capacitance vs. vD is shown in Figure 2.4. A detailed analysis of pn junction behavior can be found in Millman (1979), Yang (1988), or Gray and Meyer (1984).

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Models for Semiconductor Devices Used in Analog Electronic Systems				29
	Cj
		Cs
	Cj0
	Cd
	vD < 0	vD
	0	ψo

FIGURE 2.4

How the equivalent junction capacitance of a pn diode varies with vD. Cd is mostly depletion capacitance (see text) and Cs is due to stored minority carriers associated with forward conduction.

A forward-biased diode’s pn junction is characterized by a large, currentdependent diffusion or charge storage capacitance, which involves an entirely different mechanism from depletion capacitance. Under forward bias conditions, minority carriers are injected across the junction, where they quickly recombine. That is, electrons are injected into the p-side from the n-side and holes are injected into the n-side from the p-side. The mean lifetime of electrons in the p-side is τn and of holes in the n-side, τp. These injected minority carriers represent a stored charge, Qs, around the junction. By definition, the small-signal diffusion capacitance can be written for a symmetrical junction where τ = τp = τn:

dQ

τdi

τiDQ

τIrs exp(vDQ

VT )

C =

s

=

D

= τg =

=

farads for v

> 0 CD

(2.8)

dvD

ηVT

ηVT

D

dvD

d

D

where gd is the small-signal conductance of the forward-biased diode at its dc operating (Q) point; gd ∫ ∂iD/∂vD at Q. i.e., gd = d[Irs exp(vD/ηVT)]/dvD Q = Irs exp(vDQ/ηVT)/(ηVT) = iDQ/ηVT siemens. If the diode junction doping is asymmetrical, it can be shown (Nanavati, 1975) that the diffusion capacitance

is given by:

CD =	1	(IDnQ τn + IDpQ τp ) farads	(2.9)
	ηV
	T

The diode forward current is broken into the electron and hole injection currents. For example, calculate the diffusion capacitance of a silicon diode with a symmetrically doped step junction at 300 K, forward-biased with iDQ = 10 mA. Take τ = 1 μs, η = 2.

C =	1 E-6		∞ 1 E-2	= 0.192 μF	(2.10)
D	2	∞	0.026

© 2004 by CRC Press LLC