Chapter 2 - Analog Integrated Circuit Design by John Choma

5/25/2018 Chapter 2 - Analog Integrated Circuit Design by John Choma

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LECTURE SUPPLEMENT #2 . . .[LS #2]

CHAPTER #02

The PN Junction Diode

Dr. John ChomaProfessor of Electrical Engineering

University of Southern California

Ming Hsieh Department of Electrical Engineering

University Park: Mail Code: 0271Los Angeles, California 900890271

2137404692 [USC Office]

2137407581 [USC Fax]

8183841552 [Cell][email protected]

PRELUDE:

We address the fundamental physical properties, volt-ampere characteristics, and

circuit level models of the semiconductor PN junction diode in this chapter. While

PN junction diodes are commonly fabricated in either silicon or germanium

semiconductor technologies, most of the disclosures in this chapter focus on sili-

con diodes. Also discussed are a few practical applications of the diode, includ-

ing half wave and full wave rectifiers, logarithmic amplifiers, and limiters.

May 2013


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Chapter 2 PN Junction Diode

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2.1.0. INTRODUCTION

The semiconductor PN junction diode is the simplest of solid-state circuit elements. Its

simplicity belies its critical importance to the state of the electronics. This importance stems

from the fact that an understanding of the physical properties and operation of a junction diodepromotes an insightful comprehension of the operation and general behavior of more complex

electronic devices. The most notable of these complex solid state devices are the bipolar junc-

tion transistor (BJT) and the metal-oxide-semiconductor field effect transistor (MOSFET). How-ever, the diode is more than a mere crutch for understanding complex intricate devices in that the

PN junction diode itself boasts utility in numerous applications. Included among these applica-tions are voltage references capable of sustaining relatively constant voltages in the face of

temperature increases and/or variations in power supply voltages. Yet another application of the

diode is the rectifier (or AC -to- DC converter) that converts periodic voltage waveforms havingzero average values (and hence no DC values) to unidirectional energy offering nonzero DC

voltage levels. This conversion capability is foundational to transforming the alternating line

voltages supplied to private and commercial establishments by local power companies to theconstant, time invariant voltages mandated by non-portable electronic systems.

x

0 WnWp

H

L

Metal

Conta

cti (t)d

i (t)d

v (t)d

v (t)d

PN Junction

(a).

(b). Figure (2.1). (a).Physical abstraction of a PN junction semiconductor diode. (b).Electrical schematic

symbol of the PN junction diode.

Figure (2.1a) is a simplified physical representation of a PN junction diode, while Fig-

ure (2.1b) depicts the corresponding electrical schematic symbol. The diode is a physical struc-ture formed by p-type and n-type semiconductor crystals, such that at their common physical

boundary, which is termed the semiconductor junction, or PN junction, an almost abrupt transi-


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tion from p-type -to- n-type semiconductor is achieved. The p-type region is doped with accep-

tor impurities at an average concentration of NA(in units of atoms -per- cm3), while the n-type

volume has an average donor impurity concentration ofND. Generally, we find thatNAis around

four or five orders of magnitude smaller than is ND, which is of the order 1020

-to- 1021

atoms/cm3. Thus, the n-type region is doped very strongly and actually near its solid solubility

limit, which is to say that this region is doped to nearly its maximum impurity capacity. Al-

though the impurity concentration profiles in either diode region are rarely constant, independentof position variable x in the diagram, we can garner the salient features of diode operation by

assuming that these doping concentrations are constant at their respective average, or at least atsome appropriately weighted average, values.

The thickness of the p-region is noted as Wp, while that of the n-type crystal is Wn.

Both of these dimensions are of the order of microns1, with Wpbeing roughly half that of Wn.

Dimensions L and H define the cross section junction area, Ae, as Ae = LH. In a monolithicrealization of a diode destined for high-speed signal processing applications,Lis usually about a

micron or even smaller. Depending on the current level that we wish the diode to conduct with-

out causing thermal stress,His typically a designable variable that can be several times -to- sev-

eral tens or even hundreds of times larger thanL. The area,Ae, in question is the junction crosssection pierced orthogonally by the diode current, id(t), which flows in response to an applied

diode voltage, vd(t). We posture id(t)as a positive current that flows from the p-type region tothe n-type region, while we consider vd(t)to be a positive voltage when it raises the potential of

the p-side of the junction above that of the n-side.

Let us assume that the implanted or diffused dopants atoms of both impurities are com-

pletely ionized, which is a reasonable, but slightly flawed, presumption. The concentration offree and mobile hole charge on the p-side of the junction follows as qNA, where q is the magni-

tude of electron charge2. On the n-side of the junction, the concentration of mobile electron

charge is qND. When the diode in question is connected into a circuit that establishes vd(t) > 0,we say that the diode is forward biased. Under this biasing condition, the injection of free elec-

trons from the n-type region, across the semiconductor junction, and into the p-type volume isencouraged. Of course, hole injection from the p-side of the PN junction -to- the n-side is also

promoted. But in view of the fact that the acceptor impurity concentration is orders of magni-

tude smaller than the donor impurity concentration, the injected charge observed across a for-

ward biased PN junction is dominated by electrons. This electron dominance arguably supportshigh-speed signal processing capabilities in a diode since the mobility (ability to move in re-

sponse to biasing conditions) of transported electrons is at least twice as large as the mobility of

holes. Since the time rate of change of charge through a cross section constitutes a current andsince electrons carry negative charge, this electron injection manifests a positive diode current;

that is a current whose directional flow mirrors that of the indicated p-region -to- n-region diode

current, id(t). While hole injection is anemic in comparison to electron injection, the positive na-

ture of transported hole charge produces a diode current whose directional flow mirrors thatassociated with electron transport across the semiconductor junction. In short, a diode that oper-

ates in its forward bias regime can conduct potentially large currents because of the simultaneous(and dominant) injection of majority charge (charge due to electrons from the n-type region and

charge due to holes from the p-type volume) across the diode PN junction.

In contrast to forward diode biasing, the reverse biased diode, whose basic operating re-

1One micron (1 m)is equivalent to 104centimeter (104cm).2The magnitude of electron charge, q, is q = 1.6(1019) coulombs.


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quirement is vd(t) < 0, moves free electrons further away from the junction and toward the metal

contact at the end of the n-type region where positive potential prevails. The negative diodejunction voltage also displaces free holes away from the junction and toward the metal contact at

the end of the p-type volume where negative potential is witnessed. Reverse biasing is therefore

seen as inhibiting majority charge injection across the semiconductor junction, thereby ensuringcurrent turn off of the diode, in the sense of zero current conduction precipitated by majority

charge transport across the PN junction of the diode. Although zero diode current is usually areasonable approximation under reverse bias conditions, the reverse biased diode current, id(t), is

not quite zero. It is actually nominally constant at the small negative value, Io, where Io isknown as the diode saturationorleakage current.

The foregoing leakage current is manifested by the fact that although the mobile charge

on the p-side of the junction is dominated by holes, a small, but assuredly nonzero, concentration

of electrons prevails therein. Similarly, a small, but nonzero, concentration of holes permeatesthe n-side of the PN junction. These contentions stem from the fact that in the charge neutral re-

gions of a semiconductor volume, the product of mobile hole and mobile electron concentrations

is a constant given by the square of the intrinsic carrier concentration, ni. At a junction tempera-

ture of 27 C, parameter ni is about 10

10

carriers/cm

3

in silicon. While ni is a constant that isindependent of doping concentrations, applied voltage levels, and observed current densities, it

does vary with operating temperature, nominally doubling to quadrupling for every 10 Crise in

junction temperature. Thus, in the p-volume where the hole concentration is essentially theacceptor doping concentration,NA, the electron concentration is a small number proportional to

the square of the intrinsic carrier concentration and inversely proportional to the acceptor doping

concentration. Analogously, we can postulate that in the n-volume, the hole concentration is avery small number (because of the high donor impurity concentration) given by the ratio of the

intrinsic carrier concentration squared to the donor concentration, ND. A studious inspection of

the diode abstracted in Figure (2.1a) for the specific case of vd(t) < 0 confirms the injection

across the PN junction of minority carriers. And as is the case for forward biasing, electrons

dominate minority carrier injection because of the relatively low acceptor concentration in the p-type volume. We further note that the transport of an electron from the p-side of the junction to

the n-side supports current flowing from the n-side to the p-side, as does hole injection across thejunction from the n-side to the p-side. The superposition of the electrical effects of this low level

of minority carrier injection comprises the leakage current, Io, where the minus sign serves toremind us that currentIoflows the wrong way; that is, from the n-side of the junction to the p-

side. In short, diode current id(t)is small (because of the low concentrations of minority carriers

on either side of the PN junction), but nonzero. Specifically, id(t) Iofor values of reverse bi-ased diode voltages that are reasonable in the sense that they do not exceed device breakdown

limitations. For minimal area diodes intended for high-speed signal processing utilization,Ioisas small as several femtoamperes

3. Current Iois proportional to junction area Ae, so that larger

diodes destined for high current applications have proportionately larger Io.

2.2.0. PN JUNCTION DIODE FUNDAMENTALS

Our initial foray into the world of electrical circuits likely initiated with a stipulation ofOhms law, which effectively defined the volt-ampere characteristics of a simple two terminal

resistor. Coalescing this trivial volt-ampere relationship with a few fundamental circuit theory

3Be advised that one femtoampere (1 fA)is equivalent to 1015ampere (1015A).


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concepts, such as the Kirchhoff laws, superposition, and substitution, enabled out computation of

branch currents and voltages established by one or more energy sources applied to memorylessnetworks; that is, circuit structures that are divorced of branch capacitances and inductances. In

somewhat of an analogous fashion, our introduction into the world of electronic circuits is a

disclosure of the low frequency volt-ampere properties of the semiconductor PN junction diode.

This volt-amperecharacteristic equation is more complex than is Ohms law, if for no other rea-

son than the junction diode is an inherently nonlinear branch element. When coupled with clas-sic circuit theories and such fundamental semiconductor concepts and issues as the junction

depletion region, junction transition capacitance, diode diffusion capacitance, built-in junctionpotential, and charge transport and injection, the diode volt-ampere model enables the creative

exploitation of the PN junction diode in numerous applications, inclusive of those circuits and

systems that boast memory elements. It also helps to forge an insightful understanding of basicsemiconductor principles.

2.2.1. VOLT-AMPERE CHARACTERISTIC

If the diode depicted in either of the two diagrams of Figure (2.1) is inserted in a net-

work that is energized by only low frequency voltages or currents, the diode current, id(t), relatesto the diode voltage, vd(t), in accordance with the nonlinear expression,

d Tv (t) nV d oi (t) I 1 ,e

(2-1)

where the current,Io, is the diode saturation current introduced earlier in the context of diode re-

verse biasing. We note, in fact, that if vd(t)is strongly negative, (2-1) projects id(t) Io, as al-

rea. In (2-1), parameter n, which is known as the junction injection coefficient, is ideally oneand rarely larger than about 1.1. Moreover, the Boltzmann voltage, VT, is

TkT

V ,q

(2-2)

where k = 1.38(1023

) joules/K is Boltzmanns constant, T is the absolute temperature (meas-

ured in Kelvin degrees) of the semiconductor junction, and q is the magnitude of electron charge.At T = 27 C = 300.16 K, VT= 25.89 mV, or about 26 mV. A temperature of 27 Cis tradition-ally adopted as the reference temperature for electronic circuit analyses. It is often referred to as

room temperature, even though 27 Cis equivalent to an uncomfortably warm room held at a

temperature of about 81 F. It can be argued, however, that the internal junction of a diode,which endures the thermal stress of power dissipation imposed by the high current densities con-

ducted by very small junction cross sections, operates at temperatures of at least 10 F-to- 15 F

above room ambient.

2.2.1.1. Static Volt-Ampere Model

Under static (DC) or low signal frequency operating conditions, the time dependenceof the diode voltage, vd(t), in (2-1) can be represented by simply a time-invariant voltage, Vd.

Correspondingly, we can supplant the time domain diode current, id(t), by the constant current

notation,Id. Accordingly, (2-1) assumes the slightly simpler form,

d TV nVd oI I 1 ,e (2-3)which is plotted in Figure (2.2) for representative silicon and germanium PN junction diodes

operated at 27 C. The saturation current,Io, of the silicon diode in Figure (2.2) is taken as 2

fA, while the corresponding saturation current for the germanium unit is 1 A. For both diodes,


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parameter n is taken to be unity. The huge disparity betweenIovalues of silicon and germa-

nium diodes is typical. Observe that in both the silicon and germanium samples, the diode cur-rent remains essentially zero despite positive, but small, values of the diode voltage. For the

subject germanium device, significant current is not observed until its diode voltage reaches a

level of nominally 200 mV, which might be interpreted as a turn onvoltage, say Von, for thedevice. On the other hand, about 700 mVof forward bias turn on is required for observable

current flow in the silicon unit. These differences in the diode voltage commensurate withsignificant current flow can be attributed directly to the differences in the respective saturation

current values of the two diodes.

0

10

20

30

40

50

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Diode Voltage, V d (volts)

DiodeCurrent,Id(mA)

Silicon

Diode

Germanium

Diode

Figure (2.2). The room temperature volt-ampere characteristics of representative PN junction diodesfabricated in silicon and germanium technologies. Both diodes are presumed to project an

injection coefficient of n = 1. The silicon diode has a room temperature saturation current

of 2 fA, while its germanium cousin has a saturation current of 1 A.

We observe further that large diode currents do not require diode voltages that aresignificantly larger than their respective turn on levels. For example, Figure (2.2) confirms that

the difference in voltage of the silicon diode biased at nominally 50 mA and then operated at

roughly 50 mAis less than 100 mV, or only about 14%of the 1 mAvalue of forward diode vol-

tage. In other words, only minutely small increases in diode voltage above the turn on thresholdresult in significant increases in diode current, which is to say that the diode current is a sensitive

function of forward biasing diode potential. What a big surprise in view of the fact that diode

voltage Vd appears as an exponentiation argument in its V-I characteristic equation of (2-3)!Arguably, the forward diode voltage can therefore be viewed as remaining essentially constant at

a value very near to its turn on level for even substantive increases in diode current. In an idea-

lized sense, it might be suggested that the electrical behavior of a forward biased PN junctiondiode approximates the volt-ampere characteristics of a battery whose potential is nominally the

diode turn on voltage, Von.

For both forward and reverse biases, the situation at hand is reminiscent of the biased

electrical switch diagrammed and characterized in Figure (2.3). In particular, the switch in thesubject figure is open, whence switch currentIswis zero, for VswVon. On the other hand,Iswex-


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ceeds zerofor the closed switch condition that delivers Vsw= Von. This switch emulation can be

refined to account, albeit to first order, for the finite slope of the diode forward characteristic. Tothis end, consider Figure (2.4), which depicts a typical volt-ampere characteristic of a diode,

regardless of the semiconductor technology exploited in its fabrication. Assume that the diode in

question is inserted into a circuit that allows a maximum diode current of Idm to flow. From (2-3), this current corresponds to a forward diode voltage, Vdm, which derives from

V V :

Switch Opensw on

I = 0sw

Von

Vsw

V = V :

Switch Closedsw on

I 0sw

Von

Vsw Vsw

Isw

Von Figure (2.3). Biased switch emulation of the volt-ampere characteristics of a PN junction diode.

dm T dm T V nV V nV dm o oI I 1 I .e e (2-4)

Diode Voltage

DiodeCurrent

I dm

V dmV on

Actual Diode

Characteristic

Piecewise Linear

Characteristic

Figure (2.4). Piecewise linear approximation of the forward PN junction diode volt-ampere characteristic.

The approximation in (2-4) reflects the fact that the exponential term in the bracketed quantity issignificantly larger than unity when the diode is forward biased. Let a straight line be passed

through the coordinate, (Idm, Vdm), such that the subject line is tangent to the diode characteristic

at (Id, Vd) = (Idm, Vdm). This slope, which has units of conductance, can logically be designated


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by the inverse resistance notation, 1/rdm. Returning to (2-3),

d T dm T

d dm d dm

V nV V nV d o o dm

dm d T T T V V V V

dI I I I 1,

r dV nV nV nV

e e

(2-5)

whence

Tdmdm

nVr .I

(2-6)

We see that (2-6) projects a diode resistance that is nominally inversely proportional to the diodecurrent. It therefore becomes progressively smaller as diode currents increase. Indeed, it is quite

small for both moderately large and relatively small currents owing to the fact that the numerator

term, nVT, on the right hand side of (2-6) is only about 26 mVat room temperature. Physically, it

represents the ratio of an incrementally small change, Vd, in the diode voltage -to- an incremen-

tally small perturbation, Id, in corresponding diode current in the immediate neighborhood ofthe volt-ampere coordinate, (Idm, Vdm).

It follows that the equation of the straight line passed through the aforementioned volt-

ampere coordinate is

d dm d dmdm

1I I V V .

r (2-7)

The straight line intersects the horizontal axis in Figure (2.4) at Vd = Von. From our earlier

discussion, voltage Vonmeaningfully approximates the turn on voltage of the diode. SinceId= 0

in the context of Vd= Von, (2-7) and (2-6) provide

on dm dm dm dm T V V r I V nV . (2-8)

It is both interesting and enlightening that the turn on voltage differs from the operating voltage,

Vdm, of interest by a mere 26 mV or so at room temperature, which attests further to the pro-nounced sensitivity of diode current to forward diode voltage. In a word, the steeper the volt-

ampere characteristic curve evidenced in a PN junction diode biased strongly in its forward re-

gime, the smaller is the static diode resistance, rdm, and the closer the operating voltage, Vdm, liesto the turn on voltage, Von.

It should be understood that (2-7) is a meaningful approximation of the volt-ampere

characteristics of a PN junction diode only if the diode voltage, Vd, is at least as large as the volt-

age, Von, given by (2-8). Otherwise, the diode current is taken to be zero, which is an analyticaltack that ignores the small diode currents flowing for small positive values of diode voltage, as

well as the leakage current manifested when the diode is reverse biased. These pronouncements

lead to the piecewise linear approximation sketched in Figure (2.4), wherein a zero slope line,coincident with the voltage axis, intersects a second line, defined analytically by (2-7), at Vd=

Von. Specifically, the piecewise linear diode curve approximation is

d on

ddm d dm d on

dm

0, for V V I .1

I V V , for V Vr

(2-9)

Since each constituent of (2-9) is a linear equation, each can give rise to only a linear equivalent

circuit for the diode undergoing study. The equivalent circuit for the case of Vd< Vonis trivial in

that it is merely an open circuit, as shown in Figure (2.5a). With VdVon, a constant current

source, a constant voltage source, and a resistance are required as we diagram in the topologyoffered as Figure (2.5b).


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Id

Vd

V Vd onI = 0d

Vd(a).

Id

Vd

V Vd onId

Vd(b).

Vdm

Idm

rdm

Figure (2.5). Piecewise linear, low frequency model of the PN junction diode. (a).The piecewise li-

near (open circuit) model corresponding to a diode voltage that is at most equal to the

turn on voltage of the device. (b).The piecewise linear model for a diode voltage that is

at least the diode turn on voltage.

EXAMPLE #2.1:

In the simple circuit of Figure (2.6a), the PN junction diode, D, is fabricated insilicon semiconductor technology and operates at room temperature. Its static

volt-ampere curve is characterized by a junction injection coefficient of n = 1.03and a saturation current ofIo= 4.2 fA. The battery voltage applied to the circuit

is Vdd= 3 V, while the load resistance is Rl= 150 . Use the piecewise linearmodel of a diode to calculate the diode current,Id, the diode voltage, Vd, and the

load voltage, Vl.

Vdd

Rl

D

Vd

IdVl

(a).

Id

Vd

Vdm

Idm

rdmRl

Vl

Vdd

(b). Figure (2.6). (a).Circuit addressed in Example (2.1). (b).The piecewise linear model of the circuit in (a).

SOLUTION #2.1:(1). We can see that the supply voltage, Vdd, serves to forward bias the diode in the circuit at hand.

Accordingly, our applicable model is the structure in Figure (2.5b), which supplants diode D

in the circuit of Figure (2.6a) to establish the network given in Figure (2.6b). The practical

use of this model begins with estimating the maximum possible current, Idm, which the diode

in Figure (2.6a) can conduct. A reasonable first guess of this maximum current is Vdd/Rl,

which is the current that would flow if the diode were to behave as a short circuit; that is, a

diode boasting zero turn on voltage and infinitely large slope of the forward volt-ampere

characteristic curve. But PN junction diodes have a nonzero turn on voltage commensurate


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with the conduction of measurable current. In silicon, a reasonable approximation of the di-

ode turn on voltage is Von= 700 mV. Thus, a better second guess as to the maximum current

the circuit of Figure (2.6a) is capable of conducting is

dd ondm

l

V VI 15.33 mA .

R

(E1-1)

The diode voltage, Vdm, corresponding to this estimated maximum current is, from (2-4),

dm dmdm T T

o o

I IV nV 1 nV 771.32 mV .

I Iln ln

(E1-2)

The unity term within the parenthesized factor on the right hand side of the last equation can

be ignored comfortably in that Idm>> Io. Finally, (2-6) gives for the series diode resistance,

rdm,

Tdm

m

nVr 1.74 .

I (E1-3)

(2). Kirchhoffs current law applied to the equivalent circuit in Figure (2.6b) yields a diode cur-

rent,Id, which satisfies

dd dm l d

d dm dm

V V R I I I ,

r

(E1-4)

whence

dm dd dmd dm

dm l dm l

r V VI I 14.86 mA .

r R r R

(E1-5)

The load voltage, Vl, follows as

l l dV R I 2.23 V , (E1-6)

while the corrected diode voltage, Vd, is necessarily equal to

d dd l V V V 770.51 mV . (E1-7)

ENGINEERING COMMENTARY:

Two noteworthy points surface from this example. The first of these is that while the piece-wise linear model is an approximation in that it effectively supplants the forward bias diode

volt-ampere curve by a straight line, it nonetheless is capable of reassuringly accurate

computations. In the present case, a computer-based simulation of the network in Figure

(2.6a) reveals a diode voltage of 770.49 mV, which differs from the computed result by a

miniscule 0.002%. The same simulation confirms that the computed current is high by only

0.015%. These errors are orders of magnitude smaller than analytically deduced response er-

rors accruing from the ramifications of routine processing and manufacturing uncertainties.

Of course, the accuracy of the response results hinges largely on the estimate adopted for the

current,Idm. In particular, the closerIdmis to the actual diode current, the more accurate are

the computed results. Equation (2-7) supports the last contention, for in the limit ifIdmis esti-

mated to be the actual diode current,Id, Vdmfollows as the actual diode voltage, Vd.

The second point is that the obviously nonlinear electronic circuit at hand has been analyzed

accurately with the help of only an approximate, linear circuit model. Of course, the vehicle

fostering the propriety of this linear approximation is the applied battery voltage, Vdd, which

forces the PN junction diode to operate above its turn on threshold potential and therefore,

away from the nonlinearities implicit to the neighborhood of the turn on voltage. The lesson

worthy of learning is that traditional linear circuit theory and analytical techniques remain

crucially important to an analytical assessment of electronic networks, despite their inherent

nonlinearity. In other words, once we formulate a meaningful and realistic model for the


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nonlinear element (or nonlinear elements in more complicated networks), electronic circuit

investigations collapse to sophomoric circuit analytical undertakings.

2.2.1.2. Diode Saturation Current

The saturation current parameter,Io, has at least two important properties that influencedesign practices for electronic networks exploiting BJTs, MOSFETs operating in their subthre-

shold regimes, and, of course, PN junction diodes. The first of these characteristics is thatIois

directly proportional to the cross section area, Ae, of the semiconductor junction. Consider, forexample, the network in Figure (2.7a), which offers a static circuit wherein two PN junction dio-

des are connected in parallel with one another. We assume that the two diodes are physically the

same except for the fact that the junction cross section area of diodeD2is larger than that of di-

odeD1by a factor of k. Because of the shunt interconnection of the two diodes, the same staticvoltage, Vd, forward biases the two PN junction diodes. If the passive series resistances (termed

ohmic resistances) associated with the charge neutral regions of the p-type and n-type layers of

the diodes are their stereotypical small values, the voltage drops incurred across these resistancesby diode currents are significantly smaller than the internal junction voltages of the diodes. This

means that if the parasitic diode resistances are small, the indicated terminal voltage, Vd, is effec-

tively the forward biased junction voltage of either diode. But from (2-3), if diodeD2, whosejunction area is scaled k-timesupward from that of diode D1, shares the same internal junction

voltage that is experienced by D1, diode D2necessarily conducts a current that is k-times the

current flowing throughD1. In effect, diodeD2mirrors the current ofD1through a scale factor

of k. Resultantly, the applied voltage source, Vdd, must supply a current that is a factor of (k +1)-

timeslarger than the current conducted by diode D1. Of course, the circuit in Figure (2.7a) per-

forms similarly to the single diode circuit in Figure (2.7b), where the lone diode, D3, is fatter

than diodeD1by a factor of (k + 1).

Vdd

R

D1 D2

x k

kId1Id1

Vd(k+1)Id1

(a).

Vdd

R

D3

x (k+1)

(k+1)Id1

Vd

(k+1)Id1

(b).

x 1

Figure (2.7). (a). A circuit that has two PN junction diodes connected in shunt with one another. The

junction area of diodeD2is k-timeslarger than the junction cross section area of diode D1.

(b).The single diode equivalent of the circuit in (a) is which the utilized diode, D3, has a

junction area that is (k +1)-timeslarger than the junction area of diodeD1.

The fact that the larger area diode conducts larger current is hardly revolutionary. Forexample, prudence dictates that we use progressively thicker wire to conduct larger currents. By

selecting a wire with large cross section area, the net resistance through which current is allowedto flow is reduced. And in addition, the heat generated by any power that happens to be dissi-

pated in this resistance is effectively spread over a larger area, thereby reducing the thermal

stress per unit of cross section area. In a word, we attempt to do with both diodes of increasedcross section area and fatter wires is to sustain reasonable current densities, thereby allowing the

power dissipated by the conducting current to be distributed over wider cross section areas.

The second important property of diode saturation current Iorelates to junction operat-


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ing temperature. In contrast to the designable advantage of saturation current proportionality to

junction area, Ioexudes a troublesome temperature sensitivity that is the bane of integrated cir-cuit designers. CurrentIois, in fact, the leakage current that flows through a reverse biased di-

ode. As we argued earlier, this leakage derives from minority charge carriers injected across the

diode junction. Because the concentration of these minority carriers is directly proportional tothe square of the intrinsic carrier concentration, which rises dramatically with increases in junc-

tion operating temperature, the leakage current is itself strongly dependent on junction operatingtemperature. Accordingly, diode currents exude a positive temperature coefficient. In particular,

diode currents tend to increase over temperature, because of the dependence of current parameterIoon junction temperature.

The mathematical nature of the temperature dependence ofIocan be addressed empiri-

cally by the expression,

oT T 10o o oI (T) I (T ) P ,

(2-9)

which asserts that the diode saturation current, Io(T), at an arbitrary absolute operating tempera-ture of Tincreases from its reference value of Io(To)by a factor ofPfor each 10 Crise in junc-

tion temperature. Absolute reference temperature Tois generally taken to be 27 C,Io(T), andPis an empirical parameter ranging typically from 2-to- 5in silicon devices. Thus, for example, if

P = 3.5, the leakage or saturation current, Io(T), at a temperature that is 50 C above roomtemperature is a factor of 525.2larger than its reference temperature value. In the context of the

circuit in Figure (2.7a), suppose that the voltage, Vd, is maintained constant over a 50 C in-crease. In the face of constant diode junction voltage, the resultant current conducted by resis-

tanceRin Figure (2.7) would increase by a factor of 525.2. This means that the supply voltage,

Vdd, and thus, the power it must supply to the circuit, needs to be increased commensurately overtemperature to sustain the desired reference temperature value of the current flowing through the

shunt-connected diodes. This simple example indicates that maintaining constant diode voltage

in the face of junction temperature rises is an imprudent design option.

The disconcerting aspect of the foregoing example is that while a 50 Crise in junctiontemperature is large, it is a commonly encountered increase when small geometry devices are

compelled to conduct moderately large currents. In particular, the high current densities expe-rienced by small profile diodes incur self heating of their respective geometries. As we alluded

to earlier, this state of affairs encourages the use of relatively large geometry diodes when

correspondingly large currents must be conducted. Large cross section areas reduce current

densities, which in turn mitigate junction temperature rises incurred by unavoidable self heating.Just as a high gauge (small cross section area), extension cord becomes dangerously hot when it

connects a power hungry appliance, such as a toaster or space heater, to a power source, smalldiodes conducting large currents can likewise be damaged by the internal heat manifested by the

junction power dissipation associated with large current densities.

A casual inspection of (2-3) suggests the need of appropriately decreasing the diode

voltage to compensate for temperature-induced increases in the saturation current parameter, Io.In order to ascertain the voltage decrease commensurate with sustaining constant diode current in

the face of temperature rises, let us assume a PN junction diode energized by a static voltage that

secures strongly forward biased operation. In this case, the exponential term on the right handside of (2-3) overpowers the unity term in the bracketed quantity so that at temperature T,

d TV (T) nV d oI (T) I (T) ,e (2-10)


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while at reference temperature To,

d o ToV (T ) nV d o o oI (T ) I (T ) ,e (2-11)

where

oTo

kTV

q (2-12)

is the reference temperature value of the Boltzmann voltage, VT. Since our present goal is todetermine the amount by which the forward bias diode voltage, Vd(T), must decrease from itsreference value, Vd(To), to ensure that the diode current at temperature Tis nominally the same as

the current at temperature To, Id(T)/Id(To) = 1. Accordingly, if we divide (2-10) by (2-11) and

subsequently use (2-9), we find that

T 10 d d oT To

V (T) V (T )1 P ,

nV nV exp

(2-13)

where

oT T T (2-14)

is the temperature change experienced by the diode junction. Equations (2-12) and (2-13) and a

bit of algebra invoked on the bracketed term on the right hand side of (2-13) allows us to write

T 10 d d oT o

1 T1 P V V (T ) ,

nV Texp

(2-15)

with

d d d oV V (T) V (T ) (2-16)

understood to represent the required change in diode voltage. Upon taking the natural logarithm

of both sides of (2-15),

1 10 1 10d d o d oT Too o o

V V (T ) V (T ) TnV P nV P

T T T T ln ln

(2-17)

is readily demonstrated.

The quotient, Vd/T, in (2-17) is the average temperature rate at which diode voltage

Vdmust change to preserve constant diode current over the temperature increment, T. Observethat this average rate of voltage change is a linear function of operating temperature, T. Note

further that even if P = 1, which implies the highly idealized condition that saturation currentIo(T) is not influenced by temperature, the temperature dependence of the Boltzmann voltage

renders Vdnonzero and indeed positive for T > 0. For P > 1, the second term on the right

hand side of (2-15) is invariably larger than the first term so that Vd/Tis a negative number in

units of volts -per- C. The typical magnitude of this voltage sensitivity for T > 0 and for sili-con PN junction diodes is in the range of 1.0 mV/C-to- 3 mV/C.

Figure (2.8) portrays the pronounced temperature sensitivity of the static volt-ampere

characteristic of a PN junction diode. The curves displayed plot (2-3) for a 27 Csaturation cur-

rent value ofIo(To) = 2 fA, a junction injection coefficient of n = 1, and a temperature parameterofP = 3.5. The plot clearly conveys the necessity of decreasing junction voltage Vdif the diode

current, Id, is to be maintained nominally constant in the face of increasing junction operating

temperatures. We also offer Figure (2.9) as a companion to Figure (2.8). The latter plot dis-

plays, as a function of junction temperature, the temperature sensitivity of diode voltage, Vd/T, which is required to sustain nominally constant diode current despite increases T in the


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junction operating, T. The plot takes n = 1, To= 27 C, and a reference temperature diode vol-

tage of Vd (To) = 730 mV. As expected, these curves project temperature linearity with a negativeslope.

0

10

20

30

40

50

60

70

80

90

100

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Diode Voltage, V d (volts)

DiodeCurrent,Id(mA)

T = 27 C

T = 75 CT = 125 C

Figure (2.8). The volt-ampere characteristic of a PN junction diode for three values of

junction temperature. The considered diode has n = 1, a reference tempera-

ture, To, of 27 C, a saturation current,Io(To), at the reference temperature of2 fA, andP = 3.5.

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

-25 -15 -5 5 15 25 35 45 55 65 75 85 95 105 115 125

JunctionVoltageSensitivity

(mV/C)

Junction Temperature ( C)

Figure (2.9). The temperature rate of diode voltage change required to preserve nomi-

nally constant voltage change in light of the strong sensitivity of diode

saturation current to junction operating temperature.

EXAMPLE #2.2:

In the circuit of Figure (2.7), diode D1 has a saturation current, Io, of 5 fA atroom temperature. DiodeD2is identical toD1except that its junction area is 5-

timeslarger than that ofD1. Both diodes have a junction injection coefficient of

n = 1and a temperature factor ofP = 4. The applied voltage, Vdd, is 6 voltsand


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supplies 18 mAof current to the circuit. What is the required value of resistance

R, and what is the average rate at which Vddmust decrease if the current supplied

by Vdd is to remain 18 mA when the diodes operate at 100 C. Assume that

temperature-induced changes in the value of circuit resistance are negligible.

SOLUTION #2.2:

(1). If the supply voltage provides 18 mAof current to the circuit of Figure (2.7), and if diode D2

has a junction area that is 5-timesthe area ofD1 (k = 5), diodeD1must conduct a current of

Id1= 18 mA/(k+1) = 18 mA/6 = 3 mA . From (2-3), the voltage, Vd (To), developed across ei-

ther diode is, forId1(To) = 3 mA, Io(To) = 5 fA, n = 1, and To= 27 C,

d1 od o To

o o

I (T )V (T ) nV 1 702.11 mV .

I (T )ln

(E2-1)

(2). Kirchhoffs voltage law applied to the lone circuit loop in the subject figure yields

dd d o

d1 o

V V (T )R 294.33 .

( K 1)I (T )

(E2-2)

(3). Since resistanceRboasts ideally zero temperature coefficient, the decrease in supply voltage

must match the decrease in the diode voltage commensurate with maintaining the constant

current that the voltage source attempts to supply. With Vd(To) = 702.11 mV, P =4, To =

300.16 K, and a maximum operating temperature of T = 100C = 373.16 K, (2-17) pro-

vides

1 10dd d d o To

V V V (T )nV P 2.123 mV/C .

T T Tln (E2-3)

This computation implies that over the temperature range of 27 C -to- 100 C, the supply

voltage must diminish by (2.123)(100 27) = 154.96 mV, which amounts to a decrease of

2.58%.

ENGINEERING COMMENTARY:The bad news is that the diode currents are strongly sensitive functions of junction operating

temperatures. But the good news is that diode currents are strongly sensitive functions of di-

ode voltages to which they are exponentially dependent. Since a diode current is impacted by

temperature largely through the saturation current parameter to which a diode current is

directly proportional, only small changes in diode voltage are necessary to compensate for the

ramifications on saturation current of increasing junction operating temperature. In the

present example, less than a 2.6%drop in applied voltage stabilizes the current supplied by

the power supply in the circuit of Figure (2.7). The real engineering challenge, however, is

designing the voltage source so that it can sense temperature-induced current changes and

track automatically with requisite changes in diode voltage.

2.3.0. PHYSICAL CONSIDERATIONS OF THE PN JUNCTION

In addition to having presented the static volt-ampere characteristics of the PN junction

diode, several engineering issues surrounding these characteristics have been addressed. In-

cluded among these issues are diode conduction requirements, a circuit level assessment of thesensitivity of diode characteristics to junction temperature, and a piecewise linear model that

enables a systematic and straightforward analysis of PN junction diode circuits. Armed with a

comprehension of these issues and their associated mathematical tools, we are arguably prepared


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to analyze networks that utilize PN junction diodes and perhaps even to design simple diode

subcircuits that support particular system performance requirements. But a prerequisite for in-sightful analyses and innovative circuit design, and particularly integrated circuit design, requires

more expertise than that associated with our limited awareness of only fundamental device

properties. In effect, the understanding we have gleaned thus far has provided us with only thetraining wheels that can initiate our guarded travel along the road to innovative and creative de-

sign of reliable electronic networks. These trainers can be discarded when manual analysis suc-ceeds in meaningfully illuminating the performance attributes and shortfalls of considered cir-

cuits to the appropriate physical properties of the active devices and their interconnections. Tobe sure, many of the performance characteristics of electronic circuits may be explained in terms

of only the engineering fundamentals of semiconductor devices and their peripheral branch ele-

ments. Unfortunately, practical electronic shortcomings often ensue from the electrical effects ofsecond order, and generally undesirable and unwelcomed, phenomena that are rarely rendered

visible by discussions that are limited to only basic volt-ampere properties. At a minimum, a

consideration of these higher order effects adds realism to, and enhances our understanding of,the fruits of basic analysis. In other circumstances, the creative exploitation of these high order

phenomena can mitigate at least some of the shortfalls revealed by first order circuit analyses.

An examination of the higher order phenomena pervasive of PN junction diodes boasts the addi-tional advantage of facilitating our engineering understanding of the operation of more compli-

cated electronic devices.

The discovery of relevant high order phenomena that loom potentially significant to cir-

cuit performance compels a discussion of the physical characteristics of the semiconductor di-ode. In an attempt to forestall intimidation among those who are myopically focused on elec-

tronic circuits, as opposed to physical electronics, the discussion that follows exploits only the

essential mathematics that quantify the physics of semiconductor device operation. These

mathematical considerations are complemented by engineering discourse, which is directed to-ward assimilating the circuit level impact of relevant physical phenomena. Take comfort in the

fact that this author can hardly be viewed as an expert on semiconductor physics.

2.3.1. PN JUNCTION AT EQUILIBRIUM

Before beginning a quantitative discussion of the physical properties of a PN junctiondiode, a few basic qualitative observations prove useful. First, we can easily comprehend that a

semiconductor diode that is merely placed on the laboratory bench without any connection to a

voltage source or other source of energy cannot conduct current. In other words, if the terminals

of a diode are left open circuited, the diode is incapable of supporting current flow. But second,we may encounter difficulty understanding that this zero conduction state is seemingly

inconsistent with our elementary picture of a diode that simply projects a p-type region abutting

an n-type volume at something we have called the PN junction. In particular, there is a bunch of

positively charged holes in the p-type volume, while more bushels of negatively chargedelectrons pervade the n-type region. Since positive and negative mobile charges mutually attract

one another, presumably nothing hinders an n-volume electron from attracting a p-region holeacross the junction. Conversely, any p-region hole can encourage the transport of an n-region

electron across the junction and into the p-region. We perceive this seemingly innocent picture

as physically sound until such time that we remember that charges piercing a cross section ofsemiconductor or any other type of potentially conductive material comprise flowing electrical

current. But since the PN junction diode we are envisaging is not connected to any closedelectrical path, Kirchhoff insists that no such current flow is possible.


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The quandary with our initial vision of a PN junction diode is that the proximate

location of holes and electrons in the neighborhood of a geometric PN junction is that currentflow is physically encouraged in a Kirchhoff electrical environment that absolutely prohibits

such flow. Now, we are logically forced to conclude that some type of phenomenon takes place

in the immediate neighborhood of the PN junction to separate holes and electrons, therebyprecluding their mutual attraction. In the absence of this attraction, transport across the junction,

whence current flow through an open circuited PN junction diode is inhibited. A vague clue asto the nature of this requisite higher order phenomenon is offered by our fundamental

understanding that the presence of nonzero charge manifests an electric field, which whenappropriately directed, can hinder mobile charge transport over the region in which the field

prevails. The subsequent paragraphs consider this phenomenon and in the process, our

perception of the physical nature of the junction region of a diode is widened.

Our quantitative discussion commences with an investigation of the PN junction diodeat equilibrium; that is, the diode in Figure (2.1) is operated with no voltage, light, or other form

of external energy applied to it. The diode is effectively open circuited and consequently, zero

diode current, id(t), prevails. As alluded to earlier in this chapter, diode current is comprised of a

superposition of currents precipitated by hole and electron injection across the PN junction.Recall that holes injected from the p-type region to the n-type layer and electrons injected in the

opposite direction from the n-type region to the p-type layer combine to give rise to a currentflowing from the p-side of the junction to the n-side. Accordingly, the zero current equilibrium

condition demands that the current components due to both hole and electron injection across the

junction be shut down.

The hole current, sayIdp, is, in turn, a superposition of drift and diffusion components,which we denote herewith asIpdriftandIpdiff, respectively. In particular,

dp pdrift pdiff I I I (2-18)

where the drift component of hole current is

pdrift e pI A qp(x) (x) (x), E (2-19)

and the hole diffusion current is

pdiff e T pdp(x)

I A qV (x) .dx

(2-20)

We note that if the hole concentration gradient, dp(x)/dx, is negative, we witness a positivediffusion current, Ipdift. In the last two expressions, Ae, q, and VTare the previously introduced

junction cross section area, electron charge magnitude, and Boltzmann voltage, respectively. In

addition, p(x) is the concentration of mobile holes as a function of position x in Figure (2.1a),

p(x)symbolizes hole mobility (in units of cm2/volt-sec), and E(x)is the electric field intensity

(in units of volts/cm) prevailing at positionxin the subject diagram. The hole mobility (as wellas the electron mobility) is dependent on position x by virtue of the fact that mobility is a

function of carrier concentration, which varies with positionx. Since the hole current component

is zero at equilibrium, (2-18) through (2-20) confirm a requisite internal electric field intensity of

T

d p(x)(x) V .

dx

lnE (2-21)

In other words, a nonzero electric field must prevail to establish equilibrium. This field is

proportional to both junction temperature and the slope of the natural logarithm of the holeconcentration. As we demonstrate subsequently, the subject electric field is dominantly confined

to the immediate neighborhood of the junction.


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Analogously, the electron component,Idn, of diode current is expressible as

dn ndrift ndiff I I I , (2-22)

where the drift component of electron current is

ndrift e nI A qn(x) (x) (x), E (2-23)

and its diffusion component is

ndiff e T ndn(x)

I A qV (x) .dx

(2-24)

In (2-23) and (2-24), n(x)represents the free electron concentration, while n(x)is the position

dependent electron mobility. The last three relationships confirm that zero electron currentmandates an electric field intensity of

T

d n(x)(x) V .

dx

ln E (2-25)

The last result and (2-21) combine to stipulate that diode equilibrium is achieved when the slopesof the logarithmic profiles of the hole and electron concentrations adjust to satisfy the constraint,

d p(x) d n(x).dx dx

ln ln

(2-26)

2.3.1.1. Junction Transition Region

The simplicity of the equilibrium condition in (2-26) masks important engineering an-

swers to questions as to how an electric field that is internal to the PN junction diode can beformed to stop charge transport across the junction when no energy is applied to the diode. This

question is best addressed by examining the nature of the free carrier profiles in the PN junction

device of Figure (2.1a). To this end, assume that the p-type region is doped uniformly to anacceptor carrier concentration of NA

4. It follows that on the p-side of the junction and suffi-

ciently removed from said junction, the hole concentration is p(x) NA, assuming complete

ionization of the introduced acceptor impurity atoms. But on the n-side of the junction and farenough away from the actual PN junction, where holes are minority carriers,p(x) ni

2/ND, where

niis, of course, the intrinsic carrier concentration. The resultant hole profile is sketched in Fig-

ure (2.10a), which clearly shows that in the vicinity of the junction, the hole concentration

necessarily transitions from the large value evidenced for on the p-side of the junction for x Xno. Al-

though the transitional curve in Figure (2.10a) is delineated as a straight line, its actual

mathematical nature is an error or Gaussian function, depending on the nature of the dopingmethodology adopted in the fabrication process. The boundaries of the junction transition region

defined by Xpox Xnoare also dependent on the device fabrication scenario but in general,

the equilibrium width, Wo, of the transition region is rarely more than a half micron or so. This

declaration, coupled with the fact thatNAcommonly exceeds ni2

/NDby several orders of magni-tude implies that the slope of the transitional hole profile in the immediate vicinity of the PNjunction is both negative and very large in magnitude. It follows from Figure (2.10a) and (2-21)

that an electric field, E(x), which is established in association with the hole profile, is

commensurately large and negative near the junction and is essentially zero far enough away

from the junction. A large electric field derives from the steepness of the free hole profile in thejunction transition region, while a negative field, which implies a field directed from the n-side

4Uniform doping can be achieved through ion implantation of the dopant.


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of the junction to the p-side, accrues because of the obviously negative slope of the hole profile.

As such, this internally established built-in field exerts a force on free holes that effectivelyconfines them to the p-type volume of the PN junction diode. More about these electric fields is

offered subsequently.

x

p(x)

Xpo Xno

NA

n /Ni D2

0

(a).

x

n(x)

Xpo Xno

ND

n /Ni A2

0

(b).

W = X +X o no po

Figure (2.10).(a). Simplified depiction of the free hole profile of a PN junction diode in

equilibrium. (b).Simplified diagram of the free electron profile of a PN junc-

tion diode in equilibrium.

Analogously, the free electron profile, which we sketch in Figure (2.10b), varies from a

constant of ni2/NAforx < Xpoto a high value of approximatelyNDforx > Xno. As in the case of

the free hole curve, a straight line is adopted to represent the electron profile in the junction

transition domain. Since the slope of the indicated straight line is large and positive, (2-25) sug-gests a proportionately large and negative equilibrium electric field in the neighborhood of the

junction, while the field in regions removed from the junction approach zero. Again, more about

these electric fields is offered subsequently.

In order that a negative electric field, E(x), be established in the neighborhood of thePN junction to inhibit hole/electron transport and sustain the equilibrium condition, a positive

charge must prevail to the right of the junction, and a negative charge must be observed to the

left of the junction. This contention stems from the elementary picture of field lines emanating


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from positive charges and terminating on negative counterparts. On either end of the

semiconductor outside of the transition region abstracted in Figure (2.10), the net charge must bezero because in equilibrium, the hole and electron densities are constant, thereby forcing zero

electric field in these domains at equilibrium.

In general, the net charge, say (x), is comprised of free holes, free electrons, ionized

donor atoms, and ionized acceptor atoms. This is to say that D A(x) q p(x) n(x) N N . (2-27)In this simple relationship, p(x)is the concentration of free holes, n(x)represents the concentra-

tion of free electrons, ND symbolizes the concentration of ionized donor atoms (a positivecharge), and finally, NA is the concentration of ionized acceptor atoms (a negative charge). To

the extent that the concentration of donor and acceptor atoms is a constant, independent of

geometrical positionx, the charge concentration signified by the bracketed quantity on the righthand side of (2-27) reflects the net positive charge density at position x. Since (x) = 0outside

the transition region at equilibrium,

D A po nop(x) N n(x) N for x X and x X (2-28)

But on the p-side of the junction where Xpo x 0, there are no donor impurities, and the

concentration of electrons, which are the minority carriers in the p-side of the junction, is minis-cule in comparison to the concentration of ionized acceptor impurities. On the other hand, no

acceptor impurities reign, and the concentration of minority holes is far smaller than the

concentration of ionized donor impurity atoms, on the n-side of the transition layer. Accor-

dingly,

A po

D no

q p(x) N , X x 0(x) .

q N n(x) , 0 x X

(2-29)

From Figure (2.10), it should be noted that p(x) < NAin the transition region immediately to theleft of the junction, while in the transition region to the immediate right of the PN junction, ND>

n(x). This observation is important for it confirms the existence of a net negative charge imme-

diately to the left of the junction and a net positive charge to the right of said junction. In turn,

these negative and positive charges provide the electrostatic foundation to support the negativeelectric field in the transition region (meaning that field lines are directed from the n-side of the

junction to the p-side) that is required to sustain diode equilibrium.

In summary, the transition region that cloaks the PN junction of a semiconductor diode

is the vehicle that establishes electrical equilibrium in a diode that lies dormant as an open cir-cuited device. It accomplishes this feat by establishing a region of net negative charge on the p-

side of the junction, immediately to the left of the physical junction and a similar region of net

positive charge immediately to the right of the junction. These charge regions give rise to anelectric field directed across the PN junction from the n-side -to- the p-side. Although not expli-

citly discussed, this electric field manifests a potential, known as the built-in junction potential,across the junction and polarized positive on the n-side and negative on the p-side. Theestablishment of this voltage derives from the simple fact that voltage potential is little more than

the integrated electric field formed across the region over which the potential is developed. In

turn, the positive potential on the right side of the PN junction inhibits the transport of positively

charged holes from the p-side of the junction -to- the n-side. Moreover, said potential also servesto block negatively charged electrons from coming across the junction to the p-side from the n-

side. And when the transport of both holes and electrons are blocked, no current can flow. It is

important to understand that the condition of zero current in a PN junction diode requires stop-


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ping the transport across the junction of both holes and electrons since holes moving from the p-

side of the junction -to- n-side incur a current flow from p-side -to- n-side as do negativelycharged electrons jogging from n-side -to- p-side.

2.3.1.2. Depletion Approximation

On the p-side of the junction transition region, the observed charge is negative; it is, infact, a very good approximation of a constant (independent of positionx) negative charge. Simi-

larly, the charge to the immediate right of the PN junction is a very good approximation of a con-stant positive charge. We arrive at these approximations because on the p-side of the junction,

the concentration,p(x), of free holes drops dramatically and almost abruptly from its high value

ofNAto its significantly smaller (by several orders of magnitude) value of ni2/NDover a transi-

tion region width, Wo, which is of the order of at most a half micron. Moreover, on the n-side of

the transition region, the concentration of free holes, n(x), likewise falls precipitously as we ap-

proach the PN junction from the right. These observations, coupled with (2-29), strongly suggestthe feasibility of approximating the charge population on the p-side of the transition region

exclusively by the charge associated with ionized acceptor impurity atoms while on the n-side,

the charge population can be represented as the charge associated with ionized donor impurities.Formally, the depletion approximation reflects the presumptions, p(x)


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junction and positive ionic charge on the right side to impede the injection of mobile charge

carriers across the junction. When one side of the junction is doped more intensely than is theother side, the intrusion of the ion-charged transition layer into the more lightly doped side must

be proportionately deeper to insure a sufficient volume for the ionic charge needed to repel the

injection of mobile charge from the strongly doped side. As we noted earlier, PN junction diodesare commonly fabricated with an n-side doped more strongly than the p-side, which therefore

gives rise to a depletion region that intrudes further into the p-side of the junction than into the n-side.

2.3.1.3. Electric Field And Potential In The Depletion Layer

The simple charge model advanced by (2-30) and Figure (2.11) enables a convenientquantification of the electric field intensity and corresponding potential distribution associated

with the equilibrium condition. To this end and in accord with the depletion approximation, Fig-

ure (2.12a) portrays the approximate charge distribution as a function of position x in Figure

(2.1). Gauss law relates the electric field intensity, E(x), to this charge distribution, (x), inaccordance with

[1]

xXpo

Xno0

(a).

qNA

qND

(x)

xXpo Xno0

(b).

E(x)

E(0)

Figure (2.12).(a). Depletion approximation of the charge profile in a PN junction

operated at equilibrium. (b).The electric field corresponding to the

charge profile in (a). The electric field intensity, F(0), at the junction

is given by (2-37).


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s

d (x) (x),

dx

E (2-31)

where sis the dielectric constant of the semiconductor under consideration5. In the region, Xpo

x 0, where (x) = qNA, (2-31) implies

po po

(x) xA

s(-X ) X

qN

d (x) dx ,

E

E

E (2-32)

or

A pos

qN(x) x X ,

E (2-33)

where the indicated limits of integration exploit the previously disclosed fact that at equilibrium,

the electric field is null outside of the transition region; specifically, E(Xpo) = 0. On the n-side

of the junction,

no no(X ) X

D

s(x) x

qNd (x) dx ,

E

E

E (2-34)

which produces, with E(Xno) = 0,

D nos

qN(x) x X .

E (2-35)

Consistency atx = 0between the two electric field solutions in (2-33) and (2-35) mandates

A po D noN X N X . (2-36)

The last result mathematically confirms our previous supposition to the effect that the transition

layer about the PN junction intrudes deeper into the more lightly doped side of the junction than

it does into the heavily doped side. The results postured by (2-33), (2-35), and (2-36) give rise tothe field plot shown in Figure (2.11b), where it is understood that the electric field intensity,

E(0), at the actual junction is

A po D no

s s

qN X qN X(0) .

E (2-37)

We have thus confirmed that the electric field is negative throughout the transition region. Inother words, the electric field is always directed from the n-side of the junction, which is home to

positive ionic charge in the depletion region, -to- the p-side of the junction where negative ionic

charge reigns supreme. A maximum magnitude of field intensity is observed at the actual junc-tion of the PN structure, while zero field prevails in the charge neutral regions outside of the

transition region.

Poissons equation teaches that the presence of an electric field,E

(x), implies the exis-tence of a potential, V(x), such that

dV(x)(x) .

dx E (2-38)

This fundamental relationship can be gainfully exploited to discern the potential, V(Xpo), at thep-side boundary of the transition layer for a diode in equilibrium, and V(Xno), the corresponding

5The dielectric constant of silicon is s= 1.05 pF/cm.


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potential at the n-side boundary of the transition layer. These two voltage metrics allow for the

evaluation of the junction built-in potential, Vj, which is little more than the potential difference,

j no poV V X V X . (2-39)Recall that this built-in potential corresponds to, and arises from, the electric field plotted in Fig-

ure (2.11b).

We return now to (2-21), which delineates the requisite electric field commensurate

with zero hole current. Combining (2-38) with (2-21),

TdV(x) V d p(x) ,ln (2-40)which can be integrated over potential from an arbitrary reference potential of V(Xi)to the poten-

tial, V(x)at any positionxto stipulate

i i

V (x) ln [p(x)]

T

V(X ) ln [p(X )]

dV(x) V d p(x) .ln (2-41)

Upon carrying out the integration implicit to this relationship, we see that the potential, V(x),

necessarily satisfies

i T i

p(x)V(x) V X V .

p Xln

(2-42)

The traditional reference convention sets V(Xi)to zero with the understanding thatXiis the loca-

tion at which the free carrier concentration, p(Xi) in this case, attenuates to the intrinsicsemiconductor concentration, ni. Thus, V(Xi) = 0whenp(Xi) = ni, whence (2-42) becomes

Ti

p(x)V(x) V .

nln

(2-43)

With reference to Figure (2.10a), (2-43) identifies the equilibrium potential, measured with re-

spect to the potential at which the hole concentration profile degrades to its intrinsic value, as a

function of positionxon the p-side of the junction. Atx = Xpo,p(Xpo) = NA, which results ina potential, V(Xpo), at the p-side boundary of the PN junction transition layer of

Apo Ti

NV X V .

nln

(2-44)

Equations (2-43) and (2-44) derive from (2-21), which sets the condition for zero hole

current in a PN junction diode. On the other hand, (2-25) delineates the requirement underlying

zero electron current. If we repeat the preceding analysis and premise it on (2-25), we can easilyconfirm that

T

i

n(x)V(x) V ,

nln

(2-45)

and

Dno Ti

NV X V .

nln

(2-46)

Equation (2-45) defines the potential, referenced to the potential at the position where the free

electron concentration reduces to intrinsic value ni, on the n-side of the PN junction diode.Equation (2-46) gives the specific potential at the n-side boundary of the equilibrium transition


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layer. Upon insertion of this result and (2-44) into (2-39), the junction built-in potential, Vj,

manifested in concert with diode equilibrium is found to be

A Dj no po T 2i

N NV V X V X V .

nln

(2-47)

The built-in voltage, Vj, is a positive number since the dopant concentrations on boththe p-side and the n-side of the PN junction diode are assuredly larger than the semiconductor

intrinsic carrier concentration. This voltage, which is generally of the order of 800 mV -to- 1

volt, is polarized from the n-side transition layer boundary to the p-side transition layer boun-dary. Strangely enough, this relatively large internal voltage is established in a diode that is not

connected into a network that incorporates an energy source.

In an attempt to thwart perceptions of premature author senility, the background physi-

cal issues that attest to the foregoing (and tacitly curious) built-in potential are worthy of review.First, a diode in equilibrium conducts zero current, which requires that both the hole and electron

components of diode current be zero. Zero hole and electron current components require that an

electric field be established in the transition layer. This field is proportional to the logarithmic

slopes of the hole and electron concentrations in the immediate neighborhood of the junction.The electric field in question is directed from the n-side of the PN junction to the p-side. In turn,

this field orientation requires a concentration of positive charge in the n-side transition region

and a concentration of negative charge on the p-side of the transition layer. These chargeconcentrations support the subject built-in junction potential, whose polarization effectively

serves to reverse bias, albeit internally, the PN junction diode. The potential, Vjmight therefore

be thought of as the requisite level of reverse biasing that is just large enough to preclude holeand electron injection, and therefore null current conduction, through the PN junction under

equilibrium conditions.

2.3.1.4. Contact Potential

Adding to the puzzling nature of the built-in equilibrium junction potential, Vj, is the

fact that its direct measurement in the laboratory is all but impossible. A somewhat cavalierexplanation of this measurement dilemma is that any attempt to connect a voltmeter, oscillos-

cope, or other measurement equipment around a non-energized diode inherently destroys the

equilibrium condition that is promoted by the built-in potential. A more satisfying rationale isoffered by the concept of contact potentials.

A contact potential is an unavoidable voltage difference established between two

dissimilar materials that are in electrical contact with one another. These contact voltage drops

are precipitated by differences in the potential energies of mobile electrons on either side of thecontact. Consider, for example, the resistively shunted diode in Figure (2.13), in which the metal

contacts at either end of the diode structure are expressly delineated. Because these metalcontacts, which are typically formed of copper or aluminum, differ materially from the physical

properties of the semiconductor diode, contact potentials, Vpmand Vnmare explicitly established,as we show in the extended diode diagram. In particular, a contact potential of Vpm is forged

from the p-type semiconductor to the metal contact on the left of the device, while a contact

potential, Vnm, is analogously formed from the n-type semiconductor to the metal contact on theright. The charge neutral regions of both the p- and n-sides of the junction support zero voltage

drops because of the zero current equilibrium condition implied by the absence of a biasing

voltage. Even if diode current were to flow, these voltage drops remain near zero because the


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2A

po pos

qNV(x) V X x X .

2 (2-52)

This expression predicts a potential, V(0), at the actual junction, referenced to the potentialevidenced at the position where the free carrier concentration reduces to intrinsic level, which is

given by

2A popos

qN XV(0) V X .

2 (2-53)

On the n-side of the junction, (2-38) and (2-35) combine to offer

no noV (X ) X

Dno

sV (x) x

qNdV(x) x X dx ,

(2-54)

whence

2D

no nos

qNV(x) V X x X .

2 (2-55)

The last result posits a junction potential of

2

A nono

s

qN XV(0) V X .

2 (2-56)

This relationship for the potential at the diode junction must mirror the junction potential defined

by (2-53). If (2-53) is subtracted from (2-56) and if use is made of (2-39) and (2-36), suchconnectivity of potential at the junction forms the basis for positioning the actual boundaries of

the transition region, subject, of course, to the depletion approximation invoked earlier. Inparticular,

s j Dpo

A A D

2 V NX

qN N N

(2-57)

and

s jA Ano po

D D A D

2 VN NX X .

N qN N N

(2-58)

It follows that the equilibrium width, Wo, of the junction transition layer is

s jo no poAD

2 VW X X ,

qN (2-59)

whereNADis the effective impurity concentration,

A DAD

A D

N NN .

N N

(2-60)

We should underscore the observation here that the depletion layer width and the junction builtin potential are inextricably linked. This link is sensible since a larger Vjimplies a larger electric

field at the PN junction, which, in turn, implies more ionic charge on both sides of the junction.

Of course, larger levels of ionic charge require greater physical space to house this charge,whence a larger depletion layer width, Wo.

The positive charge prevailing on the n-side of the transition region in the PN junction

diode in Figure (2.11), the negative charge prevailing on the p-side transition volume of the junc-


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tion, and the finite, nonzero width, Wo, which separates the boundaries of the charge transition

layer combine to conjure visions of an effective parallel plate capacitance that straddles the junc-tion. The plates of this perceived capacitance are not metallic, as is the norm in traditional

capacitances. Instead, they are the heavily doped, and thus low resistivity (which synergizes

with the resistive properties of a metal), p-type and n-type charge neutral regions to whichelectrical contact is made for ultimate circuit applications of the diode structure. In concert with

our parallel plate vision, the capacitance, say Cjo, associated with the charge embedded in thetransition volume of a PN junction diode is

e s s ADjo e

o j

A q NC A .

W 2V (2-61)

This capacitance is commonly referred to as the equilibrium, or zero bias, depletion capacitance

of the PN junction. We note that this capacitance is directly proportional to the area of thejunction, which correctly hints that diodes earmarked for high-speed circuit applications must be

small enough to avoid the voltage response sluggishness implicit to significant charge storage at

the junction. Its value is also largely limited by the impurity concentration of the lightly dopedside of the junction. The latter observation mirrors engineering intuition in that we can view the

depletion capacitance straddling the entire transition layer as the series interconnection of twocapacitances. One of these series capacitances appears from the p-side boundary of the junction

transition layer to the actual PN junction, while the second of the two series capacitanceseffectively connects from the junction to the n-side boundary of the transition layer. We recall

that the net capacitance of a series interconnection of two capacitances is smaller than either of

the series capacitances.

EXAMPLE #2.3:

A certain silicon diode boasts a PN junction injection area of 80 m2, an approx-

imately constant p-side dopant concentration of 1016

atoms/cm3, and an approx-

imate n-side dopant concentration of (6.5)(1018

) atoms/cm3

. For equilibrium androom temperature conditions, calculate the built-in potential of the junction, the

width of the junction transition layer, the maximum magnitude of electric fieldinternal to the junction, and the zero bias value of the junction capacitance. As-

sume the validity of the depletion approximation and take the intrinsic carrier

concentration at room temperature to be (1.5)(1010) atoms/cm

3.

SOLUTION #2.3:

(1). At 27 C, which is 300.16 K, the Boltzmann voltage is, by (2-2), VT= 25.89 mV. From (2-

47), the built-in potential with NA= 1016atoms/cm3, ND= (6.5)(10

18) atoms/cm3, and ni=

(1.5)(1010) atoms/cm3follows as

A Dj T 2

i

N NV V 862.02 mV .

nln

(E3-1)

(2). Using (2-60), we find that the effective diode impurity concentration is

15 3A DAD

A D

N NN (9.985)(10 ) atoms/cm .

N N

(E3-2)

For Vj= 862.02 mV, a silicon dielectric constant of s= 1.05 pF/cm, and an electron charge

magnitude of q = (1.6)(1019) coulomb, (2-59) delivers an equilibrium depletion layer width


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of

s j -5o

AD

2 VW (3.37)(10 ) cm 0.337 m .

qN (E3-3)

(3). For junction cross section area ofAe= 80 m2= (80)(108) cm2, (2-61) yields an equilibrium

transition region capacitance of

e sjo

o

A C 24.95 fF .

W (E3-4)

Since the junction area of an integrated PN junction diode is a designable parameter selected

in accordance with appropriate system considerations, this capacitance is often expressed as a

capacitance density. This density is the metric, Cjo/Ae, which happens to be 31.19 nF/cm2in

this particular case.

(4). From (2-57), the intrusion of the junction transition layer into the p-side of the junction is

s j Dpo

A A D

2 V NX 0.3361 m .

qN N N

(E3-5)

For the n-side intrusion, (2-58) offersA

no poD

NX X 0.000517 m 0.517 nm .

N

(E3-6)

Appealing to (2-37), the maximum magnitude of the electric field intensity, which is ob-

served at the actual PN junction, is

A po D no

s s

qN X qN X(0) 51.22 kVolt/cm .

E (E3-7)

ENGINEERING COMMENTARY:

A notable aspect of this numerical example is the significant electric field intensity that is

established at the junction of the considered diode under equilibrium conditions. The field of

better than 50 kV/cmis comparable to the electric field established by a 12-voltautomobilebattery cable whose positive and negative leads are separated by two and a third microns of

insulation. As might be expected, the field in question is accompanied by a correspondingly

sizeable built-in potential whose magnitude in this case exceeds typical turn on voltages for

silicon diodes by better than 160 mV. Finally, observe that most of the depletion layer ex-

tends into the p-side of the junction, which is doped lighter than is the n-side. Indeed, only

0.267%of the net equilibrium width of the depletion layer intrudes to the n-side of the junc-

tion.

2.3.2. PN JUNCTION AT NON-EQUILIBRIUM

A PN junction diode operated under non-equilibrium conditions has energy applied to it

to incur either reverse bias or forward bias. The diode equilibrium conditions studied in thepreceding section are foundational for studying non-equilibrium junction dynamics. This

contention derives from earlier assertions to the effect that the equilibrium width, Wo, of thejunction depletion region is wide enough to house just the right density of positive and negative

impurity ions to repel the injection of holes and electrons across the junction. Equivalently,

these ionized impurities within the depletion layer establish an electric field that supports a built-in potential, Vj, which serves to reverse bias the intrinsic junction at precisely the level

commensurate with precluding free carrier injection across the PN junction. In a word, the


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manifested by reverse biasing voltage V incurs negligible potential drop in the ohmic, charge

neutral p-type and n-type volumes. Because vd(t) V, the built-in junction potential increasesfrom its equilibrium value of Vjto its reverse biased value of (Vj + V). At least three ramifica-

tions of this enhanced built-in potential are apparent. First, since Vjis the value of the intrinsic

junction reverse bias that is just sufficient to preclude the transport of mobile charge carriersacross the junction in equilibrium, an increase of Vjby the amount, V, serves to preclude aggres-

sively the transport of majority carriers across the junction. But it also serves to attract minoritycarrier electrons from the p-side of the junction to the n-side. Similarly, the increased intrinsic

junction potential corresponds to an increased density of ionic charge within the depletion layerthat encourages minority carrier holes from the n-side to traverse the PN junction.

The second effect of the applied reverse bias is to increase the equilibrium width, Wo,

of the depletion layer to the value, W, delineated in Figure (2.14b). In (2-59), if we replace vol-

tage Vjby its expanded value (Vj+ V), we can deduce a revised value of depletion layer widthgiven by

s j do o

AD j j

2 V V v (t)VW W 1 W 1 .

qN V V

(2-62)

Because of the increased depletion layer width, the zero bias depletion capacitance in (2-61) canbe expected to decrease from its original value of Cjoto its biased value, say Cj, such that

joe sj

do

j j

CA C ;

V v (t)W 1 1

V V

(2-63)

that is, the junction depletion capacitance decreases with increasing reverse bias, largely because

of the increased depletion layer width (or thickness of the parallel plate capacitance fantasizedearlier) incurred by this reverse bias.

The third consequence of the applied reverse bias is increased electric field intensity at

the junction. Referring to (2-37) and (2-57), the revised (reverse biased) junction electric field

intensity, say Er(0), is

AD j dr

s j

2qN V V v (t)(0) (0) 1 .

V

E E (2-64)

This increased field intensity is a concern in that the field intensity at zero bias is, as demon-

strated in Example #2.3, already surprisingly large. As a result, field-induced voltage breakdownis a possibility if the applied reverse bias, V, exceeds the reverse bias voltage rating of the de-

vice[2]

. Such breakdown, which manifests potentially appreciable negative diode current, does

not necessarily imply a catastrophic device failure. Indeed, special purpose diodes known as

Zener diodes, are commonly manufactured to operate expressly in breakdown mode. Thesecomponents deliver significant negative diode currents at a virtually constant reverse bias voltage

that is commonly referenced as the Zener voltage. Indeed, Zener diodes enjoy utility in regulatorapplications for which the system requirement is a nominally constant load voltage in the face of

perturbations in line voltages and effective load resistance. We provide more information

regarding voltage regulation later.

We can sum

Documents

Chapter 2 - Analog Integrated Circuit Design by John Choma