Lecture Notes – 14ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Outline

1 Overview 1

2 Ideal IV Characteristics 12.1 General Equation Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2.1.1 Common-Base Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Common-Emitter Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Visualization of Results for an example pnp BJT . . . . . . . . . . . . . . . . . . . . 42.2.1 Common-Emitter IV Characteristics . . . . . . . . . . . . . . . . . . . . . . . 52.2.2 Common-Base IV Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Non-Ideal Electrostatics 63.1 Base Width Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Punch Through . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Avalanche Multiplication and Breakdown . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3.1 Common Base – Breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3.2 Common Emitter – Breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.4 Current Crowding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.5 Series Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.6 Recombination-Generation Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1 Overview

In the previous lecture, the ideal electrostatics of the BJT were covered. It was shown how thelarge-signal equations can be modeled equivalently using the Ebers-Moll model, which is a usefuldescription for circuit simulations involving BJTs.

In this lecture, we continue our discussion of BJT electrostatics, focusing on non-ideal behavior.The discussion will be similar in scope to that of pn junctions.

2 Ideal IV Characteristics

2.1 General Equation Framework

In the previous lecture, we derived the following expressions for the IV relationships in an idealPNP BJT...

IE = eADB

LBpB0

cosh(

WLB

)sinh

(WLB

) (exp(eVEB

kBT

)− 1

)− 1

sinh(

WLB

) (exp(eVCB

kBT

)− 1

)+ eADE

LEnE0

(exp

(eVEB

kBT

)− 1

)

IC = eADB

LBpB0

1

sinh(

WLB

) (exp(eVEB

kBT

)− 1

)−

cosh(

WLB

)sinh

(WLB

) (exp(eVCB

kBT

)− 1

)− eADC

LCnC0

(exp

(eVCB

kBT

)− 1

)

...where the base current is defined as...

IB = IE − IC

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Lecture Notes – 14ECE 531 Semiconductor Devices

Dr. Andre Zeumault

These expressions can be rearranged to collect terms in front of the exponential pre-factors asfollows:

IE = eA

DE

LEnE0 +

DB

LBpB0

cosh(

WLB

)sinh

(WLB

)(

exp

(eVEB

kBT

)− 1

)− eADB

LBpB0

1

sinh(

WLB

) (exp

(eVCB

kBT

)− 1

)

IC = eADB

LBpB0

1

sinh(

WLB

) (exp

(eVEB

kBT

)− 1

)− eA

DC

LCnC0 +

DB

LBpB0

cosh(

WLB

)sinh

(WLB

)(

exp

(eVCB

kBT

)− 1

)

We also showed that these equations can be equivalently expressed using the so-called Ebers-Mollmodel...

IE = IF0

(exp

(eVEB

kBT

)− 1

)− αRIR0

(exp

(eVCB

kBT

)− 1

)IC = αF IF0

(exp

(eVEB

kBT

)− 1

)− IR0

(exp

(eVCB

kBT

)− 1

)

...with the following new definitions which follow directly for the grouped large signal representa-tion...

IF0 ≡ eA(DE

LEnE0 +

DB

LBpB0

cosh (W/LB)

sinh (W/LB)

)IR0 ≡ eA

(DC

LCnC0 +

DB

LBpB0

cosh (W/LB)

sinh (W/LB)

)αF IF0 = αRIR0 ≡ eA

DB

LB

pB0

sinh(

WLB

)We also defined current gain under two different configurations as follows...

Common-Base (CB) DC Current Gain: IC ≡ αDCIE + ICB0

αDC = αTγ

Common-Emitter (CE) DC Current Gain: IC ≡ βDCIB + ICE0

βDC =αDC

1− αDC

Using the Ebers-Moll model, we can relate the terminal currents in terms of other current compo-nents and gain factors.

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Lecture Notes – 14ECE 531 Semiconductor Devices

Dr. Andre Zeumault

2.1.1 Common-Base Relationships

For the common-base configuration, the collector current–that is, the output current–can be ex-pressed in terms of the emitter current–that is, the input current.

IC = αF IF0

IE + αRIR0

(exp

(eVCBkBT

)− 1)

IF0

− IR0

(exp

(eVCB

kbT

)− 1

)

= αF IE + αFαRIR0

(exp

(eVCB

kBT

)− 1

)− IR0

(exp

(eVCB

kbT

)− 1

)= αF IE + (αFαR − 1) IR0

(exp

(eVCB

kBT

)− 1

)Comparing this with the expression for the collector current that defines the common-base currentgain, it is clear that...

αF ≡ αDC

ICB0 ≡ (αFαR − 1) IR0

(exp

(eVCB

kBT

)− 1

)When the collector-base junction is reverse biased, ICB0 can be approximated as...

ICB0 ≈ (αFαR − 1) IR0(−1)

= (1− αFαR) IR0

...and is expected to be quite small as it is proportional to the reverse saturation current of thecollector-base junction. It can be shown that the quantity αR resembles the DC current gain of aBJT operated under inverted operation...

αR =

eADBLB

pB0

sinh(

WLB

)eA

(DCLCnC0 + DB

LBpB0

cosh(

WLB

)sinh

(WLB

))

=1

cosh(

WLB

)+ DC

DB

LBLC

nC0pB0

sinh(

WLB

)Comparing this expression to the DC current gain under forward active,

αF ≡ αDC =1

cosh(

WLB

)+ DE

DB

LBLE

nE0pB0

sinh(

WLB

)...it is evident that the emitter and collector have effectively switched roles, mathematically speak-ing. Thus, αR must correspond to the DC current gain under inverted operation.

2.1.2 Common-Emitter Relationships

For the common-emitter configuration, the collector current–that is, the output current–can beexpressed in terms of the base current–that is, the input current. The base current is expressed

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Lecture Notes – 14ECE 531 Semiconductor Devices

Dr. Andre Zeumault

as...

IB = (1− αF ) IF0

(exp

(eVEB

kBT

)− 1

)+ (1− αR) IR0

(exp

(eVCB

kBT

)− 1

)This expression can be re-arranged as follows...

IB =

{(1− αF )IF0 + (1− αR) IR0 exp

(−eVEC

kBT

)}exp

(eVEB

kBT

)− ((1− αF )IF0 + (1− αR)IR0)

This form of the equation is more convenient for plotting, since the output voltage is equal to theemitter-collector voltage, VEC . Similarly, the collector current can be expressed in terms of theoutput voltage...

IC =

(αF IF0 − IR0 exp

(−eVEC

kBT

))exp

(eVEB

kBT

)+ IR0 − αF IF0

Using the expression for the base-current, the collector current can be re-written in terms of thebase current as follows...

IC =

(αF IF0 − IR0 exp

(−eVEC

kBT

)) IB + ((1− αF )IF0 + (1− αR)IR0)

(1− αF )IF0 + (1− αR) IR0 exp(−eVECkBT

)+ IR0 − αF IF0

Complicated though they may be, this representation simplifies our ability to write scripts thatperform simple visualizations of expected trends, since the output is expressed in terms of theinput.

2.2 Visualization of Results for an example pnp BJT

In this section, we will plot the common-base and common-emitter characteristics to get a senseof the voltage dependencies of the various terminal currents before discussing non-ideal behavior.For device parameters, the following are assumed:

Emitter: NE = 1× 1018 cm−3

µE = 263 cm2 V−1 s−1

DE = 6.81 cm2 s−1

τE = 1× 10−7 s

LE = 8.25× 10−4 cm

Base: NB = 1.5× 1016 cm−3

µB = 437 cm2 V−1 s−1

DB = 11.3 cm2 s−1

τB = 1× 10−6 s

LB = 3.36× 10−3 cm

WB = 2.5× 10−4 cm

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Lecture Notes – 14ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Figure 1: (left) The base current (input current) and (right) collector current (output current).To generate plots for the base current, the emitter-collector voltage was stepped from 0 to 0.7 Vin 5 steps. To generate plots for the collector current, the base current was stepped in 2.5 µAincrements, starting from zero.

Collector: NC = 1.5× 1015 cm−3

µC = 1345 cm2 V−1 s−1

DC = 34.8 cm2 s−1

τC = 1× 10−6 s

LC = 5.90× 10−3 cm

2.2.1 Common-Emitter IV Characteristics

The common-emitter configuration is perhaps most common due to the fact that common-emitterhas a current gain much greater than 1. In this configuration, the input terminal is the base andthe output terminal is the collector with the emitter at common ground. In order to have gain, thebase current must be smaller than the collector current. We show this in Figure 1 . Notice thedifferent orders of magnitude of the currents flowing. The collector and emitter currents are verysimilar. This is due to a very small base current. The value of the base current is of the order ofnanoamps whereas the collector and emitter currents are a few micro-amps.

Due to the exponential dependence on the bias voltage, BJTs tend to saturate very quickly withapplied voltage. The forward active region of operation therefore closely resembles that of a voltagecontrolled current source. Often, one chooses to bias a BJT with a current rather than a voltage.The current can be varied linearly and usually provides a more straightforward interpretation ofthe device behavior. One can show this by solving for the base current explicitly and expressingthe collector current in terms of the base current rather than emitter-base voltage. (See exercise11.7 in Pierret)

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Lecture Notes – 14ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Figure 2: (left) The emitter current (input current) and (right) collector current (output current).To generate plots for the emitter current, the emitter-collector voltage was stepped from 0 to −1 Vin 5 steps. To generate plots for the collector current, the base current was stepped in 1 mAincrements, starting from zero.

2.2.2 Common-Base IV Characteristics

The common-base configuration does not have gain, but has familiar circuit usage as a buffer. Thatis because the input impedance looking into the emitter terminal is much smaller than that lookinginto the collector terminal. In this configuration, the input terminal is the emitter and the outputterminal is the collector with the base at common ground. We visualize the IV characteristics inFigure 2 . Here it can be seen that the input and output currents are roughly the same, as expecteddue to a DC current gain of approximately unity.

3 Non-Ideal Electrostatics

3.1 Base Width Modulation

When generating the ideal plots in Figure 1 and Figure 2 , it was assumed that the quasi-neutralbase width, W was equal to the physical base width WB. In general, application of a voltage acrossthe junctions leads to a voltage dependent modulation of W–that is, the depletion width extendswith voltage appreciably into the base region. This is especially true when WB is small, of theorder of a diffusion length of electrons (npn) or holes (pnp) in the base. J. Early was the first torecognize this effect and we often refer to it as the Early effect and we can define a correspondingEarly voltage. We can quantify the Early effect by explicitly writing down the expression for thebase width and substituting appropriate values for the depletion region widths. For a pnp BJT,

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Lecture Notes – 14ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Figure 3: Early effect, as indicated using the same BJT parameters as before. The Early Voltageis ≈ −118 V and is defined as the x-intercept of the linear fitted collector current.

this can be written as follows...

W = WB − xne − xnc

The quantities xne and xnc represent the depeletion width extension in the base due to the emitterand collector junction respectively. From simple pn-junction electrostatics, these can be expressedas follows:

xne =1

1 + NBNE

√2ε (φbi,eb − VEB)

e NENBNE+NB

xnc =1

1 + NBNC

√2ε (φbi,cb − VCB)

e NCNBNC+NB

A plot of the Early effect is given in Figure 3 , using the same parameters as before. It can beseen that the collector current is no longer well-saturated with the collector-emitter bias. Instead,there is a quasi-linear rise in the current with increasing collector-emitter bias. For simplicity,the depletion region expanse due to the emitter-base junction was neglected. This is justified inforward active, since the emitter-base junction is forward biased and therefore increases the quasi-neutral base width towards WB. The dashed lines represent liner fits of the collector currents. Thex-intercepts of these fits corresponds to the negative of the early voltage.

VA ≡ The negative of the x-intercept of the linear fitted collector current.

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Lecture Notes – 14ECE 531 Semiconductor Devices

Dr. Andre Zeumault

3.2 Punch Through

Punch-through can be viewed as the logical extension of base-width modulation to the extreme–thatis, when the quasi-neutral base width.

W = 0 (Punch Through)

Since the emitter-base junction is forward biased under forward-active operating conditions, punchthrough is determined by the voltage at which the collector-base junction’s depletion width touchesthe emitter.

xnc = WB − xne =1

1 + NBNC

√2ε (φbi,cb − VCB0)

e NCNBNC+NB

At punch through, the large hole concentration in the emitter is effectively shorted to thecollector. This happens due to the gradual encroachment of the large electric field within thecollector-base region towards the emitter. Eventually, holes in the emitter will become swept intothis field and a very large current will arise, effectively pinning the voltage at a certain value similarto a Zener diode.

3.3 Avalanche Multiplication and Breakdown

The injection of minority carriers in the base is governed by the input current. For the common-base configuration, the input current is the emitter current, for the common-emitter configuration,the input current is the base current.

3.3.1 Common Base – Breakdown

In the common base configuration, the base current is not fixed but can vary to accommodatechanges in emitter or collector currents. In this case, when breakdown causes additional carriersto be introduced into the base, for example, due to impact ionization, these additional carriers areextracted in the form of additional base current. The resulting breakdown characteristics of theBJT in this case are very similar to that of a pn junction.

Breakdown in a CB configuration resembles that of a pn junction diode.

3.3.2 Common Emitter – Breakdown

In the common emitter configuration, the base current is fixed by the biasing configuration andtherefore cannot vary in response to changes in emitter or collector currents. In this case, additionalminority carriers injected into the base necessarily give rise to an additional emitter current. Thenet effect is a perceived increase in the biasing current giving rise to additional carrier injectionacross the emitter-base junction. Thus, carrier multiplication in the collector-base junction leads toadditional carrier injection across the emitter-base junction–and correspondingly higher collectorcurrent–effectively serving as an internal mechanism of positive feedback.

Breakdown in a CE configuration BJT leads to additional current amplification.

This process is shown schematically in Figure 4 . The additional electrons generated due

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Lecture Notes – 14ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Figure 4: 1. Hole injection from emitter→base, and 2. from base→collector. 3. Impact ionizationcreation of electron/hole pairs. 4. Electron injection collector→base. 5. Electron transport throughbase. 6. Electron injection base→emitter.

to impact ionization in the collector-base depletion region are ultimately injected into the emitter.This has the perceived effect of an increase in the input base current. To show this, recall thedefinition of emitter efficiency...

γ =IEp

IE

This can be rearranged to show that...

IEp

IEn+ 1 =

1

1− γ

Comparing this expression to the DC current gain for a common-emitter....

βDC + 1 =1

1− γαT

Since γ ≤ γαT , it follows that...

IEp

IEn+ 1 ≥ βDC + 1

...and finally...

IEp

IEn≥ βDC

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Lecture Notes – 14ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Evidently, for each additional electron injected across the emitter-base junction, βDC more holesare injected into the base identically to an increase in bias current. This is obvious if we considerthat the bias current (i.e. base current) is largely due to IEn under normal conditions (γ → 1 andαT → 1).

IB = IE − IC= IEn + IEp − ICn − ICp

= IEn

(1 +

IEp

IEn− ICn

IEn−ICp

IEn

)= IEn

(1

1− γ− ICn

IEn−ICp

IEp

IEp

IEn

)= IEn

(1

1− γ− ICn

IEn− αT

γ

1− γ

)= IEn

(1− αTγ

1− γ− ICn

IEn

)≈ IEn

1− αTγ

1− γ≈ IEnαT

≈ IEn

...In the third to last line, it was assumed that the collector current due to electrons injected frombase-to emitter, ICn is negligible under typical reverse bias conditions of this junction (forwardactive). In the second to last line, the limit as γ → 1 was taken. This corresponds to the case whenminority carrier injection from the emitter dominates the emitter current. In the last line, the limitas αT → 1 was taken. This result implies that under normal conditions, the forward bias currentof the emitter-base junction is equal to the base current.

Accounting for breakdown mathematically, extends from our previous treatment of pn junctions.The multiplication factor due to impact ionization, M , is given by...

M =1

1−(|VA|VBr

)mHere, the applied voltage is replaced by the collector-emitter voltage, VCB, and the nominal break-down voltage is equated to the collector-base breakdown voltage...

M ≈ 1

1−(

VCEVCB0

)mWe visualize the results of including impact ionization in Figure 5 .

3.4 Current Crowding

Consider a constant current flowing through two regions with different cross-sectional areas A1 andA2. The total current is constant and does not vary with position. In order for this to be thecase, the current density must vary spatially in order to maintain a constant current. The regionwith the smallest area will necessarily have a larger current density to compensate. If the contactarea is small enough, the current density can be large enough to give rise to additional non-ideal

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Lecture Notes – 14ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Figure 5: IV characteristics of a PNP BJT including base-width modulation and impact ionization.

effects. The current is said to crowd at the contacts, leading to lateral conduction (i.e. spreadingresistance) and local Joule heating.

I = A1J1 = A2J2 = constant

The ratio of the current densities is defined by the ratio of the area of the contacts as in Figure 6...

A1

A2=J2J1

3.5 Series Resistance

Series resistances of the emitter (RE), base (RB) and collector (RC) can be added by includingthem between the terminals of the structure as shown in Figure 7 . The result is an effectivereduction in the terminal voltages from their nominal external values.

VEB = VEB0 −RBIB −REIE

VCB = VCB0 + ICRC − IBRB

VCE = VEC0 − IERE − ICRC

3.6 Recombination-Generation Current

So far, we’ve neglected recombination-generation currents when deriving the ideal BJT equations.Similar to the case of pn junctions, BJTs are also prone to R-G currents. For the emitter-base

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Lecture Notes – 14ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Figure 6: Two electrical contacts of different areas A1 and A2, flowing a constant current I betweenthem.

Figure 7: Series resistances incorporated into PNP device

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Lecture Notes – 14ECE 531 Semiconductor Devices

Dr. Andre Zeumault

junction, the additional R-G current adds to the total emitter current.

IE = IEn + IEp + IR−G,EB

The effect is to reduce the emitter efficiency compared to the nominal case.

γ =IEp

IEn + IEp + IR−G,EB≤

IEp

IEp + IEn

13