Bipolar Model

EE6604 Advanced Topics Semiconductor Devices

Bipolar Transistors 81

Base Conductivity Modulation Effect

This is another effect causing IC to decrease at high VBE. 1. At high levels of electron injection, the hole concentration in the base pp has to increase by thermal generation to maintain charge neutrality (refer page 46). 2. In addition, the injected electrons add to the negative space charge on the base side of the base collector junction. This reduces the depletion width and increases the quasineutral base width WB. Both (1) and (2) increase the base Gummel number and decrease IC.



Charge distribution in base-collector junction

Normal Forward active

High injection

Depletion region

(p-side) Depletion region (n-side)



Numerical simulation of BJT base widening

High field region shifts towards collector-subcollector

Dopant Profile



Kirk Effect (or Base Pushout)

This refers to the shifting of the base collector junction towards the collector-subcollector interface at high levels of injection. The electrons injected from the base induce holes in the collector in order to maintain charge neutrality. These holes cause the portion of collector nearest to the base to become p-type and effectively a base extension. [Analysis is difficult because the depletion approximation no longer applies.]



Nonideal Base Current

The base current consists of three components: 1. Hole injection current from base to emitter 2. Generation-recombination current 3. Tunneling current Only the first component follows the ideal transistor theory. At low currents, components (2) and (3) cause deviations.



Generation-Recombination Currents

This current component is due to interface states (x x x) at the surface of the base. They are situated at the Si-SiO2 interface. Because these states are within the Si bandgap, they can result in Shockley -Read recombination. Injected carriers recombine instead of crossing the base to the collector. Due to improvements in semiconductor processing, this is not a problem anymore.



Base Tunneling Current

Interband quantum tunneling at emitter-base junction is possible when there is heavy doping in both the emitter and the extrinsic base. Narrow depletion widths are comparable to the de-Broglie electron wavelength. Important in VLSI bipolar transistors Interband tunneling current is independent of temperature.



Band to Band Tunneling in a P-N Junction

Valence band Conduction Band



Early Effect

The collector current of bipolar transistors with a thin base often increases as the collector emitter voltage increases. This Early effect is caused by an increase in the base- collector depletion width. As a result, the quasi-neutral base width WB decreases.



Early Voltage

Collector current increases linearly with VCE in the linear region. The collector voltage at which the extrapolated IC = 0 is called the Early voltage, VA:

1−

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=+CE

CCCEA V

IIVV VA increases with base doping density. If the base is very lightly doped (small VA), the quasi-neutral base width can reduce to zero and punchthrough occurs.

E B C



Bipolar Device Models

Purpose: 1. Device models allow a circuit comprising bipolar transistors to be simulated by computer. 2. Electrical characteristics of the device are represented in the model by equivalent circuit parameters. There are two widely used DC models:

Ebers-Moll model

Gummel-Poon model



Basic DC Ebers-Moll Model

Transistor modeled as two diodes in series and two dependent current sources With BE forward biased, IF flows in BE diode and αFIF flows in the collector. When BC is forward biased, IR flows in the BC diode and Ebers Moll circuit αRIR flows in the emitter.



The dependent source αFIF models the injection of minority carriers from emitter to collector. αF is the common base current gain in the forward direction. αRIR models the injection of minority carriers from collector to emitter. αR is the common base current gain in the reverse direction.



Ebers-Moll Terminal Currents

From the equivalent circuit,

FRRE III −= α

RFFC III −= α and

RRFFB III )1()1( αα −+−= [Kirchoff: IE + IC + IB = 0]



Using the ideal p-n diode equation for IF and IR,

( )[ ]1/exp0 −= kTqVII BEFF

( )[ ]1/exp0 −= kTqVII BCRR Substituting,

( )[ ] ( )[ ]1/exp1/exp 00 −+−−= kTqVIkTqVII BCRRBEFE α

( )[ ] ( )[ ]1/exp1/exp 00 −−−= kTqVIkTqVII BCRBEFFC α



The four basic Ebers-Moll parameters are: IF0, IR0, αF, αR. Alternatively, we can rewrite IE and IC as:

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛= 1exp1exp 1211 kT

qVakT

qVaI BCBEE

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛= 1exp1exp 2221 kT

qVakT

qVaI BCBEC



where,

022

021

012

11

R

FF

RR

Fo

IaIaIa

Ia

−===

−=

αα

Note: From the reciprocity theorem for 2-port networks,

a12 = a21 Thus there are actually three independent parameters only.

αF = -a21/a11 ; αR = -a12/a22



For a given bipolar transistor, the Ebers-Moll model parameters: (IF0, IR0, αF, αR) or (a11, a12, a21, a22) can be deduced from the transistor dopant dimensions, doping profiles and material parameters.

Collector base voltage

Ic

Emitter current

Schematic of simulated E-M BJT characteristics



Base Collector Impact Ionization

If the bipolar transistor has a large VA, the collector current can still increase as the base-collector voltage is increased because of junction impact ionization. This results in carrier multiplication. Electrons injected from emitter enters collector base depletion region.

• High field causes impact ionization.

• Secondary electrons add to collector current.

• Secondary holes flow to base and reduces base current.



Effect of impact ionization

Base current and β both positive; forward active operation

Negative base current and β due to impact ionization and Ibr

High injection base widening Reduced peak field at BC junction results in a reduced impact ionization



Bipolar Breakdown Voltages

Breakdown voltages are usually measured by applying reverse bias to two transistor terminals with the third left floating (open-circuit, OC). BVCBO = collector base breakdown voltage (emitter OC) BVCEO = collector emitter breakdown voltage (base OC) BVEBO = emitter base breakdown voltage (collector OC) BVCBO and BVCEO must be sufficiently large. BVEBO is usually not of practical significance.



BV Measurement Circuits

BVCEO BVBCO



Avalanche Multiplication of Carriers When the collector base junction breaks down, avalanche multiplication (impact ionization) of carriers occurs within the collector base junction depletion region.



The electron current exiting the collector base depletion region is larger than the current entering it by a factor of M.

)()( BndBCBn WMJWWJ =+ Note: When avalanche multiplication is significant, the common base current gain, α will be increased by a numerical factor of M.

MT )(γαα = where γ is the emitter injection efficiency and αT is the base transport factor.



Emitter injection efficiency,

( )( ) )0(0

0

pn

n

JJJ

+=γ

Base transport factor,

)0()(

n

BnT J

WJ=α

For M=1 (no avalanche), Tγαα =



Relation of saturation currents in bipolar transistor

From the Ebers-Moll model, we can rewrite,

[ ] EFBCCBOC IkTqVII α−−−= 1)/exp( where ICBO = IF0(1-αFαR) is the reverse saturation current of the collector base diode with emitter on open circuit. For the BVCEO measurement, IB=0, IC= -IE, therefore:

F

CBOC

IIα−

=1



IC is in fact ICEO

F

CBOCEO

IIα−

=1

This shows that ICEO should be larger than ICBO. When breakdown occurs, ICEO becomes infinite and this corresponds to:

1=Fα

This is the avalanche breakdown condition.



Relation Between BVCEO and BVCBO

Miller found that the multiplication factor, M can often be expressed in terms of the breakdown voltage of a reverse biased diode as:

m

BVV

VM⎟⎠⎞

⎜⎝⎛−

=

1

1)(

where V is the reverse voltage and m is an empirical number between 3 - 6. For the collector base junction of the forward active transistor in the BVCBO configuration,



m

CBO

CB

CB

BVV

VM

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

1

1)(

M needs to be infinity at breakdown. For the BVCEO configuration, α =1 and VCB ~ BVCEO, at BD

1)()( == CEOTCBT BVMVM γαγα M slightly above 1 at breakdown

Combining,

mT

CBO

CEO

BVBV /1)1( γα−=



Physical Explanation of Smaller BVCEO

For each avalanche event in the base collector junction, one electron hole pair is created. Secondary electron will enter collector with the primary. Secondary hole will flow to emitter and as a result of diode action, 1/(1-α) electrons are injected into base causing rapid build up of current at a lower voltage.

E B C

Hole injection to emitter Electron injection to base



Since (1-γαT)<<1, BVCEO is substantially smaller than BVCBO The plot below shows the BVCEO and BVCBO of a number of BJTs as well as two dotted lines for a BV ratio of 2 and 4. In practice, there is a trade-off between current gain and breakdown voltage.



Bipolar transistor models

The Ebers-Moll model can only give a good prediction of terminal currents for moderate levels of injection i.e. base currents. Model does not take into account high current and other non-ideal effects. A more elaborate bipolar device models the Gummel-Poon was therefore developed in ~1970.



Gummel-Poon Model

The Gummel-Poon model is an extension of the Ebers-Moll model. It incorporates several secondary physical effects that are ignored in the Ebers Moll e.g. high injection effects. Although more accurate than the Ebers-Moll model, many device parameters (>20!) are required to model all the various physical effects.

H.K. Gummel, H.C. Poon, “An integral charge control model of bipolar transistors”, Bell Syst. Tech. J. 49, p.827 (1970).



Outline of the Gummel-Poon Model Starting point is to rewrite the Ebers-Moll equations. By comparing coefficients, the following relations can be deduced directly from the Ebers-Moll circuit equations:

F

aaα

1211 −=

R

aaα

1222 −=

022

212

11 FIaaa =+−



By treating a11, a12 and a22 as three unknowns, we can solve the above simultaneously. The Ebers-Moll coefficients can be expressed in terms of αF, αR and IF0:

RF

FIaαα−

−=1

011

RF

FF Iaαα

α−

=1

012

RRF

FF Iaααα

α)1(

022 −

−=



Now introduce a parameter called intercept current, Ii:

RF

FFi

IaIαα

α−

−=−=1

012

This current can be determined experimentally with the base-emitter short circuited, ln(Ii) is the y-intercept of ln(IE) versus VBC from VBC >> kT/q back to VBC = 0.

ln(IE)

VBC



Using the relation between α and β, we can rewrite the Ebers-Moll equation as:

⎥⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛−⎥

⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+= 1exp1exp11

kTqVI

kTqVII BC

iBE

iF

E β

⎥⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛++⎥

⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛−= 1exp111exp

kTqVI

kTqVII BC

iR

BEiC β

These are equivalent but more symmetrical forms for IE and IC.



In the Gummel-Poon model, these currents are rewritten as:

beCCE III +=

bcCCC III +−= where,

( )[ ])/exp(/exp kTqVkTqVII BCBEiCC −= is the principal component of both the emitter and collector currents.



The minor component currents are:

( )[ ]1/exp −= kTqVII BEF

ibe β

( )[ ]1/exp −= kTqVII BCR

ibc β

The key idea in the G-P model is to account for secondary effects by modifying the expression for ICC (ignore the smaller Ibe and Ibc).

( ) ( )[ ]kTqVkTqVQ

IqnI BCBEb

iGCC /exp/exp −=



Here, Qb is the total majority carrier charge in the base per unit area and nG is the base Gummel number. At low levels of injection, Qb = qnG and the equation reverts to the Ebers-Moll model. For other injection conditions, we want this form of ICC to allow additional effects to be modeled. Qb consists of (i) charge in the quasi-neutral region, (ii) space charge of the emitter-base and collector-base depletion and (iii) holes resulting from injection into the base. (See supplementary notes)



Gummel-Poon Bipolar Equivalent Circuit G-P model can predict bipolar I-V behavior more accurately. See supplementary notes for model equations which are used in circuit simulators such as SPICE.



Common Emitter Form of Ebers-Moll Model

This is a more useful form of the Ebers-Moll model for circuit analysis. It can be derived from the common base model.

R

RR

F

FF

SRSFCT

RRSR

FFSF

IIIIIII

αα

β

αα

β

αα

−=

−=

−===

1

1

IB

ISF/βF

IE

ICT

IC ISR/βR



AC Model of Bipolar Transistor When ac signals are present, capacitors have to be added to the dc model: For each junction, we need to add two capacitances: 1. Depletion layer capacitance, Cd 2. Diffusion capacitance, CD



AC Model of Bipolar Transistor (large signal)

IB

ISF/βF

IE

ICT

IC

ISR/βR

CdBE CDE

CdBC

CDC

CdCS



Small Signal Equivalent Circuit Model

Used when small signals (ss) are superposed on DC bias voltages. iB: ss base current iC: ss collector current vBE: ss base emitter voltage vCE: ss collector emitter voltage



Model parameters: 1. Transconductance, gm is the derivative of Ic with respect to VBE taken at the dc bias point.

kTqI

VIg C

BE

Cm =

∂∂

= 2. Input resistance, rπ is the derivative of VBE with respect to IB.

mBE

C

C

B

BE

B

gVI

II

VIr

βπ =⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=−− 11



3. Output resistance, r0 the reciprocal of the derivative of Ic with respect to VCE.

C

A

CE

C

IV

VIr =⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

=−1

0 The capacitances are designated by:

dBCCC =μ and

DEdBE CCC +=π



In these two capacitors, CdBC is the ss capacitance of the depletion region of the base collector junction. CdBE is the ss capacitance of the depletion region of the base emitter junction. CDE is called the emitter diffusion capacitance If parasitic resistance is significant, additional resistances should be added to this ideal small signal model (see Taur, p.57).



Diffusion capacitance

The capacitance CDE is due to all the minority carriers caused by the base-emitter forward bias.

BE

DEDE V

QC∂∂

=

Here,

BCBEBEDE QQQQQ +++= QE, QB minority carrier charge in emitter, base. QBE, QBC minority carrier in base-emitter, base-collector space charge region.



CDE accounts for the change in terminal voltages whenever there is a rearrangement of the excess charges within the transistor. For circuit modeling purposes, we often write,

( ) CBCBEBEDE IttttQ +++= where, tE: emitter delay time; tB: base transit time; tBE: base emitter space charge region transit time; tBC: base collector space charge region transit time.



Define forward transit time τF as:

( )BCBEBEF tttt +++=τ

At low collector current, τF is independent of VBE,

mFC

FBE

CFDE g

kTqI

VIC τττ ==

∂∂

=

At high collector current, τF is not constant because of the base conductivity modulation effect.



Minority Carrier Charge Distribution



Transient and AC response of BJT

The transient and AC characteristics of a BJT is determined by the charges stored in the device. When the terminal voltages are changed, the stored charges must change in response and time is required for this. Main figures of merit for high frequency performance:

1. Cutoff frequency, fT

2. Maximum oscillation frequency, fmax



Cutoff frequency, fT

Defined as the transition frequency at which the common emitter small-signal current gain at short circuit (RL=0) drops to unity, 1. Hence it is also referred to as the unity current-gain frequency. ft describes the maximum useful frequency of the transistor when used as an amplifier.



Small signal equivalent circuit for fT

From the circuit,

bebemC vCjvgi μω−=

beB vCjCjr

i ⎟⎟⎠

⎞⎜⎜⎝

⎛++= μπ

π

ωω1

gmvBE

iB Cπ

C iC E

Cμ

rπ



The frequency dependent common emitter current gain:

( ) ( )μππ

μ

ωω

ωβCCjr

Cjgm

++

−=

/1)(

It can be shown that, gm >> ωCμ, therefore at high frequencies,

( ) ( )πμωωβ

CCjgm

+≈



From this,

( )μππ CCfg

T

m

+=

21

( )μππ CCgf m

T +=

21

Note: fT depends on the forward transit time and the depletion capacitances. τF can be determined from fT.



Substituting for Cμ, Cπ, and CDE,

( )dBCdBEC

FT

CCqIkT

f++= τ

π21

This is the typically used expression for finding τF. Method: Measure fT at different bias current IC. A plot of 1/fT versus 1/IC will yield (after extrapolation) at the intercept a value for τF. Question: Why is there a deviation at small 1/Ic?



Maximum Frequency of Oscillation, fmax

This is the frequency at which the maximum available power gain of the transistor drops to 1. fmax is typically greater than fT because it takes into account voltage gain.

2

)Re()Re(

41 β⎥

⎦

⎤⎢⎣

⎡=

in

outp Z

ZG

at high frequencies, Re(Zin) is basically the base resistance, rb; Re(Zout) = 1/(2πfTCμ).

Maximum power theorem



High frequency equivalent circuit for fmax

rb Cμ

rπ Cπ

gmv’be RL

Vs

V’be

Zout Zin



By analyzing the hybrid-π equivalent circuit,

2/1

max 8 ⎟⎟⎠

⎞⎜⎜⎝

⎛=

dBCb

T

Crff

π [N.B. Proof not required] fmax can be used to estimate the maximum available power gain at other frequencies:

2

max⎟⎟⎠

⎞⎜⎜⎝

⎛=

ffGp



Maximum Available Power Gain The maximum available power is only obtained when the input and output impedances of the transistor are matched. A microwave vector network analyzer (VNA) is used to determine the maximum available power gain (MAG) as function of frequency.

40MHz-40GHz VNA



Frequency response of a bipolar transistor (MSG: Mason gain U: unilateral gain)

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Bipolar Model

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