Solid State Electronic Devices Example 5-3 Find an expression for the electron current in the n-type...

Solid State Electronic Devices

Example 5-3Example 5-3Find an expression for the electron current in the n-type material of a forward-biased p-n junction.

side. p theintojunction theacross electrons of injection theand holes, injected

theion with recombinatfor electrons of supplying theincludes expression This

)1()1()()(

is materialn in thecurrent electron theThus

)1()(

is siden on thecurrent hole The

)1(

iscurrent totalThe

//

//

/

kTqVp

n

nn

Lx

p

pnpnn

kTqVLxn

p

pnp

kTqVp

n

nn

p

p

enL

Dpe

L

DqAxIIxI

eepL

DqAxI

enL

Dp

L

DqAI

pn

pn


3. 3. Reverse Bias3. 3. Reverse Bias

Fig. 18. Reverse-biased p-n junction: minority carrier

distributions near the reverse-biased junction

pnpn

r

Lxn

p

pLxn

p

pn

nkTVq

nn

r

r

epL

DqAep

L

DqAp

pepp

q

kTV

VV

//

/)(

)(

)1(

When

n respect to with biased negatively p

If

Physically, extraction occurs because

minority carriers at the edges of the

depletion region are swept down the

barrier at the junction by the E field,

and holes in the n region diffuse toward

the junction.


Example 5-4Example 5-4

Consider a volume of n-type material of area A, with a length of one hole

diffusion length Lp. The rate of thermal generation of holes within the

volume is

Assume that each thermally generated hole diffuses out of the volume

before it can recombine. The resulting hole current is I=qALppn/τp, which is

the same as the saturation current for a p+-n junction. We conclude that

saturation current is due to the collection of minority carriers thermally

generated within a diffusion length of the junction.

p

nnnrirth

p

np

ppnng

pAL

2 since


4. Reverse-bias Breakdown4. Reverse-bias Breakdown

< Preface >< Preface >

• If the current is not limited externally, the junction can be damaged by excessive

reverse current, which overheats the device as the maximum power rating is

exceeded.

• It is important to remember, however, that such destruction of the device is not

necessarily due to mechanisms unique to reverse breakdown.

• The first mechanism, called the Zener effect, is operative at low voltages(up to a few

volts reverse bias).

• The breakdown occurs at higher voltages(from a few volts to thousands of volts), the

mechanism is avalanche breakdown.


4. 1. Zener Breakdown4. 1. Zener Breakdown

Fig. 20. The Zener effect: (a) heavily doped junction at equilibrium; (b) reverse bias with

electron tunneling from p to n; (c) I-V characteristic

Heavily doped junction → High electric fields → Tunneling effect occurs

High electric field makes steep energy band, and reverse bias makes narrower width of barrier.


4. 2. Avalanche Breakdown4. 2. Avalanche Breakdown

Fig. 21. Electron-hole pairs created by impact ionization : (a) a single ionizing collision by an

incoming electron in the depletion region of the junction; (b) primary, and tertiary collisions

• Lightly doping

• Breakdown mechanism is the impact ionization of host atoms by energetic carriers.

nbr

in

out

n

inout

VVM

P

PPPn

n

M

PPPnn

)/(1

11

1

1

tion)multiplica (Electron

...)1(

32

32


4. 2. Avalanche Breakdown4. 2. Avalanche Breakdown

Fig. 22. Variation of avalanche breakdown voltage in abrupt p+-n junctions, as a function of

donor concentration on the n side, for several semiconductors.

• In general, the critical reverse

voltage for breakdown increases

with the band gap of the material,

since more energy is required for an

ionizing collision.

• Vbr decreases as the doping

increases, as Fig. indicates.


4. 3. Rectifiers4. 3. Rectifiers

Fig. 23. Piecewise-linear approximations of junction diode characteristics : (a) the ideal

diode; (b) ideal diode with an offset voltage; (c) ideal diode with an offset voltage and a

resistance to account for slope in the forward characteristic.

• Most forward-biased diodes exhibit an offset voltage E0, which can be approximated in a

circuit model by a battery in series with the ideal diode and resister R.


4. 3. Rectifiers4. 3. Rectifiers

Fig. 24. Beveled edge and guard ring to prevent edge breakdown under reverse bias : (a)

diode with beveled edge; (b) closeup view of edge, showing reduction of depletion region

near the bevel; (c) guard ring

A short, lightly doped region → The reason of punch-through

It is possible for W to increase until it fills the entire length of this region.

→ The result of punch-through is a breakdown below the value of Vbr


4. 4. Breakdown Diode4. 4. Breakdown Diode

Fig. 26. A breakdown diode : (a) I-V characteristic; (b) application as a voltage regulator

• It is designed for a specific breakdown voltage(higher doping). Such diodes are also called

Zener diodes(several hundred voltages).

• It can be used as voltage regulators in circuits with varying inputs.


5. Transient and A-C Conditions5. Transient and A-C Conditions

< Preface >< Preface >

• Since most solid state devices are used for switching or for processing a-c

signals, we cannot claim to understand p-n junctions without knowing at

least the basics of time dependent processes.

• In this section we investigate the important influence of excess carriers in

transient and a-c problems.

• The switching of a diode from its forward state to its reverse state is

analyzed to illustrate a typical transient problem.


5. 1. Time Variation of Stored Charge5. 1. Time Variation of Stored Charge

Fig. 4-16. Current entering and leaving a volume ΔxA.

p

pp

xxx

p

x

xxJxJ

qt

p

)()(1

equation continuity The :ion recombinat andDiffusion

Jp(x)

Jp(x+Δx)Δx

Rate of increase of hole concentra- recombination

Hole buildup tion in ΔxA per unit time rate

t

txpq

txpq

x

txJ

p

p

),(),(),(

eq. above from t timeandx position at

current theof component each obtain can We

Fig. 4-16

x

ppp dx

t

txptxpqxJJ

0

),(),()()0(

obtain t toat time sidesboth integratecan We



carriersminority providing : (2) (1)

: carriers excess decreasingor Increasing (2)

/ : carriers excess ofion Recombinat (1)

isjunction n /p theacross injectedcurrent hole The

)()()(

),(),(),0()(

zero(0). iscurrent hole , xAlso,

0. xregion,n long a /n,pat current hole All

00

n

n

dt

dQ

Q

dt

tdQtQti

dxtxpt

qAdxtxpqA

txiti

p

pp

p

p

p

nnnnp

np



Fig. 27. Effects of a step turn-off transient in a p+-n diode: (a) current through the diode; (b)

decay of stored charge in the n-region; (c) excess hole distribution in the n-region as a

function of time during the transient.

Stored charges are

recombination with electrons

ptpp

p

pp

pppp

pp

p

p

p

eItQ

s

IsQ

IssQsQ

IQti

dt

tdQtQti

/)(

/1)(

)()(1

0

)0(,0)0(

)()()( :ion Tranformat Laplace

ionrecombinatafter charges stored ofon distributiCarrier



1ln)(

)()1()(),()()(

instant,any at charge stored for the have We

)()(

statesteady -Quasi assumingby obtained becan )(for solution eapproximatAn

state.steady in hasit form lexponentia convenient the

inremain not doeson distributi hole because simplenot is Obtaining

.t voltage transien theus giveeasily will Finding

)1()(

t, transien theduring 0at xion concentrat hole excess The

/

/)(

0

/

/,

/)(

n

p

pn

pn

t

np

p

p

pkTtqvnnnpn

Lxnp

Lxnn

n

n

kTtqvnn

epqAL

I

q

kTtv

qAL

tQeptptpqALdxetpqAtQ

etptxp

tv

p

p

eptp


5. 2. Reverse Recovery Transient5. 2. Reverse Recovery Transient

Fig. 28. Stored delay time in a p+-n diode: (a) circuit and input square wave; (b) hole

distribution in the n-region as a function of time during the transient; (c) variation of current

and voltage with time; (d) sketch of transient current and voltage on the device I-V

characteristic

= p(xn)-pn

t=0, p-n diode has forward-bias.

Ir=-E/R, when stored charges are

totally recombination.

It’s desirable that tsd is small

compared with the switching time.


5. 2. Reverse Recovery Transient5. 2. Reverse Recovery Transient

Fig. 28. Effects of storage delay time on switching signal: (a) switching voltage; (b) diode current

2

1

sd

sd

erf

lifetime.carrier theis gdeterminin parameter critical The

e.delay tim storage is

rf

fpsd II

Iτt

t

t


Example 5-5Example 5-5At the time t=0 the current is switched to –Ir at a forward biased p+-n diode.Apply appropriate boundary condition and quasi-steady state approximation to find the tsd.

)/1(/1)(

)()(

ms, transforLaplace Using

0,for t )()(

)(

47),-(5 Eq. From

p

r

p

pfp

pfpp

pr

pfpp

p

p

ss

I

s

IsQ

IssQsQ

s

I

IQdt

tdQtQti

r

fp

rf

rpsd

sd

trfr

p

pn

npp

trfrp

tpr

tpfp

I

I

II

It

tt

eIIIqAL

tp

tpqALtQ

eIII

eIeItQ

p

p

pp

1lnln

:obtain weand , when zero equal set to is This

)()(

52),-(5 Eq.in as )()( that Assuming

])([

)1()(

/

/

//


5. 3. Switching Diodes5. 3. Switching Diodes

• A diode with fast switching properties → either store very little charge in the

neutral regions for steady forward currents, or have a very short carrier lifetime, or

both.

The methods to improve the switching speed of a diode.The methods to improve the switching speed of a diode.

1. To add efficient recombination centers to the bulk material. For Si diodes, Au

doping is useful for this purpose. The carrier lifetime varies with the reciprocal of

the recombination center concentration.

2. To make the lightly doped neutral region shorter than a minority carrier diffusion

length. This is the narrow base diode. In this case the stored charge for forward

conduction is very small, since most of the injected carriers diffuse through the

lightly doped region to the end contact. → Very little time required to eliminate the

stored charge in the narrow neutral region.


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da

dp

da

an

apdn

da

da

da

da

NN

NNVVqAW

NN

NNqAQW

NN

NxW

NN

Nx

NqAxNqAxQ

NN

NN

q

VVW

NN

NN

q

VW

000

00

0

0

2,,

charge, sated Uncompen

2

bias, With

2

m,equilibriuIn

5. 4. Capacitance of p-n Junctions5. 4. Capacitance of p-n Junctions

1) Junction capacitance : dominant under reverse bias

2) Charge storage capacitance : dominant under forward bias

dV

dQC Junction CapacitanceJunction Capacitance



tmeasuremen ecapacitanc t viameasuremenion concentrat doping

:n Applicatio

2

2

, junction,/n pFor

assame)(2

formular capacitor plate parallel theof form theUsing

varactor:n Applicatio

ecapacitanc variableVoltage

)(

2

2)(

changeheight barrier n variatioVoltage

0

0

0

2/10

00

dj

nda

jad

adj

ad

adj

NVV

qAC

WxNN

dV

dQC

W

A

NN

NN

VV

qAC

VV

NN

NN

VV

qA

VVd

dQC

p+ n

xp0 xn0


switching good/circuit frequency -highin junction

n-p biased-forwardfor effect switchinglimit ecapacitanc storage charge The

econductanc c-a

charge stored in this changes small todue eCapacitanc

ondistributi hole injected in the charge stored

currentsteady a with biased Forward

/

/2

/

dVdQC

IkT

qe

dV

dpqAL

dV

dIG

IkT

qepAL

kT

q

dV

dQC

epqALLpqAIQ

s

kTqV

p

nps

ppkTqV

npp

s

kTqVnppnpp


Charge Storage CapacitanceCharge Storage Capacitance



Fig. 30. Depletion capacitance of a junction: (a) p+-n junction showing variation of depletion

edge on n side with reverse bias. Electrically, the structure looks like a parallel plate capacitor

whose dielectric is the depletion region, and the plates are the space charge neutral regions; (b)

variation of depletion capacitance with reverse bias.



Fig. 31. Diffusion capacitance in p-n junctions. (a) Steady-state minority carrier distribution for a

forward bias, V, and reduced forward bias, V-ΔV in a long diode; (b) minority carrier distributions

in a short diode; (c) diffusion capacitance as a function of forward bias in long and short diodes.

Solid State Electronic Devices Example 5-3 Find an expression for the electron current in the n-type...

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