Lecture 13 OUTLINE pn Junction Diodes (cont’d) – Charge control model – Small-signal model –...

Lecture 13

OUTLINE• pn Junction Diodes (cont’d)– Charge control model– Small-signal model– Transient response: turn-off

Reading: Pierret 6.3.1, 7, 8.1; Hu 4.4, 4.10-4.11

Minority-Carrier Charge Storage• Under forward bias (VA > 0), excess minority carriers are

stored in the quasi-neutral regions of a pn junction:

Pnn

x nP

LxpqA

dxxpqAQn

)(

)(

Npp

x pN

LxnqA

dxxnqAQp

)(

)(

EE130/230M Spring 2013 Lecture 13, Slide 2

Derivation of Charge Control ModelConsider the n quasi-neutral region of a forward-biased pn junction:

•The minority carrier diffusion equation is (assuming GL=0):

•Since the electric field is very small,

•Therefore

p

nnP

n p

x

pD

t

p

2

2

xp

PPnqDJ

p

nPn pq

x

J

t

pq

)(


• Integrating over the n quasi-neutral region:

• Note that

• So

n

P

npn x

np

J

xJ

P

x

n dxpqAdJAdxpqAt

1)(

)(

)()()()()(

)(

nPnPnPP

J

xJ

P xIxAJxAJAJdJAP

np

p

PnP

P QxI

dt

dQ

)(


Derivation Assuming a Long Base

We can calculate pn-junction current in 2 ways:1. From slopes of np(-xp) and pn(xn)

2. From steady-state charges QN, QP stored in each excess-minority-charge distribution:

p

PnP

QxI

τ)(

n

NpN

QxI

τ)( Similarly,

0τ

)( p

PnP

P QxI

dt

dQ


Charge Control Model

Charge Control Model for Narrow Base• For a narrow-base diode, replace p and/or n by the

minority-carrier transit time tr– time required for minority carrier to travel across the quasi-

neutral region

– For holes in narrow n-side:

– Similarly, for electrons in narrow p-side:

P

N

P

Pptr

N

nnP

nPPP

Nnn

W

x nP

D

W

I

Q

W

xpqAD

dx

pdqADAJI

WxpqAdxxpqAQN

n

2 τ

)(

)(2

1)(

2

,

N

Pntr D

W

2τ

2

,


Charge Control Model Summary• Under forward bias, minority-carrier charge is stored in the

quasi-neutral regions of a pn diode.

– Long base:

– Narrow base:

PkTqV

D

iP Le

N

nqAQ A 1/

2

NkTqV

A

iN Le

N

nqAQ A 1/

2

PkTqV

A

iN We

N

nqAQ A 1

2

1 /2

NkTqV

D

iP We

N

nqAQ A 1

2

1 /2


• The steady-state diode current can be viewed as the charge supply required to compensate for charge loss via recombination (for long base) or collection at the contacts (for narrow base).

– Long base (both sides):

– Narrow base (both sides):

where and

p

P

n

N QQI

ττ

ptr

P

ntr

N QQI

,, ττ

P

P

p

P

N

N

n

N

L

DL

L

DL

τ and

τNote that

N

Pntr D

W

2τ

2

,

P

Nptr D

W

2τ

2

,


Small-Signal Model of the Diode

kTqVkTqV eIdV

deI

dV

d

dV

dI

R/

0A

/0

AA

AA )1(1

qkT

IeI

kT

q

RG DCkTqV

/

1 /0

A

dt

dvC

R

vi aa


i

v

+

Small-signalconductance:

Charge Storage in pn Junction Diode


pn Junction Small-Signal Capacitance2 types of capacitance associated with a pn junction:

depletion capacitance due to variation of depletion charge

diffusion capacitance–due to variation of stored minority charge in the quasi-neutral regions

For a one-sided p+n junction Q = QP + QN QP so

qkT

IG

dV

dI

dV

dQC P

/

τττ DCp

pA

pA

D


A

dep

dV

dQC J

AdV

dQC D

Depletion Capacitance

What are three ways to reduce CJ?

WA

dV

dQC s

A

depJ


Total pn-Junction Capacitance

C = CD + CJ

•CD dominates at moderate to high forward biases

•CJ dominates at low forward biases, reverse biases

1/

τ /DCD kTqVAe

qkT

IC

WAC s

J


Using C-V Data to Determine Doping

NqA

VV

A

W

C Ss 2Abi

22

2

2J

)(21


ExampleIf the slope of the (1/C)2 vs. VA characteristic is -2x1023 F-2 V-1, the intercept is 0.84V, and A is 1 m2, find the dopant concentration Nl on the more lightly doped side and the dopant concentration Nh on the more heavily doped side. Solution:

ln 2i

lhbi

n

NN

q

kTV 318026.0

84.0

15

202

cm 108.1106

10

eeN

nN kT

qV

l

ih

bi

315

28121923

2

cm 106

)1010106.1102/(2

)/(2

AqslopeN sl


Small-Signal Model Summary

qkT

IG DC

/

qkT

IC

/

τ DCD

W

AC s

J

)1( /0

A kTqVDC eII

DJ CCC

Depletion capacitance

Diffusion capacitance

Conductance


Transient Response of pn Diode


• Suppose a pn-diode is forward biased, then suddenly turned off at time t = 0. Because of CD, the voltage across the pn junction depletion region cannot be changed instantaneously.

The delay in switching between

the ON and OFF states is due to the time required to change the amount of excess minority carriers stored in the quasi-neutral regions.

Turn-Off Transient• In order to turn the diode off, the excess minority

carriers must be removed by net carrier flow out of the quasi-neutral regions and/or recombination– Carrier flow is limited by the switching circuitry


Decay of Stored ChargeConsider a p+n diode (Qp >> Qn):

t

i(t)

t

vA(t)

ts

ts0 pxx

n

qAD

i

dx

dp

n

For t > 0:

pn(x)


Storage Delay Time, ts

• ts is the primary “figure of merit” used to characterize the transient response of pn junction diodes

• By separation of variables and integration from t = 0+ to t = ts, noting that

and making the approximation

We conclude that

sp

pR

p

pp ttQ

IQ

idt

dQ

0

ττ

ppF tQI τ/)0(

0)( sp ttQ

R

Fps I

It 1lnτ


Qualitative Examples

t

i(t)

ts

Increase IF

t

i(t)

ts

Increase IR

t

i(t)

ts

Decrease p


Illustrate how the turn-off transient response would change:

Lecture 13 OUTLINE pn Junction Diodes (cont’d) – Charge control model – Small-signal model –...

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