1

Overview of Silicon p-n

Junctions

Dr. David W. Graham

West Virginia UniversityLane Department of Computer Science and Electrical Engineering

©

2009 David W. Graham

2

p-n

Junctions (Diodes)

p-n

Junctions (Diodes)•

Fundamental semiconductor device

•

In every type of transistor•

Useful circuit elements (one-way valve)

•

Light emitting diodes (LEDs)•

Light sensors (imagers)

3

p-n

Junctions (Diodes)

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p-type n-type

Bring p-type and n-type material into contact

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p-n

Junctions (Diodes)

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p-type n-type

•

All the h+

from the p-type side and e-

from the n-type side undergo diffusion→ Move towards the opposite side (less concentration)

•

When the carriers get to the other side, they become minority carriers•

Recombination → The minority carriers are quickly annihilated by the large number of majority carriers

•

All the carriers on both sides of the junction are depleted from

the material leaving•

Only charged, stationary particles (within a given region)•

A net electric fieldThis area is known as the depletion region (depleted of carriers)Size of the depletion region depends on the diffusion length

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Depletion Region

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5

Charge Density

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p-type n-type

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Depletion Region

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The remaining stationary charged particles results in areas with

a net charge

Cha

rge

Den

sity

pn xxw +=

x

ρ(x)

qND

-qNA

xn

xp

x=0

6

Electric Field•

Areas with opposing charge densities creates an E-field

•

Total areas of charge are equal (but opposite)

•

E-field is the integral of the charge density

•

Poisson’s Equation

ε

is the permittivity of Silicon

Cha

rge

Den

sity

( ) ( )0ε

ρερ

SKxx

dxdE

==

x

ρ(x)

qND

-qNA

xn

xp

x=0

Ele

ctric

Fie

ld E

x

xnxp x=0

( )w

VVxqNxqNE

dxdVE

Abin

Dp

A −−=−=−=

−=

2max εε

7

Potential

( )xEdxd

−=ψ

•

E-field sets up a potential difference

•

Potential is the negative of the integral of the E-field

Cha

rge

Den

sity

x

ρ(x)

qND

-qNA

xn

xp

x=0

Ele

ctric

Fie

ld E

x

xnxp x=0

Pot

entia

l ψ

x

Vbi

xnxp x=0

Built-In Potential•Integrate the E-field within the depletion region•Use the Einstein Relation

mVq

kTU

nNN

qkTV

T

i

DAbi

9.25

ln 2

==

⎟⎟⎠

⎞⎜⎜⎝

⎛=

Let

Vbi

typically in the range of 0.6-0.7V

8

Band Diagram

•

Line up the Fermi levels•

Draw a smooth curve to connect them

EC

EfEV

Band

Dia

gram

Pot

entia

l ψ

x

Vbi

xnxp x=0

Cha

rge

Den

sity

x

ρ(x)

qND

-qNA

xn

xp

x=0

Ele

ctric

Fie

ld E

x

xnxp x=0

9

Asymmetric Doping (NA

=4ND

)

E

x

xnxp x=0

ψ

x

Vbi

xnxp x=0

x

ρ(x)

qND

-qNA

xnxp x=0

10

p-n

Junction Band Diagram

p n

VA

EC

EfEV

p-type n-type

VA

is the applied bias

p n

Depletion Region

11

p-n

Junction –

No Applied Bias

p n

VA

If VA

= 0

EC

EfEV

EC

EfEV

•

Any e-

or h+

that wanders into the depletion region will be swept to the other side via the E-field

•

Some e-

and h+

have sufficient energy to diffuse across the depletion region

•

If no applied voltageIdrift

= Idiff

p n

Depletion Region

12

p-n

Junction –

Reverse Biased

p n

VA

If VA

< 0

•

Barrier is increased•

No diffusion current occurs (not sufficient energy to cross the barrier)

•

Drift may still occur•

Any generation that occurs inside the depletion region adds to the drift current

•

All current is drift current

Reverse Biased

EC

EfEV

p n

Depletion Region

Expands

13

p-n

Junction –

Forward Biased

p n

VA

If VA

> 0

•

Barrier is reduced, so more e-

and h+

may diffuse across•

Increasing VA

increases the e-

and h+

that have sufficient energy to cross the boundary in an exponential

relationship (Boltzmann Distributions)→Exponential increase in

diffusion current•

Drift current remains the same

Forward Biased

EC

EfEV

p n

Depletion RegionShrinks

14

p-n

Junction Diode

( )10 −= TA nUVeII

Combination of drift and generation

Diffusion Drift

qkTUT = → Thermal voltage = 25.86mV

⎩⎨⎧

=21

n

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

D

i

p

p

A

i

n

n

Nn

LD

Nn

LDqAI

22

0

Cq 1910602.1 −×= A

= cross-sectional area

15

p-n

Junction Diode( )

⎩⎨⎧

−≈−=

0

00 1

IeI

eIITA

TA

nUVnUV for VA

> 0

for VA

< 0

I

-I0VA

( )1

1

0

0

−=

−=

TA

TA

nUV

nUV

eII

eII( )

( ) ( ) ( )

( ) ( )0

0

0

lnln

lnlnln

lnln

InUVI

IeI

eII

T

A

nUV

nUV

TA

TA

+=

+=

=⎟⎟⎠

⎞⎜⎜⎝

⎛

ln(I)

ln(I0)

VA

nkTq

nUT

=1

16

Curve Fitting Exponential Data (In MATLAB)

TA nUVeII 0≈

Curve Fitting Exponential Data (In MATLAB)

•

Given I and V (vectors of data)•

Use the MATLAB functions•polyfit –

function to fit a polynomial (find the coefficients)•polyval –

function to plot a polynomial with given coefficients and x values[A] = polyfit(V,log(I),1);% polyfit(independent_var,dependent_var,polynomial_order)% A(1) = slope% A(2) = intercept

[I_fit] = polyval(A,V);% draws the curve-fit line

17

Size of the Depletion Region

( ) ( )

( ) ( )

( )2

1

0

21

0

21

0

2

2

2

⎥⎦

⎤⎢⎣

⎡−

+=

+=

=

⎥⎦

⎤⎢⎣

⎡−

+=

⎥⎦

⎤⎢⎣

⎡−

+=

AbiDA

DAs

pn

nA

D

AbiDAA

Dsp

AbiDAD

Asn

VVNNNN

qK

xxw

xNN

VVNNN

Nq

Kx

VVNNN

Nq

Kx

ε

ε

ε

VA

= Applied voltage= 0 under equilibrium conditions

q

= Unit of charge= 1.602x10-19C

Ks

= Semiconductor dielectric constant= Relative permittivity= 11.8 (Silicon)

ε0

= Permittivity of free space= 8.854x10-12F/m

w

= Width of depletion region

Depends on•

Doping•

Applied voltage

Depletion region extends farther into the more lightly doped side.

18

Size of the Depletion Region

Asymmetric Doping•

One side of the p-n

junction is more heavily doped

•

Depletion region extends mostly into lightly doped side•

Common in CMOS processes

•

Ex. NA

>> ND

( )

D

A

p

n

AbiD

sn

NN

xx

VVqNKx

≈

⎥⎦

⎤⎢⎣

⎡−=

21

02 ε

19

Reverse-Biased p-n

Junction Capacitance

•

With reverse-biased diodes, there is very small current flow (neglect)

•

(Forward-biased diodes must account for movement of charge)

•

Total charge in depletion region given by width of depletion region times concentration of immobile charge

•

Capacitance definition

•

Junction CapacitancedAC ε

=

wAKC s

J0ε=

A cross-sectional area (i.e. the semiconductor has depth (3-D))

20

Small-Signal Reverse-Biased Capacitance

p n

VA

=0

VA

<0VA

<0xnxp

Small changes in VA

around a DC bias value•

Small change in charge (Q) around a baseline level of charge (on each side of the junction

•

Results in a “small-signal”

capacitance, Cj•

Baseline charge in the depletion region

( )

+−

+

=

⎥⎦

⎤⎢⎣

⎡−

+=

QQ

VVNN

NNqKQ AbiDA

DAs

21

02 ε on the n-type side

(charge must be offset)

21

Small-Signal Reverse-Biased Capacitance

Small changed in charge (for VA

= 0) is the small-signal capacitance

( )

( )DAbi

DAsj

bi

A

j

DA

DA

Abi

s

Aj

NNVNNqKC

VV

CNN

NNVV

qKdVdQC

+=

+=⎥

⎦

⎤⎢⎣

⎡+−

==+

2

12

00

02

1

0

ε

ε

Cj0 Depletion capacitance per unit area at VA=0

VA

is the reverse bias here(i.e. VA

>0 for a reverse bias)

Zero-bias junction capacitance

22

Small-Signal Reverse-Biased CapacitanceOne-Sided Junctions•

Lightly doped on one side•

Common case with substrate/well

( )

bi

Dsj

A

DA

bi

A

j

Abi

Ds

Aj

VNqKC

VNN

VV

CVVNqK

dVdQC

2

01

2

00

00

ε

ε

=

<>>

+=

−==

+

for

Smaller depletion capacitances for more lightly doped substrates

Therefore, immobile charge on each side of a reverse biased substrate

012 0 <+= Abi

Abij V

VVVCQ for

23

Small-Signal Reverse-Biased Capacitance

• Cj

is a nonlinear capacitance•

Decreases with increasing reverse bias

VA0

Cj

24

Large-Signal Reverse-Biased Junction Capacitance

•

Approximate with an average value•

Use two extreme values for the average

( ) ( )

⎟⎟⎠

⎞⎜⎜⎝

⎛+−+

−=∴

−−

==

−bibi

bijavj V

VVV

VVVC

C

VVVQVQ

VQC

12

12

0

12

12

112

(Average)

Rough

ballpark estimate

20j

avj

CC =−

25

Forward-Biased Junction Capacitance•

Appreciable amount of current flowing

•

More carriers present at the edges of the depletion region

•

Therefore, total capacitance is composed of– Cj Junction Capacitance– Cd Depletion Capacitance

T

Dtd

jd

jj

djT

UIC

CC

CC

CCC

τ=

>>

≈

+=

)(typically

where

02

τt Transit time of diodeGiven parameter for a specific process

x

xt D

L 2

=τ where x

is the more heavily doped side

26

Small-Signal Junction Resistance

Change in diode voltage as the forward-bias current changes

( )

D

Td

T

DnUV

T

nUV

DD

D

d

InUr

nUIeI

nUeI

dVd

dVdI

rTDTD

=∴

====

0011

27

Small-Signal Equivalent Model

•

For small changes around a DC bias•

For forward-biased p-n

junctions

x

xt

T

DtdjjdjT

D

Td D

LUICCCCCC

InUr

2

02 ==≈+== ττ

rd Cj Cd

28

Bipolar Junction Transistors (BJTs)•

Two back-to-back p-n

junctions

•

Either npn

or pnp•

Not the focus of this class

–However, they are inherent in CMOS processes (parasitically)–Can be useful in reference circuits–Can cause problems (latchup)

Emitter Base Collector

n np

•

Typically biased in “forward active”

mode–BE junction forward biased–BC junction reverse biased

29

BJT Band Diagram

EC

EV

E B C

•

Virtually no current from C to B•

Exponential change in current from E to B based upon VBE

(just like forward biased diodes)

•

Current flows from E to C•

B modulates how much current flows

30

BJT Equations

B

E

C

VBE+

-

IBIC

IE

B

E

CBJT Equations (Forward Active)

( )

1,

1

1

+=

−=

=

+==

==

ββα

ααβ

β

βα

α

FF

F

BC

BUV

F

SE

EFUV

sC

II

IeII

IeII

TBE

TBE

Typical Values

99.0100==

Fαβ

Almost all current is passed from E to C. Very little B current

Is

, α, β

are constants

31

Designable BJT ParametersDesignable Parameters•

Emitter Area, AE•

Base Width, W•

(These may not be designable in CMOS processes)

For npn

transistor

WNDLND

WNnqDAI

Ap

pDn

A

inEs

=

=

β

2

If pnp, swap the following subscripts• n

↔ p• D ↔ A