Chapter 4: p-n Junctions 4.2. Structure and principle of operation

2

lni

adti

n

NNV=φ

(4.2.1)

ai V−= φφ (4.2.2)

4.3. Electrostatic analysis of a p-n diode

)(2

2−+ −+−−=−= ad

ssNNnp

q

dx

dεε

ρφ

(4.3.1)

)sinh(sinh

22

2

t

F

t

F

s

iVV

nq

dx

d φφφε

φ+

−=

(4.3.2)

i

da

t

Fn

NN

V 2sinh

+− −=

φ

(4.3.3)

pnd xxx += (4.3.4)

npadad xxxNNqNNnpq ≤≤−−≅−+−= −+−+ for ,)()(ρ (4.3.5)

0)(

)(

)(

0)(

=

=

−=

=

x

qNx

qNx

x

d

a

ρ

ρ

ρ

ρ

xx

xx

xx

xx

n

n

p

p

≤

≤≤

≤≤−

−≤

for ,

0for ,

0for ,

for ,

(4.3.6)

ndn xqNQ = (4.3.7)

pap xqNQ −= (4.3.8)

npad

ssxxxxNxN

qxdx

(x)d≤≤−−≅= −+ for ,))()((

)(εε

ρE

(4.3.9)

0)(

)()(

)()(

0)(

=

−=

+−=

=

x

xxqNx

xxqNx

x

s

nd

s

pa

E

E

E

E

ε

ε

xx

xx

xx

xx

n

n

p

p

≤

≤≤

≤≤−

−≤

for ,

0for ,

0for ,

for ,

(4.3.10)

s

nd

s

pa xNqxNqx

εε−=−== )0(E

(4.3.11)

pand xNxN = (4.3.12)

da

adn NN

Nxx

+=

(4.3.13)

da

ddp NN

Nxx

+=

(4.3.14)

(x)

dxxd E−=

)(φ

(4.3.15)

s

pa

s

ndai

xNqxNqV

εεφ

22

22

+=− (4.3.16)

))(

11(

2ai

da

sd V

NNqx −+= φ

ε

(4.3.17)

)(

12ai

dad

asn V

NNNN

qx −

+= φ

ε

(4.3.18)

)(

12ai

daa

dsp V

NNNN

qx −

+= φ

ε

(4.3.19)

a

aa dV

VdQVC

)()(

∆

= (4.3.20)

da

da

ai

sj NN

NNV

qC

+−=

)(2 φε

(4.3.21)

d

sj x

Cε

= (4.3.22)

)(

212 ai

da

da

sjV

NNNN

qC−

+= φ

ε

(4.3.23)

da

da

sa

j

NNNN

qdV

Cd +−=

ε2)/1( 2

(4.3.24)

da

a

jsd NN

dV

CdqN >>−= if ,

)/1(

122ε

(4.3.25)

4.4. The p-n diode current ta VV

nnn epxxp /0)( == (4.4.1)

ta VVppp enxxn /

0)( =−= (4.4.2)

0)( nnn pwxp == (4.4.3)

0)( ppp nwxn =−= (4.4.4)

pnpn LxxLxxnnn eBeApxxp

/)(/)(0)(

−−−++=≥ (2.9.13)

pppp LxxLxxppp eDeCnxxn /)(/)(

0)( ++− ++=−≤ (2.9.14)

p

n

p

nnnn L

xxB

Lxx

Apxxp−

+−

+=≥ sinhcosh)( **0

(4.4.5)

n

p

n

pppp L

xxD

L

xxCnxxn

++

++=−≤ sinhcosh)( **

0 (4.4.6)

]sinhcothcosh)[1()(

'/

00p

n

p

n

p

nVVnnnn L

xxLw

Lxx

eppxxp ta−

−−

−+=≥ (4.4.7)

]sinhcothcosh)[1()(

'/

00n

p

n

p

n

pVVpppp L

xx

L

w

L

xxennxxn ta

++

+−+=−≤

(4.4.8)

nnn xww −=' (4.4.9)

ppp xww −=' (4.4.10)

]coshcothsinh)[1(

)(

'/0

p

n

p

n

p

nVV

p

np

pnp

Lxx

Lw

Lxx

eL

pqDdxdp

qDxxJ

ta−

−−

−−=

−=≥

(4.4.11)

]coshcoth)[sinh1(

)(

'/0

n

p

n

p

n

pVV

n

pn

npn

L

xx

L

w

L

xxe

L

nqDdxdn

qDxxJ

ta+

++

−=

=−≤

(4.4.12)

)1(])()([ / −≅+=+−== ta VVsrnppn eIJxxJxxJAI (4.4.13)

)]coth()coth([

'0'

0

p

n

p

np

n

p

n

pns L

wL

pD

L

w

L

nDAqI +=

(4.4.14)

1for ,

1tanh

1coth <<≅= x

xxx

(4.4.15)

p

nVVnnnn L

xxeppxxp ta

)(exp)1()( /

00−−

−+=≥ (4.4.16)

n

pVVpppp L

xxennxxn ta

+−+=−≤ exp)1()( /

00 (4.4.17)

p

nVV

p

npnp L

xxe

L

pqDxxJ ta

)(exp)1()( /0 −−

−=≥ (4.4.18)

n

pVV

n

pnpn L

xxe

L

nqDxxJ ta

+−=−≤ exp)1()( /0

(4.4.19)

][][ 0000

p

pn

n

np

p

np

n

pns

LpLnAq

L

pD

L

nDAqI

ττ+=+=

(4.4.20)

)1( /0 −≥ ta VV

p

nnr e

xpAqI

τ

(4.4.21)

pn Lx << (4.4.22)

2

2

2

20 and ,0

dx

pdD

dx

ndD n

pp

n == (4.4.23)

DxCpBxAn np +=+= and , (4.4.24)

−−−+= '

/00 1)1e(

n

nVVnnn

w

xxppp ta

(4.4.25)

++−+= '

/00 1)1e(

p

pVVppp w

xxnnn ta

(4.4.26)

)1(])()([ / −≅+=+−== ta VVsrnppn eIJxxJxxJAI (4.4.27)

][ '

0'

0

n

np

p

pns

w

pD

w

nDAqI +=

(4.4.28)

∫

−−− =

n

p

x

xbbbb dxUqJ

(4.4.29)

)( 2ibb nnpbU −=− (4.4.30)

taFpiiFn VVi

kTEEi

kTEEi enenenpn /2/)(/)( == −− (4.4.31)

)1()1( /2/2 −=−= ∫

−− ta

n

p

ta VVdi

x

x

VVibb exbnqdxenqJ

(4.4.32)

∫=

−

n

p

x

xSHRSHR dxUqJ

(4.4.33)

∫ −

++

−=

−

n

p

tax

x iti

VVi

SHR dx

kTEE

npn

enqJ

)cosh(2

)1(1 /2

τ

(4.4.34)

tapiin VVi

kTFEi

kTEFi enenenpn /2/)(/)( == −− (4.4.35)

)1(

2)

)cosh(2

)1(1( MAX 2/

/2

max, −≅−

++

−= ta

ta VVi

iti

VVi

SHR en

kTEE

npn

enU

ττ

(4.4.36)

max,

/2

)cosh(2

)1(1

SHR

x

x iti

VVi

U

dx

kTEE

npn

en

x

n

p

ta

∫ −++

−

=′− τ

(4.4.37)

)1(

22/

'−= ta VVi

SHR exqn

Jτ

(4.4.38)

ta VVs eJJ η/= (4.4.39)

mV/decade 59.6slope

1

slope )elog(

==tV

η

(4.4.40)

i

dta n

NVV ln2=

(4.4.41)

saa IRVV +=* (4.4.42)

adVQd

C∆

= (4.4.43)

∫ −=∆n

n

w

xnnp dxppqAQ )( 0

(4.4.44)

p

VVpsp

VVnp tata ILpAqQ τ)1e()1e( /

,/

0 −=−=∆ (4.4.45)

p

pnps L

DpAqI 0

, = (4.4.46)

t

pVV

ps

a

pVV

pspd V

I

dV

IdC

tata ττ /,

/,

,e))1e((

=−

= (4.4.47)

t

prVV

pspd V

tIC

ta ,/

,,

e=

(4.4.48)

p

ppr D

wt

2

2'

, =

(4.4.49)

4.5. Reverse bias breakdown

V/cm )10/log(1

10416

31

5

N

xbr

−=E

(4.5.1)

qNV sbr

ibr 2

2εφ

E+−=

(4.5.2)

qNx sbr

brd

εE=,

(4.5.3)

dxndn nα= (4.5.4)

∫−

=2

1

1

1x

xdx

M

α

(4.5.5)

6 2 where,

1

1<<

−

= n

VV

Mn

br

a

(4.5.6)

−=Θ

E

2/3*234

exp gE

qmh

(4.5.7)

Θ= nvqJ Rn (4.5.8)

4.6. Optoelectronic devices

phVV

s III ta −−= )1e( / (4.6.1)

inph P

hq

Iν

=max, (4.6.2)

να

hqP

RI indph )e1)(1( −−−=

(4.6.3)

fIqi ∆>=< 22 (4.6.4)

ocsc

mmVIVI

=Factor Fill (4.6.5)

1e ion amplificat Roundtrip 212 == RRgL (4.6.6)

21

1ln

21

RRLg =

(4.6.7)

)( thout II

qh

P −=ν

η (4.6.8)