ece606: solid state devices lecture 15 p-n diode ...ee606/downloads/ece606_f12_lecture15.pdf ·...

Klimeck – ECE606 Fall 2012 – notes adopted from Alam

ECE606: Solid State DevicesLecture 15

p-n diode characteristics

Gerhard [email protected]


Outline

2

1) Solution in the nonlinear regime

2) I-V in the ambipolar regime

3) Tunneling and I-V characteristics

4) Non-ideal effects: Impact ionization

5) Non-ideal effects: Junction recombination

6) Conclusion


Nonlinear Regime (3) …

3

( )( )2 2

1n pAqVpn i iT

p A n D

FFDD n nJ q e

W N W N

β−∆ ∆− = − + −

VA

2

1

3

6,7

ln(I)

( )( )0 1A pnJq V a bJI e

β− −= −

Assumption of flat Quasi-Fermi levels invalid here

Today’s lecture: Nonlinear Regime (2,3)


Flat Quasi-Fermi Level up to Junction ?

4

N N N

dnqn qD

dxµ= +J E

) )( (n n ii FF E ni N iN

EdFdnn n e qD qD

dx dxn eβ ββ− − = = −

E

Fp

Fn

n n nn n n

n D

dF J WJ F

dxn

Nµ

µ= ⇒ ∆ =

nN N

n N BN

n

dFdnqD qD

dx dx

dF D k Tq n

dx q

nβ

µµ

= −

= − = ∵

E

E

Wn

Rewrite n into non-equilibrium form, re-arrange Jn equation

New diffusion component: Plug this into original Jn equation

Drop of Quasi-Fermi level across the junction proportional to current!


Forward Bias: Nonlinear Regime …

5

n

p

n

n D

p n

n

n D

J W

N

FN

F

J W

µ

µ∆ =

∆ =

2( )(0 ) n pi

junctio

F

A

F

n

nn e

Nβ−+ =

( )( )2 2

1n pAqVpn i iT

p A n D

FFDD n nJ q e

W N W N

β−∆ ∆− = − + −

∆Fp

∆n

∆Fn

Still diffusion dominated transport? Since Quasi-Fermi levels are not flat in nonlinear regime (drift), this approximation becomes worse.

VA

( ) ( )( )2 2

(0 ) 1pA An pnqV qVi F

A

F Fi

A

Fn ne n e

N N

β β∆ ∆− − − −∆ + ∆= ⇒ ∆ = −


Outline

6






6) Conclusion


Region (2): Ambipolar Transport

7

( ) / 2ln(

2)A n pqV F Fpn A

T i Tp n B

DD qVJ q n e J

W W k T

β−∆ −∆ ≈ − + ≈

VA

2

1

3

6,7

ln(I)

Today’s lecture: Ambipolar Transport regime (2) Question: Where does the 2 come from?


Nonlinear Regime: Ambipolar Transport

8

( )

( )

/ 2

/ 2

A n p

A n p

qV F Fn in n

p p

qV F Fp ip p

n n

qD nnJ qD e

W W

qD nnJ qD e

W W

β

β

−∆ −∆

−∆ −∆

∆= − =

∆= − =

Note: junction never disappears, even for large forward bias!

( )( )( ) / 2

1A n p

A n p

q V F F

i

q V F F

i

n epn

n e

β

β

−∆ −∆

−∆ −∆≈

∆ = −∆≈

( )( )( )2

22( )( ) 1

n p

A n p

F F

i

q V F FiA i

A

np n e

nN n

Npn e

β

β

−

−∆ −∆∆

=

+ =∆+ −

Here not negligibly small. Ambipolar transport !

Excess carrier concentrations >> NA Thus…

Currents


Outline

9






6) Conclusion


Forward Bias Nonlinearity (7): Esaki Diode

I

VA

Fn Fp

Fn

Fn

Fp

Fp

X

VA

2

1

3

6,7

ln(I)

Heavy doping

1

empty

No states!

Esaki-Diode: Heavily doped diode

Tunneling in diodes.Nobel Prize (Esaki)


Reverse Bias (5): Zener Tunneling

11

2 2

4

4cosh sinh

(p.49 ADF)

= υ

=α α + − α α

I qpT

Tk

d dk

Fn

Fp

4

5

+VA

empty

empty

Tunneling (triangular barrier)

Remember: Tunneling through a triangular barrier

Zener tunneling occurs in every diode. (reverse bias)


Various Regions of I-V Characteristics

12

1. Diffusion limited

2. Ambipolar transport

3. High injection

4. R-G in depletion

5. Breakdown

6. Trap-assisted R-G

7. Esaki Tunneling

VA

2

4

5

1

3

6,7

ln(I)

MAA2

MAA2 Asad: We should redraw the figure ... Muhammad Ashrafal Alam, 1/30/2009


Applying Bias

13

EF-EV

EC-EF

qVbi

Fp-EV

EC-Fn

q(Vbi-V)

V

x

x

n,p

n,p

Equilibrium

Non-EquilibriumNet RG!

No net RG!


(4,6) Junction Recombination

14

0

W

R

nI qA dx

t

∂= −∂∫

2

1 1

( )

(

[ ]

[ ]

)

])[

(

( )τ τ−∂ = −

∂ + + +i

p n

p x

p x

n x

nn x

nn

t p

pn ττ = Ti EE = inpn == 11

]2)()([

)1( /2

i

kTqVi

nxpxn

en

t

n A

++−

−=∂∂

τ

Assume

Note: Do you remember this HW ?

What is the recombination current?

Shockley-Reed Hall

Follows from assuming midgap traps


(np) Product within the Junction

15

( ) /2

/2

( ) ( ) N P

A

F F kTi

qV kTi

n x p x n e

n e

−=

=

Mass action in non-equilibrium

For non-equilibrium at low current values.


Electron/Hole Concentrations at Junction

16

)()( xqVExE iLi −=

( )

[

(

]

)

( )

/

/

( ) N i

N iL

kTF E x

F E qV x

i

kTi

n ex

n e

n −

− +

=

=

/2

[ ] /

[ ( )] / /

( )( )N

A

N

iL

iL A

qV kTi

kTi

F E

F E qV

qV x kT qV kTi

x

n ex

e

n

pn

e− − + +

− +=

=

position


Junction Recombination

17

kT

EFU iLN

FN

−=

qkT

VU A

A /=

][

)1(AFNFN

A

UUUUU

Ui

ee

en

t

n+−−+ +

−−=

∂∂

τ

][

)(2/2/2/

2/2/2/

AFNAFNA

AAA

UUUUUUU

UUUi

eee

eeen

t

n+−−−+

−

+−−=

∂∂

⇒τ

( / 2)

[ / 2]i A

FN A

n sinh Un

t cosh U U Uτ∂

⇒ = −∂ + −

02 [ / 2]τ = − × × + −

∫W

i AR

FN A

n U dxI qA sinh

cosh U U U


Junction Recombination in Forward Bias

18

( / 2)0( ) ( )

2 FN AU

Wi A

R U U

n U dxI qA sinh

eτ + −⇒ ≈ − ∫

max( ) /( / 2 )0( ) ( )

2

Wi A

R kT qx

n U dxI qA sinh

eτ −⇒ ≈ − × ∫ E

/ 2

max2AqV kTi

Dep

nkTI qA e

q τ

⇒ = − E

( / 2)

[ / 2]τ∂

⇒ = −∂ + −

i A

FN A

n sinh Un

t cosh U U U

VA

2

45

1

3

6,7

ln(I)

Emax

Effective width Excess Carrier at mid-junction


Junction Leakage in Practice

19

n

p

d

rj rj

Insulating Layer

Junction Design ConsiderationsElectric field stronger at corners, sharp edges. � increased recombination!


Junction Recombination in Reverse Bias

20

0 2R

Win

I qA dxτ

≈ −

∫

2inn

t

∂ = −∂ τ

VA

2

45

1

3

6,7

ln(I)

2i

bi A

Wnq VA V

τ∝ −= −

W=xn+xp

(Recombination in depletion region)

Integrate…


Outline

21






6) Conclusion


Avalanche Breakdown

22

1. Diffusion limited

2. Ambipolar transport

3. High injection

4. R-G in depletion

5. Breakdown

6. Trap-assisted R-G

7. Esaki Tunneling

VA

2

4

5

1

3

6,7

ln(I)

MAA3

MAA3 Asad: We should redraw the figure ... Muhammad Ashrafal Alam, 1/30/2009


N o n l i n e a r i t y d u e t o I m p a c t -Ionization

23

Exponential current growth (Impact Ionization or Inverse Auger process)

High Reverse Bias

Reverse Bias


Outline

24






6) Conclusion


Impact-ionization and Flux Conservation

25

( ) ( ) ( ) ( )pn n pn nI I Ix dx x x dx x dxIα+ α= + +

( ) ( )( ) ( )

( )( ) ( )

n nn n p p

nn n p p

I x dx I xI x I x

dxdI x

I x I xdx

+ − = α + α

⇒ = α + α

[ ]

( )

( )( ) ( )

( )( )

np n n

nn p n p

T n

T

dI xx I xI

dxdI

I

Ix

I xdx

= α − + α

− α − α = αx

W 0

Impact Ionization probabilities

Steady state: Define IT = IN+IP (total current)

Differential equation


Impact-ionization

26

( ) (

(0) 1

( ) )n T Tp

T

n

n

p

I W I I W I

I M

I W

I

+ = ⇒ ≈

≡

( )0

'

0

11 1

x

p nW dx

ppe d

Mx

− α −α ∫− ≈ = α

∫

( )

( )( )

0

0

'

0

'

0

(0)

1

( )

x

n p

x

n p

W dx

pTn

W dxT

p

n

n

e dxII W

Ie dx

I− α −α

− α −α

∫α +

=∫

+ α − α

∫

∫

xW 0

Ip(W)~0

In(0)~0 Solution form of differential equation Reverse diffusion current

At x=W, IN has grown exponentially, and IP is now negligible.

Multiplication Factor


Impact-ionization

27

1p n pWα = α ⇒ α =

( )0

'

0

1

x

p nW dx

pe dx− α −α∫

α ≈∫

0B

p A e−α = E

( ) ( )1/ 2

0 0

20 D n D A

bi As s D A

qN x q N NV V

k k N N−

= = − + ε εE

Electric Field

Position

VA<0VA=0

VA>0

Simplify further...

from experiment and theory

Assume: Significant impact ionization

Breakdown-Field


Impact-ionization: In Practice

28

Good ….

Bad….

n

p

Insulating Layer

d

rj rj

Photon Detector

High E-fields at junction corners � Breakdown


Junction Engineering

29

E E

Reduced field for p-i-n junction, because Vbi(area under the curve) must be the same.

n

p

Insulating Layer

d

rj rj i-region

Lower E-field!

Intrinsic region: E-field has to be constant, because there are no charges!


Modern Considerations: Dead Space

30

For very small (ballistic) junctions, electrons can cross the junction without inducing impact Ionization. (Dead space too small)

W

Dead SpaceDead Space: Space you need before an electron can impact ionize.


Zener Breakdown vs. Impact Ionization

31

How do you differentiate between Zener tunneling and impact-ionization?


Conclusion

32

1) Junction recombination is often used as a diagnostic tool for

process maturity. Defects in junction arises from misplaced

donor impurities, not necessary from deep-trap impurities.

2) Impact ionization plays an important role in wide variety of

devices (e.g. avalanche photo-diodes).

3) In the next class, we will discuss AC response of p-n

junction diodes.


ECE606: Solid State Devicesp-n diode AC Response

Gerhard [email protected]


Topic Map

34

Equilibrium DC Small signal

Large Signal

Circuits

Diode

Schottky

BJT/HBT

MOSFET

Diode in Non-Equilibrium(External DC+AC voltage applied)


Why should we study AC Response?

35

Series Resistance

Conductance

Diffusion Capacitance

Junction Capacitance

www.sci-toy.com

Motivation

Radio


Outline

36

1) Conductance and series resistance

2) Majority carrier junction capacitance

3) Minority carrier diffusion capacitance

4) Conclusion

Ref. SDF, Chapter 7


Forward Bias Conductance

37

( )( )/ 1A Sq V R I moI I e −= −β

0

1

( )FBS

m

q IgR

Iβ+

+=

0

ln ( )oA S

I Iq V R I

I m

β+ = −

( )A

So

m dVR

q I I dI= −

+β

RS

G

CJ

V

2

45

1

3

6,7

ln(I)

Cdiff

m = RG (2), diff (1), Ambipolar (2)

Forward Bias Conductance


Reverse Bias Conductance

38

( )( )/

0

1A Sq V R I moI I e

I

−= −

≈ −

β

02i

bi A

qnB V V

τ− −

01

2i

RB bi A

qn B

g V Vτ=

−

02i

bi A

qnB V V

τ− −

RS

G

CJ

Cdiff

V

2

45

1

3

6,7

ln(I)

Reverse Bias Conductance


Outline

39




4) Conclusion



40

VDC

Series Resistance

Conductance



Depletion width modulation

Charge modulation

Majority carrier effect

Forward biased diode + AC signal

Fn

Fp

VA> 0

VA< 0


Majority Carrier Junction Capacitance

41

0 0

0 02 2( )

s sJ

n p s sbi A

D A

K A K AC

W W K KV V

qN qN

= =+

+ −

ε εε ε

Cj

Va

Measure

0

x

∆ρ

x

∆ρ

VA < 0

VA > 0

Series Resistance

Conductance



Majority Carrier Junction Capacitance


Measurement of Built-in Potential

42

2 20

1 2( )

( ) AJ D s

biV VC qN x K Aε

≈ −

plot CJ-2

VA

measureCJ

VA

(Assume single sided p+-n junction)

Vbi

Derive Vbi from CJmeasurements


.. And Variable Doping

43

2 20

2

( )(

1)bi

sDA

J

V VNq KC Ax ε

≈ −

plot CJ-2

VA

measureCJ

VA

Charge

VA<0VA=0

VA>0

220

2 1

(1 )( )

s AJD qK A d C

NV

xdε

=

Measure doping concentration as a function of position


Dielectric Relaxation Time (majority side)

44

1 nn n

dn dJR G

dt q dx= − +n N NJ qn E qD n= + ∇µ

VDC

( ) ( )1 ND N

d n d qn dN

dt q dx dx

µµ

∆= =

E E

( )00

D AS

d qp n n N N

dx k ε= − − ∆ + −E

( )0 0

0D

S S

NqNd n nn

dt k kεµ

εσ∆ ∆= − ∆ = −

0

00 0( ) S d

t t

kn t n e n eε τσ− −

∆ = =

0 0.1 pssd

K ετ = ≈σVery fast

How long does it take for the signal to cross the junction?

Majority side Neglect

N-side


Outline

45




4) Conclusion


Diffusion Capacitance for Minority Carriers

46

1 nN N

n dJr g

t q dx

∂ = − +∂

N N N

dnqn qD

dxµ= +J E

( ) ( )20 0

2

j t j t j tdc ac dc ac dc ac

N

n

n n n e d n n n e n n eD

t dx

∂ + ∆ + ∆ + ∆ + ∆ ∆ + ∆= −∂

ω ω ω

τ

VDC

np

x

Minority Carrier side

DC AC

ACAC


Diffusion Capacitance for Minority Carriers

47

( ) ( )20 0

2

dc dj t j t

acc dcN

j tac

n

acn e nn d nD

t d

n n

x

e n en ωω ω

τ∂ + + + + += −

∂∆∆ ∆∆∆ ∆

2 2

2 2ac dc ac

ac Ndcj t j t j

n

t

n

d dn n n nj n D

d dxe

xe eω ω ωω

τ τ∆ ∆ ∆ ∆∆ = + − −

2

2C 0D : n n

x x

dc dcN

n

L Ldcn Ae B

d n nD

dxe

τ− +∆ ∆

⇒ ∆ +− ==

( )2

2* * *AC : 0 1 n n n

x x x

L L Lac acN an

nc

d n nD j

dn Ce De C

xeωτ

τ− + −

⇒∆ ∆ ∆ = +− + →=

( ) ( )* */ 1 / 1n n n n n n nL D j jτ ωτ τ τ ωτ= + = +


AC Boundary Conditions

48

( )

2

2

( 0) 1

1

j tac

c

d

d

c V e

qVi kT

dcA

qkTj t

ac j tA ac

Vi

dc n

nn x e

N

eenp e

n

N

ω

ωω

+

∆ = = −

∆ + =

+ ∆ −∆

( )2

1dc

j tacqV q

i kT kTdc a

V e

cA

j t nn n ee e

N

ω

ω

∆ + ∆ ≈ −

2

( 0)dcqV

ac kTac

i

A

qVn x e C

kT

n

N∆ = = =

2

1 1dcqV

ackit

T

A

jqVe

n e

N kT

ω ≈ + −

xn

n

Taylor expansion


AC Current and Impedance

49

2

*0

dcqVac n i kT

ac nx n A

acVd n qD q nJ qD e

dx L kT N=

∆= − =

2

( 0)dcqV

ac i kTac

A

qV nn x e

kT NC∆ = = =

2 2

0*1

dcq

nn

Vac n i kT

acac A

J q D nY e G

V kTj

NLωτ+= = ≡

* * *( ) n n n

x x x

L L Lacn x e De eC C

− + −∆ = + →Finally…

AC Impedance

AC Current


Diffusion Conductance and Capacitance

50

1/ 22 20

1/ 22 20

1 12

1 12

D n

D n

GG

GC

τ

ω ω

ω

τ

= + +

= + −

0 1ac D D nY G j C G j= + ≡ +ω ωτ

Separate in real & imaginary parts …

DG ω∝

1/DC ω∝Product of GD and CDfrequency-independent


Conclusion

51

1) Small signal response relevant for many analog

applications.

2) Small signal parameters always refer to the DC operating

conditions, as such the parameter changes with bias

condition.

3) Important to distinguish between majority and minority

carrier capacitance. Their relative importance depends on

specific applications.

ece606: solid state devices lecture 15 p-n diode ...ee606/downloads/ece606_f12_lecture15.pdf ·...

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