Lecture 11

Lecture 11

OUTLINE• pn Junction Diodes (cont’d)

– Narrow-base diode– Junction breakdown

Reading: Pierret 6.3.2, 6.2.2; Hu 4.5

Introduction

Lecture 11, Slide 2

• The ideal diode equation was derived assuming that the lengths of the quasi-neutral p-type & n-type regions (WP’ , WN’) are much greater than the minority-carrier diffusion lengths (Ln , Lp) in these regions.

Excess carrier concentrations decay exponentially to 0. Minority carrier diffusion currents decay exponentially to 0.

• In modern IC devices, however, it is common for one side of a pn junction to be shorter than the minority-carrier diffusion length, so that a significant fraction of the “injected” minority carriers reach the end of the quasi-neutral region, at the metal contact.

Recall that p = n = 0 at an ohmic contact

In this lecture we re-derive the diode I-V equation with the boundary condition that p = 0 at a distance xc’ (rather than ) from the edge of the depletion region.

EE130/230A Fall 2013

Excess Carrier Distribution (n side)• From the minority carrier diffusion equation:

• For convenience, let’s use the coordinate system:

• So the solution is of the form:

• We have the following boundary conditions:

)1()0'( / kTqVnon

Aepxp

22

2

p

n

pp

nn

L

p

D

p

dx

pd

pp LxLxn eAeAxp /'

2/'

1)'(

Lecture 11, Slide 3

0)''( cn xxp

xc'

0 x’x’’ 0

EE130/230A Fall 2013

• Applying the boundary conditions, we have:

• Therefore

• Since this can be rewritten as

• We need to take the derivative of pn’ to obtain the hole diffusion current within the quasi-neutral n region:

'

//

/'/'/

0 '0 ,)1()'( ''

''

cLxLx

LxxLxxkTqV

nn xxee

eeepxp

PcPc

PcPc

A

0)(

)1()0(/

2/

1'

/21

''

pcpc

A

LxLxcn

kTqVnon

eAeAxp

epAAp

2sinh ee

''

'/

0 '0 ,/sinh

/'sinh)1()'( c

Pc

PckTqVnn xx

Lx

Lxxepxp A

Lecture 11, Slide 4x

xpqDJ n

pP

)(

EE130/230A Fall 2013

Thus, for a one-sided p+n junction (in which the current is dominated by injection of holes into the n-side) with a short n-side:

)1(0 kTVq AeII Pc

Pc

D

i

P

P

Lx

LxNn

LDqAI

/sinh

/cosh'0 '

'2

where

Pc

PcPkTqV

npp Lx

LxxL

epqDJ A

/sinh

/cosh1

1 /0

2cosh ee where

Lecture 11, Slide 5

Pc

PckTVq

D

i

P

p

xp Lx

Lxe

N

n

L

DqJ A

/sinh

/cosh)1(

2

0

Evaluate Jp at x=xn (x’=0) to find the injected hole current:

EE130/230A Fall 2013

0 as sinh 0 as 1cosh 2

Therefore if xc’ << LP:

For a one-sided p+n junction, then:

and

D

i

c

p

c

P

D

i

P

p

c

P

Pc

Pc

Pc

Pc

N

n

x

DqA

x

L

N

n

L

DqAI

x

L

Lx

Lx

Lx

Lx

22

0

2

/

/1

/sinh

/cosh

Lecture 11, Slide 6EE130/230A Fall 2013

pn is a linear function:

Jp is constant

(No holes are lost due to recombination as they diffuse to the metal contact.)

'/

0'

'/

0

'

'/

0

'1)1(

/

/')1(

/sinh

/'sinh)1()'(

A

A

c

kTqVn

Pc

PckTqVn

Pc

PckTqVnn

x

xep

Lx

Lxxep

Lx

Lxxepxp

A

Lecture 11, Slide 7

Excess Hole Concentration Profile

)1( / kTqVno

Aep

x'0 x'c0

slope is constant

pn(x)

If xc’ << LP:

EE130/230A Fall 2013

General Narrow-Base Diode I-V• Define WP‘ and WN’ to be the widths of the quasi-neutral regions.

• If both sides of a pn junction are narrow (i.e. much shorter than the minority carrier diffusion lengths in the respective regions):

11 /0

/2

kTqVkTqV

AP

N

DN

Pi

AA eIeNW

D

NW

DqAnI

Lecture 11, Slide 8

xJN

xn-xp

JJP

e.g. if hole injection into the n side is greater than electron injection into the p side:

EE130/230A Fall 2013

Summary: Narrow-Base Diode• If the length of the quasi-neutral region is much shorter than the

minority-carrier diffusion length, then there will be negligible recombination within the quasi-neutral region and hence all of the injected minority carriers will “survive” to reach the metal contact.– The excess carrier concentration is a linear function of distance.

For example, within a narrow n-type quasi-neutral region:

The minority-carrier diffusion current is constant within the narrow quasi-neutral region.

Shorter quasi-neutral region steeper concentration gradient higher diffusion current

)1( / kTqVno

Aep

x

pn(x)

xn0

location of metal contact(pn=0)

WN’

EE130/230A Fall 2013 Lecture 11, Slide 9

pn Junction Breakdown

A Zener diode is designed to operate in the breakdown mode:

Breakdown voltage, VBR VA


C. C. Hu, Modern Semiconductor Devices for Integrated Circuits, Figure 4-10

Review: Peak E-Field in a pn Junction

DA

DA

Si

Abi

Si

nD

Si

pA

Si

NN

NNVVqxqNxqN

dx

2 )0(

xxn-xp

E(x)

E(0)

Si

Abi NVVq

2

)0(

where N is the dopant concentration on the lightly doped side

For a one-sided junction,


Breakdown Voltage, VBR

• If the reverse bias voltage (-VA) is so large that the peak electric field exceeds a critical value ECR, then the junction will “break down” (i.e. large reverse current will flow)

• Thus, the reverse bias at which breakdown occurs is

biCRs

BR VqN

V 2

2

s

BRbiCR

VVqN

2


Avalanche Breakdown Mechanism

if VBR >> VbiHigh E-field:

Low E-field:ECR increases slightly with N:

For 1014 cm-3 < N < 1018 cm-3, 105 V/cm < ECR < 106 V/cm

qNV CRsBR 2

2


R. F. Pierret, Semiconductor Device Fundamentals, Figure 6.12

Tunneling (Zener) Breakdown Mechanism Dominant breakdown mechanism when both sides of a junction are very heavily doped.

VA = 0 VA < 0

biCRs

BR VqN

V 2

2

V/cm 106CRTypically, VBR < 5 V for Zener breakdown

Ec

Ev

Lecture 11, Slide 14EE130/230A Fall 2013C. C. Hu, Modern Semiconductor Devices for Integrated Circuits, Figure 4-12

Empirical Observations of VBR

• VBR decreases with increasing N

• VBR decreases with decreasing EG


R. F. Pierret, Semiconductor Device Fundamentals, Figure 6.11

VBR Temperature Dependence

• For the avalanche mechanism: – VBR increases with increasing T, because the mean free

path decreases

• For the tunneling mechanism: – VBR decreases with increasing T, because the flux of

valence-band electrons available for tunneling increases


Summary: Junction Breakdown• If the peak electric field in the depletion region exceeds a

critical value ECR, then large reverse current will flow.

This occurs at a negative bias voltage called the breakdown voltage, VBR:

where N is the dopant concentration on the more lightly doped side

• The dominant breakdown mechanism isavalanche, if N < ~1018/cm3

tunneling, if N > ~1018/cm3

biCRs

BR VqN

V 2

2


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Lecture 11