lecture #31 - university of california, berkeleyee130/sp03/lecture/...de dx q ε=−dv =1 c dx dε =...

1

EE130 Lecture 31, Slide 1Spring 2003

Lecture #31

ANNOUNCEMENTS• TA’s will hold a review session on Thursday May 15:

– 2-5 PM, 277 Cory

• Final Exam will take place on Friday May 23:– 12:30-3:30 PM, Sibley Auditorium (Bechtel Bldg.)– Closed book; 7 pgs of notes + calculator allowed

OUTLINE– Review of Fundamental Concepts


ni ≅ 1010 cm-3 at room temperature

Intrinsic Carrier Concentration

conduction

kTEvci

geNNn /−=

2


Charge-Carrier ConcentrationsND: ionized donor concentration (cm-3)NA: ionized acceptor concentration (cm-3)

Note: Carrier concentrations depend on net dopant concentration (ND - NA) !


Energy Band Diagram

electron kinetic energy

hole kinetic energy

Ec

Ev

incr

easi

ng e

lect

ron

ener

gy

incr

easi

ng h

ole

ener

gy

distance in semiconductor

3


Fermi Function

kTE-EeEf

eEf

FkTEE

kTEE

F

F

3 if )(

11)(

/)(

/)(

>≅

+=

−−

−


Relationship between EF and n,p

kTEEv

kTEEi

kTEEc

kTEEi

vF

Fi

Fc

iF

eN

enp

eN

enn

/)(

/)(

/)(

/)(

−

−

−

−

=

=

=

=

4


Free Carriers in Semiconductors

• Three primary types of carrier action occur inside a semiconductor:

– drift

– diffusion

– recombination-generation

qkTD =

µ


Net Generation Rate• The net generation rate is given by

levelenergy state- trap and where

)(τ)(τ/)(

1/)(

1

11

2

=≡≡

+++−=

∂∂=

∂∂

−−

T

kTEEi

kTEEi

np

i

Eenpenn

ppnnnpn

tn

tp

TiiT

5


Drift and Resistivity• Electrons and holes moving under the influence of an

electric field can be modelled as quasi-classical particles with average drift velocity

• The conductivity of a semiconductor is dependent on the carrier concentrations and mobilities

• Resistivity

|vd| = µ

σ = qnµn + qpµp

pn qpqn µµσρ

+== 11


1E14 1E15 1E16 1E17 1E18 1E19 1E20

0

200

400

600

800

1000

1200

1400

1600

Holes

Electrons

Mob

ility

(cm

2 V-1 s

-1)

Total Im purity Concenration (atom s cm -3)Total Doping Concentration NA + ND (cm-3)

impurityphonon

impurityphonon

µµµ

τττ111

111

+=

+=

Mobility Dependence on Doping

6


JN = JN,drift + JN,diff = qnµn + dxdnqDN

JP = JP,drift + JP,diff = qpµp –dxdpqDP

J = JN + JP

Total Current


Electrostatic Variables

)(1creference EE

qV −=

dxdE

qdxdVε c1=−=

dxdε=

ερ

7


Lp

p

Ln

n

Gpx

xJqt

p

Gnx

xJqt

n

)(1

)(1

+∆−∂

∂−=

∂∂

+∆−∂

∂=∂∂

τ

τContinuityEquations:

Lp

nnP

n

Ln

ppN

p

Gpx

pq

Dtp

Gn

xn

Dtn

1

2

2

2

2

+∆−∂∆∂=

∂∆∂

+∆

−∂∆∂

=∂∆∂

τ

τMinority Carrier

DiffusionEquations:


Work Function

ΦM: metal work function ΦS: semiconductor work function

Ε0: vacuum energy level

8


Schottky Diode

VA > 0 VA < 0

Fermi level splits into two levels (EFM and EFS) separated by qVA

)(2

D

Abis

qNVVW −= ε

2/

/2

A/cm 120 where

)1(kTq

S

kTqVS

B

A

eJ

eJATIΦ−=

−=A

WC sε=

ΦBn = ΦM – χ


pn Junction Electrostatics

9


pn Junction Electrostatics, VA ≠ 0• Built-in potential Vbi (non-degenerate doping):

• Depletion width W :

=

+

= 2lnlnln

i

DA

i

D

i

Abi n

NNq

kTnN

qkT

nN

qkTV

+−=+=

DAAbi

snp NN

VVq

xxW 11)(2ε

WNN

NxDA

Dp +

= WNN

NxDA

An +

=


if VBR >> Vbi

εcrit increases slightly with N:

For 1014 cm-3 < N < 1018 cm-3,

105 V/cm < εcrit < 106 V/cm

qNV crits

BR 2

2εε≈

Avalanche Breakdown Mechanism

Small E-field:

High E-field:

10


“Law of the Junction”The voltage VA applied to a pn junction falls mostly acrossthe depletion region (assuming that low-level injection conditions prevail in the quasi-neutral regions).We can draw 2 quasi-Fermi levels in the depletion region:

kTEFenn /)(i

iN −=

kTFEenp /)(i

Pi −=

kTqVenpn /2i

A =

kTFF

kTEFkTFE

PN

iNPi

en

eenpn/)(2

i

/)(/)(2i

−

−−

=

=


Excess Carrier Concentrations at –xp, xn

( )1)(

)(

)(

/

A

2

/0

A

/2

A

A

A

A

−=−∆

=

=−

=−

kTqVipp

kTqVp

kTqVi

pp

pp

eNnxn

en

Nenxn

Nxp

n-sidep-side

( )1)(

)(

)(

/

D

2

/0

D

/2

D

A

A

A

−=∆

=

=

=

kTqVinn

kTqVn

kTqVi

nn

nn

eNnxp

ep

Nenxp

Nxn

11


pLxkTVqn

p

pnpp eep

LD

qdx

xpdqDJ '0 )1(

')'(

A −−=∆−=

nLxkTVqp

n

npnn een

LDq

dxxnd

qDJ ''0 )1(

'')''(

A −−=∆

−=

pn Diode I-V Characteristic

n-side:

p-side:

)1( A

DA

2i

00

−

+=

+=+==′=′′=−=

kTVq

p

p

n

n

xpxnxxpxxn

eNL

DNL

DqnJ

JJJJJnp


pn Junction Capacitance2 types of capacitance associated with a pn junction:

1. CJ depletion capacitance

2. CD diffusion capacitance (due to variation of stored minority charge in the quasi-neutralregions)

For a one-sided p+n junction (QP >> QN ):

qkTI

GdVdI

dVdQC

/τ

ττ DCpp

Ap

AD ====

WA

dVdQ

C sε=≡A

depJ

12


Deviations from the Ideal I-V BehaviorResulting from• recombination/generation in the depletion region• series resistance• high-level injection


Transient Response of pn Diode• 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:

13


Minority-Carrier Injection & Collection• Under forward bias, minority carriers are injected

into the quasi-neutral regions of the diode.Current flowing across junction is comprised of hole and electron components

• Under reverse bias, minority carriers are collectedinto the quasi-neutral regions of the diode. (Minority carriers within a diffusion length of the depletion region will diffuse into the depletion region and then be swept across the junction by the electric field)→ Current flowing depends on the rate at which minority

carriers are supplied


MOS Band Diagrams (n-type Si)

• Inversion– VG < VT– Surface

becomes p-type

• Accumulation– VG > VFB– Electrons

accumulate at surface

• Depletion– VG < VFB– Electrons

repelled from surface

Decrease VG (toward more negative values) -> move the gate energy-bands up, relative to the Si

decrease VG decrease VG

soxFBG VVV ψ++=

14


Biasing Conditions for p-type Si

VG = VFB VG < VFB VT > VG > VFB

increase VG increase VG

A

sSi

qNW ψε2

d =


G

s

dVdQC =

MOS Charge & Capacitance (p-type Si)

VG

accumulation depletion inversion

VFB VT

slope = -Cox

VG

accumulation depletion inversionVFB VT

C

Cox

Ideal C-V curve

)( FBGoxacc VVCQ −−=

)( TGoxinv VVCQ −−=

ox

BSiABFBT C

qNVV

)2(22

ψεψ ++=

ox

FMSFB C

QV −=φ

15


VT Adjustment by Back Biasing• In some IC products, VT is dynamically adjusted by

applying a back bias:– When a MOS capacitor is biased into inversion, a pn junction

exists between the surface and the bulk.– If the inversion layer contacts a heavily doped region of the

same type, it is possible to apply a bias to this pn junction

N+ poly-Si

p-type Si

-- - - --

+ + + + + +

N+

+ +

-- -SiO2

• VG biased so surface is inverted• Inversion layer contacted by N+ region• Bias VC applied to channel

Reverse bias VB-VC applied btwn channel & body


Effect of VCB on ψs and VT• Application of reverse bias -> non-equilibrium

– 2 Fermi levels (one for n-region, one for p-region)• Separation = qVBC ψs increased by VC

• Reverse bias widens Wd, increases QdepQinv decreases with increasing VCB, for a given VGB

ox

CBBSiABCFBT C

VqNVVV

)2(22

++++=

ψεψ

lecture #31 - university of california, berkeleyee130/sp03/lecture/...de dx q ε=−dv =1 c dx dε =...

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