qualitative derivation

President University Erwin Sitompul SDP 8/1

Dr.-Ing. Erwin SitompulPresident University

Lecture 8Semiconductor Device Physics

http://zitompul.wordpress.com


Majoritycarriers

Majoritycarriers

Qualitative DerivationChapter 6 pn Junction Diodes: I-V Characteristics


Current Flow in a pn Junction DiodeChapter 6 pn Junction Diodes: I-V Characteristics

When a forward bias (VA > 0) is applied, the potential barrier to diffusion across the junction is reduced.

Minority carriers are “injected” into the quasi-neutral regions Δnp > 0, Δpn > 0.

Minority carriers diffuse in the quasi-neutral regions, recombining with majority carriers.


Ideal Diode: AssumptionsChapter 6 pn Junction Diodes: I-V Characteristics

Steady-state conditions.Non-degenerately doped step junction.One-dimensional diode.Low-level injection conditions prevail in the quasi-neutral

regions.No processes other than drift, diffusion, and thermal R–G take

place inside the diode.


N n N n N( )( ) dn d nx q n qD q n qD

dx dx

J E E

P p P p P( )( ) dp d px q p qD q p qD

dx dx

J E E

Current Flow in a pn Junction DiodeChapter 6 pn Junction Diodes: I-V Characteristics

Current density J = JN(x) + JP(x)

JN(x) and JP(x) may vary with position, but J is constant throughout the diode.


n-sidep-side

p0 p A

2i

p0 pA

( )

( )

p x N

nn xN

n0 n D

2i

n0 nD

( )

( )

n x N

np xN

p p A( )p x N n n D( )n x N

Carrier Concentrations at –xp, xn

Chapter 6 pn Junction Diodes: I-V Characteristics

Consider the equilibrium carrier concentrations at VA = 0:

If low-level injection conditions prevail in the quasi-neutral regions when VA 0, then:


i P

N i

( )i

( )i

E F kT

F E kTp n en n e

A2i qV kTnp n e

N ii P

N P

( )( )2i

( )2i

F E kTE F kT

F F kT

np n e e

n e

p nfor x x x

“Law of the Junction”Chapter 6 pn Junction Diodes: I-V Characteristics

The voltage VA applied to a pn junction falls mostly across the depletion region (assuming that low-level injection conditions prevail in the quasi-neutral regions).

Two quasi-Fermi levels is drawn in the depletion region:


p p A( )p x N

n-sidep-side

n n D( )n x N

A

2i

p pA

( ) ( 1)qV kTnn x eN

A

2i

n nD

( ) ( 1)qV kTnp x eN

Excess Carrier Concentrations at –xp, xn


A

A

2i

p pA

p0

( )

qV kT

qV kT

n en xN

n e

A

A

2i

n nD

n0

( )

qV kT

qV kT

n ep xN

p e


A 0.6 0.02586 12 3p p p0( ) 100 1.192 10 cmqV kTn x n e e

12 12 3p p p p p0( ) ( ) 1.192 10 100 1.192 10 cmn x n x n

Example: Carrier InjectionChapter 6 pn Junction Diodes: I-V Characteristics

A pn junction has NA=1018 cm–3 and ND=1016 cm–3. The applied voltage is 0.6 V.

a) What are the minority carrier concentrations at the depletion-region edges?

b) What are the excess minority carrier concentrations?

A 4 0.6 0.02586 14 3n n n0( ) 10 1.192 10 cmqV kTp x p e e

14 4 14 3n n n n n0( ) ( ) 1.192 10 10 1.192 10 cmp x p x p


An n n0( ) ( 1)qV kTp x p e

2n n

P 2p

0 , 0d p pD xdx

P Pn 1 2( ) x L x Lp x A e A e

0x 0 x

P P pL D for 0x

Excess Carrier DistributionChapter 6 pn Junction Diodes: I-V Characteristics

From the minority carrier diffusion equation,

We have the following boundary conditions:

n ( ) 0p

For simplicity, we develop a new coordinate system:

Then, the solution is given by:

• LP : hole minority carrier diffusion length


An n0( 0) ( 1)qV kTp x p e

A Pn n0( ) ( 1) , 0qV kT x Lp x p e e x

P P'/ '/n 1 2( ) x L x Lp x A e A e

NAp p0( ) ( 1) , 0x LqV kTn x n e e x

A /1 n0 ( 1)qV kTA p e 2 0A

Excess Carrier DistributionChapter 6 pn Junction Diodes: I-V Characteristics

New boundary conditions

n ( ) 0p x

From the x’ → ∞,

From the x’ → 0,

Therefore

Similarly,


A Pn PP P n0

P

( )( ) ( 1)qV kT x Ld p x Dx qD q p e edx L

J

NAp NN N p0

N

( )( ) ( 1) x LqV kTd n x Dx qD q n e e

dx L

J

n-side

p-side

p nN P N P0 0x x x x x x

J J J J J

A2 N Pi

N A P D

( 1)qV kTD Dqn eL N L N

J

pn Diode I–V CharacteristicChapter 6 pn Junction Diodes: I-V Characteristics


A0

2 N P0 i

N A P D

( 1)qV kTI I eD DI Aqn

L N L N

I A J

pn Diode I–V CharacteristicChapter 6 pn Junction Diodes: I-V Characteristics

A2 N Pi

N A P D

( 1)qV kTD DAqn eL N L N

• Shockley Equation,for ideal diode

• I0 can be viewed as the drift current due to minority carriers generated within the diffusion lengths of the depletion region


I0 can vary by orders of magnitude, depending on the semiconductor material, due to ni

2 factor. In an asymmetrically doped pn junction, the term associated

with the more heavily doped side is negligible. If the p side is much more heavily doped,

If the n side is much more heavily doped,

2 NP0 i

P D N A

DDI AqnL N L N

2 P0 i

P D

DI AqnL N

2 N0 i

N A

DI AqnL N

Diode Saturation Current I0



N P J J J

N p n N p

P p n P n

( ) ( )

( ) ( )

x x x x

x x x x

J J

J J

Diode Carrier CurrentsChapter 6 pn Junction Diodes: I-V Characteristics

• Total current density is constant inside the diode

• Negligible thermal R-G throughout depletion region dJN/dx = dJP/dx = 0


n0p

p0p

p0n

n0n

A2i qV kTnp n e

p A

n D

p Nn N

Excess minoritycarriers

Excess minoritycarriersA P

n n0( ) ( 1)qV kT x Lp x p e e NAp p0( ) ( 1) x LqV kTn x n e e

Carrier Concentration: Forward BiasChapter 6 pn Junction Diodes: I-V Characteristics

Law of the Junction

Low level injection conditions


Carrier Concentration: Reverse BiasChapter 6 pn Junction Diodes: I-V Characteristics

Deficit of minority carriers near the depletion region.Depletion region acts like a “sink”, draining carriers from the

adjacent quasineutral regions


Forward-bias current Reverse-bias current

“Breakdown”

“Slope over” No saturation

Smaller slope

Deviations from the Ideal I-V BehaviorChapter 6 pn Junction Diodes: I-V Characteristics

Si pn-junction Diode, 300 K.


Dominant breakdown mechanism is tunneling

BR 0.75B

1 VN

Empirical Observations of VBR


• VBR : breakdown voltage

VBR decreases with increasing N,

VBR decreases with decreasing EG.


2S CR A D

BR biA D2

N NV Vq N N

E

A DCR bi BR

S A D

2 N Nq V VN N

E• At breakdown, VA=–VBR

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” and large reverse current will flow

Thus, the reverse bias at which breakdown occurs is


2 A DCR BR

S A D

2 N Nq VN N

ESmall E-field:

High E-field:

Low energy, causing lattice vibration and localized heating

High energy, enabling impact ionization which causing avalanche, at doping level N < 1018 cm–3

Breakdown Mechanism: Avalanching Chapter 6 pn Junction Diodes: I-V Characteristics

• ECR : critical electric field in the depletion region


Breakdown Mechanism: Zener Process Chapter 6 pn Junction Diodes: I-V Characteristics

Zener process is the tunneling mechanism in a reverse-biased diode.Energy barrier is higher than the

kinetic energy of the particleThe particle energy remains

constant during the tunneling process

Barrier must be thin dominant breakdown mechanism when both junction sides are heavily doped

Typically, Zener process dominates when VBR < 4.5V in Si at 300 K and N > 1018 cm–3.


2i

thermal p 1 n 1R-G( ) ( )

np nnt n n p p

n

p

R-GthermalR G

x

x

nI qA dxt

T i( )1 i

E E kTn n e

Effect of R–G in Depletion RegionChapter 6 pn Junction Diodes: I-V Characteristics

R–G in the depletion region contributes an additional component of diode current IR–G.

The net generation rate is given by

i T( )1 i E E kTp n e

• ET: trap-state energy level


For reverse bias, with the carrier concentrations n and p being negligible,

n

p

2i

R-Gp 1 n 1( ) ( )

x

x

np nI qA dxn n p p

iR-G

02qAnWI

1 10 p n

i i

1 2

n pn n

where


Continuing,

• Reverse biases with VA< – few kT/q


A 2R-G i

qV kTI qAnWe


n

p

2i

R-Gp 1 n 1( ) ( )

x

x

np nI qA dxn n p p

Continuing,

For forward bias, the carrier concentrations n and p cannot be neglected,


A2 N PDIFF i

N A P D

( 1)qV kTD DI Aqn eL N L N

A

A

iR-G

0 n p 2bi A

0

( 1)2

12

qV kT

qV kT

qAnW eIV V e

kT q

DIFF R GI I I

Diffusion, ideal diode



J A SV V IR

Voltage drop, significant for high I

A S

J

( )0

0 A bi,

q V IR kT

qV kTI I e

I e V V

RS can be determined experimentally

SV IR

Sslope=R

Effect of Series ResistanceChapter 6 pn Junction Diodes: I-V Characteristics


n n0n n (for a p+n junction)

(for a pn+ junction)p p0p p

Effect of High-Level InjectionChapter 6 pn Junction Diodes: I-V Characteristics

As VA increases and about to reach Vbi, the side of the junction which is more lightly doped will eventually reach high-level injection:

This means that the minority carrier concentration approaches the majority doping concentration.

Then, the majority carrier concentration must increase to maintain the neutrality.

This majority-carrier diffusion current reduces the diode current from the ideal.


Perturbation of both minority and majority carrier

High-Level Injection EffectChapter 6 pn Junction Diodes: I-V Characteristics


Forward-bias current Reverse-bias current

Deviations from ideal I-V

Due to avalanching and Zener process

Due to high-level injection and series resistance in

quasineutral regions

Due to thermal generation in depletion region

Due to thermal recombination in depletion region

SummaryChapter 6 pn Junction Diodes: I-V Characteristics


This time no homework. Prepare well for Quiz 2.

Chapter 5 pn Junction Electrostatics

Homework

qualitative derivation

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