“o” subscript denotes the equilibrium carrier concentration. ideal diode equation

“o” subscript denotes the equilibrium carrier concentration.

Ideal diode equation

material. type-nin ion concentrat hole mequilibriu

material. type-pin ion concentrat hole mequilibriu

material. type-pin ion concentratelectron mequilibriu

material. type-nin ion concentratelectron mequilibriu

no

po

po

no

p

p

n

n

2. inpn

At equilibrium case ( no bias )


)2(.

)1(.

.

2

2

2

materialtypepnpn

materialtypennpn

npn

ipopo

inono

i


2

22

ln assuming full ionization

; majority carriers

.ln . for n-type

ln exp (3)

A Dbi

i

A po D no

po nobi no no i

i

po bibi po no

no

kT N NV

q n

N p N n

p nkTV n p n

q n

p qVkTV p p

q p kT


22

, (2)

.ln .

ln exp (4)

po nobi po po i

i

no bibi no po

po

Similarly from equation

p nkTV n p n

q n

n qVkTV n n

q n kT

This equation gives us the equilibrium majority carrier concentration.


What happens when a voltage appears across the p-n junction ?

Equations (3) and (4) still valid but you should drop (0) subscript and change Vbi with

i. Vbi – VF if a forward bias is applied.

ii. Vbi + VR if a reverse bias is applied.

VF : forward voltage

VR : reverse voltage

With these biases, the carrier densities change from equilibrium carrier densities to non- equilibrium carrier densities.


Non-equilibrium majority carrier concentration in forward bias;

( )exp

For example; for reverse bias

( )exp

bi Fp n

n

bi Rn p

q V Vp p

kT

n

q V Vn n

kT

• When a voltage is applied; the equilibrium nno changes to the non equilibrium nn.

Assumption; low level injection

• For low level injection; the number of injected minorities is much less than the number of the majorities. That is the injected minority carriers do not upset the majority carrier equilibrium densities.

pop

non

pp

nn

• Non equilibrium electron concentration in n-type when a forward bias is applied ,

( )exp non-equilibrium.bi F

n p

q V Vn n

kT


( ) (5)

exp (6)

bi Fn no no p

bino po

q V Vn n n n

kT

qVn n

kT

combining (5) and (6)

( )exp expbi F bi

p po

q V V qVn n

kT kT


poexp and subtracting n from both sides

exp 1

the excess concentration of minority electrons

over the equilibrium concentration at the edge of the DR

Fp po

p po po n

n

qVn n

kT

qVn n n

kT

Solving for non-equilibrium electron concentration in p-type material, i.e. np


Similarly ,

exp 1 the non-equilibriumn no no p

qVp p p

kT

the excess concentration of minority holes

over the equilibrium concentration at the edge of the DR

p

Forward-bias diode; injection of minority carriers across DR

+ -

p n++

--

B

pno

npo

A

nno

ppo

point A

point B

p po

n no

n n

p p

lnlp

• is the distance from DR edge into p-side

• is the distance from DR edge into n-side

p

n

l

l

When a forward bias is applied; majority carriers are injected across DR and appear as a minority carrier at the edge of DR on opposite side. These minorities will diffuse in field free opposite-region towards ohmic contact. Since ohmic contact is a long way away, minority carriers decay exponentially with distance in this region until it reaches to its equilibrium value.

Exponential decay of injected minority carriers on opposite sides

The excess injected minorities decay exponentially as

p

( ) (0)exp

( ) (0)exp

and L are diffusion lengths

for electrons and holes

pp

n

nn

p

n

lp l n

L

ln l p

L

L

Number of Injected Minority Holes Across The Depletion Region

• By means of forward-biasing a p-n junction diode, the holes diffuse from left to right accross the DR and they become minority carriers.

• These holes recombine with majority electrons when they are moving towards ohmic constants.

• So, the number of minority holes on the n-region decreases exponentially towards the ohmic contact. The number of injected minority holes;excess holes;

( ) (0)exp( )nn

p

lp l p

L

Distance into then region from the Depletion Region

Diffusion Length for holes

Number of Injected Minority Holes Across The Depletion Region

point A

( ) (0)exp( )

(0) exp( ) 1

(0) exp 1

p po

pp

n

po

no

n n

ln l n

L

qVn n

kT

qVp p

kT


• Similarly by means of forward biasing a p-n junction, the majority electrons are injected from right to left across the Depletion Region. These injected electrons become minorities at the Depletion Region edge on the p-side, and they recombine with the majority holes. When they move into the neutral p-side, the number of injected excess electrons decreases exponentially.

)exp(1)exp()(

)exp()0()(

n

ppopn

n

pnpn

L

l

kT

qVnl

L

ll

Ideal diode equationDiffusion current density for electrons ;

( ) exp 1 exp

( 0) exp 1 Minus sign shows that electron current

density is

n po pn n n p

n n

n pon p

n

D n ldn d qVJ qD qD n l q

dx dx L kT L

qD n qVJ l

L kT

in opposite direction to increasing .That is in the positive x direction.

Similarly for holes;

pl

( 0) exp 1

The total current density;

exp 1

p nop n p

p

n po p noTotal n p

n p

qD pdp qVJ l qD

dx L kT

D n D p qVJ J J q

L L kT


exp 1 exp 1

multiplying by area ;

exp 1

n po p noTotal o

n p

o

D n D p qV qVJ q J

L L kT kT

qVI I

kT


This equation is valid for both forward and reverse biases; just change the sign of V.


• Change V with –V for reverse bias. When qV>a few kT; exponential term goes to zero as

VB I0

VB ; Breakdown voltage

I0 ; Reverse saturation current

Forward Bias

Reverse Bias

oII Reverse saturation currentexp 1o

qVI I

kT

Current

Voltage

Forward bias current densities

+ -

p n

ln=0lp=0

Ohmic contact

pntotal JJJ

pJnJ

Cur

rent

den

sity

Jtotal is constant through the whole diode.

Minority current densities decreases exponentially into the the neutral sides whereas the current densities due to the majorities increase into the neutral sides.

p-n junction in reverse bias

• Depletion region gets bigger with increasing reverse bias.

• Reverse bias prevents large diffusion current to flow through the diode.

• However; reverse bias doesn’t prevent the small current flow due to the minority carrier. The presence of large electric field across the DR extracts almost all the minority holes from the n-region and minority electrons from the p-region.

• This flow of minority carriers across the junction constitudes I0, the reverse saturation current.

• These minorities are generated thermally.

- +

p n

ln=0lp=0

pntotal JJJ

pJnJCur

rent

den

sity

Car

rier

dens

ity

pn

np

pnonpo

p-n junction in reverse bias

• The flow of these minorities produces the reverse saturation current and this current increases exponentially with temperature but it is independent of applied reverse voltage.

Vb I0

VB ; Breakdown voltage

I0 ; Reverse saturation current

Forward Bias

Reverse BiasDrift current

I(current)

V(voltage)

Junction breakdown or reverse breakdown

• An applied reverse bias (voltage) will result in a small current to flow through the device.

• At a particular high voltage value, which is called as breakdown voltage VB, large currents start to flow. If there is no current limiting resistor which is connected in series to the diode, the diode will be destroyed. There are two physical effects which cause this breakdown.

1) Zener breakdown is observed in highly doped p-n junctions and occurs for voltages of about 5 V or less.

2) Avalanche breakdown is observed in less highly doped p-n junctions.

Zener breakdown

• Zener breakdown occurs at highly doped p-n junctions with a tunneling mechanism.

• In a highly doped p-n junction the conduction and valance bands on opposite side of the junction become so close during the reverse-bias that the electrons on the p-side can tunnel from directly VB into the CB on the n-side.

Avalanche Breakdown

• Avalanche breakdown mechanism occurs when electrons and holes moving through the DR and acquire sufficient energy from the electric field to break a bond i.e. create electron-hole pairs by colliding with atomic electrons within the depletion region.

• The newly created electrons and holes move in opposite directions due to the electric field and thereby add to the existing reverse bias current. This is the most important breakdown mechanism in p-n junction.

p – n junction

There is a big discontinuity in the fermi level accross the p-n junction.

EC

Eİ

EV

Ef

EC

Eİ

EV

Ef

EC

Eİ

EV

Ef

p-type n-typeEC

Eİ

EV

Ef

p-type n-type

“o” subscript denotes the equilibrium carrier concentration. ideal diode equation

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