Lecture 5: Junctions

Lecture 5Junctions!

R. TreharneNov 6th 2014

Lecture OutlineL5

● Drawing Junctions

● Junction Formation – A Physical Picture

● Abrupt Junction

● Deriving Important Junction parameters

● Deriving diode equation

● Ideality Factor - Recombination

Metal-SemiconductorL5

metal SC (n-type)

qφm

qφn

qχ

EG

EC

EV

EV

EF

Schottky Junction


metal SC (n-type)

qφm

qφn

qχ

EC

EV

EV

EF E

F

EG

One Rule!● Fermi level must be in equilibrium


metal SC (n-type)

qφm

qφn

qχ

EC

EV

EV

EF E

F

qφm

qφn

qχ

EC

EV

EV

metal SC (n-type)

EG E

G

“Intimate Contact”


metal SC (n-type)

qφm

qφn

qχ

EC

EV

EV

EF

metal SC (n-type)

EG

qφn

qχ

EC

EV

EV

EF

EG

qφm

qVbiqφ

B

Vbi = φ

m -

φ

s

φB = φ

m - χ

“Built-in Voltage”

Barrier Height

BAND BENDING!


metal SC (p-type)

qφm

qφp

qχ

EG

EC

EV

EV

EF


metal SC (p-type)

qφm

qφp

qχ

EG

EC

EV

EV

EF

qφp

qχ

EC

EV

EV

EF

EG

qφm

qVbi

qφB

metal SC (p-type)

Vbi = φ

m -

φ

p

φB = E

G/q - (φ

m - χ)


φB = φ

m - χ Why doesn't this rule hold?

Metal – SC interface effects

● Lattice Mis-match

● High density of defects

● Interface states in gap

Semiconductor-SemiconductorL5

qφp

qχp

EG

EF

p n

qφn

qχn

EV

EF

EG

p-n homojunction

EC


qφp

qχp

EG

EF

p n

qφn

qχn

EV

EF

EG

p-n homojunction

EF

EF

EC


qφp

qχp

EG

EF

p n

qφn

qχn

EV

EF

EG

p-n homojunction

qφp

qχp

EF

qφn

qχn

EG

EF

p n

EG

EV

EC

EC

EC

EV


qφp

qχp

EG

EF

p n

qφn

qχn

EV

EF

EG

p-n homojunction

EF

EF

p n

qVbi


qφp

qχp

EG

EF

p n

qφn

qχn

EV

EF

EG

p-n homojunction

EF

EF

p n

qVbi

Ei

Ei


qφp

qχp

EG

EF

p n

qφn

qχn

EV

EF

EG

p-n homojunction

EF

EF

p n

qVbi

Ei

Ei

Defining Electrostatic potential, Ψ, for a SC● Difference between Ei and Ef

qΨn

qΨp

Vbi

= Ψn - Ψ

p

Ele

c tr o

s ta

tic P

ot e

ntia

l , Ψ

Ele

c tr o

n e

ne

r gy,

E

KNOW HOW TO DRAW THIS!

Coming back here shortly to get a more physical picture


Everybody's favourite homojunction - Silicon

p-type n-type

http://electronics.howstuffworks.com/diode1.htm

Every bit of electronics you own is jam packed with silicon homojunctions

Extrinsically doped

iPhone 5:1 billion transistors!~ 45 nm wide

Mono-C solar cells~750 μm thickcell size = 1 cm2

http://electronics.howstuffworks.com/diode1.htm


qφp

qχp

EFp

p n

qφn

qχn

EFn

p-n heterojunction: Different band gaps

ΔEV

ΔEC

wide gap narrow gap

EF

1. Align the Fermi level. Leave some space for transition region.


qφp

qχp

EFp

p n

qφn

qχn

EFn


ΔEV

ΔEC

wide gap narrow gap

EF

2. Mark out ΔEC and ΔE

C at mid-way

points

ΔEC

ΔEC


qφp

qχp

EFp

p n

qφn

qχn

EFn


ΔEV

ΔEC

wide gap narrow gap

EF

3. Connect the C.B. and V.B. keeping the band gap constant in each material

Barrier to hole transport

Difference in band gaps give rise to discontinuities in band diagrams. This limits the carrier transport by introducing potential barriers at the junction


Examples of p-n heterojections

http://pubs.rsc.org/en/content/articlehtml/2013/ee/c3ee41981a#cit111

Most thin-film PV technologies

http://pubs.rsc.org/en/content/articlehtml/2013/ee/c3ee41981a#cit111


Examples of p-n heterojections

III-V multi-junction devices

Being able to control the band offsets of a material can help eliminate barriers

Physical PictureL5

Drawing band schematics for junctions all day is fun, but what is actually going on here?

http://www.youtube.com/watch?v=JBtEckh3L9Q

Excellent qualitative description6:27 – Didactic Model of Junction formation



Physical PictureL5

Equilibrium Fermi Levels – Explained – our one rule for drawing band schematics!

J=J (drift )+J (diffusion)=0

In thermal equilibrium, i.e. steady state: not net current flows

http://en.wikipedia.org/wiki/File:Pn-junction-equilibrium.png

notice position of “metallurgical” junction. Usually call this x=0

http://en.wikipedia.org/wiki/File:Pn-junction-equilibrium.png

Physical PictureL5

Lets look at hole current first:

J p = J p(drift)+J p(diffusion)

= qμ p p ξ−qDpdpdx

Physical PictureL5

Lets look at hole current first:


= qμ p p ξ−qDpdpdx

μ p=e τ

mh

mobility: hole distribution: e-field: diffusion coefficient

p=ni e(E i−E F)/ kBT

OK to use Boltzmann as approx to Fermi-Dirac

dpdx

=p

k BT ( dE i

dx−dEF

dx )

ξ=1e

dEi

dxD p=(k BT /e )μ p

Einstein relation

will use this shortly

Physical PictureL5


= qμ p p ξ−qD pdpdx

= μ p pdEi

dx−μ p p(dEi

dx−dE F

dx )0 = μ p p

dEF

dx

Condition for steady state

Physical PictureL5

dEF

dx=0

TA DA!

The same is true from consideration of net electron current, Jn

Abrupt JunctionL5

Hook & Hall, “Soild State Physics”, 2nd ed. Chap 6, p. 171

outside depletion region:p = N

A and n = N

D

Space-charge conditionN

Aw

p = N

dw

ni.e. areas of rectangles must be the same

electric field

electrostaticpotential

charge density

I've already called this Vbi

Max field (at x=0):ξ

m = qN

Dx

n/ε

s = qN

Ax

p/ε

s

ξm

Area = Vbi

V bi=ϕn−ϕp=k BTq

ln(N AN D

ni2 )

http://library.liv.ac.uk/search/?searchtype=a&SORT=D&searcharg=Hook+J+R&searchscope=8

Abrupt JunctionL5

Hook & Hall, “Soild State Physics”, 2nd ed. Chap 6, p. 171

outside depletion region:p = N

A and n = N

D

Space-charge conditionN

Aw

p = N

dw

ni.e. areas of rectangles must be the same

electric field

electrostaticpotential

charge density

I've already called this Vbi

Max field (at x=0):ξ

m = qN

Dx

n/ε

s = qN

Ax

p/ε

s

ξm

Area = Vbi

V bi=ϕn−ϕp=k BT

qln( N AN D

ni2 )

Lets derive this!(but skip if I'm being slow)

http://library.liv.ac.uk/search/?searchtype=a&SORT=D&searcharg=Hook+J+R&searchscope=8

Abrupt JunctionL5

From Earlier:

ϕ =−1e

(E i−EF )

What is in p and n regions away from junction?ϕ

Here

p=ni e(E i−E F)/ kBT n=ni e

(Ei−EF )/kBTand

ϕp≡−1e(E i−EF)|x⩽−x p

=−k BTe ( N A

ni)

ϕn≡−1e(E i−EF)|x⩾−xn

=−k BTe ( N D

ni)

S. M. Sze, “Semiconductor Devices: Physics and Technology”, Chap 4, p.90

http://library.liv.ac.uk/search~S8?/tsemiconductor+devices/tsemiconductor+devices/1,10,15,B/frameset&FF=tsemiconductor+devices&4,,4

Abrupt JunctionL5

Hence:

V bi=ϕn−ϕp=k BTq

ln( N A N D

ni2 )

Next Question:

How do I calculate in the depletion region? (And why would I want to?)

Must solve Poisson's Equation

ϕ

want to know what this is for Si? Go to: http://pveducation.org/pvcdrom/pn-junction/intrinsic-carrier-concentration

∇2ϕ=−

ρε

Go read some electrostatics if you're interested

http://pveducation.org/pvcdrom/pn-junction/intrinsic-carrier-concentration

Abrupt JunctionL5

∇2ϕ=−

ρε ρ(x)={

−N Ae+N D e

0

−wp< x<0

0< x<w n

elsewhere

ξ=−d ϕ

dx={−

N A eε ( x+w p)

N Deε ( x+wn)

−wp< x<0

0< x<wn

ϕ( x)={−N A e2ε

(x+wp)2

N D e

2ε(x+wn)

2

INTEGRATE ONCE

INTEGRATE TWICE

−wp< x<0

0< x<wn

Abrupt JunctionL5

V bi=e2ε

(N Aw p2+N Dwn

2)

Vbi

using result:

wn=√ 2 εN AV bi

eN D (N A+N D)

w p=√ 2εN DV bi

eN A(N A+N D)

and with some jiggery pokery:

W=w p+wn=√ 2ε

e (N A+N D

N A N D)

N Awp=N Dwn

V bi=ϕ(x=wn)−ϕ( x=−w p)

Under BiasL5

What happens to our ideal p-n junction if we put a forward or reverse bias across it?

Under BiasL5

p n

0 wn

-wp

W

EF

ZERO BIAS

Vbi

Under BiasL5

p n

0 wn

-wp

W

EF

FORWARD BIAS

Vbi - V

EF

+ -

● Electron energy levels in p side lowered relative to those in n side

● Energy barrier qVbi

reduced● Flow of electrons from n to p

increases● Flow of holes from p to n increases● Depletion width decreases

Under BiasL5

p n

0 wn

-wp

W

EF

REVERSE BIAS

Vbi + V

EF

- +

● Electron energy levels in p side raised relative to those in n side

● Energy barrier qVbi

increase by V● Flow of electrons from n to p

reduced● Junction width increases

L5

Under Bias

Of course, you already know all about this:

Ideal dark JV curveL2

J0 = 1 x 10-10 A/cm2

http://pveducation.org/pvcdrom/pn-junction/diode-equation

J=J 0[exp ( eVk BT )−1 ]



Ideal dark JV curveL2

J0 = 1 x 10-10 A/cm2


J=J 0[exp ( eVk BT )−1 ]

Lets derive this by considering our band schematic. Fun fun fun!



L5

Deriving Shockley Equation

Steady State again (i.e. no bias)Consider electron drift current – I

eo from p to n

L5



eo from p to n

I e0=e×(generationrate /volume)×(volumewithin Le of depletion zone )

W

Le

J e 0=e (npτ p

)Le

L5



eo from p to n

I e0=e×(generationrate /volume)×(volumewithin Le of depletion zone )

J e 0=e (npτ p

)Le

W

Le

number of electrons on p side

electron lifetime

Le=√D e τ pdiffusion length

L5



eo from p to n

W

Le

By assuming that all acceptors on p side are ionized:

np=n i

2

p p

=ni

2

N A

J e 0=eDe ni

2

LeN A

can re-write Je0

as:

L5


What happens under forward bias?

● Drift (p to n)? - Nothing. No potential barrier in this direction

● Diffusion (n to p)? - Increases by exponential factor:

exp ( eVk BT )

If we assume occupation of states in CB is given by Boltzmann distribution

L5


What happens under forward bias?

● Drift (p to n)? - Nothing. No potential barrier in this direction

● Diffusion (n to p)? - Increases by exponential factor:

exp ( eVk BT )

If we assume occupation of states in CB is given by Boltzmann distribution

Net electron current:

J e =−J e 0+J e 0 exp (eVk BT )= J e 0[exp (eVk BT )−1]

L5


Can do the same for the net hole current:

J h=J h0 [exp( eVk BT )−1] J h0=

eDH ni2

Lh N D

where

L5


Can do the same for the net hole current:

J h=J h0 [exp( eVk BT )−1] J h0=

eDH ni2

Lh N D

where

Therefore: Net Current!

J=J e+Jh=J0 [exp( eVkBT )−1] J 0=J e 0+J h0=eni

2( De

Ln N A

+D h

LhN A)

ideal diode equation reverse saturation current,a.k.a “dark current”

L5


Previous derivation ignores the recombination and generation of carriers within the depletion layer itself!

J=J 0[exp ( eVnk BT )−1]

L5


Previous derivation ignores the recombination and generation of carriers within the depletion layer itself!

J=J 0[exp ( eVnk BT )−1]

ideality factor – a complete FUDGE

Worth reading up about – but don't stress out about it.

● S. M. Sze, “Semiconductor Devices”, 2nd ed. Chap 4, pp 109 – 113● Hook & Hall “Solid State Physics”, 2nd ed. Chap 6, pp 178 – 179

I have these books if you want them.

Lecture SummaryL5

● Drawing Junctions

● Junction Formation – A Physical Picture

● Abrupt Junction

● Deriving Important Junction parameters

● Deriving diode equation

● Ideality Factor - Recombination

Lecture 5: Junctions

Science

Transcript of Lecture 5: Junctions