lecture 5: junctions
TRANSCRIPT
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Lecture 5Junctions!
R. TreharneNov 6th 2014
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Lecture OutlineL5
● Drawing Junctions
● Junction Formation – A Physical Picture
● Abrupt Junction
● Deriving Important Junction parameters
● Deriving diode equation
● Ideality Factor - Recombination
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Metal-SemiconductorL5
metal SC (n-type)
qφm
qφn
qχ
EG
EC
EV
EV
EF
Schottky Junction
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Metal-SemiconductorL5
metal SC (n-type)
qφm
qφn
qχ
EC
EV
EV
EF E
F
EG
One Rule!● Fermi level must be in equilibrium
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Metal-SemiconductorL5
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”
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Metal-SemiconductorL5
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!
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Metal-SemiconductorL5
metal SC (p-type)
qφm
qφp
qχ
EG
EC
EV
EV
EF
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Metal-SemiconductorL5
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)
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Metal-SemiconductorL5
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
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Semiconductor-SemiconductorL5
qφp
qχp
EG
EF
p n
qφn
qχn
EV
EF
EG
p-n homojunction
EC
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Semiconductor-SemiconductorL5
qφp
qχp
EG
EF
p n
qφn
qχn
EV
EF
EG
p-n homojunction
EF
EF
EC
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Semiconductor-SemiconductorL5
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
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Semiconductor-SemiconductorL5
qφp
qχp
EG
EF
p n
qφn
qχn
EV
EF
EG
p-n homojunction
EF
EF
p n
qVbi
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Semiconductor-SemiconductorL5
qφp
qχp
EG
EF
p n
qφn
qχn
EV
EF
EG
p-n homojunction
EF
EF
p n
qVbi
Ei
Ei
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Semiconductor-SemiconductorL5
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
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Semiconductor-SemiconductorL5
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
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Semiconductor-SemiconductorL5
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.
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Semiconductor-SemiconductorL5
qφp
qχp
EFp
p n
qφn
qχn
EFn
p-n heterojunction: Different band gaps
ΔEV
ΔEC
wide gap narrow gap
EF
2. Mark out ΔEC and ΔE
C at mid-way
points
ΔEC
ΔEC
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Semiconductor-SemiconductorL5
qφp
qχp
EFp
p n
qφn
qχn
EFn
p-n heterojunction: Different band gaps
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
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Semiconductor-SemiconductorL5
Examples of p-n heterojections
http://pubs.rsc.org/en/content/articlehtml/2013/ee/c3ee41981a#cit111
Most thin-film PV technologies
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Semiconductor-SemiconductorL5
Examples of p-n heterojections
III-V multi-junction devices
Being able to control the band offsets of a material can help eliminate barriers
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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
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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
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Physical PictureL5
Lets look at hole current first:
J p = J p(drift)+J p(diffusion)
= qμ p p ξ−qDpdpdx
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Physical PictureL5
Lets look at hole current first:
J p = J p(drift)+J p(diffusion)
= 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
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Physical PictureL5
J p = J p(drift)+J p(diffusion)
= 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
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Physical PictureL5
dEF
dx=0
TA DA!
The same is true from consideration of net electron current, Jn
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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 )
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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)
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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
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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
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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
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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)
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Under BiasL5
What happens to our ideal p-n junction if we put a forward or reverse bias across it?
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Under BiasL5
p n
0 wn
-wp
W
EF
ZERO BIAS
Vbi
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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
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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
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L5
Under Bias
Of course, you already know all about this:
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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 ]
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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 ]
Lets derive this by considering our band schematic. Fun fun fun!
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L5
Deriving Shockley Equation
Steady State again (i.e. no bias)Consider electron drift current – I
eo from p to n
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L5
Deriving Shockley Equation
Steady State again (i.e. no bias)Consider electron drift current – I
eo from p to n
I e0=e×(generationrate /volume)×(volumewithin Le of depletion zone )
W
Le
J e 0=e (npτ p
)Le
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L5
Deriving Shockley Equation
Steady State again (i.e. no bias)Consider electron drift current – I
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
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L5
Deriving Shockley Equation
Steady State again (i.e. no bias)Consider electron drift current – I
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:
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L5
Deriving Shockley Equation
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
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L5
Deriving Shockley Equation
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]
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L5
Deriving Shockley Equation
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
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L5
Deriving Shockley Equation
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”
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L5
Deriving Shockley Equation
Previous derivation ignores the recombination and generation of carriers within the depletion layer itself!
J=J 0[exp ( eVnk BT )−1]
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L5
Deriving Shockley Equation
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.
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Lecture SummaryL5
● Drawing Junctions
● Junction Formation – A Physical Picture
● Abrupt Junction
● Deriving Important Junction parameters
● Deriving diode equation
● Ideality Factor - Recombination