Overview of Silicon Semiconductor Device...
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1
Overview of Silicon Semiconductor Device Physics
Dr. David W. Graham
West Virginia UniversityLane Department of Computer Science and Electrical Engineering
©
2009 David W. Graham
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Silicon
Nucleus
Valence Band
Energy Bands(Shells)
Si has 14 Electrons
Silicon is the primary semiconductor used in VLSI systems
At T=0K, the highest energy band occupied by an electron is called the valence band.
Silicon has 4 outer shell / valence electrons
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Energy Bands
•
Electrons try to occupy the lowest energy band possible
•
Not every energy level is a legal state for an electron to occupy
•
These legal states tend to arrange themselves in bands
Allowed Energy States
Disallowed Energy States
Increasing Electron Energy }
}
Energy Bands
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Energy Bands
Valence Band
Conduction Band
Energy BandgapEg
EC
EVLast filled energy band at T=0K
First unfilled energy band at T=0K
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Band Diagrams
Eg
EC
EV
Band Diagram RepresentationEnergy plotted as a function of position
EC Conduction bandLowest energy state for a free electron
EV Valence bandHighest energy state for filled outer shells
EG Band gapDifference in energy levels between EC and EVNo electrons (e) in the bandgap (only above EC or below EV)EG = 1.12eV in Silicon
Increasing electron energy
Increasing voltage
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Intrinsic Semiconductor
Silicon has 4 outer shell / valence electrons
Forms into a lattice structure to share electrons
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Intrinsic Silicon
EC
EV
The valence band is full, and no electrons are free to move about
However, at temperatures above T=0K, thermal energy shakes an electron free
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Semiconductor PropertiesFor T > 0K
Electron shaken free and can cause current to flow
e–h+
•
Generation
–
Creation of an electron (e) and hole (h+) pair
•
h+
is simply a missing electron, which leaves an excess positive charge (due to an extra proton)
•
Recombination
– if an e
and an h+
come in contact, they annihilate each other
•
Electrons and holes are called “carriers”
because they are charged particles –
when they move, they carry current•
Therefore, semiconductors can conduct electricity for T > 0K …
but not much current (at room temperature (300K), pure silicon has only 1 free electron per 3 trillion atoms)
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Doping
•
Doping
–
Adding impurities to the silicon crystal lattice to increase the number of carriers
•
Add a small number of atoms to increase either the number of electrons or holes
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Periodic Table
Column 4 Elements have 4 electrons in the Valence Shell
Column 3 Elements have 3 electrons in the Valence Shell
Column 5 Elements have 5 electrons in the Valence Shell
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Donors nType MaterialDonors
•
Add atoms with 5 valenceband electrons
•
ex. Phosphorous (P)•
“Donates”
an extra e
that can freely travel around
•
Leaves behind a positively charged nucleus (cannot move)
•
Overall, the crystal is still electrically neutral
•
Called “ntype”
material (added negative carriers)
•
ND
= the concentration of donor atoms [atoms/cm3
or cm3]~10151020cm3
•
e
is free to move about the crystal (Mobility μn
≈1350cm2/V)
+
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Donors nType MaterialDonors
•
Add atoms with 5 valenceband electrons
•
ex. Phosphorous (P)•
“Donates”
an extra e
that can freely travel around
•
Leaves behind a positively charged nucleus (cannot move)
•
Overall, the crystal is still electrically neutral
•
Called “ntype”
material (added negative carriers)
•
ND
= the concentration of donor atoms [atoms/cm3
or cm3]~10151020cm3
•
e
is free to move about the crystal (Mobility μn
≈1350cm2/V)
+
+
+
+
++
+
+
+
+
+
+
+
+
++
+–
–
– –
–
––
–
–
–
–
–
––
–
–
–
+
+
nType Material
+–
+
Shorthand NotationPositively charged ion; immobileNegatively charged e; mobile;
Called “majority carrier”Positively charged h+; mobile;
Called “minority carrier”
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Acceptors Make pType Material
––
h+
Acceptors•
Add atoms with only 3 valence
band electrons
•
ex. Boron (B)•
“Accepts”
e–
and provides extra h+
to freely travel around
•
Leaves behind a negatively charged nucleus (cannot move)
•
Overall, the crystal is still electrically neutral
•
Called “ptype”
silicon (added positive carriers)
•
NA
= the concentration of acceptor atoms [atoms/cm3
or cm3]•
Movement of the hole requires breaking of a bond! (This is hard, so mobility is low, μp ≈
500cm2/V)
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Acceptors Make pType Material
Acceptors•
Add atoms with only 3 valence
band electrons
•
ex. Boron (B)•
“Accepts”
e–
and provides extra h+
to freely travel around
•
Leaves behind a negatively charged nucleus (cannot move)
•
Overall, the crystal is still electrically neutral
•
Called “ptype”
silicon (added positive carriers)
•
NA
= the concentration of acceptor atoms [atoms/cm3
or cm3]•
Movement of the hole requires breaking of a bond! (This is hard, so mobility is low, μp ≈
500cm2/V)
–
–
–
–
––
–
–
–
–
–
–
–
–
––
–+
+
+ +
+
++
+
+
+
+
+
++
+
+
+
–
–
pType Material
Shorthand NotationNegatively charged ion; immobilePositively charged h+; mobile;
Called “majority carrier”Negatively charged e; mobile;
Called “minority carrier”
–+
–
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The Fermi Function
f(E)
1
0.5
EEf
The Fermi Function•
Probability distribution function (PDF)•
The probability that an available state at an energy E will be occupied by an e
E Energy level of interestEf Fermi level
Halfway pointWhere f(E) = 0.5
k Boltzmann constant=
1.38×1023 J/K=
8.617×105 eV/KT Absolute temperature (in Kelvins)
( ) ( ) kTEE feEf −+
=1
1
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Boltzmann Distribution
f(E)
1
0.5
EEf
~Ef  4kT ~Ef + 4kT
( ) ( ) kTEE feEf −−≈
kTEE f >>−If
Then
Boltzmann Distribution•
Describes exponential decrease in the density of particles in thermal equilibrium with a potential gradient
•
Applies to all physical systems•
Atmosphere Exponential distribution of gas molecules•
Electronics Exponential distribution of electrons•
Biology Exponential distribution of ions
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Band Diagrams (Revisited)
Eg
EC
EV
Band Diagram RepresentationEnergy plotted as a function of positionEC Conduction band
Lowest energy state for a free electronElectrons in the conduction band means current can flow
EV Valence bandHighest energy state for filled outer shellsHoles in the valence band means current can flow
Ef Fermi LevelShows the likely distribution of electrons
EG Band gapDifference in energy levels between EC and EVNo electrons (e) in the bandgap (only above EC or below EV)EG = 1.12eV in Silicon
Ef
f(E)10.5
E
•
Virtually all of the valenceband energy levels are filled with e
•
Virtually no e
in the conduction band
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Effect of Doping on Fermi LevelEf
is a function of the impuritydoping level
EC
EV
Ef
f(E)10.5
E
nType Material
•
High probability of a free e
in the conduction band•
Moving Ef
closer to EC
(higher doping) increases the number of available majority carriers
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Effect of Doping on Fermi LevelEf
is a function of the impuritydoping level
EC
EV
Ef
pType Material
•
Low probability of a free e
in the conduction band•
High probability of h+
in the valence band•
Moving Ef
closer to EV
(higher doping) increases the number of available majority carriers
f(E)10.5
E
f(E)10.5
E( )Ef−1
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Equilibrium Carrier Concentrations
n = # of e
in a materialp = # of h+
in a material
ni
= # of e
in an intrinsic (undoped) material
Intrinsic silicon•
Undoped
silicon
•
Fermi level•
Halfway between Ev
and Ec•
Location at “Ei
”
Eg
EC
EV
Ef
f(E)10.5
E
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Equilibrium Carrier Concentrations
Nondegenerate Silicon•
Silicon that is not too heavily doped
•
Ef
not too close to Ev
or Ec
Assuming nondegenerate silicon
( )
( ) kTEEi
kTEEi
fi
if
enp
enn−
−
=
=
2innp =
Multiplying together
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Charge Neutrality Relationship
•
For uniformly doped semiconductor•
Assuming total ionization of dopant
atoms
0=−+− AD NNnp# of carriers # of ions
Total Charge = 0Electrically Neutral
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Calculating Carrier Concentrations
•
Based upon “fixed”
quantities•
NA
, ND
, ni
are fixed (given specific dopings for a material)
•
n, p can change (but we can find their equilibrium values)
nn
nNNNNp
nNNNNn
i
iDADA
iADAD
2
21
22
21
22
22
22
=
⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎟
⎠⎞
⎜⎝⎛ −
+−
=
⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎟
⎠⎞
⎜⎝⎛ −
+−
=
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Common Special Cases in Silicon
1.
Intrinsic semiconductor (NA
= 0, ND
= 0)2.
Heavily onesided doping
3.
Symmetric doping
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Intrinsic Semiconductor (NA
=0, ND
=0)
i
i
i
npnnpnn
====
Carrier concentrations are given by
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Heavily OneSided Doping
iADA
iDAD
nNNNnNNN
>>≈−>>≈−
This is the typical case for most semiconductor applications
iDAD nNNN >>>> ,If (Nondegenerate, Total Ionization)Then
D
i
D
Nnp
Nn2
≈
≈
iADA nNNN >>>> ,If (Nondegenerate, Total Ionization)Then
A
i
A
Nnn
Np2
≈
≈
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Symmetric Doping
Doped semiconductor where ni
>> ND
NA

•
Increasing temperature increases the number of intrinsic carriers
•
All semiconductors become intrinsic at sufficiently high temperatures
inpn ≈≈
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Determination of Ef
in Doped Semiconductor
iADAi
Afi
iDADi
Dif
nNNNnNkTEE
nNNNnNkTEE
>>>>⎟⎟⎠
⎞⎜⎜⎝
⎛=−
>>>>⎟⎟⎠
⎞⎜⎜⎝
⎛=−
,ln
,ln
for
for
Also, for typical semiconductors (heavily onesided doping)
⎟⎟⎠
⎞⎜⎜⎝
⎛−=⎟⎟
⎠
⎞⎜⎜⎝
⎛=−
iiif n
pkTnnkTEE lnln [units eV]
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Thermal Motion of Charged Particles
•
Look at drift and diffusion in silicon•
Assume 1D motion
•
Applies to both electronic systems and biological systems
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DriftDrift
→ Movement of charged particles in response to an external field (typically an electric field)
E
Efield applies forceF = qE
which accelerates the charged particle.
However, the particle does not accelerate indefinitely because of collisions with the lattice (velocity saturation)
Average velocity<vx
> ≈
µn
Ex
electrons< vx
> ≈
µp
Ex
holes
µn
→ electron mobility→ empirical proportionality constant
between E and velocityµp
→ hole mobility
µn ≈
3µp
µ↓
as T↑
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DriftDrift
→ Movement of charged particles in response to an external field (typically an electric field)
Efield applies forceF = qE
which accelerates the charged particle.
However, the particle does not accelerate indefinitely because of collisions with the lattice (velocity saturation)
Average velocity<vx
> ≈
µn
Ex
electrons< vx
> ≈
µp
Ex
holes
µn
→ electron mobility→ empirical proportionality constant
between E and velocityµp
→ hole mobility
µn ≈
3µp
µ↓
as T↑
Current Density
qpEJ
qnEJ
pdriftp
ndriftn
μ
μ
=
=
,
,
q = 1.6×1019
C, carrier densityn =
number of e
p =
number of h+
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Resistivity•
Closely related to carrier drift
•
Proportionality constant between electric field and the total particle current flow
( ) Cqpnq pn
1910602.11 −×=+
= whereμμ
ρ
nType Semiconductor
Dn Nqμρ 1=
pType Semiconductor
Ap Nqμρ 1=
•
Therefore, all semiconductor material is a resistor–
Could be parasitic (unwanted)–
Could be intentional (with proper doping) •
Typically, ptype material is more resistive than ntype material for a given amount of doping
•
Doping levels are often calculated/verified from resistivity measurements
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DiffusionDiffusion
→ Motion of charged particles due to a concentration gradient•
Charged particles move in random directions•
Charged particles tend to move from areas of high concentration to areas of low concentration (entropy –
Second Law of Thermodynamics)•
Net effect is a current flow (carriers moving from areas of high
concentration to areas of low concentration)
( )
( )dx
xdpqDJ
dxxdnqDJ
pdiffp
ndiffn
−=
=
,
,q = 1.6×1019
C, carrier densityD =
Diffusion coefficientn(x) =
e
density at position xp(x) =
h+
density at position x
→ The negative sign in Jp,diff
is due to moving in the opposite direction from the concentration gradient
→ The positive sign from Jn,diff
is because the negative from the e
cancels out the negative from the concentration gradient
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Total Current DensitiesSummation of both drift and diffusion
( )
( )
pqDqpEdx
xdpqDqpE
JJJ
nqDqnEdx
xdnqDqnE
JJJ
pp
pp
diffpdriftpp
nn
nn
diffndriftnn
∇−=
−=
+=
∇+=
+=
+=
μ
μ
μ
μ
,,
,,
pn JJJ +=Total current flow
(1 Dimension)
(3 Dimensions)
(1 Dimension)
(3 Dimensions)
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Einstein Relation
Einstein Relation
→ Relates D
and µ
(they are not independent of each other)
qkTD
=μ
UT
= kT/q→ Thermal voltage= 25.86mV
at room temperature≈
25mV
for quick hand approximations→ Used in biological and silicon applications
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Changes in Carrier Numbers
Primary “other”
causes for changes in carrier concentration•
Photogeneration
(light shining on semiconductor)
•
Recombinationgeneration
Photogeneration
Llightlight
Gtp
tn
=∂∂
=∂∂ Photogeneration
rate
Creates same # of e
and h+
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Changes in Carrier Numbers
nGR
pGR
ntn
ptp
τ
τ
Δ−=
∂∂
Δ−=
∂∂
−
−
Indirect Thermal RecombinationGeneration
e
in ptype material
h+
in ntype material n0
, p0 equilibrium carrier concentrationsn, p carrier concentrations under
arbitrary conditionsΔn, Δp change in # of e or h+ from
equilibrium conditions
Assumes lowlevel injection
material typep in material typen in
00
00
,,
pppnnnnp
≈<<Δ≈<<Δ
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Minority Carrier PropertiesMinority Carriers•
e
in ptype material•
h+
in ntype material
Minority Carrier Lifetimes•
τn The time before minority carrier electrons undergo recombinationin ptype material•
τp The time before minority carrier holes undergo recombination in ntype material
Diffusion Lengths•
How far minority carriers will make it into “enemy territory”
if they are injected into that material
ppp
nnn
DL
DL
τ
τ
=
= for minority carrier e
in ptype material
for minority carrier h+
in ntype material
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Equations of State•
Putting it all together•
Carrier concentrations with respect to time (all processes)•
Spatial and time continuity equations
for carrier concentrations
)(
)(
)(
)(
1
1
lightother
GRp
lightother
GRdiffdrift
lightother
GRn
lightother
GRdiffdrift
tp
tpJ
q
tp
tp
tp
tp
tp
tn
tnJ
q
tn
tn
tn
tn
tn
∂∂
+∂∂
+⋅∇−=
∂∂
+∂∂
+∂∂
+∂∂
=∂∂
∂∂
+∂∂
+⋅∇=
∂∂
+∂∂
+∂∂
+∂∂
=∂∂
−
−
−
−
43421
43421
Current to Related
Current to Related
40
Equations of StateMinority Carrier Equations•
Continuity equations for the special case of minority carriers•
Assumes lowlevel injection
Ln
ppn
p Gn
xn
Dtn
+Δ
−∂
Δ∂=
∂
Δ∂
τ2
2
Light generation
Indirect thermal recombination
J, assuming no Efieldx
JDJxnqD n
nnn ∂∂
→⋅∇∂∂
q1also and
Lp
nnn
n Gpx
pDtp
+Δ
−∂Δ∂
=∂Δ∂
τ2
2
np
, pn minority carriers in “other” type of material