some calculations at room temperature kt = 0.0259ev at room temperature n i for si = 1.5 x 10 10 /cm...
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ELECTRONICS II VLSI DESIGN
Fall 2013
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The Hydrogen Atom
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Allowable States for the Electron of the Hydrogen Atom
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The Periodic Table
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From Single Atoms to Solids
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Energy bands and energy gaps
Silicon
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Band Structures at ~0K
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Atomic Bonds
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Electrons and holes in intrinsic [no impurities] semiconductor materials
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Electrons and holes in extrinsic [βdopedβ] semiconductor materials
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Some Terminology and Definitions
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Electron and Hole Concentrations at Equilibrium
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Calculating Concentrations
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Some CalculationsAt room temperature kT = 0.0259eVAt room temperature ni for Si = 1.5 x 1010/cm3
Solve this equation for E = EF
Let T = 300K and EF = 0.5eV plot f(E) for 0 < E < 1
Let find f(E<EF) and f(E>EF)
EC
EV
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Fermi-Dirac plus Energy Band
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More Calculations
If Na = 2 x 1015 /cm3 find po and no
The band gap of Si at room temp is 1.1eV or EC β EV = 1.1eVWhat is the value of EC β EF for intrinsic Si at T= 300K
At room temperature kT = 0.0259eVAt room temperature ni for Si = 1.5 x 1010/cm3
The band gap of Si at room temp is 1.1eV or EC β EV = 1.1eVWhat is the value of Ei β EF if Na = 2 x 1015 /cm3 at T= 300K
The band gap of Si at room temp is 1.1eV or EC β EV = 1.1eVWhat is the value of EF β Ei if Nd = 2 x 1015 /cm3 at T= 300K
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Intrinsic Carrier ConcentrationsSEMICONDUCTOR ni
Ge 2.5 x 1013/cm3
Si 1.5 x 1010/cm3
GaAs 2 x 106/cm3
Which element has the largest Eg?
What is the value of pi for each of these elements?
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Si with 1015/cm3 donor impurity
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Conductivity
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Excess Carriers
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Photoluminescence
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Diffusion of Carriers
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Drift and Diffusion
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Diffusion Processesn(x)
n1 n2
x0
x0 - l x0 + l
ππ (π₯0 )= π2π‘
(π1βπ2)
Since the mean free path is a small differential,we can write:
(π1βπ2 )=π (π₯ )βπ (π₯+βπ₯ )
β π₯π
Where x is at the center of segment 1 and β π₯=πIn the limit of smallβ π₯
ππ (π₯ )= π2
2π‘limβ π₯β0
π (π₯ )βπ (π₯+β π₯ )β π₯
= π2
2 π‘ππ(π₯ )ππ₯
or
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Diffusion Current Equations
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Combine Drift and Diffusion
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Drift and Diffusion Currents
E(x)
n(x)
p(x)
Electron drift
Hole drift
Electron & HoleDrift current
Electron diffusion
Hole diffusion
Electron Diff current
Hole Diff current
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Energy Bands when there is an Electric Field
π (π₯ )=πΈ (π₯ )βπ
ΒΏππ (π₯ )ππ₯
E(x) ΒΏππ (π₯ )ππ₯
=β πππ₯ [ πΈπ
βπ ]= 1π π πΈπ
ππ₯E(x)
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The Einstein Relation
At equilibrium no net current flows so any concentration gradient would be accompanied by an electric field generated internally. Set the hole current equal to 0:
π½π (π₯ )=0=ππππ (π₯ )πΈ (π₯ )βππ·π
ππ (π₯ )ππ₯
ΒΏπ·π
ππ
1π(π₯ )
ππ(π₯)ππ₯
Using for p(x) π0=πππ(πΈπβπΈπΉ ) /ππ
ΒΏπ·π
ππ
1ππ (ππΈ π
ππ₯βππΈπΉ
ππ₯ ) The equilibrium Fermi Level does not vary with x.
E(x)
E(x)
0qE(x)
Finally:π·π
ππ
=πππ
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D and mu
Dn
(cm2/s)Dp mun
(cm2/V-s)mup
Ge 100 50 3900 1900
Si 35 12.5 1350 480
GaAs 220 10 8500 400
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Message from Previous AnalysisAn important result of the balance between drift and diffusion at equilibrium is that built-in fields accompany gradients in Ei. Such gradients in the bands at equilibrium (EF constant) can arise when the band gap varies due to changes in alloy composition. More commonly built-in fields result from doping gradients. For example a donor distribution Nd(x) causes a gradient in no(x) which must be balanced by a built-in electric field E(x).
Example: An intrinsic sample is doped with donors from one side such that:
π π=π0πβππ₯ Find an expression for E(x) and evaluate when a=1(ΞΌm)-1
Sketch band Diagram
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Diffusion & Recombination
x x + Ξx
Jp(x) Jp (x + Ξx)
Rate of Hole buildup =
Increase in hole concIn differential volumePer unit time
- RecombinationRate
ππππ‘ π₯βπ₯+β π₯
= 1ππ½π (π₯ )β π½π (π₯+β π₯ )
βπ₯βπΏπππ
ππΏπππ‘
=β 1ππ π½πππ₯
βπΏπππ
ππΏπππ‘
=β 1ππ π½ππ π₯
βπΏπππ
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If current is exclusively Diffusion
π½π (ππππ )=ππ·ππ πΏππ π₯
ππΏπππ‘
=π·ππ2 πΏππ π₯2
βπΏπππ
And the same for holes
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And Finally, the steady-stateDetermining Diffusion Length
ππΏπππ‘
=π·ππ2 πΏππ π₯2
βπΏπππ
=0 π2πΏπππ₯2
= πΏππ·πππ
= πΏππΏβ2 πΏπ=βπ·πππ