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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 [“doped”] 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 – EV = 1.1eVWhat is the value of EC – 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 – EV = 1.1eVWhat is the value of Ei – EF if Na = 2 x 1015 /cm3 at T= 300K
The band gap of Si at room temp is 1.1eV or EC – EV = 1.1eVWhat is the value of EF – 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
𝜑𝑛 (𝑥0 )= 𝑙2𝑡
(𝑛1−𝑛2)
Since the mean free path is a small differential,we can write:
(𝑛1−𝑛2 )=𝑛 (𝑥 )−𝑛 (𝑥+∆𝑥 )
∆ 𝑥𝑙
Where x is at the center of segment 1 and ∆ 𝑥=𝑙In the limit of small∆ 𝑥
𝜑𝑛 (𝑥 )= 𝑙2
2𝑡lim∆ 𝑥→0
𝑛 (𝑥 )−𝑛 (𝑥+∆ 𝑥 )∆ 𝑥
= 𝑙2
2 𝑡𝑑𝑛(𝑥 )𝑑𝑥
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
𝑉 (𝑥 )=𝐸 (𝑥 )−𝑞
¿𝑑𝑉 (𝑥 )𝑑𝑥
E(x) ¿𝑑𝑉 (𝑥 )𝑑𝑥
=− 𝑑𝑑𝑥 [ 𝐸𝑖
−𝑞 ]= 1𝑞 𝑑 𝐸𝑖
𝑑𝑥E(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:
𝐽𝑝 (𝑥 )=0=𝑞𝜇𝑝𝑝 (𝑥 )𝐸 (𝑥 )−𝑞𝐷𝑝
𝑑𝑝 (𝑥 )𝑑𝑥
¿𝐷𝑝
𝜇𝑝
1𝑝(𝑥 )
𝑑𝑝(𝑥)𝑑𝑥
Using for p(x) 𝑝0=𝑛𝑖𝑒(𝐸𝑖−𝐸𝐹 ) /𝑘𝑇
¿𝐷𝑝
𝜇𝑝
1𝑘𝑇 (𝑑𝐸 𝑖
𝑑𝑥−𝑑𝐸𝐹
𝑑𝑥 ) The equilibrium Fermi Level does not vary with x.
E(x)
E(x)
0qE(x)
Finally:𝐷𝑝
𝜇𝑝
=𝑘𝑇𝑞
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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:
𝑁 𝑑=𝑁0𝑒−𝑎𝑥 Find an expression for E(x) and evaluate when a=1(μm)-1
Sketch band Diagram
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Diffusion & Recombination
x x + Δx
Jp(x) Jp (x + Δx)
Rate of Hole buildup =
Increase in hole concIn differential volumePer unit time
- RecombinationRate
𝜕𝑝𝜕𝑡 𝑥→𝑥+∆ 𝑥
= 1𝑞𝐽𝑝 (𝑥 )− 𝐽𝑝 (𝑥+∆ 𝑥 )
∆𝑥−𝛿𝑝𝜏𝑝
𝜕𝛿𝑝𝜕𝑡
=− 1𝑞𝜕 𝐽𝑝𝜕𝑥
−𝛿𝑝𝜏𝑝
𝜕𝛿𝑛𝜕𝑡
=− 1𝑞𝜕 𝐽𝑛𝜕 𝑥
−𝛿𝑛𝜏𝑛
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If current is exclusively Diffusion
𝐽𝑛 (𝑑𝑖𝑓𝑓 )=𝑞𝐷𝑛𝜕 𝛿𝑛𝜕 𝑥
𝜕𝛿𝑛𝜕𝑡
=𝐷𝑛𝜕2 𝛿𝑛𝜕 𝑥2
−𝛿𝑛𝜏𝑛
And the same for holes
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And Finally, the steady-stateDetermining Diffusion Length
𝜕𝛿𝑛𝜕𝑡
=𝐷𝑛𝜕2 𝛿𝑛𝜕 𝑥2
−𝛿𝑛𝜏𝑛
=0 𝜕2𝛿𝑛𝜕𝑥2
= 𝛿𝑛𝐷𝑛𝜏𝑛
= 𝛿𝑛𝐿❑2 𝐿𝑛=√𝐷𝑛𝜏𝑛