Transcript
Page 1: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

ELECTRONICS II VLSI DESIGN

Fall 2013

Page 2: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

The Hydrogen Atom

Page 3: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

Allowable States for the Electron of the Hydrogen Atom

Page 4: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

The Periodic Table

Page 5: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

From Single Atoms to Solids

Page 6: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

Energy bands and energy gaps

Silicon

Page 7: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

Band Structures at ~0K

Page 8: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

Atomic Bonds

Page 9: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

Electrons and holes in intrinsic [no impurities] semiconductor materials

Page 10: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

Electrons and holes in extrinsic [“doped”] semiconductor materials

Page 11: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F
Page 12: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

Some Terminology and Definitions

Page 13: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

Electron and Hole Concentrations at Equilibrium

Page 14: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

Calculating Concentrations

Page 15: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

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

Page 16: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

Fermi-Dirac plus Energy Band

Page 17: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

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

Page 18: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

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?

Page 19: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

Si with 1015/cm3 donor impurity

Page 20: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

Conductivity

Page 21: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

Excess Carriers

Page 22: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

Photoluminescence

Page 23: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

Diffusion of Carriers

Page 24: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

Drift and Diffusion

Page 25: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

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

Page 26: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

Diffusion Current Equations

Page 27: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

Combine Drift and Diffusion

Page 28: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

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

Page 29: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

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)

Page 30: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

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

Page 31: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

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

Page 32: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

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

Page 33: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

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

Page 34: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

If current is exclusively Diffusion

𝐽𝑛 (𝑑𝑖𝑓𝑓 )=𝑞𝐷𝑛𝜕 𝛿𝑛𝜕 𝑥

𝜕𝛿𝑛𝜕𝑡

=𝐷𝑛𝜕2 𝛿𝑛𝜕 𝑥2

−𝛿𝑛𝜏𝑛

And the same for holes

Page 35: Some Calculations At room temperature kT = 0.0259eV At room temperature n i for Si = 1.5 x 10 10 /cm 3 Let T = 300K and E F

And Finally, the steady-stateDetermining Diffusion Length

𝜕𝛿𝑛𝜕𝑡

=𝐷𝑛𝜕2 𝛿𝑛𝜕 𝑥2

−𝛿𝑛𝜏𝑛

=0 𝜕2𝛿𝑛𝜕𝑥2

= 𝛿𝑛𝐷𝑛𝜏𝑛

= 𝛿𝑛𝐿❑2 𝐿𝑛=√𝐷𝑛𝜏𝑛


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