ELECTRONICS II VLSI DESIGN

Fall 2013

The Hydrogen Atom

Allowable States for the Electron of the Hydrogen Atom

The Periodic Table

From Single Atoms to Solids

Energy bands and energy gaps

Silicon

Band Structures at ~0K

Atomic Bonds

Electrons and holes in intrinsic [no impurities] semiconductor materials

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

Some Terminology and Definitions

Electron and Hole Concentrations at Equilibrium

Calculating Concentrations

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

Fermi-Dirac plus Energy Band

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

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?

Si with 1015/cm3 donor impurity

Conductivity

Excess Carriers

Photoluminescence

Diffusion of Carriers

Drift and Diffusion

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 )

∆ 𝑥 𝑙

Where x is at the center of segment 1 and ∆ 𝑥=𝑙In the limit of small∆ 𝑥𝜑𝑛 (𝑥 )= 𝑙2

2𝑡 lim∆ 𝑥→0

𝑛 (𝑥 )−𝑛 (𝑥+∆ 𝑥 )∆ 𝑥 =

𝑙22 𝑡

𝑑𝑛(𝑥 )𝑑𝑥

or

Diffusion Current Equations

Combine Drift and Diffusion

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

Energy Bands when there is an Electric Field

𝑉 (𝑥 )=𝐸 (𝑥 )−𝑞

¿𝑑𝑉 (𝑥 )𝑑𝑥E(x) ¿

𝑑𝑉 (𝑥 )𝑑𝑥 =− 𝑑

𝑑𝑥 [ 𝐸𝑖

−𝑞 ]= 1𝑞 𝑑 𝐸𝑖

𝑑𝑥E(x)

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 )−𝑞𝐷𝑝𝑑𝑝 (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=𝑘𝑇

𝑞

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

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

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

If current is exclusively Diffusion

𝐽𝑛 (𝑑𝑖𝑓𝑓 )=𝑞𝐷𝑛𝜕 𝛿𝑛𝜕 𝑥

𝜕𝛿𝑛𝜕𝑡 =𝐷𝑛

𝜕2 𝛿𝑛𝜕𝑥2

− 𝛿𝑛𝜏𝑛And the same for holes

And Finally, the steady-stateDetermining Diffusion Length

𝜕𝛿𝑛𝜕𝑡 =𝐷𝑛

𝜕2 𝛿𝑛𝜕 𝑥2

− 𝛿𝑛𝜏𝑛=0 𝜕2𝛿𝑛

𝜕𝑥2=

𝛿𝑛𝐷𝑛𝜏𝑛

=𝛿𝑛𝐿❑2 𝐿𝑛=√𝐷𝑛𝜏𝑛