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 Calculations At room temperature kT = 0.0259eV At room temperature ni for Si = 1.5 x 1010/cm3 Solve this equation for E = EF 𝑓 𝐸 = 1 1+ 𝑒 (𝐸− 𝐸 𝐹 )/𝑘𝑇 Let 𝑇→0𝐾 find f(E<EF) and f(E>EF) Let T = 300K and EF = 0.5eV plot f(E) for 0 < E < 1 EC EV

Fermi-Dirac plus Energy Band

More Calculations At room temperature kT = 0.0259eV At room temperature ni for Si = 1.5 x 1010/cm3 If Na = 2 x 1015 /cm3 find po and no The band gap of Si at room temp is 1.1eV or EC – EV = 1.1eV What is the value of EC – EF for intrinsic Si at T= 300K The band gap of Si at room temp is 1.1eV or EC – EV = 1.1eV What 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 = 1.1eV What is the value of EF – Ei if Nd = 2 x 1015 /cm3 at T= 300K

Intrinsic Carrier Concentrations SEMICONDUCTOR 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 Processes 𝜑 𝑛 𝑥 0 = 𝑙 2 𝑡 ( 𝑛 1 − 𝑛 2 ) n(x) n1 n2 𝜑 𝑛 𝑥 0 = 𝑙 2 𝑡 ( 𝑛 1 − 𝑛 2 ) n(x) n1 n2 Since the mean free path is a small differential, we can write: 𝑛 1 − 𝑛 2 = 𝑛 𝑥 −𝑛(𝑥+∆𝑥) ∆𝑥 𝑙 x0 Where x is at the center of segment 1 and ∆𝑥= 𝑙 x0 - l x0 + l In the limit of small ∆𝑥 𝜑 𝑛 𝑥 = 𝑙 2 2 𝑡 lim ∆𝑥→0 𝑛 𝑥 −𝑛 𝑥+∆𝑥 ∆𝑥 = 𝑙 2 2 𝑡 𝑑𝑛(𝑥) 𝑑𝑥 𝑙 2 2 𝑡 ≡ 𝐷 𝑛 or 𝐷 𝑝

Diffusion Current Equations

Combine Drift and Diffusion

Drift and Diffusion Currents Electron drift Hole drift Electron & Hole Drift current E(x) n(x) Electron diffusion Hole diffusion Electron Diff current Hole Diff current p(x)

Energy Bands when there is an Electric Field 𝑉 𝑥 = 𝐸(𝑥) −𝑞 = 𝑑𝑉(𝑥) 𝑑𝑥 = 𝑑𝑉(𝑥) 𝑑𝑥 =− 𝑑 𝑑𝑥 𝐸 𝑖 −𝑞 = 1 𝑞 𝑑 𝐸 𝑖 𝑑𝑥 E(x) E(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: 𝐽 𝑝 𝑥 =0=𝑞 𝜇 𝑝 𝑝 𝑥 𝐸 𝑥 −𝑞 𝐷 𝑝 𝑑𝑝(𝑥) 𝑑𝑥 = 𝐷 𝑝 𝜇 𝑝 1 𝑝(𝑥) 𝑑𝑝(𝑥) 𝑑𝑥 E(x) Using for p(x) 𝑝 0 = 𝑛 𝑖 𝑒 𝐸 𝑖 − 𝐸 𝐹 /𝑘𝑇 qE(x) = 𝐷 𝑝 𝜇 𝑝 1 𝑘𝑇 𝑑 𝐸 𝑖 𝑑𝑥 − 𝑑 𝐸 𝐹 𝑑𝑥 E(x) The equilibrium Fermi Level does not vary with x. 𝐷 𝑝 𝜇 𝑝 = 𝑘𝑇 𝑞 Finally:

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 Analysis An 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 = 𝑁 0 𝑒 −𝑎𝑥 Find an expression for E(x) and evaluate when a=1(μm)-1 Sketch band Diagram

Diffusion & Recombination Jp(x) Jp (x + Δx) x x + Δx Increase in hole conc In differential volume Per unit time Rate of Hole buildup Recombination Rate = - 𝜕𝑝 𝜕𝑡 𝑥→𝑥+∆𝑥 = 1 𝑞 𝐽 𝑝 𝑥 − 𝐽 𝑝 𝑥+∆𝑥 ∆𝑥 − 𝛿𝑝 𝜏 𝑝 𝜕𝛿𝑝 𝜕𝑡 =− 1 𝑞 𝜕 𝐽 𝑝 𝜕𝑥 − 𝛿𝑝 𝜏 𝑝 𝜕𝛿𝑛 𝜕𝑡 =− 1 𝑞 𝜕 𝐽 𝑛 𝜕𝑥 − 𝛿𝑛 𝜏 𝑛

If current is exclusively Diffusion 𝐽 𝑛 𝑑𝑖𝑓𝑓 =𝑞 𝐷 𝑛 𝜕𝛿𝑛 𝜕𝑥 𝜕𝛿𝑛 𝜕𝑡 = 𝐷 𝑛 𝜕 2 𝛿𝑛 𝜕 𝑥 2 − 𝛿𝑛 𝜏 𝑛 And the same for holes

And Finally, the steady-state Determining Diffusion Length 𝜕𝛿𝑛 𝜕𝑡 = 𝐷 𝑛 𝜕 2 𝛿𝑛 𝜕 𝑥 2 − 𝛿𝑛 𝜏 𝑛 =0 𝜕 2 𝛿𝑛 𝜕 𝑥 2 = 𝛿𝑛 𝐷 𝑛 𝜏 𝑛 = 𝛿𝑛 𝐿 2 𝐿 𝑛 = 𝐷 𝑛 𝜏 𝑛