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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.
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Electrons and holes in intrinsic [no impurities] semiconductor materials
Electrons and holes in extrinsic [“doped”] semiconductor materials
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 Let find f(E<EF) and f(E>EF) Let T = 300K and EF = 0.5eV plot f(E) for 0 < E < 1 EC EV
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 Which element has the largest Eg? What is the value of pi for each of these elements?
Diffusion Processes n(x) n1 n2 Since the mean free path is a small differential, we can write: x0 Where x is at the center of segment 1 and x0 + l x0 - l In the limit of small or
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)
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: Using for p(x) E(x) qE(x) 0 E(x) The equilibrium Fermi Level does not vary with x. Finally:
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: 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 Recombination Rate Rate of Hole buildup - =
If current is exclusively Diffusion And the same for holes