EE143 S06 Semiconductor Tutorial 2 -Electron Energy Band - Fermi Level -Electrostatics of device charges Professor N Cheung, U.C. Berkeley 1 EE143 S06 Semiconductor Tutorial 2 The Simplified Electron Energy Band Diagram Professor N Cheung, U.C. Berkeley 2 EE143 S06 Semiconductor Tutorial 2 Energy Band Diagram with E-field Electron Electron Energy Energy E-field E-field - + +- EC 2 1 2 1 EC EV Electric potential EV Electric potential φ(2) < φ(1) φ(2) > φ(1) x x Electron concentration n n(2) eqφ(2) / kT = = eq[φ(2)−φ(1)]/ kT n(1) eqφ(1) / kT Professor N Cheung, U.C. Berkeley 3 EE143 S06 Semiconductor Tutorial 2 The Fermi-Dirac Distribution (Fermi Function) Probability of available states at energy E being occupied f(E) = 1/ [ 1+ exp (E- Ef) / kT] -5 where Ef is the Fermi energy and k = Boltzmann constant=8.617 ∗ 10 eV/K T=0K 0.5 f(E) E -Ef Professor N Cheung, U.C. Berkeley 4 EE143 S06 Semiconductor Tutorial 2 Properties of the Fermi-Dirac Distribution Probability of electron state at energy E will be occupied (1) f(E) ≅ exp [- (E- E ) / kT] for (E- E ) > 3kT Note: f f At 300K, •This approximation is called Boltzmann approximation kT= 0.026eV (2) Probability of available states at energy E NOT being occupied 1- f(E) = 1/ [ 1+ exp (Ef -E) / kT] Professor N Cheung, U.C. Berkeley 5 EE143 S06 Semiconductor Tutorial 2 How to find Ef when n(or p) is known Ec Ef (n-type) q|ΦF| Ei Ef (p-type) Ev n = ni exp [(Ef -Ei)/kT] Let qΦF ≡ Ef -Ei ∴n = ni exp [qΦF /kT] Professor N Cheung, U.C. Berkeley 6 EE143 S06 Semiconductor Tutorial 2 Dependence of Fermi Level with Doping Concentration Ei ≡ (EC+EV)/2 Middle of energy gap When Si is undoped, Ef = Ei ; also n =p = ni Professor N Cheung, U.C. Berkeley 7 EE143 S06 Semiconductor Tutorial 2 The Fermi Energy at thermal equilibrium At thermal equilibrium ( i.e., no external perturbation), The Fermi Energy must be constant for all positions Electron energy Material A Material B Material C Material D EF Position x Professor N Cheung, U.C. Berkeley 8 EE143 S06 Semiconductor Tutorial 2 Electron Transfer during contact formation System 1 System 2 System 1 System 2 Before e contact EF1 e EF2 formation EF2 EF1 System 1 System 2 System 1 System 2 - After Net + - Net + - contact positive + - negative + charge - formation charge + - + EF EF E E Professor N Cheung, U.C. Berkeley 9 EE143 S06 Semiconductor Tutorial 2 Applied Bias and Fermi Level Fermi level of the side which has a relatively higher electric potential will have a relatively lower electron energy ( Potential Energy = -q • electric potential.) Only difference of the E 's at both sides are important, not the absolute position of the Fermi levels. E E f2 f1 q| V | qVa a E E f1 f 2 Side 1 Side 2 Side 1 Side 2 +-+ - Va > 0 Va < 0 Potential difference across depletion region = Vbi -Va Professor N Cheung, U.C. Berkeley 10 EE143 S06 Semiconductor Tutorial 2 PN junctions Thermal Equilibrium E-field N - and p A + ND and n ρ(x) is 0 Depletion region ρ(x) is 0 -- ++ Quasi-neutral Quasi-neutral -- ++ region region p-Si n-Si + - ND only NA only ρ(x) is - ρ(x) is + Complete Depletion Approximation used for charges inside depletion region + - r(x) ≈ ND (x) – NA (x) http://jas.eng.buffalo.edu/education/pn/pnformation2/pnformation2.html Professor N Cheung, U.C. Berkeley 11 EE143 S06 Semiconductor Tutorial 2 Electrostatics of Device Charges 1) Summation of all charges = 0 ρ2 • xd2 = ρ1 • xd1 2) E-field =0 outside depletion regions n-type (x) p-type ρ Semiconductor Semiconductor ρ2 - xd1 x xd2 - ρ1 x=0 E = 0 E ≠0 E = 0 Professor N Cheung, U.C. Berkeley 12 EE143 S06 Semiconductor Tutorial 2 3) Relationship between E-field and charge density ρ(x) d [ε E(x)] /dx = ρ(x) “Gauss Law” 4) Relationship between E-field and potential φ E(x) = - dφ(x)/dx Professor N Cheung, U.C. Berkeley 13 EE143 S06 Semiconductor Tutorial 2 Example Analysis : n+/ p-Si junction ρ(x) +Q' Emax =qNaxd/εs p-Si n+ Si x E (x) d x 3) ⇒ Depletion region Slope = qNa/εs Depletion region -qNa is very thin and is approximated as x=0 xd 2) ⇒ E = 0 a thin sheet charge 4) Area under E-field curve 1) ⇒ Q’ = qNaxd ⇒ = voltage across depletion region 2 = qNaxd /2εs Professor N Cheung, U.C. Berkeley 14 EE143 S06 Semiconductor Tutorial 2 Superposition Principle ρ(x) If ρ1(x) ⇒ E1(x) and V1(x) +qND ρ2(x) ⇒ E2(x) and V2(x) -xp then x ρ (x) + ρ (x) ⇒ E (x) + E (x) and -qN- 1 2 1 2 A +xn V1(x) + V2(x) x=0 ρ (x) ρ1(x) 2 +qN Q=+qNAxp D -xp x + x +x -qNA n Q=-qNAxp x=0 x=0 Professor N Cheung, U.C. Berkeley 15 EE143 S06 Semiconductor Tutorial 2 ρ1(x) ρ2(x) +qND Q=+qNAxp -xp x x -qNA +xn Q=-qNAxp x=0 x=0 E (x) 1 E2(x) -xp +xn x + x - - Slope = Slope = -qNA/εs +qND/εs x=0 x=0 Professor N Cheung, U.C. Berkeley 16 EE143 S06 Semiconductor Tutorial 2 Sketch of E(x) E(x) = E1(x)+ E2(x) -x p +xn x=0 x Slope = + qND/εs Slope = - qNA/εs - Emax = -qNA xp /εs = -qND xn /εs Professor N Cheung, U.C. Berkeley 17 EE143 S06 Semiconductor Tutorial 2 Depletion Mode :Charge and Electric Field Distributions by Superposition Principle of Electrostatics ρ(x) ρ(x) ρ(x) Q' Q' Q' MetalOxide Semiconductor Metal Oxide Semiconductor Metal Semiconductor Oxide x=xo+x x=xo+x d x x d x −ρ - Q' −ρ x=0 x=xo x=0 x=xo x=0 x=xo E(x) E(x) E(x) MetalOxide Semiconductor MetalOxide Semiconductor MetalOxide Semiconductor = x x + x x=x x=x +x o+ x x=x + x o d d o d x=0 x=x x=0 x=x x=0 x=x o o o Professor N Cheung, U.C. Berkeley 18 EE143 S06 Semiconductor Tutorial 2 Why xdmax ~ constant beyond onset of strong inversion ? VG =VFB +VOX +VSi Approximation assumes Higher than VT Picks up all VSi does not change much the changes in VG Justification: If surface electron density changes by ∆n Ei Ef ∆n ∆VOX ∝ COX δ δ ∝ ln (∆ n ) but the change of VSi changes only by kT/q [ ln (∆n)] – small! δ ∆ n ∝ e kT Professor N Cheung, U.C. Berkeley 19 EE143 S06 Semiconductor Tutorial 2 N (surface) = n (bulk) • exp [ qVSi/kT] p-Si 16 Na=10 /cm3 VSi n-surface n-bulk = 2.1 • 104/cm3 0 2.10E+04 0.1 9.84E+05 n-surface Ei 0.2 4.61E+07 0.3 2.16E+09 qVSi 0.35eV 0.4 1.01E+11 0.5 4.73E+12 Ef 0.6 2.21E+14 0.7 1.04E+16 Onset of strong inversion 0.8 4.85E+17 0.9 2.27E+19 (at VT) Professor N Cheung, U.C. Berkeley 20.