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
+-+ - V > 0 < 0 a Va
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 ρ(x) p-type 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 Semiconducto r 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
(x) E(x) E(x) Oxide MetalOxide Semiconductor MetalOxide Semiconductor E Metal Semiconductor = x x=x + x + x o x=x x o+ x x=x + x d d o d x=xo x=0 x=0 x=x x=0 x=x 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