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 2 1 Electric potential 2 1 EC 2 EV x E field 1 EC EV Electric potential 2 1 x Electron concentration n q 2 kT n 2 e q 2 1 kT q 1 kT e n 1 e 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 where Ef is the Fermi energy and k Boltzmann constant 8 617 10 5 eV K f E T 0K 0 5 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 Ef kT for E Ef 3kT This approximation is called Boltzmann approximation Note At 300K 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 q F Ef n type Ei Ev Ef p type n ni exp Ef Ei kT Let q F Ef Ei n ni exp q F 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 Before contact formation EF1 EF System 2 Net negative charge E Professor N Cheung U C Berkeley EF2 e EF2 Net positive charge System 2 System 1 e System 1 After contact formation System 2 EF1 System 2 System 1 EF E 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 q Va E f1 Side 2 Side 1 Va 0 Ef 2 E f1 q Va Ef 2 Side 2 Side 1 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 NA and E field p x is 0 ND and n Depletion region Quasi neutral region p Si NA only x is x is 0 Quasi neutral region n Si ND only 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 2 xd2 1 xd1 2 E field 0 outside depletion regions x p type Semiconductor n type Semiconductor 2 xd1 1 E 0 Professor N Cheung U C Berkeley x xd2 x 0 E 0 E 0 12 EE143 S06 Semiconductor Tutorial 2 3 Relationship between E field and charge density x d E x dx x Gauss Law 4 Relationship between E field and potential E x d x dx Professor N Cheung U C Berkeley 13 EE143 S06 Semiconductor Tutorial 2 Example Analysis n p Si junction x Q Emax qNaxd s p Si n Si x x Depletion region Depletion region qNa is very thin and is approximated as E x d x 0 3 Slope qNa s xd 2 E 0 a thin sheet charge 1 Q qNaxd Professor N Cheung U C Berkeley 4 Area under E field curve voltage across depletion region qNaxd2 2 s 14 EE143 S06 Semiconductor Tutorial 2 Superposition Principle x If 1 x E1 x and V1 x 2 x E2 x and V2 x then 1 x 2 x E1 x E2 x and V1 x V2 x qND xp x qN A xn x 0 2 x 1 x qND Q qNAxp xp x qNA x xn Q qNAxp x 0 Professor N Cheung U C Berkeley x 0 15 EE143 S06 Semiconductor Tutorial 2 1 x xp 2 x qND Q qNAxp x x xn qNA Q qNAxp x 0 x 0 E1 x E2 x xp xn x Slope qNA s x Professor N Cheung U C Berkeley x 0 Slope qND s x 0 16 EE143 S06 Semiconductor Tutorial 2 Sketch of E x E x E1 x E2 x xp x 0 Slope qNA s xn x Slope qND s Emax qNA xp s qND xn s Professor N Cheung U C Berkeley 17 EE143 S06 Semiconductor Tutorial 2 Depletion Mode Charge and Electric Field Distributions by Superposition Principle of Electrostatics x Q Metal x Q Oxide Semiconductor x xo x d x Metal x Q Oxide x 0 E x Metal Oxide Semiconductor Metal E x Oxide x xo Professor N Cheung U C Berkeley x 0 x xo Semiconductor Metal x x x x o d x xo x d x 0 Q x xo x Semiconductor x xo x d x Oxide x x 0 Metal Semiconductor x 0 x x o E x Oxide x xo Semiconductor x x x o x x 0 x x d o 18 EE143 S06 Semiconductor Tutorial 2 Why xdmax constant beyond onset of strong inversion VG VFB VOX VSi Higher than VT Picks up all the changes in VG Justification If surface electron density changes by n Ei Ef ln n n e Professor N Cheung U C Berkeley Approximation assumes VSi does not change much VOX n COX but the change of VSi changes only by kT q ln n small kT 19 EE143 S06 Semiconductor Tutorial 2 N surface n bulk exp qVSi kT p Si Na 1016 cm3 n bulk 2 1 104 cm3 n surface qVSi Ei 0 35eV Ef Onset of strong inversion at VT Professor N Cheung U C Berkeley VSi 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 n surface 2 10E 04 9 84E 05 4 61E 07 2 16E 09 1 01E 11 4 73E 12 2 21E 14 1 04E 16 4 85E 17 2 27E 19 20
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