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Prof. Ming-Jer Chen Department of Electronics Engineering National Chiao-Tung University

DEE4521 Semiconductor Device Physics Lecture 3: Electrons and Holes. Prof. Ming-Jer Chen Department of Electronics Engineering National Chiao-Tung University October 1, 2012. DOS (Density-of-States).

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Prof. Ming-Jer Chen Department of Electronics Engineering National Chiao-Tung University

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  1. DEE4521 Semiconductor Device Physics Lecture 3: Electrons and Holes Prof. Ming-Jer Chen Department of Electronics Engineering National Chiao-Tung University October 1, 2012

  2. DOS (Density-of-States) • Effective masses (ml* and mt*) featuring a singlevalley in a Brillouin zone. • Effective mass in the whole Brillouin zone accounting for allvalley minima: • DOS Effective Massm*ds • States can be thought of as available seats forelectrons in conduction band and forholes in valence band.

  3. DOS • One way to derive DOS function and hence its DOS effective mass: • Solve Schrodinger equation in real space to find • corresponding k solutions in k space • Apply Pauli exclusion principle on these k solutions • Mathematically Transform an ellipsoidal energy surface to a sphere energy surface, particularly for Si and Ge

  4. Electron DOS and Hole DOS S(E): DOS function, the number of states per unit energy per unit volume in real space. mdse*: electron DOS effective mass, which carries the information about DOS in conduction band mdsh*: hole DOS effective mass, which carries the information about DOS in valence band

  5. Relation between Valley Effective Mass and DOS Effective Mass • Conduction Band • GaAs: • mdse* = me* • Silicon and Germanium: • mdse* = g2/3(ml*mt*2)1/3 • where the degeneracy factor g is the number of ellipsoidal • constant-energy surfaces lying within the Brillouin zone. For Si, g = 6; • For Ge, g = 8/2 = 4. • 2.Valence Band – Ge, Si, GaAs • mdsh* = ((mhh*)3/2 + (mlh*)3/2)2/3 • (For brevity,we do not consider the Split-off band)

  6. Fermi-Dirac Statistics Fermi-Dirac distribution function gives the probability of occupying an energy state E. 1 - f(E): the probability of not filling state E Ef: Fermi Level

  7. Fermi level is related to one of Laws of Nature: Conservation of Charge 2-13 Extrinsic case

  8. Case of EV < Ef < EC C = (Ef – EC)/kBT Electron concentration nNC exp(C) p  NV exp(V) V= (EV – Ef)/kBT Hole concentration Effective density of states in the conduction band NC = 2(mdse*kBT/2ħ2)3/2 NV = 2(mdsh*kBT/2ħ2)3/2 Effective density of states in the valence band Note: for EV < Ef< EC, Fermi-Dirac distribution reduces to Boltzmann distribution.

  9. Case of EV < Ef < EC C = (Ef – EC)/kBT nNC exp(C) p  NV exp(V) V= (EV – Ef)/kBT NC = 2(mdse*kBT/2ħ2)3/2 NV = 2(mdsh*kBT/2ħ2)3/2 • For intrinsic case where n = p, at least four statements can be drawn: • Ef is the intrinsic Fermi level Efi • Efi is a function of the temperature T and the ratio of mdse* to mdsh* • Corresponding ni (= n = p) is the intrinsic concentration • ni is a function of the band gap Eg (= Ec- Ev)

  10. Extrinsic Semiconductors in Equilibrium (Uniform and Non-uniform Doping)

  11. Uniform Doping We first focus on Non-Degenerate semiconductors, the Case of low and moderate doping of less than 1020cm-3.

  12. Intrinsic Case (No Doping, No Impurities) 2-5 Microscopic View n = p

  13. Silicon Crystal doped with phosphorus (donor) atoms. 2-6 n > p One typical method to dope or introduce impurities: High-energy ion implant at room temperature, followed by High temperature ( 1000 oC) annealing (to eliminate the defects and to place impurities on the lattice positions correctly)

  14. Acceptors in a semiconductor An electron is excited from the valence band to the acceptor state, leaving behind a quasi-free hole. 2-8 p > n

  15. 2-13 Positioning of Fermi level can reveal the doping details

  16. 2-14 n = niexp((Ef – Efi)/KBT) = NCexp((Ef - EC)/KBT) p = niexp((Efi – Ef)/KBT) = NVexp((EV - Ef)/KBT) pn = ni2 for equilibrium extrinsic intrinsic Efi = (3/4)(KBT)ln(mdsh*/mdse*) + (Ec+Ev)/2 n = p + ND+ n = p = ni Ionized donor density

  17. n + NA- = p 2-15 Ef itself reflects the charge conservation.

  18. Compensation 2-16 n = (ND+ - NA-)+p n + NA- = p ND+ NA- ND+ > NA-

  19. 2-17 Electron distribution function n(E) Evidence of DOF = 3

  20. Energy band-gap dependence of silicon on temperature 2-18

  21. 2-19

  22. 2-20 Full ionization of impurity ni versus ND or NA Ionization energy < KBT Extrinsic temperature range for ni = ND (= ND+)

  23. Ef

  24. Non-uniform Doping Case

  25. Nonuniformly doped semiconductor Only for doping with non-uniform distribution can Einstein relationship be derived. 4-2

  26. 4-8

  27. Built-in Fields in Non-uniform Semiconductors 4-9

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