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

2 Effective masses (m l * and m t *) featuring a single valley in a Brillouin zone. Effective mass in the whole Brillouin zone accounting for all valley minima: DOS Effective Mass m* ds States can be thought of as available seats for electrons in conduction band and for holes in valence band. DOS (Density-of-States)

3 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 DOS

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

5 1.Conduction Band GaAs: m dse * = m e * Silicon and Germanium: m dse * = g 2/3 (m l *m t * 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 = Valence Band – Ge, Si, GaAs m dsh * = ((m hh *) 3/2 + (m lh *) 3/2 ) 2/3 (For brevity, we do not consider the Split-off band) Relation between Valley Effective Mass and DOS Effective Mass

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

Fermi level is related to one of Laws of Nature: Conservation of Charge Extrinsic case

8 n  N C exp(  C ) p  N V exp(  V ) N C = 2(m dse *k B T/2  ħ 2 ) 3/2 N V = 2(m dsh *k B T/2  ħ 2 ) 3/2 Effective density of states in the conduction band Effective density of states in the valence band  C = (E f – E C )/k B T  V = (E V – E f )/k B T Hole concentration Electron concentration Case of E V < E f < E C Note: for E V < E f < E C, Fermi-Dirac distribution reduces to Boltzmann distribution.

9 n  N C exp(  C ) p  N V exp(  V ) N C = 2(m dse *k B T/2  ħ 2 ) 3/2 N V = 2(m dsh *k B T/2  ħ 2 ) 3/2  C = (E f – E C )/k B T  V = (E V – E f )/k B T Case of E V < E f < E C For intrinsic case where n = p, at least four statements can be drawn: E f is the intrinsic Fermi level E fi E fi is a function of the temperature T and the ratio of m dse * to m dsh * Corresponding n i (= n = p) is the intrinsic concentration n i is a function of the band gap Eg (= E c - E v )

10 Extrinsic Semiconductors in Equilibrium Extrinsic Semiconductors in Equilibrium (Uniform and Non-uniform Doping) (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 cm -3.

Intrinsic Case (No Doping, No Impurities) Microscopic View n = p

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

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

Positioning of Fermi level can reveal the doping details

n = n i exp((E f – E fi )/K B T) = N C exp((E f - E C )/K B T) p = n i exp((E fi – E f )/K B T) = N V exp((E V - E f )/K B T) pn = n i 2 for equilibrium n = p + N D + Ionized donor density E fi = (3/4)(K B T)ln(m dsh */m dse *) + (E c +E v )/2 extrinsic intrinsic n = p = n i

E f itself reflects the charge conservation. n + N A - = p

NA-NA- ND+ND+ n + N A - = p n = (N D + - N A - )+p N D + > N A - Compensation

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

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

n i versus N D or N A Extrinsic temperature range for n i = N D (= N D + ) Full ionization of impurity Ionization energy < K B T

23 EfEf

24 Non-uniform Doping Case

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

26 4-8

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