Silicon a tetrahedron a a diamond structure
Tetrahedral bonding Hund s Rule 14Si [e] 3s 3p [e] hybridize 3sp 3 Hybridized level has higher energy for an isolated atom, but allows overall reduction in energy when forming tetrahedral bonds in a solid. 3s 3p x 3p y 3p z 1 1 1 1 s p x p y pz choose four from eight possibilities
Origin of bandgap isolated molecule pair of molecules crystal insulator: Eg kt Eg ~ 3 e semiconductor: semi-metal: 0.5 e E 3 e Eg ~ ~ g ~ 0 0.5 e metal: no E g
Band structure indirect indirect direct
Bloch s Theorem Electron wave function in a crystal can be written: r u r e i kr The Block functions are periodic: urrur R n a n a n a where: 1 1 3 3 is a lattice vector: ik rr ikr rr u rr e e r Translation by a lattice vector causes a change in phase only.
1 sinusoidal potential d q x x E x mdx Band iagram bx bn n e inx a 1 U x U0 x a 1cos Potential
1 sinusoidal potential: wave functions E=0.49 E= 0.46 E= 1.5767
Electrons and holes Migration of an electron (negative charge) vacancy in one direction (left) is equivalent to the motion of a hole (positive charge) in the opposite direction (right). time semiconductor conductivity changes with: -doping -temperature -illumination
ermi-irac distribution 1 fee e EE kt 1 Gives the probability for occupation of an electron state in equilibrium ermi energy increasing T
arrier concentrations (I) As for photons, to find the # of electrons per unit electron energy, per unit volume, we need: i) density of states ii) distribution function dn de E f E umber of states from 0 to k: 4 3 g k 4 k 3 gk L 1 3 3 L 3 3
arrier concentrations (II) ear the band edges: energy hk Ek E0 Ek E m c 0 hk m v EB k Effective masses: E 1 1 d E k m h dk k 0 1 1 d E k m h dk k 0 E g k B: B: m 0 k EE h B case: 4 m 3 3 gl E EE 3 3h d m 3 3 m 0 k E E h 0 gl E EE de 3 h 3 0 1 E EB k
arrier concentrations (III) B OS (per unit volume) : 3 E m g E 4 E E 3 L h B electron concentration (per unit energy): 3 E m g E 4 E 3 E L h 1 dn m 1 1 E f E E g E f E E 4 E E de EE kt h B OS (per unit volume) : 1 1 1 e EE kt 1 e EE kt 1 B hole concentration (per unit energy): 1 3 3 e 1 otice: So: 1 f EE f E E dp m 1 1 E 1 f E E g E f E E 4 E E de E E kt h e 1
arrier concentrations (I) p-type intrinsic n-type E g E g E g dn de dn de dn de E E E E E E E E E dp de dp de dp de g g g 3 dn 1 1 4 m E E de EE kt h e 1 3 dp 1 1 4 m E E de E E kt h e 1
arrier concentrations () Total conc. of electrons in the B (integrated over energy) 3 dn m 1 1 4 EE EE EE kt n de EE de de h 1 E E x E E ktx efine: EE kt kt de ktx dx e e E E kt x e n 8 3 x 0 mkt x dx E E kt x h e e 1 e 1 on-degenerate case, (Boltzmann approx.): 1 EE kt x e e 1 e E E kt x e E E kt E E kt x E E kt e x e dxe x0 4 n mkt 3 e E E kt n h E E kt e //non-degenerate mkt h 3 // effective B density of states n E E kt //general 4 x dx y x0 y x e e 1
arrier concentrations (I) Total conc. of holes in the B (integrated over energy) E m 3 E 1 1 h E e 1 p de 4 E E de E E kt E dp de efine: x E E kt 1 E E ktx de ktx dx EE kt EE e e kt x e 3 3 0 x E E kt x x0 EE kt x m kt x dx m kt x dx p 8 8 h e e 1 h e e 1 //integrand is even on-degenerate case, (Boltzmann approx.): 1 EE kt x e e 1 e 3 mkt h p e E E kt E E kt x e E E kt // effective B density of states //non-degenerate p E E kt //general
arrier concentrations (II) The ermi function reduces to the (simpler) Boltzmann function is sufficient when the occupation probability is low. E E kt E E kt e E E kt e E E kt Boltzmann 1 EE e kt 1 1 EE e kt 1 ermi E E kt otice: E E kt e E E kt
Semiconductors (I) We most often use an energy vs. position diagram, showing only band edges: n e E E kt energy E B edge E E g E position B edge p e E E kt
Semiconductors (II) otice: n p e E E kt Eg kt e n i n e i E g kt //intrinsic carrier conc. This is the # of electrons in B, and holes in B, respectively, of a pure semiconductor in equilibrium. We can write: nn p i n i e e i i E E kt E E kt E i E E kt E E 3kT m ln ln 4 m //intrinsic energy level If m m, then E is exactly in the center of the gap i i If E E, then n p n i (material is intrinsic).
Semiconductors (III) Somewhere above the B edge is the vacuum level: E vac (Electron is free from the solid) w //electron affinity E E E E E vac vac g E E E E E ktln ktln E n p w vac g w //work function E E ktln n E E ktln p In metals: w In semiconductors, w depends on doping.
Semiconductors: oping (I) If we substitute an atom in pure Si w/b or P, the dopant atom has one missing/extra valence electron: 14Si [e] 3sp 3 5B - [He] sp 3 15P + [e] 3sp 3 The dopant easily ionizes, either -accepting an electron (B) from the B, or -donating an electron (P) from the -These are called electron acceptors/donors
Semiconductors: oping (II) We can use the Bohr model to estimate the ionization energy of a dopant: oulomb potential: U r e e 4 r 4r 0 Bohr radius: a 40 4 0 me 0 me The ionization energy of H is: 1 Ryd m 13.6 e 0 a0 The ionization energy of the dopant atom is: n m 0 * m a m 0 1 Ryd m //permittivity of solid //carrier effective mass
Semiconductors: oping (III) n E B electrons n E ionized donors A p p E E ionized acceptors B holes donors ionized neutral ionized donor level: E E n acceptors ionized neutral ionized acceptor level: E E A p
Semiconductors: oping (I) At T=0 K, dopant levels are neutral. With a single dopant, at room temp., start by assuming all shallow acceptors/donors are ionized. doped n-type n //donor conc. p ln E E kt n i A p A doped p-type //acceptor conc. n ln n i A A E E kt E E E E Ei Ei E doped n-type E doped p-type E A E
Semiconductors: oping () opant Levels in Si B. G. Streetman, Solid State Electronic evices, 3 rd Ed., Prentice Hall (1990).
Semiconductors: oping (I) Assume: ln E E kt n Adjust the fermi level: n E E E E E E E E n Iterate: Evaluate: f E E 1 EE e kt 1 (fraction not ionized, should be small) orrect n: n 1 f E E orrect E E :
Semiconductors: oping (II) Example: Si doped with P d 16 3 10 cm 0.044 e n 16 3 n 10 cm Trick: * m m * 0 3 0 m m 0 1.1 ln E E kt 0 0 3 mkt.510 cm 19 3 E E 0.188 e 19 3.910 cm E E E E n 0.188 e 0.044 e 0.144 e 3 E f E 3.8 10 16 3 n 1 f E E 1.9910 cm nearly complete ionization: n 99.6% E E 0.188 e
harge ensity x pxnx x x A charge density holes electrons ionized donors ionized acceptors harge neutrality in a uniformly doped region: nn i e pn i i E E kt e i E E kt negative positive A fe 1 EkT e 1 n p A A 1 A f E E Algorithm: If A n p, E 1 f E E If A n p, E
harge eutrality (I) A 0 If i and n p n n i p o-doping (both acceptors and donors): If nn A i we can still assume full ionization: A A However, A of the donated electrons serve to ionize (fill) the acceptor levels: ow nn Example: n i p n n 810 cm A 16 3 10 cm 16 3 This is called compensation. na610 cm 16 3