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3 Electrons and Holes in Semiconductors

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3 Electrons and Holes in Semiconductors 3 Electrons and Holes in Semiconductors 3.1. Introduction Semiconductors : optimum bandgap Excited carriers → thermalisation Chap. 3 : Density of states, electron distribution function, doping, quasi thermal equilibrium electron and hole currents Chap. 4 : Charge carrier generation, recombination, transport equation 3.2. Basic Concepts 3.2.1. Bonds and bands in crystals Free electron approximation to band Ref. CLASSIC “Bonds and Bands in Semiconductors” J. C. Phillips, Academic 1973 Atomic orbitals → molecular orbitals → bands Valence band : HOMO Conduction band : LUMO 1 Semiconductors : 0.5 Eg 3 eV Semi-metals : 0 Eg 0.5 eV Si : sp3 orbital, diamond-like structure 3.2.2. Electrons, holes and conductivity Semiconductors At T 0 K, all electrons are in valence band – no conductivity At elevated temperature, some electrons in conduction band, and some holes in valence band 2 Conduction Band Electrons Valence Band a) a)’ Electron in CB Hole in VB b) b)’ Fig. 3.4. Electron in CB and Hole in VB 3.3. Electron States in Semiconductors 3.3.1. Band structure Electronic states in crystalline Solving Schrödinger equation in periodic potential (infinite) Bloch wavefunction ikr k, r uik r e (3.1) for crystal band i and a wavevector k, which is a good quantum number. The energy E against wavevector k is called by crystal band structure. Typical band diagram plotted along the 3 major directions in the reciprocal space. 0 0 0, X 10 0, M110, L111 3 Point k is called the Brillouin zone boundary. a 2 3 3 For zinc blende structure, 0 0 0, X 0 0, K 0, L a 2a 2a a a a 3.3.2. Conduction band At the vicinity of the minimum Ek in the conduction band k k c0 , approximation 2 2 k k c0 (3.2) Ek Ec0 * 2mc * where Ec0 Ek c0 and mc is a parameter with dimensions of mass, and defined by 2 1 1 Ec k (3.3) * 2 2 mc k is called the parabolic band approximation. * * * eg. for GaAs mc 0.067 m0 , for Si mc // 0.19 m0 mc 0.92 m0 , where m0 is the static electron mass. is determined by the atomic potentials, should actually be a tensor, but is usually treated as a scalar, and is called the effective mass of electron. Momentum of an electron and the applied force are related as dp dk F (3.4) dt dt Following eq. (3.2), the velocity and momentum can be obtained as 1 k k c0 v k Ek * (3.5) mc * p mc v k k c0 (3.6) 3.3.3. Valence band At the vicinity of the maximum in the valence band k k v0 2 2 k k v0 Ek Ev0 * (3.7) 2mv 4 * where Ev0 Ek v0 and mv is the effective mass of hole. 1 k k v0 v k Ek * (3.8) mv * p mv v k k v0 (3.9) 2 1 1 Ev k (3.10) * 2 2 mv k * * * eg. mv for GaAs mv 0.5 m0 , for Si mv 0.54 m0 , * mc is the curvature of the band. 3.3.4. Direct and indirect band gaps Whether or not minimum of conduction band and maximum of valence band occur at the same wavevector k k c0 k v0 Reciprocal lattices for (a) fcc and (b) bcc crystals 5 When the band gap is indirect, transition requires involvement of phonons for conservation of momentum. Ek (Sze) 3.3.5. Density of states According to Pauli’s exclusion principle, each quantum state only supports one electron (two, considering spins). Also, from Heisenberg’s uncertainty principle, p x ~ h ∴ k x ~ h ∴ k x ~ 2 Therefore, for a crystal of volume L L L 2 2 k ~ x L 3 L ∴ states in this unit volume. 2 Hence, the density of electron states per unit crystal volume considering spin is 2 gkd 3k d 3k (3.11) 2 3 If the band structure is spherically symmetric and isotropic around k 0 gkd 3k gk 4k 2dk (3.12) Therefore, dk gEdE gk 4k 2 dE (3.13) dE 2 dk ∴ gE 4k 2 (3.14) 2 3 dE 6 From eq. (3.2) 2m* ∴ k 2 c E E (3.15) 2 c0 Similarly, for the valence band (hole) 2m* ∴ k 2 v E E (3.15’) 2 v0 Differentiating (3.15)2 2 k k c0 Ek E 12 1 2 2m*c0 2m* dk 1 2m* k c E E c 1 2 ∴ c E E 1 2 2 c0 2 c0 dE 2 Therefore, from (3.14) 1 2 2 dk 2 1 2m* g E 4k 2 4k 2 c E E 1 2 c 3 3 2 c0 2 dE 2 2 1 2 2 2m* 1 2m* 4 c E E c E E 1 2 (3.16) 3 2 c0 2 c0 2 2 3 2 3 2 4 2m* 1 2m* c E E 1 2 c E E 1 2 3 2 c0 2 2 c0 2 2 Similarly, 3 2 1 2m* g E v E E 1 2 (3.17) v 2 2 v0 2 Question : Derive the DOS for 2D and 1D materials. (Box 3.3) Excitons A bound state of a pair of electron and hole of same k . The wavefunction can be described as ex e k,r h k,r Satisfying a hydrogenic effective mass equation 2 2 2 q * ex ex Eex ex 2 4 s r r * : reduced effective mass of e-h pair Eex : binding energy of the exciton 7 * 2 0 Ryd Eex 2 2 m0 s l 3.3.6 Electron distribution function Fermi function in semiconductors 1 f E E EF exp 1 k BT This can be expressed in terms of f k, r although the energy is only dependent on the wavevector. nr d 3k gk f k, r d 3k The total electron density is given by 3 nr gc k f k, r d k (3.20) CB k 3 pr gv k1 f k, rd k (3.21) VB k 3.3.7 Electron and hole currents Using the above formalism q 3 J n r k gc k f k, r d k (3.22) * CB k mc q 3 J p r k gv k 1 f k, r d k (3.23) * VB k mv 3.4. Semiconductor in equilibrium 3.4.1. Fermi Dirac statistics Equilibrium : no net charge exchange. In this case, it is independent of position. f k,r f0 Ek, EF , T (3.24) 1 (3.25) f 0 E, EF , T EE k T e F B 1 3.4.2. Electron and hole densities in equilibrium The number density of electrons nE with energy in the range between E and E E nEdE gEf0 E, EF , T dE (3.26) 8 Then the total density of electrons in a conduction band of minimum energy Ec is n gc Ef0 E, EF , T dE (3.27) E c and holes in a valence band with maximum (minimum) energy Ev is Ev p gv E1 f0 E, EF , T dE (3.28) 3.4.3. Boltzmann approximation When the Fermi energy EF is located far enough from the band edge such that E EF , in the conduction band, (3.25) turns into 1 E E k T F B (3.29) f0 E, EF , T EE k T e e F B 1 Similarly, in the valence band E EF leading to EF E kBT 1 e 1 EE k T F B 1 f0 E, EF , T 1 EE k T 1 E E k T E E k T e e F B 1 1 e F B 1 e F B (3.30) Using these results, (3.27), (3.28) can be integrated * 3 2 1 2mc 1 2 EF E kBT n gc E f0 E, EF , T dE E Ec0 e dE E E 2 2 c c 2 (3.31) E E F c Nc exp kBT where 9 3 2 m*k T N 2 c B (3.32) c 2 2 For holes in valence band E v Ev EF p gv E1 f0 E, EF , T dE Nv exp (3.33) kBT where 3 2 m*k T N 2 v B (3.34) v 2 2 And the product np is given as Eg kBT np Nc Nv e (3.35) This is actually a constant and the intrinsic carrier density ni suffice the relation 2 Eg kBT ni np Nc Nv e (3.36) Introducing intrinsic potential energy Ei , which is the Fermi level for intrinsic semiconductor, E E E E F i (3.37) i F (3.38) n ni exp p ni exp k BT k BT where 1 1 N c Ei Ec Ev k BT ln 2 2 N v (3.39) 1 3 m* E E k T ln c c v B * 2 4 mv Electron affinity Least amount of energy required to remove an electron from solid. → Better determined as energy difference between vacuum level and CBM. 10 Ec Evac Ev Evac Eg (3.40) EF Evac k BT ln N c Evac Eg k BT ln N v can all be derived by statistical mechanics. Validity of Boltzmann approximation The condition for the above relations to hold is E E E E F v 1 and c F 1 k BT k BT These relations hold regardless of Nc and Nv → basically, just statistical mechanical consideration.
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