Basic cell design Si cell 1
Concepts needed to describe photovoltaic device 1. energy bands in semiconductors: from bonds to bands 2. free carriers: holes and electrons, doping 3. electron and hole current: mobility, drift current, diffusion current 4. optical generation: absorption of light, direct and indirect bandgap 5. recombination of carriers: lifetime and diffusion length 6. pn junction and heterojunction 2
From bonds to bands Silicon 1s 2 electrons 2s 2 electrons 2p 6 electrons 3s 2 electrons 3p 2 electrons valence electrons 8 electrons: closed shell 3
Covalent bonding - sp 3 hybridisation Example: carbon 2sp 3 (Si: 3sp 3, Ge: 4sp 3 ) 2p 2s 2p 2s 109.5 o 2sp 3 4
Energy bands in solids energy levels in a single atom (molecule) energy bands bands: splitting of atomic orbitals number of levels in a band (2l+1)*N atomów 5
Insulators, semiconductors, metals empty conduction band (LUMO) energy energy levels in single atom (molecule) occupied valence band (HOMO) occupied bands in semiconductor energy insulator semicond. metal conduction band (empty) energy gap valence band (full) 6
Width of the energy gap depends on a distance between atoms in the lattice! distance between atoms 7
from bonds to bands Si conduction band: lowest unccupied band energy gap valence band: highest fully occupied band GaAs covalent semiconductors: 8 electrons in molecular states - closed shell 8
Compound semiconductors III-V E g [ev] GaN 3.4 GaP 2.25 GaAs 1.52 GaSb 0.81 InP 1.42 InAs 0.43 InSb 0.24 II-VI E g [ev] ZnS 3.54 ZnSe 2.7 ZnTe 2.25 CdTe 1.56 HgTe -0.01 9
Structure of solids single crystalline policrystalline policrystalline amorphous hydrogen amorphous 10
Crystal structure unit cell diamond (Si, Ge) zinc blende (ZnSe, GaAs) chalcopyrite CuInSe 2 11
Basic concepts needed to describe photovoltaic device 1. energy bands in semiconductors: from bonds to bands 2. free carriers: holes and electrons, doping, 3. electron and hole current: mobility, drift current, diffusion current 4. optical generation: absorption of light, direct and indirect bandgap 5. recombination of carriers: lifetime and diffusion length 6. pn junction and heterojunction 12
Electrons in the conduction band and holes in the valence band silicon: E g =1.09 ev germanium: E g = 0.72 ev 13
Electrons and holes thermal generation conduction band E F valence band CB VB - + hole empty place after excitation of an electron, quasi-particle with positive charge CB energy for electrons VB energy for holes 14
Concentration of free electrons in CB and holes in VB: Fermi-Dirac distribution: probability that electron (hole) has energy E prawdopodobienstwo f(e) 1.0 0.5 0.0 T=0 T 2 > T 1 T 1 >0 E F energia E T = 0 K : f(e) = 1dla E < E e f(e) = 0 dla E > E e T > 0 K 1 f(e) = e E E 1+ exp k T B F F F for holes f h =1-f e E F - Fermi-level, f(e F ) = ½ for electrons in CB: E-E F >>k B T: f e (E) - (E E exp kbt F ) for holes in VB: E F -E>> k B T: f e (E) - (E exp k F B E) T 15
Concentration of free electron and holes Valence band Valence band Valence band T increases probability of thermal excitation increases concentration of free electrons and free holes increases - (Ec E n = Ncexp kbt F ) - (EF E p = Nvexp kbt v ) 16
Intrinsic semiconductor: E F E V E c E F E g /2 conc. of free electrons in the CB = conc. of free holes in the VB n 0 =p 0 Law of mass action: n o p o =n o2 =p o2 =n i 2 n i = p i = (N C N V ) 1/2 exp(-e g /2k B T) Effective density of states in the cond. (valence) band N c, N v ~10 19 cm -3 k B T(300 K) 0.025 ev E g n i (300 K) ~0.25 ev 10 16 cm -3 InSb, PbSe ~1 ev 10 10 cm -3 Ge, Si, GaAs ~4 ev <10 10 cm -3 ZnS, SiC,GaN, 17
Free electrons and holes Intrinsic semiconductor T = 0 K T > 0 K Intrinsic free carrier concentration depends on temperature and energy gap n = p = n i = N c N v Eg exp Nc, kbt v Eg exp 2kBT E g 1 ev : n i 10 10 cm -3 ( j = nev=10 8-10 -9 Α/mm 2 ) 18
n and p-type doping Example: Si P donor (5 valence electrons) free electron B acceptor (3 valence electrons) free hole P impurity acts as donor B impurity acts as acceptor free electrons Fermi level ionised donors ionised acceptors Fermi level free holes type n type p 19
Donors and acceptors conduction band donor level E a acceptor level valence band Small energy needed for ionisation electron or hole from the dopant atom is represented by a shallow energy level in the bandgap E d and E a < 100 mev, usually 10-50 mev 20
n-type doping p-type doping E d E F + + + + + + + + + + donor level E F E a + + + + + + + + + acceptor level n=n d (all donors ionized at 300 K) n=n c exp{-(e c -E F )/k B T} majority carriers p=n a (all acceptors ionized) p=n V exp{-(e F -E V )/k B T} E F close to CB p=n i2 /N d << n minority carriers E F close to VB n=n i2 /N a << p 1 per million atom replaced by a dopant concentration of majority carriers 10 16 cm -3 >> n i concentration of minority carriers 10 4 cm -3 << n i 21
Free carriers in perfect lattice quantum mechanical description 2 2 h 2m Elektron in empty space Δ Ψ = EΨ Ψ k = Ae r r ikr k p = 2π λ h = λ hk =hk 2π de Broglie relation Electron in periodic potential 2 h r 2 r ΔΨ + UΨ = EΨ U(r) = U(R + 2m r r rr ikr Ψ(r) = Ψ(R + r) Ψ = u (r)e k k r ) u k (r) = u k (r+r) Bloch function k = const in perfect lattice p = hk quasi momentum = const 22
K vector and energy of electron in periodic lattice periodicity in space: x=x+a periodicity of the wave function Ψ ( r) = Ψ ( r a) k k + periodicity of the wave vector k=k+2π/a periodicity of electron energy E(k)=E(k+2π/a) K=2π/a limited range of k-vectors is only needed to describe an electron in the conduction band (hole in the valence band) -π/a<k<π/a 23
Band structure - model of nearly free electron: E(k) free electron in vacuum: E(p) = 2 p 2m E(k) CB electron in crystal lattice E(p) = 2 p 2m * m* - effective mass of electron in CB (hole in VB) VB k p = h hk = = hk λ 2π E(k) = 2 2 h k 2m * Effective mass of electron (hole) takes into account interaction with the lattice m* m e, usually m e <m h r dp dt r dk = h = m * dt r dv dt = r F zew 24
Free carriers in perfect lattice, F ext =0 free electron (hole) moves with k=const in the absence of external forces hk v = m* effective mass energy distribution of free carriers in the bands intrinsic semiconductor n-doped semiconductor p-doped semiconductor E av =3/2 k B T - average energy of free carrier average thermal velocity of free carrier 3k B T m * 5 v th 10 = m/s at 300 K 25