Energy band diagrams In the atoms, the larger the radius, the higher the electron potential energy Hence, electron position can be described either by radius or by its potential energy In the semiconductor crystal: the atom orbits OVERLAP; radius-based description becomes impractical. Energy-based description works well: The highest orbit filled with electrons becomes the VALENCE BAND The higher orbit (nearly empty) becomes the CONDUCTION BAND E Single atom Excited electrons cannot move Crystal Excited electrons can move (free electrons)
Energy band diagrams Free electron free electrons Conductance energy band Hole holes Valence energy band Bandgap (or forbidden energy )
Optical processes in semiconductors: radiation and absorption Related electrical processes: electron - hole pair generation and recombination
Absorption Related electrical process: electron - hole pair generation The photon with the energy exceeding the bandgap energy of semiconductor can be absorbed. The photon disappears; the photon energy excites the electron from the valence band into the conduction band. As a result, one e-h pair is being created Photon absorption E = h ν = Ε g
Radiation Related electrical process: electron - hole pair recombination When the excited electron meets the hole in the valence band, it may occupy that place. As a result the e-h pair disappear; this process is called recombination. During recombination, the electron energy is released as a photon with the energy closed to the bandgap energy of the semiconductor. Photon Photon absorption emission E = h ν = Ε h ν = Ε g g
Electron and hole concentrations under illumination We define n 0 and p 0 as the electron and hole concentrations in the absence of illumination ( dark concentrations). n and p are the additional concentrations generated by light. Note that in the equilibrium for ANY semiconductor, n p = n i 2 Under illumination: n p ( n+n 0 ) ( n+ p 0 ) > n i 2
Generation rate When the semiconductor is under CONSTANT illumination, the photons are being absorbed at a constant rate; absorbed photons GENERATE electron- hole pairs Therefore the concentration of e-h pairs MUST linearly increase with the time. The GENERATION RATE, G, is the number of electron-hole pair generated per unit time: n= p = G t; How does the semiconductor sample come to a steady-state condition under illumination?
Recombination rate The probability of electron and hole "annihilation", or the RECOMBINATION rate, is proportional to both electron AND hole concentrations: R ~ n p = Β r n p Therefore, when n and p increase due to illumination, the RECOMBINATION rate, R, also increases. The e-h concentration increases until the increasing recombination rate would compensate it. Under the steady state condition we have: G = R;
Steady-state e-h pair concentration G = R; R = Β r n p The steady-state concentration of photo-electrons (and photo-holes): G = Β r n p n p = G /Β r n p ( n+n 0 ) ( n+ p 0 ) Under strong illumination, n>>n 0 and n>> p 0 n p n 2 G = Β r n 2 : n = (G/Β r ) 1/2
Spontaneous and Excessive Recombination Rate In case of direct electron - hole recombination The spontaneous recombination rate in the equilibrium as
Electron and hole LIFETIME The recombination rate R is often expressed as R = n τ The lifetime, τ, determines the mean time an electron spends before recombining with hole. As follows from at very high excitation level, n >> n 0, p 0 At low excitation level, n << n 0, p 0
Example Optical beam irradiating an intrinsic semiconductor (GaAs) produces 0.5 10 23 cm -3 /s electron-hole pairs. The steady state concentration of photoelectrons is n = 10 14 cm -3. 1) Find the electron /hole recombination lifetime τ. 2) Find the radiative recombination coefficient B r
Solution In steady state, The recombination rate, Therefore, The lifetime G = R R = n τ G = n τ τ = n G τ = n G = 14 3 10 cm 9 = 2 10 23 3 1 0.5 10 cm s s = 2ns
Solution In GaAs, n i ~ 10 5 cm -3, therefore, n>> n 0. In this case, B r 1 1 6 3 1 = = 5 10 cm 14 3 9 τ r 10 cm 2 10 s = n s
Radiative and nonradiative recombination Nonradiative recombination typically does not produce photons; The electron energy is being transferred to phonons, i.e. into the heat.
Radiative and nonradiative recombination When both radiative and nonradiative recombination processes take place in semiconductor,