Recombination-Generation Process Chapter 5 Recombination-Generation Process

INTRODUCTION Survey of R-G Processes Recombination Processes Recombination: A process whereby electrons and holes(carriers) are annihilated or destroyed. Generation: A process whereby electrons and holes are created. Recombination Processes (1) Band-to-Band Recombination (direct thermal recombination) radiative recombination (photon emission)

(2) R-G center recombination (band-to-impurity recombination) Shockely-Read-Hall theory(indirect thermal recombination) radiative nonradiative R-G center recombination is important in indirect bandgap material with high concentration of deep or tightly bound nonhydrogenic centers. Isoelectronic N-doping in GaP (indirect bandgap) (3) Recombination via shallow levels very inefficient R-G center at room temperature. Recombination via shallow levels is important radiative process at low temperature.

(4) Recombination involving excitons free exciton bound exciton 𝐸 𝑒𝑥 ≈5 𝑚𝑒𝑉 When electrons and holes are sufficiently close, a Coulomb attraction between them can be expected because of their opposing effective charges. For sufficiently low thermal energy, the coulomb attraction results in a situation where the electrons and holes circle each other around their joint center of mass. This bound electron-hole pair is referred to as an exciton since it represent the first excited state of the one-electron energy band. The coulomb attraction lowers the energy of an electron, which would, otherwise, be in the conduction band, and produces a series of allowed energy states just below the conduction band in the forbidden bandgap. The binding energy of exciton, 𝐸 𝑒𝑥 =13.6 𝑒𝑉 1 𝑛 𝜀 𝑟 2 𝑚 𝑟 𝑚 0 2 𝑚 𝑟 = 𝑚 𝑛 ∗ 𝑚 𝑝 ∗ 𝑚 𝑛 ∗ + 𝑚 𝑝 ∗ where : reduced mass

(5) Auger recombination The energy and momentum released by recombining electron hole pair is immediately observed by another electron or hole which then dissipates this energy and momentum by emitting phonons. Collision of three particles: nonradiative recombination reverse of impact ionization Dominant in heavily doped direct band gap materials and narrow band gap materials

Generation Processes Band-to-band generation R-G center generation and photoemission from band gap center Impact ionization (opposite process of Auger recombination)

Momentum Considerations For direct band gap material Energy conservation 𝐸 𝑖 𝐸 𝑝ℎ𝑜𝑡𝑜𝑛 =ℏ𝜔 = 𝐸 𝑖 − 𝐸 𝑓 Thermal energy or light energy ℏ𝜔 ≈ 𝐸 𝑔 Momentum conservation ℏ𝜔 > 𝐸 𝑔 𝐸 𝑓 ℏ 𝑘 𝑖 − 𝑘 𝑓 + 𝑘 𝑝ℎ𝑜𝑡𝑜𝑛 =0 For example, GaAs with lattice constant, a =5.65 Å along 𝛤 → X. = 2𝜋 𝑎 Width in k-axis along (𝛤 → X) ℏ𝜔 ≈ 𝐸 𝑔 = 1.424 eV, λ= 0.87 𝜇𝑚 >> a, 𝑘 𝑝ℎ𝑜𝑡𝑜𝑛 = 𝑐 2𝜋 λ 𝑘 𝑝ℎ𝑜𝑡𝑜𝑛 ≪ 2𝜋 𝑎 very small in E – k diagram 𝑘 𝑝ℎ𝑜𝑡𝑜𝑛 ≈0

For indirect band gap material phonon absorption Thermal energy associated with lattice vibrations (phonons) ≈10 ~ 50 meV. 𝐸 𝑖 phonon emission ℏ𝜔 𝐸 𝑓 Effective phonon mass and associated momentum are comparatively large. ∴ Phonon associated transition is essentially horizontal. Energy conservation phonon absorption ℏ𝜔 = 𝐸 𝑖 − 𝐸 𝑓 ∓ 𝐸 𝑝ℎ𝑜𝑛𝑜𝑛 phonon emission Momentum conservation ℏ 𝑘 𝑖 − 𝑘 𝑓 + 𝑘 𝑝ℎ𝑜𝑡𝑜𝑛 + 𝑘 𝑝ℎ𝑜𝑛𝑜𝑛 =0 ≈0

RECOMBINATION-GENERATION STATISTICS Band-to-Band Recombination intraband scattering (~10-13 sec) 𝐸 𝑖 recombination ( 10-9 ~ 10-6 sec) photon emission ℏ𝜔 ≈ 𝐸 𝑔 light energy absorption ℏ𝜔 > 𝐸 𝑔 𝐸 𝑓 Thermal generation rate, Gt = Gt(T): function of temperature only because large number of filled electrons in the valence band and empty states in the conduction band. Recombination generation rate, R ∝ np → R(n, p,T) = r(T)np Rate constant or recombination coefficient [cm3/sec] In equilibrium, Gt(T)= R(n0, p0,T) = r(T)n0p0 = r(T)ni2 for nondegenerated case

With a steady-state nonequilibrium situation G + Gt(T)= R(n, p,T) = r(T)np When the external stimulus is turned off, G = 0, n → n0 p → p0 decay of carriers − 𝜕𝑛 𝑟 , 𝑡 𝜕𝑡 =− 𝜕𝑝 𝑟 , 𝑡 𝜕𝑡 =𝑅 𝑛, 𝑝, 𝑇 − 𝐺 𝑡 𝑇 =𝑟 𝑇 𝑛𝑝−𝑟 𝑇 𝑛 0 𝑝 0 =𝑟 𝑇 (𝑛𝑝− 𝑛 𝑖 2 ) − 𝜕𝑛 𝑟 , 𝑡 𝜕𝑡 =− 𝜕 𝑛 0 𝑟 +∆𝑛 𝑟 , 𝑡 𝜕𝑡 =− 𝜕∆𝑛 𝑟 , 𝑡 𝜕𝑡 =− 𝜕∆𝑝 𝑟 , 𝑡 𝜕𝑡 = 𝑟 𝑛 0 +∆𝑛 𝑝 0 +∆𝑝 −𝑟 𝑛 0 𝑝 0 =𝑟 𝑛 0 𝑝 0 + 𝑛 0 ∆𝑝+ 𝑝 0 ∆𝑛+∆𝑛∆𝑝 −𝑟 𝑛 0 𝑝 0 =𝑟 𝑛 0 ∆𝑝+ 𝑝 0 ∆𝑛+∆𝑛∆𝑝 Assuming charge neutrality, ∆ 𝑛=∆𝑝 − 𝜕∆𝑝 𝑟 , 𝑡 𝜕𝑡 =𝑟 𝑛 0 + 𝑝 0 +∆𝑝 ∆𝑝 then, Separating variables and integrating with time −𝑟 0 𝑡 𝑑𝑡 = ∆𝑝(0) ∆𝑝(𝑡) 𝑑∆𝑝 𝑛 0 + 𝑝 0 +∆𝑝 ∆𝑝

1 𝑛 0 + 𝑝 0 𝑙𝑛 ∆𝑝 𝑛 0 + 𝑝 0 +δ𝑝 ∆𝑝(0 ∆𝑝(𝑡) =−𝑟𝑡 𝑙𝑛 ∆𝑝(𝑡) 𝑛 0 + 𝑝 0 +∆𝑝(0) 𝑛 0 + 𝑝 0 +δ𝑝(𝑡) ∆𝑝(0) =−𝑟 (𝑛 0 + 𝑝 0 )𝑡 ∆ 𝑝(𝑡)= 𝑛 0 + 𝑝 0 ∆𝑝(0) 𝑛 0 + 𝑝 0 +∆𝑝(0) 𝑒 𝑟 (𝑛 0 + 𝑝 0 )𝑡 −∆𝑝(0) (i) Low level excitation, ∆ 𝑝 0 =∆𝑛 0 ≪ 𝑛 0 + 𝑝 0 ∆ 𝑝(𝑡)≈∆𝑝(0) 𝑒 −𝑟 (𝑛 0 + 𝑝 0 )𝑡 =∆𝑝 0 𝑒 − 𝑡 𝜏 𝑝 𝜏 𝑝 ≡ 1 𝑟 (𝑛 0 + 𝑝 0 ) where : excess carrier lifetime 𝜕∆𝑝 𝑡 𝜕𝑡 = ∆𝑝 0 𝜏 𝑝 𝑒 − 𝑡 𝜏 𝑝 =− ∆𝑝 𝑡 𝜏 𝑝 (ii) High level excitation, ∆ 𝑝 0 =∆𝑛 0 ≫ 𝑛 0 + 𝑝 0 −𝑟 0 𝑡 𝑑𝑡 ≈ ∆𝑝(0) ∆𝑝(𝑡) 𝑑∆𝑝 ∆𝑝 2 − 1 ∆𝑝(𝑡) − 1 ∆𝑝(0) =−𝑟𝑡 ∆ 𝑝 𝑡 = ∆𝑝(0) 1+𝑟𝑡∆𝑝(0) : quadratic recombination (hyperbolic decay of excess carrier with time)

Lifetime for high level excitation? From the expression in low level excitation, 𝜕∆𝑝 𝑡 𝜕𝑡 =− ∆𝑝 𝑡 𝜏 𝑝 𝜕∆𝑝 𝑡 𝜕𝑡 = 𝜕 𝜕𝑡 ∆𝑝(0) 1+𝑟𝑡∆𝑝(0) = 𝑟 ∆𝑝(0) 2 1+𝑟𝑡∆𝑝(0) 2 =𝑟 ∆𝑝(𝑡) 2 =− ∆𝑝 𝑡 𝜏 𝑝 𝜏 𝑝 𝑡 = 1 𝑟∆𝑝 𝑡 : bimolecular recombination Lifetime continually changes as system relaxes to equilibrium

RECOMBINATION-GENERATION STATISTICS Band-to-Impurity Recombination Shockely-Read-Hall Theory important in indirect band gap semiconductor with high concentration of deep or tightly bound nonhydrogenic centers. Definition of Terms 𝜕𝑛 𝜕𝑡 | 𝑅−𝐺 : Time rate in electron concentration due to both R-G center recombination and R-G center generation 𝜕𝑝 𝜕𝑡 | 𝑅−𝐺 : Time rate in hole concentration due to both R-G center recombination and R-G center generation 𝑛 𝑇 : Number of R-G centers/cm3 that are filled with electrons = 𝑁 𝑇 − for acceptor-like = 𝑁 𝑇 − 𝑁 𝑇 + for donor-like 𝑝 𝑇 : Number of empty R-G centers/cm3 𝑁 𝑇 : Total number of R-G centers/cm3 = 𝑛 𝑇 + 𝑝 𝑇

If the processes (a) and (b) have a higher probability of occurring than (c) and (d), the center acts as and electron trap. Vice versa for hole trap. The center enhances recombination(acts as a recombination center) when processes (a) and (c) have a higher probability than (b) and (d). Processes (b) and (d) should be easier for shallow center. Trapping center is most efficient when the center is near the center of the band gap. Temperature tends to slow down the recombination process through trap. 𝜕𝑛 𝜕𝑡 | 𝑅−𝐺 = 𝜕𝑛 𝜕𝑡 | (𝑎) + 𝜕𝑛 𝜕𝑡 | (𝑏) 𝜕𝑝 𝜕𝑡 | 𝑅−𝐺 = 𝜕𝑝 𝜕𝑡 | (𝑐) + 𝜕𝑝 𝜕𝑡 | (𝑑)

− 𝜕𝑛 𝜕𝑡 | 𝑎 ∝𝑛∙ 𝑁 𝑇 (1−𝑓( 𝐸 𝑇 )∝𝑛∙ 𝑝 𝑇 = 𝑐 𝑛 𝑛∙ 𝑝 𝑇 = 𝑣 𝑡ℎ 𝜎 𝑛 𝑛∙ 𝑝 𝑇 − 𝜕𝑛 𝜕𝑡 | (𝑎) Electron capture rate, proportionality constant (electron capture coefficient [cm3/sec]) thermal velocity [cm/sec] − 𝜕𝑛 𝜕𝑡 | 𝑎 ∝𝑛∙ 𝑁 𝑇 (1−𝑓( 𝐸 𝑇 )∝𝑛∙ 𝑝 𝑇 = 𝑐 𝑛 𝑛∙ 𝑝 𝑇 = 𝑣 𝑡ℎ 𝜎 𝑛 𝑛∙ 𝑝 𝑇 number of unoccupied trap states capture cross-section [cm2] free electron concentration 𝜕𝑛 𝜕𝑡 | (𝑏) Electron emission rate, proportionality constant (electron emission coefficient [1/sec]) 𝜕𝑛 𝜕𝑡 | 𝑏 = 𝑒 𝑛 ∙1∙ 𝑁 𝑇 (𝑓( 𝐸 𝑇 )= 𝑒 𝑛 𝑛 𝑇 electron emission coefficient [1/sec] number of occupied trap states because conduction band has numerus unoccupied states Hole capture rate, − 𝜕𝑝 𝜕𝑡 | (𝑐) number of holes in the valence band( number of unoccupied states in the valence band) − 𝜕𝑝 𝜕𝑡 | 𝑐 = 𝑐 𝑝 𝑁 𝑇 (𝑓( 𝐸 𝑇 )∙𝑝= 𝑐 𝑝 𝑛 𝑇 𝑝= 𝑣 𝑡ℎ 𝜎 𝑝 𝑛 𝑇 𝑝 number of occupied trap states hole capture coefficient [cm3/sec] Hole emission rate, 𝜕𝑝 𝜕𝑡 | (𝑑) because valence band has numerus filled states number of unoccupied trap states 𝜕𝑝 𝜕𝑡 | 𝑑 = 𝑒 𝑝 ∙1∙ 𝑁 𝑇 (1−𝑓( 𝐸 𝑇 )∙𝑝= 𝑒 𝑝 𝑝 𝑇 hole emission coefficient [1/sec]

The Equilibrium Simplification Net electron recombination rate, 𝑟 𝑁 +: if recombination is dominant. -: if generation is dominant. 𝑟 𝑁 ≡− 𝜕𝑛 𝜕𝑡 | 𝑅−𝐺 = 𝑐 𝑛 𝑛 𝑝 𝑇 − 𝑒 𝑛 𝑛 𝑇 Net hole recombination rate, 𝑟 𝑝 +: if recombination is dominant. -: if generation is dominant. 𝑟 𝑃 ≡− 𝜕𝑝 𝜕𝑡 | 𝑅−𝐺 = 𝑐 𝑝 𝑝 𝑛 𝑇 − 𝑒 𝑝 𝑝 𝑇 𝑟 𝑁 =0 under thermal equilibrium 𝑟 𝑃 =0 equilibrium Then, 𝑒 𝑛0 = 𝑐 𝑛0 𝑛 0 𝑝 𝑇0 𝑛 𝑇0 = 𝑐 𝑛0 𝑛 1 𝑒 𝑝0 = 𝑐 𝑝0 𝑝 0 𝑛 𝑇0 𝑝 𝑇0 = 𝑐 𝑝0 𝑝 1 and 𝑛 1 = 𝑛 0 𝑝 𝑇0 𝑛 𝑇0 = 𝑛 0 𝑁 𝑇 −𝑛 𝑇0 𝑛 𝑇0 = 𝑛 0 𝑁 𝑇 𝑛 𝑇0 −1 where from chapter 4 𝑛 𝑇0 𝑁 𝑇 =1 − 𝑁 𝑇 + 𝑁 𝑇 = 1 1+ 𝑒 𝐸 𝑇 ′ − 𝐸 𝐹 /𝑘𝑇 𝑁 𝑇 + 𝑁 𝑇 = 1 1+ 𝑔 𝑇 𝑒 𝐸 𝐹 − 𝐸 𝑇 /𝑘𝑇 = 1 1+ 𝑒 𝐸𝐹− 𝐸 𝑇 ′ /𝑘𝑇 𝐸 𝑇 ′ = 𝐸 𝑇 ±𝑘𝑇𝑙𝑛 𝑔 𝑇

The concentration of holes that would be in valence band if EF = ET’. 𝑛 𝑇0 𝑁 𝑇 = 1 1+ 𝑒 𝐸 𝑇 ′ − 𝐸 𝐹 /𝑘𝑇 , 𝑛 0 = 𝑛 𝑖 𝑒 𝐸 𝐹 − 𝐸 𝑖 /𝑘𝑇 ∴𝑛 1 = 𝑛 0 𝑁 𝑇 𝑛 𝑇0 −1 = 𝑛 𝑖 𝑒 𝐸 𝐹 − 𝐸 𝑖 /𝑘𝑇 (1+ 𝑒 𝐸 𝑇 ′ − 𝐸 𝐹 /𝑘𝑇 -1) = 𝑛 𝑖 𝑒 𝐸 𝑇 ′ − 𝐸 𝑖 /𝑘𝑇 if 𝑔 𝑇 =1, 𝐸 𝑇 ′ = 𝐸 𝑇 The concentration of electrons that would be in the conduction band if EF = ET’. Similarly, 𝑝 1 = 𝑛 𝑖 𝑒 𝐸 𝑖 − 𝐸 𝑇 ′ /𝑘𝑇 The concentration of holes that would be in valence band if EF = ET’. 𝑒 𝑛0 = 𝑐 𝑛0 𝑛 1 𝑒 𝑝0 = 𝑐 p0 𝑝 1 Assuming 𝑒 𝑛 ≅ 𝑒 𝑛0 → 𝑐 𝑛0 𝑛 1 ≅ 𝑐 𝑛 𝑛 1 𝑒 𝑝 ≅ 𝑒 𝑝0 → 𝑐 𝑝0 𝑝 1 ≅ 𝑐 𝑝 𝑝 1 nonequilibrium 𝑟 𝑁 =− 𝜕𝑛 𝜕𝑡 | 𝑅−𝐺 = 𝑐 𝑛 ( 𝑝 𝑇 𝑛− 𝑛 𝑇 𝑛 1 ) 𝑟 𝑁 ≡− 𝜕𝑛 𝜕𝑡 | 𝑅−𝐺 = 𝑐 𝑛 𝑛 𝑝 𝑇 − 𝑒 𝑛 𝑛 𝑇 𝑟 𝑃 =− 𝜕𝑝 𝜕𝑡 | 𝑅−𝐺 = 𝑐 𝑝 ( 𝑛 𝑇 𝑝− 𝑝 𝑇 𝑝 1 ) 𝑟 𝑃 ≡− 𝜕𝑝 𝜕𝑡 | 𝑅−𝐺 = 𝑐 𝑝 𝑝 𝑛 𝑇 − 𝑒 𝑝 𝑝 𝑇

Steady-State Relationship The rate of change on the filled traps under steady-state, 𝑑 𝑁 𝑇 𝑓 𝐸 𝑇 𝑑𝑡 = 𝑑 𝑛 𝑇 𝑑𝑡 =− 𝜕𝑛 𝜕𝑡 | 𝑅−𝐺 + 𝜕𝑝 𝜕𝑡 | 𝑅−𝐺 = 𝑟 𝑁 −𝑟 𝑃 =0 𝑟 𝑁 =− 𝜕𝑛 𝜕𝑡 | 𝑅−𝐺 = 𝑐 𝑛 𝑝 𝑇 𝑛− 𝑛 𝑇 𝑛 1 , 𝑟 𝑃 =− 𝜕𝑝 𝜕𝑡 | 𝑅−𝐺 = 𝑐 𝑝 ( 𝑛 𝑇 𝑝− 𝑝 𝑇 𝑝 1 ) and by using 𝑝 𝑇 = 𝑁 𝑇 − 𝑛 𝑇 𝑛 𝑇 = 𝑐 𝑛 𝑁 𝑇 𝑛+ 𝑐 𝑝 𝑁 𝑇 𝑝 1 𝑐 𝑛 𝑛+ 𝑛 1 + 𝑐 𝑝 𝑝+ 𝑝 1 Then, eliminating 𝑛 𝑇 and 𝑝 𝑇 and using 𝑛 1 𝑝 1 = 𝑛 𝑖 2 𝑅= 𝑟 𝑁 =𝑟 𝑃 = 𝑛𝑝− 𝑛 𝑖 2 1 𝑐 𝑝 𝑁 𝑇 𝑛+ 𝑛 1 + 1 𝑐 𝑛 𝑁 𝑇 𝑝+ 𝑝 1 : net steady-state recombination rate compare with band to band recombination rate at low level excitation = 𝑛𝑝− 𝑛 𝑖 2 𝜏 𝑝 𝑛+ 𝑛 1 + 𝜏 𝑛 𝑝+ 𝑝 1 − 𝜕𝑛 𝑟 , 𝑡 𝜕𝑡 =𝑟 𝑇 (𝑛𝑝− 𝑛 𝑖 2 ) →− 𝜕∆𝑛 𝜕𝑡 = ∆𝑛 𝜏 →− 𝜕∆𝑛 𝑡 𝜕𝑡 = ∆𝑛 𝑡 𝜏 𝑛 life time dimension of time where 𝜏 𝑝 ≡ 1 𝑐 𝑝 𝑁 𝑇 = 1 𝜎 𝑝 𝑣 𝑡ℎ 𝑁 𝑇 , 𝜏 𝑛 ≡ 1 𝑐 𝑛 𝑁 𝑇 = 1 𝜎 𝑝 𝑣 𝑡ℎ 𝑁 𝑇

Specialized Steady-State Relationships Assuming ∆ 𝑛=∆𝑝, 𝑛=𝑛 0 +∆𝑛, 𝑝=𝑝 0 +∆𝑝 𝑅= 𝑛𝑝− 𝑛 𝑖 2 𝜏 𝑝 𝑛+ 𝑛 1 + 𝜏 𝑛 𝑝+ 𝑝 1 = 𝑛 0 +∆𝑝 𝑝 0 +∆𝑝 − 𝑛 𝑖 2 𝜏 𝑝 𝑛 0 +∆𝑝+ 𝑛 1 + 𝜏 𝑛 𝑝 0 +∆𝑝+ 𝑝 1 = 𝑛 0 + 𝑝 0 +∆𝑝 ∆𝑝 𝜏 𝑝 𝑛 0 + 𝑛 1 + 𝜏 𝑛 𝑝 0 + 𝑝 1 + 𝜏 𝑛 + 𝜏 𝑝 ∆𝑝 = ∆𝑝 𝜏 ∴ 1 𝜏 = 𝑛 0 + 𝑝 0 +∆𝑝 𝜏 𝑝 𝑛 0 + 𝑛 1 + 𝜏 𝑛 𝑝 0 + 𝑝 1 + 𝜏 𝑛 + 𝜏 𝑝 ∆𝑝 (i) High Level Injection (∆ 𝑛=∆𝑝→∞ ) 𝑅≈ ∆𝑝 𝜏 𝑛 + 𝜏 𝑝 = ∆𝑝 𝜏 ∴𝜏= 𝜏 𝑛 + 𝜏 𝑝 (ii) Low Level Injection (∆ 𝑛= ∆𝑝≪𝑛 0 , 𝑝 0 , 𝑛 1 , 𝑝 1 ) 𝑅≈ 𝑛 0 + 𝑝 0 ∆𝑝 𝜏 𝑝 𝑛 0 + 𝑛 1 + 𝜏 𝑛 𝑝 0 + 𝑝 1 = ∆𝑝 𝜏 ∴𝜏= 𝜏 𝑝 𝑛 0 + 𝑛 1 + 𝜏 𝑛 𝑝 0 + 𝑝 1 𝑛 0 + 𝑝 0

For n-type material, 𝑛 0 ≫𝑝 0 𝑅≈ 𝑛 0 ∆𝑝 𝜏 𝑝 𝑛 0 + 𝑛 1 + 𝜏 𝑛 𝑝 1 = ∆𝑝 𝜏 ∴𝜏= 𝜏 𝑝 𝑛 0 + 𝑛 1 + 𝜏 𝑛 𝑝 1 𝑛 0 Practically the trap energy is usually near the center. 𝑛 1 ≪𝑛 0 , 𝑝 1 ≪𝑛 0 ∴𝜏≅ 𝜏 𝑝 →,𝑅= ∆𝑝 𝜏 𝑝 For p-type material, 𝑝 0 ≫𝑛 0 𝜏≅ 𝜏 𝑛 →,𝑅= ∆𝑛 𝜏 𝑛 𝜏 𝑛 ≠ 𝜏 𝑝 because 𝜎 𝑛 ≠ 𝜎 𝑝 For near intrinsic ( 𝐸 𝐹 ≈ 𝐸 𝑇 ), 𝑛 1 ≈𝑛 0 , 𝑝 1 ≈𝑝 0 𝜏≈ 𝜏 𝑝 𝑛 0 + 𝑛 1 + 𝜏 𝑛 𝑝 0 + 𝑝 1 𝑝 0 + 𝑛 0 = 𝜏 𝑝 + 𝜏 𝑛 : largest life time Physical meanings: For strong n-type case, traps are almost filled with electrons and electrons in conduction band are so numerus. One will recombine immediately with hole that is captured by a trap. ∴ Capture of holes limits the recombination process (hole life time) Vice versa for p-type case →electron life time

(iii) R−G Depletion Region R-G depletion region is formally defined to be a semiconductor volume where 𝑛≪𝑛 1 and 𝑝≪𝑝 1 →𝑛𝑝≪ 𝑛 1 𝑝 1 = 𝑛 𝑖 2 . The existence of an electrostatic depletion region requires 𝑛 and 𝑝 < 𝑁 𝐷 − 𝑁 𝐴 . A carrier deficit and associated R-G region are created inside the electric depletion region in a pn junction only under reverse bias condition. 𝑝𝑛≪ 𝑛 𝑖 2 𝑝𝑛= 𝑛 𝑖 2 at equilibrium 𝑝𝑛≫ 𝑛 𝑖 2 at forward bias 𝐸 𝑖 = 𝐸 𝑇 ′ 𝐸 𝐹𝑝 Welectrostatic WR-G < Welectrostatic (WR-G ≈ Welectrostatic at large reverse bias) WR-G 𝐸 𝐹𝑛 with 𝑛≪𝑛 1 and 𝑝≪𝑝 1 , 𝑅= 𝑛𝑝− 𝑛 𝑖 2 𝜏 𝑝 𝑛+ 𝑛 1 + 𝜏 𝑛 𝑝+ 𝑝 1 ≈ − 𝑛 𝑖 2 𝜏 𝑝 𝑛 1 + 𝜏 𝑛 𝑝 1 →𝐺=−𝑅=− 𝑛 𝑖 𝜏 𝑔 : net generation rate where 𝜏 𝑔 ≡ 𝜏 𝑝 𝑛 1 𝑛 𝑖 + 𝜏 𝑛 𝑝 1 𝑛 𝑖 :generation life time if 𝐸 𝑇 ′ = 𝐸 𝑖 . = 𝜏 𝑝 + 𝜏 𝑛

Physical View of Carrier Capture randomly distributed R-G centers concept of capture-cross section During capture process, the carrier velocity 𝑣≈ 𝑣 𝑡ℎ where 𝑣 𝑡ℎ = 3𝑘𝑇/ 𝑚 ∗ 1 2 𝑚 ∗ 𝑣 𝑡ℎ 2 = 3 2 𝑘𝑇 In a time 𝑡, an electron will travel a distance 𝑣 𝑡ℎ 𝑡 and will pass through a volume of material equal to 𝐴 𝑣 𝑡ℎ 𝑡. In this volume, there will be 𝑝 𝑇 empty R-G centers/cm3 or a total number of 𝑝 𝑇 𝐴 𝑣 𝑡ℎ 𝑡 empty R-G centers. cross-sectional area of the material normal to the electron path. The probability of the electron being captured in the volume can be determined by conceptually moving the centers to a single-plane in the middle of the volume.

If the area of the plane blocked by a single R-G center is 𝜎 𝑛 =𝜋 𝑟 2 , where 𝑟 is the radius of the R-G center, the total area blocked by empty R-G centers = 𝑝 𝑇 𝐴𝜎 𝑛 𝑣 𝑡ℎ 𝑡 The fraction of the area giving rise to capture = 𝑝 𝑇 𝐴𝜎 𝑛 𝑣 𝑡ℎ 𝑡/𝐴 = 𝑝 𝑇 𝜎 𝑛 𝑣 𝑡ℎ 𝑡 The probability of electron captured in the volume Capture rate (probability of capture/second) for a single electron = 𝑝 𝑇 𝜎 𝑛 𝑣 𝑡ℎ 𝑡/𝑡= 𝑝 𝑇 𝜎 𝑛 𝑣 𝑡ℎ The number of electrons/cm3 captured per second, given n electrons/cm3. = 𝑛𝑝 𝑇 𝜎 𝑛 𝑣 𝑡ℎ 𝜕𝑛 𝜕𝑡 | (𝑎) =−𝑛𝑝 𝑇 𝜎 𝑛 𝑣 𝑡ℎ ∴ 𝐶 𝑛 = 𝜎 𝑛 𝑣 𝑡ℎ Similarly, 𝐶 𝑝 = 𝜎 𝑝 𝑣 𝑡ℎ

SURFACE RECOMBINATION-GENERATION Interfacial traps or surface states are functionally equivalent to the R-G centers localized at the surface of a material. continuously distributed in energy throughout the semiconductor band gap

General Rate Relationships (Single Level) To stress the physical significance of surface recombination, we simplify the mathematics by considering the most efficient single level, which there are NTS surface states/cm2 at energy EIT (i.e., traps are located near midgap) Define surface recombination velocity, S, as an important parameter for surface recombination. Adding the subscript s to the corresponding bulk definitions, 𝑟 𝑁𝑠 Net electron recombination rate at surface centers (that is, the net change in the number of conduction band electrons/cm2-sec due to electron capture and emission at the single level surface centers). 𝑟 𝑃𝑠 Net hole recombination rate at the surface centers. 𝑛 𝑇𝑠 Filled surface centers/cm2 at energy EIT. 𝑝 𝑇𝑠 Empty surface centers/cm2 at energy EIT. 𝑁 𝑇𝑠 Total number of surface states/cm2; 𝑁 𝑇𝑆 = 𝑛 𝑇𝑆 + 𝑝 𝑇𝑆 𝑛 𝑠 Surface electron concentration( number per cm3); 𝑛 𝑆 = 𝑛| 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑝 𝑠 Surface hole concentration 𝑒 𝑛𝑠 , 𝑒 𝑝𝑠 Surface electron and hole emission coefficient (1/sec) 𝑐 𝑛𝑠 , 𝑐 𝑝𝑠 Surface electron and hole capture coefficient (cm3/sec)

𝑟 𝑁𝑠 = 𝑐 𝑛𝑠 𝑝 𝑇𝑠 𝑛 𝑠 − 𝑒 𝑛𝑠 𝑛 𝑇𝑠 𝑟 𝑃𝑠 = 𝑐 𝑝𝑠 𝑛 𝑇𝑠 𝑝 𝑠 − 𝑒 𝑝𝑠 𝑝 𝑇𝑠 Given the one-to-one correspondence between physical processes and parameteric quantities, 𝑟 𝑁 = 𝑐 𝑛 𝑝 𝑇 𝑛− 𝑒 𝑛 𝑛 𝑇 𝑟 𝑁𝑠 = 𝑐 𝑛𝑠 𝑝 𝑇𝑠 𝑛 𝑠 − 𝑒 𝑛𝑠 𝑛 𝑇𝑠 𝑟 𝑃𝑠 = 𝑐 𝑝𝑠 𝑛 𝑇𝑠 𝑝 𝑠 − 𝑒 𝑝𝑠 𝑝 𝑇𝑠 𝑟 𝑃 = 𝑐 𝑝 𝑛 𝑇 𝑝− 𝑒 𝑝 𝑝 𝑇 𝑟 𝑁 = 𝑐 𝑛 (𝑝 𝑇 𝑛− 𝑛 𝑇 𝑛 1 ) 𝑟 𝑁𝑠 = 𝑐 𝑛𝑠 (𝑝 𝑇𝑠 𝑛 𝑠 − 𝑛 𝑇𝑠 𝑛 1𝑠 ) 𝑟 𝑃𝑠 = 𝑐 𝑝𝑠 ( 𝑛 𝑇𝑠 𝑝 𝑠 − 𝑝 𝑇𝑠 𝑝 1𝑠 ) 𝑟 𝑃 = 𝑐 𝑝 ( 𝑛 𝑇 𝑝− 𝑝 𝑇 𝑝 1 ) where 𝑒 𝑛𝑠 = 𝑐 𝑛𝑠 𝑛 1𝑠 𝑒 𝑝𝑠 = 𝑐 𝑝𝑠 𝑝 1𝑠 Taking the surface-center degeneracy factor to be unity, 𝑛 1𝑠 = 𝑛 𝑖 𝑒 𝐸 𝐼𝑇 − 𝐸 𝑖 /𝑘𝑇 𝑛 1 = 𝑛 𝑖 𝑒 𝐸 𝑇 ′ − 𝐸 𝑖 /𝑘𝑇 if 𝑔 𝑇 =1, 𝑝 1𝑠 = 𝑛 𝑖 𝑒 𝐸 𝑖 − 𝐸 𝐼𝑇 /𝑘𝑇 𝐸 𝑇 ′ = 𝐸 𝑇

Steady-State Relationships (Single Level) Under steady-state condition, 𝑟 𝑁𝑠 = 𝑟 𝑃𝑠 ≡𝑅 𝑠 If the filled-state population of interfacial traps at EIT is assumed to change exclusively via thermal band-to-trap interactions, 𝑛 𝑇𝑠 = 𝑐 𝑛𝑠 𝑁 𝑇𝑠 𝑛 𝑠 + 𝑐 𝑝𝑠 𝑁 𝑇𝑠 𝑝 𝑠 𝑐 𝑛𝑠 𝑛 𝑠 + 𝑛 1𝑠 + 𝑐 𝑝𝑠 𝑝 𝑠 + 𝑝 1𝑠 𝑅 𝑠 = 𝑛 𝑠 𝑝 𝑠 − 𝑛 𝑖 2 1 𝑐 𝑝𝑠 𝑁 𝑇𝑠 𝑛 𝑠 + 𝑛 1𝑠 + 1 𝑐 𝑛𝑠 𝑁 𝑇𝑠 𝑝 𝑠 + 𝑝 1𝑠 : the recombination rate→[ /cm2∙sec] 𝑅= 𝑛𝑝− 𝑛 𝑖 2 1 𝑐 𝑝 𝑁 𝑇 𝑛+ 𝑛 1 + 1 𝑐 𝑛 𝑁 𝑇 𝑝+ 𝑝 1 1 𝑐 𝑝𝑠 𝑁 𝑇𝑠 , 1 𝑐 𝑛𝑠 𝑁 𝑇𝑠 : are not time constant. 𝑐 𝑝𝑠 𝑁 𝑇𝑠 ,𝑐 𝑛𝑠 𝑁 𝑇𝑠 : [cm3/sec]∙[#/cm3] = [cm/sec] : unit of velocity. 𝑐 𝑛𝑠 𝑁 𝑇𝑠 ≡ 𝑺 𝒏 : surface recombination velocity parameter for electron and hole. 𝑐 𝑝𝑠 𝑁 𝑇𝑠 ≡ 𝑺 𝒑 corresponds to the life time parameter 𝜏 𝑛 , 𝜏 𝑝

Surface Recombination Velocity p(x) 𝐽𝑝(0) ∆p(x) p0 𝑆(0) :surface recombination velocity at x = 0 𝑆(0) x x = 0 If we assume the electric field at the surface is negligible, the hole current at the surface is mainly due to diffusion. 𝐽 𝑝 0 =−𝑞 𝐷 𝑝 𝜕𝑝 0 𝜕𝑥 =−𝑞 𝐷 𝑝 𝜕∆𝑝 0 𝜕𝑥 =𝑞 𝐷 𝑝 𝜕∆ 𝑝 𝑠 𝜕𝑥 Hole flux = number of hole arriving per unit area per second, = 𝐽 𝑝 0 𝑞 From continuity equation at x = 0, 𝜕𝑝(0) 𝜕𝑡 = 𝐺 𝑠 − 𝑅 𝑠 − 1 𝑞 𝛻∙ 𝐽 𝑝 =0→ 𝐷 𝑝 𝜕∆ 𝑝 𝑠 𝜕𝑥 = 𝑅 𝑠

𝐷 𝑝 𝜕∆ 𝑝 𝑠 𝜕𝑥 = 𝑅 𝑠 = 𝑛 𝑠 𝑝 𝑠 − 𝑛 𝑖 2 1 𝑐 𝑝𝑠 𝑁 𝑇𝑠 𝑛 𝑠 + 𝑛 1𝑠 + 1 𝑐 𝑛𝑠 𝑁 𝑇𝑠 𝑝 𝑠 + 𝑝 1𝑠 = 𝑛 𝑠0 +𝑝 𝑠0 +∆ 𝑝 𝑠 𝑁 𝑇𝑠 1 𝑐 𝑝𝑠 𝑛 𝑠 + 𝑛 1𝑠 + 1 𝑐 𝑛𝑠 𝑝 𝑠 + 𝑝 1𝑠 ∆ 𝑝 𝑠 [cm2/sec] [ /cm2∙sec] 𝑅= ∆𝑝 𝜏 unit of [cm/sec] (compare with In bulk) referred to as the surface recombination velocity, S. [ /cm3∙sec] For low level excitation for n-type material, 𝑅 𝑠 = 𝑛 𝑠0 +𝑝 𝑠0 +∆ 𝑝 𝑠 𝑁 𝑇𝑠 1 𝑐 𝑝𝑠 𝑛 𝑠 + 𝑛 1𝑠 + 1 𝑐 𝑛𝑠 𝑝 𝑠 + 𝑝 1𝑠 ∆ 𝑝 𝑠 ≈ 𝑛 𝑠0 𝑁 𝑇𝑠 1 𝑐 𝑝𝑠 𝑛 𝑠 + 𝑛 1𝑠 ∆ 𝑝 𝑠 ≈ 𝑐 𝑝𝑠 𝑁 𝑇𝑠 ∆ 𝑝 𝑠 = 𝑺 𝒑 ∆ 𝑝 𝑠 For low level excitation for p-type material, 𝑅 𝑠 ≈𝑐 𝑛𝑠 𝑁 𝑇𝑠 ∆ 𝑛 𝑠 = 𝑺 𝒏 ∆ 𝑛 𝑠 S = ~ 102 cm/sec for Si, Ge 1 ~ 10 cm/sec for oxidized Si surface 104 ~ 105 cm/sec for GaAs To reduce surface recombination velocity in GaAs LED, use AlGaAs heterostructure barrier at surface.

From continuity equation at steady state, For example, G n-type p(x) x 𝐽𝑝(0) ∆p(x) p0 𝑆(0) x = 0 x From continuity equation at steady state, 𝑑 2 ∆𝑝(𝑥) 𝑑 𝑥 2 − ∆𝑝 𝑥 𝐷 𝑝 𝜏 𝑝 + 𝐺 𝐷 𝑝 =0 𝜕𝑝(0) 𝜕𝑡 =𝐺− ∆𝑝 𝜏 𝑝 − 1 𝑞 𝛻∙ 𝐽 𝑝 =0 𝐽 𝑝 𝑥 =−𝑞 𝐷 𝑝 𝑑∆𝑝 𝑑𝑥 Boundary conditions 𝐷 𝑝 𝜕∆ 𝑝 𝑠 𝜕𝑥 = 𝑺 𝒑 ∆ 𝑝 𝑠 𝐽 𝑝 (0)= −𝑞𝑺 𝒑 ∆ 𝑝 𝑠 ∆𝑝(∞) = 𝜏 𝑝 𝐺

The general solution of the differential equation 𝐴 𝑑 2 𝑓(𝑥) 𝑑 𝑥 2 − 𝑓 𝑥 𝐶 +𝐵=0 𝑓(𝑥)=𝐵𝐶+ 𝐾 1 exp − 𝑥 𝐴𝐶 + 𝐾 2 exp 𝑥 𝐴𝐶 ∆𝑝(𝑥) = 𝜏 𝑝 𝐺+ 𝐾 1 exp − 𝑥 𝐷 𝑝 𝜏 𝑝 + 𝐾 2 exp 𝑥 𝐷 𝑝 𝜏 𝑝 = 𝜏 𝑝 𝐺+ 𝐾 1 exp − 𝑥 𝐿 𝑝 + 𝐾 2 exp 𝑥 𝐿 𝑝 ∆𝑝(∞) = 𝜏 𝑝 𝐺+ 𝐾 1 (0)+ 𝐾 2 (∞) ∆𝑝(∞) = 𝜏 𝑝 𝐺 ∆𝑝(𝑥)= 𝜏 𝑝 𝐺+ 𝐾 1 exp − 𝑥 𝐿 𝑝 𝐷 𝑝 𝜕∆ 𝑝 𝑠 𝜕𝑥 = 𝐷 𝑝 𝜕∆𝑝(0) 𝜕𝑥 =𝐷 𝑝 𝐾 1 − 𝟏 𝑳 𝒑 =𝑺 𝒑 ∆ 𝑝 𝑠 = 𝑺 𝒑 ∆𝑝(0) 𝐷 𝑝 𝜕∆ 𝑝 𝑠 𝜕𝑥 = 𝑺 𝒑 ∆ 𝑝 𝑠 𝐾 1 = − 𝐷 𝑝 𝑳 𝒑 𝑺 𝒑 ∆𝑝(0) = − 𝐷 𝑝 𝑳 𝒑 𝑺 𝒑 (𝐺 𝜏 𝑝 + 𝐾 1 ) 𝐾 1 =− (𝐺 𝜏 𝑝 )( 𝑆 𝑝 𝜏 𝑝 / 𝐿 𝑝 ) 1+ 𝑆 𝑝 𝜏 𝑝 𝐿 𝑝

=𝜏 𝑝 𝐺− ( 𝜏 𝑝 𝐺)( 𝑆 𝑝 𝜏 𝑝 / 𝐿 𝑝 ) 1+ 𝑆 𝑝 𝜏 𝑝 𝐿 𝑝 𝑒 −𝑥/ 𝐿 𝑝 ∆𝑝(𝑥) =𝜏 𝑝 𝐺− ( 𝜏 𝑝 𝐺)( 𝑆 𝑝 𝜏 𝑝 / 𝐿 𝑝 ) 1+ 𝑆 𝑝 𝜏 𝑝 𝐿 𝑝 𝑒 −𝑥/ 𝐿 𝑝 ∆𝑝(𝑥) ∆𝑝(𝑥) 𝜏 𝑝 𝐺 𝑺 𝒑 =0 1 increasing 𝑺 𝒑 x 𝑺 𝒑 =∞ Ohmic contact: 𝑺 𝒑 =∞ ∆𝑝(0)=0 𝑝(0)= 𝑝 0

Steady-State Relationships (Multi-Level) Let DIT(E) be the density of interfacial traps at arbitrarily chosen energy E. [ /cm2∙eV] 𝐷 𝐼𝑇 𝐸 𝑑𝐸 : number of interfacial traps/cm2 with energies between E and E + dE. 𝑁 𝑇𝑠 𝐷 𝐼𝑇 𝐸 𝑑𝐸 single level multi-level ∴ incremental net recombination rate for energies between E and E + dE. 𝑑𝑅 𝑠 = 𝑛 𝑠 𝑝 𝑠 − 𝑛 𝑖 2 1 𝑐 𝑝𝑠 𝑛 𝑠 + 𝑛 1𝑠 + 1 𝑐 𝑛𝑠 𝑝 𝑠 + 𝑝 1𝑠 ∙𝐷 𝐼𝑇 𝐸 𝑑𝐸 𝑅 𝑠 = 𝑛 𝑠 𝑝 𝑠 − 𝑛 𝑖 2 1 𝑐 𝑝𝑠 𝑁 𝑇𝑠 𝑛 𝑠 + 𝑛 1𝑠 + 1 𝑐 𝑛𝑠 𝑁 𝑇𝑠 𝑝 𝑠 + 𝑝 1𝑠 Overall net recombination rate over all band gap energies, 𝑅 𝑠 = 𝐸 𝑉 𝐸 𝐶 𝑛 𝑠 𝑝 𝑠 − 𝑛 𝑖 2 1 𝑐 𝑝𝑠 𝑛 𝑠 + 𝑛 1𝑠 + 1 𝑐 𝑛𝑠 𝑝 𝑠 + 𝑝 1𝑠 ∙𝐷 𝐼𝑇 𝐸 𝑑𝐸

Specialized Steady-State Relationships Flat Band For low level excitation for n-type material and assuming flat band, 𝜀 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 ≈0, 𝑛 𝑠0 ≈ 𝑁 𝐷 , ∆𝑛 𝑠 = ∆𝑝 𝑠 ≪ 𝑛 𝑠0 ∴𝑛 𝑠 𝑝 𝑠 − 𝑛 𝑖 2 ≈ 𝑛 𝑠0 ∆𝑝 𝑠 𝑐 𝑛𝑠 , 𝑐 𝑝𝑠 : comparable 1 𝑐 𝑝𝑠 𝑛 𝑠 + 𝑛 1𝑠 + 1 𝑐 𝑛𝑠 𝑝 𝑠 + 𝑝 1𝑠 ≈ 1 𝑐 𝑝𝑠 𝑛 𝑠0 + 𝑛 1𝑠 + 𝑝 1𝑠 𝑐 𝑛𝑠 𝑅 𝑠 = 𝐸 𝑉 𝐸 𝐶 𝑐 𝑝𝑠 ∙𝐷 𝐼𝑇 𝐸 1+ 𝑛 1𝑠 𝑛 𝑠0 + 𝑐 𝑝𝑠 𝑐 𝑛𝑠 𝑝 1𝑠 𝑛 𝑠0 𝑑𝐸 ∆𝑝 𝑠 = 𝑺 𝒑 ∆ 𝑝 𝑠 constant; dimension of velocity, 𝑺 𝒑 . Similarly for p-type, 𝑅 𝑠 = 𝐸 𝑉 𝐸 𝐶 𝑐 𝑛𝑠 ∙𝐷 𝐼𝑇 𝐸 1+ 𝑝 1𝑠 𝑝 𝑠0 + 𝑐 𝑛𝑠 𝑐 𝑝𝑠 𝑛 1𝑠 𝑝 𝑠0 𝑑𝐸 ∆𝑝 𝑠 = 𝑺 𝒏 ∆ 𝑛 𝑠

𝐸 𝐹 ′ below 𝐸 𝑖 the band gap Usually, DIT and capture coefficients are approximately constant (energy-independent) over the middle portion of the band gap, 1+ 𝑛 1𝑠 𝑛 𝑠0 + 𝑐 𝑝𝑠 𝑐 𝑛𝑠 𝑝 1𝑠 𝑛 𝑠0 =1+ 𝑛 𝑖 𝑁 𝐷 𝑒 𝐸− 𝐸 𝑖 /𝑘𝑇 + 𝑐 𝑝𝑠 𝑐 𝑛𝑠 𝑒 𝐸 𝑖 −𝐸 /𝑘𝑇 ≈1 if 𝐸 𝐹 ′ ≤𝐸≤ 𝐸 𝐹 →∞ for 𝐸≤ 𝐸 𝐹 ′ , 𝐸≥ 𝐸 𝐹 𝐸 𝐹 ′ being the energy in the band gap where 𝑒 𝐸 𝑖 − 𝐸 𝐹 ′ /𝑘𝑇 ≈ 𝑁 𝐷 𝑛 𝑖 𝑛 𝑖 𝑁 𝐷 𝑐 𝑝𝑠 𝑐 𝑛𝑠 𝑒 𝐸 𝑖 −𝐸 /𝑘𝑇 =1 𝐸 𝐹 ′ below 𝐸 𝑖 the band gap At 𝐸=𝐸 𝐹 ′ or 𝐸= 𝐸 𝐹, 𝑒 𝐸− 𝐸 𝑖 /𝑘𝑇 + 𝑐 𝑝𝑠 𝑐 𝑛𝑠 𝑒 𝐸 𝑖 −𝐸 /𝑘𝑇 ≈ 𝑁 𝐷 𝑛 𝑖 ∴1+ 𝑛 1𝑠 𝑛 𝑠0 + 𝑐 𝑝𝑠 𝑐 𝑛𝑠 𝑝 1𝑠 𝑛 𝑠0 ≈2 𝑒 𝐸− 𝐸 𝑖 /𝑘𝑇 + 𝑐 𝑝𝑠 𝑐 𝑛𝑠 𝑒 𝐸 𝑖 −𝐸 /𝑘𝑇 ≪ 𝑁 𝐷 𝑛 𝑖 For 𝐸 𝐹 ′ ≤𝐸≤ 𝐸 𝐹 ∴1+ 𝑛 1𝑠 𝑛 𝑠0 + 𝑐 𝑝𝑠 𝑐 𝑛𝑠 𝑝 1𝑠 𝑛 𝑠0 ≈1

For 𝐸≤ 𝐸 𝐹 ′ , 𝐸≥ 𝐸 𝐹 𝑒 𝐸− 𝐸 𝑖 /𝑘𝑇 + 𝑐 𝑝𝑠 𝑐 𝑛𝑠 𝑒 𝐸 𝑖 −𝐸 /𝑘𝑇 ≫ 𝑁 𝐷 𝑛 𝑖 ∴1+ 𝑛 1𝑠 𝑛 𝑠0 + 𝑐 𝑝𝑠 𝑐 𝑛𝑠 𝑝 1𝑠 𝑛 𝑠0 ≈∞ The denominator of the 𝑺 𝒑 integrand is approximately unity in the midgap region where DIT and the capture coefficients are assumed to be approximately constant. Outside this range the denominator becomes large and the contribution to the overall integram is small. 𝑺 𝒑 ≈ 𝐸 𝐹 ′ 𝐸 𝐹 𝑐 𝑝𝑠 𝐷 𝐼𝑇 𝐸 𝑑𝐸≈ 𝑐 𝑝𝑠 𝐷 𝐼𝑇 𝐸 𝐹 − 𝐸 𝐹 ′ compare with 𝑺 𝒑 = 𝑐 𝑝𝑠 𝑁 𝑇𝑠 for single level.

Depleted Surface If non-equilibrium conditions exist such that both ns→0 and ps → 0 at the surface of a semiconductor, Rs by inspection reduces to 𝑅 𝑠 = 𝐸 𝑉 𝐸 𝐶 𝑛 𝑠 𝑝 𝑠 − 𝑛 𝑖 2 1 𝑐 𝑝𝑠 𝑛 𝑠 + 𝑛 1𝑠 + 1 𝑐 𝑛𝑠 𝑝 𝑠 + 𝑝 1𝑠 ∙𝐷 𝐼𝑇 𝐸 𝑑𝐸 ≈ 𝐸 𝑉 𝐸 𝐶 − 𝑛 𝑖 2 𝑛 1𝑠 𝑐 𝑝𝑠 + 𝑝 1𝑠 𝑐 𝑛𝑠 ∙𝐷 𝐼𝑇 𝐸 𝑑𝐸=− 𝐸 𝑉 𝐸 𝐶 𝑐 𝑛𝑠 𝑐 𝑝𝑠 𝐷 𝐼𝑇 𝐸 𝑑𝐸 𝑐 𝑛𝑠 𝑒 𝐸− 𝐸 𝑖 𝑘𝑇 + 𝑐 𝑝𝑠 𝑒 𝐸 𝑖 −𝐸 𝑘𝑇 𝑛 𝑖 = 𝐺 𝑠 dimension of velocity, called surface generation velocity, 𝑺 𝒈 . 𝑠 𝑔 ≈ −∞ ∞ 𝑐 𝑛𝑠 𝑐 𝑝𝑠 𝐷 𝐼𝑇 𝐸 𝑑𝐸 𝑐 𝑛𝑠 𝑒 𝐸− 𝐸 𝑖 𝑘𝑇 + 𝑐 𝑝𝑠 𝑒 𝐸 𝑖 −𝐸 𝑘𝑇 = 𝜋 2 𝑐 𝑛𝑠 𝑐 𝑝𝑠 𝑘𝑇 𝐷 𝐼𝑇 𝐺 𝑠 =− 𝑅 𝑠 = 𝑠 𝑔 𝑛 𝑖

Carrier Continuity Consider a uniformly doped n-type material and uniformly injected excess majority carrier density ∆𝑛 𝑡 where ∆𝑛 𝑡 ≪ 𝑛 0 . 𝛻∙ 𝜀 𝑟,𝑡 =− 𝑞 𝜖 ∆𝑛(𝑡) Poisson’s equation 𝐽 𝑛 (𝑟,𝑡)=𝑞 𝜇 𝑛 𝑛(𝑡) 𝜀 𝑟,𝑡 Current density equation 𝜕∆𝑛(𝑡) 𝜕𝑡 =− ∆𝑛 𝑟,𝑡 𝜏 𝑝 + 1 𝑞 𝛻∙ 𝐽 𝑛 (𝑟,𝑡) Continuity equation 𝜕∆𝑛(𝑡) 𝜕𝑡 =− ∆𝑛 𝑡 𝜏 𝑝 + 1 𝑞 𝛻∙ 𝐽 𝑛 (𝑟,𝑡) =− ∆𝑛 𝑟,𝑡 𝜏 𝑝 + 1 𝑞 𝛻∙𝑞 𝜇 𝑛 𝑛(𝑡) 𝜀 𝑟,𝑡 =− ∆𝑛 𝑡 𝜏 𝑝 + 𝜇 𝑛 𝑛(𝑡)𝛻∙ 𝜀 𝑟,𝑡 =− ∆𝑛 𝑡 𝜏 𝑝 − 𝑞 𝜖 𝜇 𝑛 ( 𝑛 0 +∆𝑛 𝑡 )∆𝑛(𝑡) =− 1 𝜏 𝑛 + 𝑞 𝜇 𝑛 𝑛 0 𝜖 ∆𝑛(𝑡) =− 1 𝜏 𝑛 + 𝜎 𝜖 ∆𝑛(𝑡)

If the injection process is turned off at t =0, the solution is ∆𝑛(𝑡)=∆𝑛 0 𝑒𝑥𝑝 − 1 𝜏 𝑛 + 𝜎 𝜖 𝑡 =∆𝑛 0 𝑒𝑥𝑝 − 1 𝜏 𝑛 + 1 𝜏 𝑑 𝑡 majority carrier decay where 𝜏 𝑑 = 𝜖 𝜎 = 𝜖 𝑞 𝜇 𝑛 𝑛 0 called dielectric relaxation time. 𝜏 𝑑 =𝜖𝜌=𝐶 𝑙 𝐴 ∙R 𝐴 𝑙 =𝑅𝐶 In most case 𝜏 𝑑 is much smaller than 𝜏 𝑛 and the excess majority carriers are removed from the sample at the dielectric relaxation time before they have time to recombine. Physically, the excess majority carriers produce an electric field which terminates on the contact or surface. This induced electric field than sweeps the excess majority carriers out of the sample at the dielectric relaxation time, reducing the field to zero.

Let us now continue our argument by assuming that it is possible to inject a uniform concentration of excess minority carriers ∆𝑝 𝑡 into the same uniform n-type sample. set 𝑝 0 ≪∆𝑝 𝑡 << 𝑛 0 𝛻∙ 𝜀 𝑟,𝑡 =− 𝑞 𝜖 ∆𝑝(𝑡) Poisson’s equation 𝐽 𝑝 (𝑟,𝑡)=𝑞 𝜇 𝑝 𝑛𝑝𝑡) 𝜀 𝑟,𝑡 Current density equation 𝜕∆𝑝(𝑡) 𝜕𝑡 =− ∆𝑝 𝑡 𝜏 𝑝 + 1 𝑞 𝛻∙ 𝐽 𝑝 (𝑟,𝑡) Continuity equation 𝜕∆𝑝(𝑡) 𝜕𝑡 =− ∆𝑛 𝑟,𝑡 𝜏 𝑝 + 1 𝑞 𝛻∙ 𝐽 𝑛 (𝑟,𝑡) =− 1 𝜏 𝑝 + 𝑞 𝜇 𝑝 ∆𝑝(𝑡) 𝜖 ∆𝑝(𝑡) the solution is ∆𝑝(𝑡)= ∆𝑝 0 1+ 𝜏 𝑝 𝜏 𝑑 exp 𝑡 𝜏 𝑝 − 𝜏 𝑝 𝜏 𝑑 where = 𝜖 𝑞 𝜇 𝑝 ∆𝑝(0) 𝜏 𝑑 dielectric relaxation time for the excess minority carriers

This is the concept of internal space charge neutrality. since ∆𝑝 0 is relatively small, 𝜏 𝑑 for the excess minority carriers is usually much larger than 𝜏 𝑝 . ∆𝑝(𝑡)≈∆𝑝 0 exp − 𝑡 𝜏 𝑝 The excess minority carriers decay by recombination. The injection of excess minority carriers produces an electric field that terminates on the surface or on the contact. This electric field causes excess majority carriers to move into the sample at their very short dielectric relaxation time. At this short time, ∆𝑛=∆𝑝 internally everywhere and the excess electrons and holes decay at the longer excess carrier life times. This is the concept of internal space charge neutrality. For example) injecting 1012/cm3 excess holes into an n-type GaAs with n0 = 1014/cm3. Assuming 𝜏 𝑛 = 𝜏 𝑝 ≈ 10 −9 sec, ni = 107/cm3, 𝜖 𝑟 =12.5, 𝜇 𝑛 =8× 10 3 cm2/V∙sec, 𝜇 𝑝 =5× 10 2 cm2 /V∙sec. 𝑝 0 = 𝑛𝑖 2 𝑛 0 1014/cm3 =1/ 𝑐𝑚 3 𝑝 0 <<∆𝑝(0)≪ 𝑛 0 majority carrier relaxation time, = 𝜖 0 𝜖 𝑟 𝑞 𝜇 𝑛 𝑛 0 ≈ 10 −11 𝑠𝑒𝑐 𝜏 𝑑𝑛 minority carrier relaxation time, = 𝜖 0 𝜖 𝑟 𝑞 𝜇 𝑝 ∆𝑝(0) ≈ 10 −8 𝑠𝑒𝑐 𝜏 𝑑∆𝑝 𝜏 𝑑𝑛 ≪ 𝜏 𝑛 = 𝜏 𝑝 ≪ 𝜏 𝑑∆𝑝 The material is space-charged neutral for times greater than 𝜏 𝑑𝑛 ≈ 10 −11 𝑠𝑒𝑐.