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Chem 140a Lecture Notes #8: Class #10 Recombination-Generation Statistics (R-G) (Shockley-Read-Hall statistics) <Review> Photocurrent: due to minority carrier current generated from photoinduced holes, p h n p n ND n p ns nb p If band bending is steep enough, all p cross the interface and J ph is a constant J J 0 [exp( qV / AkT ) 1] J ph Δp at Voc, J 0 So, J 0 [exp( qV / AkT ) 1] J ph if V 3kT , we can ignore "1" q J 0 [exp( qV / AkT )] J ph Voc AkT J ph ln q J0 Name of this game is to minimize J 0 to get large Voc <Review> SC/Metal interface, qb J 0 A T exp( ) kT * 2 : we want to have large b to minimize J 0 SC/Solution interface, J 0 qket [ A]ns 0 J 0, so ln J 0,m MIS, insulator induces a less than unity probability for the e- to corss the SC interface Limit of b , Max b can only be as big as band gap energy, E g Five Mechanisms Majority carrier process 1 Ecb 2 2 Tunneling 5 Minority carrier process 3 ET 1 Thermionic Emission 3 Bulk Recombination 4 Evb 4 Depletion Region Recombination 5 Surface Recombination Under light or a bias, np>ni2. So there is a driving force to return to the equilibrium. Extra holes will recombine with extra electrons Recombination If somehow we were sucking carriers out of a region of the SC, np<ni2 and carriers would be generated Generation Shockley-Read-Hall statistics: Good model for all of these minority carrier processes of recombination in the SC Bulk Recombination For recombination (loss of carriers), e1 kn 2 kn’ Ecb dn dn dn dt R dt 1 dt 2 ET kp 3 kp’ 4 p+ 1 dp dt R dp dp dt 3 dt 4 fT fraction of filled trap states Evb 1 fT fraction of empty trap states dn kn nb (1 fT ) NT dt 1 2 dn kn ' fT NT dt 2 [cm-3] 3 dp k p pb fT NT dt 3 [cm-3sec-1] [cm-3] (unitless) [cm3/sec] : has an analogy with Vth where is cross-section and Vth is velocity dp k p ' (1 fT ) NT 4 dt 4 Bulk Recombination e1 kn 2 kn’ Ecb ET kp 3 kp’ 4 p+ Evb rn = dn dn dn kn nb (1 fT ) N T kn ' fT N T dt R dt 1 dt 2 rp = dp dp dp k p pb fT N T k p ' (1 fT ) N T dt R dt 3 dt 4 Equilibrium conditions: Principle of detailed balance All fundamental processes and their inverse processes must self balance rn rp 0 electrons, kn nb ,0 (1 fT ) N T kn ' fT N T kn ' 1 fT kn nb ,0 fT Bulk Recombination For electrons, kn nb ,0 (1 fT ) N T kn ' fT N T kn ' 1 fT kn nb ,0 fT What is the ratio, 1 fT ? fT Probability that a trap is filled is given by Fermi-Dirac Statistics. fT 1 exp[( EF ET ) / kT ] 1 1 fT 1 exp[( EF ET ) / kT ] 1 1 fT exp[( EF ET ) / kT ] 1 exp[( EF ET ) / kT ] 1 1 1 1 fT exp[( EF ET ) / kT ] 1 exp[( EF ET ) / kT ] 1 1 1 exp[( EF ET ) / kT ] 1 fT 1 1 fT exp[( EF ET ) / kT ] Bulk Recombination 1 fT exp[( E F ET ) / kT ] fT kn ' kn ( 1 fT )nb ,0 kn nb ,0 exp[( EF ET ) / kT ]= kn N C exp[ ( EC ET ) / kT ] fT ( nb ,0 =NC exp[ ( EC E F ) / kT ]) kn ' doesn't depend on the Fermi level and only depends on the trap energy level For a given trap energy, NC exp[( EC ET ) / kT ] is a constant. Let n1 =NC exp[( EC ET ) / kT ] Then, kn ' kn n1 Similarly, k p ' k p p1 where p1 =N V exp[( ET EV ) / kT ] By plugging in kn ' and k p ' , rn = dn kn nb (1 fT ) N T kn n1 fT N T dt R rp = dp k p pb fT N T k p p1 (1 fT ) N T dt R Bulk Recombination Steady state relationship: Major assumption in device problems and analysis At steady state, rn rp ( Different from the equilibrium) U bulk dn dp recombination rate dt R dt R Plug in our values for rn & rp kn nb (1 fT ) N T kn n1 fT N T k p pb fT N T k p p1 (1 fT ) N T N T s cancel and solve for fT kn nb (1 fT ) kn n1 fT k p pb fT k p p1 (1 fT ) kn nb kn nb fT kn n1 fT k p pb fT k p p1 k p p1 fT fT kn nb k p p1 kn nb kn n1 k p pb k p p1 1 fT k p pb kn n1 kn nb kn n1 k p pb k p p1 Now that we have fT and 1-fT we can use them to calculate U bulk . Bulk Recombination U bulk dn dp dt R dt R = kn nb (1 fT ) N T kn n1 fT N T U bulk N T U bulk N T kn nb ( k p pb kn n1 ) kn n1 ( k n nb k p p1 ) kn ( nb n1 ) k p ( pb p1 ) kn k p ( nb pb n1 p1 ) kn ( nb n1 ) k p ( pb p1 ) n1 p1 N C exp[( EC ET ) / kT ] N V exp[ ( ET EV ) / kT ] N C N V exp[( EC EV ) / kT ] ni 2 So the recombination rate in bulk, U bulk N T kn k p ( nb pb ni 2 ) kn ( nb n1 ) k p ( pb p1 )