k BZ . Optical absorption due to interband transition therefore involves mostly vertical transitions :
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1 Interband transitins (1 Optial Absrptin Spetra a Diret Transitins We had already seen that phtn BZ. Optial absrptin due t interband transitin therefre inles mstly ertial transitins : C V Use first-rder time-dependent perturbatin thery t alulate transitin rate. H 0 = atmi Hamiltnian, H' = eletrmagneti wae Fermi s Glden Rule alid fr ntinuum f initial / final states and perturbing f the annial frm it it H( t h( e e time independent emissin absrptin Transitin prbability per unit time: P final state (eletrn in ndutin band f h' i ( f i initial state (eletrn in alene band absrptin emissin letri diple apprximatin: neglet B field i( p r t H ' ( ( er e phtn wae etr p eletri field plarizatin etr r i p eletri diple mment e 1beause p BZ as befre (in real spae : - field des nt ary appreiably er unit ell h' e ( r
2 intensity: I( n ( refratie index energy area time ( Measurable quantity: absrptin effiient ( energy absrbed per unit time and lume inming energy per unit time and area P absrptin prbability per unit time and lume Intensity prfile f inming wae in slid: Ix ( Ie (disuss later hw t arry ut aurate measurement P/ I ( T alulate, need t integrate er ertial interband transitins in the entire Brilluin zne: P ( ( 3 d I 4 BZ density f states in -spae d ( ' ( ( ( ( 3 h ( 4 n e d ( ( ( ( 3 M CV n 4 with M ( r ( nte: CV in tight-binding limit i R M e ( r r ( r R CV R with atmi rbitals an use atmi seletin rules, e.g. l 1 : transitins frm sderied bands t pderied bands lead t strng absrptin, s d transitins lead t wea absrptin et. assume MCV nst. er BZ e MCV JCV ( n d JCV ( ( ( ( jint density f states 3 4 b df 1 aluate integral using gx ( fx ( dx gx ( with fx ( 0 in [ ab, ] x x a x dx 1 ds in 3-D: JCV ( 3 4 ( ( ( ( ds : element f surfae defined by ( ( x
3 expet singularities when 0 ritial pints at high symmetry pints in BZ (3 band minimum band maximum r mre generally Parallel bands: speially signifiant in eletrnially 1-D r -D systems (wea dispersin in sme -diretins arund ritial pints, expand ( ( x y z x y z mx my m z where eletrn / hle effetie mass and 1 m m m x ex hx Tw different types f ritial pint all psitie, r all negatie (band maximum / minimum tw psitie, ne negatie, r ie ersa (saddle pint example: CV near band minimum with x y z J m m m m define qi i i x, y, z m ( spherial bands spherial rdinates 3 1 ( m JCV ( 4 q dq( q if n absrptin 3 ( m 1 q if with q 3 q q 3 ( m 3 fr J
4 ( usp singularities (saddle pints prblem set (4 0 1 ~ 0 ~ 1 Obseratin (disuss measurement tehniques later: GaAs, InSb square-rt nset, as predited wea temperature dependene Si, Ge muh weaer nset strng temperature dependene Reasn: indiret gap C V 0 0 Onset is due t indiret (nt ertial interband transitins intermediate state phtn phnn C q phnn 1 V 1
5 Tw transitins ( use send rder time-dependent perturbatin thery h '' 1 1 h ' V1 P ( ( ( 1 ( q (5 Fr absrptin f phtn f energy frm M field desribed by h, and absrptin / emissin f phnn f energy (q Hamiltnian H desribes eletrn-phnn interatin simple mdel fr H'': nulei displaed by u ptential seen by eletrns H V( r R u V( r R (nsistent with Brn-Oppenheimer apprximatin expand fr small u H u V( r R R expand u r in nrmal rdinates ( ( i q R q t q R q t u a( q e a ( q e M( q phnn phnn phnn annihilatin plarizatin reatin peratr etr peratr prefatr hsen suh that M n 1 a( q n n( q analgus t 1-D harmni sillatr in elementary quantum mehanis n 1 a ( q n n( q 1 with phnn upatin number gien by Bse-instein distributin 1 nq q/ BT e 1 Fr transitin 1 by absrptin f (thermally exited phnn: iq R h 1 n q e rv r R 1 Mq frm nulear part f wae funtin frm eletrni part f wae funtin
6 As befre fr diret transitin, we alulate the absrptin effiient expliitly near the nset f absrptin, under the assumptin that h 1 1 h 1 CT ( ( ( 1 1 (6 Is apprximately nstant arund the extrema f and bands. Nte that C(T is strngly temperature dependent: C( T n( q, T The absrptin effiient fr a press in whih a phnn is absrbed t get frm 1 t is then A e d d CT ( ( ( 1 ( q n 4 8 s that 0 BZ spin degeneray in initial state A BZ similarly, fr phnn emissin spin is nsered in phnn absrptin T d the integral, we assume a wealy dispersing phnn ( ( q nst. and spherial bands: ( ( with 3/ 3/ e m m 0 3 0n ( ( C ( T with C( T n( q 1 The ttal absrptin effiient at m m q q 1 1 m m 0 dq dq q q (4 q dq q and anmaly at larger with square -rt energy dependene due t diret transitins. The frmer tw anmalies are strngly temperature dependent, the latter nly wealy temperature dependent. 0 ( we hae / q 0 d with O 0 /16 3 ( 0 3/ 3/ A e m ( ( 0 ( m C T 3 0n 8 ( sin s sin as befre A ( ( exhibits tw anmalies with quadrati energy dependenes, fllwed by an
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