Relation between Interband Dipole and Momentum Matrix Elements

Transcription

1 Relatin between Interband Diple and Mmentum Matrix Elements Baijie Gu, Nai Kwng, Rlf Binder Cllege f Optical Sciences and Departmen f Physics The University f Arizna Reference: B. Gu, N.H. Kwng, R. Binder, Phys. Rev. B. 87, (213).

2 Infinite Vlume Nn-Vanishing Bundary Cnditins Finite Vlume Peridic Bundary Cnditins c ' r v ( ') u i u c ' v c r v u i u c v Blunt 1962 Incrrectly prven in Haug 1972 u ˆ u im u i u c p v c,v c v c pˆ v im c,v uc i uv Adams 1952, Haug 1972 Haug 1972 u pˆ u im u r u B ( ) c v c,v c v cv Yafet 1957, Peeters etal. 1993, Freman 2 c ' pˆ v c ' r v imc ',v c ' pˆ v c ' r v C im c ',v c ',v Fllws frm abve, see Gu et al. 213 This wr

3 im pˆ H, r (classically: p=mv ) 2 2 ( r) H V L 2m In crystals with peridic bundary cnditins: im c ' pˆ v c ' H r -r H v c ' rh v c ' r v v c ' H r v c c ' r v Prbably widely nwn, but nt nt equal widely applied in cntext f peridic bundary cnditins Cmparisn: infinite system, nn-vanishing bundary cnditins c ' pˆ v c ' r v imc ',v Valid in distributin sense Frmal diple matrix element is really that f the -gradient peratr -gradient may nt exist in case f degeneracy, see Za 1985, Freman 2 Diagnal element =' nt defined Prf using limiting prcedure with spatially limited wave-pacets in Gu et al. 213

4 Bul GaAs: simple Chen-Bergstresser pseudptential apprach Ga As Pseudptential: B A s-lie c-band wave fct p z -lie v-band wave fct

5 (units f p cv ) (units f p cv ) V L (r) (ev) cs β r cv r cv m cv m cv C cv C cv A t B r cv β -ip cv Crrectin term large, diple matrix element small Glbal maximum magnitude f diple matrix element: Distance between maximum f s-lie and p z -lie wave functin: Scaled mmentum matrix element p cv /m ω cv : Lattice cnstant:.36 Å.71 Å 5.81 Å 5.65 Å

6 p 21 / i m e 21, r 21, C 21 / i m e 21 p 21, i m e 21 r 21, C 21 (units f p 21 ) p 21 / i m e 21, r 21, C 21 / i m e 21 p 21, i m e 21 r 21, C 21 (units f p 21 ) Intersubband (THz) transitins in superlattices C r p Thin barrier (4Å): Depending n z, r r C an be as large as p Diple matrix element can change sign Crrectin factr can change sign There exist zer-crssing f C p z C r z Thic barrier (12Å): Large regin f very small C If bundary in barrier, C negligible (this is ften used as intuitive chice fr unit ) 2 1 unit z z +a = wave functins, nn-zer barrier thicness, left bundary at z

7 Nan-structures (e.g. quantum wells, dts): vanishing bundary cnditins Envelpe wave functin Cnvenient "zne center apprximatin": u ( r) u ( r), u ( r) u ( r) c, c, v, v, quantum dt 3 drc '( r) r v ( r) qu. dt Cmpare Burt, 1993 Alternative prf in Gu et al, 213 Generalized zne center apprximatin fr nan-structures 3 (nan) d ir z e u 3, (2 ) BZ ( ) ( ) ( r) zne center 3 nan c v c ' v all space c ' r v u i u d r ( r) ( r) Cell-envelpe factrizatin inves -gradient peratr, nt diple peratr p-r relatin fr nan-structures in generalized zne-center apprximatin: n (nan,zca) (nan,zca) im zne center c ' pˆ v ' ( c, v, ) c ' r v crrectin nan term

8 We nticed that, in the literature, p-r relatin fr peridic bundary cnditins is usually nt crrect (r defined ambiguusly) We generalized Yafet's crrectin term t the case f peridic bund. cnditins Fr bul GaAs, crrectin term fund t be large fr any lcatin f unit Fr intersubband (THz) transitins in superlattices, crrectin term fund t be small if barrier is wide and bundary is inside barrier We prvided alternative prf t Blunt's findings. This leads t vanishing crrectin term in infinite crystals but p-r relatin is in distributin sense and diple peratr is replaced by -gradient peratr Fr nan-structures, we develped alternative prf t Burt's finding that diple matrix element essentially vanishes within zne-center apprximatin We shwed that, fr nan-structures, p-r relatin admits -envelpe factrizatin, but diple matrix element replaced by -gradient matrix element B. Gu, N.H. Kwng, R. Binder, Relatin between the interband and mmentum diple matrix elements in semicnductrs, Phys. Rev. B. 87, (213). Y. Yafet, The g value in cnductin electrn spin resnance, Phys. Rev. 16, 679 (1957) A. Haug, Theretical Slid State Physics (Pergamn, Oxfrd, UK, 1972). F.M. Peeters, A. Matulis, M. Helm, T. Frmherz, and W. Hilber, Oscillatr strength and sum rule fr interssubband transitins in a superlattice, Phys. Rev. B 48, 128 (1993). M.G. Burt, The evaluatin f the matrix element fr interband ptical transitins in quantum wells using envelpe functins, J. Phys.: Cndens. Matter 5, 491 (1993). B.A. Freman, Thery f the effective Hamiltnian fr degenerate bands in an electric field, J. Phys. 12, R435 (2). J. Za, Tw classes f cmpsite energy bands in slids, Phys. Rev. Lett. 54, 175 (1985)