EE 232 Lightwave DevicesLecture 17: Polarization DependenceReading: Chuang Sec 4 142 95Reading: Chuang, Sec. 4.1-4.2, 9.5(There is also a good discussion in Coldren, Appendix 8 and 10)Instructor: Ming C. WuUniversity of California, BerkeleyElectrical Engineering and Computer Sciences DeptElectrical Engineering and Computer Sciences Dept.EE232 Lecture 17-1©2008. University of CaliforniaDetailed Band StructureBlock wavefunction: ( ) ( )At band edges, 0ik rnk nkreurkψ⋅==0Conduction Band: Remnant of atomic orbital()csur S=0Valence Band: remnants of the orbitals:() , ,itii dvpur XYZS=is symmetric in x, y, and z SX is anti-symmetric in x, and symm in y, z0SX SY SZ===00xyzSX SY SZSP X SPY SPZ=====≠EE232 Lecture 17-2©2008. University of California, , , SXYZk⋅P Method220() [ ] () () ()2nk nk n nkHr V rEk rmψψψ=− ∇+ =222() ()( ) ( ) periodic function, is lattice vectorik rnk nknk nkreururR ur Rψ⋅=+=⎡⎤⎡⎤ 22200 0() () () ()22Near 0nk n nkPkkPVrur Ek urmm mkk⎡⎤⎡⎤+⋅+ = −⎢⎥⎢⎥⎣⎦⎣⎦= can be treated as a perturbationPNear 0, kk=22000 can be treated as a perturbation() () ()2nk n nkPkHkPurEk urmm⋅⎡⎤⎡⎤+⋅ = −⎢⎥⎢⎥⎣⎦⎣⎦ 000Kane's P parameter: ziPSPZm⎣⎦⎣⎦−=EE232 Lecture 17-3©2008. University of CaliforniaEigenvectorsDfi di ti l (i )kkk CB wavefunctions:VB wavefunctions:Define z-direction along (i.e., )kkkz=ccuiSuiS=↑=↓33 1,()22233 1hhuXiY−== +↑−↓33 1,()22231 1()2hhlhuXiYuXiYZ==−↓−==+↓−↑,()222631 1,()2226lhlhuXiYZuXiYZ+↓ ↑−==−↑+↓22611 1,()223souXiYZ== +↓+↑EE232 Lecture 17-4©2008. University of California11 1,()223souXiYZ−==−↑−↓Second-Order Perturbation2Conduction Band:Second-order perturbation:222 2200()2(0)(0)cncnccnkPkEkmm E EPP hhlh≠⋅=+−∑, , , Use Kane's P parameter: cncnPuPunhh lhsoi==−0222221 1 1()ziPSPZmkEk kP=⎛⎞=+ +⎜⎟⎜⎟0()233()cggEk kPmEEEk=+ +⎜⎟⎜⎟+Δ⎝⎠=22 2 2 2232gEkkP k⎛⎞+Δ+=⎜⎟⎜⎟EE232 Lecture 17-5©2008. University of California()cEk=*023()2gg emEE m+=⎜⎟⎜⎟+Δ⎝⎠Eigenvalues22 2 2 22Conduction Band:32()gEkkP kEk⎛⎞+Δ=+ =⎜⎟⎜⎟*0()23()2Vl B dcgg eEkmEE m=+ =⎜⎟⎜⎟+Δ⎝⎠220Valence Band:HH: ( ) (incorrect in this approx)2hhkEkm=022 2 2 22*02LH: ( )23 2lhglhkkP kEkmE m=− =−2202SO: ( )2sokkEkm=−Δ+ −()22 22*23sogPkmE=−+ΔEE232 Lecture 17-6©2008. University of CaliforniaWavefunctions in General CoordinatesWh i t l di ti (i )kkk sin cos sin sin coskk xk yk zθφ θφ θ=++ When is not along z direction (i.e., )kkkz≠zkThe new wavefunctions are now linear combinations of new orbital functionszθcombinations of new orbital functions ' , ' , ' . They can be transormed back to the orbital functions in the fixed XYZykθφcoordinate through the followingCoordination Transformation:⎡⎤⎡ ⎤⎡⎤xkφ' cos cos cos sin sin'sincos 0'sin cos sin sin cosXXYYZZθφθφ θθφθφ θφ θ−⎡⎤⎡ ⎤⎡⎤⎢⎥⎢ ⎥⎢⎥=−⎢⎥⎢ ⎥⎢⎥⎢⎥⎢ ⎥⎢⎥⎣⎦⎣ ⎦⎣⎦xEE232 Lecture 17-7©2008. University of Californiasin cos sin sin cosZZθφ θφ θ⎢⎥⎢ ⎥⎢⎥⎣⎦⎣ ⎦⎣⎦Example: Heavy Hole Wavefunction'coscoscossin sin'sincos 0'sin cos sin sin cosXXYYZZθφ θφ θθφθφ θφ θ−⎡⎤⎡ ⎤⎡⎤⎢⎥⎢ ⎥⎢⎥=−⎢⎥⎢ ⎥⎢⎥⎢⎥⎢ ⎥⎢⎥⎣⎦⎣ ⎦⎣⎦'sin cos sin sin cos33 1 1'('')' ('')'ZZXiY XiYθφ θφ θ⎢⎥⎢ ⎥⎢⎥⎣⎦⎣ ⎦⎣⎦−−+↑ + ↑,'('')' ('')'22221(cos cos sin ) (cos sin cos ) (sin ) '2hhuXiY XiYiX iY Zθφ φ θφ φ θ==+↑=+↑−=−++−↑'233 1'('Xi−− 1') ' ( ' ') 'YXiY−↑+↑,'('222hhuXi==−') ' ( ' ') '21(cos cos sin ) (cos sin cos ) (sin ) '2YXiYiX iY Zθφ φ θφ φ θ↑=+↑=++−−↑EE232 Lecture 17-8©2008. University of California2C-HH Matrix Element'00Optical Matrix Element:22enba hmcveA eAHebPaepI=− ⋅ =− ⋅ ⋅ 00''22There are 4 possible terms for C-HH transistion: 1CHHCHHmmMp−−=−⎡⎤ ''1(cos cos sin ) (cos sin cos ) (sin )2chh x y zuPu i Px i Py Pzθφφ θφ φ θ⎡⎤=−++−⎣⎦= (cos cos sin)(cos sin cos)(sin)xPix iyzθφ φθφφθ−⎡⎤−+ + −⎣⎦ ''()()()20 ' ' 0hhcyuPuφφ φ φ⎣⎦=↑↓=∵ ''''0 (cos cos sin ) (cos sin cos ) (sin )chhxuPuPuPu i x i y zθφ φ θφ φ θ=⎡⎤=++⎣⎦ EE232 Lecture 17-9©2008. University of California(cos cos sin ) (cos sin cos ) (sin )2chhuPu i x i y zθφ φ θφ φ θ⎡⎤=++−−⎣⎦C-HH Matrix Element 22To find polarziation Dependence Integrate over all possible and :1sinxp d dππθφθθ φ⇒∫∫()()002222 2 22 2sin41 cos cos sin cos cos sinCHHxxxp d dPPθθ φπθφ φθφφ−⋅= ⋅⋅⎧⎫+++⎨⎬∫∫()() 2222 21sin4CHHxp d dππφφ φφθθ φπ−⎨⎬⎩⎭⋅= ⋅∫∫()2222 2 2cos cos sin23xxbPPMθφ φ⎧⎫⋅+=≡⎨⎬⎩⎭004π 2223Similarly,yp M⎩⎭⋅=22bCHHbCHHyp Mzp M−−=⋅= EE232 Lecture 17-10©2008. University of CaliforniaC-HH transistion in Bulk Semiconductor is Polarization Independent!C-LH Matrix Element 22Similarly:xp M= 22bCLHbCLHxp Myp M−−⋅=⋅=22C-LH transistion in Bulk Semiconductor is Polarization Independent!bCLHzp M−⋅= Bulk Semiconductor is Polarization Independent!⇒EE232 Lecture 17-11©2008. University of CaliforniaValues of Matrix Element222 2*To find polarziation Dependence Integrate over all possible and :()11EEPθφ⇒⎛⎞⎜⎟+Δ⎛⎞22 2220022*0()111233 32( )3ggxebegEEPmmmMPmmE⎜⎟+Δ⎛⎞== = −⎜⎟⎜⎟⎝⎠⎜⎟+Δ⎝⎠*0211()emmEE E⎛⎞−≈⎜⎟⎝⎠Δ202**()26( )3gg gbeegmEE EMmmE+Δ≈∝+ΔEpGaAs 25.7 eV2006bemmMm≈0*()26()ggPEEmEE⎛⎞⎜⎟+Δ→⎜⎟⎜⎟+ΔInP 20.7 eVInAs22 2eVEE232 Lecture 17-12©2008. University of Californiae()3gE⎜⎟+Δ⎝⎠InAs22.2 eVC-HH Matrix Element in Quantum Well222Integrate over only1Pπφ⎧⎫zk ()()222 20221cos cos sin2231cosxCHHPxp dMφθφ φπθ−⎧⎫⋅= ⋅ +⎨⎬⎩⎭+∫ykθφ() ()22222 21cos41cos sin cos22bxCHHMPyp dπθφθφ φπ−=+⎧⎫⋅= ⋅
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