EE"232"Lightwave"Devices Lecture"15:"Polarization"Dependence

Transcription

1 EE"3"Lightwave"Devices Lecture"15:"Polarization"Dependence Reading:"Chuang,"Sec."4.1C4.,"9.5 (There"is"also"a"good"discussion"in"Coldren," Appendix"8"and"1) Instructor:"Ming"C."Wu University"of"California,"Berkeley Electrical"Engineering"and"Computer"Sciences"Dept. EE3$Lecture$15-1

2 Detailed(Band(Structure Block wavefunction: ψ nk (r ) = e ik At and edges, k = r u nk (r) Conduction Band: Remnant of s atomic orital u c (r) = S Valence Band: remnants of the p oritals: u v (r) = X, Y, Z S is symmetric in x, y, and z X is anti-symmetric in x, and symm in y, z S X = S Y = S Z = S P x X = S P y Y = S P z Z S, X, Y, Z EE3$Lecture$15-

3 u c Wavefunctions for.electrons.and.various. Holes.(heavy,.light.holes) Define z-direction along k (i.e., k = k z ) CB wavefunctions: VB wavefunctions: = is u c = is " u hh = 3, 3 = 1 (X + iy ) uhh = 3, 3 = 1 (X iy ) u lh = 3, 1 = 1 6 (X + iy ) Z ulh = 3, 1 = 1 6 (X iy ) +Z u so = 1, 1 = 1 3 (X + iy ) +Z EE3$Lecture$15-3 uso = 1, 1 = 1 3 (X iy ) Z

4 QuantumWellMatrixElement TE TM All Polarizations e = x or y e = z (xte+tm) C6HH Transition 3 1 cos 4 3 sin M ( + ) M ( ) 3M C6LH Transition " 5 3 cos % $ M 4 4 ( " 1 3 cos $ + % M 3M SumRule: HH+LH M M 6M EE3$Lecture$15-4

5 Angular(Factor Angular factor : cos θ = k z k = k z k t + k z In quantum wells, k z is quantized: E en = k z cos θ = k t k z m e m + k z e m e = k t can e estimated from k t E en m e + E en k t m r At and edge, k t = cos θ =1 m e = ω - E g E en E hm EE3$Lecture$15-5

6 QuantumWellMatrixElementfor Light5MatterInteractionNearBendEdges TE TM All Polarizations e = x or y e = z (xte+tm) C6HH Transition 3 M 3M C6LH Transition 1 M M 3M SumRule: HH+LH M M 6M EE3$Lecture$15-6

7 Transition)Strength)versus)Transverse) Wavevector EE3$Lecture$15-7

8 Appendix: " Method-and-Derivation-of-Matrix- Elements EE3$Lecture$15-8

9 kp Method Hψ nk (r ) = [ +V ]ψ m nk (r) = E n (k)ψ nk (r) ψ nk (r ) = e ik r u nk (r) " " u nk (r + R) = u nk (r) periodic function, R is lattice vector $ P + " $ k P +V (r))u % m m nk (r) = E n (k ) k )u ( % m nk (r) ( " Near k =, k P can e treated as a perturation $ H + " $ k P)u % m nk (r) = E n (k ) k )u ( % m nk (r) ( Kanes P parameter: P = i m S P z Z EE3$Lecture$15-9

10 SecondOrderPerturation Conduction Band: Second-order perturation: " E c (k ) = k + k Pcn m m n c E c () E n () " " Pcn = u c P u n, n = hh, lh, so Use Kanes P parameter: P = i m S P z Z E c (k ) = k " + k P % m $ 3 E g 3 E g + Δ E c (k ) = k + k P " 3E g + Δ % m 3 $ E g (E g + Δ) = k m e EE3$Lecture$15-1

11 Eigenvalues Conduction Band: E c (k ) = k + k P m 3 " $ 3E g + Δ % E g (E g + Δ) = k m e Valence Band: HH: E hh (k ) = k (incorrect in this approx) m LH: E lh (k ) = k k P = k m 3E g m lh SO: E so (k ) = Δ + k m k P 3 E g + Δ ( ) = Δ + k m so EE3$Lecture$15-11

12 k z Wavefunctions in-general-coordinates When k is not along z direction (i.e., k k z ) k = k sinθ cosφ x + k sinθ sinφ y + k cosθ z The new wavefunctions are now linear k y cominations of new orital functions X, Y, Z. They can e transormed ack to the orital functions in the fixed coordinate through the following k x Coordination Transformation: X $ cosθ cosφ cosθ sinφ sinθ Y = sinθ cosφ Z " % sinθ cosφ sinθ sinφ cosθ " $ % " X Y Z $ % EE3$Lecture$15-1

13 " X Y Z Example:)Heavy)Hole)Wavefunction $ = % " cosθ cosφ cosθ sinφ sinθ sinθ cosφ sinθ cosφ sinθ sinφ cosθ $ % " X Y Z $ % u hh = 3, 3 = 1 (X + iy ) = 1 (X + iy ) = 1 (cosθ cosφ isinφ)x + (cosθ sinφ + icosφ)y (sinθ)z uhh = 3, 3 = 1 (X iy ) = 1 (X + iy ) = 1 (cosθ cosφ + isinφ)x + (cosθ sinφ icosφ)y (sinθ)z EE3$Lecture$15-13

14 C"HH$Matrix$Element Optical Matrix Element: H a = ea e " P a = ea e " en pcv I m m hm " There are 4 possile terms for C-HH transistion: M " u c P u hh " C HH = pc HH = 1 (cosθ cosφ isinφ)p x + (cosθ sinφ + icosφ)p x y y (sinθ)p z z $% ( = P x (cosθ cosφ isinφ)x + (cosθ sinφ + icosφ) y (sinθ)z $% ( " P uhh = = u c uc " P u hh = uc " P uhh = P x (cosθ cosφ + isinφ)x + (cosθ sinφ icosφ)y (sinθ)z $% ( EE3$Lecture$15-14

15 C"HH$Matrix$Element To find polarziation Dependence Integrate over all possile θ and φ : x " π π 1 pc HH = sinθ dθ dφ 4π 1 % P x ( cos θ cos φ + sin φ) + P ) x ( cos θ cos φ + sin φ) ( + x " π π 1 % P ) pc HH = sinθ dθ dφ x ( 4π cos θ cos φ + sin φ) ( + = P x 3 M Similarly, " y p C HH = M z " pc HH = M C-HH transistion in Bulk Semiconductor is Polarization Independent EE3$Lecture$15-15

16 C"LH%Matrix%Element Similarly: x " p = M C LH " y p C LH z " pc LH = M = M C-LH transistion in Bulk Semiconductor is Polarization Independent Bulk Semiconductor is Polarization Independent EE3$Lecture$15-16

17 ValuesofMatrixElement To find polarziation Dependence Integrate over all possile θ and φ : $ M = P x 3 = 1 m 3 P = 1 m 1 m $ E e 3 g (E g + Δ) " m % m e (E g + " 3 Δ) % 1 m $ e 1 " % m M m E g (E g + Δ) 6m e (E g + 3 Δ) M m 6 " m m e E g m e $ E g (E g + Δ) (E g + 3 Δ) % m 6 E P GaAs InP InAs E p 5.7)eV.7)eV.)eV EE3$Lecture$15-17

18 k z C"HH$Matrix$Element$in$Quantum$Well Integrate over φ only k x k y Quantum$well$is$ polarization$dependent$ x " pc HH " y p C HH z " pc HH = 1 π π dφ " $ % ( cos θ cos φ + sin φ) P x = 3 ( 4 1+ cos θ)m = 1 π π dφ " $ % ( cos θ sin φ + cos φ) P x = 3 ( 4 1+ cos θ)m = 1 π π dφ " $ % ( sin θ) P x = 3 ( sin θ)m EE3$Lecture$15-18

19 C"LH%Matrix%Element%in%Quantum%Well k x k z k y Quantum%well%is% polarization%dependent% Integrate over φ only : x " pc LH " y p C HH z " pc HH 5 = 4 3 $ 4 cos θ " % 5 = 4 3 $ 4 cos θ " % 1 = + 3 $ cos θ " % M M M EE3$Lecture$15-19