EE 232 Lightwave DevicesLecture 9: Intersubband Absorption iQ t Wllin Quantum WellsInstructor: Ming C. WuUniversity of California, BerkeleyElectrical Engineering and Computer Sciences DeptElectrical Engineering and Computer Sciences Dept.EE232 Lecture 9-1©2008. University of CaliforniaIntersubband Transition in Quantum Wells• Transition between quantized energy levels in a quantum well by absorption or emission of a photon0V =∞EbEor emission of a photon• Absorption–Infraredphotodetectors1eE2eEbEaEInfrared photodetectors– Thremal imager: blackbody radiation of human body ~ 10 µm–3 to 5 and 8 to 10 µm wavelengthFor a 10-nm QW0z =zL= or xykk1e–3 to 5 and 8 to 10 µm wavelength bands particularly interesting• Emission2121For a 10-nm QW in GaAseeeeEEE=−– Gain media for quantum cascaded lasers (QCL)– Long wavelength emission 121224 meV 56 meV168 meVeee=−=EE232 Lecture 9-2©2008. University of Californiacoincides with molecular vibration spectra1.247.40.168mλμ==Optical Matrix Element for Intersubband Transition1() () ()tikacearur zAρψφ⋅==GJGGGQuantum Well Wavefunction'2'() () ()tikbcebrur zAHEρψφ⋅==GJGGGJGJGEnvelopWavefunction'ttbababaik ikHEberaeeρρμμ⋅⋅=−⋅=GJG GJGJG G*21() () ( ) ( )ttcceeurur dAAzer zdrρρρφφ≈⋅∫∫GG JGGGSlowly VaryingEnvelop Approx'2121,1ttkkzδμ=⋅∫GG AtomicWavefunctionpppEE232 Lecture 9-3©2008. University of CaliforniaAbsorption Coefficient for Intersubband Transition()202() ( )tbaba abkrgEE effnc Vωαω ω μεπ=⋅ −−⋅ −∑GJG ==()02212( ) (Lorentzian Lineshape)2tkrgEEπΓΔ=Δ+Γ222121 only when (TM polarization) , independent of baetba eeezEEE kμμ⋅= =−=JG G2212102() ( )eeargE fnc Vωαωπωμε=−==()tbkf−∑G22121120() ( ) ( )eergE N Nncωαεπωωμ=− −==EE232 Lecture 9-4©2008. University of CaliforniaAbsorption Coefficient for Intersubband Transition2212112012() ( ) ( )(1) eergE N NncEFEωαω ωμεπ=− −<<==122221211()0( ) ( ) eeNgE Nωαω ωμπ≈=−==121102is proportional to doping concentration(2) erncEFε<12***11 1222()ln(1 ) ( )BFEkTeB eB ezzBzmkT mkT mFENe FELLkTLπππ−−=+≈ =−===*2emN ≈22*2()zFELmππω−=EE232 Lecture 9-5©2008. University of California222121 120() ( )is a constant, independent of doping concentrationeeeeerzmgE Enc Lπωαωωμεπ=−===Intersubband Transition in P-Doped QW (1): Fermi-Level Below Second Subband0V =∞E1eE2eE21eeE (Fermi Level)F0z =zL=kyk1e22121101() ( )N-type doping concentrationeergE NncNωαω ωμεπ=−===xk1hEE2hE1 Ntype doping concentrationNProportional to doping concentration2hE1hEEE232 Lecture 9-6©2008. University of CaliforniaIntersubband Transition in N-Doped QW (2): Fermi-Level Above Second Subband0V =∞E1eE2eE21eeE22() ( )egEπωαω ωμ=−== (Fermi Level)F0z =zL=kyk1e1210*212() ( ) ereeegEncmELαω ωμεπ=⋅===xk1hEE2hEzLπ=Independent of doping concentration2hE1hEEE232 Lecture 9-7©2008. University of CaliforniaIntersubband Transition in P-Doped QW (1): Fermi-Level Above Second Subband0V =∞E1eE2eE21eeE0z =zL=kyk1e212211() ( )hhgE Pωαω ωμπ=−==xk1hEE2hE(il)221101() ( ) P-type doping concentrationhrgE PncPαω ωμε===2hE1hE (Fermi Level)FEE232 Lecture 9-8©2008. University of CaliforniaIntersubband Transition in P-Doped QW (2): Fermi-Level Below Second Subband0V =∞E1eE2eE21eeEIndependent of doping 0z =zL=kyk1e21221() ( )hhgEωαωπωμ=−==ppgconcentrationxk1hEE2hE2210*122() ( ) hrhhhgncmELαωωμεπ⋅===2hE1hE (Fermi Level)FzLπ=EE232 Lecture 9-9©2008. University of CaliforniaIntersubband Dipole Moment12() sinzzLLπφ⎛⎞=⎜⎟⎝⎠222() sinzzzzLLzzLLπφ⎝⎠⎛⎞=⎜⎟⎝⎠21 2 10() ()22zzzLLezzzdzeμφφππ⎝⎠=⋅⋅⎛⎞ ⎛ ⎞∫∫022sin sin16zLzz zezz zdzLL LeLππ⎛⎞ ⎛ ⎞=⋅⋅⎜⎟ ⎜ ⎟⎝⎠ ⎝ ⎠=−∫29zeLπEE232 Lecture 9-10©2008. University of CaliforniaExample221056 meVzLnmEπ=⎛⎞==⎜⎟=1*222*56 meV22224 meVeezeEmLEπ==⎜⎟⎝⎠⎛⎞==⎜⎟=2*2118 32168 meVN=10 cmeezeemLE−⎜⎟⎝⎠=1,maxN = 10 cmFirst, determine if the second subband is occupied. Find , the electron concentration in the first subbaN nd *2183 1831,max 12when the Fermi level is at the bottom of the second subband:4.7 10 cm > N = 10 cmeeemNEL−−==×=EE232 Lecture 9-11©2008. University of California1,max 122So the second subband is not occupied. N 0ezLπ==Absorption Spectra2212016() ( )9eezrgE eL Nncωαω ωεππ⎛⎞=−⎜⎟⎝⎠==021max42Peak absorption116 cm210rzeL Nωαπ−⎛⎞=≈⎜⎟⎛⎞⎝⎠×max0292zrncεππ⎜⎟Γ⎛⎞⎝⎠⎟⎜⎝⎠42104×α0icm1−01104×EE232 Lecture 9-12©2008. University of California5 6 7 8 9 10