EE 232 Lightwave DevicesLecture 3: Density of States, Quantum Wells dWiand WiresInstructor: Ming C. WuUniversity of California, BerkeleyElectrical Engineering and Computer Sciences DeptElectrical Engineering and Computer Sciences Dept.EE232 Lecture 3-1©2008. University of CaliforniaReview of Quantum MechanicsSchrodinger Equation() ()Hrti rt∂Ψ=ΨGG=2(,)(,)(,) Hamiltonian = Kinetic + Potential Energy Hrti rttPHVrtΨ=Ψ∂=+=G2(,)gy2( , ) WavefunctionmrtΨGGGG G2*( , ) ( , ) ( , ) Probability of finding particle at Momrt rt rt rPiΨ=Ψ⋅Ψ=− ∇GGG GJG= entum operator**( , ) ( , ) Average Momentum() () AveragePositionP i rt rtdrrrtrrtdr=− Ψ ∇Ψ=Ψ Ψ∫∫JG G G G=GGGGGEE232 Lecture 3-2©2008. University of California(,)(,) Average Positionrrtrrtdr=Ψ Ψ∫Electron Plane Wave()ik r i trt eω⋅−ΨGGG22(,) LHS: (,) (,) (,) (,)2rt ekHrt rt Vrt rtmΨ=Ψ= Ψ+ ΨGGGG=2RHS: () ()mPkirt rtω=∂ΨΨ=GG==22RHS: (,)(,)irt rttωΨ=Ψ∂⇒==22(,)2kVrtmω+=G==EE232 Lecture 3-3©2008. University of CaliforniaExample: Infinite Potential Wellit22(,) () Solve Eigenvalue initzt zeEdωφω−Ψ== ==0V =∞22() ()() ()22For0 () () 0dzVzz EzmdzdmEzL z zφφφφφ−+=<< + ==2E22For 0, () () 0sin( )( ) ()zL z zdzkzzkφφφ<< + =⎧=⎨⎩=1Ecos()B.C. ( 0) ( ) 02kzzzLnφφπ⎨⎩== ==⎛⎞0x = xL=Time Independent Potential222() sinnnzzLLnπφπ⎛⎞=⎜⎟⎝⎠⎛⎞=0 for 0() for 0 or zLVzzzL<<⎧=⎨∞< >⎩EE232 Lecture 3-4©2008. University of California2nnEmLπ⎛⎞=⎜⎟⎝⎠=Typical Examples*In GaAs 0 067mm01In GaAs, 0.067For a 10-nm-wide potential well ( 10 )56 meVemmLnmE===1214 224 meVEE==EE232 Lecture 3-5©2008. University of CaliforniaComplete Wavefunction for Infinite Potential Well(,) '(, )() Electron confined in but free initrt xy zezxyωφφ−Ψ=GEElectron confined in , but free in , Plane wave in x and y1'( )xyik x ikyzxyφ+⇒2n='(,) A: area (normalization const)xyyxyeAφ=12n221(,) sinxyik x ik ynrt e zLLAπ+⎛⎞Ψ=⎜⎟⎝⎠⎛⎞G2⎡⎤yk1n =2222nxynEkkmLπ⎛⎞=++⎜⎝⎠=2 Energy quantized only in directionk⎡⎤⎢⎥⎟⎢⎥⎣⎦xkEE232 Lecture 3-6©2008. University of CaliforniaEnergy quantized only in directionzk2-d Density of StatesykV∞Ekkk+Δ0V=∞2n =xkkE2E1n=0x =xL=1Eyk1n=xkEE232 Lecture 3-7©2008. University of California2-d Density of StatesCidthl tbdfit(1)Consider the lowest band first (n=1):Number of electron states between and per unit volumekkk+Δper unit volume22 2()222kzkdk kkdk dkVLπρπππ=⋅ =222*()xyLLEk kπ⎡⎤⎛⎞=+⎢⎥⎜⎟=*22()221() ()edkEk kmLdk kEdEkdEρρ+⎢⎥⎜⎟⎝⎠⎢⎥⎣⎦⎛⎞==⎜⎟dE22() ()2dkzdkddE Lmρρπ⎜⎟⎝⎠=**edkmEE232 Lecture 3-8©2008. University of California22()edzmELρπ==2-d DOS for Multiple Energy Levels⎧12*0()0dEE Emρ⎧<< =⎪⎪⎪2()dEρ1222*()2edzmEEE ELmρπ⎪<< =⎪⎪⎨⎪=2()dρ2322*2()3()edzemEEE ELmEEE Eρπρ⎪<<=⎪⎪⎪<< =⎪=1E2E3E3422()dzEEE ELρπ<< =⎪⎩=*22In general() ( )ednmEHEELρπ∞=−∑=EE232 Lecture 3-9©2008. University of California1nzLπ=∑=Step Function2-d Electron/Hole ConcentrationElectron and hole concentrations:∞Example: 10 nm wide GaAs quantum well,2() ()CVnedEEnfE EdEρ∞=∫110-nm-wide GaAs quantum well quasi-Fermi energy is 100 meV above E,2() ()VphdpfE EdEρ−∞=∫2-d electron concentrationm_e 0.067 m0⋅:= m_e 6.104 1032−× kg=()121,21 2At T = 0K, and for ()nedEEEnFE EEEρ<<=−⋅ <<Lz 10nm:= Lz 1 108−× m=ρ2dm_eπhbar2⋅Lz⋅:= ρ2d 1.747 1044×s2kgm5⋅=()*12enzmnFELπ=−=πh_barLzkgmn100meVρ2d⋅:= n2.7951018×1cm3⋅=1EE232 Lecture 3-10©2008. University of Californian_s n Lz⋅:= n_s 2.795 1012×1cm2⋅=1-d Density of Statesz2222,*22()2mn z zemnEk kmL Lkππ⎛⎞⎛⎞⎛⎞=++⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠===yx,**() 2222zmn z z z zeezkdE k k dk dkmmdkndk∞∞=⋅===∑∑∫∫==,,0*22zmn mnxyzndkVLLLmππ−∞∞==⎛⎞⎜⎟⎝⎠∑∑∫∫∑∫2,0*2211emnxy zemdELL kmdEπ∞==∑∫∑∫=2,0*12211()mnxymx nyeDdELLEE EmELLπρ=−−=∑∫==∑EE232 Lecture 3-11©2008. University of California12()DxyLLEρπ=,mnmx
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