Modeling of Transport and Gain in Quantum Cascade Lasers

Modeling of Transport and Gain in Quantum Cascade Lasers Andreas Wacker in collaboration with: M.P. Pereira Jr., NMRC, Cork p.1

Introduction p.2

The Challenge: Intersubband Lasing tunneling Kazarinov and Suris, 1971 hν p.3

The Challenge: Intersubband Lasing tunneling Kazarinov and Suris, 1971 hν Advantage: Photon energy tunable by well width: hν 100 mev±?? molecular spectroscopy p.3

The Challenge: Intersubband Lasing tunneling Kazarinov and Suris, 1971 hν Advantage: Photon energy tunable by well width: hν 100 mev±?? molecular spectroscopy Problems: Fast non-radiative transitions (Phonon scattering) Escape to continuum p.3

The Quantum Cascade Laser First realized by Faist, Capasso et al: 1994 minigap Energy hν miniband active region injector Growth direction active region p.4

The Quantum Cascade Laser First realized by Faist, Capasso et al: 1994 Energy hν active region minigap miniband injector Growth direction active region Key design features: 3 levels in active region Injector region efficient electron transfer into level 3 prevents escape into continuum Cascading: 30 repetitions of (active region + injector) p.4

The Quantum Cascade Laser First realized by Faist, Capasso et al: 1994 Energy hν active region minigap miniband injector Growth direction Current state of the art: active region Key design features: 3 levels in active region Injector region efficient electron transfer into level 3 prevents escape into continuum Cascading: 30 repetitions of (active region + injector) Wavelength: 3.5µm < λ < 24µm and 60µm Energy: 350 mev> hν > 50 mev and 16 mev Operation temperature: > 300 K for IR, > 77 K for THz p.4

Transport Models Semiclassical rate equation Donovan, Harrison, and Kelsall: JAP 2001 p.5

Transport Models Semiclassical rate equation MC-simulation for f n (k) electric field (kv/cm) 75 50 25 0 Donovan, Harrison, and Kelsall: JAP 2001 without carrier-carrier with carrier-carrier T = 77 K 0 1 2 3 4 5 6 7 8 current density (ka/cm 2 ) Iotti and Rossi: PRL 2001 (c) 0.3 0.2 0.1 0.0 applied bias per period (V) Voltage (V) 8 6 4 2 Quantum transport Theory Structure A 77 K 0 0 2 4 6 8 Current (A) Lee and Wacker: PRB 2002 Good agreement with experiment (Sirtori at al.: PRB 1998) p.5

The Quantum Transport Model p.6

Why Quantum Transport? Interplay between Tunneling transitions Scattering processes Optical transitions = Operation of QCL Broadening of tunneling resonances and optical transitions by scattering Electronic distribution modified by transitions modified scattering rates Combine quantum evolution, scattering, and carrier distribution Quantum Transport Theory p.7

Nonequilibrium Green Functions Start with (single particle) basis Ψ α (z)e i(k xx+k y y) Ĥ = Ĥo + Ĥscatt with Ĥ o = α,β,k H o αβ(k)a αk a βk Aim: f αk (t) = a αk (t)a αk(t), J = e V dẑ dt = ie V [Ĥ, ẑ] =... p.8

Nonequilibrium Green Functions Start with (single particle) basis Ψ α (z)e i(k xx+k y y) Ĥ = Ĥo + Ĥscatt with Ĥ o = α,β,k H o αβ(k)a αk a βk Aim: f αk (t) = a αk (t)a αk(t), J = e dẑ = ie V dt V Concept: Green functions [Ĥ, ẑ] =... G < α,β (k; t, t ) = i a βk (t )a αk (t) steady state = de 2π G< α,β (k, E)e ie(t t )/ Dyson equation ( E H o (k) Σ ret (k, E) ) G ret (k, E) = 1 Keldysh relation G < (k, E) = G ret (k, E)Σ < (k, E)G adv (k, E) p.8

Nonequilibrium Green Functions Dyson equation ( E H o (k) Σ ret (k, E) ) G ret (k, E) = 1 Keldysh relation G < (k, E) = G ret (k, E)Σ < (k, E)G adv (k, E) Self-energies (in self-consistent Born approx.) Σ ret,imp (k, E) = k Σ <,phon (k, E) = p V(k, k )G ret (k, E)V(k, k) imp M( p)n B G < (k + p, E ω p )M l ( p) + M( p) [n B + 1] G < (E + ω p )M l ( p) Temperature enters only via phonon occupations n B ( ω p ) p.8

Calculation Details Scattering by optical phonons, interface roughness, charged impurities, artificial acoustic phonon Electron-Electron interaction in mean field Momentum-independent scattering matrix elements (Calculated for typical momentum transfer) Self-energies Σ α,β (E) do not depend on k p.9

Numerical Implementation Initial guess Σ ret (E), Σ < (E) Evaluate G ret (E, k), G < (E, k) Evaluate Σ ret/new (E), Σ </new (E) and n new = P R k de G < (E, k) Σnew (E) Σ old (E) < ɛ and n new n 0 < ɛ Evaluate current density New guess for Σ ret (E) and Σ < (E) trying different strategies: linear mixing, Broyden method, Pulay method,... p.10

Transport and Electron Distribution p.11

Current-voltage characteristic Page et al. APL01 a)=77 K, c)=233 K p.12

Current-voltage characteristic Page et al. APL01 20 15 Theory Structure B (b) Voltage (V) 10 5 77 K 233 K a)=77 K, c)=233 K 0 0 5 10 15 20 25 30 J (ka/cm 2 ) Quantitative agreement p.12

THz-QCL operating at 77 K Data for sample of Kumar et al. APL 84, 2494 (2004) 12 10 Bias (V) 8 6 4 2 T=78 K Theory T=30 K Theory T=78 K Exp T=5 K Exp 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Current (A) See APL 86, 041108 (2005) p.13

Spatial and energetical resolution density localized in wells p.14

Spatial and energetical resolution density localized in wells see cond-mat/0507410 Current through the structure p.14

Calculation of Gain p.15

Linear response We know G ret (E), G < (E) in the nonequilibrium state p.16

Linear response We know G ret (E), G < (E) in the nonequilibrium state Perturbation δu(t) = eδf (ω)z α,β e iωt p.16

Linear response We know G ret (E), G < (E) in the nonequilibrium state Perturbation δu(t) = eδf (ω)z α,β e iωt Result within linear response δg ret (ω, E) = G ret (E + ω) [ δu(ω) + δσ ret (ω, E) ] Gret (E) δg < (ω, E) = G ret (E + ω)δu(ω) G < (E) + G < (E + ω)δu(ω) G adv (E) + G ret (E + ω)δσ ret (ω, E) G < (E) +... p.16

Result for Gain 30 T=78 K Full theory efd=65 mev g (1/cm) 20 10 Gain in the THz region Exp: 3.2 THz 13.2 mev 0 0.01 0.015 0.02 hω (ev) p.17

Width of Gain Peak Compare with simple model g(ω) ij z ij 2 (n i n j ) Γ i + Γ j (E i E j ω) 2 + (Γ i + Γ j ) 2 /4 p.18

Width of Gain Peak Compare with simple model g(ω) ij z ij 2 (n i n j ) g (1/cm) 30 20 10 T=78 K Full theory Simple approach efd=65 mev Γ i + Γ j (E i E j ω) 2 + (Γ i + Γ j ) 2 /4 0 0.01 0.015 0.02 hω (ev) p.18

Width of Gain Peak Compare with simple model g(ω) ij z ij 2 (n i n j ) g (1/cm) 30 20 10 T=78 K Full theory Simple approach efd=65 mev Γ i + Γ j (E i E j ω) 2 + (Γ i + Γ j ) 2 /4 0 0.01 0.015 0.02 hω (ev) Significantly reduced width See APL 86, 041108, (2005) p.18

Origin of main effect δg < (ω, E) = G ret (E+ ω)δu(ω) G < (E)+ G < (E+ ω)δu(ω) G adv (E) gives simple model p.19

Origin of main effect δg < (ω, E) = G ret (E+ ω)δu(ω) G < (E)+ G < (E+ ω)δu(ω) G adv (E) gives simple model Main correction + G ret (E + ω)δσ < (ω, E) G ret (E), in particular δσ < 12k (E) k V imp 11 (k k )V imp 22 (k k) }{{} = 2 γ 1212 /(Am) δg < 12k (E) p.19

Origin of main effect δg < (ω, E) = G ret (E+ ω)δu(ω) G < (E)+ G < (E+ ω)δu(ω) G adv (E) gives simple model Main correction + G ret (E + ω)δσ < (ω, E) G ret (E), in particular provides δσ < 12k (E) k V imp 11 (k k )V imp 22 (k k) }{{} = 2 γ 1212 /(Am) δg < 12k (E) g(ω) z 12 2 (n 2 n 1 ) Γ 1 + Γ 2 2γ 1212 (E 2 E 1 ω) 2 + (Γ 1 + Γ 2 2γ 1212 ) 2 /4 similar to Ando, J. Phys. Soc. Jpn. 54, 2671 (1985) p.19

When is this of importance? We have (for impurity scattering): γ 1212 V imp 11 (k k )V imp 22 (k k) imp Γ 1 γ 1111 V imp 11 (k k )V imp 11 (k k) imp γ 1212 comparable to Γ 1, Γ 2 if scattering environment is correlated for states 1 and 2 p.20

When is this of importance? We have (for impurity scattering): γ 1212 V imp 11 (k k )V imp 22 (k k) imp Γ 1 γ 1111 V imp 11 (k k )V imp 11 (k k) imp 150 efd=65mev 100 γ 1212 comparable to Γ 1, Γ 2 if scattering environment is correlated for states 1 and 2 E (mev) 50 1 2 0-20 0 20 40 p.20

Sample with less correlation Standard QCL from Sirtori et al, APL 73, 3486 (1998) Energy [mev] 400 200 0 1 efd=260 mev 2 g (1/cm) 200 150 100 50 0 200 mev Full 200 mev Simple 260 mev Full 260 mev Simple T=77 K -20-10 0 10 20 z [nm] -50 0.1 0.11 0.12 0.13 0.14 0.15 0.16 hω (ev) p.21

Coulomb Shifts p.22

Coulomb-corrections in Hartree Fock Coulomb corrections in Hartree-Fock: Depolarization Shift (Hartree, dynamic) Subband shifts (Fock, static) Exchange correction (Fock, dynamic, like excitonic coupling) Absorption (1/micron) 0.1 0.3 V/period Full Hartree Fock Free carriers 0.05 0 0.6 0.4 0.2 0 0.3 0.12 V/period Full hartree Fock 0.2 Free carriers 0.1 25 75 125 Energy (mev) 0.2 V/period Full Hartree Fock Free carriers QCL from Sirtori et al, APL 73, 3486 (1998) p.23

Resolved Gain Spectrum Absorption (1/cm) 10 20 30 40 0.3 V (6,7) Transition Full Hartree Fock Sub band shift only Depolarisation only Exchange only Free carriers Absorption (1/cm) 60 120 180 240 0.3 V (6,8) Transition Full Hartree Fock Sub band shift only Depolarisation only Exchange only Free carriers 110 130 150 Energy (mev) p.24

Resolved Absorption Spectrum 0.08 0.06 0.04 0.02 (7,9) Transition 0.3 V/period Full Hartree Fock Sub band shift only Depolarisation only Exchange only Free carriers Absorption (1/micron) 0.60 0.40 0.20 (7,8) Transition 0.2 V/period Full Hartree Fock Sub band shift only Depolarisation only Exchange only Free carriers 0.24 0.18 0.12 0.06 (6,7) Transition 0.12 V/period Full Hartree Fock Sub band shift only Depolarisation only Exchange only Free carriers 10 20 30 40 Energy (mev) p.25

Different sample Absorption (1/cm) 1200 1000 800 600 400 200 209 V/period with many-body effects 210 V/period with many-body effects 209 V/period free carriers 210 V/period free carriers 0 0 20 40 60 80 100 120 140 Energy (mev) QCL from Gmachl et al., APL 72, 1430 (1998) p.26

Conclusion Development of a transport theory for Quantum cascade lasers using Nonequilibrium Green functions p.27

Conclusion Development of a transport theory for Quantum cascade lasers using Nonequilibrium Green functions Quantitative results for current-voltage characteristics and gain profile p.27

Conclusion Development of a transport theory for Quantum cascade lasers using Nonequilibrium Green functions Quantitative results for current-voltage characteristics and gain profile Reduction of spectral width due to correlations p.27

Conclusion Development of a transport theory for Quantum cascade lasers using Nonequilibrium Green functions Quantitative results for microscopic understanding of current-voltage the device characteristics and gain profile Reduction of spectral width due to correlations Many-Particle effects: Blue shift in absorption PRB 66, 245314 (2002); PRB 69, 205310 (2004) APL 86, 41108 (2005); cond-mat/0507410 p.27