Columnar quantum dots (QD) in polarization insensitive SOA and non-radiative Auger processes in QD: a theoretical study
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1 olumnar quantum dots (QD) in polarization insensitive SOA and non-radiative Auger processes in QD: a theoretical study J. Even, L. Pedesseau, F. Doré, S. Boyer-Richard, UMR FOTON 68 NRS, INSA de Rennes, France Jacky.even@insa-rennes.fr BIQUINIS Supported by the French ANR project BIQUINIS 8 th International onference on Numerical Simulation of Optoelectronic Devices, Nottingham, 4 th September 8
2 Outline I- k.p axial approximation II- olumnar quantum dots (QD) in polarization insensitive semiconductor optical amplifiers (SOA) III- Non radiative Auger process in QD IV- onclusion BIQUINIS Soline Boyer-Richard, NUSOD 8, Nottingham, 4 th September 8
3 Quantum dot with axial symmetry Geometry of quantum dot : v symmetry A new description in cylindrical coordinates : r, φ, z Some references for k.p axial approximation : 1 P. Enders and M. Woerner, SST (1996) : k-dependent 4x4 block-diag. of 8x8 Ham. (Bulk). Y. P. hao and S. L. huang, PRB (199) : k-ind. block-diag. of 6x6 Ham. (QW) 3 Y. M. Mu and S. S. Pei, JAP (4) : k-ind. block-diag. of 8x8 Ham. (QW) 4 K. J. Vahala and P.. Sercel, PRL, PRB (199) : k-ind. block-diag. of 8x8 Ham. (Qwire and Spherical QD) 5 M. Tadic, F.M. Peeters and K. J. Janssens, JAP, PRB (, 4) k-ind. block-diag. of 6x6 Ham. (QD)
4 Axial approximation for mechanical properties Acoustic phonon band warping in cubic materials A test : sound velocities in InAs ubic material Surfaces : angular dependence of the longitudinal velocity Even et al, APL (7) P4-P1 3D isotropic approx. d = d =.5 d = 1 Transverse isotropic approx. = d
5 Axial approximation for mechanical properties Description in cylindrical coordinates : r, φ, z ε, εϕϕ, ε, ε rr zz rz Transverse isotropic approx. for cubic materials ' ' ' 1 ' ' = + ' = 4 parameters Back to cartesian coordinates ε cos ( ϕ ) ε + sin ( ϕ ) xx = rr ε sin ( ϕ ) ε + cos ( ϕ ) yy = rr z ε ϕϕ ε ϕϕ ε = cos( ϕ ) xz ε rz ε = sin( ϕ ) yz ε rz ε xy sin( ϕ ) = rr ( ε ) ε ϕϕ ε rr ε rz Example : InAs/GaP QD r
6 Axial approximation for the 8-band k.p strained Hamiltonian Unstrained part : Strained part : (new proposition) R ε h [ γ ( k k ) iγ k k ] 3 k h R = 3 γ b = x y 3 x y m m 3 d b 3 ε εϕϕ cos(ϕ ) ε εϕϕ sin(ϕ ) ε ε i rr i rr rr ϕϕ e ( ) ( ) ( ) ϕ b 3 1 b 3 d = + Even et al, PRB (8) A P ( ε rr + ε ϕϕ ε zz ) ( ε + ε ε ) InAs GaAs InP b b b ε = a c + ε rr + ε ϕϕ ε = a v rr ϕϕ + Q = ε b ε zz zz d = 1.58eV = 1.8eV d = 1.56eV =.5eV d = 1.73eV =.5eV R S ε b = dε ϕ ( ε ) e ε = 3 ε i rr ϕϕ iϕ rze Hydrostatic strain Biaxial strain Shear strain
7 Axial approximation for the 8-band k.p strained Hamiltonian Block diagonalization of the Hamiltonian for each Fz value : A good quantum number : total angular momentum F = J + z z L z Development of the wavefunction : J, J z L z = F z J z Basis of 8 Bloch functions u i (i=1 8) 8 enveloppe functions of (r,z) J Z 1/ -1/ B -1/ -3/ 3/ HH 1/ 1/ -1/
8 Axial approximation for the 8-band k.p strained Hamiltonian Excited p and d states splitting predicted from symmetry analysis gap
9 Outline I- k.p axial approximation II- olumnar quantum dots (QD) in polarization insensitive semiconductor optical amplifiers (SOA) III- Non radiative Auger process in QD IV- onclusion BIQUINIS Soline Boyer-Richard, NUSOD 8, Nottingham, 4 th September 8
10 olumnar QD for polarization insensitive SOA InAs/GaAs Kita, Phys. Stat. Sol. (3) Saito, Physica E (5)
11 olumnar QD for polarization insensitive SOA InAs/InP Transitions between states 1 / TE absorption : ΔF = F 1 Fz z = ± 1 TM absorption : ΔF = F 1 Fz z = In 1 QD : Electron GS : F Z = ±1/, Hole GS : F Z = ±3/ TE transitions ross over Kawagushi, Phys. Stat. Sol. (6) This model
12 olumnar QD for polarization insensitive SOA 7 layers 8 layers 9 layers 1 layers 11 layers TM TE TE TM rossing
13 Outline I- k.p axial approximation II- olumnar quantum dots (QD) in polarization insensitive semiconductor optical amplifiers (SOA) III- Non radiative Auger process in QD IV- onclusion BIQUINIS Soline Boyer-Richard, NUSOD 8, Nottingham, 4 th September 8
14 Auger processes in narrow gap QD Screened interaction Barrier material B Quantum dot Δ HH LH Even et al, Physica B (7) SO
15 Auger processes in narrow gap QD =-3/ =-1/ =3/ =1/ =-3/ =-1/ =3/ =1/ =1/ =1/ hν (-H) + relaxation =3/ =3/ Radiative process in QD Non-radiative process in QD =-1/ =-5/ =1/ =5/ =-1/ =-5/ =1/ =5/
16 Auger processes in narrow gap QD =-3/ =-1/ =3/ =1/ =-3/ =-1/ =3/ =1/ =1/ =1/ B relaxation in QD (-) =3/ =3/ =-1/ =-5/ =1/ =5/ =-1/ =-5/ =1/ =5/
17 Auger processes in narrow gap QD =-3/ =-1/ =3/ =1/ =-3/ =-1/ =3/ =1/ =1/ =1/ Hole relaxation in QD (H-H) =3/ =3/ =-1/ =-5/ =1/ =5/ =-1/ =-5/ =1/ =5/
18 Auger processes in narrow gap QD B B B - Bulk relaxation processes L-H S-H Δ HH Δ HH Δ HH LH LH LH SO SO SO
19 Auger processes in narrow gap QD Example : InAs/Q1.18/InP truncated cone QD E H =.77eV (gap) E =3meV E HH =16meV N Bulk =1 18 cm -3 Non-radiative processes : τ H =.μs τ HLH =1.1ms τ HSH =14ns B relaxation in QD : τ =.74ps τ LH =4ps τ SH = Hole relaxation in QD : τ HH =1.ps τ HHLH =ps τ HHSH =
20 onclusion : further studies - New axially symmetric strained nanostructures - Auger effects (gap influence, comparison WL/bulk, barrier materials, hydrostatic pressure ) - Beyond the 8-band k.p approximation Soline Boyer-Richard, NUSOD 8, Nottingham, 4 th September 8
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