A T-matrix many-particle theory for coherently coupled superlattice optics

The high- order Coulomb correlations described by T-matrix diagrams in both carrier occupation and polarization functions are studied here with a Keldysh- Green’s Functions formalism. Numerical applications for low dimensional semiconductor systems illustrate the relevance of the effects and their i...

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Дата:	2004
Автори:	Schmielau, T., Pereira Jr., M.F., Henneberger, K.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України 2004
Назва видання:	Semiconductor Physics Quantum Electronics & Optoelectronics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/118176
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	A T-matrix many-particle theory for coherently coupled superlattice optics / T. Schmielau, M.F. Pereira Jr, K. Henneberger // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 2. — С. 141-146. — Бібліогр.: 11 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-1181762017-05-30T03:03:22Z A T-matrix many-particle theory for coherently coupled superlattice optics Schmielau, T. Pereira Jr., M.F. Henneberger, K. The high- order Coulomb correlations described by T-matrix diagrams in both carrier occupation and polarization functions are studied here with a Keldysh- Green’s Functions formalism. Numerical applications for low dimensional semiconductor systems illustrate the relevance of the effects and their importance for realistic nonlinear optical spectra calculations. A frequency and momentum resolved numerical demonstration of the Mott transition at the T-matrix level is presented. 2004 Article A T-matrix many-particle theory for coherently coupled superlattice optics / T. Schmielau, M.F. Pereira Jr, K. Henneberger // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 2. — С. 141-146. — Бібліогр.: 11 назв. — англ. 1560-8034 PACS: 78.67.Pt http://dspace.nbuv.gov.ua/handle/123456789/118176 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The high- order Coulomb correlations described by T-matrix diagrams in both carrier occupation and polarization functions are studied here with a Keldysh- Green’s Functions formalism. Numerical applications for low dimensional semiconductor systems illustrate the relevance of the effects and their importance for realistic nonlinear optical spectra calculations. A frequency and momentum resolved numerical demonstration of the Mott transition at the T-matrix level is presented.
format	Article
author	Schmielau, T. Pereira Jr., M.F. Henneberger, K.
spellingShingle	Schmielau, T. Pereira Jr., M.F. Henneberger, K. A T-matrix many-particle theory for coherently coupled superlattice optics Semiconductor Physics Quantum Electronics & Optoelectronics
author_facet	Schmielau, T. Pereira Jr., M.F. Henneberger, K.
author_sort	Schmielau, T.
title	A T-matrix many-particle theory for coherently coupled superlattice optics
title_short	A T-matrix many-particle theory for coherently coupled superlattice optics
title_full	A T-matrix many-particle theory for coherently coupled superlattice optics
title_fullStr	A T-matrix many-particle theory for coherently coupled superlattice optics
title_full_unstemmed	A T-matrix many-particle theory for coherently coupled superlattice optics
title_sort	t-matrix many-particle theory for coherently coupled superlattice optics
publisher	Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
publishDate	2004
url	http://dspace.nbuv.gov.ua/handle/123456789/118176
citation_txt	A T-matrix many-particle theory for coherently coupled superlattice optics / T. Schmielau, M.F. Pereira Jr, K. Henneberger // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 2. — С. 141-146. — Бібліогр.: 11 назв. — англ.
series	Semiconductor Physics Quantum Electronics & Optoelectronics
work_keys_str_mv	AT schmielaut atmatrixmanyparticletheoryforcoherentlycoupledsuperlatticeoptics AT pereirajrmf atmatrixmanyparticletheoryforcoherentlycoupledsuperlatticeoptics AT hennebergerk atmatrixmanyparticletheoryforcoherentlycoupledsuperlatticeoptics AT schmielaut tmatrixmanyparticletheoryforcoherentlycoupledsuperlatticeoptics AT pereirajrmf tmatrixmanyparticletheoryforcoherentlycoupledsuperlatticeoptics AT hennebergerk tmatrixmanyparticletheoryforcoherentlycoupledsuperlatticeoptics
first_indexed	2025-07-08T13:31:03Z
last_indexed	2025-07-08T13:31:03Z
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fulltext	141© 2004, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine Semiconductor Physics, Quantum Electronics & Optoelectronics. 2004. V. 7, N 2. P. 141-146. PACS: 78.67.Pt T-matrix many-particle theory for coherently coupled superlattice optics T. Schmielaua, M.F. Pereira Jrb, and K. Hennebergera aFachbereich Physik, Universität Rostock, D-18051 Rostock, Germany bNMRC, University College Cork, prospect Row, Lee Maltings, Cork, Ireland Abstract. The high-order Coulomb correlations described by T-matrix diagrams in both carrier occupation and polarization functions are studied here with the Keldysh�Green�s Functions formalism. Numerical applications for low dimensional semiconductor systems illustrate the relevance of the effects and their importance for realistic nonlinear optical spectra calculations. A frequency and momentum resolved numerical demonstration of the Mott transition at the T-matrix level is presented. Keywords: semiconductors, superlattices, T-matrix, many body effects, Green�s functions. Paper received 12.02.04; accepted for publication 17.06.04. 1. Introduction Coulomb effects are now well established as the origin of the near band-gap optical nonlinearities in semiconductors [1], and many- body techniques are required for a realistic description of their optical and transport properties, due to the large number of charged carriers created. Among the current possibilities, non-equilibrium Keldysh�Green�s functions allow the consistent description of both equi- librium and non equilibrium situations [2]. As the temperature is reduced, and the effective Cou- lomb interaction between charged carriers increases, espe- cially in systems designed to enhance those effects, like low dimensional semiconductors, many-particle corrections of increasingly higher order are required to explain the elec- tronic and highly non-linear optical properties of such sys- tems, e.g. corrections beyond the Random Phase (RPA) and GW approximations must be considered [3,4,5,6]. Numerical results for deviations beyond RPA and GW have been given for the first time in Ref. 3, where exciton- induced satellites in the one-particle spectral functions of charged carriers in a bulk material are discussed. In low-dimensional systems, quantum-confinement and bandstructure effects add a further complication for realis- tic numerics, and here we use as a model system, coher- ently coupled semiconductor superlattices, in which such complications are dealt with by means of an effective dimensionality formalism that describes the evolution of the system from quasi- two to quasi- three dimensions [7,8]. We discuss under which conditions the T-matrix leads to deviations in the selfenergies that characterize the rel- evant quasi-particles of the system, and show a frequency and momentum resolve numerical demonstration of the Mott transition at the T-matrix level. The paper is organized as follows: In Section 2 the main non-equilibrium Green�s functions expressions are summarized. In Section 3 the dephasing computation is outlined. Numerical results are presented in Section 4, which is closed by a brief Summary. 2. Nonequilibrium Green�s functions expressions The interacting quasi- particles, which describe the excited semiconductor in our Keldysh�Green�s, function ap- proach, namely, carriers (G), photons (D), and plasmons (W) [9], have their time-dependence dictated by Dyson equations (sum over repeated indices is assumed), [ ] [ ] [ ] ),12()32()13()13( ),12()32()13()13( ),12()32()13()13( 1 0 1 0 1 ,0 δ δ δ =− =− =Σ− − − − WpW DPD GG T abcbacac (1) where δT is the transverse delta function. The labels a,b,c � denote generically denote generically the sev- eral conduction and valence sub bands. The inverse free propagators read 142 SQO, 7(2), 2004 T. Schmielau et al.: T-matrix many-particle theory for coherently coupled superlattice � ).12( 1 )12( ),12( 4 )12( ),12()1()12( 2 1 2 21 1 0 12 01 0 1 1 ,0 δ δ π ε δ         ∂ ∂−∆= ∆−=       − ∂ ∂= − − − tc D e W h t iG effab h (2) Here ε0 is the static dielectric function. The effective one-particle Hamiltonian in the equation for the free-carrier propagator reads, [ ] ),1()1()1()1()1( 0 0 ∇⋅+Φ+= rrh effabeffeff A cm ie Hh δ (3) where effΦ and effA r denote, respectively, the expectation values of the scalar and vector potentials. The selfenergies, Σ, P and p are called, respectively the carrier selfenergy, the transverse and longitudinal polarization functions. Detailed band-structure and quantum-confinement effects are in- cluded in the theory through effh in the free-carrier propa- gator, and also in the optical transition selection rules de- scribed by the matrix elements of the velocity operator, ( ) ( ) 0 * 2/)2()1(2/)2()1()12( im∇−∇=Π+Π=Π rv h rrr The carrier self-energy Σ leads to bandgap renorma- lization, includes dynamic effects such as corrections be- yond Hartree-Fock, scattering rates in the carriers kinetics, and enables the description of bound states (excitons) in the spectral density of carriers, defining the degree of ioni- zation. The longitudinal polarization function p is responsi- ble for (plasmon) screening of the Coulomb interaction. Furthermore, it describes dynamical screening, screening by excitons, plasmon kinetics and the build up of screen- ing, although, these issues will not be addressed in this paper. The transverse polarization function P yields the excitation dependent absorption coefficient and refractive index, and defines scattering rates (generation/recombina- tion, respectively absorption/emission) in the photon ki- netics. It is responsible for the inclusion of bound states (excitons) in the photons spectral density. Functional derivative or diagrammatic techniques allow the consideration of increasingly higher order Coulomb corrections in the self-energies. In this paper, we consider T-matrix correlation contributions (c) beyond the Random Phase (RPA) and GW approximations. For the carrier and photon self-energies, we obtain [4,5], ),43()1324()12( ),12()12()12( ),12()12()12( " GiT WGi ab c aa GW aa c aa GW aaaa −=Σ −=Σ Σ+Σ=Σ h (4) ),62()52()3456()14()13( )12()12( ,0 hheeehhhee eheh GGTGG PP h− −= (5) where we have introduced the electron-hole quantity, ),24()13()1234( hheeeh GGiP h−= (6) Within screened ladder approximation, the T-matrix, ),7834()5678()1256( 1 )1234()1234( ,0,0 ,0 eheheh eheh PTP PP h + += (7) satisfies the equation, ).5634()26()15()12( )24()13()12()1234( ehhheeeh eheh TGGiW WT − −= δδ (8) The transverse and longitudinal polarization func- tions are connected through )2( 4 )12( 2 2 = c e Pab ππ π rr (1)p(11±2). (9) 3. Dephasing The possibility of computing the dephasing and the cor- responding bandgap shift that follows by Kramers-Kronig allows for further predictability in the theory, in contrast with a previous approach where formulas that simulate the fully computed dephasing have been used [10]. They ap- pear directly in the spectral function G, entering all equa- tions presented above. After suitable Fourier transforma- tions, ( ) 222 ),( aa a E kG Γ+− Γ = hh h ω ω , (10) where { }),( ωε kE r aaaa Σℜ−= h , and { }),( ωkr aaa Σ−ℑ=Γh denote, respectively, the renormalized energies and the dephasing rate. The correlation functions are given by )(),(),( ωωω << = aaaaa fkGkG , (11) and )(ω< af is a Fermi function. Algebraic manipulations lead to the GW dephasing, [ ] [ ]),()( )()( 2 ),( Ωℑ+× ×Ω++ΩΩ−=Γ ∑ ∫ +∞ ∞− qWqkG fn d k aa aB q a rrr r r ω π ω (12) where )(ΩBn is the Bose distribution function. Using the Kramers-Kroning relation, the real part of the retarded selfenergy is given by { } ωπ ω −Ω ΩΓΩ=Σℜ ∫ +∞ ∞− ),( 2 ),( kd Pk nr aa r r . (13) Here ∫P denotes a Cauchy principal value integral. Using the Optical theorem for the retarded potential, { } { }),(),(),( , 2 ωωω kpkWkW r bb heb aaaa rrr ℑ=ℑ ∑ = , (14) together with the retarded longitudinal polarization function, , T. Schmielau et al.: T-matrix many-particle theory for coherently coupled superlattice � 143SQO, 7(2), 2004 , )()( ),(),( 22 ),( 21 21 21 21 δωωω ωω ωω π ω π ω ω i ff kqGqG dd kp bb bbbb q r bb ++− − × ×+−= << +∞ ∞− ∑ ∫ rrrr r the self energies can be computed iteratively together with the corresponding quasi-chemical potentials, which char- acterize the distribution functions. In order to understand the full process and make a connection to less advanced iteration schemes, note that the average number of parti- cles in a system of particles of type a (electrons or holes) reads dRRRN aaa ∫ ΨΨ= + )()( , (16) which can be written as ( ) ( ) )( )( 2 222 ω ωπ ω h hh hh r <⋅ Γ+− Γ = ∑ a aa a k a f kE d N . (17) In the quasi-particle approximation 0→Γah , we ob- tain the usual ))((2 kEfN a k aa ∑= v . At this point we include higher-order T-matrix cor- rections, following the prescription presented in the pre- vious section. We must then include the terms below in the iterative scheme. The Fourier-transformed ladder or correlation T-matrix self energy has Keldysh components given by ,),(),,,( 2 2 ),( 2 " 1 , 2 Ω⋅Ω+′Ω= =Σ < < ∑ pGPppT d i p bab p c aa vrrr r r ω π ω (18) with analogous expressions for the �>� Keldysh compo- nents. For cases in which effective masses can be defined, as in the superalattice data used for the numerics used here, 21 p mm m p mm m p ba a ba b rrr + − + = , and 21 ppP rrr += . In the statically screened ladder approximation, the Fou- rier-transformed T-matrix, ),,',( ωPppT r ab vrr , satisfies the Bether-Salpeter equation, ,),,',"(),,"()"( )'(),,',( " ωω ω PppTPpgppW ppWPppT r ab r ab p ab ab r ab rrrrrrr rrrrr r ⋅⋅−+ +−= ∑ (19) where the uncorrelated two-particle Green�s function reads, ),(),( 2 ),,( 21 Ω⋅Ω−Ω= ><< ∫ pGpG d iPpg ba rrrr ω π ω . (20) 4. Numerical results and discussion In this section, we apply our theory for coherently coupled superlattices, treated within the anisotropic medium ap- proach, in which the motion of electrons and holes is char- acterized by angle-averaged effective masses. Structures designed to increase the carriers confinement lead to larger effective masses [7,8]. Figure 1 displays the GW dephasing of electrons ob- tained from the imaginary part of the corresponding retarded selfenergy for a II�VI superlattice. Higher order corrections are due to T-matrix corrections and are depicted in Fig. 2. The GW dephasing follows the electronic dispersion (free carrier energy plus Fock and GW self energies), while the T-matrix dephasing is mainly centered about the excitonic dispersion. The correlations described by the T-matrix also lower the potential energy of the unbound carriers, thus increasing the damping on the low-energy side of the free dispersion and decreasing it on the high- energy side. This effectively shifts the spectral weight to lower energies. In all plots the energy axis is the detuning with respect to the fundamental band gap (including the quantum confinement correction) and measured in units of the superlattice binding energy. Figure 3 illustrates the bleaching of the excitonic absorption (Mott transition) at the T-matrix level for (15) 0 0.1 0.2 0.3 �2 �1 0 0 1 12 2 3 3 4 4 k a 1 / [ ] w R y d[ ] G w ( , ) R y dk [ ]R PA 0 5 Fig. 1. GW dephasing for a 5Å ZnSe / 75Å ZnMgSe superlattice at T = 77 K and N = 1015 carriers/cm3. 0 0.05 �2 �1 0 0 1 12 2 3 3 4 4 w R y d[ ] G w ( , ) Ry dk [ ]T k a 1 / [ ]0 5 Fig. 2. T-matrix dephasing for the superlattice in Fig. 1, evalu- ated with the same parameters. 144 SQO, 7(2), 2004 T. Schmielau et al.: T-matrix many-particle theory for coherently coupled superlattice � 15Å / 5Å GaAs superlattice at 150 K. The lowest density plot clearly displays contributions from the 1s bound excitonic state, as well as the combined contribution of the 2s and 2p states, which already overlaps with the continuum. The k-dependence of these features reflects the corresponding hydrogen wavefunctions, thus the 2s and 2p features have a maximum at k > 0. While the 2p contribution is suppressed in the optical properties like the polarization function, it does appear in the T-mat- rix in forward scattering, which enters the T-matrix self- energy. It occurs in our numerical calculations only be- cause no further angle averaging is performed upon the T-matrix elements. As the carrier density increases from N = 1015 to 1018 carriers/cm3, the band-gap shrinkage shifts the continuum towards lower energies, where it joins the excitonic features (Mott transition). Our calculations are capable of displaying a frequency- and momentum resolved numerical demonstration of the Mott transition. The sign change in the T-matrix at 1018 carriers/cm3 corresponds to the appearance of gain in the optical spectra. Figure 4 shows distribution functions for a 5Å / 15Å ZnSe-ZnMgSe superlattice. The top, central and lower plots are, respectively for T = 77, 150 and 300K. On the left and right, N = 1015 and 1016 carriers/cm3. The solid curves and circles are, respectively, for Fermi and Wigner distributions. For comparison, exciton distributions are also shown as dot- dashed curves. They are defined as the convolution of the squared 1s wavefunction with the center-of-mass Boltzmann distribution. Note that, as the temperature increases, the Fermi and Wigner distributions become indistinguishable. In other words, in this case, we can compute optical spectra without the T-matrix diagram in the carrier�s self-energy, which justifies successful calculations by the present au- thors and several others in the literature for high tem- peratures considering Fermi distributions only. Figure 5 displays the optical absorption with the T-ma- trix included on both polarization and self-energy diagrams. No phenomenological parameters are needed here in order to give a consistent broadening to the low-density spectra. Even if fully computed within the GW approximation, the broadening would be unrealistically small at low densities and an arbitrary dephasing would have to be added. In other words, our calculations add further predictability to previous approaches, extending rather successful ap- proaches, which at high densities could reproduce several important experimental findings [11], to the low-density re- gime dominated by excitonic features. In summary, the microscopic theory for the nonlinear optical properties of semiconductors presented here pro- vides a technique to study Coulomb effects beyond RPA and GW, by analyzing their influence on optical spec- tra. Unphysical features, like a spurious absorption for Fig. 3. Evolution with increasing density of the T-matrix for a 5Å /15Å ZnSe-ZnMgSe superlattice. �2 �2 �2�2 �3 �3 �3 �5 �3 �1 �1 �1�1 0 0 00 0 0 00 0 0 0 0 5000 100 500 10000 200 5 1000 15000 300 10 1500 2000 1 1 11 1 1 11 2 2 22 2 2 22 3 3 33 3 3 33 4 4 44 4 4 44 w, Ryd w, Ryd w, Ryd w, Ryd n = 10 / cm n = 10 / cm n = 10 / cm n = 10 / cm 15 A ZnSe / 5A ZnMgSE superlattice, = 77 KT k, 1/a k, 1/a k, 1/a k, 1/a T k( , ), Rydw T k( , ), Rydw T k( , ), Rydw T k( , ), Rydw 0 0 0 0 15 16 17 18 3 3 3 3 T. Schmielau et al.: T-matrix many-particle theory for coherently coupled superlattice � 145SQO, 7(2), 2004 0.008 0.008 0.006 0.006 0.004 0.004 0.002 0.002 0 00.5 0.51 11.5 1.52 2 k 1/a[ ] k 1/a[ ]T = 77 K Fig. 4. Distributions functions for a 5Å / 15Å (left) and 5Å / 75Å ZnSe-ZnMgSe superlattices. The top, central and lower plots are, respectively for T = 77, 150 and 300 K. The carrier density is N = 1015, and 1016 carriers/cm3. Solid: Fermi distributions; Dot-dashed: Exciton distributions; Circles: Wigner distributions. 0.0030.003 0.0020.002 0.0010.001 00 0.50.5 11 1.51.5 22 k 1/a[ ]k 1/a[ ] T = 30 0 K 0.004 0.004 0.003 0.003 0.002 0.002 0.001 0.001 0 0 0.5 0.5 1 1 1.5 1.5 2 2 k 1/a[ ]k 1/a[ ] T = 15 0 K 146 SQO, 7(2), 2004 T. Schmielau et al.: T-matrix many-particle theory for coherently coupled superlattice � photon frequencies below those in the gain range do not appear, since our polarization function satisfies the KMS sum rule. The numerical results show that the actual car- rier occupation functions (Wigner distributions) differ from the commonly used Fermi distributions for suffi- ciently low carrier densities and temperatures. Our iter- ated GW and T-matrix dephasing adds further predict- ability to the approach and for multi-sub-band quantum wells provides important insight on high-density gain operation, which may be important for high-power semi- conductor laser applications. The numerics presented can also be used as the start- ing point for the realistic simulation of more complicated light emitting and processing devices. Acknowledgements The authors thank Science Foundation Ireland (SFI) and the Deutsche Forschungsgemeinschaft (DFG) for finan- cial support of this work. References 1. H. Haug, S.W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, World Scientific, Singapore (1990). 2. Progress in Nonequilibrium Green�s Functions, Edited by M. Bonitz, World Scientific, Singapore, (2000). 3. R. Zimmermann, Many-Particle Theory of Highly Excited Semiconductors, �Teubner Texte zur Physik�, Band 18, Leip- zig (1987), ISBN 3-322-00493-7. 4. R. Schepe, T. Schmielau, D. Tamme, and K. Henneberger // Phys. Stat. Sol. (b), 206, pp. 273 (1998). 5. M. Pereira Jr. and K. Henneberger // Phys. Rev. B, 58, pp. 2064 (1998). 6. T. Schmielau, G. Manzke, D. Tamme, and K. Henneberger, Phys. Stat. Sol. (b), 221, pp. 215 (2000). 7. M. Pereira Jr., I. Galbraith, S. Koch, and G. Duggan // Phys. Rev. B, 42, pp. 7084 (1990). 8. M. Pereira Jr. // Phys. Rev. B, 52, pp. 1978 (1995). 9. K. Henneberger and H. Haug // Phys. Rev. B, 38, (1988). 10. M.F. Pereira Jr., R. Binder and S.W. Koch // Appl. Phys. Lett., 64, pp. 279 (1994). 11. P. Michler, M. Vehse, J. Gutowski, M. Behringer, and D. Hommel, M.F.Pereira Jr. and K. Henneberger // Phys. Rev. B, 58, pp. 2055 (1998). 100 10 1 0.1 0.01 �1.5 �1 �0.5 0 0.5 1 1 0 /cm 1 0 /cm 1 0 /cm 1 0 /cm 15 3 3 3 3 16 17 118 Fig. 5. Optical absorption for a 5Å / 15Å ZnSe-ZnMgSe superlattice at T = 300K. The solid, long-dashed, short-dashed and dotted curves are, respectively, for N = 1015, 1016, 1017, and 1018 carriers/cm3.

A T-matrix many-particle theory for coherently coupled superlattice optics

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