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Influence of the magnetic field on formation and spectrum of the exciton–polariton in a microcavity
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Physica B 378–380 (2006) 1049–1050 www.elsevier.com/locate/physb
Influence of the magnetic field on formation and spectrum of the exciton–polariton in a microcavity N.E. Kaputkinaa,, Yu.E. Lozovikb, M. Willanderc a
Moscow Institute for Steel, Alloys, Leninskii prosp. 4, 117936, Moscow, Russia Institute of Spectroscopy, Academy of Sciences of Russia,142092 Troitsk, Moscow region, Russia c Laboratory of Physical Electronics and Photonics, Department of Physics, Go¨teborg University, S-412 96 Go¨teborg, Sweden b
Abstract Magnetic field effect on properties of exciton–polaritons in an optical microcavity and on direct and indirect excitons in single and coupled quantum wells is considered. Magnetic field controls exciton dispersion curve and its intersection with microcavity photon dispersion curve. This gives the possibility to control exciton–polariton formation, polariton splitting and polariton dispersion. Excitons and exciton–polaritons on higher Landau levels are discussed. r 2006 Elsevier B.V. All rights reserved. PACS: 78.67.de; 73.21.fg Keywords: Exciton–polariton; Magnetic field
1. Introduction
2. Polariton formation criteria
The system with single or coupled quantum wells (QWs) in a microcavity can demonstrate interesting characteristics and possible new effects. These systems were recently the objects of theoretical and experimental research [1–9]. The interaction of two-dimensional excitons (direct and indirect) with photons and possible polariton formation is of particular interest. Polaritons in microcavities essentially change radiation characteristics of QW in microcavities [5–9] and the spectrum of light scattering in the region of the exciton–polariton Rabi splitting [1–4,9]. We consider magnetic field effect on the properties of direct and indirect excitons in single and coupled QWs and on exciton–polaritons in an optical microcavity. Applying an external magnetic field gives the possibility to tune polariton formation for exciton–photon interaction in a microcavity. The influence of external magnetic field on polariton dispersion law is considered. Excitons and exciton–polaritons on high-Landau levels are also discussed.
For consideration of collective properties of excitons we use the second quantization representation X 1X þ þ H^ ¼ ef b þ bnf ; bmg ; bmg bnf f 0 g0 jV^ nm jfg : nf þ bnf þ 2
Corresponding author. Fax: +7 95 2372127.
E-mail address: [email protected] (N.E. Kaputkina). 0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.01.399
where we summarize on all quantum numbers of the system (Landau levels and magnetic momenta). For weak exciton–photon interaction polariton effects are essential if exciton and cavity dispersion curves intersect. For isolated excitons, polaritons are formed if exciton energy is greater than or equal to cavity photon mode at k ¼ 0. For very low exciton density, N0, ideal gas spectrum coincides with Bogolubov spectrum of weakly non-ideal Bose gas with repulsion. The Bogolon polariton formation condition is e0 ¼ Am ðm2 M 2 m2 Þ=ð4m2 2M 2 Þ, where e0 is the difference between exciton and photon mode energy at k ¼ 0; A ¼ 4pN 0 _2 a=m , M ¼ p_n2 =cL, n is effective refractive coefficient, L is cavity width, a is
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N.E. Kaputkina et al. / Physica B 378–380 (2006) 1049–1050
exciton–exciton scattering amplitude, m* is effective exciton mass. Binding energies and effective masses of indirect magnetoexcitons depend on magnetic field H (and interlayer distance D for indirect exciton). By a magnetic field one can control exciton and cavity photon branches intersection, and thus the possibility of exciton–polariton formation and properties of the exciton–polariton. For low (but not extra low) effective exciton density, when N0 a*2 oo 1 (where a ¼ a ðH; D) is effective radius of magnetoexciton) adequate approximation for interexciton interaction corresponds to the summation of ladder diagrams (see Ref. [10] and references herein). One can make mapping the transition from two-dimensional system with magnetic field with potential UðRÞ ¼ e2 D2 =R3 to the system of indirect magnetoexcitons without magnetic field but with exciton mass equal to effective mass of magnetoexcitons. For strong magnetic field when magnetic length rH1/H1/2 is much smaller than the barrier width D effective mass of magnetoexciton can be estimated as MHD3/e2r4H. Contrary to three-dimensional rare system for the two-dimensional system of magnetoexcitons characteristic magnetic momentum k is not equal to zero. This peculiarity of two-dimensional Bose system is connected with logarithmic divergence of two-dimensional scattering amplitude in zero. Collective excitation spectrum is the acoustic one for small magnetic momentum region eðPÞ ¼ C s P, C S ¼ ðm=mH Þ1=2 is sound velocity, m is chemical potential. As it was demonstrated [11–13] for excited Landau levels effective magnetic mass of indirect magnetoexciton can be negative for small magnetic momentum region for certain range of magnetic field and barrier width and roton-type minima at exciton dispersion curve can appeared. In consequence of that energy dependence for excited exciton–polariton (dispersion law) is also not monotony for certain range of parameters of coupled QWs, microcavity and magnetic field and can contain several minima. Magnetic field controls the depth and position of the minima. The presence of such peculiarities of exciton–polariton spectra and dispersion laws in magnetic field induce new qualitative effects—new states of the system and new resonances in absorption and reflection spectra. One of the most interesting collective effects for system of interacting exciton–polaritons in microcavity is Bose condensation, and Kosterlitz–Thouless transition [10]. Moreover, in the presence of several minima several different condensates with different characteristic temperature can be formed. Magnetic field can control that effect also.
3. Conclusion The condition of exciton–polariton formation is determined through the value of the binding energy E0 and effective mass m* of the exciton which depend on external magnetic field (when other parameters are determined by the microcavity and single or coupled QWs are fixed). So application of external magnetic field gives partly the possibility to control the polariton formation, and partly to control polariton spectra.
Acknowledgments The work is supported by Russian Foundation of Basic Research and Russian Ministry of Education and by grant of the President of the Russian Federation for the State Support of Young Russian Scientists MK-2626.2005.2 and the Swedish Strategic Research Foundation.
References [1] C. Weisbuch, M. Nishioka, A. Ishikawa, Y. Arakawa, Phys. Rev. Lett. 69 (1992) 3314; T.B. Norris, J.-K. Rhee, C.-Y. Sung, Y. Arakawa, M. Nishioka, C. Weisbuch, Phys. Rev. B 50 (1994) 14663. [2] Y. Zhu, D.J. Gauthier, S.E. Morin, Q. Wu, H.J. Carmichael, T.W. Mossberg, Phys. Rev. Lett. 64 (1990) 2499. [3] V. Savona, Z. Hradil, A. Quattropani, P. Schwendimann, Phys. Rev. B 49 (1994) 8774; V. Savona, L.C. Andreane, P. Schwendimann, A. Quattropani, Solid State Commun. 93 (1995) 733; V. Savona, F. Tassone, Solid State Commun. 95 (1995) 673. [4] S. Jorda, Phys. Rev. B 50 (1994) 18690; S. Jorda, Phys. Rev. B 51 (1995) 10185. [5] D.S. Citrin, Phys. Rev. B. 47 (1993) 3832. [6] S. Pau, H. Cao, J. Jacobson, G. Bjork, Y. Yamamoto, A. Imamoglu, Phys. Rev. A 54 (1996) R1789; H. Cao, S. Pau, J.M. Jacobson, G. Bjork, Y. Yamamoto, A. Imamoglu, Phys. Rev. A 55 (1997) 4632. [7] G. Malpuech, A. Kavokin, A.Di. Carlo, J.J. Baumberg, Phys. Rev. B 65 (2002) 153310. [8] A.I. Tartakovskii, D.N. Krizhanovskii, G. Malpuech, M. EmamIsmail, A.V. Cherneko, A.V. Kavokin, V.D. Kulakovskii, M.S. Skolnick, J.S. Roberts, Phys. Rev. B 67 (2003) 165302. [9] A.V. Kavokin, I.A. Shelykh, M.M. Glazov, Phys. Stat. Sol. (c) 2 (2) (2005) 914. [10] Y.E. Lozovik, O.L. Berman, V.G. Tsvetus, Phys. Rev. B 59 (1999) 5627; O.L. Berman, Y.E. Lozovik, D.W. Snoke, R.D. Coalson, Phys. Rev. B 70 (2004) 235310. [11] Y.E. Lozovik, N.E. Kaputkina, Phys. Stat. Solidi B 207 (1998) 147. [12] Y.E. Lozovik, A.M. Ruvinsky, Phys. Lett. A 227 (1997) 271. [13] N.E. Kaputkina, Y.E. Lozovik, Physica E 12 (2002) 323.
Physica B 378–380 (2006) 1049–1050 www.elsevier.com/locate/physb
Influence of the magnetic field on formation and spectrum of the exciton–polariton in a microcavity N.E. Kaputkinaa,, Yu.E. Lozovikb, M. Willanderc a
Moscow Institute for Steel, Alloys, Leninskii prosp. 4, 117936, Moscow, Russia Institute of Spectroscopy, Academy of Sciences of Russia,142092 Troitsk, Moscow region, Russia c Laboratory of Physical Electronics and Photonics, Department of Physics, Go¨teborg University, S-412 96 Go¨teborg, Sweden b
Abstract Magnetic field effect on properties of exciton–polaritons in an optical microcavity and on direct and indirect excitons in single and coupled quantum wells is considered. Magnetic field controls exciton dispersion curve and its intersection with microcavity photon dispersion curve. This gives the possibility to control exciton–polariton formation, polariton splitting and polariton dispersion. Excitons and exciton–polaritons on higher Landau levels are discussed. r 2006 Elsevier B.V. All rights reserved. PACS: 78.67.de; 73.21.fg Keywords: Exciton–polariton; Magnetic field
1. Introduction
2. Polariton formation criteria
The system with single or coupled quantum wells (QWs) in a microcavity can demonstrate interesting characteristics and possible new effects. These systems were recently the objects of theoretical and experimental research [1–9]. The interaction of two-dimensional excitons (direct and indirect) with photons and possible polariton formation is of particular interest. Polaritons in microcavities essentially change radiation characteristics of QW in microcavities [5–9] and the spectrum of light scattering in the region of the exciton–polariton Rabi splitting [1–4,9]. We consider magnetic field effect on the properties of direct and indirect excitons in single and coupled QWs and on exciton–polaritons in an optical microcavity. Applying an external magnetic field gives the possibility to tune polariton formation for exciton–photon interaction in a microcavity. The influence of external magnetic field on polariton dispersion law is considered. Excitons and exciton–polaritons on high-Landau levels are also discussed.
For consideration of collective properties of excitons we use the second quantization representation X 1X þ þ H^ ¼ ef b þ bnf ; bmg ; bmg bnf f 0 g0 jV^ nm jfg : nf þ bnf þ 2
Corresponding author. Fax: +7 95 2372127.
E-mail address: [email protected] (N.E. Kaputkina). 0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.01.399
where we summarize on all quantum numbers of the system (Landau levels and magnetic momenta). For weak exciton–photon interaction polariton effects are essential if exciton and cavity dispersion curves intersect. For isolated excitons, polaritons are formed if exciton energy is greater than or equal to cavity photon mode at k ¼ 0. For very low exciton density, N0, ideal gas spectrum coincides with Bogolubov spectrum of weakly non-ideal Bose gas with repulsion. The Bogolon polariton formation condition is e0 ¼ Am ðm2 M 2 m2 Þ=ð4m2 2M 2 Þ, where e0 is the difference between exciton and photon mode energy at k ¼ 0; A ¼ 4pN 0 _2 a=m , M ¼ p_n2 =cL, n is effective refractive coefficient, L is cavity width, a is
ARTICLE IN PRESS 1050
N.E. Kaputkina et al. / Physica B 378–380 (2006) 1049–1050
exciton–exciton scattering amplitude, m* is effective exciton mass. Binding energies and effective masses of indirect magnetoexcitons depend on magnetic field H (and interlayer distance D for indirect exciton). By a magnetic field one can control exciton and cavity photon branches intersection, and thus the possibility of exciton–polariton formation and properties of the exciton–polariton. For low (but not extra low) effective exciton density, when N0 a*2 oo 1 (where a ¼ a ðH; D) is effective radius of magnetoexciton) adequate approximation for interexciton interaction corresponds to the summation of ladder diagrams (see Ref. [10] and references herein). One can make mapping the transition from two-dimensional system with magnetic field with potential UðRÞ ¼ e2 D2 =R3 to the system of indirect magnetoexcitons without magnetic field but with exciton mass equal to effective mass of magnetoexcitons. For strong magnetic field when magnetic length rH1/H1/2 is much smaller than the barrier width D effective mass of magnetoexciton can be estimated as MHD3/e2r4H. Contrary to three-dimensional rare system for the two-dimensional system of magnetoexcitons characteristic magnetic momentum k is not equal to zero. This peculiarity of two-dimensional Bose system is connected with logarithmic divergence of two-dimensional scattering amplitude in zero. Collective excitation spectrum is the acoustic one for small magnetic momentum region eðPÞ ¼ C s P, C S ¼ ðm=mH Þ1=2 is sound velocity, m is chemical potential. As it was demonstrated [11–13] for excited Landau levels effective magnetic mass of indirect magnetoexciton can be negative for small magnetic momentum region for certain range of magnetic field and barrier width and roton-type minima at exciton dispersion curve can appeared. In consequence of that energy dependence for excited exciton–polariton (dispersion law) is also not monotony for certain range of parameters of coupled QWs, microcavity and magnetic field and can contain several minima. Magnetic field controls the depth and position of the minima. The presence of such peculiarities of exciton–polariton spectra and dispersion laws in magnetic field induce new qualitative effects—new states of the system and new resonances in absorption and reflection spectra. One of the most interesting collective effects for system of interacting exciton–polaritons in microcavity is Bose condensation, and Kosterlitz–Thouless transition [10]. Moreover, in the presence of several minima several different condensates with different characteristic temperature can be formed. Magnetic field can control that effect also.
3. Conclusion The condition of exciton–polariton formation is determined through the value of the binding energy E0 and effective mass m* of the exciton which depend on external magnetic field (when other parameters are determined by the microcavity and single or coupled QWs are fixed). So application of external magnetic field gives partly the possibility to control the polariton formation, and partly to control polariton spectra.
Acknowledgments The work is supported by Russian Foundation of Basic Research and Russian Ministry of Education and by grant of the President of the Russian Federation for the State Support of Young Russian Scientists MK-2626.2005.2 and the Swedish Strategic Research Foundation.
References [1] C. Weisbuch, M. Nishioka, A. Ishikawa, Y. Arakawa, Phys. Rev. Lett. 69 (1992) 3314; T.B. Norris, J.-K. Rhee, C.-Y. Sung, Y. Arakawa, M. Nishioka, C. Weisbuch, Phys. Rev. B 50 (1994) 14663. [2] Y. Zhu, D.J. Gauthier, S.E. Morin, Q. Wu, H.J. Carmichael, T.W. Mossberg, Phys. Rev. Lett. 64 (1990) 2499. [3] V. Savona, Z. Hradil, A. Quattropani, P. Schwendimann, Phys. Rev. B 49 (1994) 8774; V. Savona, L.C. Andreane, P. Schwendimann, A. Quattropani, Solid State Commun. 93 (1995) 733; V. Savona, F. Tassone, Solid State Commun. 95 (1995) 673. [4] S. Jorda, Phys. Rev. B 50 (1994) 18690; S. Jorda, Phys. Rev. B 51 (1995) 10185. [5] D.S. Citrin, Phys. Rev. B. 47 (1993) 3832. [6] S. Pau, H. Cao, J. Jacobson, G. Bjork, Y. Yamamoto, A. Imamoglu, Phys. Rev. A 54 (1996) R1789; H. Cao, S. Pau, J.M. Jacobson, G. Bjork, Y. Yamamoto, A. Imamoglu, Phys. Rev. A 55 (1997) 4632. [7] G. Malpuech, A. Kavokin, A.Di. Carlo, J.J. Baumberg, Phys. Rev. B 65 (2002) 153310. [8] A.I. Tartakovskii, D.N. Krizhanovskii, G. Malpuech, M. EmamIsmail, A.V. Cherneko, A.V. Kavokin, V.D. Kulakovskii, M.S. Skolnick, J.S. Roberts, Phys. Rev. B 67 (2003) 165302. [9] A.V. Kavokin, I.A. Shelykh, M.M. Glazov, Phys. Stat. Sol. (c) 2 (2) (2005) 914. [10] Y.E. Lozovik, O.L. Berman, V.G. Tsvetus, Phys. Rev. B 59 (1999) 5627; O.L. Berman, Y.E. Lozovik, D.W. Snoke, R.D. Coalson, Phys. Rev. B 70 (2004) 235310. [11] Y.E. Lozovik, N.E. Kaputkina, Phys. Stat. Solidi B 207 (1998) 147. [12] Y.E. Lozovik, A.M. Ruvinsky, Phys. Lett. A 227 (1997) 271. [13] N.E. Kaputkina, Y.E. Lozovik, Physica E 12 (2002) 323.