- Email: [email protected]
Balance between incoming and outgoing cavity polaritons in a disordered organic microcavity
ARTICLE IN PRESS
Journal of Luminescence 122–123 (2007) 418–420 www.elsevier.com/locate/jlumin
Balance between incoming and outgoing cavity polaritons in a disordered organic microcavity M. Litinskayaa,b,, P. Reinekera a
Department of Theoretical Physics, University of Ulm, 89069 Ulm, Germany b Institute of Spectroscopy RAS, Troitsk, Moscow Region, 142190 Russia Available online 10 March 2006
Abstract We examine the population of cavity polaritons at non-resonant pumping. In our model the energy relaxes by means of emission of large-energy vibrational quanta into low-energy polaritonic states, which eventually escape the cavity through the mirrors. We discuss the ways to achieve large polariton population in the low-energy states. r 2006 Elsevier B.V. All rights reserved. PACS: 71.36.+c; 71.35.Aa; 78.55.Kz Keywords: Cavity polaritons; Energy relaxation; Intramolecular vibrations
The exciton–polariton excitation spectrum in planar organic microcavities made from J-aggregates of cyanine dyes [1] has an unusual structure [2]. It consists of two coexisting parts (see Fig. 1). Coherent (i.e., characterized by a well-defined in-plane wave vector) polaritonic states are formed for restricted intervals of the wave vector, e.g., ðLÞ for qðLÞ min oqoqmax for the lower branch polaritons (LP). The other excited states are incoherent (IS) and are similar to the electronic excitation of individual J-aggregates uncoupled with light. Their number N inc exceeds strongly the number of polaritonic states, and this determines the decisive role of IS in energy relaxation processes. Under non-resonant excitation of the microcavity, the energy relaxes mainly into IS. From there it can be transferred to polaritonic states. One possibility is that the ISs decay radiatively within the time tPL , and the emitted light is absorbed by polaritons. After this, the energy is redistributed along the polaritonic branches by means of different interaction processes (dynamical or kinematical interaction, acoustic phonon-assisted relaxation). This stage is qualitatively similar to the relaxation in inorganic microcavities. As known, there the relaxation towards the Corresponding author: Tel./fax: +7 095 334 02 24.
E-mail address: [email protected] (M. Litinskaya). 0022-2313/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2006.01.175
bottom of the LP band is hindered due to the so-called bottleneck [3] (the region near the ‘‘knee’’ of the LP branch, where the radiative escape of LPs through the cavity mirrors becomes faster than the rate of relaxation processes, since there LPs change their character from exciton-like to photon-like). The bottleneck can be overcome at very high pump densities (400 W/cm2 [4]). In organic cavities, due to the large values of the Rabi splitting energy D (80–500 meV at room temperature), intramolecular vibrations become involved into the relaxation processes, allowing for one (or two) step relaxation with large energy exchange [5,6]. This can possibly be used to overcome the bottleneck and to populate directly the photon-like part of the band. This happens on the timescale tvib about tens of picoseconds [6], which is comparable with tPL 20 ps. Polaritons are approximately bosons. It means that as soon as the population of the lowest energy state is N 0 ¼ 1, bosonic stimulation can start. This can allow eventually to collect a macroscopic population in one state, resulting in interesting physics and possible applications [7]. Organic structures with fast vibration-assisted relaxation are especially interesting in this connection. On the other hand, the pump power in J-aggregate cavities is limited by irreversible photodegradation processes. Also, the quality
ARTICLE IN PRESS M. Litinskaya, P. Reineker / Journal of Luminescence 122–123 (2007) 418–420
E
Incoherent localized excited states (IS)
min
Incoherent hybrid sates ("inc")
dðE E L ðqÞ E vib Þ, where g is the exciton–photon coupling constant, N is the total number of molecules, and jbðqÞj2 ¼ 1 jaðqÞj2 is the exciton weight in the LP state. Here qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E L ðqÞ ¼ ðE cav ðqÞ þ E 0 Þ=2 D2 þ ðE cav ðqÞ E 0 Þ2 =4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is the energy of LP (E cav ðqÞ ¼ E c 1 þ ðq=kz Þ2 is the energy of the cavity photon, kz ¼ p=Lc is the z-component of the total wave vector, Lc is the cavity thickness, pffiffiffiffi E c ¼ E cav ð0Þ ¼ ð_ckz = ec Þ, ec is the background dielectric constant). We write the system of equations for the populations N q and dN I ðEÞ, denoting respectively the number of LP states with the wave vector q, and the number of IS between E and E þ dE: Z N_ q ¼ dN I ðEÞ ð1 þ N q Þ W vib ðE; E L ðqÞÞ tL ðqÞ1 N q ,
upper polariton
q(U)
E0
E0 - Evib
q (L)
min
q
*
lower polariton (LP)
(L) qmax
419
q
dN_ I ðEÞ ¼ ain ðEÞdE
of the existing mirrors is such that the lifetime of the cavity photon inside the cavity tc is as short as 100 fs. The question thus arises if realistic pump powers can provide a LP population close to unity. We have considered a simple model situation, where the emission of the vibrational quanta is the only decay channel for ISs. The model is adequate for zero detuning between the cavity photon and the electronic transition, when the main part of the ISs is not in resonance with polaritonic states and the radiative pumping of them becomes energetically impossible, or for the case tPL btvib . The lifetime of a polariton within the mirrors tL ðqÞ ¼ tc =jaðqÞj2 (jaðqÞj2 is the photon weight in the polaritonic state) is very short. It varies for typical numbers between 2 ps for qqðLÞ max , and 200 fs for small wave vectors (at zero detuning). We neglect all polariton–polariton interaction processes, supposing that they are slower than the escape trough the mirrors. We suppose that the material of the cavity possesses an intramolecular vibration with energy E vib D. Let the energy be pumped directly into ISs and each IS relaxes into an energy matched LP state emitting one vibrational quantum. The number of incoming excitations ain ðEÞdE into the ISs between E and E þ dE is proportional to the pump intensity P and to the distribution function of IS rðEÞ, which we take to be Gaussian with a width s and the central energy E 0 : ain ðEÞ ¼ PrðEÞ=E 0 . From the microscopic theory of Ref. [6] we find the mean probability of the decay of an IS with the energy E into a LP state with the energy E L ðqÞ as W vib ðE; E L ðqÞÞ ¼ 2pg2 E 2vib jbðqÞj2 =ðN_Þ
dN I ðEÞ
q
ðLÞ qmin ,
Fig. 1. Excitation spectrum in a disordered organic microcavity. ðLÞ qmax , qðUÞ min restrict the coherent (polaritonic) part of the spectrum. At the right, the localized incoherent excited states (IS) similar to purely electronic excitations of individual molecules are drafted. Near the bottoms of the polariton bands, there are a number of incoherent states of mixed exciton–photon character (inc). Arrow shows the vibrationassisted transition from an IS to a LP state.
X
ð1 þ N q Þ W vib ðE; E L ðqÞÞ.
ð1Þ
Solving these equations in the steady-state regime, one finds Nq ¼
tL ðqÞ ain ðE L ðqÞ þ E vib Þ , f L ðE L ðqÞÞ
(2)
where f L ðE L ðqÞÞ is the LP density of states given by f L ðE L ðqÞÞ ¼ qS=ð2pÞðdE L ðqÞ=dqÞ1 : f L ðE L ðqÞÞ ¼
S E cav ðqÞk2z S k2z . 2 2 2p E c jaðqÞj 2p E c jaðqÞj2
(3)
The maximal population N ðmaxÞ gathers at the wave q vector q such that E L ðq Þ þ E vib ¼ E 0 . This population value is 2tc L2c P . (4) p3=2 s S ˚ s¼ For typical numbers (tc ¼ 100 fs, Lc ¼ 2000 A, ðmaxÞ 2 20 meV, P=S ¼ 10 W=cm ) we find: N q ¼ 0:05. The value N ðmaxÞ ¼ 1 for these parameters can be reached for q P=S ¼ 200 W=cm2 , which is just the upper possible limit for the cyanine dyes based microcavities [8]. From this estimate it is clear that special efforts to improve the quality of the microcavity mirrors are strongly desirable. Another technological direction important for non-linear processes is to produce a microcavity utilizing an organic material with large transition dipole moment, allowing for higher excitation powers. One possibility could be an anthracene crystal. In conclusion, we note that the physically most interesting lowest energy states at the very bottom of the LP band are not coherent [2]. These states, marked by inc in Fig. 1, are not characterized by a wave vector, but, in contrast to IS, they are still hybridized exciton–photon states. An interesting question is if the vibration-assisted energy relaxation from IS to these states can allow to gather a ¼ N ðmaxÞ q
ARTICLE IN PRESS 420
M. Litinskaya, P. Reineker / Journal of Luminescence 122–123 (2007) 418–420
macroscopic population in these lowest energy states, and how the population will be distributed within them. To answer this question, one has to know their wave functions and density distribution. This problem is under investigation. We thank V. M. Agranovich for many discussions. M.L. acknowledges also the support of the Alexander von Humboldt Foundation.
[2] [3] [4] [5] [6]
References [1] D.G. Lidzey, D.D.C. Bradley, M.S. Skolnick, T. Virgili, S. Walker, D.M. Whittaker, Nature 395 (1998) 53; A.I. Tartakovskii, M. Emam-Ismail, D.G. Lidzey, M.S. Skolnick, D.D.C. Bradley, S. Walker, V.M. Agranovich, Phys. Rev. B 63 (2001) 121302;
[7] [8]
N. Takada, T. Kamata, D.D.C. Bradley, Appl. Phys. Lett. 82 (2003) 1812. V.M. Agranovich, M. Litinskaya, D.G. Lidzey, Phys. Rev. B 67 (2003) 085311. F. Tassone, C. Piermarocchi, V. Savona, A. Quattropani, P. Schwendimann, Phys. Rev. B 56 (1997) 7554. A.I. Tartakovskii, D.N. Krizhanovskii, D.A. Kurysh, V.D. Kulakovskii, M.S. Skolnick, J.S. Roberts, Phys. Rev. B 65 (2002). V.M. Agranovich, H. Benisty, C. Weisbuch, Solid State Commun. 102 (1997) 631. M. Litinskaya, P. Reineker, V.M. Agranovich, J. Lumin. 110 (2004) 364. A. Kavokin, G. Malpuech, Cavity Polaritons, Elsevier, Academic Press, New York, 2003. D.G. Lidzey, private communication.
Journal of Luminescence 122–123 (2007) 418–420 www.elsevier.com/locate/jlumin
Balance between incoming and outgoing cavity polaritons in a disordered organic microcavity M. Litinskayaa,b,, P. Reinekera a
Department of Theoretical Physics, University of Ulm, 89069 Ulm, Germany b Institute of Spectroscopy RAS, Troitsk, Moscow Region, 142190 Russia Available online 10 March 2006
Abstract We examine the population of cavity polaritons at non-resonant pumping. In our model the energy relaxes by means of emission of large-energy vibrational quanta into low-energy polaritonic states, which eventually escape the cavity through the mirrors. We discuss the ways to achieve large polariton population in the low-energy states. r 2006 Elsevier B.V. All rights reserved. PACS: 71.36.+c; 71.35.Aa; 78.55.Kz Keywords: Cavity polaritons; Energy relaxation; Intramolecular vibrations
The exciton–polariton excitation spectrum in planar organic microcavities made from J-aggregates of cyanine dyes [1] has an unusual structure [2]. It consists of two coexisting parts (see Fig. 1). Coherent (i.e., characterized by a well-defined in-plane wave vector) polaritonic states are formed for restricted intervals of the wave vector, e.g., ðLÞ for qðLÞ min oqoqmax for the lower branch polaritons (LP). The other excited states are incoherent (IS) and are similar to the electronic excitation of individual J-aggregates uncoupled with light. Their number N inc exceeds strongly the number of polaritonic states, and this determines the decisive role of IS in energy relaxation processes. Under non-resonant excitation of the microcavity, the energy relaxes mainly into IS. From there it can be transferred to polaritonic states. One possibility is that the ISs decay radiatively within the time tPL , and the emitted light is absorbed by polaritons. After this, the energy is redistributed along the polaritonic branches by means of different interaction processes (dynamical or kinematical interaction, acoustic phonon-assisted relaxation). This stage is qualitatively similar to the relaxation in inorganic microcavities. As known, there the relaxation towards the Corresponding author: Tel./fax: +7 095 334 02 24.
E-mail address: [email protected] (M. Litinskaya). 0022-2313/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2006.01.175
bottom of the LP band is hindered due to the so-called bottleneck [3] (the region near the ‘‘knee’’ of the LP branch, where the radiative escape of LPs through the cavity mirrors becomes faster than the rate of relaxation processes, since there LPs change their character from exciton-like to photon-like). The bottleneck can be overcome at very high pump densities (400 W/cm2 [4]). In organic cavities, due to the large values of the Rabi splitting energy D (80–500 meV at room temperature), intramolecular vibrations become involved into the relaxation processes, allowing for one (or two) step relaxation with large energy exchange [5,6]. This can possibly be used to overcome the bottleneck and to populate directly the photon-like part of the band. This happens on the timescale tvib about tens of picoseconds [6], which is comparable with tPL 20 ps. Polaritons are approximately bosons. It means that as soon as the population of the lowest energy state is N 0 ¼ 1, bosonic stimulation can start. This can allow eventually to collect a macroscopic population in one state, resulting in interesting physics and possible applications [7]. Organic structures with fast vibration-assisted relaxation are especially interesting in this connection. On the other hand, the pump power in J-aggregate cavities is limited by irreversible photodegradation processes. Also, the quality
ARTICLE IN PRESS M. Litinskaya, P. Reineker / Journal of Luminescence 122–123 (2007) 418–420
E
Incoherent localized excited states (IS)
min
Incoherent hybrid sates ("inc")
dðE E L ðqÞ E vib Þ, where g is the exciton–photon coupling constant, N is the total number of molecules, and jbðqÞj2 ¼ 1 jaðqÞj2 is the exciton weight in the LP state. Here qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E L ðqÞ ¼ ðE cav ðqÞ þ E 0 Þ=2 D2 þ ðE cav ðqÞ E 0 Þ2 =4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is the energy of LP (E cav ðqÞ ¼ E c 1 þ ðq=kz Þ2 is the energy of the cavity photon, kz ¼ p=Lc is the z-component of the total wave vector, Lc is the cavity thickness, pffiffiffiffi E c ¼ E cav ð0Þ ¼ ð_ckz = ec Þ, ec is the background dielectric constant). We write the system of equations for the populations N q and dN I ðEÞ, denoting respectively the number of LP states with the wave vector q, and the number of IS between E and E þ dE: Z N_ q ¼ dN I ðEÞ ð1 þ N q Þ W vib ðE; E L ðqÞÞ tL ðqÞ1 N q ,
upper polariton
q(U)
E0
E0 - Evib
q (L)
min
q
*
lower polariton (LP)
(L) qmax
419
q
dN_ I ðEÞ ¼ ain ðEÞdE
of the existing mirrors is such that the lifetime of the cavity photon inside the cavity tc is as short as 100 fs. The question thus arises if realistic pump powers can provide a LP population close to unity. We have considered a simple model situation, where the emission of the vibrational quanta is the only decay channel for ISs. The model is adequate for zero detuning between the cavity photon and the electronic transition, when the main part of the ISs is not in resonance with polaritonic states and the radiative pumping of them becomes energetically impossible, or for the case tPL btvib . The lifetime of a polariton within the mirrors tL ðqÞ ¼ tc =jaðqÞj2 (jaðqÞj2 is the photon weight in the polaritonic state) is very short. It varies for typical numbers between 2 ps for qqðLÞ max , and 200 fs for small wave vectors (at zero detuning). We neglect all polariton–polariton interaction processes, supposing that they are slower than the escape trough the mirrors. We suppose that the material of the cavity possesses an intramolecular vibration with energy E vib D. Let the energy be pumped directly into ISs and each IS relaxes into an energy matched LP state emitting one vibrational quantum. The number of incoming excitations ain ðEÞdE into the ISs between E and E þ dE is proportional to the pump intensity P and to the distribution function of IS rðEÞ, which we take to be Gaussian with a width s and the central energy E 0 : ain ðEÞ ¼ PrðEÞ=E 0 . From the microscopic theory of Ref. [6] we find the mean probability of the decay of an IS with the energy E into a LP state with the energy E L ðqÞ as W vib ðE; E L ðqÞÞ ¼ 2pg2 E 2vib jbðqÞj2 =ðN_Þ
dN I ðEÞ
q
ðLÞ qmin ,
Fig. 1. Excitation spectrum in a disordered organic microcavity. ðLÞ qmax , qðUÞ min restrict the coherent (polaritonic) part of the spectrum. At the right, the localized incoherent excited states (IS) similar to purely electronic excitations of individual molecules are drafted. Near the bottoms of the polariton bands, there are a number of incoherent states of mixed exciton–photon character (inc). Arrow shows the vibrationassisted transition from an IS to a LP state.
X
ð1 þ N q Þ W vib ðE; E L ðqÞÞ.
ð1Þ
Solving these equations in the steady-state regime, one finds Nq ¼
tL ðqÞ ain ðE L ðqÞ þ E vib Þ , f L ðE L ðqÞÞ
(2)
where f L ðE L ðqÞÞ is the LP density of states given by f L ðE L ðqÞÞ ¼ qS=ð2pÞðdE L ðqÞ=dqÞ1 : f L ðE L ðqÞÞ ¼
S E cav ðqÞk2z S k2z . 2 2 2p E c jaðqÞj 2p E c jaðqÞj2
(3)
The maximal population N ðmaxÞ gathers at the wave q vector q such that E L ðq Þ þ E vib ¼ E 0 . This population value is 2tc L2c P . (4) p3=2 s S ˚ s¼ For typical numbers (tc ¼ 100 fs, Lc ¼ 2000 A, ðmaxÞ 2 20 meV, P=S ¼ 10 W=cm ) we find: N q ¼ 0:05. The value N ðmaxÞ ¼ 1 for these parameters can be reached for q P=S ¼ 200 W=cm2 , which is just the upper possible limit for the cyanine dyes based microcavities [8]. From this estimate it is clear that special efforts to improve the quality of the microcavity mirrors are strongly desirable. Another technological direction important for non-linear processes is to produce a microcavity utilizing an organic material with large transition dipole moment, allowing for higher excitation powers. One possibility could be an anthracene crystal. In conclusion, we note that the physically most interesting lowest energy states at the very bottom of the LP band are not coherent [2]. These states, marked by inc in Fig. 1, are not characterized by a wave vector, but, in contrast to IS, they are still hybridized exciton–photon states. An interesting question is if the vibration-assisted energy relaxation from IS to these states can allow to gather a ¼ N ðmaxÞ q
ARTICLE IN PRESS 420
M. Litinskaya, P. Reineker / Journal of Luminescence 122–123 (2007) 418–420
macroscopic population in these lowest energy states, and how the population will be distributed within them. To answer this question, one has to know their wave functions and density distribution. This problem is under investigation. We thank V. M. Agranovich for many discussions. M.L. acknowledges also the support of the Alexander von Humboldt Foundation.
[2] [3] [4] [5] [6]
References [1] D.G. Lidzey, D.D.C. Bradley, M.S. Skolnick, T. Virgili, S. Walker, D.M. Whittaker, Nature 395 (1998) 53; A.I. Tartakovskii, M. Emam-Ismail, D.G. Lidzey, M.S. Skolnick, D.D.C. Bradley, S. Walker, V.M. Agranovich, Phys. Rev. B 63 (2001) 121302;
[7] [8]
N. Takada, T. Kamata, D.D.C. Bradley, Appl. Phys. Lett. 82 (2003) 1812. V.M. Agranovich, M. Litinskaya, D.G. Lidzey, Phys. Rev. B 67 (2003) 085311. F. Tassone, C. Piermarocchi, V. Savona, A. Quattropani, P. Schwendimann, Phys. Rev. B 56 (1997) 7554. A.I. Tartakovskii, D.N. Krizhanovskii, D.A. Kurysh, V.D. Kulakovskii, M.S. Skolnick, J.S. Roberts, Phys. Rev. B 65 (2002). V.M. Agranovich, H. Benisty, C. Weisbuch, Solid State Commun. 102 (1997) 631. M. Litinskaya, P. Reineker, V.M. Agranovich, J. Lumin. 110 (2004) 364. A. Kavokin, G. Malpuech, Cavity Polaritons, Elsevier, Academic Press, New York, 2003. D.G. Lidzey, private communication.