Bose glass and superfluid phase transitions of exciton–polaritons in GaN microcavities

Solid State Communications 144 (2007) 390–394 www.elsevier.com/locate/ssc

Bose glass and superfluid phase transitions of exciton–polaritons in GaN microcavities D.D. Solnyshkov a,∗ , H. Ouerdane a , M.M. Glazov b , I.A. Shelykh c , G. Malpuech a a LASMEA, CNRS-Universit´e Blaise Pascal, 24 Avenue des Landais, 63177 Aubi`ere Cedex, France b A.F. Ioffe Physico-Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia c ICCMP, Universidade de Bras`ılia, 70919-970 Bras`ılia-DF, Brazil

Received 6 March 2007; accepted 1 July 2007 by the Guest Editors Available online 30 August 2007

Abstract Microcavity exciton–polaritons within GaN-based structures are the object of the present work. The impact of the structural imperfections on the properties of the two-dimensional polariton gas is investigated through the calculation of its phase diagram. We demonstrate that the presence of disorder first induces a quasi-phase transition of the polariton system towards a Bose-glass phase before it becomes superfluid as its density increases. Calculations of the density of states as well as the condensate wavefunction and the related spectrum of elementary excitations in the framework of the Gross–Pitaevskii theory provide further insight into the properties of exciton–polaritons in GaN-based microcavities. c 2007 Elsevier Ltd. All rights reserved.

PACS: 71.36.+c; 71.35.Lk; 03.75.Mn Keywords: A. Disorder; A. GaN; D.Microcavity polaritons; D. Phase diagram

1. Introduction GaN-based microcavities are currently the object of intense theoretical and experimental research activities [1–3] since there is great hope that they will be at the basis of the next generation of coherent light emitters. Excitons in GaN-based quantum wells (QW) have a very large binding energy as well as a large oscillator strength which favors strong coupling to light. Moreover, because of their light effective mass (typically 10−4 times the free electron mass) polaritons show extremely small critical density and high critical temperatures that can be larger than room temperature in some cases. These properties clearly make GaN-based systems promising candidates for the demonstration of the feasibility of a polariton laser. However, GaN, as a material that belongs to the nitride subgroup of the III–V compounds, is also known for being very problematic as far as epitaxy of high quality monolayers is concerned because of its combination of electronic, chemical, thermal and ∗ Corresponding author.

E-mail address: [email protected] (D.D. Solnyshkov). c 2007 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2007.07.046

mechanical properties [1]. The samples produced may suffer from strong disorder induced by structural imperfections and it is therefore of great importance to investigate its impact on the properties of the polariton gas within GaN-based microcavities. Recent work [4,5] shows that when the condensation temperature is of the order of the Rabi splitting energy, the composite nature of polaritons should not be neglected. This result is based on the use of a Dicke Hamiltonian [6] derived first for atomic systems and hence, in principle, relevant for the study of quantum dots in cavities. But in Refs. [4,5,7] Dicke’s model is nonetheless applied to polaritons in a planar cavity. We believe that essential information about microcavity polaritons can be derived from models that treat them as weakly interacting structureless bosons in equilibrium [8,9]. In fact, they have a short radiative life time (∼10−12 s) that in principle makes the system non-equilibrium, but stimulated scattering towards their ground state has been demonstrated [10,11], and quasi-thermal equilibrium was recently reported [12]. Within this framework phase diagrams of clean systems have been established [13,14]. Non-equilibrium properties of the polariton gas induced by various dephasing processes (interactions with phonons and scattering by impurities) as well as the necessary

D.D. Solnyshkov et al. / Solid State Communications 144 (2007) 390–394

external pumping to sustain a steady state have been treated in Ref. [7]. While the observation of a Kosterlitz–Thouless (KT) phase transition towards superfluidity [15] can be expected, no signature of it has been reported thus far. This situation originates from the disorder induced by structural imperfections of the sample to which the polariton system is subjected. In fact, it is the strong localization of the condensate, characteristic of a Bose-glass phase [16], which is indeed observed as evidenced by experimental data [12]. In a recent work [17], we established the phase diagram of polaritons in a disordered CdTe cavity, predicting that with increasing density the polariton system first enters the Bose-glass phase before it becomes superfluid. The Bose-glass picture was found to be in good agreement with experimental data [12] obtained for a CdTe sample for which the disorder energy was of the order of 1 meV. In the case of GaN-based system, the disorder energy is at least 10 meV and it is therefore crucial to perform simulations that yield relevant information on the impact of disorder on the phase diagram of GaN cavity polaritons. To this end, we present a new phase diagram for GaN cavity polaritons and calculate the condensate wavefunctions as well as the spectra of elementary excitations from the Gross–Pitaevskii equation [18] including disorder. The paper is organized as follows: in Section 2, our approach to the study of the polariton system in a disordered medium is presented; then, the related numerical results are discussed in Section 3. The object of Section 4 is the calculation of the polariton phase diagram in disordered GaN-based microcavity, which is discussed before we give our conclusions in Section 5. 2. Model of the polariton gas in a disordered medium In this section we give the necessary information about our model presented in Ref. [17]. The polaritons are moving in a random potential V (r) generated by uncontrolled structural imperfections of the sample. The mean amplitude and root mean square fluctuation are given by hV (r)i = 0 and p 2 hV (r)i = V0 q respectively. The correlation length of this R potential is R0 = hV (r)V (0)idr/V02 . According to Ref. [19], two types of polariton states can be defined in a disordered sample: the free propagating states and the localized states with energy E < E c , where E c is the critical “delocalization” energy. The localization radius scales like a(E) ∝ a0 V0s /(E c − E)s , s being a critical index q and a0 = h¯ 2 /mV0 [20]. In two dimensions E c is of the order of mean potential energy (i.e. 0 in our case), and s ≈ 0.75 [19]. The quasi-classical density of states is D0 (E) ∼ = M/4π h¯ 2 [1 + erf(E/V0 )] [21]. It is shown in Fig. 1 together with the localization radius. One needs to note that because of the photonic component, it is not possible to trap a polariton in a state with a localization radius smaller than the wavelength of the incident light, and hence the resulting effective density of states shows an abrupt cut-off at E = E 0 which is selfconsistently determined from a(E 0 ) = ac : D(E) = D0 (E) for E > E 0 and D(E) = 0 otherwise. Even in the presence of disorder, BEC cannot strictly speaking take place for cavity polaritons, but it is possible

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Fig. 1. (Color online) Localization radius (black) and density of states (red) as functions of energy.

to define a quasi-phase transition which takes place in finite systems as shown in Ref. [13], and the critical density is given by the total number of polaritons which can be accommodated in all the energy levels of the disorder potential V (r) except the ground one [13]: X n c (T ) = f B (E i , E 0 , T ), (1) i6=0

where f B (T, µ, T ) is the Bose–Einstein distribution function, µ is the chemical potential. To evaluate the critical density n c (T, R), the discrete sum is replaced by an integral in Eq. (1), and, assuming D(E) is a smooth function, we find n c (T ) ≈ D(E 0 )kB T ln[1/(1 − eδ/kB T )], where δ represents the typical energy difference between the ground and excited states of the finite-size system. Above this density all additional particles accumulate in the ground state and the concentration of condensed particles n 0 satisfies n 0 ≥ (n−n c ), where n is the total density of polaritons. It is not a real phase transition since the system has a discrete energy spectrum and the value of the chemical potential never becomes strictly equal to E 0 . Interactions between particles start to become dominant once the polaritons begin to accumulate in the ground state. The situation can be qualitatively described as follows: particles start to fill the lowest energy state which is therefore blue shifted because of interactions (µ − E 0 > 0). Thus, for some occupation number of the condensate the chemical potential reaches the energy of another localized state, and this state starts in turn to populate and to blue shift. The condensate, like a liquid, fills several minima of the potential. This gives rise to the spatial and reciprocal space pictures of Refs. [12, 15]. A few localized states, covering about 20% of the surface of the emitting spot are all emitting light at the same energy and are strongly populated. This characterizes a Bose-glass [16] phase, which exists up to the achievement of the condition µ = E c . This condition should be accompanied by a percolation of the condensate which at this stage should cover 50% or more of the sample (in the semiclassical representation). The delocalized condensate becomes a KT superfluid. More

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precisely, the different sides of the finite-size system are linked by the phase coherent path. Therefore we predict two quasiphase transitions driven by temperature and particle density: first, with an increase in the polariton density beyond n c (T ) the system enters the Bose-glass phase, then with a further increase in the density the polariton system becomes superfluid. The critical condition µ = E c is valid only at low temperature where the thermal depletion of the condensate is negligible. The quantitative analysis can be carried out in the framework of the Gross–Pitaevskii equation for the exciton–polariton condensate wavefunction Ψ (r, t) which reads ! ∂ h¯ 2 2 ih¯ Ψ (r, t) = − 4 + V (r) + g |Ψ (r, t)| Ψ (r, t), (2) ∂t 2M where g is a constant characterizing the weak repulsive interaction between polaritons. We use the exciton–polariton basis which has proven to be a good approximation for CdTe cavities. A more precise approach is to work in exciton–photon basis [22,23], but in our case it is not necessary since we consider the states close to k = 0. To calculate the quasiparticle spectra, we introduce a single-particle Green’s function which takes the form G ω (r, r0 ) =

X Ψ j (r)Ψ Ďj (r0 ) j

h¯ ω − E j

,

(3)

where E j and Ψ j (r) are energies and eigenfunctions of the elementary excitations [24], found numerically from Eq. (2). It is also instructive to analyze both the variation in the emission pattern and the quasi-particle spectrum in comparison with the behavior of the superfluid fraction of the polariton system. The latter quantity can be calculated using the twisted boundary condition method [25]. Imposing such boundary conditions implies that the condensate wavefunction acquires a phase between the boundaries, namely Ψθ (r + Li ) = eiθ Ψθ (r),

(4)

where Li (i = x, y) are the vectors which form the rectangle confining the polaritons and θ is the twisting parameter. The superfluid fraction of the condensate is given by [25] fs =

ns 2M L 2 (µθ − µ0 ) , = lim θ →0 n h¯ 2 nθ 2

(5)

where µθ is the chemical potential corresponding to the boundary conditions Eq. (4) and µ0 is the chemical potential corresponding to the periodic boundary conditions (θ = 0). Eqs. (2) and (4) allow us to study the depletion of the superfluid fraction for arbitrary disorder, but in the high temperature regime, since thermal depletion of the condensate is a much more efficient process than the one induced by the disorder, one can treat the disorder perturbatively and obtain an analytical expression for the n dn as a response of the system for the motion of the disorder potential (which acts as a wall for the superfluid in the capillary). The correction to the condensate wavefunction Ψ (r, t) due to the perturbation V (r − vt) is

denoted δΨ (r, t) and satisfies (to the lowest order in V ): ! 2 h ¯ 2 E 4 − µ + 2gΨ0 δΨ (r, t) ih¯ v · ∇δΨ (r, t) = − 2M + gΨ02 δΨ ∗ (r, t) + V (r)Ψ0 ,

(6)

whose Fourier transform is: ! h¯ 2 k 2 −h¯ k · v + + µ δΨk + µ(δΨ−k )∗ + Vk Ψ0 = 0, (7) 2M since δΨk∗ = (δΨ−k )∗ . The complex conjugate of Eq. (7) is simply obtained by the substitution of k into −k and vice versa. The correction δΨk is then readily found:
Vk h¯ k · v + h¯ 2 k 2 /2M Ψ0 δΨk = − 2 . (8)
h¯ k 2 /2M h¯ 2 k 2 /2M + 2µ − (h¯ k · v)2 Taking v → 0, the part of the normal component of the polariton fluid due to disorder can be written as: Z X E 2 (k) S d ∗ nn = µhVk V−k idk, (9) hδΨk δΨk i = 4π 2 g 4 (k) k q where (k) = (E(k) + µ)2 − µ2 is the Bogoliubov’s dispersion, E(k) = h¯ 2 k 2 /2M is the bare polariton dispersion, and h. . .i denotes averaging over disorder. Note that Eq. (9) has exactly the same form as the expression found by other means in Ref. [26]. 3. Numerical results The solution of the Gross–Pitaevskii equation may take the following form Ψ (r, t) = Ψ0 (r) exp (−iµt/h¯ ). Top panels (a), (b), and (c) of Fig. 2 show the real space distribution of the polaritons obtained from the solution of the Gross–Pitaevskii equation. We choose to consider a GaN microcavity similar to the one of [3], but containing Nqw = 18 instead of 6 in the existing structure. The expected Rabi Splitting is therefore of the order of 60 meV. The polariton mass is m = 3 × 10−5 m 0 , where m 0 is the free electron mass, and the interaction constant g = 3E b aB2 /Nqw , where E b is the exciton binding energy (about 50 meV in thin GaN QWs), ˚ is the exciton Bohr radius and Nqw = 18 is aB = 20 A the number of wells embedded in the microcavity. Although the excitonic component of the exciton–polariton can be much stronger localized than the photonic one [27], it is the photonic disorder that is manifested experimentally in the emission from the localized condensate [12,28]. In our calculations we have included an effective random disorder potential with V0 = 5 meV and R0 = 4 µm (typical photonic disorder scale). Fig. 2(a) corresponds to the non-condensed situation. The spatial profile is given all P by statistical averaging over 2 . Here occupied states, n(r) = f (E , T, µ(T ))|Ψ (r)| B j j j the temperature is set to T = 300 K. In this case the total number of particles is small and thus nonlinear terms in Gross–Pitaevskii equation can be neglected. Even though the

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Fig. 2. (Color online) Spatial images (top panels) and quasi-particle spectra (bottom panels) for a realistic disorder potential. The figures shown correspond to densities 0, 3 × 1012 and 9 × 1012 cm−2 . The colormap is the same for all three top panels.

on the lower panels of Fig. 2. Only the upper Bogoliubov branch is shown. The left panel 2(d) shows the excitations which correspond to the localized states. The middle panel 2(e) shows parabolic dispersion with a flat part produced by the localization of the condensate. The Bogoliubov spectrum on the right panel 2(f) is the distinct feature of the superfluid state of the system. Using the twisted boundary conditions technique described in Section 2, we have obtained that the superfluid fraction is 10% for the Bose-glass phase shown on Fig. 2(b),(e) and 80% for the superfluid phase (Fig. 2(c),(f)). 4. Phase diagram of GaN cavity polaritons

Fig. 3. (Color online) Polariton phase diagram for a GaN microcavity containing 18 QWs. The horizontal and vertical dashed lines show the limiting temperatures and densities where the strong coupling holds. The lower solid line shows the critical density for the transition from normal to Bose-glass phase. The upper solid line shows the critical density for the transition from the Bose glass to the superfluid phase. The dashed part of the line shows the temperature range where the validity of our approximations ceases.

disorder is relatively strong, at this temperature it does not affect the spatial distribution, which stays rather uniform. Once the quasi-condensate is formed, and for moderate temperatures, one can neglect the thermal occupation of the excited states and the spatial image of the polariton distribution is given by the ground state wavefunction which corresponds to solution of Eq. (2). We show the resulting density below and above the percolation threshold on Fig. 2(b) and (c) respectively. As expected, the condensate is localized in a few minima of the random potential as shown on Fig. 2(b). On Fig. 2(c) the condensate wave function still exhibits some spatial fluctuations connected to disorder, but the condensate is nonetheless well delocalized, covering the whole sample area. The spectrum of elementary excitations is given by the poles of the Green’s function in the (k, ω) representation, and shown

We concentrate now on the calculation of the cavity polariton phase diagram. As in previous works [17], a temperature and density domain where the strong coupling is supposed to hold, is roughly defined. The limits are shown on Fig. 3 as thick dotted lines.1 The transition from normal to Bose-glass phase can be calculated from Eq. (1) and a realistic realization of disorder. The lower solid line on Fig. 3 shows n c (T ) for the same realization of disorder as for Fig. 1. The free polariton dispersion is calculated using the geometry of Ref. [3]. We now calculate the density for the transition between the Bose glass and the superfluid phase. In the low temperature domain, this density is approximately given by the percolation threshold µ = E c and does not depend significantly on temperature. This condition corresponds with good accuracy to the abrupt change in the superfluid fraction f s . However, at higher temperature the thermal depletion of the condensate becomes the dominant effect. In that case the chemical potential of the condensate is much higher than the percolation energy 1 The edge temperature is assumed to be 550 K equal to the exciton binding energy. The maximum polariton density is taken 36 times larger than bleaching exciton density which is assumed to be 5 × 1011 cm−2 in GaN QWs.

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E c and the depletion induced by disorder can be neglected compared to the thermal depletion of the superfluid. The normal density then reads Z ∂ f B [(k), µ = 0, T ] 2 n 0n (T ) = − dk, (10) E(k) 2 ∂ (2π ) and the superfluid density in the system is given by n s (T ) = n − n 0n (T ),

(11)

which can be substituted into the Kosterlitz–Nelson formula [29] to obtain a self-consistent equation for the transition temperature: h¯ 2 πn s (TK T ) . (12) 2M The joint solution of Eqs. (12) and (11) allows us to determine the superfluid phase transition temperature TK T (n). The result of this procedure is shown on Fig. 3. Below 350 K the critical density is given by the percolation threshold and there is no temperature dependence. Above 600 K the superfluid depletion is determined solely by the thermal effects. In the intermediate regime the crossover between the thermal and disorder contributions takes place and our approximations are no longer justified. We also find that the superfluid transition takes place very close to the weak to strong coupling threshold. This suggests that the experimental observation of this phenomenon remains a great challenge, however it may still be feasible. TK T =

5. Conclusion We have established the phase diagram of disordered GaN cavity polaritons. The disorder is modelled as a random potential accounting for the effect of structural imperfections of the sample. We found that with increasing density the polariton system first enters the Bose-glass phase before becoming superfluid. The Bose-glass picture is in good agreement with recent experimental data [12]. The condensate wavefunctions as well as the spectra of elementary excitations were obtained from the solutions of the Gross–Pitaevskii equation including disorder. We also showed the density of states and localization radius. We can conclude that while disorder strongly affects the occurrence of the superfluid phase transition, the presence of disorder has no significant impact on the occurrence of a bosonic phase transition for polaritons. This explains why this phenomenon has been observed in a rather disordered system like CdTe, and there is an experimental evidence suggesting such phase transition also in even more disordered system like GaN [28]. It should be stressed that the superfluid

threshold obtained in the present work remains below the Mott transition threshold, and so the observation of superfluidity of polariton condensates in GaN remains a feasible, though rather challenging task. Acknowledgments We acknowledge the support of the STREP “STIMSCAT” 517769 and the Chair of Excellence program of the ANR. References [1] B. Gil (Ed.), Low-Dimensional Nitride Semiconductors, Oxford Science Publications, 2002. [2] R. Butt´e, E. Feltin, J. Dorsaz, G. Christmann, J.-F. Carlin, N. Grandjean, M. Ilegems, Jpn. J. Appl. Phys. Part 1 44 (2005) 7207. [3] G. Christmann, D. Simeonov, R. Butt´e, E. Feltin, J.-F. Carlin, N. Grandjean, Appl. Phys. Lett. 89 (2006) 261101. [4] J. Keeling, P.R. Eastham, M.H. Szymanska, P.B. Littlewood, Phys. Rev. Lett. 93 (2004) 206403. [5] J. Keeling, P.R. Eastham, M.H. Szymanska, P.B. Littlewood, Phys. Rev. B 72 (2005) 115320. [6] R.H. Dicke, Phys. Rev. 93 (1954) 99. [7] M.H. Szymanska, J. Keeling, P.B. Littlewood, Phys. Rev. Lett. 96 (2006) 230602. [8] A. Kavokin, G. Malpuech, Cavity Polaritons, Elsevier, 2003. [9] A. Imamoglu, J.R. Ram, Phys. Lett. A 214 (1996) 193. [10] L.S. Dang, et al., Phys. Rev. Lett. 81 (1998) 3920. [11] H. Deng, et al., Science 298 (2002) 199. [12] J. Kasprzak, et al., Nature 443 (2006) 409. [13] G. Malpuech, et al., Microcavities, in: J.J. Baumberg, L. Vi˜na (Eds.), Semicond. Sci. Technol. S 395 (2003) (special issue). [14] J. Keeling, Phys. Rev. B 74 (2006) 155325. [15] J.M. Kosterlitz, D.J. Thouless, J. Phys. C 6 (1973) 1181. [16] M.P.A. Fisher, P.B. Weichman, G. Grinstein, D.S. Fisher, Phys. Rev. B 40 (1989) 546. [17] G. Malpuech, D.D. Solnyshkov, H. Ouerdane, M.M. Glazov, I. Shelykh, Phys. Rev. Lett. 98 (2007) 206402. [18] L. Pitaevskii, S. Stringari, Bose–Einstein Condensation, Oxford University Press, 2003. [19] A.L. Efros, B.I. Shklovskii, Electronic Properties of Doped Semiconductors, Springer, Heidelberg, 1989. [20] Here and below we disregard the non-parabolicity effects on the cavity polariton dispersion which are known to be small provided the temperature and polariton number is not too high [14]. [21] We disregard the spin of polaritons and omit the spin degeneracy factor here and below. [22] Iacopo Carusotto, Cristiano Ciuti, Phys. Rev. Lett. 93 (2004) 166401. [23] I.A. Shelykh, et al., Phys. Rev. Lett. 97 (2006) 066402. [24] E.M. Lifshitz, L.P. Pitaevskii, Statistical Physics, Part 2, Pergamon Press, New York, 1980. [25] A.J. Leggett, Phys. Rev. Lett. 25 (1970) 1543. [26] O.L. Berman, Y.E. Lozovik, D.W. Snoke, R.D Coalson, Phys. Rev. B 70 (2004) 235310. [27] V.M. Agranovich, Yu.N. Gartstein, Phys. Rev. B 75 (2007) 075302. [28] S. Christopoulos, et al., Phys. Rev. Lett. 98 (2007) 126405. [29] D.R. Nelson, J.M. Kosterlitz, Phys. Rev. Lett. 39 (1977) 1201.