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Biexcitons and dark states in semiconductor microcavities
JOURNAL
OF
LUMINESCENCE Journal
ELSEWIER
Biexcitons
of Lummescence
76&77
II9981
Ihl
I67
and dark states in semiconductor
microcavities
Abstract We analyze the formation of two-dimensional biexcitons and the role of cxciton dark states decoupled from the cavity mode in semiconductor microcavities containing a few resonating quantum wells. A general formalism is presented for the treatment of excitons. photon modes and biexcitons in resonating microcavities. Scattering by disorder is also considered. The interface disorder seen by cavity polaritons is shown to be reduced by averaging along the cavity axis. The spectral narrowing of cavity polariton lines is discussed. Results for the biexciton binding energy. radius and damping are discussed in the physically relevant cases. The cavity biexciton photoluminescence spectra arc found to bc qualitatively different depending on the relative magnitudes of the exciton-cavity detuning. the bicxciton bindinpcncrgy. the exciton photon coupling and the damping. (‘ 1998 Elsevier Science B.V. All rights reserved. K~~~~~vo~cls: Biexcitons;
Dark states: Microcavities
1. Introduction In the present paper, we address the problem of the formation of 2D biexcitons in cavity embedded quantum wells (in the following, we will refer to them as cavity biexcitons). As multiple quantum wells are frequently used to reach the strong coupling regime [l-2]. we explicitly take into account also the exciton dark states which are orthogonal to the cavity polaritons. The influence of polaritonic effects on quantum well biexcitons has been recently considered [3] and, in particular. an increase of the binding energy has been found. The presence of the resonant microcavity drastically alters the photon density of states and brings about a completely different physical picture, affecting, in particular. the biexciton luminescence. Depending *Corresponding
2. General formalism The system we consider comprises a few equivalent quantum wells (not electronically coupled) embedded in a i/2 resonating microcavity. The 2D exciton and photon eigenmodes are taken to be polarized perpendicular to the growth direction. The fundamental mode of the ideal planar resonator is given as a function of the 2D wave vector k by w,(k) = w,(0)v,’ 1 +
(kt!7r)Z,
(1)
where L is the effective cavity thickness and (e),(O) = (cz.‘nL) is the cutoff frequency. II being the
author
()(p?-~i13 9X $19.00 t PII S002~-2’I3(97~00143-9
on the values of the exciton-cavity detuning. the exciton-exciton interaction, the exciton-photon coupling and the damping. we predict qualitatively different spectra.
1998 Elselier
Science B.V. All rights reserved
V.M. Agranmich
162
background index dispersion is taken
et al. / Journal
of refraction. The 2D exciton to be parabolic:
ho,(k) = hw,(O) + g,
(2)
where M is the total mass for in-plane motion and at resonance o,(O) = w,(O). No distinction is made between T and L exciton modes that selectively couple to s and p polarized light, respectively [46]. For in-plane polarized excitations (such as the 2D heavy hole excitons of GaAs), no photon density of state is available for radiative decay below the cutoff frequency o,(O). As a consequence, the physics changes altogether with respect to the case in which no cavity is present [7]. All quantum wells are taken to be equivalent and placed at the cavity center, because each of them is approximately affected by the same value of the electric field when the total width of the multiple quantum-well structure is small compared to the light wavelength in the cavity. The Hamiltonian of the exciton-photon system is Ho = 1
hw,(k)&
k
a
by
c
VQ
BikBztq&k-QBolq+Q,
(4)
n.k.q.Q
where .d is a normalization factor (in plane number of unit cells) and VQ = V(lQl) the Fourier transform of the effective attractive interaction V(r) between excitons having the appropriate spin combination
77 (1998)
161-167
[3]. The potential V can be chosen to have a convenient analytical form such to reproduce the bare 2D biexciton binding energy EE as obtained from accurate microscopic calculations [S]. The B’s are assumed to satisfy the boson commutation relations, which is a valid approximation for a low exciton density. The total Hamiltonian H = Ho + H,, includes both the cavity polariton effects and the exciton-exciton interactions. In the following, the quadratic Hamiltonian Ho is diagonalized exactly and the hamiltonian H,, is then written in terms of the new operators corresponding to the eigenstates of Ho, i.e. the cavity polaritons and dark excitons. First of all, it is convenient to change the basis of exciton states to isolate the totally symmetric combination blk = (Blk + Bzk + ... + Bwk)/fi which is the only one coupled to the cavity mode, the remaining N - 1 new exciton states bik being orthogonal to blk bJk =xB3fkT,,,i, 1 where
+ 1 !?m,(k)B,t&,
where the index c( = 1, 2 ,.... N, labels the N different quantum wells, the B’s and a’s are, respectively, 2D exciton and cavity photon operators and 6 is proportional to the transition dipole matrix element (taken to be independent of k and CC) and gives the single well Rabi splitting at resonance. We use a model description of the many-body interactions responsible for the biexciton formation: each exciton is treated as a point particle interacting with the others in the same quantum well through a two-body effective potential given
He, = ;
cf Luminescence 76&
x,j
= 1,2 ,...,
T is an orthogonal
N,
matrix
(5) and for all x’s
T 2.1 = l/J’%. Physically, the bj’s represent exciton states delocalized over the N quantum wells according to specific coherent linear combinations of the original single-well exciton states. In the new basis, the entire oscillator strength is taken up by the b, optically active state and all other b states are dark, as for a linear chain of oscillators [9]. Then, Ho takes the form H,, = c k
-
h,(k)&&
+ hw,(k)b:kblk
c
S(bIkak
+
b,kaL)
+
C j>
h~~e(k)bjkbjk 1
3
(6)
1
where A = 6fi is the Rabi splitting at resonance and it is sufficient to take the appropriate linear . combmations of uk and b,,, i.e. the upper- and lower-cavity polariton States P1k and P2k, to COmplete the diagonalization. This is accomplished by setting hi, = F:(k)p,, uk
=
F(;(k)Plk
+ F:(k)P,,> +
F:(k)Pzk,
(7)
where the F’s (real, satisfying Fh,Fi + F”,F”, = d,,) are obtained solving the system of eigenvalue equations (to,(k) - lg,j(k))F;(k)
- a;(k)
= 0
= ((tjc(k) - Opj(k))Fy(k) - $Ft(k),
(8)
withj = 1, 2 and wP,, wPz giving the upperlower-cavity polariton dispersion laws tn,,,.#)
=
tM)
+ (o,(k) 2
_t ;\.(w,(k) Finally,
and
- to,(k))’ + AZ/h’.
Ho is written
in diagonal
(9)
=
f ~Jiml_ q.k.
(16)
'2
The formation of cavity biexcitons is formally described by the total Hamiltonian as given in Eq. (15) fully including the exciton-photon interaction. The predictions resulting from the present model are discussed in Section 4. In the following, we will consider the significance of the dark excitons and of the scattering by disorder or phonons.
form as 3. Dark states and phonon or disorder scattering
Now, H,, can be expressed in terms of the eigenstates of H,,, making use of the transformation
+
F?(k)PAL)+ 1 Tz.jbjk.
(11)
j=2
In order to simplify to define (‘lh = .i
=
Pla-
(‘2.
54 ,...,
=
the notations,
(‘Jk
PZk,
E
b(j-
I)k.
N + 1,
S,.,(k) = T,. 1f?Cd,
(12) S,.,(k) 5 T,. , F;(k)>
S,,i(k) E T,.j-,,j
= 3,4,...,N
E,(k) = htupl(k),
E,(k) E ho,,(k),
Ej(k) s h,(k),
it is convenient
+ 1,
(13)
j = 3.4, . . . , N + 1.
The total Hamiltonian
can then be written
(14) as
The dark excitons described above by the 11;s withj = 2,3, . , N do not affect directly the optical properties and they have been usually altogether neglected [l,lO]. Nevertheless, as evident from Eq. (15), the exciton-exciton interaction couples them with the cavity polaritons and, in general, they do take part in the formation of cavity biexcitons. The optically active state h, is given by the totally symmetric combination of single-well excitons because in our model the electronically decoupled quantum wells are all equivalent. Therefore, any perturbation selectively disturbing one particular quantum well (e.g., alloy composition or well-width fluctuations [Ill]) can partially mix up the h states which are coherently delocalized over all the quantum wells. The intrinsic exciton-phonon scattering affecting all the wells on an equal footing, however, does not couple the different dark states among themselves or with the cavity polariton states [12]. It can mix only the two cavity polariton states represented by cI and c2 1131. The effects of scattering by disorder in each quantum well (e.g., due to well-width fluctuations) can be described by an interaction of the form
Iv+1
H = Ho + He, = C i=l
1 Ei(k)C]kC’ik k
Hed = c z.k.4
C’f:B;,B,,
.v+ 1 = J, M;i(k. Y)‘.:kcjrp
(17)
where
Mij(k3 Y) =
1
u!s?J*.i(k)Sz.j(q).
(18)
The disorder may give rise to localized exciton states having radii small compared to the corresponding light wavelength. These states would not be influenced by polaritonic effects, but could be resonant with the cavity polariton extended states and, thus, scatter them efficiently. Apart from such nonperturbative behavior, the disorder will in general lead to a broadening of the cavity polariton states. Although the disorder mixes all branches, in the strong coupling regime for small wave vectors, only scattering within a single polariton branch is important. Considering the lower-cavity polariton the disorder Hamiltonian reduces to
(19) The disorder potentials U’“’ in each well can be taken to be independent and characterized in real space by the same correlation length I,,, that is ( U/‘“‘(r)U’“)(O)) = Ui exp ( - ~‘/1&5,.~;
(20)
then, it follows
With respect to the disorder potential U’(r) seen by a quantum well exciton, the effective disorder (~ZUi’“‘(r))/N seen by a cavity polariton is therefore reduced by averaging over all the wells because such a state is delocalized over a new quantum length given by the cavity width. Furthermore, considering states with vanishing wave vectors and a resonant condition (i.e., (~~(0) = co&O)),the additional factor F’j(k)Fi(q) reduces to t, simply because the exciton and photon components have equal weight in the polariton state. These two effects conspire to significantly reduce the line width of a cavity polariton, as can be estimated with a detailed calculation [14] employing the above formalism.
If the cavity mirrors and the multiple quantumwell structure are of high quality, the cavity polariton and dark exciton states are well defined. Introducing their phenomenological line widths to account for all broadening mechanisms, we can specify the condition for the strong coupling regime as follows. If only one well is present (i.e., no dark state), the upper and lower polaritons will split when (y, + ;~,)/2 < 6. This condition is the one usually considered in the literature. However, we emphasize that when two or more wells are present, one or more dark states appear and the problem of the separation between cavity polariton and dark exciton states arises. The upper polariton will separate from the dark states only if (;,, + ;,,)/2 d A/2, the lower one only if (;; + 7,)/2 d A/2. Here, ;qU, ;‘/ and ;‘r indicate, respectively, the upper polariton, the lower polariton and the dark exciton line widths. From the analysis carried out above, we can infer that the dark state line width is comparable to that of a quantum-well exciton, while the cavity polariton line width is reduced by averaging effects. We also point out that since the upper polariton has an additional relaxation channel, namely the phonon-induced decay into a lower one [13], in general, ;‘” > ~1,. With several quantum wells, a semistrony coupling regime can thus be achieved in which the lower polariton splits off from the unresolved dark exciton and upper polariton states (Fig. 1). Finally, when considering excitation dynamics and thermalization [ 151, the dark states should also be taken into account. In general, the luminescence rise and decay times will depend on dynamical equilibrium between dark excitons and cavity polaritons.
4. Cavity biexcitons We will focus on two principal aspects of the physics of cavity biexcitons: the character of the bound states (i.e., binding energy and radius) and their luminescence spectrum. Given the restructuring of the photon and exciton states brought about by the presence of the resonating cavity, one might be tempted to believe that all of the above properties should be drastically modified. However, whereas the biexciton luminescence is qualitatively
V.M. Agranorich
et al. / Journal
qf Luminescence
written
715~377 (1998)
16/--167
165
as
where the 2D wave vector x is associated to the center of mass translation of the biexciton and Q to the relative motion of the excitons. Exploiting the boson commutation relations among the c operators, it is a straightforward calculation to show that the eigenvalue equation HI@,) = ,!$%)I@,) reduces to (Ei(lX,‘2 +
PI) + Ej(lX/2 - Ql) - E(x))@~(Q) (23)
F-
6
k
Fig. 1. Strong (a), semistrong (b) and weak (c) coupling regime in microcavity with several quantum wells (W,,(O)= U>,(O)).Solid lines: cavity polaritons, dashed line: dark states, dotted line: bar cavity photon. The broadening is indicated only for states with !i = 0.
altered, as will be discussed in Section 5, the bound state structure is not strongly modified as described in the following. A two-particle eigenstate can be
A bound state solution of the above equation represents, in general, a combination of cavity polariton and dark exciton states for which no simple analytical expression is available. Nevertheless, it is possible to understand the main features of such cavity biexcitons in physically relevant limiting cases even without resorting to numerical calculations [ 161. As mentioned above, the qualitatively different strong and weak coupling regimes are realized depending on the relation between the Rabi splitting A and the damping ;‘. However, for cavity biexcitons an additional energy scale, namely the bare 2D biexciton binding energy Eg, appears and plays an important role. We distinguish the following relevant cases: strong coupling regime and n > Et (case A), strong coupling and A < Ez (case B), weak coupling and ;’ > EE (case C), weak coupling and ;’ < EE (case D). The polaritonic effects, in particular, the mixing between the optically active exciton h, and the cavity mode u. are limited to small k’s (i.e., k z 2n/i, = n/L), whereas the bare biexciton wave function and the attractive interaction VLextend to large k’s (i.e.. k z l/ubD, where the bare 2D biexciton radius a: is comparable with the exciton Bohr radius). Starting from case A, we assume perfect resonance Q,(O) = o,(O) = w,,. As the Rabi splitting is in this instance larger than the bare binding energy, the lowest bound state of Eq. (23) with x 2 0 is formed by two lower cavity polaritons p?.
It is easy to show (see [17]) that for this case the small and it is binding energy Eb” is exponentially negligible compared to the unavoidable damping; the presence of such a loosely bound state is a mathematical consequence of the bidimensional character of the problem [18,19]. A similar reasoning also holds for a bound state comprising two upper polaritons with total energy 2hto,, + A - E{’ as well as for those obtained from one polariton and one dark state, for instance with total energy 2tic1~ - A/2 - E{2.d. On the contrary, the biexciton states comprising two dark excitons have a much larger reduced mass (i.e., M/2) and a wave function extending to large k values for which the excitonphoton mixing may only lead to small corrections that for the present discussion are not important. All bound states (made of dark as well as optically active excitons) having a radius comparable to the bare radius u: and a wave function extending to large k’s will not be much affected by cavity polariton effects. As a result, their energies can be taken to be 2htv0 - EE. In the cases B, C, D and out of resonance (i.e., when the exciton-cavity detuning to,(O) - (U,(O) is large compared to the exciton-photon coupling), the cavity biexciton states are not significantly modified with respect to the bare ones. Nevertheless, in all cases, the presence of the cavity drastically changes the biexciton photoluminescence properties, as discussed below.
5. Cavity biexciton luminescence
spectra
The decay of cavity biexcitons occurs through their dissociation into two of the quasiparticles described above (cavity polaritons and dark states in the strong coupling regime, cavity photon and quantum-well excitons in the weak coupling regime or out of resonance). Such process is allowed only when the total wave vector and energy are conserved. When at least one of the dissociation products has a finite radiative life time (i.e., it eventually escapes from the cavity as a photon), cavity biexciton luminescence is observed. Thus, in the strong coupling regime, the biexciton may decay radiatively when its dissociation produces at least one cavity polariton having a small wave vector; in the weak
coupling regime or out of resonance, when its dissociation produces at least one cavity photon having a small wave vector. Assuming at first the cavity mode to be detuned from the exciton resonance, we have two situations: (0, > cc), and (0, < IU,. In the former case the biexciton luminescence is forbidden because energy conservation cannot be fulfilled; in the latter one, instead, for tZcu,(O)< hco,(O) - Eg, the biexciton luminescence is resonantly enhanced when the cavity mode corresponds to the energy hco of the emitted photon (as given by Eq. (24) below). In the resonant case (o,(O) = (a,(O)) and at low temperature we can distinguish two most interesting cases: (a) the products of biexciton dissociation are two lowest energy cavity polaritons; this take place if binding energy EE B A, (b) the products of biexciton dissociation are dark state and lower energy cavity polariton; this take place if EE 2 A/2 (more detail see in Ref. [17]). From this and previous consideration it follows that the biexciton luminescence is always superimposed on that of cavity mode (in weak coupling) or on that of cavity polaritons (in strong coupling). This is different with respect to bulk or QW case, where biexciton luminescence is at lower energy compared to the exciton and can be distinguished in photoluminescence. It means that in microcavity the biexciton contribution to photoluminescence can be distinguished only due to its nonlinear dependence on pumping. Even when the two particle biexciton radiative decay is forbidden, it is possible a direct transition to the ground state with the emission of a photon taking up the entire biexciton energy, because, such a process for two-dimensional biexcitons could be superradiant [20].
6. Conclusions The cavity eigenstates can be described in the weak and strong coupling regimes, respectively, as N quantum-well excitons plus one Fabry-Perotlike photon mode and as N - 1 dark excitons plus two Rabi split cavity polaritons. In the latter case (for N > I), the dark excitons are coherent linear superpositions of 2D excitons of all the quantum wells. They are orthogonal to the cavity polaritons
and can be populated in competition with the cavity polaritons. We have considered the effects of scattering by disorder and phonons and have shown, in particular. that the disorder mixes all states. In some cases, however. the disorder scattering is significantly reduced by averaging over all the wells. The exciton-exciton attraction gives rise to biexcitons which. in the strong coupling regime, can be formed by the combination of polaritonic and dark states. The drastic modification of the radiative decay channels brought about by the presence of the cavity leads to qualitatively different scenarios for the biexciton luminescence. We have discussed the main features of the corresponding radiative emission spectra depending on the relative size of the exciton-cavity detuning. the exciton-exciton attractive potential, the exciton-photon coupling and the damping. We have shown under which conditions the cavity biexciton luminescence is forbidden (possibly leading to long lifetimes). It remains to be studied how the presence of the exciton dark states and,:or the cavity biexcitons may affect experimentally accessible properties (e.g., excitation thermalization. Raman scattering, nonlinear optics) besides the luminescence spectra.
References 111 E.
Bursteln. C. Weisbuch
Photons: York.
New
Laser
Raton.
K. Ujihara
Oscillations
and Ncu
M1croca\ltieh.
Emission
CRC.
Boca
thereln.
131 A.L. I\anov, H. Haug. Phys. Re\. Lett. 74 (1995) 33X. M l-. Tassone. F. Bassani. L.C. Andreani. Nuovo Cimento D I2 (1990) 1673.
ISI L.C. Andrean). Phys. Lett. A 192 (19941 99. C61 D.S. Citrin. Phqs. Rev. B 50 (1994) 5397. 171 t or polarization along the growth dircctlon.
the funda-
mental mode of the cavity has the same dispersion as a free photon and a situation
similar to that of Ref. [IO]
would
result.
PI D.A. KleInman. Phys. Rev. B 28 (1983) 871. c91 R.M. Hochstrasser. J.D. Whiteman. J. Chcm.
( 1971)
Phys. 56
5945.
[lOI V. Savona, F. Tassone. C. Piermarocchl. I’. Schwendimann.
Phys. Rev. B 53
A. Quattropani.
I 1996)
13051.
r111 D.S. Citrin. Phys. Rev. B 47 (1993) 3831. 1121The Interface and confined phonon modes associated to the multiple quantum
well structure may lead to a partial
mixing of the h states because they affect the lirst and last wells differently 1131 V.M.
Commun. V.M.
than those in the middle.
Agranokich.
H. Benisty. C. Weisbuch.
Agranovich
et al., in press.
F. Tassone. C. Plermarocchi.
biexciton
Solid State
101 (1997) 631. V. Savona. A. Quattropani.
Phys. Rev. B 53 (1996) R7642.
It would be interesting
V.M.A. is thankful to Scuola Normale Superiore and FORUM-INFM (Pisa, Italy) for hospitality and support. He also acknowledges partial support through Grant 96-0334049 of the Russian Foundation of Basic Researches, Grant l-044 of the Russian Ministry of Science and Technology and INTAS Grant 93-461.
Plenum.
(Eds.). Spontaneous in
1995. and reference\
P. Schwendlmann.
Acknowledgements
Electrons
1995. and references therein.
H. Yokoyama. and
(Eds.), Confined
Physics and Applications.
to calculate the corrections
to the
energy in the presence of the cavity along the
lines followed in Ref. [3] for a free quantum
well.
G.C. La Rocc;l. I-‘. Bassani. V.M.
JOSA.
1997.
Mechanics.
Non-
Agranovlch.
\ubmitted. L.D.
Landau.
relati\lstlc
E.M.
Theory.
LifshitL. Quantum
SectIon 45. Pergamon
Press. Ouftrrd.
1958. pp. 153- 156. B. Simon. Ann. Phys. 97 (1976) 779. V.M.
155.
Agranovlch,
S. Mukamel.
Phy\
Lett. A 147 (19901
OF
LUMINESCENCE Journal
ELSEWIER
Biexcitons
of Lummescence
76&77
II9981
Ihl
I67
and dark states in semiconductor
microcavities
Abstract We analyze the formation of two-dimensional biexcitons and the role of cxciton dark states decoupled from the cavity mode in semiconductor microcavities containing a few resonating quantum wells. A general formalism is presented for the treatment of excitons. photon modes and biexcitons in resonating microcavities. Scattering by disorder is also considered. The interface disorder seen by cavity polaritons is shown to be reduced by averaging along the cavity axis. The spectral narrowing of cavity polariton lines is discussed. Results for the biexciton binding energy. radius and damping are discussed in the physically relevant cases. The cavity biexciton photoluminescence spectra arc found to bc qualitatively different depending on the relative magnitudes of the exciton-cavity detuning. the bicxciton bindinpcncrgy. the exciton photon coupling and the damping. (‘ 1998 Elsevier Science B.V. All rights reserved. K~~~~~vo~cls: Biexcitons;
Dark states: Microcavities
1. Introduction In the present paper, we address the problem of the formation of 2D biexcitons in cavity embedded quantum wells (in the following, we will refer to them as cavity biexcitons). As multiple quantum wells are frequently used to reach the strong coupling regime [l-2]. we explicitly take into account also the exciton dark states which are orthogonal to the cavity polaritons. The influence of polaritonic effects on quantum well biexcitons has been recently considered [3] and, in particular. an increase of the binding energy has been found. The presence of the resonant microcavity drastically alters the photon density of states and brings about a completely different physical picture, affecting, in particular. the biexciton luminescence. Depending *Corresponding
2. General formalism The system we consider comprises a few equivalent quantum wells (not electronically coupled) embedded in a i/2 resonating microcavity. The 2D exciton and photon eigenmodes are taken to be polarized perpendicular to the growth direction. The fundamental mode of the ideal planar resonator is given as a function of the 2D wave vector k by w,(k) = w,(0)v,’ 1 +
(kt!7r)Z,
(1)
where L is the effective cavity thickness and (e),(O) = (cz.‘nL) is the cutoff frequency. II being the
author
()(p?-~i13 9X $19.00 t PII S002~-2’I3(97~00143-9
on the values of the exciton-cavity detuning. the exciton-exciton interaction, the exciton-photon coupling and the damping. we predict qualitatively different spectra.
1998 Elselier
Science B.V. All rights reserved
V.M. Agranmich
162
background index dispersion is taken
et al. / Journal
of refraction. The 2D exciton to be parabolic:
ho,(k) = hw,(O) + g,
(2)
where M is the total mass for in-plane motion and at resonance o,(O) = w,(O). No distinction is made between T and L exciton modes that selectively couple to s and p polarized light, respectively [46]. For in-plane polarized excitations (such as the 2D heavy hole excitons of GaAs), no photon density of state is available for radiative decay below the cutoff frequency o,(O). As a consequence, the physics changes altogether with respect to the case in which no cavity is present [7]. All quantum wells are taken to be equivalent and placed at the cavity center, because each of them is approximately affected by the same value of the electric field when the total width of the multiple quantum-well structure is small compared to the light wavelength in the cavity. The Hamiltonian of the exciton-photon system is Ho = 1
hw,(k)&
k
a
by
c
VQ
BikBztq&k-QBolq+Q,
(4)
n.k.q.Q
where .d is a normalization factor (in plane number of unit cells) and VQ = V(lQl) the Fourier transform of the effective attractive interaction V(r) between excitons having the appropriate spin combination
77 (1998)
161-167
[3]. The potential V can be chosen to have a convenient analytical form such to reproduce the bare 2D biexciton binding energy EE as obtained from accurate microscopic calculations [S]. The B’s are assumed to satisfy the boson commutation relations, which is a valid approximation for a low exciton density. The total Hamiltonian H = Ho + H,, includes both the cavity polariton effects and the exciton-exciton interactions. In the following, the quadratic Hamiltonian Ho is diagonalized exactly and the hamiltonian H,, is then written in terms of the new operators corresponding to the eigenstates of Ho, i.e. the cavity polaritons and dark excitons. First of all, it is convenient to change the basis of exciton states to isolate the totally symmetric combination blk = (Blk + Bzk + ... + Bwk)/fi which is the only one coupled to the cavity mode, the remaining N - 1 new exciton states bik being orthogonal to blk bJk =xB3fkT,,,i, 1 where
+ 1 !?m,(k)B,t&,
where the index c( = 1, 2 ,.... N, labels the N different quantum wells, the B’s and a’s are, respectively, 2D exciton and cavity photon operators and 6 is proportional to the transition dipole matrix element (taken to be independent of k and CC) and gives the single well Rabi splitting at resonance. We use a model description of the many-body interactions responsible for the biexciton formation: each exciton is treated as a point particle interacting with the others in the same quantum well through a two-body effective potential given
He, = ;
cf Luminescence 76&
x,j
= 1,2 ,...,
T is an orthogonal
N,
matrix
(5) and for all x’s
T 2.1 = l/J’%. Physically, the bj’s represent exciton states delocalized over the N quantum wells according to specific coherent linear combinations of the original single-well exciton states. In the new basis, the entire oscillator strength is taken up by the b, optically active state and all other b states are dark, as for a linear chain of oscillators [9]. Then, Ho takes the form H,, = c k
-
h,(k)&&
+ hw,(k)b:kblk
c
S(bIkak
+
b,kaL)
+
C j>
h~~e(k)bjkbjk 1
3
(6)
1
where A = 6fi is the Rabi splitting at resonance and it is sufficient to take the appropriate linear . combmations of uk and b,,, i.e. the upper- and lower-cavity polariton States P1k and P2k, to COmplete the diagonalization. This is accomplished by setting hi, = F:(k)p,, uk
=
F(;(k)Plk
+ F:(k)P,,> +
F:(k)Pzk,
(7)
where the F’s (real, satisfying Fh,Fi + F”,F”, = d,,) are obtained solving the system of eigenvalue equations (to,(k) - lg,j(k))F;(k)
- a;(k)
= 0
= ((tjc(k) - Opj(k))Fy(k) - $Ft(k),
(8)
withj = 1, 2 and wP,, wPz giving the upperlower-cavity polariton dispersion laws tn,,,.#)
=
tM)
+ (o,(k) 2
_t ;\.(w,(k) Finally,
and
- to,(k))’ + AZ/h’.
Ho is written
in diagonal
(9)
=
f ~Jiml_ q.k.
(16)
'2
The formation of cavity biexcitons is formally described by the total Hamiltonian as given in Eq. (15) fully including the exciton-photon interaction. The predictions resulting from the present model are discussed in Section 4. In the following, we will consider the significance of the dark excitons and of the scattering by disorder or phonons.
form as 3. Dark states and phonon or disorder scattering
Now, H,, can be expressed in terms of the eigenstates of H,,, making use of the transformation
+
F?(k)PAL)+ 1 Tz.jbjk.
(11)
j=2
In order to simplify to define (‘lh = .i
=
Pla-
(‘2.
54 ,...,
=
the notations,
(‘Jk
PZk,
E
b(j-
I)k.
N + 1,
S,.,(k) = T,. 1f?Cd,
(12) S,.,(k) 5 T,. , F;(k)>
S,,i(k) E T,.j-,,j
= 3,4,...,N
E,(k) = htupl(k),
E,(k) E ho,,(k),
Ej(k) s h,(k),
it is convenient
+ 1,
(13)
j = 3.4, . . . , N + 1.
The total Hamiltonian
can then be written
(14) as
The dark excitons described above by the 11;s withj = 2,3, . , N do not affect directly the optical properties and they have been usually altogether neglected [l,lO]. Nevertheless, as evident from Eq. (15), the exciton-exciton interaction couples them with the cavity polaritons and, in general, they do take part in the formation of cavity biexcitons. The optically active state h, is given by the totally symmetric combination of single-well excitons because in our model the electronically decoupled quantum wells are all equivalent. Therefore, any perturbation selectively disturbing one particular quantum well (e.g., alloy composition or well-width fluctuations [Ill]) can partially mix up the h states which are coherently delocalized over all the quantum wells. The intrinsic exciton-phonon scattering affecting all the wells on an equal footing, however, does not couple the different dark states among themselves or with the cavity polariton states [12]. It can mix only the two cavity polariton states represented by cI and c2 1131. The effects of scattering by disorder in each quantum well (e.g., due to well-width fluctuations) can be described by an interaction of the form
Iv+1
H = Ho + He, = C i=l
1 Ei(k)C]kC’ik k
Hed = c z.k.4
C’f:B;,B,,
.v+ 1 = J, M;i(k. Y)‘.:kcjrp
(17)
where
Mij(k3 Y) =
1
u!s?J*.i(k)Sz.j(q).
(18)
The disorder may give rise to localized exciton states having radii small compared to the corresponding light wavelength. These states would not be influenced by polaritonic effects, but could be resonant with the cavity polariton extended states and, thus, scatter them efficiently. Apart from such nonperturbative behavior, the disorder will in general lead to a broadening of the cavity polariton states. Although the disorder mixes all branches, in the strong coupling regime for small wave vectors, only scattering within a single polariton branch is important. Considering the lower-cavity polariton the disorder Hamiltonian reduces to
(19) The disorder potentials U’“’ in each well can be taken to be independent and characterized in real space by the same correlation length I,,, that is ( U/‘“‘(r)U’“)(O)) = Ui exp ( - ~‘/1&5,.~;
(20)
then, it follows
With respect to the disorder potential U’(r) seen by a quantum well exciton, the effective disorder (~ZUi’“‘(r))/N seen by a cavity polariton is therefore reduced by averaging over all the wells because such a state is delocalized over a new quantum length given by the cavity width. Furthermore, considering states with vanishing wave vectors and a resonant condition (i.e., (~~(0) = co&O)),the additional factor F’j(k)Fi(q) reduces to t, simply because the exciton and photon components have equal weight in the polariton state. These two effects conspire to significantly reduce the line width of a cavity polariton, as can be estimated with a detailed calculation [14] employing the above formalism.
If the cavity mirrors and the multiple quantumwell structure are of high quality, the cavity polariton and dark exciton states are well defined. Introducing their phenomenological line widths to account for all broadening mechanisms, we can specify the condition for the strong coupling regime as follows. If only one well is present (i.e., no dark state), the upper and lower polaritons will split when (y, + ;~,)/2 < 6. This condition is the one usually considered in the literature. However, we emphasize that when two or more wells are present, one or more dark states appear and the problem of the separation between cavity polariton and dark exciton states arises. The upper polariton will separate from the dark states only if (;,, + ;,,)/2 d A/2, the lower one only if (;; + 7,)/2 d A/2. Here, ;qU, ;‘/ and ;‘r indicate, respectively, the upper polariton, the lower polariton and the dark exciton line widths. From the analysis carried out above, we can infer that the dark state line width is comparable to that of a quantum-well exciton, while the cavity polariton line width is reduced by averaging effects. We also point out that since the upper polariton has an additional relaxation channel, namely the phonon-induced decay into a lower one [13], in general, ;‘” > ~1,. With several quantum wells, a semistrony coupling regime can thus be achieved in which the lower polariton splits off from the unresolved dark exciton and upper polariton states (Fig. 1). Finally, when considering excitation dynamics and thermalization [ 151, the dark states should also be taken into account. In general, the luminescence rise and decay times will depend on dynamical equilibrium between dark excitons and cavity polaritons.
4. Cavity biexcitons We will focus on two principal aspects of the physics of cavity biexcitons: the character of the bound states (i.e., binding energy and radius) and their luminescence spectrum. Given the restructuring of the photon and exciton states brought about by the presence of the resonating cavity, one might be tempted to believe that all of the above properties should be drastically modified. However, whereas the biexciton luminescence is qualitatively
V.M. Agranorich
et al. / Journal
qf Luminescence
written
715~377 (1998)
16/--167
165
as
where the 2D wave vector x is associated to the center of mass translation of the biexciton and Q to the relative motion of the excitons. Exploiting the boson commutation relations among the c operators, it is a straightforward calculation to show that the eigenvalue equation HI@,) = ,!$%)I@,) reduces to (Ei(lX,‘2 +
PI) + Ej(lX/2 - Ql) - E(x))@~(Q) (23)
F-
6
k
Fig. 1. Strong (a), semistrong (b) and weak (c) coupling regime in microcavity with several quantum wells (W,,(O)= U>,(O)).Solid lines: cavity polaritons, dashed line: dark states, dotted line: bar cavity photon. The broadening is indicated only for states with !i = 0.
altered, as will be discussed in Section 5, the bound state structure is not strongly modified as described in the following. A two-particle eigenstate can be
A bound state solution of the above equation represents, in general, a combination of cavity polariton and dark exciton states for which no simple analytical expression is available. Nevertheless, it is possible to understand the main features of such cavity biexcitons in physically relevant limiting cases even without resorting to numerical calculations [ 161. As mentioned above, the qualitatively different strong and weak coupling regimes are realized depending on the relation between the Rabi splitting A and the damping ;‘. However, for cavity biexcitons an additional energy scale, namely the bare 2D biexciton binding energy Eg, appears and plays an important role. We distinguish the following relevant cases: strong coupling regime and n > Et (case A), strong coupling and A < Ez (case B), weak coupling and ;’ > EE (case C), weak coupling and ;’ < EE (case D). The polaritonic effects, in particular, the mixing between the optically active exciton h, and the cavity mode u. are limited to small k’s (i.e., k z 2n/i, = n/L), whereas the bare biexciton wave function and the attractive interaction VLextend to large k’s (i.e.. k z l/ubD, where the bare 2D biexciton radius a: is comparable with the exciton Bohr radius). Starting from case A, we assume perfect resonance Q,(O) = o,(O) = w,,. As the Rabi splitting is in this instance larger than the bare binding energy, the lowest bound state of Eq. (23) with x 2 0 is formed by two lower cavity polaritons p?.
It is easy to show (see [17]) that for this case the small and it is binding energy Eb” is exponentially negligible compared to the unavoidable damping; the presence of such a loosely bound state is a mathematical consequence of the bidimensional character of the problem [18,19]. A similar reasoning also holds for a bound state comprising two upper polaritons with total energy 2hto,, + A - E{’ as well as for those obtained from one polariton and one dark state, for instance with total energy 2tic1~ - A/2 - E{2.d. On the contrary, the biexciton states comprising two dark excitons have a much larger reduced mass (i.e., M/2) and a wave function extending to large k values for which the excitonphoton mixing may only lead to small corrections that for the present discussion are not important. All bound states (made of dark as well as optically active excitons) having a radius comparable to the bare radius u: and a wave function extending to large k’s will not be much affected by cavity polariton effects. As a result, their energies can be taken to be 2htv0 - EE. In the cases B, C, D and out of resonance (i.e., when the exciton-cavity detuning to,(O) - (U,(O) is large compared to the exciton-photon coupling), the cavity biexciton states are not significantly modified with respect to the bare ones. Nevertheless, in all cases, the presence of the cavity drastically changes the biexciton photoluminescence properties, as discussed below.
5. Cavity biexciton luminescence
spectra
The decay of cavity biexcitons occurs through their dissociation into two of the quasiparticles described above (cavity polaritons and dark states in the strong coupling regime, cavity photon and quantum-well excitons in the weak coupling regime or out of resonance). Such process is allowed only when the total wave vector and energy are conserved. When at least one of the dissociation products has a finite radiative life time (i.e., it eventually escapes from the cavity as a photon), cavity biexciton luminescence is observed. Thus, in the strong coupling regime, the biexciton may decay radiatively when its dissociation produces at least one cavity polariton having a small wave vector; in the weak
coupling regime or out of resonance, when its dissociation produces at least one cavity photon having a small wave vector. Assuming at first the cavity mode to be detuned from the exciton resonance, we have two situations: (0, > cc), and (0, < IU,. In the former case the biexciton luminescence is forbidden because energy conservation cannot be fulfilled; in the latter one, instead, for tZcu,(O)< hco,(O) - Eg, the biexciton luminescence is resonantly enhanced when the cavity mode corresponds to the energy hco of the emitted photon (as given by Eq. (24) below). In the resonant case (o,(O) = (a,(O)) and at low temperature we can distinguish two most interesting cases: (a) the products of biexciton dissociation are two lowest energy cavity polaritons; this take place if binding energy EE B A, (b) the products of biexciton dissociation are dark state and lower energy cavity polariton; this take place if EE 2 A/2 (more detail see in Ref. [17]). From this and previous consideration it follows that the biexciton luminescence is always superimposed on that of cavity mode (in weak coupling) or on that of cavity polaritons (in strong coupling). This is different with respect to bulk or QW case, where biexciton luminescence is at lower energy compared to the exciton and can be distinguished in photoluminescence. It means that in microcavity the biexciton contribution to photoluminescence can be distinguished only due to its nonlinear dependence on pumping. Even when the two particle biexciton radiative decay is forbidden, it is possible a direct transition to the ground state with the emission of a photon taking up the entire biexciton energy, because, such a process for two-dimensional biexcitons could be superradiant [20].
6. Conclusions The cavity eigenstates can be described in the weak and strong coupling regimes, respectively, as N quantum-well excitons plus one Fabry-Perotlike photon mode and as N - 1 dark excitons plus two Rabi split cavity polaritons. In the latter case (for N > I), the dark excitons are coherent linear superpositions of 2D excitons of all the quantum wells. They are orthogonal to the cavity polaritons
and can be populated in competition with the cavity polaritons. We have considered the effects of scattering by disorder and phonons and have shown, in particular. that the disorder mixes all states. In some cases, however. the disorder scattering is significantly reduced by averaging over all the wells. The exciton-exciton attraction gives rise to biexcitons which. in the strong coupling regime, can be formed by the combination of polaritonic and dark states. The drastic modification of the radiative decay channels brought about by the presence of the cavity leads to qualitatively different scenarios for the biexciton luminescence. We have discussed the main features of the corresponding radiative emission spectra depending on the relative size of the exciton-cavity detuning. the exciton-exciton attractive potential, the exciton-photon coupling and the damping. We have shown under which conditions the cavity biexciton luminescence is forbidden (possibly leading to long lifetimes). It remains to be studied how the presence of the exciton dark states and,:or the cavity biexcitons may affect experimentally accessible properties (e.g., excitation thermalization. Raman scattering, nonlinear optics) besides the luminescence spectra.
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V.M.A. is thankful to Scuola Normale Superiore and FORUM-INFM (Pisa, Italy) for hospitality and support. He also acknowledges partial support through Grant 96-0334049 of the Russian Foundation of Basic Researches, Grant l-044 of the Russian Ministry of Science and Technology and INTAS Grant 93-461.
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