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Microcavities$

Alexey Kavokin, Jeremy J. Baumberg, Guillaume Malpuech, and Fabrice P. Laussy

Print publication date: 2007

Print ISBN-13: 9780199228942

Published to Oxford Scholarship Online: May 2008

DOI: 10.1093/acprof:oso/9780199228942.001.0001

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(p.399) APPENDIX B SCATTERING RATES OF POLARITON RELAXATION

(p.399) APPENDIX B SCATTERING RATES OF POLARITON RELAXATION

Source:
Microcavities
Publisher:
Oxford University Press

This appendix complements Chapter 8 with detailed calculations of the rates of polariton scattering with phonons, electrons and other polaritons in microcavities.

B.1 Polariton–phonon interaction

The theoretical description of carrier–phonon or of exciton–phonon interaction has received considerable attention throughout the history of semiconductor heterostructures. Here we present a simplified picture that is, however, well suited to our problem. Cavity polaritons are two-dimensional particles with only an inplane dispersion. They are scattered by phonons that are in the QWs we consider, mainly three-dimensional (acoustic phonons) or two-dimensional (optical phonons). Scattering events should conserve the wavevector in the plane. We call q the phonon wavevector and q , q z the inplane and z-component of q:

(B.1)
q = ( q , q z ) .
ωq will denote the phonon energy.

Using Fermi's golden rule, the scattering rate between two discrete polariton states of wavevector k and k′ reads:

(B.2)
W k k phon = 2 π q | M ( q ) | 2 ( θ ± + N q = k k +q z ) δ ( E ( k ) E ( k ) ± ω q ) ,
where N q is the phonon distribution function and θ ± is a quantity whose sign matches the one in the delta function corresponding to phonon emission and absorption, and is defined as θ + = 1 and θ = 0. In the case of an equilibrium phonon distribution, N q follows the Bose distribution. The sum of eqn (B.2) is over phonon states.

M is the matrix element of interaction between phonons and polaritons. If one considers polariton states with a finite energy width γ k, the function can be replaced by a Lorentzian and eqn (B.2) becomes:

(B.3)
W k k phon = 2 π q | M ( q ) | 2 ( θ ± + N q = k k +q z ) γ k / π ( E ( k ) E ( k ) ± ω q ) 2 + γ k / π 2 .

Wavevector conservation in the plane actually limits the sum of eqn (B.3) to the z-direction. In the framework of the Born approximation the matrix element of interaction reads:

(B.4)
| M ( q ) | 2 = | < ψ k pol | H q pol phon | ψ k pol > | 2 = | χ k | 2 | χ k | 2 | < ψ k ex | H q ex phon | ψ k ex > | 2 ,
(p.400) where | ψ k pol > is the polariton wavefunction and | ψ k ex > the exciton wavefunction, and χ k is the exciton Hopfield coefficient (which squares to the exciton fraction). The exciton wavefunction reads:
(B.5)
ψ k ex ( r e , r h ) = U e ( z e ) U h ( z h ) 1 S e i k( β e r e + β h r h ) 2 π 1 a B 2 D exp ( | r e r h | a B 2 D ) ,
where z e, z h are electron and hole coordinates along the growth axis and re and rh their coordinates in the plane, U e and U h are the electron and hole wavefunctions in the growth direction, a B 2 D is the two-dimensional exciton Bohr radius, β e, h = m e, h/(m e + m h).

B.1.1 Interaction with longitudinal optical phonons

This interaction is mainly mediated by the Frölich interaction (Frohlich 1937). In three dimensions the exciton–LO-phonon matrix element reads:

(B.6)
M LO ( q ) = e q 4 π ω LO S L ε 0 ( 1 ε 1 ε s ) = M 0 LO q S L ,
where ωLO is the energy for creation of a LO-phonon, ε is the optical dielectric constant, εs the static dielectric constant, L is the dimension along the growth axis and S the normalization area. In two dimensions one should consider confined optical phonons with quantized wavevector in the z-direction. L becomes the QW width and q z m = m π / L with m an integer. Moreover, the overlap integral between exciton and phonon wavefunctions quickly vanishes while m increases. Therefore, we consider only the first confined phonon state and the matrix element (B.6) becomes:
(B.7)
M LO ( q ) = M 0 LO | q | 2 + ( π / L ) 2 S L .

The wavevectors exchanged in the plane are typically much smaller than π/L and eqn (B.7) can be approximated by:

(B.8)
M LO ( q ) = M 0 LO π L S .

Considering a dispersionless phonon dispersion for LO-phonons, the LO-phonon contribution to Eq. (B.3) reads:

(B.9)
W k k phon LO = 2 L π 2 S | χ k | 2 | χ k | 2 | M 0 LO | 2 × ( θ ± + 1 exp ( ω LO / k B T ) 1 ) γ k ( E ( k ) E ( k ) ± ω LO ) 2 + γ k 2 .
Optical phonons interact very strongly with carriers. They allow fast exciton formation. Their energy of formation ωLO is, however, of the order of 20 to 90 meV, depending (p.401) on the nature of the semiconductor involved. An exciton with a kinetic energy smaller than 20 meV can no longer emit an optical phonon. The probability of absorbing an optical phonon remains extremely small at low temperature. This implies that an exciton gas cannot cool down to temperature lower than 100–200 K by interacting only with optical phonons. Optical phonons are therefore extremely efficient at relaxing a hot-carrier gas (optically or electrically created) towards an exciton gas with a temperature 100–300 K in a few picoseconds. The final cooling of this exciton gas towards the lattice temperature should, however, be assisted by acoustical phonons or other scattering mechanisms. The semiconductor currently used to grow microcavities, and where optical phonons play the largest role, is CdTe. In such a material, ωLO is only 21 meV, namely larger than the exciton binding energy in CdTe-based QWs. Moreover, the Rabi splitting is of the order of 10–20 meV in CdTe-based cavities. This means that the direct scattering of a reservoir exciton towards the polariton ground-state is a possible process that may play an important role.

B.1.2 Interaction with acoustic phonons

This interaction is mainly mediated by the deformation potential. The exciton–acoustic phonon matrix element reads:

(B.10)
M ac ( q ) = q μ 2 ρ c s S L G ( q , q z ) ,
where μ is the reduced mass of electron–hole relative motion, ρ is the density and c s is the speed of sound in the medium. Assuming isotropic bands, G reads
(B.11)
G ( q , q z ) = D e I e ( q z ) I e ( q ) D h I h ( q z ) I h ( q ) D e I e ( q ) D h I h ( q ) .
D e, D h are the deformation coefficients of the conduction and valence band, respectively, and I e(h) ( ) are the overlap integrals between the exciton and phonon mode in the growth direction and in the plane, respectively:
(B.12a)
I e(h) ( q ) = ( 2 π 1 a B 2 D ) 2 exp ( 2 r a B 2 D ) exp ( i m h(e) m e + m h q r ) d r = ( 1+ ( m h(e) q a B 2 D 2 M x ) ) 2 ,
(B.12b)
I e(h) ( q z ) = | f e(h) ( z ) | 2 e i q z z d z 1.

Using this matrix element and moving to the thermodynamic limit in the growth direction, the scattering rate (B.3) becomes:

(B.13)
W k k phon = 2 π L 2 π | χ k | 2 | χ k | 2 q z q 2 ρ c s S L | G ( k- k ) | 2 ( θ ± + N k k +qz phon ) γ k / π ( E ( k ) E ( k ) ± ω k- k +q z ) 2 + γ k 2 .

(p.402) Moving to the thermodynamic limit means that we let the system size in a given direction (here the z-direction) go to infinity, substituting the summation with an integral, using the formula q z ( L / 2 π ) d q z .

Equation (B.13) can be easily simplified:

(B.14)
W k k phon = | G ( k- k | 2 2 π Z ρ c s | χ k | 2 | χ k | 2 q z | k- k + q z | ( θ ± + N k k +qz phon ) γ k / π ( E ( k ) E ( k ) ± ω k- k +q z ) 2 + γ k 2 .

B.2 Polariton–electron interaction

The polariton–electron scattering rate is calculated using Fermi's golden rule as

(B.15)
W k k el = 2 π q | M q , k k el | 2 | χ k | 2 | χ k | 2 N q e ( 1 N q + k k e ) γ k ( E ( k ) - E ( k ) + ħ 2 2 m e ( q 2 - | q + k - k | 2 ) ) 2 + γ k 2 ,
where N q e is the electron distribution function and m e the electron mass. If one considers electrons at thermal equilibrium, it is given by the Fermi–Dirac electron distribution function with a chemical potential
(B.16)
μ e = k B T ln ( exp ( 2 n e π k B T m e ) 1 ) ,
where n e is the electron concentration. M el is the matrix element of interaction between an electron and an exciton. A detailed calculation of the electron–exciton matrix element has been given by Ramon et al. (2003). M el is composed of a direct contribution and of an exchange contribution:
(B.17)
M el = M dir el ± M exc el .
The + sign corresponds to a triplet configuration (parallel electron spins) and the – to a singlet configuration (antiparallel electron spins). If both electrons have the same spin, the total exciton spin is conserved through the exchange process. However, if both electron spins are opposite, an active exciton state of spin +1, for example, will be scattered towards a dark state of spin +2 through the exchange process. Here, and in what follows, we shall consider only the triplet configuration for simplicity.

In order to calculate M el we adopt the Born approximation and obtain: (p.403)

(B.18a)
M der el = ϕ k * ( r e , r h ) f q * ( r e ) [ V ( | r e r e | ) - V ( | r h - r e ' | ) ] ϕ k ( r e , r h ) f q+k- k * ( r e ) d r e d r h d r e ,
(B.18b)
M exc el = ϕ k * ( r e , r h ) f q * ( r e ) [ V ( | r e r e | ) V ( | r h r e | ) V ( | r h r e | ) ] ϕ k ( r e , r h ) f q+k- k * ( r e ) d r e d r h d r e ,
with Coulomb potential V(r) = e 2/(4πεε 0 r) with ε the dielectric susceptibility of the QW.

The free-electron wavefunction f is given by:

(B.19)
f q ( r e ) = 1 S e i q r e .

Integrals (B.18) can be calculated analytically. One finds:

(B.20)
M dir el = e 2 2 S ε ε 0 | k k | [ ( 1 + ξ h 2 ) 3 / 2 ( 1 + ξ e 2 ) 3 / 2 ] ,
(B.21)
M dir dir = 2 e 2 S ε ε 0 [ ( 1 + ξ c 2 ) 3 / 2 ( 1 4 ξ h 2 ) 3 / 2 ( a 2 + | q- β e k | 2 ) 1 / 2 ( 1 + 4 ξ h 2 ) 3 / 2 ( a 2 + | k -k-q- β e k | 2 ) 1 / 2 ] ,
where
(B.22)
ξ e,h = 1 2 β e,h | k k | a B 2 D and ξ c = | β e k + k k q | a B 2 D .

Passing to the thermodynamic limit, eqn (B.15) becomes

(B.23)
W k k el = S 2 π q | M dir el + M exc el | 2 | χ k | 2 | χ k | 2 N q e ( 1 N q+ k -k e ) × γ k ' ( E ( k ) - E ( k ) + ħ 2 2 m e ( q 2 - | q + k - k | 2 ) ) 2 + γ k 2 .

The polariton–electron interaction is a dipole-charge interaction that takes place on a picosecond time scale. An equilibrium electron gas can thermalize a polariton gas quite efficiently. A more complex effect may, however, take place such as trion formation or exciton dephasing.

(p.404) B.3 Polariton–polariton interaction

The polariton–polariton scattering rate reads:

(B.24)
W k k pol = 2 π q | M ex | 2 | χ k | 2 | χ k | 2 | χ q | 2 | χ q+ k -k | 2 N q pol ( 1 N q+ k -k pol ) × γ k ( E ( k ) E ( k ) + E ( q + k -k)- E ( q ) ) 2 + γ k 2 .

The exciton–exciton matrix element of interaction is also composed of a direct and an exchange term. It has been investigated and calculated by Ciuti et al. (1998) and recently by Combescot et al. (2007). Here, and in what follows, we use a numerical estimate provided by Tassone and Yamamoto (1999) that we further assume constant over the whole reciprocal space:

(B.25)
M ex 6 ( a B 2 D ) 2 S E b = 1 S M exc 0 ,
where E b is the exciton binding energy. Passing to the thermodynamic limit in the plane, eqn (B.24) becomes:
(B.26)
W k k pol = 1 2 π | M ex 0 | 2 | χ k | 2 | χ k | 2 | χ q | 2 | χ q+ k -k | 2 N q pol ( 1 + N q+ k -k pol ) × γ k ( E ( k ) E ( k ) + E ( q + k -k)- E ( q ) ) 2 + γ k 2 .
As one can see, the a priori unknown polariton distribution function is needed to calculate scattering rates. This means that in any simulation these scattering rates should be updated dynamically throughout the simulation time, which can be extremely time consuming.

Polariton–polariton scattering has been shown to be extremely efficient when a microcavity is resonantly excited. It also plays a fundamental role in the case of non-resonant excitation. Depending on the excitation condition and on the nature of the semiconductor used, the exciton–exciton interaction may be strong enough to self-thermalize the exciton reservoir at a given temperature.

B.3.1 Polariton decay

There are three different regions in reciprocal space: [0, k sc], ]k sc, k L] and ]k L, ∞[.

  • [0, k sc] is the region where the exciton–photon anticrossing takes place. Cavity mirrors reflect the light only within a finite angular cone, which corresponds to an inplane wavevector k sc that depends on the detuning. In this central region the polariton decay is mainly due to the finite cavity photon lifetime Γk = ∣C k2/τ c, where C k is the photon Hopfield coefficient (which squares to the photon fraction of the polariton) and τ c is the cavity photon lifetime. k sc values are typically of the order of 4 to 8 × 106 m−1 and τ c is in the range between 1 and 10 ps.

  • (p.405)
  • ]k sc, k L] where k L is the wavevector of light in the medium. In this region excitons are only weakly coupled to the light and polariton decay is Γk = Γ0, which is the radiative decay rate of QW excitons.

  • ]k L, ∞[. Beyond k L excitons are no longer coupled to light. They only decay non-radiatively with a decay rate Γnr. We do not wish to enter into the details of the mechanism involved in this decay, which we consider as constant in the whole reciprocal space. This quantity is given by the decay time measured in time-resolved luminescence experiments, and is typically in the range between 100 ps and 1 ns.

B.4 Polariton–structural-disorder interaction

This scattering process is mainly associated with the excitonic part of polaritons. Structural disorder induces coherent elastic (Rayleigh) scattering with a typical timescale of about 1ps. It couples very efficiently all polaritons situated on the same “elastic circle” in reciprocal space (see the results of Freixanet et al. (1999) and Langbein and Hvam (2002)). This allows us to simplify the description of polariton relaxation by assuming cylindrical symmetry of the polariton distribution function. Also, disorder induces a broadening of the polariton states, which should be accounted for when scattering rates are calculated. (p.406)