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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.409) APPENDIX D LANDAU QUANTIZATION AND RENORMALIZATION OF RABI SPLITTING

(p.409) APPENDIX D LANDAU QUANTIZATION AND RENORMALIZATION OF RABI SPLITTING

Source:
Microcavities
Publisher:
Oxford University Press

This appendix addresses the renormalization of the exciton binding energy and oscillator strength in a quantum well subjected to an external magnetic field, resulting in the variation of the Rabi splitting in a microcavity.

Consider an exciton confined in a QW and subject to a magnetic field normal to the QW plane. Separating the exciton centre of mass motion and relative electron–hole motion in the QW plane, and assuming that the exciton does not move as a whole in the QW plane, we obtain the following exciton Hamiltonian:

(D.1)
H ^ = H ^ e + H ^ h + H ^ ex ,
where
(D.2)
H ^ v = - 2 2 m v 2 z v 2 + V v ( z v ) - μ B g v S v B + ( l v + 1 / 2 ) ω c v , = e , h , H ^ ex = - 2 2 m [ 1 ρ ρ ( ρ ρ ) - ρ 2 4 L 4 ] - e 2 4 π e 0 e ρ 2 + ( z e - z h ) 2 ,
ρ is the coordinate of electron–hole relative motion in the QW plane, L = ħ / ( e B ) is the so-called magnetic length, B is the magnetic field, se(h) is the electron (hole) spin, m e(h) is the electron (hole) effective mass in normal to the plane direction, μ is the reduced mass of electron–hole motion in the QW plane, V e(h) is the QW potential for an electron (hole), g e(h) and ω c e(h) are the electron (hole) g-factor and cyclotron frequency, respectively, l = 0, 1, 2, … and ε is the dielectric constant. Hereafter, we neglect the heavy–light hole mixing.

The excitonic Hamiltonian (D.2) was first derived by Russian theorists Gor'kov and Dzialoshinskii (1967). It contains a parabolic term dependent on the magnetic field. If the field increases, the magnetic length L decreases, which leads to the shrinkage of the wavefunction of the electron–hole relative motion. Thus, the probability of finding the electron and hole at the same point increases, leading to an increase of the exciton oscillator strength. In order to estimate this effect, let us solve the Schrödinger equation (3.1) variationally for the wavefunction Ψexc(z e, z h, ρ) chosen as an approximate (or trial) expression equal to

(D.3)
ψ exc ( z e , z h , ρ ) = U e ( z e ) U h ( z h ) f ( ρ ) ;
where z e(h) is the electron (hole) coordinate in the direction normal to the plane and ρ is the coordinate of electron–hole inplane relative motion. If the conduction-band offsets (p.410) are large in comparison to the exciton binding energy, which is the case in conventional GaAs/AlGaAs QWs, we find U e,h(z e,h) as a solution of single-particle problems in a rectangular QW. We separate variables in the excitonic Schrödinger equation, and choose
(D.4)
f ( ρ ) = 2 π 1 a e ρ / a ,
where a is a variational parameter. Substituting this trial function into the Schrödinger equation for electron–hole relative motion with Hamiltonian (D.1), we obtain the exciton binding energy
(D.5)
E B = 3 16 ħ 2 a 2 μ L 4 ħ 2 2 μ a 2 + 4 a 2 0 ρ d ρ e 2 ρ / a V ( ρ ) ( l e + 1 / 2 ) ħ ω c e ( l h + 1 / 2 ) ħ ω c h ,
where
(D.6)
V ( ρ ) = e 2 4 π ε 0 ε d z e d z h U e 2 ( z e ) U h 2 ( z h ) ρ 2 + ( z e z h ) 2 .

The parameter a should maximize the binding energy. Differentiating eqn (D.5) with respect to a we obtain:

APPENDIX D LANDAU QUANTIZATION AND RENORMALIZATION OF RABI SPLITTING

Fig. D.1: Typical “fan-diagram” of an InGaAs/GaAs QW. Circles show the resonances in transmission spectra of the sample associated with the heavy-hole exciton transition. In the limit of strong fields Landau quantization dominates over the Coulomb interaction of electron and hole, and the energies of excitonic transitions increase linearly with field. Square and diamond correspond to the light-hole exciton transitions. From Seisyan et al. (2001).

(p.411)
(D.7)
ħ 2 8 μ [ 1 a 4 2 L ] = 0 ρ d ρ ( 1 ρ a ) e 2 ρ / a V ( ρ ) .

The exciton radiative damping rate Γ0, defined in Chapter 3, can be expressed in terms of exciton parameters as

(D.8)
Γ 0 = ω 0 c ω LT ε a B 2 a 2 J eh 2 .

Here, J eh = ∫ U e(z)U h(z) dz, ω 0 is the exciton resonance frequency and ω LT and a B are the longitudinal-transverse splitting and Bohr radius of the bulk exciton, respectively.116 The vacuum-field Rabi splitting in a microcavity is

(D.9)
Ω Γ 0 1 a .

Shrinkage of the wavefunction of electron–hole relative motion in the magnetic field becomes essential if the magnetic length L is comparable to the exciton Bohr radius a B, i.e., for magnetic fields of about 3 T and more in the case of GaAs QWs. Taking into account the fact that L ≈ 70 Å at B = 10 T, the exciton Bohr radius can be realistically reduced by a factor of two. (p.412)

Notes:

(116) Typical parameters for GaAs are ω LT = 0.08 meV and a B = 14 nm.