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Europhysics Letters (EPL)

Novel approach to nonlinear susceptibility in semiconductors

Monique Combescot, Odile Betbeder-Matibet, K. Cho, Hiroshi Ajiki

Published Oct 13, 2005• 21 citations
Publisher

Abstract

We here propose a new procedure to derive a general expression of the third-order susceptibility valid for any semiconductor Hamiltonian. It relies on our theory for composite exciton many-body effects which is quite appropriate to this problem since all optical nonlinearities in semiconductors come from exchange scatterings with the virtual excitons coupled to photons. Our expression of χ (3) is sample volume free, as physically expected. It also contains a Pauli term (exchange without Coulomb) which is missed by all bosonized exciton approaches. This term, dominant at large detuning, is a direct manifestation of the exciton composite nature.

DOI
10.1209/epl/i2005-10282-0

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We here propose a new procedure to derive a general expression of the third-order susceptibility valid for any semiconductor Hamiltonian. It relies on our theory for composite exciton many-body effects which is quite appropriate to this problem since all optical nonlinearities in semiconductors come from exchange scatterings with the virtual excitons coupled to photons. Our expression of χ (3) is sample volume free, as physically expected. It also contains a Pauli term (exchange without Coulomb) which is missed by all bosonized exciton approaches. This term, dominant at large detuning, is a direct manifestation of the exciton composite nature.
We revisit the approach proposed by Mukamel and coworkers to describe interacting excitons through infinite series of composite-boson operators for both, the system Hamiltonian and the exciton commutator -- which, in this approach, is properly kept different from its elementary boson value. Instead of free electron-hole operators, as used by Mukamel's group, we here work with composite-exciton operators which are physically relevant operators for excited semiconductors. This allows us to get \emph{all} terms of these infinite series explicitly, the first terms of each series agreeing with the ones obtained by Mukamel's group when written with electron-hole pairs. All these terms nicely read in terms of Pauli and interaction scatterings of the composite-exciton many-body theory we have recently proposed. However, even if knowledge of these infinite series now allows to tackle NNN-body problems, not just 2-body problems like third order nonlinear susceptibility χ(3)χ^{(3)}χ(3), the necessary handling of these two infinite series makes this approach far more complicated than the one we have developed and which barely relies on just four commutators.
The interplay of exciton statistics and Coulomb interactions in the optical response of semiconductors is studied by derivation of an effective Hamiltonian written in terms of exciton operators, starting with a two-band model. Statistical effects are incorporated through the nonboson commutation relations of exciton operators, which contain an exciton-size-dependent parameter q that may vary from q = 0 (boson statistics) through q ≪ 1 (Wannier excitons) to q = 2 (Frenkel excitons or Pauli statistics). A unified Green’s function expression for χ(3) that applies to excitons of an arbitrary nature is derived.
We revisit the approach proposed by Mukamel and co-workers to describe interacting excitons through infinite series of composite-boson operators for both the system Hamiltonian and the exciton commutator---which, in this approach, is properly kept different from its elementary-boson value. Instead of free-electron-hole operators, as used by Mukamel's group, we here work with composite-exciton operators which are physically relevant operators for excited semiconductors. This allows us to get all terms of these infinite series explicitly, the first terms of each series agreeing with the ones obtained by Mukamel's group when written with electron-hole pairs. All these terms nicely read in terms of Pauli and interaction scatterings of the composite-exciton many-body theory we have recently proposed. However, even if knowledge of these infinite series now allows us to tackle NNN-body problems, not just two-body problems like third-order nonlinear susceptibility ${\ensuremath{\chi}}^{(3)}$, the necessary handling of these two infinite series makes this approach far more complicated than the one we have developed and which barely relies on just four commutators.
The calculation of the third order susceptibility still is a long standing fundamental problem of particular importance in nonlinear nanooptics: Indeed, cancellation of size-dependent terms coming from uncorrelated excitations is expected, but up to now shown for very simple Hamiltonians only. Using a many-body theory recently developed to handle interacting close-to-bosons, we prove it here for \emph{arbitrary} H. This new formalism actually provides the first clean way to calculate nonlinear susceptibilities, with results different from previous ones.
The calculation of the third order susceptibility still is a long standing fundamental problem of particular importance in nonlinear nanooptics: Indeed, cancellation of size-dependent terms coming from uncorrelated excitations is expected, but up to now shown for very simple Hamiltonians only. Using a many-body theory recently developed to handle interacting close-to-bosons, we prove it here for \emph{arbitrary} H. This new formalism actually provides the first clean way to calculate nonlinear susceptibilities, with results different from previous ones.
We derive a bosonized Hamiltonian describing two-exciton correlation of semiconductor-photon coupled systems with a bosonization method, which takes into full account an effect of deviation of the excitons from ideal bosons. This deviation effect stems from the fact that excitons are composite particles, whose character appears clearly in the case where the excitons overlap each other. We call this effect a composite-particle effect (CPE). To our knowledges this effect was not considered completely in previous theoretical works on exciton-exciton interaction. The Hamiltonian introduced in this paper includes the results of the previous works as low-order terms of the CPE. After the introduction of a general theory of the bosonization method for arbitrary dimension and electron-hole mass ratio, we also demonstrate an application to a semiconductor bulk system coupled with a photon field in the heavy-hole limit. The bosonized Hamiltonian shows that the CPE brings about an enhancement of the exciton-exciton scattering strength and a qualitative change of the photo transition amplitude. It is also shown that the Hamiltonian describes two-exciton bound and scattering states.