Resonant electronic Raman scattering: A BCS-like system

Leonarde N. Rodrigues, A. Arantes, C. Schüller, M. J. V. Bell, and V. Anjos

Phys. Rev. B 93, 205409 – Published 6 May 2016

Abstract

In this paper we investigate the resonant intersubband Raman scattering of two-dimensional electron systems in GaAs-AlGaAs single quantum wells. Self-consistent calculations of the polarized and depolarized Raman cross sections show that the appearance of excitations at the unrenormalized single-particle energy are related to three factors: the extreme resonance regime, the existence of degeneracy in intersubband excitations of the electron gas, and, finally, degeneracy in the interactions between pairs of excitations. It is demonstrated that the physics that governs the problem is similar to the one that gives rise to the formation of the superconducting state in the BCS theory of normal metals. Comparison between experiment and theory shows an excellent agreement.

Received 4 December 2015
Revised 12 April 2016

DOI:https://doi.org/10.1103/PhysRevB.93.205409

Physics Subject Headings (PhySH)

Quantum wells Superconductors

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Leonarde N. Rodrigues¹, A. Arantes¹, C. Schüller², M. J. V. Bell¹, and V. Anjos^1,*

¹Laboratório de Espectroscopia de Materiais, Departamento de Física, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora, MG, Brazil
²Department of Physics, University of Regensburg, 93040 Regensburg, Germany

^*virgilio@fisica.ufjf.br; http://www.ufjf.br/gem/

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Vol. 93, Iss. 20 — 15 May 2016

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Images

Figure 1
Illustrations of (a) an electron gas second-order inelastic light scattering mechanism in quantum wells which involves two interband optical transitions and (b) a third-order inelastic light scattering mechanism which involves interband optical transitions (first and third step) and intersubband transitions induced by Coulomb interactions between the electron-hole pairs (or excitons) and the Fermi sea (second step). (c) Resultant excitation in the near resonance regime and (d) in the extreme resonance regime.
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Figure 2
Newton's pendulum. Mechanical analog of the Raman scattering in an extreme resonance regime of a 2D electron gas. The gravitational potential energy corresponds to the laser pump. When this energy is transferred to the system, two kinds of oscillations appear: (i) $N - 1$ ones with small amplitudes—they are identified as single-particle excitations; and (ii) one which receives most of the energy and transferred momentum. The analog of this excitation is the two-dimensional plasmon (collective excitation: CDE or SDE).
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Figure 3
(a) Experimental and (b) calculated polarized resonant Raman spectra of collective (CDE) and single-particle (SPE) excitations.
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Figure 4
Calculated polarized spectra with and without taking into account the coupling of the CDE with the optical phonons. The coupling is introduced via the frequency-dependent dielectric function.
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Figure 5
(a) Experimental and (b) calculated depolarized resonant Raman spectra showing the spectral lines of the collective (SDE) and single-particle (SPE) excitations for an incident laser energy of 1.6 eV. The estimation of the exchange-correlation effects by the redshift of the SPE is also shown.
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Figure 6
Correspondence between resonant Raman scattering and the formation of the superconducting state in the BCS theory. (a) Second-order inelastic light scattering mechanism involving the split-off valence band (VB) and conduction band (CB) of GaAs. The electronic transitions $α \to β (δ \to γ)$ are degenerate $(E)$ and coupled by a Coulomb interaction $(U_{β α, δ γ})$ . The coupling is the same for any two pairs of transitions. (b) Formation of the superconducting state in the BCS theory where $k (k^{'})$ represent states and $↑ (↓)$ spin states. The electron-phonon interaction $(U_{- k^{'} k^{'}, - kk})$ in the Fermi surface is responsible for a degenerate coupling between the two electron pairs (Copper pairs). (c) The schematic matrix common to both situations. The diagonalization results in a single level separated from the others with energy given by $(N - 1) U$ (superconducting $state \equiv CDE$ or SDE) and $(N - 1)$ levels with degenerate energy (normal states of the $metal \equiv SPE$ ). From this analogy, SPEs are in reality collective unrenormalized excitations.
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