Coulomb-induced synchronization of intersubband coherences in highly doped quantum wells and the formation of giant collective resonances

Mikhail Tokman, Maria Erukhimova, Yongrui Wang, and Alexey Belyanin

Phys. Rev. B 107, 245403 – Published 2 June 2023

Abstract

Many-body Coulomb interactions drastically modify the optical response of highly doped semiconductor quantum wells leading to a merger of all intersubband transition resonances into one sharp peak at the frequency substantially higher than all single-particle transition frequencies. Starting from standard density matrix equations for the gas of pairwise interacting fermions within Hartree-Fock approximation, we show that this effect is due to Coulomb-induced synchronization of the oscillations of coherences of all $N$ intersubband transitions and sharp collective increase in their coupling with an external optical field. In the high-doping limit, the dynamics of light-matter interaction is described by the analytic theory of $N$ coupled oscillators which determines new collective normal modes of the system and predicts the frequency and strength of the blueshifted collective resonance.

Received 13 March 2023
Accepted 23 May 2023

DOI:https://doi.org/10.1103/PhysRevB.107.245403

Physics Subject Headings (PhySH)

Synchronization

Quantum wells Two-dimensional electron gas

Density matrix methods Hartree-Fock methods

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalNonlinear Dynamics

Authors & Affiliations

Mikhail Tokman¹, Maria Erukhimova², Yongrui Wang ³, and Alexey Belyanin ³

¹Department of Electrical and Electronic Engineering, Schlesinger Knowledge Center for Compact Accelerators and Radiation Sources, Ariel University, Ariel
²Biraghigasse 8, 1130 Vienna, Austria
³Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843, USA

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Vol. 107, Iss. 24 — 15 June 2023

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Images

Figure 1
Mechanical model of the “Coulomb springs” effect in a high-density regime. The oscillations of dipole moments at populated intersubband transitions (here the number of active transitions $N_{t} = 3$ ) are modeled by vertical vibrations of masses $m$ on green springs with corresponding spring constants $k_{j}$ . The allowed motion is one dimensional (along $x$ axis), and only the oscillations with immobile center of mass of each oscillator are considered. The Coulomb coupling is modeled by grey springs which tie each “upper” mass with all “lower” masses and vice versa. These springs are characterized by spring constant $K$ , which increases with increasing electron density in a QW. Each oscillator can be independently excited by an external force. Under the condition $K N_{t} ≫ k_{j}$ , the collective in-phase oscillation with frequency defined by the spring constant of the “Coulomb spring” and proportional to $\sqrt{N_{t}}$ is excited most efficiently; see the solution in Appendix pp3.
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Figure 2
The calculated absorption spectra of a $L = 18.5$ nm quantum well with different 2D electron densities $N_{2 D}$ . From bottom to top: $N_{2 D} = 1 \times 10^{11}$ , $1 \times 10^{12}$ , $5 \times 10^{12}$ , $1 \times 10^{13}$ , and $2.2 \times 10^{13} {cm}^{- 2}$ . The phenomenological broadening of transitions (full width at half maximum) is 10 meV. The temperature is 300 K. Red continuous lines are the absorption spectra, calculated with Eq. (25) taking into account Hartree modification of energies and Coulomb coupling of oscillations at different intersubband transitions described by Eq. (9). Blue dashed lines are the absorption spectra calculated with Eq. (18) obtained without taking into account Coulomb coupling. The insets present the band structure, Hartree energy levels, and square moduli of the Hartree wave functions. The Fermi energy corresponding to electron densities in each case is indicated by a (violet) dashed line.
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Figure 3
The calculated absorption spectra of QWs with different thicknesses; from bottom to top: $L = 10$ , 20, and 30 nm, and the same 2D electron density of $2 \times 10^{12} {cm}^{- 2}$ . The phenomenological broadening of transitions (full width at half maximum) is 10 meV. The insets present the band structure, Hartree energy levels, and square moduli of the Hartree wave functions. The notations and material parameters are the same as in Fig. 2.
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Figure 4
The calculated absorption spectra with exchange interaction taken into account perturbatively. The Hartree ground state is treated as unperturbed. Different perturbing terms are taken into account independently. Case 1: without any Coulomb coupling; case 2: with Fock energy renormalization; case 3: with Hartree coupling (coincides almost exactly with case 6); case 4: with Hartree coupling and Fock energy renormalization; case 5: with Fock coupling and Fock energy renormalization; case 6: with Hartree-Fock coupling and Fock energy renormalization, i.e., with all effects included. The following parameters are used: $L = 18.5$ nm and $N_{2 D} = 2.2 \times 10^{13} {cm}^{- 2}$ . The phenomenological broadening of transitions (FWHM) is 10 meV.
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