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Excitons

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Semiconductor Physics

Abstract

Optical band-to-band absorption can produce an electron and a hole in close proximity which attract each other via Coulomb interaction and can form a hydrogen-like bond state, the exciton. The spectrum of free Wannier–Mott excitons in bulk crystals is described by a Rydberg series with an effective Rydberg constant given by the reduced effective mass and the dielectric constant. A small dielectric constant and large effective mass yield a localized Frenkel exciton resembling an excited atomic state. Excitons increase the absorption slightly below the band edge significantly. The interaction of photons and excitons creates a mixed state, the exciton–polariton, with photon-like and exciton-like dispersion branches. An exciton can bind another exciton or carriers to form molecules or higher associates of excitons. Free charged excitons (trions) and biexcitons have a small binding energy with respect to the exciton state. The binding energy of all excitonic quasiparticles is significantly enhanced in low-dimensional semiconductors. Basic features of confined excitons with strongest transitions between electron and hole states of equal principal quantum numbers remain similar. The three-dimensional confinement of quantum dots allows for forming stable antibinding exciton associates. Excitonic states of quantum dots are a prominent physical basis for the realization of photonic qubits. The exciton emission is deployed for producing single photons on demand with qubits encoded into the polarization or another degree of freedom. Pairs of entangled photons are obtained from the biexciton-exciton cascade. Photons used as flying qubits enable inherently secure data transmission; semiconductor-based quantum optics is a highly active field of research and development.

Karl W. Böer: deceased.

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Notes

  1. 1.

    This causes the breakdown of the adiabatic approximation. The error in this approximation is on the order of the fourth root of the mass ratio. For hydrogen this is \( {\left({m}_n/{M}_{\mathrm{H}}\right)}^{1/4}\cong 10\% \) and is usually acceptable. For excitons, however, the error is on the order of 1 and is no longer acceptable. This is relevant for the estimation of exciton molecule formation discussed in Sect. 1.4.

  2. 2.

    When a quasi-free charge carrier (electron or hole) moves through a crystal with strong lattice polarization, it is surrounded by a polarization cloud. Carrier plus polarization form a polaron, a quasiparticle with an increased effective mass (see Sect. 1.2 of Chap. 22, “Carrier-Transport Equations”).

  3. 3.

    It is, however, influenced by the gradient of an electric field or by strain; see, e.g., Tamor and Wolfe 1980.

  4. 4.

    The ionization energy is also referred to as binding energy or Rydberg energy.

  5. 5.

    \( {\phi}_n(0)\ne 0 \) applies only for S states.

  6. 6.

    Strictly, such transitions cannot occur at k = 0; however, a slight shift because of the finite momentum of the photon permits the optical transition to occur because of a weak electric quadrupole coupling (Elliott 1961). Such transitions can also be observed under a high electric field using modulation spectroscopy (Washington et al. 1977). Dipole-forbidden transitions are easily detected with Raman scattering (Sect. 1.3) or two-photon absorption (for Cu2O, see Uihlein et al. 1981), which follow different selection rules.

  7. 7.

    With a correspondingly large exciton Bohr radius of 1.04 μm for n = 25, compared to ~1 nm for n = 1.

  8. 8.

    The analysis of the measured reflection spectrum as a function of the wavelength and incident angle is rather involved. A relatively simple method for measuring the central part of the exciton–polariton spectrum in transmission through a prismatic crystal was used by Broser et al. (1981) (see Fig. 15).

  9. 9.

    A state close to an actual biexciton state (Sect. 1.4) which immediately decays into other states.

  10. 10.

    Deviations from a pure quadratic dependence are due to the short radiative lifetime for the involved species in direct-bandgap semiconductors, preventing a thermal equilibrium of the population.

  11. 11.

    Still a significant broadening of exciton transitions (of single quantum dots) well above the natural linewidth is observed due to the interaction of the quantum dot with its environment. The interaction with acoustic phonons (deformation potential coupling) and optical phonons (Fröhlich coupling) leads to broad transitions at increased temperature (Rudin et al. 1990); in addition, randomly fluctuating electrical fields of charged defects in the vicinity of the dots lead to a spectral jitter of the transitions on a very short time scale (spectral diffusion) even at low temperature (Türck et al. 2000).

  12. 12.

    We consider only the heavy-hole exciton due to its lower energy in the generally compressively strained epitaxial quantum dots and their larger oscillator strength.

  13. 13.

    Pure states with |M| = 2 are generally not observed in luminescence experiments (in absence of magnetic fields and confinements with reduced symmetry which can mix them with |M| = 1 states), since a j =  ± 1 photon cannot induce a radiative transition to the M = 0 ground state.

  14. 14.

    The three triplet states XXT+3, XXT0, and XXT-3, with parallel electron and hole spin are distinguished by the projection of their the total angular momentum (spin + orbital) on the symmetry axis of the quantum dot given by the growth direction.

  15. 15.

    The Bloch sphere is a generalization of the representation of a complex number \( z=x+ iy \) with \( {\left|z\right|}^2={z}^{\ast }z=1 \) as a point in the complex plane on the unit circle: \( {z}^{\ast }z=\left(x- iy\right)\left(x+ iy\right)={x}^2+{y}^2 \).

  16. 16.

    A simple classical example is a code word with three bits: 000 and 111; a (single) bit flip can be corrected by a majority vote: 010→000.

  17. 17.

    Cluster states are formed by an entangled network of qubits (Raussendorf and Briegel 2001; Nielsen 2004).

  18. 18.

    Meaning g(2)(0) ≪ 0.5, ideally 0; there must actually be only a single photon (and not more) present at a time.

  19. 19.

    A substitutional nitrogen atom with a vacancy at an adjacent lattice position (Kurtsiefer et al. 2000).

  20. 20.

    There are four (maximally entangled) two-qubit states designated as Bell states. In addition to |Φ+〉 noted in the text these are |Φ-〉 (with “+” replaced by “-” in the sum) and |Ψ+/-〉 with α and β being 0 and 1 in the first and 1 and 0 in the second qubit and the sum built with either “+” or “−.”

  21. 21.

    Optically excited single-photon emission with g(2)(0) = 0.33 at room temperature was achieved using a core-shell GaN/AlGaN nanowire quantum-dot (Holmes et al. 2014).

  22. 22.

    The luminescence of the dark exciton is shown in Fig. 31b.

  23. 23.

    With a π pulse, the system changes, e.g., from “qubit state” |0〉 to |1〉; with 2π, it returns to its initial state.

  24. 24.

    The circularly polarized transitions are a superposition of two cross-polarized linear transitions. The fine-structure splitting is negligible if it is smaller than the radiadive linewidth.

  25. 25.

    Even if the biexciton binding-energy gets zero for larger quantum dots (Fig. 29b), a nonzero fine-structure splitting of such dots leads usually to photon pairs of different energies.

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    Udo W. Pohl

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Böer, K.W., Pohl, U.W. (2023). Excitons. In: Semiconductor Physics. Springer, Cham. https://doi.org/10.1007/978-3-031-18286-0_14

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