Second harmonic generation of cuprous oxide in magnetic fields

Patric Rommel, Jörg Main, Andreas Farenbruch, Johannes Mund, Dietmar Fröhlich, Dmitri R. Yakovlev, Manfred Bayer, and Christoph Uihlein

Phys. Rev. B 101, 115202 – Published 25 March 2020

Abstract

Recently, second harmonic generation (SHG) for the yellow exciton series in cuprous oxide was demonstrated [Phys. Rev. B 98, 085203 (2018)]. Assuming perfect $O_{h}$ symmetry, SHG is forbidden along certain high-symmetry axes. Perturbations can break this symmetry and forbidden transitions may become allowed. We investigate theoretically the effect of external magnetic fields on the yellow exciton lines of cuprous oxide. We identify two mechanisms by which an applied magnetic field can induce a second harmonic signal in a forbidden direction. First of all, a magnetic field by itself generally lifts the selection rules. In the Voigt configuration, an additional magneto-Stark electric field appears. This also induces certain SHG processes differing from those induced by the magnetic field alone. Complementary to the paper by Farenbruch et al. [Phys. Rev. B 101, 115201 (2020)], we perform a full numerical diagonalization of the exciton Hamiltonian including the complex valence-band structure. Numerical results are compared with experimental data.

Received 20 November 2019
Revised 27 February 2020
Accepted 27 February 2020

DOI:https://doi.org/10.1103/PhysRevB.101.115202

Physics Subject Headings (PhySH)

Electronic structure Excitons Magneto-optical spectra

Crystal structures Wide band gap systems

Group theory Luttinger–Kohn model Optical second-harmonic generation Symmetries in condensed matter

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Patric Rommel and Jörg Main

Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany

Andreas Farenbruch¹, Johannes Mund ¹, Dietmar Fröhlich¹, Dmitri R. Yakovlev^1,2, Manfred Bayer ^1,2, and Christoph Uihlein¹

¹Experimentelle Physik 2, Technische Universität Dortmund, 44221 Dortmund, Germany
²Ioffe Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia

Magneto-Stark and Zeeman effect as origin of second harmonic generation of excitons in ${Cu}_{2} O$

A. Farenbruch, J. Mund, D. Fröhlich, D. R. Yakovlev, M. Bayer, M. A. Semina, and M. M. Glazov

Phys. Rev. B 101, 115201 (2020)

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Vol. 101, Iss. 11 — 15 March 2020

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Images

Figure 1
Scheme of a second harmonic generation process. The ground state of the crystal is denoted by $| g 〉$ , the resonantly stimulated exciton state by $| f 〉$ , and the virtual intermediate state by $| i 〉$ .
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Figure 2
Comparison of experimental (black) and numerical (color palette) SHG spectra in arbitrary units with $E^{in, 2} ∥ [110], E^{out, 2} ∥ [001]$ for different values of $A$ as defined in Eq. (31). The wave vector points along the $[1 \bar{1} 0]$ axis and the magnetic field is applied in the Voigt geometry in the [110] direction and has a strength of $B = 6 T$ . The main features shown belong to principal quantum numbers $n = 3$ and $n = 4$ . The color encodes the value of $A$ , which parametrizes the relative strength of dipole-emission processes to quadrupole-emission processes. We show the comparison for (a) negative and (b) positive values of $A$ .
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Figure 3
Same as Fig. 2 but with a fixed value of $A = 0.4$ . With this value, the numerical spectrum matches the experimental spectrum reasonably well. Choosing a higher value for $A$ leads to the $3 S$ peak being excessively high, whereas a lower value results in a too-weak $n = 4$ manifold. For further discussion, see the text.
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Figure 4
Experimental SHG spectra in arbitrary units with (a) $E^{in, 1} ∥ [11 \sqrt{2}], E^{out, 1} ∥ [110]$ and (b) $E^{in, 2} ∥ [110], E^{out, 2} ∥ [001]$ . The wave vector points along the $[1 \bar{1} 0]$ axis and the magnetic field is applied in the [110] direction. The spectra on the left-hand side are mediated through the magneto-Stark effect and the spectra on the right are mediated by the Zeeman effect. The main features visible belong to excitons with principal quantum numbers $n = 3$ and $n = 4$ . The corresponding numerically calculated spectra [see Eq. (14)] are shown in panels (c) and (d).
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Figure 5
Experimental SHG spectra (blue lines) with $B | | [001], K | | [1 \bar{1} 0]$ and (a) $E^{in} | | [11 \sqrt{2}], E^{out} | | [\bar{1} \bar{1} \sqrt{2}]$ and (b) $E^{in} | | [110], E^{out} | | [110]$ . The corresponding numerically simulated spectra (grayscale) have been shifted by $- 0.5 meV$ to allow for a better comparison. The main visible features belong to excitons with principal quantum numbers $n = 3$ and 4. The feature visible at $E \approx 2.162 eV, B \approx 8 - 10 T$ in the numerical spectrum in panel (a) has an intensity exceeding the color palette scale and is most likely due to some numerical artifact.
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Figure 6
Experimental spectra (blue lines) with $K ∥ [111], E^{in} ∥ E^{out} ∥ [11 \bar{2}]$ in (a) Faraday configuration with $B ∥ [111]$ and (b) Voigt configuration with $B ∥ [1 \bar{1} 0]$ . The corresponding numerically simulated spectra (shifted by $- 0.5 meV$ ) are shown in grayscale. The main features visible belong to excitons with principal quantum numbers $n = 3$ and 4. Note that the $3 D$ line in panel (a) exceeds the upper limit of the gray scale.
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