Composite two-mode resonant state in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ mesa terahertz emitters: locking mechanism, $\pi/2$ phase selection, and mode-control conditions

Rectangular mesas patterned on Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (Bi-2212) single crystals behave as stacks of intrinsic Josephson junctions whose synchronized oscillations emit coherent terahertz radiation, with the mesa itself acting as the resonant cavity that locks them. Low-temperature scanning laser microscopy of one such mesa revealed a 'composite' two-mode resonant state — two cavity modes, one with and one without a transverse half-period across the mesa width, oscillating at a common frequency with a relative $\pi/2$ phase shift — that is corroborated, but not explained, by later polarization measurements on similar devices. The physical mechanism that locks the two modes, the origin of the $\pi/2$ phase selection, the quantitative role of electrode losses and c-axis boundary conditions, and the device conditions favoring composite over single-mode operation all remain open. The problem is to establish a predictive physical account of this composite resonant state.

Background

The layered cuprate superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (Bi-2212) behaves, for current flow along the crystallographic c-axis, as a natural stack of 'intrinsic' Josephson junctions (IJJs): superconducting CuO$_2$ double layers separated by insulating sheets, each pair forming a tunnel junction of atomic height. By the ac Josephson effect, a dc voltage $V$ across a junction drives an oscillation at frequency $f_J = V/\Phi_0$, where $\Phi_0 = 2.07\times10^{-15}$ Wb is the magnetic flux quantum. A single junction radiates negligibly, but if several hundred junctions oscillate in phase the combined source emits coherent continuous-wave radiation in the terahertz range — the technologically under-served 'terahertz gap' (Ozyuzer et al. 2007). In the standard device geometry, a rectangular 'mesa' etched from a Bi-2212 single crystal, the mesa acts as an electromagnetic cavity: a resonant cavity mode, with the ac electric field along the c-axis and standing-wave structure set by the mesa dimensions, captures the Josephson oscillations and synchronizes them, and the synchronized current sheet radiates from the mesa sidewalls (Klemm et al. 2011). Rectangular-mesa cavity modes are labeled $(n, m_x, m_y)$, where $n$ is the c-axis index and $m_x$, $m_y$ count half-wavelengths across the width and along the length; theory and angular-distribution measurements identify modes with a transverse component ($m_x = 1$) as the most efficient radiators.

The main experimental window into the excited cavity mode is low-temperature scanning laser microscopy (LTSLM): a focused laser beam locally heats a biased mesa, and the resulting local change of the dc voltage maps the standing-wave pattern of the excited mode, because the local perturbation of material parameters (Josephson critical current density, quasiparticle conductivities, London penetration depth) couples to the mode amplitude at the beam position. A long-standing ambiguity — whether oscillating LTSLM patterns reflect electromagnetic cavity resonances or static inhomogeneous temperature distributions ('hot spots') — was resolved for one device by combining LTSLM with direct surface thermometry (Benseman et al. 2019).

In this context, Benseman et al. (2019) studied a 60×360 μm$^2$, 640-junction Bi-2212 mesa emitting a single narrow line at 396 GHz (about 1.4 μW total power) at 75 K. Thermal maps showed no hot spots, and the LTSLM pattern showed, for the first time, the single half-period across the mesa width expected for a transverse ($m_x=1$) mode component. However, the pattern oscillated strongly along the mesa centerline and only weakly near the edges — inconsistent with the LTSLM response of any single cavity mode, which either peaks at the edges (for c-axis parameter perturbations) or oscillates with equal amplitude at center and edge (for in-plane parameter perturbations). The observed pattern could be reproduced only by a 'composite mode': two locked cavity modes — one without ($m_x=0$) and one with ($m_x=1$) a transverse half-period across the width — oscillating at the same frequency with a relative phase shift of $\pi/2$, so that their time-averaged response patterns add. Notably, the $\pi/2$ phase was inferred from pattern matching, not measured phase-sensitively, and the device emitted a single narrow line while both mode components were present. A second mesa of slightly different dimensions showed the same behavior, suggesting the composite state is intrinsic rather than a device accident.

A superposition of two phase-shifted modes with different transverse structure is also polarization-relevant: two orthogonal field components oscillating $\pi/2$ out of phase produce elliptically polarized radiation, and indeed full Stokes-parameter polarimetry (the standard decomposition of a polarization state into intensity and phase-related parameters) on similar rectangular mesas shows elliptically polarized emission with a non-zero phase difference between the field components along the width and length directions (Tsujimoto et al. 2020). Microstrip-antenna models describe the mesa modes (Kuwano et al. 2021; Rain et al. 2021), but the c-axis boundary condition at the mesa top and bottom — whether the ac electric field sees a node or an antinode — depends on the thin (30 nm, lossy) Au top electrode required for LTSLM versus the thick electrode of high-power devices, and is only approximately known. What is missing is the physics that selects and stabilizes the composite two-mode state.

Problem Statement

What physical mechanism locks two cavity modes of different transverse structure in a Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ mesa terahertz emitter to a common frequency with a relative $\pi/2$ phase shift — a value fitted from the LTSLM pattern rather than measured phase-sensitively — and what determines this phase? The goal is a predictive physical account of this composite two-mode resonant state. A resolving account must address all of the following interrelated aspects, without restricting the admissible mechanisms: identifying the locking interaction — candidate mechanisms discussed in the literature include the nonlinear Josephson medium, in-plane dissipation, the lossy symmetry-breaking electrode environment, geometric asymmetry, and external-circuit coupling, but the mechanism is not restricted to these — and deriving or simulating the resulting two-mode state; specifying the c-axis electromagnetic boundary conditions (node versus antinode of the ac electric field) set by the thin versus thick metallic contacts, a role that the same group's later work still treats as an assumption rather than a determined input; and characterizing the device conditions — mesa geometry and electrode configuration — under which the composite two-mode state is excited rather than a single cavity mode. The source work reported the composite state, framed single-mode design as important for maximizing the emitted terahertz power, and left the locking mechanism and the phase selection unexplained.

References

1. T. M. Benseman, A. E. Koshelev, V. Vlasko-Vlasov, Y. Hao, U. Welp, W.-K. Kwok, B. Gross, M. Lange, D. Koelle, R. Kleiner, H. Minami, M. Tsujimoto, and K. Kadowaki, 'Observation of a two-mode resonant state in a Bi2Sr2CaCu2O8+δ mesa device for terahertz emission', Physical Review B 100, 144503 (2019), DOI: 10.1103/PhysRevB.100.144503, https://doi.org/10.1103/PhysRevB.100.144503.
2. L. Ozyuzer, A. E. Koshelev, C. Kurter, N. Gopalsami, Q. Li, M. Tachiki, K. Kadowaki, T. Yamamoto, H. Minami, H. Yamaguchi, T. Tachiki, K. E. Gray, W.-K. Kwok, and U. Welp, 'Emission of Coherent THz Radiation from Superconductors', Science 318, 1291–1293 (2007), DOI: 10.1126/science.1149802, https://doi.org/10.1126/science.1149802.
3. R. A. Klemm, E. R. LaBerge, D. R. Morley, T. Kashiwagi, M. Tsujimoto, and K. Kadowaki, 'Cavity mode waves during terahertz radiation from rectangular Bi2Sr2CaCu2O8+δ mesas', Journal of Physics: Condensed Matter 23, 025701 (2011), DOI: 10.1088/0953-8984/23/2/025701, https://doi.org/10.1088/0953-8984/23/2/025701, arXiv:1010.4924, https://arxiv.org/abs/1010.4924.
4. M. Tsujimoto, S. Fujita, G. Kuwano, K. Maeda, A. Elarbi, J. Hawecker, J. Tignon, J. Mangeney, S. S. Dhillon, and I. Kakeya, 'Mutually synchronized macroscopic Josephson oscillations demonstrated by polarization analysis of superconducting terahertz emitters', Physical Review Applied 13, 051001 (2020), DOI: 10.1103/PhysRevApplied.13.051001, arXiv:2004.07422, https://arxiv.org/abs/2004.07422.
5. Y. Saito, I. Kakeya, and Y. Takano, 'Polarization analysis of terahertz emission from Bi-2212 cross-whisker intrinsic Josephson junction devices and their refractive index', Applied Physics Letters 121, 212601 (2022), DOI: 10.1063/5.0123290, https://doi.org/10.1063/5.0123290 (preprint: arXiv:2209.08506, https://arxiv.org/abs/2209.08506, titled 'Polarization analysis of terahertz emission from Bi-2212 cross-whisker intrinsic Josephson junction and its refractive index').
6. R. Kobayashi, K. Hayama, and I. Kakeya, 'Circuit models of simultaneously biased intrinsic Josephson junction stacks for terahertz radiations in high-bias regime', Applied Physics Express 15, 093002 (2022), DOI: 10.35848/1882-0786/ac8597, https://doi.org/10.35848/1882-0786/ac8597.
7. K. J. Kihlstrom, K. C. Reddy, S. Elghazoly, T. E. Sharma, A. E. Koshelev, U. Welp, Y. Hao, R. Divan, M. Tsujimoto, K. Kadowaki, W.-K. Kwok, and T. M. Benseman, 'Powerful Terahertz Emission from a Bi2Sr2CaCu2O8+δ Mesa Operating Above 77 K', Physical Review Applied 19, 034055 (2023), DOI: 10.1103/PhysRevApplied.19.034055, https://doi.org/10.1103/PhysRevApplied.19.034055.
8. R. Wieland, O. Kizilaslan, N. Kinev, E. Dorsch, S. Guénon, Z. Song, Z. Wei, H. Wang, P. Wu, D. Koelle, V. P. Koshelets, and R. Kleiner, 'Terahertz emission from mutually synchronized standalone Bi2Sr2CaCu2O8+x intrinsic-Josephson-junction stacks', Physical Review Applied 22, 044022 (2024), DOI: 10.1103/PhysRevApplied.22.044022, arXiv:2405.20742, https://arxiv.org/abs/2405.20742.
9. M. Zhang, S. Nakagawa, Y. Enomoto, Y. Kuzumi, R. Kikuchi, Y. Yamauchi, T. Hattori, R. A. Klemm, K. Kadowaki, T. Kashiwagi, and K. Delfanazari, 'Terahertz source-on-a-chip with decade-long stability using layered superconductor elliptical microcavities' (2025), arXiv:2506.22811, https://arxiv.org/abs/2506.22811.
10. J. R. Rain, P. Cai, A. Baekey, M. A. Reinhard, R. I. Vasquez, A. C. Silverman, C. L. Cain, and R. A. Klemm, 'Wave functions for high-symmetry, thin microstrip antennas, and two-dimensional quantum boxes', Physical Review A 104, 062205 (2021), DOI: 10.1103/PhysRevA.104.062205, arXiv:2108.07916, https://arxiv.org/abs/2108.07916.
11. G. Kuwano, M. Tsujimoto, Y. Kaneko, K. Nagayama, T. Imai, Y. Ono, S. Kusunose, T. Yuhara, H. Minami, T. Kashiwagi, and K. Kadowaki, 'Experimental validation of a microstrip antenna model for high-Tc superconducting terahertz emitters', Journal of Applied Physics 129, 223905 (2021), DOI: 10.1063/5.0054018, https://doi.org/10.1063/5.0054018.