Photoexcitation of electron wave packets in quantum spin Hall edge states: Effects of chiral anomaly from a localized electric pulse

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Photoexcitation of electron wave packets in quantum spin Hall edge states: Effects of chiral anomaly from a localized electric pulse

Fabrizio Dolcini, Rita Claudia Iotti, Arianna Montorsi, and Fausto Rossi

Phys. Rev. B 94, 165412 – Published 13 October 2016

Abstract

We show that, when a spatially localized electric pulse is applied at the edge of a quantum spin Hall system, electron wave packets of the helical states can be photoexcited by purely intrabranch electrical transitions, without invoking the bulk states or the magnetic Zeeman coupling. In particular, as long as the electric pulse remains applied, the photoexcited densities lose their character of right and left movers, whereas after the ending of the pulse they propagate in opposite directions without dispersion, i.e., maintaining their space profile unaltered. Notably we find that, while the momentum distribution of the photoexcited wave packets depends on the temperature $T$ and the chemical potential $μ$ of the initial equilibrium state and displays a nonlinear behavior on the amplitude of the applied pulse, in the mesoscopic regime the space profile of the wave packets is independent of $T$ and $μ$ . Instead, it depends purely on the applied electric pulse, in a linear manner, as a signature of the chiral anomaly characterizing massless Dirac electrons. We also discuss how the photoexcited wave packets can be tailored with the electric pulse parameters, for both low and finite frequencies.

Received 9 August 2016

DOI:https://doi.org/10.1103/PhysRevB.94.165412

Physics Subject Headings (PhySH)

Dirac fermions Mesoscopics Optoelectronics Photoconductivity

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Fabrizio Dolcini^1,2,*, Rita Claudia Iotti¹, Arianna Montorsi¹, and Fausto Rossi¹

¹Dipartimento di Scienza Applicata e Tecnologia del Politecnico di Torino, I-10129 Torino, Italy
²CNR-SPIN, Monte S.Angelo - via Cinthia, I-80126 Napoli, Italy

^*fabrizio.dolcini@polito.it

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Issue

Vol. 94, Iss. 16 — 15 October 2016

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Images

Figure 1
Spatially localized electric pulse is applied to a quantum spin Hall bar, thereby photoexciting counterpropagating electronic wave packets in the helical edge states. The wave packets propagate without dispersion, due to the Dirac linear spectrum of the edge states, with a well defined spin orientation, due to their helical nature. Inset: the two branches $\pm$ of the linear Dirac spectrum of the helical edge states emerging in the topological phase of a QSH system.
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Figure 2
Space profile of the electron density $Δ n = Δ n_{+} + Δ n_{-}$ photoexcited by the pulse (49) in the regime of low frequency, namely for $Ω = 10 GHz, τ = 50 fs$ , and $Δ = 50 nm$ , is shown at three different snapshots: $t = 0.01 ps$ (thin solid curve), $t = 0.5 ps$ (dashed curve), and $t = 1.5 ps$ (thick solid curve). In this regime, the electronic density profiles reproduce the space profile of the electric pulse, with, however, a renormalized space extension parameter $Δ_{e l}$ given by Eq. (54). The profile is independent of the temperature and chemical potential of the initial electronic equilibrium state, and is linear in the amplitude $E_{0}$ of the applied pulse, as is emphasized by the vertical axis label. The two photoexcited wave packets counterpropagate with velocity $v_{F} = 5 \times 10^{5} m / s$ .
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Figure 3
Space profile of the electron density $Δ n = Δ n_{+} + Δ n_{-}$ photoexcited by the pulse (49) in the regime of high frequency, namely for $Ω = 40 THz, τ = 150 fs$ , and $Δ = 50 nm$ , is shown at three different snapshots: $t = 0.2 ps$ (thin solid curve), $t = 1.0 ps$ (dashed curve), and $t = 2.0 ps$ (thick solid curve). In this regime, the electronic wave packet exhibits space and time oscillations, characterized by a frequency $Ω_{e l}$ , given by Eq. (56), renormalized with respect to the pulse frequency $Ω$ . Again, the profile is independent of the temperature and chemical potential of the initial electronic equilibrium state, and is linear in the amplitude $E_{0}$ of the applied pulse. The propagation velocity is $v_{F} = 5 \times 10^{5} m / s$ .
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Figure 4
Momentum distribution $Δ f_{+} (k)$ of the right-moving electron wave packet photoexcited by the pulse (49) with $Δ = 100 nm$ and $τ = 100 fs$ and $Ω = 10 GHz$ is shown as a function of the wave vector deviation $k - k_{F}$ from the Fermi wave vector $k_{F}$ , for different values of the electric pulse amplitude $E_{0} = 2 kV / m$ (dotted curve), $E_{0} = 5 kV / m$ (dashed curve), $E_{0} = 10 kV / m$ (thin solid curve), and $E_{0} = 25 kV / m$ (thin solid curve). The initial equilibrium state is characterized by a temperature $T = 2 K$ and a chemical potential $μ = ℏ v_{F} k_{F}$ , with $v_{F} = 5 \times 10^{5} m / s$ . Differently from the space profile of the photoexcited density shown in Figs. 2 and 3, $Δ f_{+} (k)$ depends nonlinearly on $E_{0}$ .
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Figure 5
Momentum distribution $Δ f_{+} (k)$ of the right-moving electron wave packet photoexcited by the pulse (49) with $E_{0} = 25 kV / m, Δ = 100 nm$ , and $τ = 100 fs$ and $Ω = 10 GHz$ is shown as a function of the wave vector deviation $k - k_{F}$ from the Fermi wave vector $k_{F}$ , for different values of temperature $T$ of the initial equilibrium state: $T = 100 mK$ (thick solid curve), $T = 1 K$ (dashed curve), and $T = 4 K$ (thin solid curve). The Fermi velocity is $v_{F} = 5 \times 10^{5} m / s$ . While sharp peaks and oscillations related to the localization length scale $Δ$ of the applied pulse arise at low temperature, with increasing $T$ the oscillations are washed out by thermal fluctuations and the peaks of $Δ f_{+} (k)$ decrease and broaden.
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