Delocalization of charge and current in a chiral quasiparticle wave packet

Subhajit Sarkar

Phys. Rev. B 97, 115112 – Published 7 March 2018

Abstract

A chiral quasiparticle wave packet (c-QPWP) is defined as a conventional superposition of chiral quasiparticle states corresponding to an interacting electron system in two dimensions (2D) in the presence of Rashba spin-orbit coupling (RSOC). I investigate its internal structure via studying the charge and the current densities within the first-order perturbation in the electron-electron interaction. It is found that the c-QPWP contains a localized charge which is less than the magnitude of the bare charge and the remaining charge resides at the system boundary. The amount of charge delocalized turns out to be inversely proportional to the degenerate Fermi velocity $v_{0} (= \sqrt{α^{2} + 2 μ / m})$ when RSOC (with strength $α$ ) is weak, and therefore externally tunable. For strong RSOC, the magnitudes of both the delocalized charge and the current further strongly depend on the direction of propagation of the wave packet. Both the charge and the current densities consist of an anisotropic $r^{- 2}$ tail away from the center of the wave packet. Possible implications of such delocalizations in real systems corresponding to 2D semiconductor heterostructure are also discussed within the context of particle injection experiments.

Received 27 September 2017
Revised 18 January 2018

DOI:https://doi.org/10.1103/PhysRevB.97.115112

Physics Subject Headings (PhySH)

Fermions Quasiparticles & collective excitations Spin-orbit coupling

Fermi liquid theory Feynman diagrams

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Subhajit Sarkar^*

Institute of Physics, P.O.: Sainik School, Bhubaneswar 751005, Odisha, India

^*sbhjt72@gmail.com, subhajit@iopb.res.in

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Issue

Vol. 97, Iss. 11 — 15 March 2018

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Images

Figure 1
(a) Dispersion relation ${\bar{ξ}}_{k, s}$ vs ${\bar{k}}_{x}$ and ${\bar{k}}_{y}$ , where the $z$ axis represents ${\bar{ξ}}_{k, s} = \frac{ξ_{k, s}}{m α^{2}}$ , and ${\bar{k}}_{x} = \frac{k_{x}}{m α}$ and ${\bar{k}}_{y} = \frac{k_{y}}{m α}$ . (b) Fermi surfaces are concentric circles, $k_{F}^{+}$ is the radius corresponding to $s = + 1$ , and $k_{F}^{-}$ is the radius corresponding to $s = - 1$ . Quasiparticle picture is valid only near the Fermi surface, quasiparticle being schematically shown as red circle and blue dot on the respective Fermi surfaces.
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Figure 2
Scattering processes leading to (A1). (a) Electron-hole pair annihilation by $c_{k^{'} - q, s}^{†} c_{k^{'}, s^{'}}$ , (b) annihilation of a chiral hole $k_{2}, s_{3}$ and a chiral electron $k_{1} - p, s_{1}$ by $c_{k^{'} - q, s}^{†} c_{k^{'}, s^{'}}$ . The dashed line denoted by $M_{s, s^{'}} (k^{'}, q)$ represents the form factor (or the matrix element) $\frac{1}{2} [1 + s s^{'} e^{- i (θ_{k^{'} - q} - θ_{k^{'}})}]$ corresponding to the charge density operator (8). This illustrates how the charge density operator takes part in the scattering processes. Conservation of momentum and chirality index at each vertex has been taken into account. Points to note that when $M_{s, s^{'}} (k^{'}, q)$ (rather a vector quantity in nature) is taken to be $\frac{1}{m} (k^{'} - \frac{q}{2}) \frac{1}{2} [1 + s s^{'} e^{- i (θ_{k^{'} - q} - θ_{k^{'}})}]$ , the scattering processes lead to (A3), and when taken to be $\frac{α}{2} [s^{'} e^{i θ_{k^{'}}} (\hat{x} - i \hat{y}) + s e^{- i θ_{k^{'} - q}} (\hat{x} + i \hat{y})]$ , these lead to (A5).
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Figure 3
(a) Schematic picture of the delocalization effect where deep blue dot corresponds to the peak of the Gaussian distribution and the blue shaded area around the dot signifies the Gaussian distribution of the localized charge within the spread of the wave packet; delocalized charge is indicated in the figure. The delocalized charge resides at the boundary of a finite system or in the case of an infinite system at a large distance away from the center of the c-QPWP. (b) Spatial distribution of the localized charge, $n_{0} {(r) |}_{loc} = \frac{1 + f_{0}}{2 π a^{2}} e^{- r^{2} / a^{2}}$ .
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Figure 4
Self-energy diagrams.
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Figure 5
Other scattering processes.
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Figure 6
$v_{0} f_{0}^{\pm}$ as a function of the dimensionless electron gas parameter $r_{s}$ . In the limit of $r_{s} \to 0$ , both the values of $v_{0} f_{0}^{+}$ and $v_{0} f_{0}^{-}$ diverge to $- \infty$ , however, in this limit of $r_{s}$ the electron density becomes too high and the system becomes a homogeneous electron gas (instead of being a Fermi liquid) and the Coulomb interaction becomes very small compared to the kinetic energy of individual particles. Typical value of $r_{s}$ ranges [41] from 1 to 20, however (up to $r_{s} = 5$ is shown), even for InGaAs 2DEG with $r_{s} = 0.18$ one finds $f_{0}^{+} \approx \frac{- 5.4}{v_{0}}$ and $f_{0}^{-} \approx \frac{- 6.83}{v_{0}}$ as mentioned above.
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