Spin polarization of entangled and mixed electron states in a beam splitter geometry coupled to an electron reservoir

Luis A. González-Árraga, Bertrand Berche, and Ernesto Medina

Phys. Rev. B 86, 155312 – Published 16 October 2012

Abstract

We study the spin polarization of mixed and entangled electron states in a four-probe/beam splitter geometry with local Rashba and Dresselhaus interactions. A pair of maximally entangled electrons collides with the beam splitter and enters into two perpendicular branches of length $L$ , composed of spin-orbit active materials (gate-confined two-dimensional electron gas). One of the branches is connected to an electron reservoir that acts as a source of decoherence by either behaving as a voltage probe or as a controlled source or sink of current at fixed voltage. Such a decoherence source is used to modify the entropy of an unpolarized incoming state in order to generate electron polarization at one or both output branches. The degree of entanglement of the global state and the spin polarization is computed for the outgoing electrons as a function of the coupling to the electron reservoir. Experimentally available spin-orbit strength at the beam splitter arms, for arm lengths of a few micrometers, is able to modulate spin polarization up to 80 $%$ in particular spin axes. The Dresselhaus and Rashba coefficients play a symmetric role in modulating the polarization. Significantly less polarization is achieved for incoming mixed states due to the local operation of the reservoir.

Received 16 May 2012

DOI:https://doi.org/10.1103/PhysRevB.86.155312

Authors & Affiliations

Luis A. González-Árraga

Laboratorio de Física Estadística de Sistemas Desordenados, Centro de Física, Instituto Venezolano de Investigaciones Científicas (IVIC), Apartado 21827, Caracas 1020 A, Venezuela

Bertrand Berche and Ernesto Medina

Laboratorio de Física Estadística de Sistemas Desordenados, Centro de Física, Instituto Venezolano de Investigaciones Científicas (IVIC), Apartado 21827, Caracas 1020 A, Venezuela and Statistical Physics Group, P2M, Institut Jean Lamour, Nancy Université, BP70239, F-54506 Vandœuvre les Nancy, France

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Issue

Vol. 86, Iss. 15 — 15 October 2012

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Images

Figure 1
Sketch of the spin-polarizing device. Incoming electrons traveling through leads 1 and 2 are scattered into lead 3 and through the four-port beam splitter QPC implemented within a SO active region of the Rashba and Dresselhaus type. Lead 4 is connected to a voltage probe model electron reservoir. The electrons are collected at detectors $D_{3}$ and $D_{4}$ . $x$ denotes the position of the reservoir junction with respect to the QPC.Reuse & Permissions
Figure 2
Plot of the concurrence (shades of gray) and linear entropy (dashed line) of the outgoing electrons as a function of coupling to the reservoir $ε$ for the spin-orbit strengths indicated ( $β = 0.004$ a.u). The linear entropy is evaluated for the smallest value of the SO interaction. The energy of the injected electrons is $E = 0.2$ a.u. at the Fermi energy of the reservoir at a reference temperature of $90$ K. The inset shows the decoherence current as a function of $ε$ from Eq. (20).Reuse & Permissions
Figure 3
Concurrence versus the Rashba coupling parameter, for $β = 0.004$ a.u., Fermi energy $E_{F} = 0.21 a . u .$ (top panel) $E_{F} = 0.19$ a.u. (bottom panel) while the electron energy in the interferometer arm is $E = 0.2 a . u .$ . Depending on the electron energy, temperature can either enhance concurrence oscillations (top) or reduce them (bottom). The temperature range has been chosen rather large to enhance the visibility of the effect. Arrow indicates the direction of temperature increase. $T = 3.157 \times 10^{5} K = 1$ a.u..Reuse & Permissions
Figure 4
The $z$ component of the polarization for the outgoing electrons at detector $D_{4}$ as a function of coupling to the reservoir $ɛ$ and for representative values of Rashba spin-orbit coupling. The output state is completely unpolarized for $ε \to 0$ . As current is leaked from or to the reservoir, a polarization arises that depends on the spin-orbit strength of the SO active region. The Dresselhaus interaction is set to 0.004 a.u. The inset shows the $z$ component of the polarization at the other detector for $α = 0.0019$ .Reuse & Permissions
Figure 5
The $z$ component of the polarization for the detector $D_{4}$ as a function of the Rashba SO parameter at two different couplings to the reservoir indicated in the figure. When current is injected to the beam splitter arm (before nodal point in Fig. 3, also inset in Fig. 2) the oscillation is $π$ out of phase compared to the situation when current is drawn.Reuse & Permissions
Figure 6
The polarization of the outgoing state as a function of coupling to the electron reservoir $ε$ for different values of the Rashba parameter. The energy is fixed at the Fermi energy of the reservoir and the temperature of the reservoir is 90 K. The position of the reservoir junction is set to $x = 1 μ$ m (see Fig. 1).Reuse & Permissions

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