Spin relaxation and g-factor manipulation in quantum dots

Abstract

Phonon-induced spin relaxation rates and electron g-factor tuning of quantum dots are studied as function of in-plane and perpendicular magnetic fields for different dot sizes. We consider Rashba and Dresselhaus spinorbit mixing in wide and narrow-gap semiconductors, and show how Zeeman sublevels can relax via piezoelectric (GaAs) and deformation (InSb) potential coupling to acoustic phonons. We find that strong confinement may induce minima in the rates at particular values of the magnetic field (due to a magnetic field-induced cancellation of the spin-orbit effects), where spin relaxation times can reach seconds. We also report on g-factor anisotropy. We obtain good agreement with available experimental values.Keywords: g-factor; Phonon-induced spin relaxation; Quantum dotsSince the proposal of a qubit based on the electron spin of quantum dots (QDs) [1], much work has been done to understand the processes that may cause their relaxation, since long coherence times are required. One of those processes is related to the phonon-induced spin-flip rates of Zeeman sublevels in QDs in magnetic fields, where the spin purity of the levels is broken by the spin-orbit (SO) interaction. A recent experiment [2] has shown a spin relaxation time ≈ 0.55 ms at an in-plane field of 10 T in a GaAs QD defined in a 2DEG. In general, SO effects have been considered via perturbation theory [3], although exact treatments have also been presented [4,5]. The perturbative approach, which includes only a few states, has been called into question by the demonstration that a larger basis is needed in order to achieve convergence even for the lowest QD states when the QD vertical width is narrow [4], as a complex interplay between different energy scales can be present [6]. Insights on the purity of the spin degree of freedom of electrons in QDs can also be extracted from measurements of their effective g-factor, e.g., by means of capacitance [7] and energy [8] spectroscopies.In this work we study spin-flip rates and g-factor tuning in QDs under SO influence. Our goal is to compare wide and narrow-gap materials under in-plane and perpendicular magnetic fields, at different QD sizes.The QD is defined by an in-plane parabolic confinement, V (r) = mω 2 0 r 2 /2, where m (ω 0 = E 0 / ) is the electronic effective mass (confinement frequency); the QD lateral length is l 0 = /(mω 0 ). The vertical confinement V (z) is strong enough so that only the state in the first quantum well subband is relevant, and its function is ϕ z (z) = 2/z 0 sin (πz/z 0 ) if a hard wall is assumed, z 0 being the QD vertical well thickness. In a magnetic field B, the unperturbed Hamiltonian, H 0 = 2 k 2 /2m+V (r)+H Z , has the well-known FockDarwin (FD) solution, where H Z = g 0 µ B B · σ/2 is the Zeeman term, g 0 is the bulk g-factor, and k is the kinetic momentum that includes the magnetic vector potential. We include all SO terms in 2D zincblende QDs, namely Rashba * present address: Physics Department, University of Ottawa, Ottawa, Ontario K1N6N5, Canada.and Dresse...

Two- and Four-Level Systems in Magnetic Fields Restricted in Time

“…where µ B is the Bohr magneton and g 1 and g 2 are effective g-factors for the corresponding spins (see for example [15]), one can show that the evolution operator Û (t) for the equation ( 13) can be reduced to an evolution operator û (t) (4) for the Schrödinger equation of a two-level system [8]. Such a reduction is given by the equation…”

“…Although the expression ( 18) allows to construct a variety of external fields with some particular characteristic, the solution for a general field can hardly be constructed in this manner. In this section we will analyze a special case for a specific external parallel field (15) restricted in time. Consider a four-level system in which the field difference B − (17) varies adiabatically with time, i.e., a variation that met the adiabaticity criteion [16], while the interaction function is constant.…”

“…In addition, it is reasonable to assume that if the only quantity that varies is B − , and B + ≫ B − , the interaction function will remain constant [14]. With regard to the variation of B − , there are some proposals for the application of localized magnetic fields [17] and some techniques that permit the manipulation of the g-factor by changing the size of the dots or by the application of external electromagnetic fields [15,18].…”

Four-level systems and a universal quantum gate

“…It is possible to couple two different QD [17] in such a way that g 1 = g 2 . In addition, some techniques permit the manipulation of the g-factor by changing the size of the dots or by the application of external electromagnetic fields [18,19]. The hyperfine coupling between the electron spin and the nuclear spins of the semiconductor material can also be explored to obtain a field gradient by producing a differential Overhauser field [20].…”

Four‐level systems and a universal quantum gate

“…It is possible to couple two different QD [17] in such a way that g 1 = g 2 . In addition, some techniques permit the manipulation of the g-factor by changing the size of the dots or by the application of external electromagnetic fields [18,19]. The hyperfine coupling between the electron spin and the nuclear spins of the semiconductor material can also be explored to obtain a field gradient by producing a differential Overhauser field [20].…”