Two-electron dephasing in single Si and GaAs quantum dots

We study the dephasing of two-electron states in a single quantum dot in both GaAs and Si. We investigate dephasing induced by electron-phonon coupling and by charge noise analytically for pure orbital excitations in GaAs and Si, as well as for pure valley excitations in Si. In GaAs, polar optical phonons give rise to the most important contribution, leading to a typical dephasing rate of $\sim$5.9 GHz. For Si, intervalley optical phonons lead to a typical dephasing rate of $\sim$140 kHz for orbital excitations and $\sim$1.1 MHz for valley excitations. For harmonic, disorder-free quantum dots, charge noise is highly suppressed for both orbital and valley excitations, since neither has an appreciable dipole moment to couple to electric field variations from charge fluctuators. However, both anharmonicity and disorder break the symmetry of the system, which can lead to increased dipole moments and therefore faster dephasing rates.

Figure 1

Calculated singlet-triplet dephasing rates of two electrons in a single dot via the phonon mechanisms considered in Sec. 3 and the charge noise mechanism considered in Sec. 4. (a) Plot of dephasing rates versus the lateral extent of the electron wave functions for the mechanisms pertinent for GaAs: polar optical phonons [Eq. (24)], deformation potential phonons [Eq. (18)], transverse piezoelectric phonons [Eq. (22)], and longitudinal piezoelectric phonons [Eq. (20)]. (b) Plot of singlet-triplet dephasing rates versus lateral wave function extent for different mechanisms in Si: intervalley phonons for valley excitations [Eq. (33)], and intervalley phonons [Eq. (32)], longitudinal acoustic phonons [Eq. (29)], and transverse acoustic phonons [Eq. (30)] for orbital excitations. The dephasing rates due to charge noise plotted for both materials systems are determined from Eq. (40), using the energy fluctuations for orbital excitations in a single dot [Eq. (36)], and assuming zero dipole moment. In all cases, the vertical extent of the wave function $d$ is set to 3 nm.

Figure 2

Cartoon of the absolute value of the Fourier transform of the difference between the triplet and singlet charge distributions, ${\left|A\right|}^2$ [Eq. (13)] in Si. Here, the shaded regions indicate the $\mathbf{k}$ values that contribute significantly to dephasing. Phonons that couple electrons in the same valley (intravalley processes) lie near the origin, while phonons that couple electrons in different valleys (intervalley processes) lie near $\mathbf{k}=\pm 2k_0{\widehat{k}}_z$. (a) The contribution resulting from an orbital-like first excited state [Eq. (25)]. (b) The contribution resulting from a valley-like first excited state [Eq. (26)].

Figure 3

Singlet-triplet dephasing rate ${\Gamma }_{\mathit{ST}}$ due to charge noise and electron-phonon coupling as a function of effective dipole moment $p_0$. In this plot, a constant quadrupole contribution of $1.3$ MHz [Eq. (36)] is added to the dipole contribution, which is the estimated dephasing rate from charge noise for a dot with a purely harmonic confinement potential, for which $p_0/2$ is zero. Here, we have set the lateral electron confinement to be $L=40$ nm, the vertical confinement to be $d=3$ nm, and have assumed an orbital excited state. The charge noise curve for Si is estimated using a dielectric constant ${\epsilon }_0^{\mathrm{Si}}=11.7{\epsilon }_{\mathrm{vac}}$. In a perfectly harmonic dot, $p_0\approx 0$, but anharmonicity and disorder can introduce a dipole moment of $p_0\lesssim eL$. In GaAs we expect phonon-mediated dephasing to be most important, but in Si quantum dots charge noise can easily dominate. The circle marker indicates the dephasing due to charge noise at $p_0/e=1.8$ nm, an estimated dipole moment for realistic devices (Refs. 52 and 53).