Spin decoherence in graphene quantum dots due to hyperfine interaction

Carbon-based systems are prominent candidates for a solid-state spin qubit due to weak spin-orbit and hyperfine interactions in combination with a low natural abundance of spin-carrying isotopes. We consider the effect of the hyperfine interaction on the coherence of an electron spin localized in a graphene quantum dot. It is known that the hyperfine interaction in these systems is anisotropic promising interesting physics. We calculate the dynamics of an electron spin surrounded by a bath of nuclear spins in a non-Markovian approach using a generalized master equation. Considering a general form of the hyperfine interaction, we are able to extend the range of validity of our results to other systems beyond graphene. For large external magnetic fields, we find within Born approximation that the electron spin state is conserved up to small corrections, which oscillate with a frequency determined by the hyperfine interaction. The amplitude of these oscillations decays with a power law, where its initial value depends on the specific form of the anisotropy. Analyzing this in more detail, we identify two distinct classes of anisotropy, which can be both found in graphene depending on the orientation of the external magnetic field with respect to the carbon layer.

Figure 1

Schematic illustration of a two-dimensional quantum dot. An electron spin $\mathbf{S}$ is localized in a rotational symmetric QD, where it is in contact with bath of nuclear spins ${\mathbf{I}}_k$. Due to the confinement, the spatial distribution of the electron in the ground state of the QD may be described by a Gaussian envelope function $\phi \left(\mathbf{r}\right)$ given in Eq. (13), which in turn leads to a nonuniform HI between the electron spin and the nuclear spins. Finally, an external magnetic field is applied either perpendicular (${\mathbf{B}}_{\perp }$) or parallel (${\mathbf{B}}_{\parallel }$) with respect to the plane of the dot.

Figure 2

Illustration of the contour integral in the complex plane with the analytic features of the denominator functions $D_{+}\left(s\right)$ and $D_{-}\left(s\right)$ consisting of branch cuts (red dashed lines) and poles (blue and green circles). The integral in Eq. (63) is completed to a closed contour by an exponentially vanishing integral over the great circle, integrals along the upper and lower branch cuts, $B_{\beta ,\lambda }^{\alpha }\left(t\right)$, and an integral along the imaginary axis $I_{\beta ,\lambda }\left(t\right)$. Within this closed contour lie the poles $s_{j,\pm }$, $j=1,3$, whereas the poles $s_{2,\pm }$ are not encircled and, thus, do not contribute to the integral. Note that the poles $s_{j,-}$ are relevant only for the calculation of $\langle S_{2,+}\rangle \left(t\right)$.

Figure 3

Real part of $P_{+,1}\left(t\right)$ (solid, black) as a function of time obtained by numerical integration. For times $t\gtrsim {\tau }_{\text{HI}}$, it is asymptotically described by an oscillating function (dashed, red), whose amplitude decays with $\sim t^{-1}$ (dotted, blue), where the amplitude is proportional to the polarization $p$. The oscillations show a frequency of $f={\tau }_{\text{HI}}^{-1}\propto \left|{\lambda }_z\right|A$ determined by the HI.

Figure 4

Imaginary part of $P_{+,1}\left(t\right)$ (solid, black) showing a similar behavior as the real part depicted in Fig. 3. In contrast to the real part, the amplitude of the imaginary part does not depend on the polarization.

Figure 5

Real part of $P_{+,2}\left(t\right)$ (solid, black) as a function of time obtained by numerical integration. For times $t\gtrsim {\tau }_{\text{HI}}$, it is asymptotically described by an oscillating power law decay (dashed, red; dotted, blue) $\sim t^{-1}$, where the amplitude is proportional to the polarization $p$ and the frequency is given by $f={\tau }_{\text{HI}}^{-1}$.

Figure 6

Real part of $P_{z,2}\left(t\right)$ (solid, black) as a function of time obtained by numerical integration showing a similar behavior as $\mathrm{Re}\left[P_{+,1}\left(t\right)\right]$ depicted in Fig. 3. In contrast to this, however, its amplitude does not depend on the polarization.

Figure 7

Imaginary part of $P_{z,2}\left(t\right)$ (solid, black) as a function of time obtained by numerical integration showing a similar behavior as $\mathrm{Im}\left[P_{+,1}\left(t\right)\right]$ depicted in Fig. 4. Deviating from this, the amplitude depends on the polarization of the nuclear bath.