Ultralong spin decoherence times in graphene quantum dots with a small number of nuclear spins

We study the dynamics of an electron spin in a graphene quantum dot, which is interacting with a bath of less than ten nuclear spins via the anisotropic hyperfine interaction. Due to substantial progress in the fabrication of graphene quantum dots, the consideration of such a small number of nuclear spins is experimentally relevant. This choice allows us to use exact diagonalization to calculate the long-time average of the electron spin as well as its decoherence time. We investigate the dependence of spin observables on the initial states of nuclear spins and on the position of nuclear spins in the quantum dot. Moreover, we analyze the effects of the anisotropy of the hyperfine interaction for different orientations of the spin quantization axis with respect to the graphene plane. Interestingly, we then predict remarkable long decoherence times of more than 10 ms in the limit of few nuclear spins.

Figure 1

(a) The graphene [$(x_1,x_2,x_3)$] and quantization axis [$(x,y,z)$] reference frames. (b) A graphene QD (red sites) for a Gaussian envelope function with $K=10$ uniformly random distributed ${}^{13}\mathrm{C}$ atoms (blue squares) carrying a nuclear spin $1/2$. The extent of the dot over the graphene lattice is defined via the electron envelope function, where all sites within the dot obey the cutoff relation defined in Eq. (2). (c) The envelope function for fixed $x_1=0$ and $x_1=22a_{NN}$, respectively. The dashed line indicates the cutoff defined in Eq. (2).

Figure 2

Exemplary time evolution of the longitudinal electron spin component $\langle S_z\rangle \left(t\right)$ as a function of time for out-of-plane orientation $\beta =0$, cf. Fig. 1. For a certain range of time $[T_{\text{min}},T_{\max }]$ using a resolution $\Delta T$, we calculate the long-time average ${\langle S_z\rangle }_T$ and the standard deviation ${\sigma }_{S_z}$ and find the maximal deviation $\Delta S_z$ around this value. These quantities as well as the details of the oscillations including the beating structure depend on the choice of the parameters. The decoherence time $T_D$ is determined by a constant threshold $C_S$.

Figure 3

Plot of the long-time average ${\langle S_z\rangle }_T$ for out-of-plane orientation $\beta =0$ and $K=3$ and 6 nuclear spins without an external magnetic field. We considered 51 different RC initial states and 51 random configurations. For both numbers of nuclear spins, the electron spin looses roughly one half of its amplitude to ${\langle S_z\rangle }_T\approx -0.22\hslash$. The horizontal stripes for $K=3$ indicate the importance of the initial states for few nuclear spins, while configurations seem less relevant signaled by weaker vertical structures. This changes for $K=6$ nuclear spins, where the configurations dominate over initial states indicated by the vertical structures in the right plot.

Figure 4

Same plot as in Fig. 3 but for in-plane orientation $\beta =\pi /2$. For all parameters, the long-time average of the longitudinal electron spin component sharply saturates at ${\langle S_z\rangle }_T\approx 0$. In contrast to the out-of-plane case, the results seem independent of the initial state even in the few spin regime. For certain configurations, however, we find a non-negligible dependence on the initial state.

Figure 5

Plot of the long-time average ${\langle S_z\rangle }_T$, the standard deviation ${\sigma }_{S_z}$ and, the sample range $\Delta S_z$ as a function of the number of nuclear spins $K$ for in-plane ($\beta =\pi /2$, red) and out-of-plane orientation ($\beta =0$, black). The values are obtained by averaging over 51 RC initial states and 51 different configurations, see Figs. 3 and 4. Error bars are given by the standard deviation with respect to averaging over all $51\times 51$ results. While the long-time average is almost constant, the averaged standard deviation ${\sigma }_{S_z}$ as well as the averaged sample range $\Delta S_z$ strongly decrease for larger $K$ indicating the reduction of fluctuations and of the occurrence of beating or recurrence events.

Figure 6

(a) Eigenvalues ${\lambda }_i$ of the AHI Hamiltonian for for $K=6$ nuclear spins and configuration $C=10$ in normalized units of $A_{\mathrm{HI}}{\left|\varphi \left(r_K\right)\right|}^2$. (b) Eigenvalues ${\lambda }_i$ for $K=6$ and $C=3$. (c) Number of distinct eigenvalues ${\lambda }_i$ as a function of the relative probabilities $|\varphi \left(r_{K-1}\right){|}^2/{\left|\varphi \left(r_K\right)\right|}^2$ and $|\varphi \left(r_{K-2}\right){|}^2/{\left|\varphi \left(r_K\right)\right|}^2$ for $K=6$ nuclear spins. If both $|\varphi \left(r_{K-1}\right){{|}^2/\varphi \left(r_K\right)|}^2\approx 1$ and $|\varphi \left(r_{K-2}\right){{|}^2/\varphi \left(r_K\right)|}^2\approx 1$, at least three nuclear spins are strongly interacting with the electron spin causing a spectrum with many different eigenvalues as depicted in (b). If only the most central nuclear spin couples strongly to the electron spin (lower left part), the spectrum is highly degenerate showing only three different eigenvalues as shown in (a). The upper limit for the number of 15 has no deeper meaning besides distinguishing both types of spectra.

Figure 7

Dependence of the long-time average ${\langle S_z\rangle }_T\left(\beta \right)$ on the orientation of the quantization axis with respect to the graphene plane for $K=6$ nuclear spins. The example shown here was calculated for the configuration $C=3$, whose continuous spectrum is presented in Fig. 6. The only parameter used to fit the numerical values to the analytic curve ${\langle S_z\rangle }_T\left(\beta \right)={\langle S_z\rangle }_T\left(0\right){cos}^2\left(\beta \right)$ is the out-of-plane value ${\langle S_z\rangle }_T\left(0\right)$.

Figure 8

Normalized decoherence time $T_D·{\left|\varphi \left(r_K\right)\right|}^2$ as a function of 51 RC initial states and 51 random configurations for $K=6$ nuclear spins in in-plane and out-of-plane orientation. While the decoherence time is almost the same for different initial states, it strongly depends on the configurations showing deviations over several orders of magnitude. White spaces indicate the total lack of decoherence up to absolute times of $0.1\mathrm{s}$ given a threshold of $C_S=-0.325\hslash$. For special configurations, $C=10,32$, and 35, and $\beta =0$ there is no decoherence at all, but coherent oscillations of the electron spin.

Figure 9

Normalized decoherence time $T_D·{\left|\varphi \left(r_K\right)\right|}^2$ as a function of the number of distinct eigenvalues of the AHI Hamiltonian for $K=6$ nuclear spins in out-of-plane orientation. For this plot all out-of-plane results presented in Fig. 8 are considered. Obviously, long decoherence times occur only for a small number of distinct eigenvalues. Together with the results of Fig. 6 the importance of the relative coupling strengths $\propto$$|\varphi \left(r_{K-1}\right){|}^2/{\left|\left(\varphi \left(r_K\right)\right)\right|}^2,\cdots$ and, hence, of the relative position of different nuclei becomes evident.

Figure 10

Relative number of absolute decoherence times $T_D$ falling in a certain time interval $[T_{\min },T_{\max }]$ for different numbers $K$ of nuclei in in-plane and out-of-plane orientation. For each $K$ 51 RC initial states and 51 configurations were considered leading to $N_{\text{calc}}=2601$ calculations in total. (a) In the out-of-plane case, long decoherence times are clearly dominating for few nuclear spins. Increasing $K$ leads to a quick decay of the decoherence times such that short times $T_D$ are common. Very short decoherence times start to become relevant for $K>6$. (b) In the in-plane case, even for few nuclear spins short relaxation times are the rule. For increasing $K$, the percentage of short decoherence times is growing further.