Nuclear spin diffusion in quantum dots: Effects of inhomogeneous hyperfine interaction

We study the effect of contact hyperfine interaction on the nuclear spin diffusion coefficients in semiconductor quantum dots. The diffusion coefficients are calculated with both the method of moment and density matrix. We show that nuclear spin diffusion is strongly suppressed by the nonuniform hyperfine coupling resulting from the confined electron wave function. Our results agree with the observed suppression of nuclear spin diffusion in these structures in recent experiments, and clarify the degree of validity of the method of moment in an inhomogeneous system.

Figure 1

The ratio of NSD coefficients $D^{\mu \mu }∕D_0^{\mu \mu }$ as a function of spatial coordinate $z$ for various Fock-Darwin radii, where $z$ is the perpendicular distance from the center of the dot. The left panel (a) shows the results using moment method while the right panel (b) represents those obtained with density matrix method. In all these calculations we assume a quantum dot with thickness $z_0=10\mathrm{nm}$. The solid line (upward triangle) represents $l_0=30\mathrm{nm}$ for $D^{zz}$ $\left(D^{xx}\right)$ and the dashed line (downward triangle) describes $l_0=80\mathrm{nm}$ for $D^{zz}$ $\left(D^{xx}\right)$.

Figure 2

The ratio of NSD coefficients $D^{\mu \mu }∕D_0^{\mu \mu }$ as a function of spatial coordinate $r$ for various quantum dot thickness, where $r$ is the radial displacement in the 2D plane. The origin of the coordinate system is located at the center of the dots. The left panel (a) are the results of moment method and the right panel (b) shows those from density matrix method. In all these calculations we have used a quantum dot with Fock-Darwin radius $l_0=50\mathrm{nm}$. The solid line (upward triangle) describes $z_0=5\mathrm{nm}$ for $D^{zz}$ $\left(D^{xx}\right)$ while the dashed line (downward triangle) shows $z_0=15\mathrm{nm}$ for $D^{zz}$ $\left(D^{xx}\right)$.

Figure 3

The ratio of NSD coefficients $D^{xx}∕D_0^{xx}$ at the center of the dot as a function of Fock-Darwin radius $l_0$ with three different dot thickness. The results are obtained with the density matrix method.

Figure 4

The ratio of NSD coefficients $D^{xx}∕D_0^{xx}$ (calculated using the density matrix method) as a function of spatial coordinate $r$ for strongly confined dots [panel (a)] and quantum wells [panel (b)] with Gaussian shaped carrier spin polarization (see discussion in the text). In the case of quantum dots we assume $z_0=10\mathrm{nm}$ and $l_0=25\mathrm{nm}$. In the calculation of quantum well with thickness of $15\mathrm{nm}$, we assume the effective radius of the Gaussian shaped carrier spin polarization is $12\mu \mathrm{m}$. In both panels the thinner lines show the variations of hyperfine coupling strength as a function of $r$. In panel (b), the two solid and dashed lines represent different polarization magnitude.