Microscopic Origin of the Effective Spin-Spin Interaction in a Semiconductor Quantum Dot Ensemble

We present a microscopic model for a singly charged quantum dot (QD) ensemble to reveal the origin of the long-range effective interaction between the electron spins in the QDs. Wilson’s numerical renormalization group (NRG) is used to calculate the magnitude and the spatial dependency of the effective spin-spin interaction mediated by the growth-induced wetting layer. Surprisingly, we found an antiferromagnetic Heisenberg coupling for very short inter-QD distances that is caused by the significant particle-hole asymmetry of the wetting layer band at very low filling. Using the NRG results obtained from realistic parameters as input for a semiclassical simulation for a large QD ensemble, we demonstrate that the experimentally reported phase shifts in the coherent spin dynamics between single- and two-color laser pumping can be reproduced by our model, solving a long-standing open problem of the microscopic origin of the inter-QD electron spin-spin interaction.

Figure 1

Sketch of two quantum dots that are linked by the InAs wetting layer.

Figure 2

NRG results: (a) $⟨{\overrightarrow{S}}_1{\overrightarrow{S}}_2⟩(T,R)$ vs $T$ for two distances $R$ and the fit to an interacting two-spin model for extracting $J$. (b) $J_{\mathrm{RKKY}}^{\mathrm{NRG}}(R)/({\mathrm{\Gamma }}_0^2{ϵ}_F)$ vs $R{k}_F/\pi$ for different ${\mathrm{\Gamma }}_0$ and ${ϵ}_F$. We added a fit to Eq. (5) as the solid magenta line. The graphs corresponding to $U=4$ meV are calculated with ${\epsilon }^d=-1.5$ meV, the red graph corresponding to $U=40$ meV with ${\epsilon }^d=-15$ meV. NRG parameters $\mathrm{\Lambda }=5$, high energy cutoff $D=1\mathrm{eV}$ and $\overline{\beta }=40$.

Figure 3

Semiclassical simulation of the QD ensemble. (a) Traces of the electron spin dynamics using the frozen Overhauser field approximation (FOA) with ${\mathrm{\Gamma }}_0=0.14$ meV, ${\epsilon }^d=-1.5$ meV, and $U=4\mathrm{meV}$. A phase shift $\varphi$ in the coherent electron spin oscillations between a single-color pumping reference curve and a two-color laser pumped QD ensemble occurs. The incidence time of the second pump is marked as a black circle. (b) Comparison of the experimental phase shift $\varphi$ to various simulations. The simulations are for a FOA, for $A_k=\text{const}$ (box model), a distribution of the $A_k$, including some disorder of the $x_j$ as well as the second set of parameters with ${\mathrm{\Gamma }}_0=1.25$ meV, ${\epsilon }^d=-15$ meV, and $U=40\mathrm{meV}$. The experimental data are taken from Ref. [10].