RKKY interaction in a disordered two-dimensional electron gas with Rashba and Dresselhaus spin-orbit couplings

We study theoretically the statistical properties of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between localized magnetic moments in a disordered two-dimensional electron gas with both Rashba and Dresselhaus spin-orbit couplings. Averaging over disorder, the static spin susceptibility tensor is evaluated diagrammatically in the mesoscopic (phase-coherent) regime. The disorder-averaged susceptibility leads to a twisted exchange interaction suppressed exponentially with distance, whereas the second-order correlations, which determine the fluctuations (variance) of the RKKY energy, decay with the same power law as in the clean case. We obtain analytic expressions in the limits of large/small spin-orbit interactions and for equal Rashba and Dresselhaus couplings. Beside these limiting cases, we study numerically the variance of the RKKY interaction in the presence of pure Rashba spin-orbit coupling. Our results are relevant for magnetic impurities or nuclear moments embedded in III-V two-dimensional heterostructures or in contact with surface states of metals and metal alloys, which can display a sizable Rashba spin-orbit coupling.

Figure 1

Diffuson (left) and cooperon (right) diagrams contributing to the second-order correlations of the RKKY interaction tensor given in Eq. (15).

Figure 2

(Color online) Main plot: dependence of $A_0$ of Eq. (35) on $R/\lambda$ (red solid curve). The horizontal dashed line is the asymptotic value 1/4. The coefficient $A_0^{\prime }$ of Eq. (45) almost perfectly coincides with $A_0$. The blue dashed curve is the small difference $\delta A_0=A_0-A_0^{\prime }$ (multiplied by a factor of 100). The inset shows $A_0-1/4$ (dots) on a semilogarithmic plot. A linear fit gives $\text{ln}\left(A_0-1/4\right)\simeq 1.73–2.57\left(R/\lambda \right)$.

Figure 3

(Color online) Main plot: the four solid red overlapping curves show the dependence on $R/\lambda$ of the rotation angles $\Phi$, ${\Phi }^{\prime }$, and ${\Phi }^{\pm }$ defined in Eqs. (38, 44, 49), respectively. The dashed line indicates the linear dependence $\Phi =R/\lambda =m\alpha R$. Inset: the small differences ${\Phi }^{\prime }-\Phi$ (purple, solid), ${\Phi }^{+}-\Phi$ (blue, long dashed), and ${\Phi }^{-}-\Phi$ (red, short dashed) are plotted as functions of $R/\lambda$.

Figure 4

(Color online) Dependence on $R/\lambda$ of $A_x$ (purple, solid), $A_y$ (blue, long dashed), and $A_z$ (red, short dashed).

Figure 5

(Color online) Dependence on $R/\lambda$ of $B_x^{+}$ (purple, solid), $B_y^{+}$ (blue, long dashed), and $B_z^{+}$ (red, short dashed).

Figure 6

(Color online) Dependence on $R/\lambda$ of $B_x^{-}$ (purple, solid), $B_y^{-}$ (blue, long dashed), and $B_z^{-}$ (red, short dashed).

Figure 7

(Color online) Thick black solid line: plot of $B_0=2B_x^{+}$ as function of $R/\lambda$. This can be compared to the coefficients $2B_i^{+}$ (purple solid lines, see also Fig. 5) and $2B_i^{-}$ (short dashed red lines, see also Fig. 6). Thick black dashed line: plot of $A_0-B_0$, which can be compared to the $A_i$ coefficients (long dashed blue lines, see also Fig. 4).

Figure 8

Diffuson (left) and cooperon (right) ladder diagrams corresponding to Eqs. (A1, A2), respectively.