Ferromagnetic order of nuclear spins coupled to conduction electrons: A combined effect of electron-electron and spin-orbit interactions

We analyze the ordered state of nuclear spins embedded in an interacting two-dimensional electron gas (2DEG) with Rashba spin-orbit interaction (SOI). Stability of the ferromagnetic nuclear-spin phase is governed by nonanalytic dependences of the electron spin susceptibility ${\chi }^{ij}$ on the momentum ($\tilde{\mathbf{q}}$) and on the SOI coupling constant ($\alpha$). The uniform ($\tilde{q}=0$) spin susceptibility is anisotropic (with the out-of-plane component ${\chi }^{zz}$ being larger than the in-plane one ${\chi }^{xx}$ by a term proportional to $U^2\left(2k_F\right)\left|\alpha \right|$, where $U\left(q\right)$ is the electron-electron interaction). For $\tilde{q}\le 2m^{*}\left|\alpha \right|$, corrections to the leading $U^2\left(2k_F\right)\left|\alpha \right|$ term scale linearly with $\tilde{q}$ for ${\chi }^{xx}$ and are absent for ${\chi }^{zz}$. This anisotropy has important consequences for the ferromagnetic nuclear-spin phase: (i) the ordered state—if achieved—is of an Ising type and (ii) the spin-wave dispersion is gapped at $\tilde{q}=0$. To second order in $U\left(q\right)$, the dispersion is a decreasing function of $\tilde{q}$, and anisotropy is not sufficient to stabilize long-range order. However, renormalization in the Cooper channel for $\tilde{q}\ll 2m^{*}\left|\alpha \right|$ is capable of reversing the sign of the $\tilde{q}$ dependence of ${\chi }^{xx}$ and thus stabilizing the ordered state. We also show that a combination of the electron-electron and spin-orbit (SO) interactions leads to a new effect: long-wavelength Friedel oscillations in the spin (but not charge) electron density induced by local magnetic moments. The period of these oscillations is given by the SO length $\pi /m^{*}\left|\alpha \right|$.

Figure 1

A normalized dispersion of the out-of-plane spin-wave mode $\tilde{\omega }\left(\tilde{q}\right)=\omega \left(\tilde{q}\right)/(A^{2}I{\chi }_0/4n_s)$ as a function of the momentum. To second order in interaction (lower curve) $\omega \left(\tilde{q}\right)$ is necessarily negative for $m\left|\alpha \right|\ll \tilde{q}\ll k_F$, and thus LRO is unstable. Solid parts of the curves corresponds to actual calculations; dashed parts are interpolations between various asymptotic regimes. Renormalization effects in the Cooper channel reverse the slope of $\omega \left(\tilde{q}\right)$ (upper curve) and stabilize LRO.

Figure 2

Diagram 1. Top: small-momentum transfer part. Bottom: $2k_F$-momentum transfer part. $K\text{and}s$ denotes a fermion from Rashba subband $s=\pm 1$ with “four-momentum” $K=({\omega }_k,\mathbf{k})$.

Figure 3

Diagram 2. Top: small-momentum transfer part. Bottom: $2k_F$-momentum transfer part.

Figure 4

Top: diagram 3. Bottom: diagram 4. The momentum transfer $q$ in both diagrams can be either small or close to $2k_F$.

Figure 5

Diagram 5. The momentum transfer $q$ is close to zero and $|\mathbf{k}-\mathbf{p}|=2k_F$.

Figure 6

Diagram 6. The momentum transfer $q$ is close to zero and $|\mathbf{k}-\mathbf{p}|=2k_F$.

Figure 7

Diagram $7a$ (upper figure) and diagram $7b$ (lower figure). The transferred momenta are $q=$ and $|\mathbf{k}-\mathbf{p}|=2k_F$.

Figure 8

The nonanalytic part of the electron spin susceptibility in units of $(2/3\pi )u_{2k_F}^2\left(\right|\alpha |/v_F){\chi }_0$ as a function of the momentum in units of $q_{\alpha }=2m^{*}\left|\alpha \right|$. $i=x,z$. Solid lines are the exact results (3.13a) and (3.13b). Dashed lines are the approximate results (3.14a) and (3.14b) valid near the singularity at $q=q_{\alpha }$.

Figure 9

Diagram 1 to third order in electron-electron interaction at large momentum transfer; here, the in-plane component is shown.

Figure 10

(Left) A skeleton diagram for the free energy in the presence of the Cooper renormalization; $\tilde{\Gamma }$ is a renormalized Cooper vertex. (Middle) The effective scattering amplitude ${\Gamma }_{s_1{s}_2;s_3{s}_4}^{\left(1\right)}(\mathbf{k},{\mathbf{k}}^{\prime };\mathbf{p},{\mathbf{p}}^{\prime })$ in the chiral basis. (Right) A generic $n$th order ladder diagram in the Cooper channel, ${\Gamma }_{s_1{s}_2;s_3{s}_4}^{\left(n\right)}(\mathbf{k},-\mathbf{k};\mathbf{p},-\mathbf{p})$.