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

Abstract

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 ($\tq=0$) 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 $U^2(2k_F)|\alpha |$, where $U(q)$ is the electron-electron interaction). For $\tq \leq 2m^*|α|$, corrections to the leading, $U^2(2k_F)|\alpha |$, term scale linearly with $\tq$ 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 $\tq=0$. To second order in $U(q)$, the dispersion a decreasing function of $\tq$, and anisotropy is not sufficient to stabilize long-range order. However, renormalization in the Cooper channel for $\tq\ll2m^*|α|$ is capable of reversing the sign of the $\tq$-dependence of ${\chi }^{xx}$ and thus stabilizing the ordered state. We also show that a combination of the electron-electron and 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 $\pi /m^{\ast }|\alpha |$.

22 pages, 15 figures

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• Spin-echo of a single electron spin in a quantum dot
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• Magnetic Ordering of Nuclear Spins in an Interacting 2D Electron Gas
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• Nonanalytic paramagnetic response of itinerant fermions away and near a ferromagnetic quantum phase transition
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• Temperature dependence of spin susceptibility in two-dimensional Fermi liquid systems
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