Competing Hyperfine and Spin-Orbit Couplings: Spin Relaxation in a Quantum Hall Ferromagnet

Spin relaxation in a quantum Hall ferromagnet, where filling is $\nu=1, 1/3, 1/5,...$, can be considered in terms of spin wave annihilation/creation processes. Hyperfine coupling with the nuclei of the GaAs matrix provides spin non-conservation in the two-dimensional electron gas and determines spin relaxation in the quantum Hall system. This mechanism competes with spin-orbit coupling channels of spin-wave decay and can even dominate in a low-temperature regime where $T$ is much smaller than the Zeeman gap. In this case the spin-wave relaxation process occurs non-exponentially with time and does not depend on the temperature. The competition of different relaxation channels results in crossovers in the dominant mechanism, leading to non-monotonic behavior of the characteristic relaxation time with the magnetic field. We predict that the relaxation times should reach maxima at $B\simeq 18\,$T in the $\nu=1$ Quantum Hall system and at $B\simeq 12\,$T for that of $\nu=1/3\,$. We estimate these times as $\sim10\,-\,30\,\mu$s and $\sim2\,-\,5\,\mu$s, respectively.