Extrinsic spin-valley Hall effect and spin-relaxation anisotropy in magnetized and strained graphene

The implementation of the quantum spin-valley Hall effect is a critical challenge for the spintronics and valleytronic experimentalists because it requires breaking both time-reversal symmetry (T) and spatial inversion symmetry (P) while preserving the joint symmetry O=TP. Here, we demonstrate an extrinsic spin-valley Hall effect by the magnetic field and temperature modulation of the nonlocal resistance in a Hall bar device consisting of magnetized and strained graphene. Besides, we achieve a striking crossover from positive to negative nonlocal magnetoresistance owing to the magnetic field dependence of spin-valley relaxation instead of the usual Hanle spin precession. Moreover, we microscopically derive a large and tunable spin-relaxation anisotropy of the magnetized graphene within the Born-Markov and the Weiss-field approximations. Our findings offer fascinating opportunities to manipulate the spin and valley degrees of freedom and design novel electronic devices.