contradiction is lifted if one takes into account the localized states of oxygen vacancies in the MgO tunnel barrier. Extensive solid-state tunnelling spectroscopy experiments across MgO MTJs revealed in 2014 that the electronic retention on the ground and excited states of an oxygen vacancy, which is temperature-dependent, determines the tunnelling barrier height for electrons of a given symmetry, and thus crafts the effective TMR ratio and its temperature dependence. This low barrier height in turn enables the high current densities required for spin-transfer torque, discussed hereafter. Spin-transfer torque in magnetic tunnel junctions (MTJs) The effect of spin-transfer torque has been studied and applied widely in MTJs, where there is a tunnelling barrier sandwiched between a set of two ferromagnetic electrodes such that there is (free) magnetization of the right electrode, while assuming that the left electrode (with fixed magnetization) acts as spin-polarizer. This may then be pinned to some selecting transistor in a magnetoresistive random-access memory device, or connected to a preamplifier in a hard disk drive application. The spin-transfer torque vector, driven by the linear response voltage, can be computed from the expectation value of the torque operator: where is the gauge-invariant nonequilibrium density matrix for the steady-state transport, in the zero-temperature limit, in the linear-response regime, and the torque operator is obtained from the time derivative of the spin operator: Using the general form of a 1D tight-binding Hamiltonian: where total magnetization (as macrospin) is along the unit vector and the Pauli matrices properties involving arbitrary classical vectors , given by it is then possible to first obtain an analytical expression for (which can be expressed in compact form using , and the vector of Pauli spin matrices ). The spin-transfer torque vector in general MTJs has two components: a parallel and perpendicular component: A parallel component: And a perpendicular component: In symmetric MTJs (made of electrodes with the same geometry and exchange splitting), the spin-transfer torque vector has only one active component, as the perpendicular component disappears: . Therefore, only vs. needs to be plotted at the site of the right electrode to characterise tunnelling in symmetric MTJs, making them appealing for production and characterisation at an industrial scale. Note: In these calculations the active region (for which it is necessary to calculate the retarded Green's function) should consist of the tunnel barrier + the right ferromagnetic layer of finite thickness (as in realistic devices). The active