In this work, exchange bias was used as a probe to characterise the temperature profile induced by the inelastic relaxation of electrons tunnelling across a MgO barrier. Thermally assisted magnetic random access memory (TA-MRAM) cells comprising a magnetic tunnel junction (MTJ) with a reference pinned layer and a FeMn exchange biased storage layer were used. The pinning direction of the ferromagnetic storage layer is reversed when heated above the blocking temperature of the antiferromagnetic layer (FeMn). The power density required to reach this blocking temperature in the FeMn layer depends on the current polarity, indicating that the heat source term associated with the current flowing through the barrier depends itself on the current direction in contrast to simple Joule heating. This effect is due to the mechanism of energy dissipation in tunnelling. The tunnelling itself is ballistic i.e., without dissipation. However, after tunnelling, the hot electrons very quickly relax to the Fermi energy thereby loosing their excess energy in the receiving electrode. Therefore, the heat is essentially generated on one side of the barrier so that the whole profile of temperature throughout the pillar depends on the current direction. Full 3D thermal simulations also confirmed the temperature profile asymmetry. The proper choice of heating current direction (i.e., voltage polarity applied to the MTJ) can yield a reduction of about 10% in the heating power density required to enable writing in thermally assisted MRAM cells.