Density-of-states tunnelling magnetoresistance model ($\mathrm{TMR}=2P_1{P}_2/(1-P_1{P}_2)$)

Assume spin-conserving, low-voltage tunnelling and a spin-independent matrix element. Then parallel conductance is proportional to $D_{1\uparrow }{D}_{2\uparrow }+D_{1\downarrow }{D}_{2\downarrow }$, while antiparallel conductance has the crossed products. Define $P_i=(D_{i\uparrow }-D_{i\downarrow })/(D_{i\uparrow }+D_{i\downarrow })$. The two conductances are $G_0(1\pm P_1{P}_2)$; taking their reciprocals and using the resistance-ratio convention proves the displayed formula. Orbital filtering and interface-dependent matrix elements can invalidate a bulk-density-of-states estimate.