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We show that bilayer graphene exhibits magneto-electric effects that areformally similar to those commonly seen in band insulators with brokeninversion and time-reversal symmetries. Three unusual features characterize themagneto-electric responses exhibited by bilayer graphene: (i) unlike most othermagneto-electric media, bilayer graphene is a conductor, (ii) bilayer graphenehas a non-quantized magneto-electric coupling even though its electronicstructure does not break parity and time-reversal symmetry, and (iii) themagnitude of the magneto-electric coupling in bilayer graphene is determined bythe valley-isospin density, which can be manipulated experimentally. This lastproperty also enables a purely electric measurement of valley-isospindensities. While our theoretical arguments use bilayer graphene as an example,they are generally valid for any material with similar symmetries.

Magnetoelectric effect in bilayer graphene controlled by valley-isospin density