Direct measurement of quantum phases in graphene via photoemission spectroscopy

Abstract

Quantum phases provide us with important information for understanding the fundamental properties of a system. However, the observation of quantum phases, such as Berry's phase and the sign of the matrix element of the Hamiltonian between two nonequivalent localized orbitals in a tight-binding formalism, has been challenged by the presence of other factors, e.g. , dynamic phases and spin or valley degeneracy, and the absence of methodology. Here, we report a way to directly access these quantum phases, through polarization-dependent angle-resolved photoemission spectroscopy (ARPES), using graphene as a prototypical two-dimensional material. We show that the momentum- and polarization-dependent spectral intensity provides direct measurements of (i) the phase of the band wavefunction and (ii) the sign of matrix elements for nonequivalent orbitals. Upon rotating light polarization by $\ensuremath{\pi}/2$, we found that graphene with a Berry's phase of $n\ensuremath{\pi}$ ($n=1$ for single- and $n=2$ for double-layer graphene for Bloch wavefunction in the commonly used form) exhibits the rotation of ARPES intensity by $\ensuremath{\pi}/n$, and that ARPES signals reveal the signs of the matrix elements in both single- and double-layer graphene. The method provides a technique to directly extract fundamental quantum electronic information on a variety of materials.