Measurement of collective dynamical mass of Dirac fermions in graphene - 964437055609826967_measurement

Individual electrons in graphene behave as massless quasiparticles1–8. Unexpectedly, it is inferred from plasmonic investigations9–12 that electrons in graphene must exhibit a non-zero mass when collectively excited. The inertial acceleration of the electron collective mass is essential to explain the behaviour of plasmons in this material, and may be directly measured by accelerating it with a time-varying voltage and quantifying the phase delay of the resulting current. This voltage–current phase relation wouldmanifest as a kinetic inductance, representing the reluctance of the collective mass to accelerate. However, at optical (infrared) frequencies, phase measurements of current are generally difficult, and, at microwave frequencies, the inertial phase delay has been buried under electron scattering13–15. Therefore, to date, the collective mass in graphene has defied unequivocal measurement. Here, we directly and precisely measure the kinetic inductance, and therefore the collective mass, by combining device engineering that reduces electron scattering and sensitive microwave phase measurements. Specifically, the encapsulation of graphene between hexagonal boron nitride layers16, one-dimensional edge contacts17 and a proximate top gate configured as microwave ground18,19 together enable the inertial phase delay to be resolved from the electron scattering. Beside its fundamental importance, the kinetic inductance is found to be orders of magnitude larger than the magnetic inductance, which may be utilized to miniaturize radiofrequency integrated circuits. Moreover, its bias dependency heralds a solid-state voltage-controlled inductor to complement the prevalent voltage-controlled capacitor. The collective excitation of massless fermions in graphene exhibits a non-zero mass. This fact is subsumed under the general theoretical framework of graphene plasmonics9, yet it can be considered simply. Let electrons in graphene (width W, unit length) be subjected to a voltage difference V across the length. The resulting translation of the Fermi disk in two-dimensional k-space by Dk≪ F (where F is the Fermi wavenumber) from disk A to B (Fig. 1a,b) yields a collective momentum per unit length, P1⁄4 0 ÿ− Dk (n0 is the electron density and h ÿ− the reduced Planck constant). The corresponding collective kinetic energy per unit length (E) is obtained by subtracting the sum of single electron energies 1 1⁄4 h ÿ− F (vF is the Fermi velocity) over disk A from that over disk B. Because E is minimal at Dk1⁄4 0, we must have E / (Dk) / P for small Dk (Fig. 1c), from which the collective mass per unit length (M) emerges, satisfying E1⁄4 P/2M. Detailed calculation indeed shows E1⁄4W1F/2p× (Dk)2 (1F is the Fermi energy), yielding M1⁄4 pWn0h ÿ− /1F (Supplementary Section 1). This emergence of a non-zero collective mass in graphene from its massless individual electrons contrasts sharply with the case of typical conductors, where the non-zero individual electron masses m* simply sum to the collective mass. Yet, as an analogy to m*, we can consider an effective collective mass per electron for graphene, m* c;M/(Wn0)1⁄4 (h ÿ− /vF) p (pn0). In fact, this is an insightfully defined theoretical entity called ‘plasmon mass’ in graphene10,12,20. However, the collective mass of graphene electrons, which we set out to measure here, is an observable physical reality that proves the existence of the plasmon mass beyond the theoretical model. The collective current I associated with the Fermi disk shift (that is, the inertial acceleration of the collective mass M) has an inductive phase relationship with the voltage V causing the acceleration. The associated inductance of kinetic origin can be evaluated by