Textbook physics from a cutting-edge material

Graphene has garnered significant attention for its unusual massless electronic dispersion and the ability to realize exotic electronic phenomena such as the integer [1, 2] and fractional [3, 4] quantum Hall effects in a new condensed matter system. However, graphene is also a conceptually simple two-dimensional electronic system, which makes it ideal as a testbed to demonstrate textbook condensed matter phenomena. In a recent publication in Physical Review Letters, Dmitri Efetov and Philip Kim at Columbia University, US, do just that, studying the temperature-dependent scattering of electrons by phonons in graphene [5]. They demonstrate that the boundary between high-temperature and lowtemperature behavior in the electron-phonon scattering is set not by the Debye temperature—the characteristic phonon energy scale—as in conventional metals with large Fermi surfaces, but rather by the Bloch-Grüneisen temperature, a characteristic electronic energy scale for metals with small Fermi surfaces, such as graphene [6] and doped semiconductors [7]. Efetov and Kim demonstrate tuning of the Bloch-Grüneisen temperature by almost an order of magnitude by varying the Fermi energy in graphene over a wide range of more than ±1 eV by applying a voltage to a gate. The resistivity of metals due to electron-phonon scattering is a basic problem in condensed matter physics. Scattering of electrons by phonons at finite temperature is an unavoidable phenomenon, and this “intrinsic resistivity” is typically the dominant source of resistivity in metals at room temperature. The familiar result is that at high temperature the resistivity of a metal ρ is proportional to temperature T. This reflects the bosonic nature of the phonons that scatter the electrons: at temperatures greater than the Debye temperature θD, the characteristic temperature at which all phonon modes of a crystal are excited, the phonon population in any given mode is proportional to T, hence the number of scatterers and the resistivity are proportional to T. Below the Debye temperature, the phonon modes begin to “freeze out,” and in a typical metal the resistivity drops much more rapidly. For a three-dimensional metal, the resistivity is expected to drop as ρ(T) ∼ T5 (T4 for a two-dimensional metal), the so-called Bloch-Grüneisen regime [8, 9]. Figure 1 illustrates this effect and the factors that lead to the T5 or T4 dependences.