Breakdown of the adiabatic Born–Oppenheimer approximation in graphene

Abstract

The Born-Oppenheimer approximation (BO) [1] is the standard ansatz to describe the interaction between electrons and nuclei. BO assumes that the lighter electrons adjust adiabatically to the motion of the heavier nuclei, remaining at any time in their instantaneous ground-state. BO is well justified when the energy gap between ground and excited electronic states is larger than the energy scale of the nuclear motion. In metals, the gap is zero and phenomena beyond BO (such as phonon-mediated superconductivity or phonon-induced renormalization of the electronic properties) occur [2]. The use of BO to describe lattice motion in metals is, therefore, questionable [3,4]. In spite of this, BO has proven effective for the accurate determination of chemical reactions [5], molecular dynamics [6, 7] and phonon frequencies [9,8,10] in a wide range of metallic systems. Graphene, recently discovered in the free state [11,12], is a zero band-gap semiconductor [13], which becomes a metal if the Fermi energy is tuned applying a gate-voltage V g [14,12]. Graphene electrons near the Fermi energy have twodimensional massless dispersions, described by Dirac cones. Here * acf26@eng.cam.ac.uk † francesco.mauri@impmc.jussieu.fr 1 we show that a change in V g induces a stiffening of the Raman G peak (i.e. the zone-center E 2g optical phonon [15,16]), which cannot be described within BO. Indeed, the E 2g vibrations cause rigid oscillations of the Dirac-cones in the reciprocal space [17]. If the electrons followed adiabatically the Dirac-cone oscillations, no change in the phonon frequency would be observed. Instead, since the electron-momentum relaxation near the Fermi level [18,19,20] is much slower than the phonon motion, the electrons do not follow the Dirac-cone displacements. This invalidates BO and results in the observed phonon stiffening. This spectacular failure of BO is quite significant since BO has been the fundamental paradigm to determine crystal vibrations from the early days of quantum mechanics [1,9,21,8,10]. Graphene samples are prepared by micromechanical cleavage of bulk graphite at the surface of an oxidized Si wafer with a 300 nm thick oxide layer, following the procedures described in Ref. [11]. This allows us to obtain graphene monocrystals exceeding 30 microns in size. By using photolithography, we then make Au/Cr electrical contacts, which enable the application of a gate voltage, V g , between the Si wafer and graphene (Fig. 1A,B). The resulting devices are characterized by electric-field-effect measurements [12,14,22], yielding a charge carrier mobility µ of 5,000 to 10,000 cm 2 /Vs at 295K and a zero-bias (V g =0) doping of ∼10 12 cm −2 [23]. This is reflected in the existence of a finite gate voltage V n at which the Hall resistance is zero and the longitudinal resistivity reaches its maximum. Accordingly, a positive (negative) V g -V n induces an electron (hole) doping, having an excess-electron surface-concentration of n=η(V g -V n ). The coefficient η ≈7.2 10 10 cm −2 /V is found from Hall effect measu...

Janus graphene from asymmetric two-dimensional chemistry

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Probing the Intrinsic Properties of Exfoliated Graphene: Raman Spectroscopy of Free-Standing Monolayers

“…[13][14][15][16] Our experiment did not permit us to calibrate the Raman measurements with a systematic variation of doping, since no gate was present in our sample geometry.…”

Photothermoelectric and Photoelectric Contributions to Light Detection in Metal–Graphene–Metal Photodetectors

“…This allows monitoring defects, 42−45 as well as local doping. 6,42,46,47 The doping of the p−n junction can be determined by measuring the back gate voltage dependence of the photoresponse. Figure 3a compares the photovoltage in dependence of back gate voltage, V g , with the resistance, at an incident light wavelength of 633 nm.…”