Accessing the Spectral Function in a Current-Carrying Device

The presence of an electrical transport current in a material is one of the simplest and most important realizations of nonequilibrium physics. The current density breaks the crystalline symmetry and can give rise to dramatic phenomena, such as sliding charge density waves, insulator-to-metal transitions, or gap openings in topologically protected states. Almost nothing is known about how a current influences the electron spectral function, which characterizes most of the solid’s electronic, optical, and chemical properties. Here we show that angle-resolved photoemission spectroscopy with a nanoscale light spot provides not only a wealth of information on local equilibrium properties, but also opens the possibility to access the local nonequilibrium spectral function in the presence of a transport current. Unifying spectroscopic and transport measurements in this way allows simultaneous noninvasive local measurements of the composition, structure, many-body effects, and carrier mobility in the presence of high current densities. In the particular case of our graphene-based device, we are able to correlate the presence of structural defects with locally reduced carrier lifetimes in the spectral function and a locally reduced mobility with a spatial resolution of 500 nm.

Figure 1

Characterization of a $\text{graphene}/h\text{−}\mathrm{BN}$ device. (a) Sketch of the device with the UV beam focused onto it using a zone plate. (b) Optical image of the device. The white dot gives the size of the UV light spot used (diameter of 500 nm) with respect to the device. The inset shows the entire device as it is mounted on a chip carrier. (c) Atomic force microscopy image of the device together with an outline of the $h\text{−}\mathrm{BN}$ flake (dashed green line) and graphene (gray line). The white dashed lines follow rows of protrusions identified on the surface by AFM (see SM [6]). The size of the graphene flake is determined by nanoARPES; see (e). (d) Spectrum taken from the middle of the device (photoemission intensity as a function of binding energy $E_b$ and $k_y$), showing the Dirac cone of graphene and the valence band maximum of $h\text{−}\mathrm{BN}$ at $E_b\approx 2.7\mathrm{eV}$. The sketch to the left shows the Brillouin zone together with the scan direction (red line). (e) Integrated photoemission intensity in the gray window around the Dirac cone in (d), marking the location of graphene. (f) Photoemission intensity from the green area in (d), emphasizing the Au contacts.

Figure 2

Graphene spectra taken in different locations of the device, as given by the corresponding marker in (e). (a) Reference spectrum. (b)–(d) Spectra taken at different positions along with the difference to the reference spectrum below (blue negative, red positive). From the difference plots, it can be concluded that the spectrum in (b) is shifted upward in energy, the spectrum in (c) is shifted toward negative $k_y$ values, and the spectrum in (d) is broader than the reference spectrum, as seen by the intensity decrease in the middle of the branches and the increase in the tails. (e) Intensity of the Dirac cone across the device, obtained by the fitting procedure also used for Fig. 3, with the locations for the spectra in (a)–(d) given by colored markers.

Figure 3

Detailed characterization of the device properties obtained by scanning the light spot across the device and fitting the spectra measured at every point. The upper parts of the panels schematically show the property encoded in the measured parameter. (a) Energy of the Dirac point $E_D$, mapping the local variation in doping across the device. The markers correspond to the locations for the spectra in Fig. 2. (b) Position of the Dirac cone in $k_y$. A shift is primarily assigned to azimuthal rotations between domains on the sample. The inset demonstrates how a domain rotation from a reference direction leads to a shift of the Dirac cone dispersion with respect to the fixed scan range in $k$ space (red line). Note that the rotation between domains in the sketch (8°) is much larger than observed here in order to illustrate the effect. (c) Imaginary part of the self-energy ${\mathrm{\Sigma }}^{\prime \prime }$ measuring the width of the spectra or, equivalently, the inverse lifetime of the state.

Figure 4

Analysis of current-carrying device. (a) Series of spectra along a line connecting the Au electrodes [see inset in (b)] for a device current of 0.5 mA. The Fermi level on the right Au electrode is used as a zero for the energy scale. (b) Sketch of the device along with the potential $\varphi$ along the red line, determined from the local Fermi energy. The black markers give the location of the spectra shown in (a). $\varphi$ shows a linear change over the graphene region and steep drops at the interfaces to the Au, corresponding to the contact resistances. (c) Map of the potential $\varphi$ across the device for a current of 0.5 mA, also determined from the local Fermi energy. The energy zero is the Fermi energy on the right-hand side. The markers correspond to the locations for the spectra in Fig. 2. (d) Magnitude of the electric field $|\mathbf{E}|=|\nabla \varphi |$ across the device. (e) Local conductivity $\sigma$ calculated from $\varphi$. (f) Local mobility $\mu$, calculated by combining $\sigma$ with the hole density derived from the Dirac point energy $E_D$ in Fig. 3.