Direct measurement of quantum phases in graphene via photoemission spectroscopy

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 $\pi /2$, we found that graphene with a Berry's phase of $n\pi$ ($n=1$ for single- and $n=2$ for double-layer graphene for Bloch wavefunction in the commonly used form) exhibits the rotation of ARPES intensity by $\pi /n$, 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.

Figure 1

(a) A LEEM image using an electron energy of 3.5 eV over $4\mu$m $\times$ $4\mu$m range. (b) Reflectivity spectra for the three regions (1, 2, and 3) specified in (a). (c) Post-processed image of (a) showing the regions covered by single-, double-, and triple-layer graphene. (d) A histogram showing the fractions of single-, double-, and triple-layer graphene in our sample used for double-layer graphene measurements.

Figure 2

(a) $X$-polarization geometry. (b) $Y$-polarization geometry. A beam of monochromatic lights with energy $\hslash \omega =$50 eV and polarization vector $\vec{\varepsilon }$ is incident on a sample. The light polarizations in $X$- and $Y$-polarization geometries are almost parallel to the $x$ and $y$ axes, respectively. (c), (d) Measured intensity maps of single-layer graphene at energy $E=E_{\mathrm{F}}$ with $X$- and $Y$-polarized lights, respectively. Intensity maxima are denoted by white arrows and the electronic band structure of single-layer graphene is drawn in the cartoon. (e) Constant-energy ARPES intensity maps for single-layer graphene at $E_{\mathrm{F}}$ with $X$- and $Y$-polarized light. (f) The angle-dependent intensity profiles of single-layer graphene are obtained by integrating the constant-energy intensity map along the radial direction from the Dirac point, in which the solid and dashed lines denote the experimental data for $X$- and $Y$-polarized lights, respectively. The angle $\theta$ is measured from the $+k_x$ direction. The plotted quantities are with respect to the intensity minimum.

Figure 3

(a), (b) Measured intensity maps of double-layer graphene at energies $E=E_{\mathrm{D}}+0.25$ eV$(=E_{\mathrm{F}})$ and $E=E_{\mathrm{D}}-0.95$ eV. Intensity maxima are denoted by white arrows and the electronic band structure of double-layer graphene is drawn in the cartoon. (c) Measured intensity maps of double-layer graphene at energy $E=E_{\mathrm{F}}$. (d) The angle-dependent intensity profiles of double-layer graphene at energy $E=E_{\mathrm{F}}$, in which the solid and dashed lines denote the experimental data for $X$- and $Y$-polarized lights, respectively. The plotted quantities are with respect to the intensity minimum.

Figure 4

(a) Schematic of single- and double-layer graphene. (b) The Brillouin zone. Here, ${\mathbf{b}}_1=b(0,1)$, ${\mathbf{b}}_2=b(-\frac{\sqrt{3}}{2},-\frac{1}{2})$, and ${\mathbf{b}}_3=b(\frac{\sqrt{3}}{2},-\frac{1}{2})$ are the three vectors connecting the in-plane nearest-neighbor atoms where the intercarbon distance $b=1.42$ Å, and the lattice constant is $a=\sqrt{3}b$. The positions of the K and K${}^{\prime }$ points are $(\frac{4\pi }{3a},0)$ and $(-\frac{4\pi }{3a},0)$, respectively.

Figure 5

(a), (b) Calculated intensity maps of single-layer graphene for $X$- and $Y$-polarized lights, respectively. The insets are the results of calculations (Ref.14) using the simplified momentum operator instead of the correct velocity operator. An arbitrary energy broadening of 0.10 eV has been used. Intensity maxima are denoted by white arrows. (c), (d) The angle-dependent intensity map of single-layer graphene for $X$- and $Y$-polarized lights, respectively, in which the solid black, solid red, and dashed blue lines denote the experimental data, the calculated results obtained by assuming the actual light polarization used in the experiment, and the calculated results obtained by assuming perfectly $Y$-polarized light, respectively. The theory results shown in (a) and (b) have adopted the light polarization used in the actual experiment shown in Figs. 2a and 2b. The plotted quantities are with respect to the intensity minimum.

Figure 6

(a), (b) Calculated intensity maps of double-layer graphene for $X$- and $Y$-polarized lights, respectively. An arbitrary energy broadening of 0.10, 0.10, and 0.15 eV have been used for bands 1, 2, and 3, respectively. Panels denoted by “Ideal DG” show results obtained by considering a double-layer graphene alone and those denoted by “SG$+$DG” show results considering the contribution from some single-layer fraction of the sample as well (see text). Intensity maxima are denoted by white arrows. (c), (d) The angle-dependent intensity map of single-layer graphene for $X$- and $Y$-polarized lights, respectively, in which the solid black, solid red, and dashed green lines denote the experimental data, the calculated results obtained by considering the contribution from single-layer graphene portion of the sample, and the calculated results for ideal double-layer graphene, respectively. The theory results shown in (a) and (b) have adopted the light polarization used in the actual experiment as shown in Figs. 2a and 2b. The plotted quantities are with respect to the intensity minimum.

Figure 7

(a) Schematic of a one-dimensional crystal having one $s$-type orbital per unit cell. The nearest-neighbor hopping integral is $t^{\prime }$. (b) Calculated electron energy band structure of the system depicted in (a) with different choices for the sign of $t^{\prime }$. (c) Schematic of a one-dimensional crystal having two nonequivalent $s$-type orbitals per unit cell (i.e. , having different on-site energies). We assume that all the distances between the centers of the nearest-neighbor orbitals are the same and, hence, the corresponding hopping integrals, are denoted by $t^{\prime \prime }$. (d) Calculated electronic band structure of the system depicted in (c) with different choices for the sign of $t^{\prime \prime }$.

Figure 8

(a)–(d) Calculated intensity maps for four different choices of the signs of nearest-neighbor in-plane and vertical interlayer hopping integrals $t_0$ and $t_1$, respectively. Case (c) agrees with the experimental results.