Two-Dimensional Gas of Massless Dirac Fermions in Graphene

TL;DR

This study reports an experimental study of a condensed-matter system (graphene, a single atomic layer of carbon) in which electron transport is essentially governed by Dirac's (relativistic) equation and reveals a variety of unusual phenomena that are characteristic of two-dimensional Dirac fermions.

Abstract

Electronic properties of materials are commonly described by quasiparticles that behave as non-relativistic electrons with a finite mass and obey the Schroedinger equation. Here we report a condensed matter system where electron transport is essentially governed by the Dirac equation and charge carriers mimic relativistic particles with zero mass and an effective "speed of light" c* ~10^6m/s. Our studies of graphene - a single atomic layer of carbon - have revealed a variety of unusual phenomena characteristic of two-dimensional (2D) Dirac fermions. In particular, we have observed that a) the integer quantum Hall effect in graphene is anomalous in that it occurs at half-integer filling factors; b) graphene's conductivity never falls below a minimum value corresponding to the conductance quantum e^2/h, even when carrier concentrations tend to zero; c) the cyclotron mass m of massless carriers with energy E in graphene is described by equation E =mc*^2; and d) Shubnikov-de Haas oscillations in graphene exhibit a phase shift of pi due to Berry's phase.

Figures (4)

• Figure 1: Electric field effect in graphene. a, Scanning electron microscope image of one of our experimental devices (width of the central wire is$0.2 \mu \mathrm{~m}$ ). False colours are chosen to match real colours as seen in an optical microscope for larger areas of the same materials. Changes in graphene's conductivity $\sigma$ (main panel) and Hall coefficient $R_{\mathrm{H}}$ (b) as a function of gate voltage $V_{\mathrm{g}} \cdot \sigma$ and $R_{\mathrm{H}}$ were measured in magnetic fields $B=0$ and 2T, respectively. The induced carrier concentrations $n$ are described by [2] $n / V_{\mathrm{g}}=\varepsilon_{0} \varepsilon / t e$ where $\varepsilon_{0}$ and $\varepsilon$ are permittivities of free space and $\mathrm{SiO}_{2}$, respectively, and $t \approx 300 \mathrm{~nm}$ is the thickness of $\mathrm{SiO}_{2}$ on top of the Si wafer used as a substrate. $R_{\mathrm{H}}=1 /$ ne is inverted to emphasize the linear dependence $n \propto V_{\mathrm{g}} .1 / R_{\mathrm{H}}$ diverges at small $n$ because the Hall effect changes its sign around $V_{\mathrm{g}}=0$ indicating a transition between electrons and holes. Note that the transition region ( $R_{\mathrm{H}} \approx 0$ ) was often shifted from zero $V_{\mathrm{g}}$ due to chemical doping [2] but annealing of our devices in vacuum normally allowed us to eliminate the shift. The extrapolation of the linear slopes $\sigma\left(V_{\mathrm{g}}\right)$ for electrons and holes results in their intersection at a value of $\sigma$ indistinguishable from zero. c, Maximum values of resistivity $\rho=1 / \sigma$ (circles) exhibited by devices with different mobilites $\mu$ (left y -axis). The histogram (orange background) shows the number $P$ of devices exhibiting $\rho_{\text{max }}$ within $10 \%$ intervals around the average value of $\approx h / 4 e^{2}$. Several of the devices shown were made from 2 or 3 layers of graphene indicating that the quantized minimum conductivity is a robust effect and does not require "ideal" graphene.
• Figure 2: Quantum oscillations in graphene. SdHO at constant gate voltage$V_{\mathrm{g}}$ as a function of magnetic field $B$ (a) and at constant $B$ as a function of $V_{\mathrm{g}}(\mathbf{b})$. Because $\mu$ does not change much with $V_{\mathrm{g}}$, the constant- $B$ measurements (at a constant $\omega_{\mathrm{c}} \tau=\mu B$ ) were found more informative. Panel $\mathbf{b}$ illustrates that SdHO in graphene are more sensitive to $T$ at high carrier concentrations. The $\Delta \sigma_{x x}$-curves were obtained by subtracting a smooth (nearly linear) increase in $\sigma$ with increasing $V_{\mathrm{g}}$ and are shifted for clarity. SdHO periodicity $\Delta V_{\mathrm{g}}$ in a constant $B$ is determined by the density of states at each Landau level ( $\alpha \Delta V_{\mathrm{g}}=f B / \phi_{0}$ ) which for the observed periodicity of $\approx 15.8 \mathrm{~V}$ at $B=12 \mathrm{~T}$ yields a quadruple degeneracy. Arrows in $\mathbf{a}$ indicate integer $v$ (e.g., $v=4$ corresponds to 10.9 T ) as found from SdHO frequency $B_{\mathrm{F}} \approx 43.5 \mathrm{~T}$. Note the absence of any significant contribution of universal conductance fluctuations (see also Fig. 1) and weak localization magnetoresistance, which are normally intrinsic for 2D materials with so high resistivity.
• Figure 3: Dirac fermions of graphene. a, Dependence of$B_{\mathrm{F}}$ on carrier concentration $n$ (positive $n$ correspond to electrons; negative to holes). $\mathbf{b}$, Examples of fan diagrams used in our analysis [2] to find $B_{\mathrm{F}} . N$ is the number associated with different minima of oscillations. Lower and upper curves are for graphene (sample of Fig. 2a) and a 5 -nm-thick film of graphite with a similar value of $B_{\mathrm{F}}$, respectively. Note that the curves extrapolate to different origins; namely, to $N= \frac{1}{2}$ and 0 . In graphene, curves for all $n$ extrapolate to $N =1 / 2$ (cf. [2]). This indicates a phase shift of $\pi$ with respect to the conventional Landau quantization in metals. The shift is due to Berry's phase [9,15]. c, Examples of the behaviour of SdHO amplitude $\Delta$ (symbols) as a function of $T$ for $m_{\mathrm{c}} \approx 0.069$ and $0.023 m_{0}$; solid curves are best fits. d, Cyclotron mass $m_{\mathrm{c}}$ of electrons and holes as a function of their concentration. Symbols are experimental data, solid curves the best fit to theory. e, Electronic spectrum of graphene, as inferred experimentally and in agreement with theory. This is the spectrum of a zero-gap 2D semiconductor that describes massless Dirac fermions with $c_{*} 300$ times less than the speed of light.
• Figure 4: Quantum Hall effect for massless Dirac fermions. Hall conductivity$\sigma_{\mathrm{xy}}$ and longitudinal resistivity $\rho_{\mathrm{xx}}$ of graphene as a function of their concentration at $B=14 \mathrm{~T} . \sigma_{\mathrm{xy}}=\left(4 e^{2} / h\right) v$ is calculated from the measured dependences of $\rho_{x y}\left(V_{g}\right)$ and $\rho_{x x}\left(V_{g}\right)$ as $\sigma_{x y}=\rho_{x y} /\left(\rho_{x y}+\rho_{x x}\right)^{2}$. The behaviour of $1 / \rho_{x y}$ is similar but exhibits a discontinuity at $V_{g} \approx 0$, which is avoided by plotting $\sigma_{\mathrm{xy}}$. Inset: $\sigma_{\mathrm{xy}}$ in "two-layer graphene" where the quantization sequence is normal and occurs at integer $\nu$. The latter shows that the half-integer QHE is exclusive to "ideal" graphene.