Drude conductivity of Dirac fermions in graphene

Jason Horng, Chi-Fan Chen, Baisong Geng, Caglar Girit, Yuanbo Zhang, Zhao Hao, Hans A. Bechtel, Michael Martin, Alex Zettl, Michael F. Crommie, Y. Ron Shen, and Feng Wang

Phys. Rev. B 83, 165113 – Published 15 April 2011

Abstract

Electrons moving in graphene behave as massless Dirac fermions, and they exhibit fascinating low-frequency electrical transport phenomena. Their dynamic response, however, is little known at frequencies above one terahertz (THz). Such knowledge is important not only for a deeper understanding of the Dirac electron quantum transport, but also for graphene applications in ultrahigh-speed THz electronics and infrared (IR) optoelectronics. In this paper, we report measurements of high-frequency conductivity of graphene from THz to mid-IR at different carrier concentrations. The conductivity exhibits Drude-like frequency dependence and increases dramatically at THz frequencies, but its absolute strength is lower than theoretical predictions. This anomalous reduction of free-electron oscillator strength is corroborated by corresponding changes in graphene interband transitions, as required by the sum rule.

Received 9 January 2011

DOI:https://doi.org/10.1103/PhysRevB.83.165113

Authors & Affiliations

Jason Horng¹, Chi-Fan Chen¹, Baisong Geng¹, Caglar Girit¹, Yuanbo Zhang², Zhao Hao^3,4, Hans A. Bechtel³, Michael Martin³, Alex Zettl^1,5, Michael F. Crommie^1,5, Y. Ron Shen^1,5, and Feng Wang^1,5,*

¹Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
²Department of Physics, Fudan University, Shanghai 200433, China
³Advanced Light Source Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
⁴Earth Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
⁵Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

^*fengwang76@berkeley.edu.

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Vol. 83, Iss. 16 — 15 April 2011

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Images

Figure 1
Properties of CVD-grown graphene device. (a) Optical microscope image of CVD-grown graphene transferred to a SiO $_{2}$ /Si substrate. (b) Graphene dc conductivity as a function of gate voltage. The conductivity minimum at $V_{g}$ $=$ 14 V defines the charge neutral point (CNP). (c) Gate-induced change of ir transmittance Δ $T$ / $T$ through graphene at $V_{g}$ $=$ −70 V (hole doped) compared to transmittance at the CNP. The spectrum shows an increase of free-carrier absorption at low wave numbers and a reduction of interband absorption at higher wave numbers. (d) An illustration of intraband (i.e., free-carrier absorption, dashed arrow) and interband (solid arrow) transitions in hole-doped graphene. Intraband absorption increases with carrier doping, while interband transitions up to $2 E_{F}$ become forbidden due to empty initial states, as observed in (c).Reuse & Permissions
Figure 2
Free-carrier responses in graphene. (a),(b) Gate-induced change of ac conductivity $Δ σ^{'}$ in hole- and electron-doped graphene, respectively, for 30 < ω < 450 cm $^{- 1}$ (dashed lines). The frequency dependence of ac conductivity can be fit by the Drude model (solid lines). (c) The scattering rate $Γ$ and Drude weight $D$ at different electron and hole concentrations. Surprisingly, the Drude weight is substantially lower than theoretical predictions based on the Boltzmann transport theory [blue line in (d)] and shows an electron-hole asymmetry. (e) dc conductivity in the form of $D$ /π $Γ$ obtained from the Drude fit of terahertz/far-IR measurements (triangles) agrees well with the values from direct dc transport measurements (squares).Reuse & Permissions
Figure 3
Interband transitions in graphene. (a,b) Gate-induced change of interband ac conductivity $Δ σ^{'}$ in hole- and electron-doped graphene, respectively, over 600 < ω < 6000 cm $^{- 1}$ . The ac conductivity decreases due to empty initial states (hole doping) or filled final states (electron doping) for interband transitions below $2 E_{F}$ . (The sharp features around 1200 cm $^{- 1}$ are extrinsic and arise from SiO $_{2}$ phonon resonances of the substrate.) Observed interband conductivity decrease is appreciably less than the theoretical predicted value of $π e^{2} / 2 h$ (horizontal dashed line), and shows electron-hole asymmetry. (c) Upon charge-carrier doping, integrated conductivity change has an increase from intraband transitions (squares) that equals the decrease from interband transitions (circles). The two values agree quantitatively at all gate voltages, as required by the oscillator strength sum rule.Reuse & Permissions

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