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JASON HORNG et al. PHYSICAL REVIEW B 83, 165113 (2011) at V = 0 and a hole mobility around 2700 cm 2 /Vs.As -50 0 (a) (b) 5%. The sample had an initial hole doping of 1.05 × 10 12 cm −2 electron doping. This electron-hole concentration dependence 40 g seen from Fig. 1(b), the dc conductivity for hole doping is 30 reasonably linear with |n|, and there is a large asymmetry between electron and hole conductivity at the same |n|. In 20 previous studies, conclusions have frequently been made on 10 carrier scattering from such dc conductivity data based on σ dc = D/πƔ and the assumption that D = (v F e 2 / ¯h) √ π|n| 0 holds exactly as theory predicts. 23–25 However, the validity of 50 V g (V) 100 this approach has never been tested. 5 2E has two characteristic features: a large absorption increase at F 0 lower wave numbers, which reaches over 15% at terahertz 0 1500 3000 4500 6000 frequencies for a graphene monolayer, and an absorption ω(cm -1 ) reduction over a broad range of higher wave numbers. These (c) 20 (d) An independent determination of D and Ɣ can be achieved through ac conductivity measurements using IR spectroscopy. 15 Figure 1(c) shows a difference IR absorption spectrum of 10 hole-doped graphene (V g =−70 V) in reference to absorption at the CNP (V g = 14 V). This difference absorption spectrum features can be understood qualitatively from the gate-induced FIG. 1. (Color online) Properties of CVD-grown graphene changes in intraband and interband electronic transitions in device. (a) Optical microscope image of CVD-grown graphene graphene due to carrier doping [Fig. 1(d)]. Free-carrier conductivity (from intraband transitions) increases dramatically transferred to a SiO 2 /Si substrate. (b) Graphene dc conductivity as a function of gate voltage. The conductivity minimum at V g = with the hole doping. It peaks at zero frequency and gives rise 14 V defines the charge neutral point (CNP). (c) Gate-induced change to a substantial absorption increase at low wave numbers. At of ir transmittance T/T through graphene at V g =−70 V (hole doped) compared to transmittance at the CNP. The spectrum shows the same time, however, the interband transitions up to energy an increase of free-carrier absorption at low wave numbers and a 2E F [arrows in Fig. 1(d)] become forbidden due to empty reduction of interband absorption at higher wave numbers. (d) An initial states, leading to a reduction of absorption in the broad illustration of intraband (i.e., free-carrier absorption, dashed arrow) spectral range below 2E F [dashed line in Fig. 1(c)]. and interband (solid arrow) transitions in hole-doped graphene. The gate-induced change of ac conductivity (referred to as Intraband absorption increases with carrier doping, while interband the CNP value), ˜σ = ˜σ − ˜σ CNP , can be obtained readily from transitions up to 2E F become forbidden due to empty initial states, the difference IR absorption spectra (see auxiliary material). as observed in (c). Figures 2(a) and 2(b) show the real part of the ac conductivity change σ ′ in the low-wave-number range (
DRUDE CONDUCTIVITY OF DIRAC FERMIONS IN GRAPHENE PHYSICAL REVIEW B 83, 165113 (2011) (a) Δσ' (e 2 /h) (c) Γ(cm -1 ) (e) σ DC (e 2 /h) 25 20 15 10 5 0 0 150 300 450 ω(cm -1 ) 180 150 120 90 60 Hole Electron 30 regime regime 0 -100 -50 0 50 V g -V cnp (V) 40 30 20 10 Hole regime D/πΓ σ DC 0 -100 -50 0 V g -V cnp (V) 50 V g -V cnp -80V -50V -20V 0V (b) Δσ' (e 2 /h) (d) D (e 2 /h*1000cm -1 ) 25 20 15 10 5 0 0 150 300 450 ω(cm -1 ) 15 12 9 6 3 Hole regime Electron regime Theory Experiment Electron regime 0 -100 -50 0 V g -V cnp (V) 50 V g -V cnp 50V 20V 0V FIG. 2. (Color online) Free-carrier responses in graphene. (a),(b) Gate-induced change of ac conductivity σ ′ in hole- and electrondoped graphene, respectively, for 30
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