Complex Optical Conductivity of Graphene Measured by Ultra-broadband THz Time-domain Spectroscopic Ellipsometry Sho Ikeda1,2, Masatsugu Yamashita1, and Chiko Otani1,2 1 RIKEN Center for Advanced Photonics, 519-1399 Aoba, Aramaki, Aoba, Sendai, 980-0845, Japan 2 Grad. Sch. of Sci. Tohoku Univ., 6-3 Aoba, Aramaki, Aoba, Sendai, 980-0845, Japan Abstract—Carrier dynamics in a single-layered graphene has been studied by reflection-type THz time-domain spectroscopic ellipsometry (THz-TDSE). THz-TDSE determines the ellipsometric parameter ρ=rp/rs by measuring the p- and s-polarized THz waveforms after the reflection form sample. We successfully obtained the frequency-dependent complex sheet conductivity of monolayer graphene from 0.3 to 20 THz. By fitting with Drude model, the sheet resistance, sheet carrier density and mobility of graphene were estimated. THz-TDSE can be a novel tool for the non-contact and non-destructive characterization of the carrier transport property in graphene samples. R I. INTRODUCTION ECENT progress of ultra-broadband THz time domain spectroscopy (THz-TDS) has enabled to extend the phase sensitive spectroscopic technology from THz to mid-infrared (MIR) region [1,2], which is a powerful tool for the study of the carrier dynamics and the determination of the dielectric function. Monolayer graphene is a gapless two-dimensional material with linear electron and hole bands. The charge carriers in this system behave like massless Dirac fermions and have high Fermi velocity and mobility. Because of such unique properties, graphene has many possibilities of application for electronic or optoelectronic devices. In order to realize such high performance devices [3,4], non-contact and non-destructive characterization method for carrier transport property of graphene is necessary. Here, we report the direct measurement of the frequency dependent optical conductivity of a monolayer graphene by a reflection type ultra-broadband THz time-domain spectroscopic ellipsometry (THz-TDSE) utilizing GaP and GaSe crystals as THz emitters . THz or MIR pulses reflected from the sample transmit the free-standing WP(B) with the computer-controlled rotation stage. The system detector is a low-temperature grown GaAs (LT-GaAs) epitaxial layer-transferred photoconductive switch (ELT-PCS). The ELT-PCS detector has a LT-GaAs epitaxial layer with a thickness of 2 μm which is bonded to a Si substrate using epoxy-based adhesive. The transmission angle of WP(C), which is fabricated on a high-density polyethylene (HDPE) film, is set at a 45° angle against the p- and s-polarized waves balancing the sensitivity of the ELT-PCS between the p- and s-polarized THz or MIR pulses. The extinction ratios of WP(A) and WP(B) are below 1 ×10-4 in the 1.2–30 THz range, and the extinction ratio of WP(C) is below 1×10-2 in the 0.3–30 THz range. II. EXPERIMENT & RESULTS Figure 1 shows a schematic of the ultra-broadband THz-TDSE system. We used a 200-μm thick GaP (110) crystal and a 100-μm thick z-cut GaSe crystal as the THz and MIR emitters, respectively. These emitters are excited by femtosecond laser pulses (300 mW average power, 800-nm center wave-length, 75-MHz repetition rate) with a 10-fs pulse duration obtained by compensating the group velocity dispersion(GVD) with negative GVD mirrors. The emitters can be switched without requiring additional optical alignment by moving the computer-controlled emitter stage. The generated THz or MIR pulses transmit the ultra-broadband free-standing wire-grid polarizer (WP) (A) fixing the polarization angle at 45°and illuminate the sample at a 60° incident angle. To reduce the uncertainty of the incidence angle due to the opening angle of the incident waves, the THz or MIR pulses are loosely focused by a 2-in. off-axis parabolic mirror (M 2) with a 6-in. focal length. The p- or s-polarized FIG. 1. Schematic diagram of the ultra-broadband THz-TDSE. The systemutilizes GaP and GaSe crystals for generating the THz and MIR pulses,respectively, and an ELT-PCS detector. The measurable frequency range can be switched between 0.5–7.8 THz and 7.8–30 THz by moving the computer-controlled emitter stage. A sample measured is a mono-layer polycrystalline graphene on a polyethylene terephthalate (PET) substrate. The sheet resistance is 800 Ω/sq. Figure 2(a) and (b) show the temporal waveforms of the p- and s-polarized THz and MIR pulses after the reflection off the graphene layer, respectively, and the obtained ellipsometric parameter ρ (=rp/rs) are shown in Fig. 2(c). By using the thin film approximation, the complex reflection coefficient of graphene for p- and s-polarized electromagnetic pulses are given by the following formula (CGS units) using optical conductivity of graphene. ⎛ ⎛ ε1 ⎜σ − c ⎜ ε 2 − ⎜⎜ ⎜ 4π cos θ ε 1 − ε 2 sin 2 θ ⎝ ⎝ rp = ⎛ ⎛ ε1 ⎜σ + c ⎜ ε 2 − ⎜⎜ ⎜ 4π cos θ ε 1 − ε 2 sin 2 θ ⎝ ⎝ c ⎛ ⎜σ + 4π ⎝ rs = − c ⎛ ⎜σ + 4 π ⎝ (ε (ε 1 1 determination of the carrier transport property of graphene samples on various substrates. ⎞⎞ ⎟⎟ ⎟ ⎟⎟ ⎠⎠ ⎞⎞ ⎟⎟ ⎟ ⎟⎟ ⎠⎠ III. SUMMARY (1), ) ) ⎞ − ε 2 sin 2 θ − cos θ ε 2 ⎟ ⎠ ⎞ − ε 2 sin 2 θ + cos θ ε 2 ⎟ ⎠ where, σ is the in-plain complex optical conductivity of graphene, ε1 is the dielectric constant of PET substrate, ε2 is the dielectric constant of air, and θ is the incident angle of THz and MIR pulses. By using the eq.(1), the complex sheet conductivity of graphene are obtained from the ellipsometric parameter ρ=rp/rs as shown in Fig. 2(d). We performed the fitting of the optical complex conductivity with the Drude model for charge carriers in graphene. EF e2 σ (ω ) = 2 π= (Γ − iω ) (2), where Γ is the average scattering rate for momentum changing collisions of charge carriers, and EF is Fermi energy. By fitting the data with the eq. (2), EF and Γ are determined as 0.35 eV and 27 THz, respectively. The estimated sheet resistance is 740 Ω/sq. The sheet carrier density N and carrier mobility μ are obtained as 7.48×1012 /cm2 and 1279 cm2/Vs, respectively. The transmission measurements of thin film samples suffer from the acquisition of the accurate phase information due to the error of the substrate thickness. The THz-TDSE enables the phase-sensitive reflection measurement and is useful to the Fig. 2 The p- and s-polarized temporal waveforms of (a) THz and (b) MIR pulses after the reflection off the graphene sample. (c) The ellipsometric parameter ρ=rp/rs and (d) The complex optical conductivity σ(ω) of the graphene obtained from ρ and the eq. (1). We have successfully measure the frequency-dependent of a single layer graphene by using ultra-broadband THz-TDSE. Since graphene shows the Drude-like sheet conductivity, we could estimate the sheet carrier density and carrier mobility by analysis. The THz-TDSE enables the phase-sensitive reflection measurement and can be applied to determine the carrier transport property of graphene samples on various substrates with no-contact and no-destructive. REFERENCES  M. Yamashita, H. Takahashi, and C. Otani, "Ultra-broadband THz time-domain spectroscopic ellipsometry, " Appl. Phys. Lett., vol.104, pp. 051103, 2014.  N. Vieweg, B. M. Fischer, and P. U. Jepsen, "Ultrabroadband terahertz spectroscopy of a liquid crystal," Opt. Express, vol.20, pp.28249, 2012.  S. Boubanga-Tombet, S. Chan, and T. 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