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Robust mapping of electrical properties of graphene from terahertz time-domain spectroscopy with timing jitter correction

Patrick R. Whelan, Krzysztof Iwaszczuk, Ruizhi Wang, Stephan Hofmann, Peter Bøggild, and Peter Uhd Jepsen

Open Access Open Access

Abstract

We demonstrate a method for reliably determining the electrical properties of graphene including the carrier scattering time and carrier drift mobility from terahertz time- domain spectroscopy measurements (THz-TDS). By comparing transients originating from directly transmitted pulses and the echoes from internal reflections in a substrate, we are able to extract electrical properties irrespective of random time delays between pulses emitted in a THz-TDS setup. If such time delays are not accounted for they can significantly influence the extracted properties of the material. The technique is useful for a robust determination of electrical properties from THz-TDS measurements and is compatible with substrate materials where transients from internal reflections are well-separated in time.

© 2017 Optical Society of America

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Figures (4)
Fig. 1
Fig. 1 (a) Waveform of THz pulses after transmission through a Si substrate without and with graphene. The inset shows how different echoes are generated by internal reflections in the substrate. (b) Sheet conductivity spectra for graphene on Si extracted from the first echo together with fits to the Drude model. (c) Optical map of the graphene sample combined with map of σ1 from the directly transmitted pulse averaged from 0.8 to 0.9 THz. White arrows indicate the scanning direction during the THz-TDS measurement and the grey box illustrates the area chosen for performing timing jitter corrections. White crosses mark different areas used as sample reference points. (d) A representative Raman spectrum from the sample shown in (c) with a histogram of Raman I(D)/(G) peak ratios in the inset.

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Fig. 2
Fig. 2 (a) Time delay between individual waveforms measured in dry air. Inset shows one waveform. (b) Sheet conductivity spectra for graphene on Si extracted from the directly transmitted pulse and the first echo. (c) Sheet conductivity spectra from same pixel as (b) after correction for timing jitter. (d) Timing jitter delay in each pixel from the highlighted area in Fig. 1(c).

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Fig. 3
Fig. 3 Histograms of (a-c) τsc and (d-f) µdrift for graphene extracted from the directly transmitted pulse from the area highlighted in Fig. 1(c). The five colors corresponds to properties calculated using five different reference areas on the sample, respectively and are plotted on top of each other. (a,d) Full fit after correction for timing jitter, (b,e) Full fit uncorrected, (c,f) Real-part fit after correction for timing jitter. Average values and standard deviations shown on plots cover all five data sets.

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Fig. 4
Fig. 4 Comparisons of electrical properties of graphene extracted from THz-TDS measurements using a full fit (FF) to the Drude model and only fitting to the real-part fit (RF) including non-corrected (ucorr) data and data after correction (corr) for timing jitter. (0) refers to data from directly transmitted pulses and (1) to the first reflected echo. (a) σDC. (b) τsc. (c) Ns. (d) µdrift. The bars show the average value based on measurements from five different reference areas with error bars showing the standard deviation.

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Equations (6)

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T ˜ f i l m ( 0 ) ( ω ) = E ˜ f i l m ( 0 ) ( ω ) E ˜ s u b ( 0 ) ( ω ) = E 0 t ˜ f i l m t ˜ s u b , a i r e − i δ E 0 t ˜ a i r , s u b t ˜ s u b , a i r e − i δ = t ˜ f i l m t ˜ a i r , s u b ,
σ ˜ s ( 0 ) ( ω ) = 1 Z 0 ( n A T ˜ ( 0 ) f i l m ( ω ) − n A ) ,
σ ˜ s ( 1 ) ( ω ) = ± n A n A 2 + 4 n A 2 n B T ˜ ( 1 ) f i l m ( ω ) + 4 n B 2 T ˜ ( 1 ) f i l m ( ω ) − n A 2 − 2 n A n B T ˜ ( 1 ) f i l m ( ω ) 2 n B Z 0 T ˜ ( 1 ) f i l m ( ω ) ,
σ ˜ s ( ω ) = σ D C 1 − i ω τ s c .
N s = π ℏ 2 e 4 ν F 2 ( σ D C τ s c ) 2 ,
μ d r i f t = σ D C e N s ,
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