J Milli Terahz Waves DOI 10.1007/s10762-017-0362-5

Calculation and Study of Conductivity Based on Terahertz

Xiaodong Feng1 & Min Hu 1 & Jun Zhou 1 & Shenggang Liu 1

Received: 4 December 2016 /Accepted: 22 January 2017 # Springer Science+Business Media New York 2017

Abstract Based on terahertz time-domain spectroscopy system and two-dimensional scan- ning control system, terahertz transmission and reflection intensity mapping images on a graphene film are obtained, respectively. Then, graphene conductivity mapping images in the frequency range 0.5 to 2.5 THz are acquired according to the calculation formula. The conductivity of graphene at some typical regions is fitted by Drude-Smith formula to quanti- tatively compare the transmission and reflection measurements. The results show that terahertz reflection spectroscopy has a higher signal-to-noise ratio with less interference of impurities on the back of substrates. The effect of a red laser excitation on the graphene conductivity by terahertz time-domain transmission spectroscopy is also studied. The results show that the graphene conductivity in the excitation region is enhanced while that in the adjacent area is weakened which indicates carriers transport in graphene under laser excitation. This paper can make great contribution to the study on graphene electrical and optical properties in the terahertz regime and help design graphene terahertz devices.

Keywords Graphene conductivity. Terahertz spectroscopy. Drude-Smith formula . Laser excitation

1 Introduction

Terahertz (THz) radiation with frequencies typically from 0.1 to 30 THz, which occupies a middle ground between and infrared waves, is becoming a hot topic in the world [1–5]. Graphene with the structure of a two-dimensional, single-atomic-scale, honeycomb lattice, has been receiving more and more attention in the terahertz science field such as

* Min Hu [email protected]

1 University of Electronic Science and Technology of China, Terahertz Science and Technology Research Center, Chengdu, China J Infrared Milli Terahz Waves terahertz sources [6, 7], terahertz modulators [8–10], terahertz detectors [11–13], and terahertz absorbers [14–16] since it was isolated and characterized by A. K. Geim et al. [17–19]. As graphene chemical potential, carrier density, carrier mobility, and relaxation time can be achieved from graphene conductivity [20, 21], researches related to graphene conductivity make full sense. Most works like Ref. [21, 22] have been carried out for graphene conductivity calculation using terahertz transmission spectroscopy [23–27] but few based on terahertz reflection spectroscopy. However, not all substrates allow THz waves to transmit with high transmittance. Ref. [28] also images of a single-layer graphene film based on terahertz transmission spectroscopy without graphene conductivity fitting by Drude or Drude-Smith model. Ref. [29] studies gate-induced change of graphene conductivity without mapping. Ref. [30] measures of the optical absorption spectra of epitaxial graphene from terahertz to visible without mapping, too. Ref. [31] presents an identification method of the dynamic range of the detectable absorption coefficient in the analysis of transmission and reflection THz time- domain spectroscopy data, which does not cover any experimental measurement of graphene conductivity. In this paper, graphene conductivity mapping images on a damaged monolayer graphene/ quartz sample based on THz transmission and reflection spectroscopy are obtained and Drude- Smith formula is adopted to fit the calculated conductivity in the 0.5–2.5-THz frequency range. The results by transmission and reflection measurements are compared. In addition, the effect of a red laser excitation on the graphene conductivity is studied. The red laser is a cylindrical, low-power, red (632.8 nm) Helium-Neon laser with output power range from 2 to 5mW. The paper is organized as follows: Section 2 introduces the graphene conductivity algo- rithm in detail by transmission and reflection measurements respectively; Section 3 presents experimental measurement, transmission and reflection mapping images of graphene/quartz sample, and discontinuity analysis; Section 4 gives graphene conductivity images, Drude- Smith fitting, and comparison between the transmission and reflection data as well as the graphene conductivity change mapping images with laser excitation; Section 5 concludes the paper.

2 Graphene Conductivity Calculation Algorithm

When a THz pulse passes through the graphene/substrate sample, a portion of the pulse transmits while some is reflected at the interfaces as shown in Fig. 1. Graphene is treated as an infinitely thin conducting film and the THz pulse is treated as vertically incident in both transmission and reflection measurements as incident angles are both very small. The first- order transmission coefficient through the air-graphene interface is given by Ref. [22]

tG ¼ 2=ðÞnS þ 1 þ Z0σ ð1Þ where nS is the of substrates, σ is the graphene conductivity, and Z0 =377Ω is the vacuum wave impedance. From Fresnel formula, we can get the first-order transmission coefficient at the air/substrate interface given by

t0 ¼ 2=ðÞ1 þ nS ð2Þ J Infrared Milli Terahz Waves

Fig. 1 THz pulses pass through THz pulse THz pulse the graphene/substrate sample

1st-order 1st-order 2nd-order 2nd-order Graphene

Substrate

2nd-order 2nd-order

1st-order 1st-order

Combined with Eqs. (1) and (1), we get the calculation formula of graphene conductivity given by

σ ¼ ðÞt0=tG−1 ðÞ1 þ nS =Z0 ð3Þ where t0/tG is the relative transmittance of the first-order transmitted pulses for areas without and with graphene coverage. From the relationship between the transmission and reflection coefficients, we get first- order reflection coefficient at the air-graphene interface is given by

rG ¼ tG−1 ¼ ðÞ1−nS−Z0σ =ðÞ1 þ nS þ Z0σ ð4Þ and the first-order reflection coefficient at the air/substrate interface is given by

r0 ¼ ðÞ1−nS =ðÞ1 þ nS ð5Þ Combined with Eqs. (3)and(4), we get the graphene conductivity expression given by

ðÞ1−n ðÞ1 þ n ðÞ1−r =r σ ¼ S S G 0 ð6Þ ðÞðÞ1−nS rG=r0 þ ðÞ1 þ nS Z0

where rG/r0 is the relative reflectance of the 1-th transmitted pulses for areas with and without graphene coverage. Using Eqs. (2) and (5) and the relative transmittance/reflectance in frequency domain, we can obtain the graphene conductivity via the transmission as well as reflection measurement.

3 Experimental Measurement of Transmission and Reflection Images

For comparison between transmission and reflection measurement, a long time placed dam- aged commercial monolayer graphene film and another less damaged commercial monolayer are prepared with dimensions of about 1 × 1 cm while the quartz substrate is about 3 mm in thickness and 1 in. in diameter. The refractive index of quartz is 1.96 in the terahertz frequency range 0.5 to 2.5 THz measured by terahertz spectroscopy. J Infrared Milli Terahz Waves

Terahertz transmission and reflection time domain signals (TDS) are obtained with a THz Spectroscope(TPS 3000 assembled by the British company TeraView Ltd) by means of a two- dimensional raster scanning on the graphene/quartz sample in 0.1 mm steps between the fiber coupled emitter and receiver through transmission and reflection configurations separately. The scanning step is 0.1 mm. By Fourier transform of TDS, their corresponding terahertz transmission and reflection frequency-domain signals (FDS) range from 0.5 to 2 THz with a frequency resolution of 0.05 THz are acquired. As time delay for second-order transmission and reflection signals are at least 2nd t ¼ ð7Þ c where n is the refractive index of the substrate, d is the thickness of the substrate and c is the speed of light in vacuum. For the graphene/quartz sample, the time delay for second-order signal is at least 39.2 ps. Since only short time signals of 10 ps are chosen, so the second-order signals or other high-order signals are ignored. The chosen measured transmission and reflection signals for example in time domain and corresponding frequency domain signals by Fourier transformation for graphene/quartz sample are shown in Fig. 2.

(a) (b) 0.4 quartz 1.0 quartz graphene/quartz graphene/quartz 0.2 0.8

0.0 0.6

0.4 -0.2 Electric field/a.u. Transmittance/a.u. 0.2 -0.4

0.0 0 246810 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Time delay/ps Frequency/

(c) (d) quartz 1.0 0.20 quartz graphene/quartz graphene/quartz 0.15 0.8 0.10 0.6 0.05

0.00 0.4 Electric field/a.u. -0.05 Reflectance/a.u. 0.2 -0.10

-0.15 0.0 0246810 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Time delay/ps Frequency/THz Fig. 2 Transmission and reflection signals in time domain and frequency domain. a transmission signals in time domain. b Transmission signals in frequency domain. c Reflection signals in time domain. d Reflection signals in frequency domain J Infrared Milli Terahz Waves

Fig. 3 (color online) THz transmission (up) and reflection (down) intensity mapping images of graphene/quartz sample at 1.0, 1.5, and 2.0 THz (from left to right, normalized). Scale bars are1mm.Theareaenclosedbythered rectangle is about where graphene locates and the area enclosed by the white ellipse is about where the crack locates

As shown in Fig. 2, the transmittance is lower for quartz with graphene coverage while the reflectance is higher, which can be derived from Eqs. (1), (1), (3), and (4). Besides, since THz pulses transmit the sample which increases the time delay as the velocity of THz pulses is slower in the sample than that in the air, it can be seen that the transmission signals own a longer time delay resulting in a narrower bandwidth in frequency as shown in Fig. 2,which also has an impact on the results achieved by the two measurements.

Fig. 4 (color online) Graphene conductivity mapping images (units: mS) at 1.0, 1.5, 2.0 THz (from left to right)by using terahertz transmission FDS (up) and reflection FDS (down). Scale bars are 1 mm. The labeled regions like T1 or R1 with size of about 1 × 1 mm2 are chosen for comparing graphene average conductivity got by the two methods J Infrared Milli Terahz Waves (a) (b) 0.9 0.55

0.8 0.50 0.45 0.7 0.40 0.6 0.35 mS 0.5 mS 0.30 0.4 Data T1 0.25 Data T2 0.3 Data R1 Data R2 Fit T1 0.20 Fit T2 0.2 Fit R1 Fit R2 0.15 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 Frequency/THz Frequency/THz (c) (d) 0.55 0.55

0.50 0.50

0.45 0.45

0.40 0.40 0.35

mS mS 0.35

0.30 0.30 Data T4 0.25 Data T3 0.25 Data R3 Data R4 Fit T4 0.20 Fit T3 0.20 Fit R3 Fit R4 0.15 0.15 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 Frequency/THz Frequency/THz Fig. 5 Graphene average conductivity of different areas obtained through transmission and reflection measure- ments and their corresponding fitting curves by Drude-Smith formula. a, b, c, d correspond to different chosen areas

The transmission and reflection intensity mapping images of graphene/quartz sample at 1.0, 1.5 and 2.0 THz for instance are shown in Fig. 3 (All mapping images in this paper are obtained by Origin software using Contour-Color Fill with initial or calculated data). It is easy to find that the graphene is not continuous with an obvious crack as shown in Fig. 3.

4 Results and Discussion

4.1 Results by Transmission and Reflection TDS

With Eqs. (2) and (5), graphene conductivity mapping images by transmission and reflection FDS separately in the frequency range 0.5–2.5 THz are obtained. Conductivity images at

Table 1 Fitting parameters

T1 R1 T2 R2 T3 R3 T4 R4

D (mS) 1.25 1.26 0.91 0.95 0.94 0.98 0.82 0.91 b −1 −0.80 −0.82 −0.90 −0.87 −0.95 −0.75 −0.97 τ (fs) 85 68 92 96 86 107 62 107 Adj.R-Square 0.755 0.997 0.840 0.971 0.970 0.961 0.928 0.962 J Infrared Milli Terahz Waves

Fig. 6 (color online) Graphene conductivity mapping images without laser excitation (up) and with laser excitation (down) at 0.5, 1.0 and 1.5 THz (from left to right). Scale bars are 1 mm.Thearea in the red rectangle is the rough position of graphene while the area in the red ellipse is about where laser excites on. Areas labeled of about6×1.5,6×2and6×2mm2 (up to down) are chosen for analysis frequencies of 1.0, 1.5, and 2.0 THz for examples are shown in Fig. 4. It is clear that the graphene conductivity is not spatially uniform. It seems that a higher signal to noise ratio is achieved by reflection measurement. Four regions of graphene (shown in Fig. 4)arechosento compare the graphene conductivity in detail by the two methods. As graphene on quartz is non-uniform and Drude-Smith formula [21, 29] characterizes the semi-continuity or disconti- nuity, Drude-Smith formula is adopted to fit the obtained graphene conductivity in the frequency range 0.5–2.5 THz. The expression of the Drude-Smith formula depicting the conductivity of graphene is given by Ref. [32]  D b σ ¼ 1 þ ð8Þ 1−jωτ 1−jωτ

The express of the real part is

D DbðÞ1−ω2τ 2 Re½Š¼σ þ ð9Þ 1 þ ω2τ2 ðÞ1 þ ω2τ 2 2 where D is a Drude weight, ω is the angular frequency, τ is the carrier relaxation time, b is a backscattering parameter. For simplicity, the imaginary part of graphene conductivity is ignored as that is very small [33], which does not have an impact on our comparison. The fitting results by the real part of Drude-Smith formula are shown in Fig. 5 and fitting parameters are listed in Table 1. Figure 5a shows that graphene conductivity on quartz substrate achieved by reflection FDS is much better fitted by Drude-Smith formula than that by transmission FDS for region 1, which can be explained by some pollutants on the quartz J Infrared Milli Terahz Waves (a) (b)

1.05 Data A1 1.05 Data A2 Data A2 Data B2 1.00 Fit A1 1.00 Fit A2 Fit B2 Fit A2 0.95 0.95

0.90 0.90 / mS / mS 0.85 0.85

0.80 0.80

0.75 0.75

0.70 0.70 0.50 0.75 1.00 1.25 1.50 1.75 2.00 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Frequency/THz Frequency/THz (c) 1.10 Data A3 1.05 Data B3 Fit A3 1.00 Fit B3

0.95

0.90 / mS 0.85

0.80

0.75

0.70 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Frequency/THz

Fig. 7 Drude-Smith fit of the graphene average conductivity of chosen areas back introducing much more noise leading to inaccurate transmission results of T1 area. Figure 5b, c indicates that both results can be well fitted by Drude-Smith fitting formula for region 2 and region 3 with a little difference. Figure 5d shows that for region 4 the result got by transmission measurement is a little lower indicating that some contaminants on the back lead to a lower transmittance. As shown in Fig. 5, the results obtained by reflection measurement possess a higher signal to noise with fewer deviations for regions 1, 2, and 4 as a result of little influence of impurities on the quart’s back. Adj.R-Square is an adjusted coefficient of determination, which represents the goodness of fit. The better the fit is, the closer to the value of 1 it is. From Table 1, it is easy to find that for regions 1, 2, and 4, the results got by reflection FDS are much better fitted than that by transmission FDS. As for region 3, there is little difference between the two fitting qualities. The table also indicates that the calculated conductivity from reflection measurement is with higher signal to noise ratio. From the values of backscattering parameter b, we can find all b is between −0.75 and −0.98, which implies a predominance of carrier backscattering.

4.2 Results With and Without Laser Excitation by Transmission TDS

The graphene conductivity of another graphene/quartz sample mapping images without and with laser excitation are shown in Fig. 6. Through comparison at each frequency it is easy to find the enhanced conductivity of the excitation region (the upper part), the weakened J Infrared Milli Terahz Waves

Table 2 Fitting parameters

A1 B1 A2 B2 A3 B3

D (mS) 1.60 1.79 1.72 1.60 1.81 1.80 b −0.61 −0.58 −0.53 −0.56 −0.54 −0.52 τ (fs)837579767372 conductivity of the neighborhood (the middle part) and the almost unchanged conductivity of the far-zone from excitation (the bottom part). Different parts of graphene are chosen to make sure the change of the average conductivity and the real part of Drude-Smith formula is adopted to fit the graphene conductivity at different frequencies as shown in Figs. 6 and 7 separately. Fitting parameters are listed in Table 2. Figure 7 and Table 2 show that the conductivity of laser excitation area increased, the conductivity of the adjacent area decreased and the bottom part almost stays the same. It is attributed to laser induced electrons moving to adjacent places since graphene is initially p- type hole doping as a result of absorbing water, NO2,N2,O2 in the air [34–38]aswellas surface transfer doping [39–43] and other factors. Laser-induced electrons moving to the near places leading to the increased number of holes in A1 area and decreased number of holes in A2 area as a consequence of electron-hole recombination. Only very little spare electrons transport to the bottom part, so the bottom part hardly changes at all.

5 Conclusions

To sum up, graphene transmission and reflection mapping intensity images as well as conductivity mapping images are experimentally achieved based on terahertz transmission and reflection TDS. Graphene conductivity is compared between both methods by Drude- Smith formula fitting. The results show that reflection TDS can achieve graphene conductivity with little noise of substrates. Also, the effect of laser excitation on graphene conductivity is studied, which demonstrates the carriers transport in graphene with laser excitation. Study on THz graphene electrical and optical properties and design graphene THz devices can benefit from our work.

Acknowledgements This work is supported by National Key Program of Fundamental Research of China under Contract No.2014CB339801, National Natural Science Foundation of China under Contract No.11305030 and Contract No.61231005, and Fundamental Research Funds for the Central Universities (FRFCU) under Contract No.ZYGX2016KYQD113.

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