Abstract

We demonstrate wafer-scale, non-contact mapping of essential carrier transport parameters, carrier mobility (µdrift), carrier density (Ns), DC sheet conductance (σdc), and carrier scattering time (τsc) in CVD graphene, using spatially resolved terahertz time-domain conductance spectroscopy. σdc and τsc are directly extracted from Drude model fits to terahertz conductance spectra obtained in each pixel of 10 × 10 cm2 maps with a 400 µm step size. σdc- and τsc-maps are translated into µdrift and Ns maps through Boltzmann transport theory for graphene charge carriers and these parameters are directly compared to van der Pauw device measurements on the same wafer. The technique is compatible with all substrate materials that exhibit a reasonably low absorption coefficient for terahertz radiation. This includes many materials used for transferring CVD graphene in production facilities as well as in envisioned products, such as polymer films, glass substrates, cloth, or paper substrates.

© 2015 Optical Society of America

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2015 (5)

F. Pizzocchero, B. S. Jessen, P. R. Whelan, N. Kostesha, S. Lee, J. D. Buron, I. Petrushina, M. B. Larsen, P. Greenwood, W. J. Cha, K. Teo, P. U. Jepsen, J. Hone, P. Bøggild, and T. J. Booth, “Non-destructive electrochemical graphene transfer from reusable thin-film catalysts,” Carbon 85, 397–405 (2015).
[Crossref]

V. L. Nguyen, B. G. Shin, D. L. Duong, S. T. Kim, D. Perello, Y. J. Lim, Q. H. Yuan, F. Ding, H. Y. Jeong, H. S. Shin, S. M. Lee, S. H. Chae, Q. A. Vu, S. H. Lee, and Y. H. Lee, “Seamless stitching of graphene domains on polished copper (111) foil,” Adv. Mater. 27(8), 1376–1382 (2015).
[Crossref] [PubMed]

C. Cervetti, E. Heintze, B. Gorshunov, E. Zhukova, S. Lobanov, A. Hoyer, M. Burghard, K. Kern, M. Dressel, and L. Bogani, “Sub-terahertz frequency-domain spectroscopy reveals single-grain mobility and scatter influence of large-area graphene,” Adv. Mater. 27(16), 2635–2641 (2015).
[Crossref] [PubMed]

J. D. Buron, F. Pizzocchero, P. U. Jepsen, D. H. Petersen, J. M. Caridad, B. S. Jessen, T. J. Booth, and P. Bøggild, “Graphene mobility mapping,” Sci. Rep. 5, 12305 (2015).
[Crossref] [PubMed]

D. M. A. Mackenzie, J. D. Buron, P. R. Whelan, B. S. Jessen, A. Silajdźić, A. Pesquera, A. Centeno, A. Zurutuza, P. Bøggild, and D. H. Petersen, “Fabrication of CVD graphene-based devices via laser ablation for wafer-scale characterization,” 2D Mater. 2(4), 045003 (2015).
[Crossref]

2014 (6)

J. D. Buron, F. Pizzocchero, B. S. Jessen, T. J. Booth, P. F. Nielsen, O. Hansen, M. Hilke, E. Whiteway, P. U. Jepsen, P. Bøggild, and D. H. Petersen, “Electrically continuous graphene from single crystal copper verified by terahertz conductance spectroscopy and micro four-point probe,” Nano Lett. 14(11), 6348–6355 (2014).
[Crossref] [PubMed]

M. R. Lotz, M. Boll, O. Hansen, D. Kjær, P. Bøggild, and D. H. Petersen, “Revealing origin of quasi-one dimensional current transport in defect rich two dimensional materials,,” Appl. Phys. Lett. 105, 053115 (2014).
[Crossref]

M. Boll, M. R. Lotz, O. Hansen, F. Wang, D. Kjær, P. Bøggild, and D. H. Petersen, “Sensitivity analysis explains quasi-one-dimensional current transport in two-dimensional materials,” Phys. Rev. B 90(24), 245432 (2014).
[Crossref]

T. Ma, W. Ren, Z. Liu, L. Huang, L.-P. Ma, X. Ma, Z. Zhang, L.-M. Peng, and H.-M. Cheng, “Repeated growth-etching-regrowth for large-area defect-free single-crystal graphene by chemical vapor deposition,” ACS Nano 8(12), 12806–12813 (2014).
[Crossref] [PubMed]

J.-H. Lee, E. K. Lee, W.-J. Joo, Y. Jang, B.-S. Kim, J. Y. Lim, S.-H. Choi, S. J. Ahn, J. R. Ahn, M.-H. Park, C.-W. Yang, B. L. Choi, S.-W. Hwang, and D. Whang, “Wafer-scale growth of single-crystal monolayer graphene on reusable hydrogen-terminated germanium,” Science 344(6181), 286–289 (2014).
[Crossref] [PubMed]

S. Rahimi, L. Tao, S. F. Chowdhury, S. Park, A. Jouvray, S. Buttress, N. Rupesinghe, K. Teo, and D. Akinwande, “Toward 300 mm wafer-scalable high-performance polycrystalline chemical vapor deposited graphene transistors,” ACS Nano 8(10), 10471–10479 (2014).
[Crossref] [PubMed]

2013 (1)

G. Jnawali, Y. Rao, H. Yan, and T. F. Heinz, “Observation of a transient decrease in terahertz conductivity of single-layer graphene induced by ultrafast optical excitation,” Nano Lett. 13(2), 524–530 (2013).
[Crossref] [PubMed]

2012 (6)

I. Maeng, S. Lim, S. J. Chae, Y. H. Lee, H. Choi, and J.-H. Son, “Gate-controlled nonlinear conductivity of Dirac fermion in graphene field-effect transistors measured by terahertz time-domain spectroscopy,” Nano Lett. 12(2), 551–555 (2012).
[Crossref] [PubMed]

L. Tao, J. Lee, M. Holt, H. Chou, S. J. McDonnell, D. A. Ferrer, M. G. Babenco, R. M. Wallace, S. K. Banerjee, R. S. Ruoff, and D. Akinwande, “Uniform wafer-scale chemical vapor deposition of graphene on evaporated Cu (111) film with quality comparable to exfoliated monolayer,” J. Phys. Chem. C 116(45), 24068–24074 (2012).
[Crossref]

M. J. Paul, J. L. Tomaino, J. W. Kevek, T. DeBorde, Z. J. Thompson, E. D. Minot, and Y.-S. Lee, “Terahertz imaging of inhomogeneous electrodynamics in single-layer graphene embedded in dielectrics,” Appl. Phys. Lett. 101(9), 091109 (2012).
[Crossref]

J. D. Buron, D. H. Petersen, P. Bøggild, D. G. Cooke, M. Hilke, J. Sun, E. Whiteway, P. F. Nielsen, O. Hansen, A. Yurgens, and P. U. Jepsen, “Graphene conductance uniformity mapping,” Nano Lett. 12(10), 5074–5081 (2012).
[Crossref] [PubMed]

J. Tomaino, A. Jameson, M. Paul, J. Kevek, A. van der Zande, R. Barton, H. Choi, P. McEuen, E. Minot, and Y.-S. Lee, “High-contrast imaging of graphene via time-domain terahertz spectroscopy,” J. Infrared Millim. Terahertz Waves 33(8), 839–845 (2012).
[Crossref]

L. Ren, Q. Zhang, J. Yao, Z. Sun, R. Kaneko, Z. Yan, S. Nanot, Z. Jin, I. Kawayama, M. Tonouchi, J. M. Tour, and J. Kono, “Terahertz and infrared spectroscopy of gated large-area graphene,” Nano Lett. 12(7), 3711–3715 (2012).
[Crossref] [PubMed]

2011 (3)

J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B 83(16), 165113 (2011).
[Crossref]

P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging – Modern techniques and applications,” Laser Photonics Rev. 5(1), 124–166 (2011).
[Crossref]

J. Náhlík, I. Kašpárková, and P. Fitl, “Study of quantitative influence of sample defects on measurements of resistivity of thin films using van der Pauw method,” Measurement 44(10), 1968–1979 (2011).
[Crossref]

2010 (1)

S. Bae, H. Kim, Y. Lee, X. Xu, J.-S. Park, Y. Zheng, J. Balakrishnan, T. Lei, H. R. Kim, Y. I. Song, Y.-J. Kim, K. S. Kim, B. Özyilmaz, J.-H. Ahn, B. H. Hong, and S. Iijima, “Roll-to-roll production of 30-inch graphene films for transparent electrodes,” Nat. Nanotechnol. 5(8), 574–578 (2010).
[Crossref] [PubMed]

2009 (5)

K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, and B. H. Hong, “Large-scale pattern growth of graphene films for stretchable transparent electrodes,” Nature 457(7230), 706–710 (2009).
[Crossref] [PubMed]

L. A. Ponomarenko, R. Yang, T. M. Mohiuddin, M. I. Katsnelson, K. S. Novoselov, S. V. Morozov, A. A. Zhukov, F. Schedin, E. W. Hill, and A. K. Geim, “Effect of a high-κ environment on charge carrier mobility in graphene,” Phys. Rev. Lett. 102(20), 206603 (2009).
[Crossref] [PubMed]

G. Ng, D. Vasileska, and D. K. Schroder, “Calculation of the electron Hall mobility and Hall scattering factor in 6H-SiC,” J. Appl. Phys. 106(5), 053719 (2009).
[Crossref]

H. Choi, F. Borondics, D. A. Siegel, S. Y. Zhou, M. C. Martin, A. Lanzara, and R. A. Kaindl, “Broadband electromagnetic response and ultrafast dynamics of few-layer epitaxial graphene,” Appl. Phys. Lett. 94(17), 172102 (2009).
[Crossref]

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

2008 (3)

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys. 4(7), 532–535 (2008).
[Crossref]

J. M. Dawlaty, S. Shivaraman, J. Strait, P. George, M. Chandrashekhar, F. Rana, M. G. Spencer, D. Veksler, and Y. Chen, “Measurement of the optical absorption spectra of epitaxial graphene from terahertz to visible,” Appl. Phys. Lett. 93(13), 131905 (2008).
[Crossref]

D. H. Petersen, O. Hansen, R. Lin, and P. F. Nielsen, “Micro-four-point probe Hall effect measurement method,” J. Appl. Phys. 104(1), 013710 (2008).
[Crossref]

2007 (2)

E. H. Hwang, S. Adam, and S. D. Sarma, “Carrier transport in two-dimensional graphene layers,” Phys. Rev. Lett. 98(18), 186806 (2007).
[Crossref] [PubMed]

K. Nomura and A. H. MacDonald, “Quantum transport of massless Dirac fermions,” Phys. Rev. Lett. 98(7), 076602 (2007).
[Crossref] [PubMed]

2006 (1)

T. Ando, “Screening effect and impurity scattering in monolayer graphene,” J. Phys. Soc. Jpn. 75(7), 074716 (2006).
[Crossref]

1998 (1)

G. Rutsch, R. P. Devaty, W. J. Choyke, D. W. Langer, and L. B. Rowland, “Measurement of the Hall scattering factor in 4H and 6H SiC epilayers from 40 to 290 K and in magnetic fields up to 9 T,” J. Appl. Phys. 84(4), 2062–2064 (1998).
[Crossref]

1958 (1)

L. J. Van Der Pauw, “A method of measuring the resistivity and Hall coefficient on lamellae of arbitrary shape,” Philips Tech. Rev. 20, 220–224 (1958).

Adam, S.

E. H. Hwang, S. Adam, and S. D. Sarma, “Carrier transport in two-dimensional graphene layers,” Phys. Rev. Lett. 98(18), 186806 (2007).
[Crossref] [PubMed]

Ahn, J. R.

J.-H. Lee, E. K. Lee, W.-J. Joo, Y. Jang, B.-S. Kim, J. Y. Lim, S.-H. Choi, S. J. Ahn, J. R. Ahn, M.-H. Park, C.-W. Yang, B. L. Choi, S.-W. Hwang, and D. Whang, “Wafer-scale growth of single-crystal monolayer graphene on reusable hydrogen-terminated germanium,” Science 344(6181), 286–289 (2014).
[Crossref] [PubMed]

Ahn, J.-H.

S. Bae, H. Kim, Y. Lee, X. Xu, J.-S. Park, Y. Zheng, J. Balakrishnan, T. Lei, H. R. Kim, Y. I. Song, Y.-J. Kim, K. S. Kim, B. Özyilmaz, J.-H. Ahn, B. H. Hong, and S. Iijima, “Roll-to-roll production of 30-inch graphene films for transparent electrodes,” Nat. Nanotechnol. 5(8), 574–578 (2010).
[Crossref] [PubMed]

K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, and B. H. Hong, “Large-scale pattern growth of graphene films for stretchable transparent electrodes,” Nature 457(7230), 706–710 (2009).
[Crossref] [PubMed]

Ahn, S. J.

J.-H. Lee, E. K. Lee, W.-J. Joo, Y. Jang, B.-S. Kim, J. Y. Lim, S.-H. Choi, S. J. Ahn, J. R. Ahn, M.-H. Park, C.-W. Yang, B. L. Choi, S.-W. Hwang, and D. Whang, “Wafer-scale growth of single-crystal monolayer graphene on reusable hydrogen-terminated germanium,” Science 344(6181), 286–289 (2014).
[Crossref] [PubMed]

Akinwande, D.

S. Rahimi, L. Tao, S. F. Chowdhury, S. Park, A. Jouvray, S. Buttress, N. Rupesinghe, K. Teo, and D. Akinwande, “Toward 300 mm wafer-scalable high-performance polycrystalline chemical vapor deposited graphene transistors,” ACS Nano 8(10), 10471–10479 (2014).
[Crossref] [PubMed]

L. Tao, J. Lee, M. Holt, H. Chou, S. J. McDonnell, D. A. Ferrer, M. G. Babenco, R. M. Wallace, S. K. Banerjee, R. S. Ruoff, and D. Akinwande, “Uniform wafer-scale chemical vapor deposition of graphene on evaporated Cu (111) film with quality comparable to exfoliated monolayer,” J. Phys. Chem. C 116(45), 24068–24074 (2012).
[Crossref]

Ando, T.

T. Ando, “Screening effect and impurity scattering in monolayer graphene,” J. Phys. Soc. Jpn. 75(7), 074716 (2006).
[Crossref]

Babenco, M. G.

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J. D. Buron, F. Pizzocchero, B. S. Jessen, T. J. Booth, P. F. Nielsen, O. Hansen, M. Hilke, E. Whiteway, P. U. Jepsen, P. Bøggild, and D. H. Petersen, “Electrically continuous graphene from single crystal copper verified by terahertz conductance spectroscopy and micro four-point probe,” Nano Lett. 14(11), 6348–6355 (2014).
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M. Boll, M. R. Lotz, O. Hansen, F. Wang, D. Kjær, P. Bøggild, and D. H. Petersen, “Sensitivity analysis explains quasi-one-dimensional current transport in two-dimensional materials,” Phys. Rev. B 90(24), 245432 (2014).
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J. D. Buron, D. H. Petersen, P. Bøggild, D. G. Cooke, M. Hilke, J. Sun, E. Whiteway, P. F. Nielsen, O. Hansen, A. Yurgens, and P. U. Jepsen, “Graphene conductance uniformity mapping,” Nano Lett. 12(10), 5074–5081 (2012).
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J. D. Buron, F. Pizzocchero, P. U. Jepsen, D. H. Petersen, J. M. Caridad, B. S. Jessen, T. J. Booth, and P. Bøggild, “Graphene mobility mapping,” Sci. Rep. 5, 12305 (2015).
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J. D. Buron, F. Pizzocchero, P. U. Jepsen, D. H. Petersen, J. M. Caridad, B. S. Jessen, T. J. Booth, and P. Bøggild, “Graphene mobility mapping,” Sci. Rep. 5, 12305 (2015).
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J. D. Buron, F. Pizzocchero, B. S. Jessen, T. J. Booth, P. F. Nielsen, O. Hansen, M. Hilke, E. Whiteway, P. U. Jepsen, P. Bøggild, and D. H. Petersen, “Electrically continuous graphene from single crystal copper verified by terahertz conductance spectroscopy and micro four-point probe,” Nano Lett. 14(11), 6348–6355 (2014).
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J. D. Buron, D. H. Petersen, P. Bøggild, D. G. Cooke, M. Hilke, J. Sun, E. Whiteway, P. F. Nielsen, O. Hansen, A. Yurgens, and P. U. Jepsen, “Graphene conductance uniformity mapping,” Nano Lett. 12(10), 5074–5081 (2012).
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J. D. Buron, F. Pizzocchero, P. U. Jepsen, D. H. Petersen, J. M. Caridad, B. S. Jessen, T. J. Booth, and P. Bøggild, “Graphene mobility mapping,” Sci. Rep. 5, 12305 (2015).
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F. Pizzocchero, B. S. Jessen, P. R. Whelan, N. Kostesha, S. Lee, J. D. Buron, I. Petrushina, M. B. Larsen, P. Greenwood, W. J. Cha, K. Teo, P. U. Jepsen, J. Hone, P. Bøggild, and T. J. Booth, “Non-destructive electrochemical graphene transfer from reusable thin-film catalysts,” Carbon 85, 397–405 (2015).
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H. Choi, F. Borondics, D. A. Siegel, S. Y. Zhou, M. C. Martin, A. Lanzara, and R. A. Kaindl, “Broadband electromagnetic response and ultrafast dynamics of few-layer epitaxial graphene,” Appl. Phys. Lett. 94(17), 172102 (2009).
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L. Tao, J. Lee, M. Holt, H. Chou, S. J. McDonnell, D. A. Ferrer, M. G. Babenco, R. M. Wallace, S. K. Banerjee, R. S. Ruoff, and D. Akinwande, “Uniform wafer-scale chemical vapor deposition of graphene on evaporated Cu (111) film with quality comparable to exfoliated monolayer,” J. Phys. Chem. C 116(45), 24068–24074 (2012).
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J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B 83(16), 165113 (2011).
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J. M. Dawlaty, S. Shivaraman, J. Strait, P. George, M. Chandrashekhar, F. Rana, M. G. Spencer, D. Veksler, and Y. Chen, “Measurement of the optical absorption spectra of epitaxial graphene from terahertz to visible,” Appl. Phys. Lett. 93(13), 131905 (2008).
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C. Cervetti, E. Heintze, B. Gorshunov, E. Zhukova, S. Lobanov, A. Hoyer, M. Burghard, K. Kern, M. Dressel, and L. Bogani, “Sub-terahertz frequency-domain spectroscopy reveals single-grain mobility and scatter influence of large-area graphene,” Adv. Mater. 27(16), 2635–2641 (2015).
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L. Tao, J. Lee, M. Holt, H. Chou, S. J. McDonnell, D. A. Ferrer, M. G. Babenco, R. M. Wallace, S. K. Banerjee, R. S. Ruoff, and D. Akinwande, “Uniform wafer-scale chemical vapor deposition of graphene on evaporated Cu (111) film with quality comparable to exfoliated monolayer,” J. Phys. Chem. C 116(45), 24068–24074 (2012).
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J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B 83(16), 165113 (2011).
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J. M. Dawlaty, S. Shivaraman, J. Strait, P. George, M. Chandrashekhar, F. Rana, M. G. Spencer, D. Veksler, and Y. Chen, “Measurement of the optical absorption spectra of epitaxial graphene from terahertz to visible,” Appl. Phys. Lett. 93(13), 131905 (2008).
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J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B 83(16), 165113 (2011).
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C. Cervetti, E. Heintze, B. Gorshunov, E. Zhukova, S. Lobanov, A. Hoyer, M. Burghard, K. Kern, M. Dressel, and L. Bogani, “Sub-terahertz frequency-domain spectroscopy reveals single-grain mobility and scatter influence of large-area graphene,” Adv. Mater. 27(16), 2635–2641 (2015).
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F. Pizzocchero, B. S. Jessen, P. R. Whelan, N. Kostesha, S. Lee, J. D. Buron, I. Petrushina, M. B. Larsen, P. Greenwood, W. J. Cha, K. Teo, P. U. Jepsen, J. Hone, P. Bøggild, and T. J. Booth, “Non-destructive electrochemical graphene transfer from reusable thin-film catalysts,” Carbon 85, 397–405 (2015).
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J. D. Buron, F. Pizzocchero, B. S. Jessen, T. J. Booth, P. F. Nielsen, O. Hansen, M. Hilke, E. Whiteway, P. U. Jepsen, P. Bøggild, and D. H. Petersen, “Electrically continuous graphene from single crystal copper verified by terahertz conductance spectroscopy and micro four-point probe,” Nano Lett. 14(11), 6348–6355 (2014).
[Crossref] [PubMed]

M. R. Lotz, M. Boll, O. Hansen, D. Kjær, P. Bøggild, and D. H. Petersen, “Revealing origin of quasi-one dimensional current transport in defect rich two dimensional materials,,” Appl. Phys. Lett. 105, 053115 (2014).
[Crossref]

M. Boll, M. R. Lotz, O. Hansen, F. Wang, D. Kjær, P. Bøggild, and D. H. Petersen, “Sensitivity analysis explains quasi-one-dimensional current transport in two-dimensional materials,” Phys. Rev. B 90(24), 245432 (2014).
[Crossref]

J. D. Buron, D. H. Petersen, P. Bøggild, D. G. Cooke, M. Hilke, J. Sun, E. Whiteway, P. F. Nielsen, O. Hansen, A. Yurgens, and P. U. Jepsen, “Graphene conductance uniformity mapping,” Nano Lett. 12(10), 5074–5081 (2012).
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[Crossref]

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys. 4(7), 532–535 (2008).
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Lee, S. Y.

K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, and B. H. Hong, “Large-scale pattern growth of graphene films for stretchable transparent electrodes,” Nature 457(7230), 706–710 (2009).
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Lee, Y.

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V. L. Nguyen, B. G. Shin, D. L. Duong, S. T. Kim, D. Perello, Y. J. Lim, Q. H. Yuan, F. Ding, H. Y. Jeong, H. S. Shin, S. M. Lee, S. H. Chae, Q. A. Vu, S. H. Lee, and Y. H. Lee, “Seamless stitching of graphene domains on polished copper (111) foil,” Adv. Mater. 27(8), 1376–1382 (2015).
[Crossref] [PubMed]

I. Maeng, S. Lim, S. J. Chae, Y. H. Lee, H. Choi, and J.-H. Son, “Gate-controlled nonlinear conductivity of Dirac fermion in graphene field-effect transistors measured by terahertz time-domain spectroscopy,” Nano Lett. 12(2), 551–555 (2012).
[Crossref] [PubMed]

Lee, Y.-S.

J. Tomaino, A. Jameson, M. Paul, J. Kevek, A. van der Zande, R. Barton, H. Choi, P. McEuen, E. Minot, and Y.-S. Lee, “High-contrast imaging of graphene via time-domain terahertz spectroscopy,” J. Infrared Millim. Terahertz Waves 33(8), 839–845 (2012).
[Crossref]

M. J. Paul, J. L. Tomaino, J. W. Kevek, T. DeBorde, Z. J. Thompson, E. D. Minot, and Y.-S. Lee, “Terahertz imaging of inhomogeneous electrodynamics in single-layer graphene embedded in dielectrics,” Appl. Phys. Lett. 101(9), 091109 (2012).
[Crossref]

Lei, T.

S. Bae, H. Kim, Y. Lee, X. Xu, J.-S. Park, Y. Zheng, J. Balakrishnan, T. Lei, H. R. Kim, Y. I. Song, Y.-J. Kim, K. S. Kim, B. Özyilmaz, J.-H. Ahn, B. H. Hong, and S. Iijima, “Roll-to-roll production of 30-inch graphene films for transparent electrodes,” Nat. Nanotechnol. 5(8), 574–578 (2010).
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Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys. 4(7), 532–535 (2008).
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J.-H. Lee, E. K. Lee, W.-J. Joo, Y. Jang, B.-S. Kim, J. Y. Lim, S.-H. Choi, S. J. Ahn, J. R. Ahn, M.-H. Park, C.-W. Yang, B. L. Choi, S.-W. Hwang, and D. Whang, “Wafer-scale growth of single-crystal monolayer graphene on reusable hydrogen-terminated germanium,” Science 344(6181), 286–289 (2014).
[Crossref] [PubMed]

Lim, S.

I. Maeng, S. Lim, S. J. Chae, Y. H. Lee, H. Choi, and J.-H. Son, “Gate-controlled nonlinear conductivity of Dirac fermion in graphene field-effect transistors measured by terahertz time-domain spectroscopy,” Nano Lett. 12(2), 551–555 (2012).
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V. L. Nguyen, B. G. Shin, D. L. Duong, S. T. Kim, D. Perello, Y. J. Lim, Q. H. Yuan, F. Ding, H. Y. Jeong, H. S. Shin, S. M. Lee, S. H. Chae, Q. A. Vu, S. H. Lee, and Y. H. Lee, “Seamless stitching of graphene domains on polished copper (111) foil,” Adv. Mater. 27(8), 1376–1382 (2015).
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D. H. Petersen, O. Hansen, R. Lin, and P. F. Nielsen, “Micro-four-point probe Hall effect measurement method,” J. Appl. Phys. 104(1), 013710 (2008).
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T. Ma, W. Ren, Z. Liu, L. Huang, L.-P. Ma, X. Ma, Z. Zhang, L.-M. Peng, and H.-M. Cheng, “Repeated growth-etching-regrowth for large-area defect-free single-crystal graphene by chemical vapor deposition,” ACS Nano 8(12), 12806–12813 (2014).
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C. Cervetti, E. Heintze, B. Gorshunov, E. Zhukova, S. Lobanov, A. Hoyer, M. Burghard, K. Kern, M. Dressel, and L. Bogani, “Sub-terahertz frequency-domain spectroscopy reveals single-grain mobility and scatter influence of large-area graphene,” Adv. Mater. 27(16), 2635–2641 (2015).
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M. R. Lotz, M. Boll, O. Hansen, D. Kjær, P. Bøggild, and D. H. Petersen, “Revealing origin of quasi-one dimensional current transport in defect rich two dimensional materials,,” Appl. Phys. Lett. 105, 053115 (2014).
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M. Boll, M. R. Lotz, O. Hansen, F. Wang, D. Kjær, P. Bøggild, and D. H. Petersen, “Sensitivity analysis explains quasi-one-dimensional current transport in two-dimensional materials,” Phys. Rev. B 90(24), 245432 (2014).
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T. Ma, W. Ren, Z. Liu, L. Huang, L.-P. Ma, X. Ma, Z. Zhang, L.-M. Peng, and H.-M. Cheng, “Repeated growth-etching-regrowth for large-area defect-free single-crystal graphene by chemical vapor deposition,” ACS Nano 8(12), 12806–12813 (2014).
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T. Ma, W. Ren, Z. Liu, L. Huang, L.-P. Ma, X. Ma, Z. Zhang, L.-M. Peng, and H.-M. Cheng, “Repeated growth-etching-regrowth for large-area defect-free single-crystal graphene by chemical vapor deposition,” ACS Nano 8(12), 12806–12813 (2014).
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T. Ma, W. Ren, Z. Liu, L. Huang, L.-P. Ma, X. Ma, Z. Zhang, L.-M. Peng, and H.-M. Cheng, “Repeated growth-etching-regrowth for large-area defect-free single-crystal graphene by chemical vapor deposition,” ACS Nano 8(12), 12806–12813 (2014).
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K. Nomura and A. H. MacDonald, “Quantum transport of massless Dirac fermions,” Phys. Rev. Lett. 98(7), 076602 (2007).
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D. M. A. Mackenzie, J. D. Buron, P. R. Whelan, B. S. Jessen, A. Silajdźić, A. Pesquera, A. Centeno, A. Zurutuza, P. Bøggild, and D. H. Petersen, “Fabrication of CVD graphene-based devices via laser ablation for wafer-scale characterization,” 2D Mater. 2(4), 045003 (2015).
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I. Maeng, S. Lim, S. J. Chae, Y. H. Lee, H. Choi, and J.-H. Son, “Gate-controlled nonlinear conductivity of Dirac fermion in graphene field-effect transistors measured by terahertz time-domain spectroscopy,” Nano Lett. 12(2), 551–555 (2012).
[Crossref] [PubMed]

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J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B 83(16), 165113 (2011).
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H. Choi, F. Borondics, D. A. Siegel, S. Y. Zhou, M. C. Martin, A. Lanzara, and R. A. Kaindl, “Broadband electromagnetic response and ultrafast dynamics of few-layer epitaxial graphene,” Appl. Phys. Lett. 94(17), 172102 (2009).
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Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys. 4(7), 532–535 (2008).
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L. Tao, J. Lee, M. Holt, H. Chou, S. J. McDonnell, D. A. Ferrer, M. G. Babenco, R. M. Wallace, S. K. Banerjee, R. S. Ruoff, and D. Akinwande, “Uniform wafer-scale chemical vapor deposition of graphene on evaporated Cu (111) film with quality comparable to exfoliated monolayer,” J. Phys. Chem. C 116(45), 24068–24074 (2012).
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J. Tomaino, A. Jameson, M. Paul, J. Kevek, A. van der Zande, R. Barton, H. Choi, P. McEuen, E. Minot, and Y.-S. Lee, “High-contrast imaging of graphene via time-domain terahertz spectroscopy,” J. Infrared Millim. Terahertz Waves 33(8), 839–845 (2012).
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J. Tomaino, A. Jameson, M. Paul, J. Kevek, A. van der Zande, R. Barton, H. Choi, P. McEuen, E. Minot, and Y.-S. Lee, “High-contrast imaging of graphene via time-domain terahertz spectroscopy,” J. Infrared Millim. Terahertz Waves 33(8), 839–845 (2012).
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M. J. Paul, J. L. Tomaino, J. W. Kevek, T. DeBorde, Z. J. Thompson, E. D. Minot, and Y.-S. Lee, “Terahertz imaging of inhomogeneous electrodynamics in single-layer graphene embedded in dielectrics,” Appl. Phys. Lett. 101(9), 091109 (2012).
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L. A. Ponomarenko, R. Yang, T. M. Mohiuddin, M. I. Katsnelson, K. S. Novoselov, S. V. Morozov, A. A. Zhukov, F. Schedin, E. W. Hill, and A. K. Geim, “Effect of a high-κ environment on charge carrier mobility in graphene,” Phys. Rev. Lett. 102(20), 206603 (2009).
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L. A. Ponomarenko, R. Yang, T. M. Mohiuddin, M. I. Katsnelson, K. S. Novoselov, S. V. Morozov, A. A. Zhukov, F. Schedin, E. W. Hill, and A. K. Geim, “Effect of a high-κ environment on charge carrier mobility in graphene,” Phys. Rev. Lett. 102(20), 206603 (2009).
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G. Ng, D. Vasileska, and D. K. Schroder, “Calculation of the electron Hall mobility and Hall scattering factor in 6H-SiC,” J. Appl. Phys. 106(5), 053719 (2009).
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V. L. Nguyen, B. G. Shin, D. L. Duong, S. T. Kim, D. Perello, Y. J. Lim, Q. H. Yuan, F. Ding, H. Y. Jeong, H. S. Shin, S. M. Lee, S. H. Chae, Q. A. Vu, S. H. Lee, and Y. H. Lee, “Seamless stitching of graphene domains on polished copper (111) foil,” Adv. Mater. 27(8), 1376–1382 (2015).
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J. D. Buron, F. Pizzocchero, B. S. Jessen, T. J. Booth, P. F. Nielsen, O. Hansen, M. Hilke, E. Whiteway, P. U. Jepsen, P. Bøggild, and D. H. Petersen, “Electrically continuous graphene from single crystal copper verified by terahertz conductance spectroscopy and micro four-point probe,” Nano Lett. 14(11), 6348–6355 (2014).
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D. H. Petersen, O. Hansen, R. Lin, and P. F. Nielsen, “Micro-four-point probe Hall effect measurement method,” J. Appl. Phys. 104(1), 013710 (2008).
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K. Nomura and A. H. MacDonald, “Quantum transport of massless Dirac fermions,” Phys. Rev. Lett. 98(7), 076602 (2007).
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A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
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S. Bae, H. Kim, Y. Lee, X. Xu, J.-S. Park, Y. Zheng, J. Balakrishnan, T. Lei, H. R. Kim, Y. I. Song, Y.-J. Kim, K. S. Kim, B. Özyilmaz, J.-H. Ahn, B. H. Hong, and S. Iijima, “Roll-to-roll production of 30-inch graphene films for transparent electrodes,” Nat. Nanotechnol. 5(8), 574–578 (2010).
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Park, S.

S. Rahimi, L. Tao, S. F. Chowdhury, S. Park, A. Jouvray, S. Buttress, N. Rupesinghe, K. Teo, and D. Akinwande, “Toward 300 mm wafer-scalable high-performance polycrystalline chemical vapor deposited graphene transistors,” ACS Nano 8(10), 10471–10479 (2014).
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M. J. Paul, J. L. Tomaino, J. W. Kevek, T. DeBorde, Z. J. Thompson, E. D. Minot, and Y.-S. Lee, “Terahertz imaging of inhomogeneous electrodynamics in single-layer graphene embedded in dielectrics,” Appl. Phys. Lett. 101(9), 091109 (2012).
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T. Ma, W. Ren, Z. Liu, L. Huang, L.-P. Ma, X. Ma, Z. Zhang, L.-M. Peng, and H.-M. Cheng, “Repeated growth-etching-regrowth for large-area defect-free single-crystal graphene by chemical vapor deposition,” ACS Nano 8(12), 12806–12813 (2014).
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V. L. Nguyen, B. G. Shin, D. L. Duong, S. T. Kim, D. Perello, Y. J. Lim, Q. H. Yuan, F. Ding, H. Y. Jeong, H. S. Shin, S. M. Lee, S. H. Chae, Q. A. Vu, S. H. Lee, and Y. H. Lee, “Seamless stitching of graphene domains on polished copper (111) foil,” Adv. Mater. 27(8), 1376–1382 (2015).
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A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
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D. M. A. Mackenzie, J. D. Buron, P. R. Whelan, B. S. Jessen, A. Silajdźić, A. Pesquera, A. Centeno, A. Zurutuza, P. Bøggild, and D. H. Petersen, “Fabrication of CVD graphene-based devices via laser ablation for wafer-scale characterization,” 2D Mater. 2(4), 045003 (2015).
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D. M. A. Mackenzie, J. D. Buron, P. R. Whelan, B. S. Jessen, A. Silajdźić, A. Pesquera, A. Centeno, A. Zurutuza, P. Bøggild, and D. H. Petersen, “Fabrication of CVD graphene-based devices via laser ablation for wafer-scale characterization,” 2D Mater. 2(4), 045003 (2015).
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M. R. Lotz, M. Boll, O. Hansen, D. Kjær, P. Bøggild, and D. H. Petersen, “Revealing origin of quasi-one dimensional current transport in defect rich two dimensional materials,,” Appl. Phys. Lett. 105, 053115 (2014).
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M. Boll, M. R. Lotz, O. Hansen, F. Wang, D. Kjær, P. Bøggild, and D. H. Petersen, “Sensitivity analysis explains quasi-one-dimensional current transport in two-dimensional materials,” Phys. Rev. B 90(24), 245432 (2014).
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J. D. Buron, D. H. Petersen, P. Bøggild, D. G. Cooke, M. Hilke, J. Sun, E. Whiteway, P. F. Nielsen, O. Hansen, A. Yurgens, and P. U. Jepsen, “Graphene conductance uniformity mapping,” Nano Lett. 12(10), 5074–5081 (2012).
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D. H. Petersen, O. Hansen, R. Lin, and P. F. Nielsen, “Micro-four-point probe Hall effect measurement method,” J. Appl. Phys. 104(1), 013710 (2008).
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F. Pizzocchero, B. S. Jessen, P. R. Whelan, N. Kostesha, S. Lee, J. D. Buron, I. Petrushina, M. B. Larsen, P. Greenwood, W. J. Cha, K. Teo, P. U. Jepsen, J. Hone, P. Bøggild, and T. J. Booth, “Non-destructive electrochemical graphene transfer from reusable thin-film catalysts,” Carbon 85, 397–405 (2015).
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J. D. Buron, F. Pizzocchero, P. U. Jepsen, D. H. Petersen, J. M. Caridad, B. S. Jessen, T. J. Booth, and P. Bøggild, “Graphene mobility mapping,” Sci. Rep. 5, 12305 (2015).
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J. D. Buron, F. Pizzocchero, B. S. Jessen, T. J. Booth, P. F. Nielsen, O. Hansen, M. Hilke, E. Whiteway, P. U. Jepsen, P. Bøggild, and D. H. Petersen, “Electrically continuous graphene from single crystal copper verified by terahertz conductance spectroscopy and micro four-point probe,” Nano Lett. 14(11), 6348–6355 (2014).
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Ponomarenko, L. A.

L. A. Ponomarenko, R. Yang, T. M. Mohiuddin, M. I. Katsnelson, K. S. Novoselov, S. V. Morozov, A. A. Zhukov, F. Schedin, E. W. Hill, and A. K. Geim, “Effect of a high-κ environment on charge carrier mobility in graphene,” Phys. Rev. Lett. 102(20), 206603 (2009).
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S. Rahimi, L. Tao, S. F. Chowdhury, S. Park, A. Jouvray, S. Buttress, N. Rupesinghe, K. Teo, and D. Akinwande, “Toward 300 mm wafer-scalable high-performance polycrystalline chemical vapor deposited graphene transistors,” ACS Nano 8(10), 10471–10479 (2014).
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J. M. Dawlaty, S. Shivaraman, J. Strait, P. George, M. Chandrashekhar, F. Rana, M. G. Spencer, D. Veksler, and Y. Chen, “Measurement of the optical absorption spectra of epitaxial graphene from terahertz to visible,” Appl. Phys. Lett. 93(13), 131905 (2008).
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G. Jnawali, Y. Rao, H. Yan, and T. F. Heinz, “Observation of a transient decrease in terahertz conductivity of single-layer graphene induced by ultrafast optical excitation,” Nano Lett. 13(2), 524–530 (2013).
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L. Ren, Q. Zhang, J. Yao, Z. Sun, R. Kaneko, Z. Yan, S. Nanot, Z. Jin, I. Kawayama, M. Tonouchi, J. M. Tour, and J. Kono, “Terahertz and infrared spectroscopy of gated large-area graphene,” Nano Lett. 12(7), 3711–3715 (2012).
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Ren, W.

T. Ma, W. Ren, Z. Liu, L. Huang, L.-P. Ma, X. Ma, Z. Zhang, L.-M. Peng, and H.-M. Cheng, “Repeated growth-etching-regrowth for large-area defect-free single-crystal graphene by chemical vapor deposition,” ACS Nano 8(12), 12806–12813 (2014).
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G. Rutsch, R. P. Devaty, W. J. Choyke, D. W. Langer, and L. B. Rowland, “Measurement of the Hall scattering factor in 4H and 6H SiC epilayers from 40 to 290 K and in magnetic fields up to 9 T,” J. Appl. Phys. 84(4), 2062–2064 (1998).
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L. Tao, J. Lee, M. Holt, H. Chou, S. J. McDonnell, D. A. Ferrer, M. G. Babenco, R. M. Wallace, S. K. Banerjee, R. S. Ruoff, and D. Akinwande, “Uniform wafer-scale chemical vapor deposition of graphene on evaporated Cu (111) film with quality comparable to exfoliated monolayer,” J. Phys. Chem. C 116(45), 24068–24074 (2012).
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S. Rahimi, L. Tao, S. F. Chowdhury, S. Park, A. Jouvray, S. Buttress, N. Rupesinghe, K. Teo, and D. Akinwande, “Toward 300 mm wafer-scalable high-performance polycrystalline chemical vapor deposited graphene transistors,” ACS Nano 8(10), 10471–10479 (2014).
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G. Rutsch, R. P. Devaty, W. J. Choyke, D. W. Langer, and L. B. Rowland, “Measurement of the Hall scattering factor in 4H and 6H SiC epilayers from 40 to 290 K and in magnetic fields up to 9 T,” J. Appl. Phys. 84(4), 2062–2064 (1998).
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L. A. Ponomarenko, R. Yang, T. M. Mohiuddin, M. I. Katsnelson, K. S. Novoselov, S. V. Morozov, A. A. Zhukov, F. Schedin, E. W. Hill, and A. K. Geim, “Effect of a high-κ environment on charge carrier mobility in graphene,” Phys. Rev. Lett. 102(20), 206603 (2009).
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Schroder, D. K.

G. Ng, D. Vasileska, and D. K. Schroder, “Calculation of the electron Hall mobility and Hall scattering factor in 6H-SiC,” J. Appl. Phys. 106(5), 053719 (2009).
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Shen, Y. R.

J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B 83(16), 165113 (2011).
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Shin, B. G.

V. L. Nguyen, B. G. Shin, D. L. Duong, S. T. Kim, D. Perello, Y. J. Lim, Q. H. Yuan, F. Ding, H. Y. Jeong, H. S. Shin, S. M. Lee, S. H. Chae, Q. A. Vu, S. H. Lee, and Y. H. Lee, “Seamless stitching of graphene domains on polished copper (111) foil,” Adv. Mater. 27(8), 1376–1382 (2015).
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D. M. A. Mackenzie, J. D. Buron, P. R. Whelan, B. S. Jessen, A. Silajdźić, A. Pesquera, A. Centeno, A. Zurutuza, P. Bøggild, and D. H. Petersen, “Fabrication of CVD graphene-based devices via laser ablation for wafer-scale characterization,” 2D Mater. 2(4), 045003 (2015).
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2D Mater. (1)

D. M. A. Mackenzie, J. D. Buron, P. R. Whelan, B. S. Jessen, A. Silajdźić, A. Pesquera, A. Centeno, A. Zurutuza, P. Bøggild, and D. H. Petersen, “Fabrication of CVD graphene-based devices via laser ablation for wafer-scale characterization,” 2D Mater. 2(4), 045003 (2015).
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ACS Nano (2)

S. Rahimi, L. Tao, S. F. Chowdhury, S. Park, A. Jouvray, S. Buttress, N. Rupesinghe, K. Teo, and D. Akinwande, “Toward 300 mm wafer-scalable high-performance polycrystalline chemical vapor deposited graphene transistors,” ACS Nano 8(10), 10471–10479 (2014).
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V. L. Nguyen, B. G. Shin, D. L. Duong, S. T. Kim, D. Perello, Y. J. Lim, Q. H. Yuan, F. Ding, H. Y. Jeong, H. S. Shin, S. M. Lee, S. H. Chae, Q. A. Vu, S. H. Lee, and Y. H. Lee, “Seamless stitching of graphene domains on polished copper (111) foil,” Adv. Mater. 27(8), 1376–1382 (2015).
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Appl. Phys. Lett. (4)

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L. Tao, J. Lee, M. Holt, H. Chou, S. J. McDonnell, D. A. Ferrer, M. G. Babenco, R. M. Wallace, S. K. Banerjee, R. S. Ruoff, and D. Akinwande, “Uniform wafer-scale chemical vapor deposition of graphene on evaporated Cu (111) film with quality comparable to exfoliated monolayer,” J. Phys. Chem. C 116(45), 24068–24074 (2012).
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[Crossref] [PubMed]

I. Maeng, S. Lim, S. J. Chae, Y. H. Lee, H. Choi, and J.-H. Son, “Gate-controlled nonlinear conductivity of Dirac fermion in graphene field-effect transistors measured by terahertz time-domain spectroscopy,” Nano Lett. 12(2), 551–555 (2012).
[Crossref] [PubMed]

J. D. Buron, F. Pizzocchero, B. S. Jessen, T. J. Booth, P. F. Nielsen, O. Hansen, M. Hilke, E. Whiteway, P. U. Jepsen, P. Bøggild, and D. H. Petersen, “Electrically continuous graphene from single crystal copper verified by terahertz conductance spectroscopy and micro four-point probe,” Nano Lett. 14(11), 6348–6355 (2014).
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Figures (4)

Fig. 1
Fig. 1 (a) The sample is a 100 mm wide CVD graphene film on a 4-inch layered wafer substrate, consisting of 525 µm high resistivity silicon, 50 nm p + -doped poly-Si, and 165 nm LPCVD Si3N4. An approximately 10 mm wide area near one edge of the silicon wafer is kept free from graphene in the transfer process. The sample is raster scanned in the focal plane of the THz beam to form spatially resolved maps. (b) Examples of THz time-domain transients recorded after transmission through the sample (c) σs spectra (real part) extracted from time-domain measurements at two different graphene locations, showing different τsc. (d) vdP devices are formed after THz-TDS mapping by shadow mask deposition of Ti/Au contacts, and patterning of the graphene film by picosecond laser to form (e) 5 mm wide, square devices. µHall and Ns,Hall is determined from measurements of resistances in the A-, B-, and C-configurations in a constantly applied externally magnetic field of 255 mT.
Fig. 2
Fig. 2 Spatial maps of σdc and τsc obtained by raster-scanned THz-TDS analysis. Histograms show distributions of σdc and τsc within the dashed squares marked on the maps.
Fig. 3
Fig. 3 (a) Maps of Ns and µdrift derived from σdc and τsc-maps in Fig. 2. Distributions of Ns and µdrift within the dashed squares marked on the maps. (b) Distribution of Ns and µdrift relative to their average values.
Fig. 4
Fig. 4 Comparison of transport properties measured by THz-TDS and dual-configuration Hall effect method (a) σs;THz vs σs;vdP (b) µdrift vs. µHall (c) Ns. The full diagonal lines represent 1:1 correlation, while shaded areas represent 2:1 correlation or better. Error bars represent the standard deviation of the 36 THz-TDS pixels per device. Colored markers correspond to areas where the vdP dual configuration measurements indicate highly homogeneous / continuous conductance.

Equations (9)

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T ˜ ( ν )= E ˜ graphene ( ν ) E ˜ ref ( ν ) = 1+ n sub 1+ n sub + Z 0 σ ˜ s ( ν ) .
R A = I 12 V 34 , R A ' = I 21 V 43 , R B = I 24 V 31 , R B ' = I 42 V 13 , R C = I 23 V 41 , R C ' = I 32 V 14
exp[ π R A / R s ]+exp[ π R C / R s ]=1,
R Hall = ( R B R B ' ) /2 ,
N s,Hall = B ext / ( q R Hall ) ,
μ Hall =1/ ( e N s,Hall R S ) ,
σ ˜ s ( ω )= σ s,dc 1iωτ ,
σ s,dc = q 2 v F τ sc N S / ( π ) ,
N s = π 2 q 4 v F 2 ( σ s,dc τ sc ) 2 , μ drift = σ s,dc e N s = q 3 v F 2 π 2 τ sc 2 σ s,dc = q v F π τ sc N s

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