Abstract

In this letter, we report optical pump terahertz (THz) near-field probe (n-OPTP) and optical pump THz near-field emission (n-OPTE) experiments of graphene/InAs heterostructures. Near-field imaging contrasts between graphene and InAs using these newly developed techniques as well as spectrally integrated THz nano-imaging (THz s-SNOM) are systematically studied. We demonstrate that in the near-field regime (λ/6000), a single layer of graphene is transparent to near-IR (800 nm) optical excitation and completely “screens” the photo-induced far-infrared (THz) dynamics in its substrate (InAs). Our work reveals unique frequency-selective ultrafast dynamics probed at the near field. It also provides strong evidence that n-OPTE nanoscopy yields contrast that distinguishes single-layer graphene from its substrate.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (8)

M. C. Giordano, S. Mastel, C. Liewald, L. L. Columbo, M. Brambilla, L. Viti, A. Politano, K. Zhang, L. Li, A. G. Davies, E. H. Linfield, R. Hillenbrand, F. Keilmann, G. Scamarcio, and M. S. Vitiello, “Phase-resolved terahertz self-detection near-field microscopy,” Opt. Express 26(14), 18423–18435 (2018).
[Crossref] [PubMed]

H. T. Stinson, A. Sternbach, O. Najera, R. Jing, A. S. Mcleod, T. V. Slusar, A. Mueller, L. Anderegg, H. T. Kim, M. Rozenberg, and D. N. Basov, “Imaging the nanoscale phase separation in vanadium dioxide thin films at terahertz frequencies,” Nat. Commun. 9(1), 3604 (2018).
[Crossref] [PubMed]

C. Zhou, Y. P. Liu, Z. Wang, S. J. Ma, M. W. Jia, R. Q. Wu, L. Zhou, W. Zhang, M. K. Liu, Y. Z. Wu, and J. Qi, “Broadband terahertz generation via the interface inverse Rashba-Edelstein effect,” Phys. Rev. Lett. 121(8), 086801 (2018).
[Crossref] [PubMed]

M. B. Jungfleisch, Q. Zhang, W. Zhang, J. E. Pearson, R. D. Schaller, H. Wen, and A. Hoffmann, “Control of terahertz emission by ultrafast spin-charge current conversion at Rashba interfaces,” Phys. Rev. Lett. 120(20), 207207 (2018).
[Crossref] [PubMed]

M. B. Jungfleisch, W. Zhang, and A. Hoffmann, “Perspectives of antiferromagnetic spintronics,” Phys. Lett. A 382(13), 865–871 (2018).
[Crossref]

J. Zhang, X. Chen, S. Mills, T. Ciavatti, Z. Yao, R. Mescall, H. Hu, V. Semenenko, Z. Fei, H. Li, V. Perebeinos, H. Tao, Q. Dai, X. Du, and M. Liu, “Terahertz nanoimaging of graphene,” ACS Photonics 5(7), 2645–2651 (2018).
[Crossref] [PubMed]

G. X. Ni, A. S. McLeod, Z. Sun, L. Wang, L. Xiong, K. W. Post, S. S. Sunku, B. Y. Jiang, J. Hone, C. R. Dean, M. M. Fogler, and D. N. Basov, “Fundamental limits to graphene plasmonics,” Nature 557(7706), 530–533 (2018).
[Crossref] [PubMed]

S. S. Sunku, G. X. Ni, B. Y. Jiang, H. Yoo, A. Sternbach, A. S. McLeod, T. Stauber, L. Xiong, T. Taniguchi, K. Watanabe, P. Kim, M. M. Fogler, and D. N. Basov, “Photonic crystals for nano-light in moiré graphene superlattices,” Science 362(6419), 1153–1156 (2018).
[Crossref] [PubMed]

2017 (5)

S. N. Gilbert Corder, X. Chen, S. Zhang, F. Hu, J. Zhang, Y. Luan, J. A. Logan, T. Ciavatti, H. A. Bechtel, M. C. Martin, M. Aronson, H. S. Suzuki, S. I. Kimura, T. Iizuka, Z. Fei, K. Imura, N. K. Sato, T. H. Tao, and M. Liu, “Near-field spectroscopic investigation of dual-band heavy fermion metamaterials,” Nat. Commun. 8(1), 2262 (2017).
[Crossref] [PubMed]

S. N. Gilbert Corder, J. Jiang, X. Chen, S. Kittiwatanakul, I.-C. Tung, Y. Zhu, J. Zhang, H. A. Bechtel, M. C. Martin, G. L. Carr, J. Lu, S. A. Wolf, H. Wen, T. H. Tao, and M. Liu, “Controlling phase separation in vanadium dioxide thin films via substrate engineering,” Phys. Rev. B 96(16), 161110 (2017).
[Crossref]

J. B. Héroux and M. Kuwata-Gonokami, “Photoexcited carrier dynamics in InAs, GaAs, and InSb probed by terahertz excitation spectroscopy,” Phys. Rev. Appl. 7(5), 054001 (2017).
[Crossref]

M. B. Lundeberg, Y. Gao, R. Asgari, C. Tan, B. Van Duppen, M. Autore, P. Alonso-González, A. Woessner, K. Watanabe, T. Taniguchi, R. Hillenbrand, J. Hone, M. Polini, and F. H. L. Koppens, “Tuning quantum nonlocal effects in graphene plasmonics,” Science 357(6347), 187–191 (2017).
[Crossref] [PubMed]

P. Klarskov, H. Kim, V. L. Colvin, and D. M. Mittleman, “Nanoscale laser terahertz emission microscopy,” ACS Photonics 4(11), 2676–2680 (2017).
[Crossref]

2016 (2)

L. Jiang, Z. Shi, B. Zeng, S. Wang, J.-H. Kang, T. Joshi, C. Jin, L. Ju, J. Kim, T. Lyu, Y.-R. Shen, M. Crommie, H.-J. Gao, and F. Wang, “Soliton-dependent plasmon reflection at bilayer graphene domain walls,” Nat. Mater. 15(8), 840–844 (2016).
[Crossref] [PubMed]

H. R. Seren, J. Zhang, G. R. Keiser, S. J. Maddox, X. Zhao, K. Fan, S. R. Bank, X. Zhang, and R. D. Averitt, “Nonlinear terahertz devices utilizing semiconducting plasmonic metamaterials,” Light Sci. Appl. 5(5), e16078 (2016).
[Crossref] [PubMed]

2015 (1)

G. X. Ni, H. Wang, J. S. Wu, Z. Fei, M. D. Goldflam, F. Keilmann, B. Özyilmaz, A. H. Castro Neto, X. M. Xie, M. M. Fogler, and D. N. Basov, “Plasmons in graphene moiré superlattices,” Nat. Mater. 14(12), 1217–1222 (2015).
[Crossref] [PubMed]

2014 (3)

Z. Shi, C. Jin, W. Yang, L. Ju, J. Horng, X. Lu, H. A. Bechtel, M. C. Martin, D. Fu, J. Wu, K. Watanabe, T. Taniguchi, Y. Zhang, X. Bai, E. Wang, G. Zhang, and F. Wang, “Gate-dependent pseudospin mixing in graphene/boron nitride moiré superlattices,” Nat. Phys. 10(10), 743–747 (2014).
[Crossref]

M. Eisele, T. L. Cocker, M. A. Huber, M. Plankl, L. Viti, D. Ercolani, L. Sorba, M. S. Vitiello, and R. Huber, “Ultrafast multi-terahertz nano-spectroscopy with sub-cycle temporal resolution,” Nat. Photonics 8(11), 841–845 (2014).
[Crossref]

D. N. Basov, M. M. Fogler, A. Lanzara, F. Wang, and Y. Zhang, “Colloquium: graphene spectroscopy,” Rev. Mod. Phys. 86(3), 959–994 (2014).
[Crossref]

2013 (2)

T. Kampfrath, M. Battiato, P. Maldonado, G. Eilers, J. Nötzold, S. Mährlein, V. Zbarsky, F. Freimuth, Y. Mokrousov, S. Blügel, M. Wolf, I. Radu, P. M. Oppeneer, and M. Münzenberg, “Terahertz spin current pulses controlled by magnetic heterostructures,” Nat. Nanotechnol. 8(4), 256–260 (2013).
[Crossref] [PubMed]

J. L. Garcia-Pomar, A. Y. Nikitin, and L. Martin-Moreno, “Scattering of graphene plasmons by defects in the graphene sheet,” ACS Nano 7(6), 4988–4994 (2013).
[Crossref] [PubMed]

2012 (2)

R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S. H. Oh, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of gold,” Phys. Rev. B Condens. Matter Mater. Phys. 86(23), 235147 (2012).
[Crossref]

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[Crossref] [PubMed]

2011 (1)

2010 (2)

2008 (4)

E. J. Nicol and J. P. Carbotte, “Optical conductivity of bilayer graphene with and without an asymmetry gap,” Phys. Rev. B Condens. Matter Mater. Phys. 77(15), 155409 (2008).
[Crossref]

G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103(6), 064302 (2008).
[Crossref]

H.-G. von Ribbeck, M. Brehm, D. W. van der Weide, S. Winnerl, O. Drachenko, M. Helm, and F. Keilmann, “Spectroscopic THz near-field microscope,” Opt. Express 16(5), 3430–3438 (2008).
[Crossref] [PubMed]

A. J. Huber, F. Keilmann, J. Wittborn, J. Aizpurua, and R. Hillenbrand, “Terahertz near-field nanoscopy of mobile carriers in single semiconductor nanodevices,” Nano Lett. 8(11), 3766–3770 (2008).
[Crossref] [PubMed]

2007 (1)

A. J. Huber, D. Kazantsev, F. Keilmann, J. Wittborn, and R. Hillenbrand, “Simultaneous IR material recognition and conductivity mapping by nanoscale near-field microscopy,” Adv. Mater. 19(17), 2209–2212 (2007).
[Crossref]

2006 (1)

L. F. J. Piper, T. D. Veal, M. J. Lowe, and C. F. McConville, “Electron depletion at InAs free surfaces: Doping-induced acceptor like gap states,” Phys. Rev. B Condens. Matter Mater. Phys. 73(19), 195321 (2006).
[Crossref]

2004 (1)

K. Wang, D. M. Mittleman, N. C. J. Van Der Valk, and P. C. M. Planken, “Antenna effects in terahertz apertureless near-field optical microscopy,” Appl. Phys. Lett. 85(14), 2715–2717 (2004).
[Crossref]

2003 (2)

H.-T. Chen, R. Kersting, and G. C. Cho, “Terahertz imaging with nanometer resolution,” Appl. Phys. Lett. 83(15), 3009–3011 (2003).
[Crossref]

T. Taubner, R. Hillenbrand, and F. Keilmann, “Performance of visible and mid-infrared scattering-type near-field optical microscopes,” J. Microsc. 210, 311–314 (2003).
[Crossref] [PubMed]

2001 (1)

R. Hillenbrand, B. Knoll, and F. Keilmann, “Pure optical contrast in scattering-type scanning near-field microscopy,” J. Microsc. 202, 77–83 (2001).
[Crossref] [PubMed]

Aizpurua, J.

A. J. Huber, F. Keilmann, J. Wittborn, J. Aizpurua, and R. Hillenbrand, “Terahertz near-field nanoscopy of mobile carriers in single semiconductor nanodevices,” Nano Lett. 8(11), 3766–3770 (2008).
[Crossref] [PubMed]

Alonso-González, P.

M. B. Lundeberg, Y. Gao, R. Asgari, C. Tan, B. Van Duppen, M. Autore, P. Alonso-González, A. Woessner, K. Watanabe, T. Taniguchi, R. Hillenbrand, J. Hone, M. Polini, and F. H. L. Koppens, “Tuning quantum nonlocal effects in graphene plasmonics,” Science 357(6347), 187–191 (2017).
[Crossref] [PubMed]

Anderegg, L.

H. T. Stinson, A. Sternbach, O. Najera, R. Jing, A. S. Mcleod, T. V. Slusar, A. Mueller, L. Anderegg, H. T. Kim, M. Rozenberg, and D. N. Basov, “Imaging the nanoscale phase separation in vanadium dioxide thin films at terahertz frequencies,” Nat. Commun. 9(1), 3604 (2018).
[Crossref] [PubMed]

Andreev, G. O.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[Crossref] [PubMed]

Aronson, M.

S. N. Gilbert Corder, X. Chen, S. Zhang, F. Hu, J. Zhang, Y. Luan, J. A. Logan, T. Ciavatti, H. A. Bechtel, M. C. Martin, M. Aronson, H. S. Suzuki, S. I. Kimura, T. Iizuka, Z. Fei, K. Imura, N. K. Sato, T. H. Tao, and M. Liu, “Near-field spectroscopic investigation of dual-band heavy fermion metamaterials,” Nat. Commun. 8(1), 2262 (2017).
[Crossref] [PubMed]

Asgari, R.

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Zhang, K.

Zhang, L. M.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
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Zhang, Q.

M. B. Jungfleisch, Q. Zhang, W. Zhang, J. E. Pearson, R. D. Schaller, H. Wen, and A. Hoffmann, “Control of terahertz emission by ultrafast spin-charge current conversion at Rashba interfaces,” Phys. Rev. Lett. 120(20), 207207 (2018).
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M. B. Jungfleisch, Q. Zhang, W. Zhang, J. E. Pearson, R. D. Schaller, H. Wen, and A. Hoffmann, “Control of terahertz emission by ultrafast spin-charge current conversion at Rashba interfaces,” Phys. Rev. Lett. 120(20), 207207 (2018).
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C. Zhou, Y. P. Liu, Z. Wang, S. J. Ma, M. W. Jia, R. Q. Wu, L. Zhou, W. Zhang, M. K. Liu, Y. Z. Wu, and J. Qi, “Broadband terahertz generation via the interface inverse Rashba-Edelstein effect,” Phys. Rev. Lett. 121(8), 086801 (2018).
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D. N. Basov, M. M. Fogler, A. Lanzara, F. Wang, and Y. Zhang, “Colloquium: graphene spectroscopy,” Rev. Mod. Phys. 86(3), 959–994 (2014).
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Z. Shi, C. Jin, W. Yang, L. Ju, J. Horng, X. Lu, H. A. Bechtel, M. C. Martin, D. Fu, J. Wu, K. Watanabe, T. Taniguchi, Y. Zhang, X. Bai, E. Wang, G. Zhang, and F. Wang, “Gate-dependent pseudospin mixing in graphene/boron nitride moiré superlattices,” Nat. Phys. 10(10), 743–747 (2014).
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Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
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Adv. Mater. (1)

A. J. Huber, D. Kazantsev, F. Keilmann, J. Wittborn, and R. Hillenbrand, “Simultaneous IR material recognition and conductivity mapping by nanoscale near-field microscopy,” Adv. Mater. 19(17), 2209–2212 (2007).
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Nano Lett. (1)

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L. Jiang, Z. Shi, B. Zeng, S. Wang, J.-H. Kang, T. Joshi, C. Jin, L. Ju, J. Kim, T. Lyu, Y.-R. Shen, M. Crommie, H.-J. Gao, and F. Wang, “Soliton-dependent plasmon reflection at bilayer graphene domain walls,” Nat. Mater. 15(8), 840–844 (2016).
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Nat. Nanotechnol. (1)

T. Kampfrath, M. Battiato, P. Maldonado, G. Eilers, J. Nötzold, S. Mährlein, V. Zbarsky, F. Freimuth, Y. Mokrousov, S. Blügel, M. Wolf, I. Radu, P. M. Oppeneer, and M. Münzenberg, “Terahertz spin current pulses controlled by magnetic heterostructures,” Nat. Nanotechnol. 8(4), 256–260 (2013).
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Nat. Photonics (1)

M. Eisele, T. L. Cocker, M. A. Huber, M. Plankl, L. Viti, D. Ercolani, L. Sorba, M. S. Vitiello, and R. Huber, “Ultrafast multi-terahertz nano-spectroscopy with sub-cycle temporal resolution,” Nat. Photonics 8(11), 841–845 (2014).
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Nat. Phys. (1)

Z. Shi, C. Jin, W. Yang, L. Ju, J. Horng, X. Lu, H. A. Bechtel, M. C. Martin, D. Fu, J. Wu, K. Watanabe, T. Taniguchi, Y. Zhang, X. Bai, E. Wang, G. Zhang, and F. Wang, “Gate-dependent pseudospin mixing in graphene/boron nitride moiré superlattices,” Nat. Phys. 10(10), 743–747 (2014).
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Nature (2)

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
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G. X. Ni, A. S. McLeod, Z. Sun, L. Wang, L. Xiong, K. W. Post, S. S. Sunku, B. Y. Jiang, J. Hone, C. R. Dean, M. M. Fogler, and D. N. Basov, “Fundamental limits to graphene plasmonics,” Nature 557(7706), 530–533 (2018).
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Opt. Express (4)

Phys. Lett. A (1)

M. B. Jungfleisch, W. Zhang, and A. Hoffmann, “Perspectives of antiferromagnetic spintronics,” Phys. Lett. A 382(13), 865–871 (2018).
[Crossref]

Phys. Rev. Appl. (1)

J. B. Héroux and M. Kuwata-Gonokami, “Photoexcited carrier dynamics in InAs, GaAs, and InSb probed by terahertz excitation spectroscopy,” Phys. Rev. Appl. 7(5), 054001 (2017).
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Phys. Rev. B (1)

S. N. Gilbert Corder, J. Jiang, X. Chen, S. Kittiwatanakul, I.-C. Tung, Y. Zhu, J. Zhang, H. A. Bechtel, M. C. Martin, G. L. Carr, J. Lu, S. A. Wolf, H. Wen, T. H. Tao, and M. Liu, “Controlling phase separation in vanadium dioxide thin films via substrate engineering,” Phys. Rev. B 96(16), 161110 (2017).
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Phys. Rev. B Condens. Matter Mater. Phys. (3)

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L. F. J. Piper, T. D. Veal, M. J. Lowe, and C. F. McConville, “Electron depletion at InAs free surfaces: Doping-induced acceptor like gap states,” Phys. Rev. B Condens. Matter Mater. Phys. 73(19), 195321 (2006).
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R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S. H. Oh, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of gold,” Phys. Rev. B Condens. Matter Mater. Phys. 86(23), 235147 (2012).
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C. Zhou, Y. P. Liu, Z. Wang, S. J. Ma, M. W. Jia, R. Q. Wu, L. Zhou, W. Zhang, M. K. Liu, Y. Z. Wu, and J. Qi, “Broadband terahertz generation via the interface inverse Rashba-Edelstein effect,” Phys. Rev. Lett. 121(8), 086801 (2018).
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M. B. Jungfleisch, Q. Zhang, W. Zhang, J. E. Pearson, R. D. Schaller, H. Wen, and A. Hoffmann, “Control of terahertz emission by ultrafast spin-charge current conversion at Rashba interfaces,” Phys. Rev. Lett. 120(20), 207207 (2018).
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Science (2)

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Figures (6)

Fig. 1
Fig. 1 Schematic of the near field optical-pump THz-probe (n-OPTP) setup, equally capable of performing optical pump near-field THz emission (n-OPTE) experiments. ①: 800 nm, 300 mW pump pulses pass through the ITO then go through the center of, while parallel to, the THz beam ②. Optical pump ① and THz probe ② are focused onto the AFM tip apex using an off-axis parabolic mirror. ③: THz gate (detection) beam. Tip scattered THz signals can be mapped out in the time domain by modifying the arrival time of THz gate beam ③ to the THz photoconductive antenna detector, thus changing the delay between ② and ③ (t1). Changing the delay between ① and ③ (t2) by modifying arrival time of pump pulse ① probes the photo-excited ultrafast dynamics of the sample (see main text), where in this work, t1 is fixed at the peak position of the scattered THz electric field while t2 is varied to yield n-OPTE (with THz probe ② blocked) and n-OPTP (with THz probe ② unblocked) measurements.
Fig. 2
Fig. 2 IR and THz s-SNOM of a graphene/InAs heterostructure. (a): AFM topography using IR near-field scanning system. (b): IR s-SNOM obtained simultaneously with (a). The bare InAs substrate, single layer graphene (SLG) and bilayer graphene (BLG) are easily distinguishable. The wavelength of the IR light is ~10.8 μm. (c): AFM topography using THz near-field system on the same sample area with (a). (d): THz s-SNOM image obtained simultaneously with (c). The regions covered by graphene, regardless of number of layers, show almost identical near-field (S2) contrast.
Fig. 3
Fig. 3 Near-field optical pump THz probe (n-OPTP) measurements of a graphene/InAs heterostructure. (a): Schematic of the experimental setup. SLG: single layer graphene. BLG: bilayer graphene. The tip-scattered signal includes 800 nm pump induced THz emission and near-field reflection of the incident THz pulses under photo-excitation. (b): The n-OPTP imaging of the InAs-SLG/InAs boundary at the same region as shown in Fig. 2. The time delay t2 between the gate beam and pump is fixed at when the THz signal reaches its maximum value (indicated by the green circles in panels (c) and (d)). Scale bar: 5 μm. (c): n-OPTP dynamics of bare InAs substrate taken with the tip located at the blue cross in (b). The dotted black line is a guide to the eye, showing the rise in the baseline of THz near-field reflectivity after InAs is pumped. (d): n-OPTP dynamics of SLG/InAs taken with the tip located at the red cross in (b). The dotted line shows that there is no notable change for the baseline of THz reflection signal before and after the pumping of SLG/InAs.
Fig. 4
Fig. 4 (a): Schematic of the near-field optical pump THz emission (n-OPTE) experiment. The tip scattered THz signal is induced by an 800 nm pump in InAs [9]. (b): n-OPTE imaging of graphene/InAs heterostructure and bare InAs substrate at the same region as in Fig. 2 and Fig. 3. t2 is fixed at the peak position of the THz emission, indicated by the green circles in (c) and (d). (c) and (d): n-OPTE dynamics of InAs and SLG/InAs, respectively. The signals are comparable, with the peak slightly lower in the case of SLG/InAs. Graphene does not screen the THz emission from the underlying InAs substrate.
Fig. 5
Fig. 5 Simulated s-SNOM S2 signals vs incident light frequency (ω) for different carrier mobilities and carrier densities in InAs, and different Fermi energies in graphene. Dotted curves are for bare InAs and solid curves for InAs covered by SLG. The same columns are for the same carrier mobility in InAs and the same rows are for the same carrier density in InAs. (a)-(e): μ InAs =2.5× 10 4 cm2/Vs. (f)-(j), μ InAs =3.5× 10 3 cm2/Vs. (a) and (f): InAs carrier density Nd = 1015cm−3. (b) and (g): Nd = 1016 cm−3. (c) and (h): Nd = 1017 cm−3. (d) and (i): Nd = 1018 cm−3. (e) and (j): Nd = 1019 cm−3. Note that both axes are in logarithmic scale. The y-axes appear in different scale for different rows. All the signals are normalized to those of bare gold under the same conditions. The dielectric permittivity of the gold reference was calculated using the Drude formula with a plasma frequency of ωp = 8.45 eV and a scattering time of ν−1 = 14 fs [35].
Fig. 6
Fig. 6 Solid lines: frequency dependence of the S2 signal from the InAs surface calculated for different doping levels Nd using Eq. (5) for the E-field reflectance, normalized to the S2 signal the signals from InAs and from gold are calculated using the Eq. (7) for the E-field reflection. The radius of the cylindrical or spherical tip is taken to be a = 30 nm. The dielectric permittivity of InAs and gold are modeled by the Drude formula with the parameters stated in the main text.

Equations (23)

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e(r,t)= 4p(t)[ r XZ h XZ ] ( r XZ h XZ ) 4 ( r XZ h XZ ) 2p(t) ( r XZ h XZ ) 2
v XZ =v( v y ^ )v, y ^ =(0,1,0)
E(r,t)= k=M M e(rkd,t)
E x (x,t)= e iωt j=N N ε 0j e i q j x , D z (x,t)= e iωt j=N N Δ 0j e i q j x
E (0) = j=N N a j [ i κ 0j 0 q j ]exp( iωti q j x κ 0j z )
κ 0j = q j 2 + ε 0 ω 2 / c 2
E (l) = e iωt j=N N e i k 0 a l,j x { A l,j [ γ l,j 0 α l,j ] e i b 0 γ l,j z + B l,j [ γ l,j 0 α l,j ] e i b 0 γ l,j (z+ h l ) }
D zj (1) E xj (1) = ε 1 α 1,j γ 1,j × Ξ j
E x (0) = E x (1) , D z (0) D z (1) =4πσ
σ t + j x =0
j ω = λ ω E xω (0)
λ ω = e 2 k B T π 2 (iω+ν) ( μ c k B T +2log( e μ c k B T +1 ) )+ e 2 ( ωiν ) iπ 2 0 f d (E) f d (E) ( ωiν ) 2 4 ( E/ ) 2 dE
f d ( E )= 1 1+exp[ ( E μ c )/ k B T ]
Imλ( ω )=+ 2ω π 0 Re[ λ( ω )λ( ) ] ω ' 2 ω 2 dω'
a j = ε 0j ( Ξ j ε 1 γ 1j + 4π λ ω ω b 0 ) Δ 0j α 0j b 0 ( ε 0 Ξ j ε 1 i κ 0,j γ 1j b 0 4π λ ω ω i κ 0,j )
E 0 sc =β p 0
p 0 =α E tot , α= a 2 ε 0 2 ε ε 0 ε+ ε 0
p 0 = α E 0 1αβ
E 0 sc = ε 1 ε 0 ε 0 ( ε 1 + ε 0 ) p 0 2 2 h 2
p 0 = ε 0 a 2 /2 1 ε 1 ε 0 ( ε 1 + ε 0 ) a 2 4 h 2 E 0
p 0 '= ε 0 a 3 1 ε 1 ε 0 ( ε 1 + ε 0 ) a 3 4 h 3 E 0
z(t)= z min + z max 2 + z min z max 2 cosΩt
S n = Ω π π Ω π Ω p 0 (t)cosnΩtdt

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