Tunable dual-band graphene-based infrared reflectance filter

Michael D. Goldflam; Isaac Ruiz; Stephen W. Howell; Joel R. Wendt; Michael B. Sinclair; David W. Peters; Thomas E. Beechem

doi:10.1364/OE.26.008532

Tunable dual-band graphene-based infrared reflectance filter

Michael D. Goldflam, Isaac Ruiz, Stephen W. Howell, Joel R. Wendt, Michael B. Sinclair, David W. Peters, and Thomas E. Beechem

Optics Express
Vol. 26,
Issue 7,
pp. 8532-8541
(2018)
•https://doi.org/10.1364/OE.26.008532

Get Citation
Copy Citation Text
Michael D. Goldflam, Isaac Ruiz, Stephen W. Howell, Joel R. Wendt, Michael B. Sinclair, David W. Peters, and Thomas E. Beechem, "Tunable dual-band graphene-based infrared reflectance filter," Opt. Express 26, 8532-8541 (2018)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Get PDF (5101 KB)
Set citation alerts
Save article

Related Topics
Table of Contents Category
- Plasmonics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Filters (120.2440)
- Plasmonics (250.5403)
- Filters, absorption (350.2450)

About this Article
History
- Original Manuscript: February 1, 2018
- Revised Manuscript: March 8, 2018
- Manuscript Accepted: March 8, 2018
- Published: March 23, 2018

Accessible

Open Access

Abstract

We experimentally demonstrated an actively tunable optical filter that controls the amplitude of reflected long-wave-infrared light in two separate spectral regions concurrently. Our device exploits the dependence of the excitation energy of plasmons in a continuous and unpatterned sheet of graphene on the Fermi-level, which can be controlled via conventional electrostatic gating. The filter enables simultaneous modification of two distinct spectral bands whose positions are dictated by the device geometry and graphene plasmon dispersion. Within these bands, the reflected amplitude can be varied by over 15% and resonance positions can be shifted by over 90 cm⁻¹. Electromagnetic simulations verify that tuning arises through coupling of incident light to graphene plasmons by a grating structure. Importantly, the tunable range is determined by a combination of graphene properties, device structure, and the surrounding dielectrics, which dictate the plasmon dispersion. Thus, the underlying design shown here is applicable across a broad range of infrared frequencies.

Full Article | PDF Article

OSA Recommended Articles

Independently tunable dual-band plasmonically induced transparency based on hybrid metal-graphene metamaterials at mid-infrared frequencies

Chen Sun, Zhewei Dong, Jiangnan Si, and Xiaoxu Deng
Opt. Express 25(2) 1242-1250 (2017)

Tunable mid-infrared dual-band and broadband cross-polarization converters based on U-shaped graphene metamaterials

Fang Zeng, Longfang Ye, Li Li, Zhihui Wang, Wei Zhao, and Yong Zhang
Opt. Express 27(23) 33826-33839 (2019)

Designing graphene absorption in a multispectral plasmon-enhanced infrared detector

Michael D. Goldflam, Zhe Fei, Isaac Ruiz, Stephen W. Howell, Paul S. Davids, David W. Peters, and Thomas E. Beechem
Opt. Express 25(11) 12400-12408 (2017)

References

View by:
Article Order
|
Year
|
Author
|
Publication

Y. W. Huang, H. W. Lee, R. Sokhoyan, R. A. Pala, K. Thyagarajan, S. Han, D. P. Tsai, and H. A. Atwater, “Gate-tunable conducting oxide metasurfaces,” Nano Lett. 16, 5319–5325 (2016).
[Crossref] [PubMed]
J. Park, J. H. Kang, X. Liu, and M. L. Brongersma, “Electrically tunable epsilon-near-zero (ENZ) metafilm absorbers,” Sci. Reports 5, 15754 (2015).
[Crossref]
T. Driscoll, S. Palit, M. M. Qazilbash, M. Brehm, F. Keilmann, B. G. Chae, S. J. Yun, H. T. Kim, S. Y. Cho, N. M. Jokerst, D. R. Smith, and D. N. Basov, “Dynamic tuning of an infrared hybrid-metamaterial resonance using vanadium dioxide,” Appl. Phys. Lett. 93, 024101 (2008).
[Crossref]
M. D. Goldflam, M. K. Liu, B. C. Chapler, H. T. Stinson, A. J. Sternbach, A. S. McLeod, J. D. Zhang, K. Geng, M. Royal, B.-J. Kim, R. D. Averitt, N. M. Jokerst, D. R. Smith, H. T. Kim, and D. N. Basov, “Voltage switching of a VO2 memory metasurface using ionic gel,” Appl. Phys. Lett. 105, 041117 (2014).
[Crossref]
M. J. Dicken, K. Aydin, I. M. Pryce, L. A. Sweatlock, E. M. Boyd, S. Walavalkar, J. Ma, and H. A. Atwater, “Frequency tunable near-infrared metamaterials based on VO2 phase transition,” Opt Express 17, 18330–18339 (2009).
[Crossref] [PubMed]
T. B. Hoang and M. H. Mikkelsen, “Broad electrical tuning of plasmonic nanoantennas at visible frequencies,” Appl. Phys. Lett. 108, 183107 (2016).
[Crossref]
D. Shrekenhamer, W. C. Chen, and W. J. Padilla, “Liquid crystal tunable metamaterial absorber,” Phys. Rev. Lett. 110, 177403 (2013).
[Crossref] [PubMed]
M. C. Sherrott, P. W. C. Hon, K. T. Fountaine, J. C. Garcia, S. M. Ponti, V. W. Brar, L. A. Sweatlock, and H. A. Atwater, “Experimental demonstration of >230 degrees phase modulation in gate-tunable graphene-gold reconfigurable mid-infrared metasurfaces,” Nano Lett. 17, 3027–3034 (2017).
[Crossref] [PubMed]
Y. C. Jun and I. Brener, “Electrically tunable infrared metamaterials based on depletion-type semiconductor devices,” J. Opt. 14, 114013 (2012).
[Crossref]
J. Kim, E. G. Carnemolla, C. DeVault, A. M. Shaltout, D. Faccio, V. M. Shalaev, A. V. Kildishev, M. Ferrera, and A. Boltasseva, “Dynamic control of nanocavities with tunable metal oxides,” Nano Lett. 18, 740–746 (2018).
[Crossref]
M. S. Jang, V. W. Brar, M. C. Sherrott, J. J. Lopez, L. Kim, S. Kim, M. Choi, H. A. Atwater, and et al., “Tunable large resonant absorption in a midinfrared graphene salisbury screen,” Phys. Rev. B 90, 165409 (2014).
[Crossref]
Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. García de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7, 2388–2395 (2013).
[Crossref] [PubMed]
V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, “Highly confined tunable mid-infrared plasmonics in graphene nanoresonators,” Nano Lett. 13, 2541–2547 (2013).
[Crossref] [PubMed]
S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, “Electronic transport in two-dimensional graphene,” Rev. Mod. Phys. 83, 407–470 (2011).
[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, 109–162 (2009).
[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, 82–85 (2012).
[Crossref] [PubMed]
Z. Fei, M. D. Goldflam, J. S. Wu, S. Dai, M. Wagner, A. S. McLeod, M. K. Liu, K. W. Post, S. Zhu, G. C. Janssen, M. M. Fogler, and D. N. Basov, “Edge and surface plasmons in graphene nanoribbons,” Nano Lett. 15, 8271–8276 (2015).
[Crossref] [PubMed]
A. Woessner, M. B. Lundeberg, Y. Gao, A. Principi, P. Alonso-Gonzalez, M. Carrega, K. Watanabe, T. Taniguchi, G. Vignale, M. Polini, J. Hone, R. Hillenbrand, and F. H. Koppens, “Highly confined low-loss plasmons in graphene-boron nitride heterostructures,” Nat. Mater. 14, 421–425 (2015).
[Crossref]
M. D. Goldflam, G. X. Ni, K. W. Post, Z. Fei, Y. Yeo, J. Y. Tan, A. S. Rodin, B. C. Chapler, B. Ozyilmaz, A. H. Castro Neto, M. M. Fogler, and D. N. Basov, “Tuning and persistent switching of graphene plasmons on a ferroelectric substrate,” Nano Lett. 15, 4859–4864 (2015).
[Crossref] [PubMed]
J. Chen, M. Badioli, P. Alonso-Gonzalez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovic, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. Garcia de Abajo, R. Hillenbrand, and F. H. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature. 487, 77–81 (2012).
[Crossref] [PubMed]
V. W. Brar, M. S. Jang, M. Sherrott, S. Kim, J. J. Lopez, L. B. Kim, M. Choi, and H. Atwater, “Hybrid surface-phonon-plasmon polariton modes in graphene/monolayer h-bn heterostructures,” Nano Lett. 14, 3876–3880 (2014).
[Crossref] [PubMed]
M. M. Jadidi, A. B. Sushkov, R. L. Myers-Ward, A. K. Boyd, K. M. Daniels, D. K. Gaskill, M. S. Fuhrer, H. D. Drew, and T. E. Murphy, “Tunable terahertz hybrid metal-graphene plasmons,” Nano Lett. 15, 7099–7104 (2015).
[Crossref] [PubMed]
T. Mueller, F. Xia, M. Freitag, J. Tsang, and P. Avouris, “Role of contacts in graphene transistors: A scanning photocurrent study,” Phys. Rev. B 79, 245430 (2009).
[Crossref]
C. Casiraghi, S. Pisana, K. S. Novoselov, A. K. Geim, and A. C. Ferrari, “Raman fingerprint of charged impurities in graphene,” Appl. Phys. Lett. 91, 233108 (2007).
[Crossref]
J. D. Fowler, M. J. Allen, V. C. Tung, Y. Yang, R. B. Kaner, and B. H. Weiller, “Practical chemical sensors from chemically derived graphene,” ACS Nano 3, 301–306 (2009).
[Crossref] [PubMed]
C. W. Chen, F. Ren, G.-C. Chi, S. C. Hung, Y. P. Huang, J. Kim, I. Kravchenko, and S. J. Pearton, “Effects of semiconductor processing chemicals on conductivity of graphene,” J. Vac. Sci. & Technol. B, Nanotechnol. Microelectron. Materials, Process. Meas. Phenom. 30, 040602 (2012).
V. E. Dorgan, M.-H. Bae, and E. Pop, “Mobility and saturation velocity in graphene on SiO2,” Appl. Phys. Lett. 97, 082112 (2010).
[Crossref]
J. Robertson, “High dielectric constant oxides,” The Eur. Phys. J. Appl. Phys. 28, 265–291 (2004).
[Crossref]
E. Hwang and S. D. Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75, 205418 (2007).
[Crossref]
M. D. Goldflam, Z. Fei, I. Ruiz, S. W. Howell, P. S. Davids, D. W. Peters, and T. E. Beechem, “Designing graphene absorption in a multispectral plasmon-enhanced infrared detector,” Opt. Express 25, 12400 (2017).
[Crossref] [PubMed]
A. Pirkle, J. Chan, A. Venugopal, D. Hinojos, C. W. Magnuson, S. McDonnell, L. Colombo, E. M. Vogel, R. S. Ruoff, and R. M. Wallace, “The effect of chemical residues on the physical and electrical properties of chemical vapor deposited graphene transferred to SiO2,” Appl. Phys. Lett. 99, 122108 (2011).
[Crossref]
L. Novotny and B. Hecht, Principles of nano-optics (Cambridge University, 2012).
[Crossref]
Z. Fei, G. O. Andreev, W. Bao, L. M. Zhang, A. S. McLeod, C. Wang, M. K. Stewart, Z. Zhao, G. Dominguez, M. Thiemens, M. M. Fogler, M. J. Tauber, A. H. Castro-Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Infrared nanoscopy of dirac plasmons at the graphene-SiO2 interface,” Nano Lett. 11, 4701–4705 (2011).
[Crossref] [PubMed]
W. Gannett, W. Regan, K. Watanabe, T. Taniguchi, M. F. Crommie, and A. Zettl, “Boron nitride substrates for high mobility chemical vapor deposited graphene,” Appl. Phys. Lett. 98, 242105 (2011).
[Crossref]
L. Banszerus, M. Schmitz, S. Engels, J. Dauber, M. Oellers, F. Haupt, K. Watanabe, T. Taniguchi, B. Beschoten, and C. Stampfer, “Ultrahigh-mobility graphene devices from chemical vapor deposition on reusable copper,” Sci. Adv. 1, e1500222 (2015).
[Crossref] [PubMed]
J. Yota, H. Shen, and R. Ramanathan, “Characterization of atomic layer deposition HfO2, Al2O3, and plasma-enhanced chemical vapor deposition Si3N4 as metal-insulator-metal capacitor dielectric for GaAs HBT technology,” J. Vac. Sci. & Technol. A: Vacuum, Surfaces, Films 31, 01A134 (2013).
[Crossref]
M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009).
[Crossref]
P. Khomyakov, G. Giovannetti, P. Rusu, G. v. Brocks, J. Van den Brink, and P. J. Kelly, “First-principles study of the interaction and charge transfer between graphene and metals,” Phys. Rev. B 79, 195425 (2009).
[Crossref]
S. Kim, J. Nah, I. Jo, D. Shahrjerdi, L. Colombo, Z. Yao, E. Tutuc, and S. K. Banerjee, “Realization of a high mobility dual-gated graphene field-effect transistor with Al2O3 dielectric,” Appl. Phys. Lett. 94, 062107 (2009).
[Crossref]
S. Jandhyala, G. Mordi, B. Lee, G. Lee, C. Floresca, P. R. Cha, J. Ahn, R. M. Wallace, Y. J. Chabal, M. J. Kim, L. Colombo, K. Cho, and J. Kim, “Atomic layer deposition of dielectrics on graphene using reversibly physisorbed ozone,” ACS Nano 6, 2722–2730 (2012).
[Crossref] [PubMed]
S. W. Howell, I. Ruiz, P. S. Davids, R. K. Harrison, S. W. Smith, M. D. Goldflam, J. B. Martin, N. J. Martinez, and T. E. Beechem, “Graphene-insulator-semiconductor junction for hybrid photodetection modalities,” Sci. Reports 7, 14651 (2017).
[Crossref]

2018 (1)

J. Kim, E. G. Carnemolla, C. DeVault, A. M. Shaltout, D. Faccio, V. M. Shalaev, A. V. Kildishev, M. Ferrera, and A. Boltasseva, “Dynamic control of nanocavities with tunable metal oxides,” Nano Lett. 18, 740–746 (2018).
[Crossref]

2017 (3)

M. C. Sherrott, P. W. C. Hon, K. T. Fountaine, J. C. Garcia, S. M. Ponti, V. W. Brar, L. A. Sweatlock, and H. A. Atwater, “Experimental demonstration of >230 degrees phase modulation in gate-tunable graphene-gold reconfigurable mid-infrared metasurfaces,” Nano Lett. 17, 3027–3034 (2017).
[Crossref] [PubMed]

M. D. Goldflam, Z. Fei, I. Ruiz, S. W. Howell, P. S. Davids, D. W. Peters, and T. E. Beechem, “Designing graphene absorption in a multispectral plasmon-enhanced infrared detector,” Opt. Express 25, 12400 (2017).
[Crossref] [PubMed]

S. W. Howell, I. Ruiz, P. S. Davids, R. K. Harrison, S. W. Smith, M. D. Goldflam, J. B. Martin, N. J. Martinez, and T. E. Beechem, “Graphene-insulator-semiconductor junction for hybrid photodetection modalities,” Sci. Reports 7, 14651 (2017).
[Crossref]

2016 (2)

T. B. Hoang and M. H. Mikkelsen, “Broad electrical tuning of plasmonic nanoantennas at visible frequencies,” Appl. Phys. Lett. 108, 183107 (2016).
[Crossref]

Y. W. Huang, H. W. Lee, R. Sokhoyan, R. A. Pala, K. Thyagarajan, S. Han, D. P. Tsai, and H. A. Atwater, “Gate-tunable conducting oxide metasurfaces,” Nano Lett. 16, 5319–5325 (2016).
[Crossref] [PubMed]

2015 (6)

J. Park, J. H. Kang, X. Liu, and M. L. Brongersma, “Electrically tunable epsilon-near-zero (ENZ) metafilm absorbers,” Sci. Reports 5, 15754 (2015).
[Crossref]

Z. Fei, M. D. Goldflam, J. S. Wu, S. Dai, M. Wagner, A. S. McLeod, M. K. Liu, K. W. Post, S. Zhu, G. C. Janssen, M. M. Fogler, and D. N. Basov, “Edge and surface plasmons in graphene nanoribbons,” Nano Lett. 15, 8271–8276 (2015).
[Crossref] [PubMed]

A. Woessner, M. B. Lundeberg, Y. Gao, A. Principi, P. Alonso-Gonzalez, M. Carrega, K. Watanabe, T. Taniguchi, G. Vignale, M. Polini, J. Hone, R. Hillenbrand, and F. H. Koppens, “Highly confined low-loss plasmons in graphene-boron nitride heterostructures,” Nat. Mater. 14, 421–425 (2015).
[Crossref]

M. D. Goldflam, G. X. Ni, K. W. Post, Z. Fei, Y. Yeo, J. Y. Tan, A. S. Rodin, B. C. Chapler, B. Ozyilmaz, A. H. Castro Neto, M. M. Fogler, and D. N. Basov, “Tuning and persistent switching of graphene plasmons on a ferroelectric substrate,” Nano Lett. 15, 4859–4864 (2015).
[Crossref] [PubMed]

L. Banszerus, M. Schmitz, S. Engels, J. Dauber, M. Oellers, F. Haupt, K. Watanabe, T. Taniguchi, B. Beschoten, and C. Stampfer, “Ultrahigh-mobility graphene devices from chemical vapor deposition on reusable copper,” Sci. Adv. 1, e1500222 (2015).
[Crossref] [PubMed]

M. M. Jadidi, A. B. Sushkov, R. L. Myers-Ward, A. K. Boyd, K. M. Daniels, D. K. Gaskill, M. S. Fuhrer, H. D. Drew, and T. E. Murphy, “Tunable terahertz hybrid metal-graphene plasmons,” Nano Lett. 15, 7099–7104 (2015).
[Crossref] [PubMed]

2014 (3)

V. W. Brar, M. S. Jang, M. Sherrott, S. Kim, J. J. Lopez, L. B. Kim, M. Choi, and H. Atwater, “Hybrid surface-phonon-plasmon polariton modes in graphene/monolayer h-bn heterostructures,” Nano Lett. 14, 3876–3880 (2014).
[Crossref] [PubMed]

M. S. Jang, V. W. Brar, M. C. Sherrott, J. J. Lopez, L. Kim, S. Kim, M. Choi, H. A. Atwater, and et al., “Tunable large resonant absorption in a midinfrared graphene salisbury screen,” Phys. Rev. B 90, 165409 (2014).
[Crossref]

M. D. Goldflam, M. K. Liu, B. C. Chapler, H. T. Stinson, A. J. Sternbach, A. S. McLeod, J. D. Zhang, K. Geng, M. Royal, B.-J. Kim, R. D. Averitt, N. M. Jokerst, D. R. Smith, H. T. Kim, and D. N. Basov, “Voltage switching of a VO2 memory metasurface using ionic gel,” Appl. Phys. Lett. 105, 041117 (2014).
[Crossref]

2013 (4)

D. Shrekenhamer, W. C. Chen, and W. J. Padilla, “Liquid crystal tunable metamaterial absorber,” Phys. Rev. Lett. 110, 177403 (2013).
[Crossref] [PubMed]

Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. García de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7, 2388–2395 (2013).
[Crossref] [PubMed]

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, “Highly confined tunable mid-infrared plasmonics in graphene nanoresonators,” Nano Lett. 13, 2541–2547 (2013).
[Crossref] [PubMed]

J. Yota, H. Shen, and R. Ramanathan, “Characterization of atomic layer deposition HfO2, Al2O3, and plasma-enhanced chemical vapor deposition Si3N4 as metal-insulator-metal capacitor dielectric for GaAs HBT technology,” J. Vac. Sci. & Technol. A: Vacuum, Surfaces, Films 31, 01A134 (2013).
[Crossref]

2012 (5)

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, 82–85 (2012).
[Crossref] [PubMed]

C. W. Chen, F. Ren, G.-C. Chi, S. C. Hung, Y. P. Huang, J. Kim, I. Kravchenko, and S. J. Pearton, “Effects of semiconductor processing chemicals on conductivity of graphene,” J. Vac. Sci. & Technol. B, Nanotechnol. Microelectron. Materials, Process. Meas. Phenom. 30, 040602 (2012).

J. Chen, M. Badioli, P. Alonso-Gonzalez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovic, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. Garcia de Abajo, R. Hillenbrand, and F. H. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature. 487, 77–81 (2012).
[Crossref] [PubMed]

Y. C. Jun and I. Brener, “Electrically tunable infrared metamaterials based on depletion-type semiconductor devices,” J. Opt. 14, 114013 (2012).
[Crossref]

S. Jandhyala, G. Mordi, B. Lee, G. Lee, C. Floresca, P. R. Cha, J. Ahn, R. M. Wallace, Y. J. Chabal, M. J. Kim, L. Colombo, K. Cho, and J. Kim, “Atomic layer deposition of dielectrics on graphene using reversibly physisorbed ozone,” ACS Nano 6, 2722–2730 (2012).
[Crossref] [PubMed]

2011 (4)

S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, “Electronic transport in two-dimensional graphene,” Rev. Mod. Phys. 83, 407–470 (2011).
[Crossref]

A. Pirkle, J. Chan, A. Venugopal, D. Hinojos, C. W. Magnuson, S. McDonnell, L. Colombo, E. M. Vogel, R. S. Ruoff, and R. M. Wallace, “The effect of chemical residues on the physical and electrical properties of chemical vapor deposited graphene transferred to SiO2,” Appl. Phys. Lett. 99, 122108 (2011).
[Crossref]

Z. Fei, G. O. Andreev, W. Bao, L. M. Zhang, A. S. McLeod, C. Wang, M. K. Stewart, Z. Zhao, G. Dominguez, M. Thiemens, M. M. Fogler, M. J. Tauber, A. H. Castro-Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Infrared nanoscopy of dirac plasmons at the graphene-SiO2 interface,” Nano Lett. 11, 4701–4705 (2011).
[Crossref] [PubMed]

W. Gannett, W. Regan, K. Watanabe, T. Taniguchi, M. F. Crommie, and A. Zettl, “Boron nitride substrates for high mobility chemical vapor deposited graphene,” Appl. Phys. Lett. 98, 242105 (2011).
[Crossref]

2010 (1)

V. E. Dorgan, M.-H. Bae, and E. Pop, “Mobility and saturation velocity in graphene on SiO2,” Appl. Phys. Lett. 97, 082112 (2010).
[Crossref]

2009 (7)

J. D. Fowler, M. J. Allen, V. C. Tung, Y. Yang, R. B. Kaner, and B. H. Weiller, “Practical chemical sensors from chemically derived graphene,” ACS Nano 3, 301–306 (2009).
[Crossref] [PubMed]

T. Mueller, F. Xia, M. Freitag, J. Tsang, and P. Avouris, “Role of contacts in graphene transistors: A scanning photocurrent study,” Phys. Rev. B 79, 245430 (2009).
[Crossref]

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009).
[Crossref]

P. Khomyakov, G. Giovannetti, P. Rusu, G. v. Brocks, J. Van den Brink, and P. J. Kelly, “First-principles study of the interaction and charge transfer between graphene and metals,” Phys. Rev. B 79, 195425 (2009).
[Crossref]

S. Kim, J. Nah, I. Jo, D. Shahrjerdi, L. Colombo, Z. Yao, E. Tutuc, and S. K. Banerjee, “Realization of a high mobility dual-gated graphene field-effect transistor with Al2O3 dielectric,” Appl. Phys. Lett. 94, 062107 (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, 109–162 (2009).
[Crossref]

M. J. Dicken, K. Aydin, I. M. Pryce, L. A. Sweatlock, E. M. Boyd, S. Walavalkar, J. Ma, and H. A. Atwater, “Frequency tunable near-infrared metamaterials based on VO2 phase transition,” Opt Express 17, 18330–18339 (2009).
[Crossref] [PubMed]

2008 (1)

T. Driscoll, S. Palit, M. M. Qazilbash, M. Brehm, F. Keilmann, B. G. Chae, S. J. Yun, H. T. Kim, S. Y. Cho, N. M. Jokerst, D. R. Smith, and D. N. Basov, “Dynamic tuning of an infrared hybrid-metamaterial resonance using vanadium dioxide,” Appl. Phys. Lett. 93, 024101 (2008).
[Crossref]

2007 (2)

C. Casiraghi, S. Pisana, K. S. Novoselov, A. K. Geim, and A. C. Ferrari, “Raman fingerprint of charged impurities in graphene,” Appl. Phys. Lett. 91, 233108 (2007).
[Crossref]

E. Hwang and S. D. Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75, 205418 (2007).
[Crossref]

2004 (1)

J. Robertson, “High dielectric constant oxides,” The Eur. Phys. J. Appl. Phys. 28, 265–291 (2004).
[Crossref]

Adam, S.

S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, “Electronic transport in two-dimensional graphene,” Rev. Mod. Phys. 83, 407–470 (2011).
[Crossref]

Ahn, J.

Ajayan, P. M.

Allen, M. J.

J. D. Fowler, M. J. Allen, V. C. Tung, Y. Yang, R. B. Kaner, and B. H. Weiller, “Practical chemical sensors from chemically derived graphene,” ACS Nano 3, 301–306 (2009).
[Crossref] [PubMed]

Alonso-Gonzalez, P.

Andreev, G. O.

Atwater, H.

Atwater, H. A.

Averitt, R. D.

Avouris, P.

T. Mueller, F. Xia, M. Freitag, J. Tsang, and P. Avouris, “Role of contacts in graphene transistors: A scanning photocurrent study,” Phys. Rev. B 79, 245430 (2009).
[Crossref]

Aydin, K.

Badioli, M.

Bae, M.-H.

V. E. Dorgan, M.-H. Bae, and E. Pop, “Mobility and saturation velocity in graphene on SiO2,” Appl. Phys. Lett. 97, 082112 (2010).
[Crossref]

Banerjee, S. K.

Banszerus, L.

Bao, W.

Basov, D. N.

Beechem, T. E.

Beschoten, B.

Boltasseva, A.

Boyd, A. K.

Boyd, E. M.

Brar, V. W.

Brehm, M.

Brener, I.

Y. C. Jun and I. Brener, “Electrically tunable infrared metamaterials based on depletion-type semiconductor devices,” J. Opt. 14, 114013 (2012).
[Crossref]

Brocks, G. v.

Brongersma, M. L.

J. Park, J. H. Kang, X. Liu, and M. L. Brongersma, “Electrically tunable epsilon-near-zero (ENZ) metafilm absorbers,” Sci. Reports 5, 15754 (2015).
[Crossref]

Buljan, H.

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009).
[Crossref]

Camara, N.

Carnemolla, E. G.

Carrega, M.

Casiraghi, C.

C. Casiraghi, S. Pisana, K. S. Novoselov, A. K. Geim, and A. C. Ferrari, “Raman fingerprint of charged impurities in graphene,” Appl. Phys. Lett. 91, 233108 (2007).
[Crossref]

Castro Neto, A. H.

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, 109–162 (2009).
[Crossref]

Castro-Neto, A. H.

Centeno, A.

Cha, P. R.

Chabal, Y. J.

Chae, B. G.

Chan, J.

Chapler, B. C.

Chen, C. W.

Chen, J.

Chen, W. C.

D. Shrekenhamer, W. C. Chen, and W. J. Padilla, “Liquid crystal tunable metamaterial absorber,” Phys. Rev. Lett. 110, 177403 (2013).
[Crossref] [PubMed]

Chi, G.-C.

Cho, K.

Cho, S. Y.

Choi, M.

Colombo, L.

Crommie, M. F.

Dai, S.

Daniels, K. M.

Das Sarma, S.

S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, “Electronic transport in two-dimensional graphene,” Rev. Mod. Phys. 83, 407–470 (2011).
[Crossref]

Dauber, J.

Davids, P. S.

DeVault, C.

Dicken, M. J.

Dominguez, G.

Dorgan, V. E.

V. E. Dorgan, M.-H. Bae, and E. Pop, “Mobility and saturation velocity in graphene on SiO2,” Appl. Phys. Lett. 97, 082112 (2010).
[Crossref]

Drew, H. D.

Driscoll, T.

Elorza, A. Z.

Engels, S.

Faccio, D.

Fang, Z.

Fei, Z.

Ferrari, A. C.

C. Casiraghi, S. Pisana, K. S. Novoselov, A. K. Geim, and A. C. Ferrari, “Raman fingerprint of charged impurities in graphene,” Appl. Phys. Lett. 91, 233108 (2007).
[Crossref]

Ferrera, M.

Floresca, C.

Fogler, M. M.

Fountaine, K. T.

Fowler, J. D.

J. D. Fowler, M. J. Allen, V. C. Tung, Y. Yang, R. B. Kaner, and B. H. Weiller, “Practical chemical sensors from chemically derived graphene,” ACS Nano 3, 301–306 (2009).
[Crossref] [PubMed]

Freitag, M.

T. Mueller, F. Xia, M. Freitag, J. Tsang, and P. Avouris, “Role of contacts in graphene transistors: A scanning photocurrent study,” Phys. Rev. B 79, 245430 (2009).
[Crossref]

Fuhrer, M. S.

Gannett, W.

Gao, Y.

Garcia, J. C.

Garcia de Abajo, F. J.

García de Abajo, F. J.

Gaskill, D. K.

Geim, A. K.

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, 109–162 (2009).
[Crossref]

C. Casiraghi, S. Pisana, K. S. Novoselov, A. K. Geim, and A. C. Ferrari, “Raman fingerprint of charged impurities in graphene,” Appl. Phys. Lett. 91, 233108 (2007).
[Crossref]

Geng, K.

Giovannetti, G.

Godignon, P.

Goldflam, M. D.

Guinea, F.

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, 109–162 (2009).
[Crossref]

Halas, N. J.

Han, S.

Harrison, R. K.

Haupt, F.

Hecht, B.

L. Novotny and B. Hecht, Principles of nano-optics (Cambridge University, 2012).
[Crossref]

Hillenbrand, R.

Hinojos, D.

Hoang, T. B.

T. B. Hoang and M. H. Mikkelsen, “Broad electrical tuning of plasmonic nanoantennas at visible frequencies,” Appl. Phys. Lett. 108, 183107 (2016).
[Crossref]

Hon, P. W. C.

Hone, J.

Howell, S. W.

Huang, Y. P.

Huang, Y. W.

Hung, S. C.

Huth, F.

Hwang, E.

E. Hwang and S. D. Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75, 205418 (2007).
[Crossref]

Hwang, E. H.

S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, “Electronic transport in two-dimensional graphene,” Rev. Mod. Phys. 83, 407–470 (2011).
[Crossref]

Jablan, M.

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009).
[Crossref]

Jadidi, M. M.

Jandhyala, S.

Jang, M. S.

Janssen, G. C.

Jo, I.

Jokerst, N. M.

Jun, Y. C.

Y. C. Jun and I. Brener, “Electrically tunable infrared metamaterials based on depletion-type semiconductor devices,” J. Opt. 14, 114013 (2012).
[Crossref]

Kaner, R. B.

J. D. Fowler, M. J. Allen, V. C. Tung, Y. Yang, R. B. Kaner, and B. H. Weiller, “Practical chemical sensors from chemically derived graphene,” ACS Nano 3, 301–306 (2009).
[Crossref] [PubMed]

Kang, J. H.

J. Park, J. H. Kang, X. Liu, and M. L. Brongersma, “Electrically tunable epsilon-near-zero (ENZ) metafilm absorbers,” Sci. Reports 5, 15754 (2015).
[Crossref]

Keilmann, F.

Kelly, P. J.

Khomyakov, P.

Kildishev, A. V.

Kim, B.-J.

Kim, H. T.

Kim, J.

Kim, L.

Kim, L. B.

Kim, M. J.

Kim, S.

Koppens, F. H.

Kravchenko, I.

Lau, C. N.

Lee, B.

Lee, G.

Lee, H. W.

Liu, M. K.

Liu, X.

J. Park, J. H. Kang, X. Liu, and M. L. Brongersma, “Electrically tunable epsilon-near-zero (ENZ) metafilm absorbers,” Sci. Reports 5, 15754 (2015).
[Crossref]

Liu, Z.

Lopez, J. J.

Lundeberg, M. B.

Ma, J.

Ma, L.

Magnuson, C. W.

Martin, J. B.

Martinez, N. J.

McDonnell, S.

McLeod, A. S.

Mikkelsen, M. H.

T. B. Hoang and M. H. Mikkelsen, “Broad electrical tuning of plasmonic nanoantennas at visible frequencies,” Appl. Phys. Lett. 108, 183107 (2016).
[Crossref]

Mordi, G.

Mueller, T.

T. Mueller, F. Xia, M. Freitag, J. Tsang, and P. Avouris, “Role of contacts in graphene transistors: A scanning photocurrent study,” Phys. Rev. B 79, 245430 (2009).
[Crossref]

Murphy, T. E.

Myers-Ward, R. L.

Nah, J.

Ni, G. X.

Nordlander, P.

Novoselov, K. S.

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, 109–162 (2009).
[Crossref]

C. Casiraghi, S. Pisana, K. S. Novoselov, A. K. Geim, and A. C. Ferrari, “Raman fingerprint of charged impurities in graphene,” Appl. Phys. Lett. 91, 233108 (2007).
[Crossref]

Novotny, L.

L. Novotny and B. Hecht, Principles of nano-optics (Cambridge University, 2012).
[Crossref]

Oellers, M.

Osmond, J.

Ozyilmaz, B.

Padilla, W. J.

D. Shrekenhamer, W. C. Chen, and W. J. Padilla, “Liquid crystal tunable metamaterial absorber,” Phys. Rev. Lett. 110, 177403 (2013).
[Crossref] [PubMed]

Pala, R. A.

Palit, S.

Park, J.

J. Park, J. H. Kang, X. Liu, and M. L. Brongersma, “Electrically tunable epsilon-near-zero (ENZ) metafilm absorbers,” Sci. Reports 5, 15754 (2015).
[Crossref]

Pearton, S. J.

Peres, N. M. R.

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, 109–162 (2009).
[Crossref]

Pesquera, A.

Peters, D. W.

Pirkle, A.

Pisana, S.

C. Casiraghi, S. Pisana, K. S. Novoselov, A. K. Geim, and A. C. Ferrari, “Raman fingerprint of charged impurities in graphene,” Appl. Phys. Lett. 91, 233108 (2007).
[Crossref]

Polini, M.

Ponti, S. M.

Pop, E.

V. E. Dorgan, M.-H. Bae, and E. Pop, “Mobility and saturation velocity in graphene on SiO2,” Appl. Phys. Lett. 97, 082112 (2010).
[Crossref]

Post, K. W.

Principi, A.

Pryce, I. M.

Qazilbash, M. M.

Ramanathan, R.

Regan, W.

Ren, F.

Robertson, J.

J. Robertson, “High dielectric constant oxides,” The Eur. Phys. J. Appl. Phys. 28, 265–291 (2004).
[Crossref]

Rodin, A. S.

Rossi, E.

S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, “Electronic transport in two-dimensional graphene,” Rev. Mod. Phys. 83, 407–470 (2011).
[Crossref]

Royal, M.

Ruiz, I.

Ruoff, R. S.

Rusu, P.

Sarma, S. D.

E. Hwang and S. D. Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75, 205418 (2007).
[Crossref]

Schlather, A.

Schmitz, M.

Shahrjerdi, D.

Shalaev, V. M.

Shaltout, A. M.

Shen, H.

Sherrott, M.

Sherrott, M. C.

Shrekenhamer, D.

D. Shrekenhamer, W. C. Chen, and W. J. Padilla, “Liquid crystal tunable metamaterial absorber,” Phys. Rev. Lett. 110, 177403 (2013).
[Crossref] [PubMed]

Smith, D. R.

Smith, S. W.

Sokhoyan, R.

Soljacic, M.

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009).
[Crossref]

Spasenovic, M.

Stampfer, C.

Sternbach, A. J.

Stewart, M. K.

Stinson, H. T.

Sushkov, A. B.

Sweatlock, L. A.

Tan, J. Y.

Taniguchi, T.

Tauber, M. J.

Thiemens, M.

Thongrattanasiri, S.

Thyagarajan, K.

Tsai, D. P.

Tsang, J.

T. Mueller, F. Xia, M. Freitag, J. Tsang, and P. Avouris, “Role of contacts in graphene transistors: A scanning photocurrent study,” Phys. Rev. B 79, 245430 (2009).
[Crossref]

Tung, V. C.

J. D. Fowler, M. J. Allen, V. C. Tung, Y. Yang, R. B. Kaner, and B. H. Weiller, “Practical chemical sensors from chemically derived graphene,” ACS Nano 3, 301–306 (2009).
[Crossref] [PubMed]

Tutuc, E.

Van den Brink, J.

Venugopal, A.

Vignale, G.

Vogel, E. M.

Wagner, M.

Walavalkar, S.

Wallace, R. M.

Wang, C.

Wang, Y.

Watanabe, K.

Weiller, B. H.

J. D. Fowler, M. J. Allen, V. C. Tung, Y. Yang, R. B. Kaner, and B. H. Weiller, “Practical chemical sensors from chemically derived graphene,” ACS Nano 3, 301–306 (2009).
[Crossref] [PubMed]

Woessner, A.

Wu, J. S.

Xia, F.

T. Mueller, F. Xia, M. Freitag, J. Tsang, and P. Avouris, “Role of contacts in graphene transistors: A scanning photocurrent study,” Phys. Rev. B 79, 245430 (2009).
[Crossref]

Yang, Y.

J. D. Fowler, M. J. Allen, V. C. Tung, Y. Yang, R. B. Kaner, and B. H. Weiller, “Practical chemical sensors from chemically derived graphene,” ACS Nano 3, 301–306 (2009).
[Crossref] [PubMed]

Yao, Z.

Yeo, Y.

Yota, J.

Yun, S. J.

Zettl, A.

Zhang, J. D.

Zhang, L. M.

Zhao, Z.

Zhu, S.

ACS Nano (3)

J. D. Fowler, M. J. Allen, V. C. Tung, Y. Yang, R. B. Kaner, and B. H. Weiller, “Practical chemical sensors from chemically derived graphene,” ACS Nano 3, 301–306 (2009).
[Crossref] [PubMed]

Appl. Phys. Lett. (8)

V. E. Dorgan, M.-H. Bae, and E. Pop, “Mobility and saturation velocity in graphene on SiO2,” Appl. Phys. Lett. 97, 082112 (2010).
[Crossref]

C. Casiraghi, S. Pisana, K. S. Novoselov, A. K. Geim, and A. C. Ferrari, “Raman fingerprint of charged impurities in graphene,” Appl. Phys. Lett. 91, 233108 (2007).
[Crossref]

T. B. Hoang and M. H. Mikkelsen, “Broad electrical tuning of plasmonic nanoantennas at visible frequencies,” Appl. Phys. Lett. 108, 183107 (2016).
[Crossref]

J. Opt. (1)

Y. C. Jun and I. Brener, “Electrically tunable infrared metamaterials based on depletion-type semiconductor devices,” J. Opt. 14, 114013 (2012).
[Crossref]

J. Vac. Sci. & Technol. A: Vacuum, Surfaces, Films (1)

J. Vac. Sci. & Technol. B, Nanotechnol. Microelectron. Materials, Process. Meas. Phenom. (1)

Nano Lett. (9)

Nat. Mater. (1)

Nature. (2)

Opt Express (1)

Opt. Express (1)

Phys. Rev. B (5)

E. Hwang and S. D. Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75, 205418 (2007).
[Crossref]

T. Mueller, F. Xia, M. Freitag, J. Tsang, and P. Avouris, “Role of contacts in graphene transistors: A scanning photocurrent study,” Phys. Rev. B 79, 245430 (2009).
[Crossref]

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009).
[Crossref]

Phys. Rev. Lett. (1)

D. Shrekenhamer, W. C. Chen, and W. J. Padilla, “Liquid crystal tunable metamaterial absorber,” Phys. Rev. Lett. 110, 177403 (2013).
[Crossref] [PubMed]

Rev. Mod. Phys. (2)

S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, “Electronic transport in two-dimensional graphene,” Rev. Mod. Phys. 83, 407–470 (2011).
[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, 109–162 (2009).
[Crossref]

Sci. Adv. (1)

Sci. Reports (2)

J. Park, J. H. Kang, X. Liu, and M. L. Brongersma, “Electrically tunable epsilon-near-zero (ENZ) metafilm absorbers,” Sci. Reports 5, 15754 (2015).
[Crossref]

The Eur. Phys. J. Appl. Phys. (1)

J. Robertson, “High dielectric constant oxides,” The Eur. Phys. J. Appl. Phys. 28, 265–291 (2004).
[Crossref]

Other (1)

L. Novotny and B. Hecht, Principles of nano-optics (Cambridge University, 2012).
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.

Click here to see a list of articles that cite this paper

Fig. 1 Characteristic transport curves obtained from each device during application of variable bias (V_G) across the gate dielectric with a constant source-drain voltage of 50 mV. (Inset) Schematic of the device showing the various layers. The sample is probed from the grating side during FTIR measurements. Dimensions are not to scale.

Download Full Size | PPT Slide | PDF

Fig. 2 (a) Simulated reflectance demonstrating the effects of each additional element. Blue: Grating structure in the absense of graphene and with a dispersionless (n = 1.3) and lossless (k = 0) dielectric in place of SiO₂. Green: Grating structure in the absence of graphene with realistic dielectric optical responses. Black: Simulated full device response including graphene and dispersive dielectrics. (b) Comparison of graphene absorption in full device (red) and total absorption (black) demonstrating increased graphene absorption in bands of tunability.

Download Full Size | PPT Slide | PDF

Fig. 3 (a) Measured voltage and frequency dependent reflectance map obtained from the smaller device. Black dots track the minima of each resonance. (b) Experimental voltage and frequency dependent reflectance map obtained from the larger device (c) Measured reflectance for the smaller device at voltages of smallest and largest graphene conductivity. (d) Measured reflectance of the larger device at voltages of smallest and largest graphene conductivity. (e) Measured (black) and simulated (blue) differential reflectance between V_G = 1.75 V and −6 V and 0.3 and 0.7 eV respectively for the smaller device. (f) Measured (black) and simulated (blue) differential reflectance between V_G = −6 V and 6 V and 0.3 and 0.7 eV for the larger device.

Download Full Size | PPT Slide | PDF

Fig. 4 Maps of the real part of the y-component of the electric field near the graphene in one period of the larger device. Yellow rectangles represent the gold grating while the black dotted line corresponds to the location of the graphene. (a) Field map at 1053 cm⁻¹ (9.5 μm) and E_F = 0.4 eV. (b) Field map at 1053 cm⁻¹ (9.5 μm) and E_F = 0.8 eV. (c) Field map at 1250 cm⁻¹ (8 μm) and E_F = 0.4 eV. (d) Field map at 1250 cm⁻¹ (8 μm) and E_F = 0.8 eV.

Download Full Size | PPT Slide | PDF

Fig. 5 Simulated reflectance map for the larger device assuming ideal behavior. Black dots track the location of minima.

Download Full Size | PPT Slide | PDF

Fig. 6 Graphene plasmon dispersion calculated for Fermi levels of 0.4 eV (black) and 0.7 eV (red).

Download Full Size | PPT Slide | PDF

Equations (1)

Equations on this page are rendered with MathJax. Learn more.

E_{F} = \sqrt{\frac{∊ ∊_{0} ℏ^{2} v_{F}^{2} π | V_{G} |}{e d}}