Letter Cite This: Nano Lett. 2018, 18, 1373−1378
pubs.acs.org/NanoLett
Quantum Control of Graphene Plasmon Excitation and Propagation at Heaviside Potential Steps Dongli Wang,†,‡,# Xiaodong Fan,†,‡,# Xiaoguang Li,§ Siyuan Dai,∥ Laiming Wei,†,‡ Wei Qin,† Fei Wu,†,‡ Huayang Zhang,†,‡ Zeming Qi,*,⊥ Changgan Zeng,*,†,‡ Zhenyu Zhang,† and Jianguo Hou†
Nano Lett. 2018.18:1373-1378. Downloaded from pubs.acs.org by IOWA STATE UNIV on 01/06/19. For personal use only.
†
International Center for Quantum Design of Functional Materials (ICQD), Hefei National Laboratory for Physical Sciences at the Microscale, and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China ‡ CAS Key Laboratory of Strongly-Coupled Quantum Matter Physics and Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, China § Institute for Advanced Study, Shenzhen University, Shenzhen, Guangdong 518060, China ∥ Department of Physics, University of California, San Diego, La Jolla, California 92093, United States ⊥ National Synchrotron Radiation Laboratory, University of Science and Technology of China, Hefei, Anhui 230029, China S Supporting Information *
ABSTRACT: Quantum mechanical effects of single particles can affect the collective plasmon behaviors substantially. In this work, the quantum control of plasmon excitation and propagation in graphene is demonstrated by adopting the variable quantum transmission of carriers at Heaviside potential steps as a tuning knob. First, the plasmon reflection is revealed to be tunable within a broad range by varying the ratio γ between the carrier energy and potential height, which originates from the quantum mechanical effect of carrier propagation at potential steps. Moreover, the plasmon excitation by freespace photos can be regulated from fully suppressed to fully launched in graphene potential wells also through adjusting γ, which defines the degrees of the carrier confinement in the potential wells. These discovered quantum plasmon effects offer a unified quantum-mechanical solution toward ultimate control of both plasmon launching and propagating, which are indispensable processes in building plasmon circuitry. KEYWORDS: Quantum plasmon, graphene, quantum transmission, scanning near-field optical microscopy
Q
relativistic linear energy dispersion.25 Here we exploit the effective manipulation of plasmon excitation and propagation in graphene by tuning the quantum transmission of carriers at potential steps and thus realize the quantum control of plasmon behaviors. Results and Discussion. Practically, the Heaviside potential steps in graphene are realized by transferring graphene onto a SiO2/Si substrate prepatterned by aminopropyltriethoxysilane (APTES) microribbons,26 as shown in Figure 1c and Figure 2a (see more details in the Experimental Section). The amine groups in the APTES molecules donate additional electrons to the atop graphene, leading to electron doping in graphene.26 Therefore, a Heaviside potential step is established at the boundary between graphene on APTES and that on SiO2. A field effect transistor device was made on such patterned graphene, and the resistance (R) as a function of the back gate voltage (Vg) for the device shown in Figure 1c is displayed in Figure 1d. The R-Vg curve reveals two distinct
uantum plasmonics has attracted a lot of research interests recently to pursue the quantum characteristics and quantum control of plasmons.1−6 In particular, the collective plasmonic properties of materials can be effectively tuned by the quantum mechanical effects of the constituent single particles.1−6 For example, it has been revealed that quantum size effect,2 quantum tunneling,3,4 and quantum nonlocality5,6 substantially shift the plasmon resonances or even induce new plasmon mode. On the other hand, textbook theories predict another quantum mechanical effect, that is, the quantum transmission of carriers across a one-dimensional (1D) potential step can be effectively tuned from 0 to 1 continuously by regulating γ = E/ U0 (E is the carrier energy and U0 the potential height),7 as shown in Figure 1a,b. How such single-particle quantum effect affects the collective plasmon behaviors, especially the plasmon excitation and propagation, remains unexploited. The challenge relies on the tunability of the carrier energy, which is inaccessible for conventional metals.8 Graphene, an exceptional material to exploit two-dimensional (2D) plasmons,9−24 offers an ideal platform to study this important issue because the carrier energy in graphene can be readily tuned thanks to its © 2018 American Chemical Society
Received: December 3, 2017 Revised: January 11, 2018 Published: January 16, 2018 1373
DOI: 10.1021/acs.nanolett.7b05085 Nano Lett. 2018, 18, 1373−1378
Letter
Nano Letters
Figure 1. Heaviside potential step and its construction in graphene. (a) Schematic of a single electron propagating at a 1D potential step along either direction. (b) Electron reflectance at a potential step as a function of γ = E/U0 for classic and quantum cases. (c) AFM image of the graphene/ APTES/SiO2/Si device. The coverage area of graphene is also illustrated with the black dashed curve indicating the graphene edge. The scale bar is 200 nm. (d) Resistance R as a function of Vg for the device shown in (c), revealing two Dirac points. (e) Schematic of the scanning near-field measurements. The black arrows indicate the incident and reflected light. The orange circles represent the graphene plasmon waves launched by the AFM tip.
Figure 2. Near-field characterization of graphene plasmons. (a) Zoom-in AFM image of the graphene/APTES/SiO2/Si device in Figure 1d. The coverage area of graphene is illustrated, with the black dashed curve indicating the graphene edge. Height profile along the arrowed white line is also shown. (b) Normalized near-field images for the same area displayed in (a) at various Vgs. The white dashed curves denote the APTES edge underneath the graphene. The scale bars in (a) and (b) are 200 nm. (c) Averaged line profiles of the normalized near-field signal across the potential step at different Vgs along the arrowed white lines in (b). The fringe signal close to the potential step signifies plasmon reflectance, and is strongly dependent on Vg.
various Vgs for the area shown in Figure 1c are displayed in Figure 2b. Strong interference fringes are evidenced at the graphene edges for graphene on both APTES and SiO2. Quantitatively, the images in Figure 2b show the normalized signals (s − sg)/(se − sg), where s is the site-dependent nearfield signal, se is the signal of the bright fringe near the graphene edge (on the APTES side), and sg is the signal of graphene interior (on top of the APTES) where no plasmonic features are registered. Interestingly, interference fringes are also registered at the boundary between graphene on APTES and that on SiO2, suggesting that the potential step established at this boundary can indeed reflect plasmons. As detailed in Section S1 and Figure S1 in the Supporting Information, the variations in the fringe width and correspondingly the plasmon wavelength in response to the changes in the incident light
peaks at VD1 and VD2 (VD1 < VD2) corresponding to the two Dirac points of graphene on the APTES and SiO2 regions, confirming the establishment of electronic potential steps in the patterned graphene. The potential height U0 is derived from the relation U0 = |EF1 − EF2|,27 where EF1,F2 = sgn(Vg − VD1,D2)ℏvF πCg|Vg − VD1,D2| are the Fermi energies of graphene on the APTES and SiO2 regions, sgn(x) is the sign function, vF ≈ 1 × 106 m/s is the Fermi velocity,25 and Cg ≈ 7.2 × 1010 cm−2/V is the gate capacitance for graphene on the 300 nm thick SiO2.25,26 Here we adopt the Dirac point of graphene on the SiO2 region as the reference point of carrier energy. The plasmon transmission/reflection at the potential steps is exploited by adopting scattering-type scanning near-field optical microscopy (SNOM), as depicted in Figure 1e (see more details in the Experimental Section). The near-field images at 1374
DOI: 10.1021/acs.nanolett.7b05085 Nano Lett. 2018, 18, 1373−1378
Letter
Nano Letters
similar behavior as for the ideal 1D case of nonrelativistic particles, except that it decays slower with increasing γ. More strikingly, the decay behavior of the experimental plasmon reflectance at a potential step with increasing γ resembles the predicted quantum behavior of single particles. The slight differences between the experimental and theoretical reflectances may originate from the deviation of the realistic smooth potential step from the ideal sharp one. Nevertheless, the overall consistency strongly suggests that the γ-dependent plasmon reflectance is ultimately determined by the quantum nature of the constituent carriers. A thorough understanding of this consistency may require quantitative quantum-level treatment of the plasmon propagation with many-body effect taken into account, which is beyond the scope of the present study. Nevertheless, here we offer a phenomenological understanding on the connection between the plasmon propagation and the constituent carrier propagation at a potential step as follows: Plasmon wave in graphene is longitudinal oscillations of charge density. Analogous with the mechanical longitudinal propagating wave, the transmission of plasmon wave at a boundary depends mainly on the behaviors of its constituent carriers within the vicinity of the boundary. Generally, the direction of the wave propagation is determined by the oscillation phase difference between the constituent particles at adjacent positions.31 When the constituent carriers of the plasmon wave encounter a potential step, quantum mechanical theory predicts that they will be reflected with a certain possibility. Meanwhile, these reflected carriers acquire extra phase shifts in their oscillating states,31 leading to a sign change of the adjacent phase difference, driving the plasmon propagating in the opposite direction, namely, plasmon reflection. In contrast, those constituent carriers transmitting through the potential step do not acquire additional phase shift, leading to plasmon transmission. Therefore, the plasmon propagation at a potential step is largely dependent on the quantum transmissions of its constituent carriers. Previously, the reflection of graphene plasmon at a boundary was attributed to different optical conductivities.17,20,22−24 Both the optical conductivity and the plasmon reflection should depend on the quantum behaviors of electrons in the potential landscape, therefore the scenarios in the present and previous studies should essentially root in the same principles of quantum mechanics. It is noted that the APTES layer is about 3 nm higher than the SiO2/Si substrate, as evidenced from the atomic force microscopy (AFM) images collected simultaneously (Figure 2a). Therefore, structural steps in graphene are also formed in addition to the electronic potential steps at the boundary between graphene on APTES and that on SiO2. Previous studies revealed that the potential changes across the structural steps are negligible,32 and such structural steps are ineffective in reflecting plasmon waves for continuous graphene.18 To quantitatively exploit the influence of the structural step on the plasmon reflection, we performed near-field measurements on a graphene/SiO2/Si device with a structural step of ∼8 nm height. As shown in Figure S4b, no reflection of plasmons is observed near the structural step, suggesting that the revealed plasmon reflection in Figure 2b arises from the quantum reflection at the electronic potential step rather than the structural step. Strikingly, the revealed quantum picture of γ-dependent reflection at a potential step can be further adopted to control
wavelength further confirm the plasmonic origin of the observed fringes at the potential steps. It is noted that the plasmonic interference fringes on the graphene/APTES side is more evident than that on the graphene/SiO2 side (see Figure 2b and Figure S2). Such difference was observed previously and was attributed to the different wavelengths and damping rates of graphene plasmons in different regions.20,24 In the present study, we focus on the graphene/APTES side in the holedoping regime for simplicity. When Vg is swept from −80 to 15 V, with the Fermi level of graphene correspondingly moving toward the Dirac points, the normalized fringe signal near the potential step increases monotonically as revealed in Figure 2b. This is reflected more clearly in the line profiles as displayed in Figure 2c. The normalized fringe signal near the potential step represents the plasmon reflectance at the potential step,18,23 and its Vg dependence is displayed in Figure 3. When Vg increases from
Figure 3. Tunability of plasmon reflection at a potential step. Experimentally measured plasmon reflectance as a function of γ = E/ U0 is denoted by the red circles. Theoretically calculated quantum reflectances for the 2D relativistic particles (red curve) and 1D nonrelativistic particles (blue curve) are also shown. Interband excitations arise in the region marked in yellow where the Fermi level is close to the Dirac points, and the plasmon excitation is suppressed substantially.
−80 to 15 V, γ is estimated to decrease from 13 to 4, while the reflectance increases from 0.15 to 0.37. On the other hand, plasmon interference is absent at the potential step when 25 V < Vg < 75 V. In this range, the plasmons become substantially Landau damped due to interband excitations.15,16 Similar tunability of the plasmon reflectance in a separate sample is shown in Figure S2 in the Supporting Information, confirming the reproducibility of the main findings. Next we demonstrate that the reflection of graphene plasmon at a potential step is similar to that of single particles. In 1D ideal case where the carrier incident direction is perpendicular to the potential step, the relativistic Dirac carriers in graphene can completely transmit through the potential step (namely, Klein tunneling28). In contrast, the quantum transmission of carriers at a potential step constructed in realistic 2D graphene is more complicated, because the incident angles of carriers are randomly distributed.29,30 Taking into account this factor, the reflectance of 2D relativistic particles can be calculated by angle averaging29,30 (see Section S2 in the Supporting Information for detailed calculations). As depicted in Figure 3 and Figure S3 in the Supporting Information, the estimated γ-dependent reflectance of 2D relativistic particles still maintains qualitatively 1375
DOI: 10.1021/acs.nanolett.7b05085 Nano Lett. 2018, 18, 1373−1378
Letter
Nano Letters
Figure 4. Tunability of plasmon excitations in potential wells. (a) Typical scanning electron microscope image of the graphene transferred onto the SiO2/Si substrate prepatterned by the periodic APTES microribbons. The dark (light) gray ribbons represent the APTES (SiO2) regions, respectively. (b) Resistance R as a function of Vg for the device shown in (a), revealing two Dirac points. (c) Optical absorption spectra at various Vg’s, with the light polarization along the potential steps. A typical fitting curve based on the Drude model is also shown. (d) Optical absorption spectra at various Vgs, with the light polarization perpendicular to the potential steps. The fitting curves based on a linearly mixed Drude-Lorentz model are also shown. (e) Coefficient α of the plasmon absorption as a function of Vg extracted from the fittings in (d).
from −50 to 20 V, the Fermi level approaches the Dirac points from the hole doping side and then moves away to the electron doping regime, leading to the observation that the Drude absorption decreases initially and then increases. The weak feature arising around 270 cm−1 originates from the phonon modes in SiO2,33 which had also been observed previously.12,34 In sharp contrast, for incident light polarized perpendicular to the potential steps, the Vg-dependent absorption behavior is qualitatively different as revealed in Figure 4d. For Vg = −50 V, the optical absorption curve still shows the typical Drude shape without any sign of plasmon excitation. As Vg increases, the single-particle Drude absorption is gradually suppressed, accompanied by the emergence of a peak located at ∼120 cm−1,an evident feature attributed to the collective plasmon resonance.12 When Vg reaches −20 V, corresponding to the case where the Fermi energy is very close to the Dirac points, the free-carrier Drude response is overwhelmed by the plasmon excitation. When Vg further increases to 5 V, the system enters the electron-doping regime, and the Drude absorption recovers gradually at the expense of the plasmon absorption. The optical responses for the electron-doping and hole-doping regimes are approximately symmetric around the Dirac points. These observations convincingly demonstrate that the plasmon excitation in the graphene with patterned potential wells can be effectively manipulated via electrostatic gating. Continuous tunability of plasmon excitation in graphene potential wells with width of 3 μm is also demonstrated in Figure S5 in the Supporting Information. For comparison, the optical spectra of the bare APTES microribbons with different polarization direction were also measured as shown in Figure S6 in the Supporting Information, and both plasmon and intraband absorptions are absent. To gain more understanding of the absorption spectra, we recall that the plasmon absorption line shape is traditionally
the tunability of plasmon excitation. Traditionally, to launch graphene plasmons by free-space photons, proper micro/nanoor grating structures have to be tailored to compensate the large momentum mismatch between graphene plasmons and freespace photons.9−14 Here, if we construct potential wells with limited potential heights and further adjust γ, the carrier reflection at each step and thus the carrier confinement in the potential wells defined by neighboring potential steps can be effectively adjusted. This would result in highly tunable carrier localization in the potential wells and eventually highly tunable degree of compensation to the momentum mismatch between photons and the plasmons formed by the confined carriers in the potential wells. Correspondingly, the degree of plasmon excitation by free-space light can be regulated continuously. The tunable plasmon excitation is exploited in graphene potential wells constructed by periodic APTES microribbons, with the ribbon width and spacing both to be 3.5 μm (Figure 4a), and the potential steps are confirmed by the two Dirac points revealed in the R−Vg curve (Figure 4b). Polarized Fourier transform infrared spectroscopy was employed to study the far-field excitation of plasmons (see more details in the Experimental Section). Optical absorption is directly related to 1 − T/Ts, where T and Ts are the light transmissions of the sample and the bare substrate, respectively.13 When the light polarization is parallel to the potential steps, the optical absorption increases monotonically with decreasing frequency in the terahertz range at all the applied Vg as shown in Figure 4c, similar to that for pristine graphene.10 The spectral shape follows the Drude model Im[−1/(ω + iΓD)], where ΓD is the free electron scattering rate. The Drude model describes the intraband excitation of individual electrons,10 and the absence of plasmon excitation can be attributed to the momentum mismatch between the plasmons and incident photons along the potential steps. As shown in Figure 4c, when Vg is swept 1376
DOI: 10.1021/acs.nanolett.7b05085 Nano Lett. 2018, 18, 1373−1378
Letter
Nano Letters formulated by a damped Lorentz model as Im[−ω/(ω2 − ω2p + iωΓp)], where ωp is the plasmon frequency and Γp is the plasmon damping rate.12 Here, there are two types of potential wells with different carrier densities for graphene on SiO2 and APTES, and thus there should be two plasmon resonances in the absorption spectra. We adopt a linearly mixed DrudeLorentz model to quantitatively characterize the Vg-dependent absorption behaviors. Within this model, the absorption can be expressed as (1−α)Im[−1/(ω + iΓD)] + (α/2)Im[−ω/ (ω2−ω2p1 + iωΓp)] + (α/2)Im[−ω/(ω2−ω2p2+iωΓp)], where α and 1 − α represent the coefficients of the plasmon and intraband Drude absorption, respectively. The details of the two-mode fitting are given in Section S3 in the Supporting Information. As shown in Figure 4d, the fitting curves agree well with the experimental data. The coefficient of the plasmon absorption is subsequently extracted as plotted in Figure 4e. The single-particle intraband excitations dominate when the Fermi level is far away from the Dirac points, while the plasmon excitations dominate when the Fermi level is close to the Dirac points. When Vg = −50 V, E is much larger than U0 (γ = 4.9), and the intraband excitation prevails with α = 0. For such a high γ value, the carriers can readily transmit through the potential steps, namely, the electrons are nearly free in the graphene, resulting in the dominant intraband absorption. When Vg = −35 V, E is still larger than U0, but γ is reduced to 3, and the possibility that the carriers are reflected by the potential steps and confined in the potential wells increases accordingly, leading to coexistence of both the plasmon and intraband absorptions, with α = 0.63. When Vg = −10 V, E is lower than U0 (γ = 0.37), and the carriers are mainly confined in the potential wells (see more discussion in Section S2 in the Supporting Information), giving rise to the dominant plasmon absorption with α = 0.98. Collectively, the present work validates experimentally the quantum mechanical effect of quantum propagation at potential steps using graphene plasmon as a probe. More importantly, we demonstrate the quantum control of plasmon propagation at potential steps and of plasmon excitation in potential wells, by adopting this quantum mechanical effect as an effective tuning knob. The demonstrated quantum plasmon effects, that is, the ultimate control of plasmon excitation and propagation at the quantum level offer a reliable quantum solution in future design of elemental building blocks, for example, transistors and switches, in future plasmonic circuits operating at room temperature. Experimental Section. Sample Fabrication. The graphene/APTES/SiO2/Si devices were fabricated as follows: First, photoresist was spin coated onto doped Si substrate with a thermally grown silicon dioxide layer (300 nm) and then patterned into microribbons using standard optical lithography. The APTES layers were subsequently deposited on the SiO2/Si substrate following a solution-based self-assembled monolayer coating technique.26 After removal of the photoresist, APTES microribbons were developed on the surface. Next, monolayer graphene was transferred onto the SiO2/Si substrate covered by the APTES microribbons. Mechanically exfoliated monolayer graphene sheets (a few tens of μm in size) were adopted in the SNOM experiments. On the other hand, large-scale monolayer graphene films (∼7 mm in size) from chemical vapor deposition were adopted to form periodic potential wells in the far-field measurements. For the graphene/SiO2/Si device, the SiO2 structural step (∼8 nm height) was fabricated using optical lithography and ion beam etching, and subsequently
monolayer graphene was mechanically exfoliated onto the top of the SiO2. Near-Field Infrared Measurement. The near-field measurements were performed at atmosphere in a scattering-type SNOM (Neaspec) system based on an AFM. The scatteringtype SNOM is operated in the tapping mode with the tapping frequency 265 kHz and tapping amplitude 60 nm. The PtIr5coated AFM tip was illuminated by a CO2 laser with the incident wavelength 10.532 μm unless otherwise clarified. Under far-field illumination, radical plasmon waves can be launched with momentum compensation from the metal tip (the momentums centered at 1/a, where a is the tip radius). The demodulated fourth harmonic component s(ω) was recorded to suppress the background signal. The width of the APTES microribbons is 10 μm for the near-field measurements. The fringe amplitude is stronger at the graphene/APTES side than at the graphene/SiO2 side. This is arising from the fact that the graphene plasmons couple strongly with the surface polar phonon modes of SiO2 in the mid-infrared regime,13 while such coupling is absent for graphene on APTES. Far-Field Terahertz Measurement. The terahertz transmission measurements were performed with a Fourier transform infrared spectrometer (Bruker IFS 66v) on the infrared beamline (BL01B) at the National Synchrotron Radiation Laboratory of China. The measurements were carried out in the vacuum of 0.2 mbar at room temperature.
■
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.7b05085. Additional discussions and figures (PDF)
■
AUTHOR INFORMATION
Corresponding Authors
*E-mail:
[email protected]. *E-mail:
[email protected]. ORCID
Siyuan Dai: 0000-0001-7259-7182 Laiming Wei: 0000-0002-8808-9374 Wei Qin: 0000-0003-1035-1130 Changgan Zeng: 0000-0001-8630-845X Zhenyu Zhang: 0000-0001-5844-3558 Author Contributions #
D.W. and X.F. contributed equally to this work.
Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS This work was supported in part by the National Basic Research Program of China (Grant 2014CB921102), National Key R&D Program of China (Grant 2017YFA0403600), National Natural Science Foundation of China (Grants 11434009, 11374279, 11461161009, 11634011, 61434002, and 11304299), the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (Grant XDB01020000), the Fundamental Research Funds for the Central Universities (Grant WK2030020027), and Anhui Provincial Natural Science Foundation (Grant 1708085MF136). This work was partially carried out at the USTC Center for Micro and Nanoscale 1377
DOI: 10.1021/acs.nanolett.7b05085 Nano Lett. 2018, 18, 1373−1378
Letter
Nano Letters
Keilmann, F.; Janssen, G. C. A. M.; Zhu, S.-E.; Jarillo-Herrero, P.; Fogler, M. M.; Basov, D. N. Nat. Nanotechnol. 2015, 10, 682−686. (22) Jiang, B.-Y.; Ni, G. X.; Pan, C.; Fei, Z.; Cheng, B.; Lau, C. N.; Bockrath, M.; Basov, D. N.; Fogler, M. M. Phys. Rev. Lett. 2016, 117, 086801. (23) Jiang, L.; Shi, Z.; Zeng, B.; Wang, S.; Kang, J.-H.; Joshi, T.; Jin, C.; Ju, L.; Kim, J.; Lyu, T.; Shen, Y.-R.; Crommie, M.; Gao, H.-J.; Wang, F. Nat. Mater. 2016, 15, 840−844. (24) Fei, Z.; Ni, G.-X.; Jiang, B.-Y.; Fogler, M. M.; Basov, D. N. ACS Photonics 2017, 4, 2971−2977. (25) Das Sarma, S.; Adam, S.; Hwang, E. H.; Rossi, E. Rev. Mod. Phys. 2011, 83, 407−470. (26) Baltazar, J.; Sojoudi, H.; Paniagua, S. A.; Kowalik, J.; Marder, S. R.; Tolbert, L. M.; Graham, S.; Henderson, C. L. J. Phys. Chem. C 2012, 116, 19095−19103. (27) Dubey, S.; Singh, V.; Bhat, A. K.; Parikh, P.; Grover, S.; Sensarma, R.; Tripathi, V.; Sengupta, K.; Deshmukh, M. M. Nano Lett. 2013, 13, 3990−3995. (28) Katsnelson, M. I.; Novoselov, K. S.; Geim, A. K. Nat. Phys. 2006, 2, 620−625. (29) Cheianov, V. V.; Fal’ko, V. I. Phys. Rev. B: Condens. Matter Mater. Phys. 2006, 74, 041403. (30) Young, A. F.; Kim, P. Annu. Rev. Condens. Matter Phys. 2011, 2, 101−120. (31) Serway, R. A.; Beichner, R. J. Physics for Scientists and Engineers, 5th ed.; Harcourt Brace: Orlando, FL, 2000. (32) Ji, S.-H.; Hannon, J. B.; Tromp, R. M.; Perebeinos, V.; Tersoff, J.; Ross, F. M. Nat. Mater. 2012, 11, 114−119. (33) Swainson, I. P.; Dove, M. T.; Palmer, D. C. Phys. Chem. Miner. 2003, 30, 353−365. (34) Yan, H.; Xia, F.; Zhu, W.; Freitag, M.; Dimitrakopoulos, C.; Bol, A. A.; Tulevski, G.; Avouris, P. ACS Nano 2011, 5, 9854−9860.
Research and Fabrication, and we thank Xiaolei Wen for her help on near-field measurement.
■
ABBREVIATIONS 1D, one-dimensional 2D, two-dimensional APTES, aminopropyltriethoxysilane SNOM, scanning near-field optical microscopy AFM, atomic force microscopy
■
REFERENCES
(1) Tame, M. S.; McEnery, K. R.; Ö zdemir, S. K.; Lee, J.; Maier, S. A.; Kim, M. S. Nat. Phys. 2013, 9, 329−340. (2) Scholl, J. A.; Koh, A. L.; Dionne, J. A. Nature 2012, 483, 421− 427. (3) Tan, S. F.; Wu, L.; Yang, J. K. W.; Bai, P.; Bosman, M.; Nijhuis, C. A. Science 2014, 343, 1496−1499. (4) Savage, K. J.; Hawkeye, M. M.; Esteban, R.; Borisov, A. G.; Aizpurua, J.; Baumberg, J. J. Nature 2012, 491, 574−577. (5) Ciracì, C.; Hill, R. T.; Mock, J. J.; Urzhumov, Y.; FernándezDomínguez, A. I.; Maier, S. A.; Pendry, J. B.; Chilkoti, A.; Smith, D. R. Science 2012, 337, 1072−1074. (6) Lundeberg, M. B.; Gao, Y.; Asgari, R.; Tan, C.; Van Duppen, B. V.; Autore, M.; Alonso-González, P.; Woessner, A.; Watanabe, K.; Taniguchi, T.; Hillenbrand, R.; Hone, J.; Polini, M.; Koppens, F. H. L. Science 2017, 357, 187−191. (7) Griffiths, D. J. Introduction to Quantum Mechanics, 2nd ed.; Pearson Prentice Hall: Upper Saddle River, NJ, 2005. (8) Kittel, C. Introduction to Solid State Physics, 8th ed.; John Wiley & Sons: New York, 2005. (9) Grigorenko, A. N.; Polini, M.; Novoselov, K. S. Nat. Photonics 2012, 6, 749−758. (10) Low, T.; Avouris, P. ACS Nano 2014, 8, 1086−1101. (11) Caldwell, J. D.; Vurgaftman, I.; Tischler, J. G.; Glembocki, O. J.; Owrutsky, J. C.; Reinecke, T. L. Nat. Nanotechnol. 2016, 11, 9−15. (12) Ju, L.; Geng, B.; Horng, J.; Girit, C.; Martin, M.; Hao, Z.; Bechtel, H. A.; Liang, X.; Zettl, A.; Shen, Y. R.; Wang, F. Nat. Nanotechnol. 2011, 6, 630−634. (13) Yan, H.; Low, T.; Zhu, W.; Wu, Y.; Freitag, M.; Li, X.; Guinea, F.; Avouris, P.; Xia, F. Nat. Photonics 2013, 7, 394−399. (14) Brar, V. W.; Jang, M. S.; Sherrott, M.; Lopez, J. J.; Atwater, H. A. Nano Lett. 2013, 13, 2541−2547. (15) Chen, J.; Badioli, M.; Alonso-González, P.; Thongrattanasiri, S.; Huth, F.; Osmond, J.; Spasenović, M.; Centeno, A.; Pesquera, A.; Godignon, P.; Elorza, A. Z.; Camara, N.; García de Abajo, F. J.; Hillenbrand, R.; Koppens, F. H. L. Nature 2012, 487, 77−81. (16) Fei, Z.; Rodin, A. S.; Andreev, G. O.; Bao, W.; McLeod, A. S.; Wagner, M.; Zhang, L. M.; Zhao, Z.; Thiemens, M.; Dominguez, G.; Fogler, M. M.; Castro Neto, A. H.; Lau, C. N.; Keilmann, F.; Basov, D. N. Nature 2012, 487, 82−85. (17) Fei, Z.; Rodin, A. S.; Gannett, W.; Dai, S.; Regan, W.; Wagner, M.; Liu, M. K.; McLeod, A. S.; Dominguez, G.; Thiemens, M.; Castro Neto, A. H.; Keilmann, F.; Zettl, A.; Hillenbrand, R.; Fogler, M. M.; Basov, D. N. Nat. Nanotechnol. 2013, 8, 821−825. (18) Chen, J.; Nesterov, M. L.; Nikitin, A. Y.; Thongrattanasiri, S.; Alonso-González, P.; Slipchenko, T. M.; Speck, F.; Ostler, M.; Seyller, T.; Crassee, I.; Koppens, F. H. L.; Martin-Moreno, L.; García de Abajo, F. J.; Kuzmenko, A. B.; Hillenbrand, R. Nano Lett. 2013, 13, 6210− 6215. (19) Gerber, J. A.; Berweger, S.; O’Callahan, B. T.; Raschke, M. B. Phys. Rev. Lett. 2014, 113, 055502. (20) Woessner, A.; Lundeberg, M. B.; Gao, Y.; Principi, A.; AlonsoGonzález, P.; Carrega, M.; Watanabe, K.; Taniguchi, T.; Vignale, G.; Polini, M.; Hone, J.; Hillenbrand, R.; Koppens, F. H. L. Nat. Mater. 2015, 14, 421−425. (21) Dai, S.; Ma, Q.; Liu, M. K.; Andersen, T.; Fei, Z.; Goldflam, M. D.; Wagner, M.; Watanabe, K.; Taniguchi, T.; Thiemens, M.; 1378
DOI: 10.1021/acs.nanolett.7b05085 Nano Lett. 2018, 18, 1373−1378
