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Efficient Fizeau Drag from Dirac electrons in monolayer graphene

Fizeau demonstrated in 1850 that the speed of light can be modified when it is propagating in moving media. Can we achieve such control of the light speed efficiently with a fast-moving electron media by passing electrical current? Because the strong electromagnetic coupling between the electron and light leads to the collective excitation of plasmon polaritons, it will manifest as the plasmonic Doppler effect. Experimental observation of the plasmonic Doppler effect in electronic system has been challenge because the plasmon propagation speed is much faster than the electron drift velocity in conventional noble metals. Here, we report direct observation of Fizeau drag of plasmon polaritons in strongly biased graphene by exploiting the high electron mobility and the slow plasmon propagation of massless Dirac electrons. The large bias current in graphene creates a fast drifting Dirac electron medium hosting the plasmon polariton. It results in nonreciprocal plasmon propagation, where plasmons moving with the drifting electron media propagate at an enhanced speed. We measure the Doppler-shifted plasmon wavelength using a cryogenic near-field infrared nanoscopy, which directly images the plasmon polariton mode in the biased graphene at low temperature. We observe a plasmon wavelength difference up to 3.6% between plasmon moving along and against the drifting electron media. Our findings on the plasmonic Doppler effect open new opportunities for electrical control of non-reciprocal surface plasmon polaritons in nonequilibrium systems.


Main text:
Surface plasmon polariton (SPP), coupled electromagnetic and electron oscillation mode, has the unique capability to confine and manipulate light at subwavelength scale [2][3][4][5] . The strong light-matter interaction enhanced by the plasmon plays a key role in nanophotonics 6 , ranging from nanoscale nonlinear optics 7 , quantum optics 8 , to flat optics 9 . Electrical control of the plasmon polariton is highly desirable in such applications. An intriguing possibility for plasmon control is through the Fizeau drag 1 , where the moving electron media modulate the propagation speed of the plasmon polariton. This can be viewed as a plasmonic Doppler effect, where counter-propagating plasmons can have different speed that depends on the moving electron media.
The plasmonic Doppler effect is negligibly small in conventional noble metals because the plasmon velocity is more than a million times larger than the highest drift velocity achievable in those metals 3 . Recent theories predict that the two-dimensional (2D) Dirac electrons in graphene provide an ideal platform to realize strong plasmonic Doppler effect due to a combination of low carrier density, high electron mobility, and strong plasmon polariton confinement [10][11][12][13][14][15][16][17] . Compared with conventional metals, the electron drift velocity (v d ) in graphene is orders of magnitude higher and can reach a value of 3×10 5 m/s 18 . At the same time, the graphene plasmon features an ultra-high field confinement 14,[19][20][21][22] (λ /λ 0~1 /150), resulting in a plasmon group velocity (v p ) at around 2×10 6 m/s, two orders smaller than the speed of light. A significant plasmonic Doppler effect can emerge in electrically biased graphene when the electron drift velocity v d reaches a substantial fraction of the plasmon velocity 18,23 . Such a Doppler effect has been predicted to break the time reversal symmetry in the graphene optical response in the nonlocal limit and create non-reciprocal surface plasmon propagations [24][25][26][27][28][29][30][31] .
Here we report the first experimental observation of the plasmonic Doppler effect in monolayer graphene. Taking advantage of the cryogenic scanning near-field infrared nanoscopy, we can measure the Doppler-induced wavelength shift in real space even under a large bias current in graphene. Our two-terminal graphene device is composed of ultraclean monolayer graphene fully encapsulated in hexagonal boron nitride (hBN) and a nanofabricated gold nanobar as the integrated plasmon launcher. The plasmon launched by the gold nanobar is imaged by the near-field infrared nanoscopy, and it exhibits significant modulation by the electrical bias current (and therefore the electron drift velocity). We quantify the plasmonic Doppler effect by monitoring the plasmon wavelength change for positive and negative electrical current and observe a Doppler-induced wavelength modulation as large as 3.6% for a bias current density of ± 0.8 mA/μm. Our experimental results agree well with the existing theoretical model 30 . This strong plasmonic Doppler effect opens new doors for exploring nonequilibrium plasmons [32][33][34][35] and non-reciprocal plasmonic [36][37][38] phenomena in graphene and other high mobility 2D materials. Figure 1a displays the schematic of our device fabricated on a SiO 2 /Si substrate. Ultraclean monolayer graphene was fully encapsulated by two hBN flakes using the dry-transfer technique (details of sample fabrication in Methods). Source-drain electrodes with low contact resistance are fabricated using the one-dimensional edge contact method 39 . The narrow gold nanobar (~ 500 nm in width and 9.5 m in length) in the middle of the device is used to excite the plasmon in the graphene sheet (5 m in width and 15 m in length). In order to access to the graphene plasmon efficiently, we use a very thin top hBN layer with a thickness of ~ 2 nm. Figure 1b illustrates the Doppler effect of graphene plasmon launched by the gold nanobar under positive and negative carrier flow directions. The sharp edge of the gold nanobar serves as an efficient launcher of the plasmon in graphene 12,20 , which propagates away from the nanobar. The presence of a carrier flow breaks time reversal symmetry ( ω ≠ ω − ) and results in different plasmonic group velocities depending on the direction of the external driving current 26 . This leads to a stretched (compressed) plasmon wavelength for plasmons propagating along (against) the carrier flow direction. In Figure 1b, the wavelength change is exaggerated for better illustrating the idea.

Doppler-induced wavelength shift
We probe the graphene plasmon with a home-built near-field infrared nanoscopy setup at cryogenic temperature (details in the Methods). The base temperature of the sample is fixed at 25 K, which helps to dissipate the unintended Joule heating caused by the large current through the graphene channel. In addition, the phonon scattering of the graphene plasmon is strongly suppressed at low temperature, resulting in a higher plasmon quality factor 3,12 and thus a better accuracy in determining the plasmon wavelength. To probe the graphene plasmon, a 10.6 m CO 2 laser was focused on the sample by an aspheric ZnSe lens with a spot size around 10 m.
This guarantees a uniform light illumination on the nanobar and atomic force microscope (AFM) tip when scanning around the bar area (within 1 m range), with negligibly small intensity and phase inhomogeneity. Basically, the graphene plasmon excited by the gold nanobar propagates away from the bar edge. The propagating plasmon is later scattered by the tip apex, which interferes with a background scattered light and gets detected by a HgCdTe (MCT) detector in the far field. By scanning along the plasmon propagating direction and recording the near-field interference fringes, we can determine the plasmon wavelength accurately 12,[19][20][21]40 . Figure 1c shows the gate-dependent resistance of a representative Doppler device at 25 K. This specific device has a 2 m thick SiO 2 dielectrics to minimize the doping change induced by the bias voltage (V b ) required in the Doppler measurements. Only the hole doping data is measured because such device with thick SiO 2 dielectrics breaks down easily under a high positive backgate voltage V g (See Methods). The device resistance decreases quickly with increased doping away from the charge neutral point at V g = -380 V. The small resistance peak at V g = -780 V is presumably due the graphene/boron nitride moiré superlattice. There is little hysteresis between the forward (blue trace) and backward gate voltage scan (red trace). Figure 1d shows the channel current of the same device as a function of V b at V g = -1050 V, corresponding to a carrier density of |n| = 7.2×10 12 cm −2 based on the capacitance model (see Methods). Doppler measurements are carried out at this doping, where high quality plasmons at mid-infrared wavelengths are present. The two-terminal resistance at high doping is dominated by the electrical contact. Using the one-dimensional edge contact, we achieve a low contact resistance of ~ 870 Ω (slope of the linear fit of Fig.1d), which allows us to drive large electrical current through the graphene device. Figure 2 shows the near-field signal under different driving currents. The data is collected from scanning the same line along the plasmon propagating direction which is perpendicular to the nanobar. At different currents, the graphene plasmons behave quite differently not only because of the Doppler-induced wavelength shift, but also because of a temperature change due to the strong Joule heating. The temperature change has two effects. First, plasmon damping increases with temperature which leads to a decreased near-field signal in amplitude; second, thermal expansion and other effects can modify the background light scattering and thus modulate the near-field interference signal. In order to eliminate those complexities induced by the thermal effects, we directly compare the plasmon wavelengths for positive and negative currents of the same magnitude, which has the same thermal load in the device but generates opposite Doppler shift. Figure 2a shows the 2D near-field signal images obtained from stacking 30 consecutive scans of a single line at + 2 mA and -2 mA. Figure 2d shows the line profile obtained by averaging 30 scans to improve the signal-to-noise ratio. The wavelength at + 2 mA is stretched by 1.2% compared with the -2 mA case, which is determined by averaging the wavelength shifts at the interference peaks and dips in the line profile in Fig. 2d. Figures 2b and 2e show the measured near-filed signals when the current increases to about ± 3 mA. The difference between positive and negative currents induced shift becomes more prominent. We observe a total ~ 2.2% Doppler-induced wavelength shift between + 3 mA and -3 mA. The largest wavelength shift we can observe reaches ~ 3.6% between + 4 mA and -4 mA (Figs. 2c and 2f). At currents higher than ± 4 mA, the plasmons become very weak and are beyond the detection limit of our setup, presumably due to the significant thermal heating. A similar Doppler shift has been observed in other graphene devices (See the Methods). In principle, Doppler-induced wavelength shift can be larger at low carrier density, because a larger carrier drift velocity can be realized at low doping with the same thermal load. Experimentally, however, the graphene plasmon becomes quite weak at low carrier doping and accurate measurements of the plasmons are very challenging with 10.6 m laser excitation.

Non-uniform doping effects
Next, we rule out the possibility of doping-induced wavelength shift in our experimental observation. In order to achieve large carrier drift velocity, the applied bias (V b ) is on the order of few Volts. The switching of the current direction by switching the sign of V b will result in some changes in the local graphene doping level, which in turn gives rise to a finite wavelength shift. Figure 3a illustrates the V b -induced doping change in graphene channel with positive and negative V b . In some devices, we used SiO 2 /Si substrate with a 2 μm thick oxide layer to significantly reduce this side effect. We experimentally investigate the graphene near-field response with changing the V g (as schematically illustrated in Fig. 3b). The graphene channel is fixed at -3 mA while the V g is varied by adding an offset of ± 2.66 V (the bias voltages to achieve about ± 3 mA current) to -1050 V. Figure 3c  We note that the doping effect for Si/SiO 2 devices with 285nm SiO 2 will be stronger, but the Doppler wavelength shift will still be significantly larger than the doping induced wavelength change.

Nonequilibrium plasmon response
The nonequilibrium plasmon response of the current-carrying monolayer graphene can be calculated using the linear response theory 30 . The density-density response function of current-carrying states can be analytically approximated by employing the random phase approximation 30   ). The green line in Figure 4c shows the theoretical prediction, which is obtained by calculating the difference between the upstream and downstream dispersion curves in Fig. 4b.    was used to etch hBN to expose graphene edges with saw-tooth edge shapes. A separate EBL process was then carried out to design the electrode patterns of a two-terminal device and a narrow nanobar structure in the middle of graphene. Immediately before the metal deposition, the samples were treated with mild oxygen plasma to expose a clean graphene edges. Cr/Au (typically 5 nm/75 nm) electrode was made using an e-beam evaporator equipped with a water cooling system at high vacuum (< 1×10 -6 Torr). The surfaces of the devices were cleaned by a mild hydrogen plasma treatment (250 °C , 10 sccm H 2 ) for about 30 min. The treated devices were kept in a conductive case for 1 or 2 days to avoid potential charging issue before any near-field infrared nanoscopy measurements.

Cryogenic near-field infrared nanoscopy measurements
Our cryogenic near-field infrared nanoscope was based on a home-made AFM which has the capability to work at high vacuum and low temperature. The whole AFM setup was built inside a closed cycle cryostat, and the AFM head was connected to the cold plate by a soft copper braid in order to damp the vibration from the pulse tube. The lowest sample temperature achieved in our AFM system was 25 K. A CO 2 laser was coupled into the vacuum chamber through an aspheric ZnSe lens with 0.45 NA. The position of the lens was controlled by a vacuum compatible stage. The back scattered light from the tip was collected by MCT in a self-homodyne configuration, the near-field signal was demodulated at the third harmonic of the tapping frequency to suppress the background. In order to increase the signal to noise ratio, the time constant was set to be 10 mS. During the scanning, the turbo pump was turned off in order to minimize the mechanical vibration. The background vacuum level remains below 1×10 -6 mbar through all the measurements.

Estimation of carrier density in graphene
The dielectrics in the plate capacitator formed between graphene and Si backgate were composed of the SiO 2 dielectrics (medium 1) and the bottom hBN layer (medium 2). The total geometric capacitance per unit area was calculated from the expression , where C 1 = ϵ 0 ε r1 d 1 ⁄ and C 2 = ϵ 0 ε r2 d 2 ⁄ are the capacitance per unit area of SiO 2 and hBN, ϵ 0 is the vacuum permittivity, ε r1 = ε r2 = 3.9. For the device measured in Fig. 2  As we increase the current to be ± 1.2 mA, the plasmon wavelength difference becomes more prominent (~ 2.3% in Extended Data Fig. 2f). When the current reaches + 1.7 mA and -1.9 mA, the wavelength shift becomes ~ 3.1%. The wavelength shift is extracted by averaging the shift at the interference extrema.
On the right side of the gold nanobar (Extended Data Fig. 3 Fig. 5 and therefore we limit our measurements of graphene plasmon at the hole doping side by applying negative backgate voltages. Since the graphene holds particle-hole symmetry, the Doppler effect should be the same for the electron side 24,30 .
Formation of graphene/hBN moiré superlattice in the device The small resistance peak in our device as shown in Fig. 1c in the manuscript is most likely due to the formation of graphene/hBN moiré superlattice. It has been well established in transport studies of hBN-encapsulated graphene that a second small resistance peak can be observed when the graphene and hBN happen to align within a small angle (< 2°) 45 . This alignment can often be inferred from the alignment of straight edges between the graphene and hBN layers. To enhance the graphene and hBN contrast on thick SiO 2 dielectric layer, Extended Data Fig. 6 shows the optical microscope picture of our device, in which we have adjusted the contrast to observe the graphene and hBN layers. As shown in the photo, the graphene straight edge (white dashed line) nearly aligns with an edge of top hBN (yellow dashed line), which can potentially form graphene/hBN moiré superlattice with long period.
The carrier density at the small resistance peak in our device is n s~3 .98 × 10 12 cm −2 , which corresponds to a moiré period of ~10.3 nm and alignment angle of ~ 0.93°4 5,46 .
Nonequilibrium response of the plasmons in a current-carrying graphene The optical properties of a current-carrying graphene sheet can be calculated by using linear response theory. We follow the model in Ref. 30 is the normalized electron drift velocity v d relative to Fermi velocity v F . μ is the chemical potential of the graphene. γ = +1(−1) denotes 'upstream' ('downstream') plasmon propagation. D(μ) is the two dimensional (2D) massless Dirac fermion (MDF) density of states.
In the long wavelength limit k ̅ ≪ 1, the plasmon dispersion can be further reduced to