Broadband impedance match to two-dimensional materials in the terahertz domain

The coupling of an electromagnetic plane wave to a thin conductor depends on the sheet conductance of the material: a poor conductor interacts weakly with the incoming light, allowing the majority of the radiation to pass; a good conductor also does not absorb, reflecting the wave almost entirely. For suspended films, the transition from transmitter to reflector occurs when the sheet resistance is approximately the characteristic impedance of free space (Z 0 = 377 Ω). Near this point, the interaction is maximized, and the conductor absorbs strongly. Here we show that monolayer graphene, a tunable conductor, can be electrically modified to reach this transition, thereby achieving the maximum absorptive coupling across a broad range of frequencies in terahertz (THz) band. This property to be transparent or absorbing of an electromagnetic wave based on tunable electronic properties (rather than geometric structure) is expected to have numerous applications in mm wave and THz components and systems.


Supplementary Note 1 -Impedance, Resistance, Conductance
In circuits, impedance means the ratio of the voltage across a given element to the current through a given element, and can be complex (i.e. the current and voltage may be out of phase). An "element" is a physical object with dimensions much smaller than the wavelength at the frequency of interest. The resistance, reactance, admittance, susceptance, and conductance all are various representations of the real or imaginary part of the impedance or its inverse. However, in our experiments, there are no "elements" (all of our structures are of order the wavelength in size), hence one must consider the electromagnetics of waves, not lumped elements.
In electromagnetic waves, the wave impedance means the ratio of the electric field to the magnetic field, and can also be complex (i.e. the electric field and magnetic field may be out of phase). This is also sometimes referred to as the characteristic impedance or the wave impedance of the medium. In cases where the medium is not lossy, the electric field and magnetic field are in phase, and hence the wave impedance is purely real. For example, in vacuum, the wave impedance is 377 . Even though there is no imaginary component, one still generally refers to this as the "characteristic impedance of free space". For all of our experiments, the medium is not absorbing and so the wave impedance is always real. In a medium with index of refraction n (assuming n real as is the case for our experiments), the wave impedance (or also called the characteristic impedance of the medium) is given by 377 /n. Note that a similar definition holds for the characteristic impedance of a transmission line, which is a distributed inductance and capacitance (and perhaps conductance).
In the context of conducting media such as a metal, the wave impedance has significant imaginary components. In this case, the wave impedance is referred to as the "surface impedance", 1 even for cases where the waves are propagating and there is no physical "surface" in sight. We avoid this definition of "surface impedance" as it does not directly apply to our case.
Instead, we use the concept of sheet resistance. The sheet resistance of a thin film is the ratio of the voltage through a film of width W and length L to the current through that film, divided by the number of squares, i.e. L/W. The concept of sheet impedance is similar; if the voltage and current are out of phase the sheet impedance can have an imaginary component. The sheet conductance is defined similarly. While we consider the case of an imaginary component in the supplementary info for our experiments, we find little evidence for it experimentally in our frequency range and so the main text focuses the sheet resistance only.
The purpose of this paper is to show that, when a plane wave is incident on a thin film, the transmission, reflection, and absorption coefficients of that plane wave depend on the ratio of the characteristic impedance of the wave to the sheet resistance of the thin film.
The quantitative calculation of the transmission, reflection, and absorption coefficients is simplified dramatically using a transmission line equivalent circuit model that captures the key electromagnetic wave phenomena but requires only the use of discreet or disturbed circuit elements, and is described in detail. The model was validated theoretically and experimentally 2 .

Supplementary Note 2 -Transmission Line Model
To analyze the THz transmittance of metallic films on dielectric substrates, a transmission-line model can provide accurate values of the transmittance, absorptance, and reflectance over a wide frequency range 2 . The primary requirement is that the incident beam be much wider in extent than the thickness of the substrate + metallic film, and that the beam be at, or near, a "waist". The secondary requirement is that the metallic film be much thinner than a wavelength, which is quite easy to satisfy in the THz region. The "waist" condition means that the constant-phase surface for the beam is a plane perpendicular to the direction of propagation. Hence, because the transmittance depends heavily on the interference effects of propagation in the film + substrate structure, the propagation can be handled as a transverse electromagnetic (TEM) mode. And TEM modes are the basis of the transmission-line model. From a practical standpoint, it is actually more realistic than the standard "plane-wave" model in physical optics because the latter assumes the beam has not only a planar constant-phase surface but also infinite lateral extent -something impossible to achieve in real optical systems.
The equivalent-circuit diagram of the transmission-line model is shown in Supplementary Fig. 2b. The incident beam is modelled as a TEM ideal voltage source having phasor amplitude vs and source resistance 0 -the intrinsic impedance of free space. The thin-film-on -substrate structure is represented by the parallel combination of a complex impedance ZG and a possibly lossy transmission line (to account for substrate absorption effects) of characteristic impedance Z0. The free space region after the structure is represented by the free-space intrinsic (load) resistance 0.

Supplementary Figure 2: a) Beam propagation through a thin metallic film on substrate b) Transmission-line equivalent-circuit model
The combination of transmission line plus free load resistance can be written where s is the substrate thickness and  is the complex propagation constant, and s is the intrinsic impedance of the substrate = 0 /(r) 1/2 . Supplementary Eq. (1) comes from the basic theory of lossy TEM transmission lines such that (s) represents the complex impedance "seen" by a generator connected to the input port 3 .
Given the circuit in Supplementary Fig. 2b, we can calculate the transmitted power, the power dissipated in the thin-film-on-substrate structure, and the reflected power as follows. We first calculate the phasor currents flowing out of the generator i1, into the transmission line i2, and into the thin film i3, where i1 = i2 + i3 by the Kirchoff current law. By voltage division, we then have (2) where the last two follow from current division. Substitution for i1 then yields and, (4) The average power dissipated in the thin-film-on-substrate structure is (assuming Ohm's law and sinusoidal phasor) (5) and the transmittance T through the metal film is just P2 divided by the "available power" from the source, |vs| 2 /80. Hence, we can write By similar reasoning the average power dissipated in ZG is Such that the thin film absorptance is In the special case of a resistive film, ZG = RG, Supplementary Eq. (10) -(12) can be greatly simplified algebraically:

Supplementary Figure 3: The T, R, A values versus sheet resistance for a suspended, purely resistive film.
Interestingly none of these depends on frequency unless RG is dispersive (more on that later). Historically, Supplementary Eq. (14) is known as the Wolsterdorff equation 4 .
Special Case#2: Lossless Substrate; Purely Resistive Film Because of the difficulty of suspending thin metal films in free space, a more practical case is a substrate of finite thickness but negligible loss. This can be readily achieved even in the THz region by highly-resistive substrates having zero or very low polarity, such as high-resistivity silicon. The low polarity is necessary to make the optical phonons weakly interacting with electromagnetic radiation (silicon having near-zero polarity). In this case, we can re-write Supplementary Eq. impedance matched to their load. But there are special cases well known from microwave transformer theory that simplify the analysis once again.
For a "half-wave" transformer, ·s = m(2/)·/2 = mwhere m is any integer (including zero) and  is the wavelength in the transmission-line medium having dielectric constant rand index of refraction, nr   This condition makes tan(·s) → 0 for all m, so according to Supplementary Eq. (16),(s) = 0 . And then we recover the same values of T, A, and R, as given by Supplementary Eq. (13) -(15). For a given s, the half-wave condition is satisfied by a set of periodic frequencies  = m·c/(2ns), where m = any positive integer (including zero). For a "quarter-wave" transformer, ·s = (2/)*/4 = , so that tan(s) →, and (s) = (s) 2 /0.

Supplementary Note 3 -Transmission Line Model for Calculating T, R, A
Using the transmission line model, the effect of different substrate dielectric constant/index of refraction can be easily accounted for, and provides a convenient "device design tool". For our experiments where graphene is mounted on a silicon substrate, ≈ 11.66 , and a substrate thickness of L = 400 m is used for the calculations of T, R, and A. These values are compared to a suspended film, and a graphene film on substrate in Supplementary Fig. 4. In order to investigate the frequency dependence, in the figure below, the scattering time is set as, = 50 fs, as this value is widely reported in the literature (see table below). On half-wave resonance (~ 110 GHz for L = 400 m), which occur at even integer multiples of = (4 * * ) , or when kL = N, the transmittance, reflectance, absorptance values from the transmission line model reduce to the Woltersdorff 4 values. In contrast, at odd integer multiples = (4 * * ) , the device is at quarter-wave resonance (~ 164 GHz for L = 400 m), and mostly reflecting. The equations for T, R, A are listed below. For device design, in the limit where the substrate index of refraction n = 1, and thickness of L = 0, the frequency dependent Fabry-Perot fringes of T, A, R decrease in amplitudes, and approaches the ideal case of a suspended film where the T, R, A values are generally flat with broadband, 50% absorption, matching the Woltersdorff values.

Half-Wave Resonance at Transmittance Peak
It is important to note that, even though the transmission line model matches the Woltersdorff values at the half-wave resonance values (Supplementary Fig. 4d,h), the half-wave resonance condition does not always occur at a transmittance peak maxima. The half-wave resonance condition occurs at a transmittance maxima up to a certain critical sheet resistance value (i.e. when the sheet resistance is large), but because of a phase shift when the graphene film becomes reflecting (i.e. when the sheet resistance value is small), the half-wave resonance condition occurs at the transmittance minima 2 . This phenomenon is obvious in Supplementary Fig. 4e; for sheet resistance values greater than 156 /sq, the half-wave resonance condition occurs at the transmittance maxima, but for sheet resistance values less than 156 /sq, the half-wave resonance condition occur at the transmittance minima. In this work, we never reach this critical sheet resistance threshold, and thus, in the manuscript, the half-wave resonance condition is referred to at the transmittance peak maxima. The opposite situation occurs for quarter-wave resonance, that is, for high sheet resistance the quarter-wave resonance occurs at a transmittance minima, but after the critical sheet resistance is reached, the quarter-wave resonance flips and occurs at the transmittance maxima.

Supplementary Note 4 -THz Incident on Substrate
The case where the THz beam is incident on the substrate first (with a graphene film on the backside) was investigated analytically with similar results as ref [ 6 ]. Contrary to the experimental setup used in this work, that is, where the THz beam is incident on the graphene film, if the THz beam is incident on the substrate side, the transmittance through the etalon remains the same (compared to incidence on the graphene side), but the reflectance and absorptance are different. The absorptance is plotted (with n = 3.41) varying the graphene sheet resistance and kL varying from 0 -2 in Supplementary  Fig. 5a Although the values T, R, A, remain identical at half-wave resonance when kL = N, at quarter-wave resonance, as clearly evident at kL = M, where M is odd integer values, the total reflectance can decrease, resulting in increased absorbance. For this case (which is different from the experimental setup in this work) the absorptance values can be greater than 50% for n > 1. The peak absorption found at very low sheet resistance values calculated here were not apparent in similar plots in ref [ 6 ] (the sheet resistance where absorption begins to decrease was beyond the range analyzed), but here we emphasize that a sheet resistance dependent maximum absorption does indeed occur. The maximum absorptance, and the sheet resistance value at which it occurs can vary depending on the substrate index of refraction. These values were computed in Supplementary Fig. 5b. Here, it is obvious that the largest sheet resistance value for maximum absorbance occurs at 377/2 sq, when n = 1. Absorptance values greater than 90% can be achieved when n > 3 with the sheet resistance value required decreasing to values < 100 sq.

T, R, A (Incidence on Substrate Side)
T = 4 0 | gn(n sin(kL) − icos(kL)) (1 + g + gn 2 ) 0 sin(kL) − i(1 + 2g)n 0 cos(kL) | 2 2 0 1 + n 2 − cos(2kL) (n 2 − 1)  Supplementary Fig. 6. The ODTS SAM is vacuum deposited on the wafer prior graphene transfer. During the transfer step, BI can be introduced on the bottom of the film to intentionally dope the graphene. Scanning electron microscopy (SEM) imaging of the transferred graphene film was performed to determine the surface film quality following transfer, shown in Supplementary Fig. 6b. Although there are small topological variations, such as bumps and wrinkles created during the transfer process, we observe no detrimental defects (such as holes or tears) in the graphene film. As demonstrated in depletion curves shown in Supplementary Fig. 7b- Supplementary Fig. 7a shows the transmittance versus frequency of a substrate with a SAM modified surface. The THz transmittance (half-wave resonance) nearly reaches unity across the measured frequency range. In the most extreme circumstance, the transmittance is not completely lossless, but is nevertheless over 90%. This value corresponds to a sheet resistance value of ~ 3500 /sq. Because the SAM resistance is in parallel with the graphene film, the large resistance of the SAM has limited effect when the graphene sheet resistance is low, and results in a difference of less than 15%. The influence of the SAM plays a more significant role when the graphene sheet resistance is comparable (such as when the graphene film is gated to ~ 2000 /sq in Fig. 3f). This small absorption contribution from the SAM may potentially explain why the data point (d) deviates from the theory line in Fig. 4c.

Supplementary Note 7 -Electrical Modulation of Transmission
The broadband transmittance was measured for a single device at various gate voltages to test the performance as a THz modulator. The peak transmittance (centered at ~ 694 GHz) versus gate voltage is plotted on the same chart as the device resistance, shown below. On a different device (best), the depth of modulation (using +20 Vg and -7 Vg) versus frequency is displayed, and shows frequency variation of the DoM, with a maximum value of ~ 52%. Variations may arise from inhomogeneous gating using the high resistivity silicon substrate. The use of a high resistivity substrate limits the device switching speed. Experimentally, we measured a modulation frequency of < 2 Hz.

Supplementary Figure 8: a) The resistance versus gate voltage is plotted with the voltage dependent peak transmittance value (located at ~ 694 GHz). b) The depth of modulation of the transmittance peaks for a device under -7 and 20 Vg gate biasing. The maximum value (~ 52%) occurs around ~ 320 GHz, although significant modulation is seen across the broadband frequency range measured.
Tmax is the maximum transmittance value at a peak of the spectra. Then the transmission variation is given by, where Z0=377 Ω is the impedance of free space, εs is the dielectric constant of the Si substrate and σg is the conductance of graphene film. Assuming the device is mounted on a silicon substrate ( ≈ 11.66), the room temperature values from Banzerus et. al 7 , (mobility ~ 145,000 cm 2 /Vs) approaches 100% depth of modulation with only ~10 V on the gate.

Supplementary Note 8 -Characterizing Doping of Graphene Films
We measured the depletion curve to investigate the effects of doping on the electrical properties of the graphene films. Supplementary Fig. 10 shows the depletion curves of samples with and without BI doping on an ODTS substrate. After doping, the Dirac point voltage is subsequently shifted to > 48 Vg, indicative that the graphene film becomes hole doped. We found that with the use of BI doping, we could achieve a (zero gate biased) DC sheet resistance below the impedance of free-space. Even with BI doping, the sheet resistance values would vary from sample to sample. The sheet resistance differences from sample to sample could be explained by spatial variations in and EF after doping. To further investigate the effects of doping graphene, we employed Raman spectroscopy. Raman spectroscopy was performed using a Renishaw Invia Spectrometer with a 532 nm laser. Raman mapping was performed using the Streamline High Resolution (HR) mapping function over a 3 x 3 mm area with 60 um step size. The peak analysis and fitting was performed using the Wire 3.6 Renishaw software package. Mapping images are processed using Igor 6 plotting software. Figure 11: Representative Raman spectra for samples with and without BI doping on an ODTS SAM modified substrate. Before doping, the [I]2D/[I]G ratio is much larger than after doping. Also the peak position of the g-peak becomes blue-shifted after doping. Supplementary Fig. 11 shows representative Raman spectra of graphene samples with and without BI doping during transfer. Notably, after the addition of BI doping, we observe a decrease in the intensity ratio, [I]2D/[I]G. The [I]2D/[I]G ratio is known to decrease when both the scattering and Fermi energy of the graphene film increase in magnitude [8][9][10] , and thus provides a convenient parameter to assess the BI doping of graphene films. Supplementary Fig. 12 shows the Raman mapping histograms and images of the [I]2D/[I]G with and without BI doping. We confirm the general trend (lower average in histrogram) of decreased [I]2D/[I]G ratio, but we also observe spatial variance of the graphene film after doping. This supports our belief that after doping, the graphene sheet resistance is not dominated merely by changes in EF, but instead, by a combination of both EF and The extracted AC conductivity parameters (outlined below) further support this hypothesis, since under chemical doping, the lowest sheet resistance devices are not the largest EF, but instead, a combination of moderate EF and long 

Supplementary Note 9 -Transmission Matrix Method for Calculating AC Conductance
The transmission matrix method is used 5 for calculating the conductance of graphene films from the transmittance measurement on a substrate of index of refraction, n, and thickness, L. Assuming the THz beam is normally incident on the graphene surface, the transmittance, T = S21 2 21 = 1 2 exp (− ) 1 + 1 2 (−2 ) (30) where ks is the wave vector inside the substrate, and is equal to ks = 2√ / is the dielectric constant of the substrate, silicon, and is the free-space wavelength. For devices on a silicon substrate, n = √ , with ≈ 11.66 in the THz frequencies. r1, and t1, are the coefficients of reflectance and transmittance at the air-Si interface, while r2, and t2, are the coefficients of reflectance and transmittance at the Si-graphene/graphene-air interface. These are given by, Each transmittance peak (consisting of a maximum, and two minimum values) is fitted to calculate the real and imaginary admittance value using a least squares fit procedure. During fitting, the substrate thickness, L, is adjusted for each device being analyzed.

Supplementary Note 10 -Comparison Plot
We now discuss the relationship of this work to prior work on graphene in the THz domain. Supplementary Fig. 13 shows the range of conductance versus frequency for other monolayer graphene devices previously measured in the literature. Each trend line is plotted using equations (1) and (2), using data points supplied from each reference. The labeled and colored hashed (solid) regions (lines) indicate frequency domain measurements, whereas the transparent grey regions (lines) are for time-domain systems. The frequency range measured for this paper is one of the broadest (with 500 MHz spectral resolution) investigating graphene-THz coupling, complemented by an extensive range of conductance values, including the achievement of surpassing the free-space impedance threshold. The table below shows both the conductance range and the frequency range of the references used in Supplementary Fig. 13. The optical conductance values are those measured and reported from each reference, and τ and EF are estimated to reflect this value. In reference 8, τ is defined as τ = ħ/Γ, where τ is the scattering time, ħ is the Planck constant, and Γ is the phenomenological scattering parameter, whereas in our calculations, τ is defined as, τ = ħ/2Γ. The absorptance versus sheet resistance is plotted using the general absorptance Eq. (7) -(9) in the case where the THz beam is incident on the graphene side for three values of the n (substrate index of refraction), while varying kL from 0 to 2When n = 1 the maximum absorptance is 50%. The maximum absorptance value is also 50% with when n = 3.41 (substrate used in this work) at the half-wave resonance values (kL = N), with a minimum value of ~ 8% in the quarter-wave resonance case. Although Eq. (7) -(9) can yield absorptance values greater than 50%, this is only satisfied for values of n < 1, which is not considered in this manuscript, but nonetheless useful for cases such as metamaterials. Hence, 50% is considered the maximum for incidence on the graphene side.

THz Mobility
Supplementary Figure 15: The mobility versus carrier density calculated 11 using the scattering time and optical conductivity values from this work, and those reported in Supplementary  The effective thickness of each device is held constant during calculation and fitting using the values listed above. The peak-to peak spacing of each transmittance peak of all devices were analyzed, and the average values are also listed in Supplementary Table 3. Small deviations in the peak-to-peak spacing are apparent. This variation could be due to the common issue collimating the THz beam, where, when the device is not absolutely parallel leads to a spread in the effective substrate thickness. Nonetheless, the peak-to peak spacing agrees with the different effective thickness from the various device. Furthermore, the instrument used is a fiber-based frequency domain photomixing spectrometer; insertion of a sample into the THz beam may shift the interference fringes on the THz arm, leading to a slight off-balance with respect to the optical arm. Therefore, some irregular frequency points in the measured transmittance may be contributed by "instrument drift". To mitigate the experimental uncertainty from both the instrument drift and beam collimation, post experimental data analysis was employed. Here, we performed fitting to every individual interference pattern around every peak which contains up to hundreds of frequency-transmission points. From the extracted conductance at those individual peaks, we obtain EF and  with the Drude model, and compare the calculated DC conductance to the measured DC value, as seen in Fig. 4a-b. Here, we find that the two values agree with each other, and supports the reliability of our data.

Supplementary Note 13 -Imaginary Contributions
A Drude-like roll off is expected for frequencies greater than 1/ as the imaginary contributions become significant, especially when is long. During fitting of the measured transmittance peaks, both the real and imaginary contributions were calculated. However, when fitting for  and EF only the real contribution is used. Because the Kramers-Koenig relation relates the real and imaginary contributions, a fit on only the real yields valid results 5 . Supplementary Fig. 16 shows the real and imaginary values calculated, and the calculated Drude trend from the (real part) fitted EF and values for the best device of 250 /sq. The predicted line generally agrees the measured conductance trend. Though the error of the imaginary fit is larger than that on the real part, this error can potentially be improved from a direct measurement of the phase. This would improve the fitting of , and would provide a more accurate range of the Drudeshaped roll off (1/. For our best device (~ 250 /sq), the high frequency