Plasmon transport and its guiding in graphene

To start, we show results for the plasmon transport in the uniform gate device. Figure 1(a) shows the current through the D2 detector ${{I}_{{\rm D}2}}\left( t \right)$ for $\Delta {{V}_{{\rm g}}}=100\;{\rm V}$ and 40 V (${{I}_{{\rm D}1}}\left( t \right)$ and ${{I}_{{\rm D}3}}\left( t \right)$ are presented in section 2 of the supplementary information). The sharp peak at zero time delay is due to direct crosstalk between high-frequency lines. The plasmon signal appears as a broad peak with a time delay. The slowly decaying tail is due to the time constant of the detection and, as will be discussed below, nonlinear plasmon dispersion induced by the resistance R of graphene. The time delay, which roughly corresponds to the time of flight of the plasmon pulse (section 3 of the supplementary information), increases with decreasing $\Delta {{V}_{{\rm g}}}$. The red trace in figure 1(b) shows the propagation velocity of the plasmon wave packet obtained from the time of flight and the path length as a function of $\Delta {{V}_{{\rm g}}}$. When $\Delta {{V}_{{\rm g}}}$ is large, the velocity is of the order of ${{10}^{5}}\;{\rm m}\;{{{\rm s}}^{-1}}$ . The velocity decreases with decreasing $\left| \Delta {{V}_{{\rm g}}} \right|$ and varies almost symmetrically with respect to the CNP ($\Delta {{V}_{{\rm g}}}=0\;{\rm V}$).

The $\Delta {{V}_{{\rm g}}}$ dependence of the plasmon velocity can be explained by a distributed constant circuit model. The response of charge carriers in graphene to the high-frequency electric field is characterized by the impedance ${{Z}_{{\rm graphene}}}=R+i\omega L$, where L is the kinetic inductance arising from the inertia of charge carriers [19]. Since the carrier dynamics depends on n, L can be modified by changing n as $L\propto m/n\propto {{n}^{-1/2}}\propto {{\left| \Delta {{V}_{{\rm g}}} \right|}^{-1}}$ with the effective mass $m\propto {{n}^{1/2}}$ in graphene. Including the capacitive coupling to the metal gate (inset of figure 1(b)), the wave equation for the propagation along the x axis becomes

$$\begin{eqnarray}&&\left( \frac{{{\partial }^{2}}}{\partial {{t}^{2}}}-\frac{1}{LC}\frac{{{\partial }^{2}}}{\partial {{x}^{2}}}+\frac{R}{L}\frac{\partial }{\partial t} \right)V\left( x,t \right)=0,\end{eqnarray} \tag{ 1 }$$

where C is the capacitance to the metal gate and V is the potential induced by excess carriers in graphene. This gives the plasmon dispersion,

$$\begin{eqnarray}&&\omega =\sqrt{\frac{{{k}^{2}}}{LC}-\frac{{{R}^{2}}}{4{{L}^{2}}}}.\end{eqnarray} \tag{ 2 }$$

Using equation (2), we simulate a propagation of a plasmon wave packet (details of the simulation are given in section 4 of the supplementary information). For the simulation, we assumed a Gaussian wave packet with the full width at half maximum of 666 ps. We used $R=340+3.7\times {{10}^{6}}/\left( 22+\Delta V_{{\rm g}}^{2} \right)$ $\Omega$ and $C=2.1\times {{10}^{-4}}\;{\rm F}\;{{{\rm m}}^{-2}}$; R is obtained by a phenomenological Lorenzian fitting to the measured resistance (blue trace in figure 1(b)), while C is determined by the geometry. Then, by setting $L=1.8\times {{10}^{-5}}|\Delta {{V}_{{\rm g}}}{{|}^{-1}}$, the calculated velocity reproduces the experimental result well (black line in figure 1(b)). When $\Delta {{V}_{{\rm g}}}$ is large, $\frac{{{k}^{2}}}{LC}\gg \frac{{{R}^{2}}}{4{{L}^{2}}}$ at a typical wave number of our experimental condition $k=5\times {{10}^{4}}$ ${{m}^{-1}}$(section 5 of the supplementary information). In this regime, the dispersion is almost linear $\omega \sim \frac{k}{\sqrt{LC}}$ and $v\sim \frac{1}{\sqrt{LC}}$ depends on n as $v\propto {{L}^{-1/2}}\propto {{n}^{1/4}}$ (dashed trace in figure 1(b)), consistent with [3, 20]. For $\Delta {{V}_{{\rm g}}}<50$ V, on the other hand, the main component of k in the initial wave packet satisfies $\frac{{{k}^{2}}}{LC}<\frac{{{R}^{2}}}{4{{L}^{2}}}$. In such a regime, charge excitations propagate diffusively rather than as traveling waves and thus the charge velocity is strongly reduced. It should be noted that the value of L used in the simulation is about 100 times larger than the value expected by $L=\frac{\hbar }{4{{v}_{F}}{{e}^{2}}\sqrt{\pi n}}$, where ${{v}_{F}}={{10}^{6}}\;{\rm m}\;{{{\rm s}}^{-1}}$ is the Fermi velocity and e is the electron charge. We suggest that the enlargement of L, or correspondingly the suppression of $v\propto 1/\sqrt{L}$, is due to the screening of the electron charge by the interface state between the graphene and SiC substrate. By the strong but not perfect screening, the effective charge ${{e}_{*}}$ for plasmons is reduced by a factor $C/{{C}_{i}}$, where ${{C}_{i}}$ is the quantum capacitance of the interface state. Using ${{C}_{i}}$ deduced in [18], ${{e}_{*}}$ is estimated to be one order of magnitude smaller than e [21]. This is consistent with the enhancement of $L\propto e_{*}^{-2}$.

An important implication of the distributed RLC circuit model is that the Z obtained from equation (1) also depends on L and thus n as

$$\begin{eqnarray}&&Z=\sqrt{\frac{L}{C}\left( \frac{2L\omega -iR}{2L\omega +iR} \right)}.\end{eqnarray} \tag{ 3 }$$

Z increases with decreasing n: when R is small, it varies as $Z\propto {{L}^{1/2}}\propto {{n}^{-1/4}}$. Since the reflection coefficient between two media increases with the difference in Z of the media, the tunability of Z suggests that it is possible to form a plasmonic waveguide by tailoring n.

Next, we show results for a sample with three parallel top gates (figure 2(a)). The two side gates (guiding gates) and the center gate (channel gate) serve to define the channel for the plasmon transport and change the properties of guided plasmons, respectively. By changing the guiding gate bias $\Delta {{V}_{{\rm gg}}}$ and the channel gate bias $\Delta {{V}_{{\rm cg}}}$ independently, spatial distribution of n can be tuned. The gap between the gates is $10\;\mu {\rm m}$, in which n is fixed at the value for ${{V}_{{\rm g}}}=0\;{\rm V}$. Before demonstrating the guiding effect, we evaluate how the gap between the gates affects the plasmon transport. This can be done by measuring the current at the three detectors while applying the same gate bias to the guiding and channel gates ($\Delta {{V}_{{\rm gg}}}=\Delta {{V}_{{\rm cg}}}$). The results are shown in figures 2(b)–(d). At a constant bias, ${{I}_{{\rm D}3}}$ is largest, while ${{I}_{{\rm D}1}}$ and ${{I}_{{\rm D}2}}$ are almost the same, consistent with the difference in the path length between the injector and each detector. As the bias is decreased, the time position of the current peak for all the detectors shifts to larger delays and the current pulse becomes broad, similar to the behavior observed in the sample with the uniform gate. These results indicate that the gap between the gates hardly affects the plasmon transport. This is reasonable because the gap is much smaller than the typical plasmon wavelength of $100\;\mu {\rm m}$.

When the spatial modulation is induced by applying different biases to the guiding and channel gates, the behavior becomes qualitatively different. Figures 2(e)–(g) show the current at the three detectors as a function of $\Delta {{V}_{{\rm gg}}}$ at a fixed channel gate bias $\Delta {{V}_{{\rm cg}}}=94\;{\rm V}$. When the guiding-gate region is tuned closer to the CNP, the plasmon signal in ${{I}_{{\rm D}2}}\left( t \right)$ becomes larger, while that in ${{I}_{{\rm D}1}}\left( t \right)$ and ${{I}_{{\rm D}3}}\left( t \right)$ almost disappears. This indicates that plasmons are guided in the channel defined by gates from the injector to detector D2. Figure 2(h) shows the plasmon velocity in the channel derived from ${{I}_{{\rm D}2}}\left( t \right)$ (${{v}_{{\rm D}2}}$) and in the guiding-gate region derived from ${{I}_{{\rm D}1}}\left( t \right)$ (${{v}_{{\rm D}1}}$) as a function of $\Delta {{V}_{{\rm gg}}}$. ${{v}_{{\rm D}2}}$ is about $2\times {{10}^{5}}\;{\rm m}\;{{{\rm s}}^{-1}}$ and depends on $\Delta {{V}_{{\rm gg}}}$ only weakly. On the other hand, ${{v}_{{\rm D}1}}$ decreases with decreasing $|\Delta {{V}_{{\rm gg}}}|$, reflecting the n dependence of the velocity in the guiding-gate region. These results demonstrate that Z is modulated spatially and a plasmonic waveguide can be formed by applying different voltages to local gates.

To evaluate the yield of the guiding quantitatively, we calculated the current integral $A=\int _{0}^{80\,{\rm ns}}I\left( t \right){\rm d}t$, which represents the amount of plasmon charge arriving at each detector (figure 2(i)). Then the guiding yield is defined as ${{Y}_{{\rm g}}}={{A}_{{\rm D}2}}/{{A}_{{\rm total}}}$, where ${{A}_{{\rm total}}}$ is the total injected charge; since the sample has bilateral symmetry, ${{I}_{{\rm D}1^{\prime} }}\left( t \right)$ [${{I}_{{\rm D}3^{\prime} }}\left( t \right)$] is expected to be the same as ${{I}_{{\rm D}1}}\left( t \right)$ [${{I}_{{\rm D}3}}\left( t \right)$] and thus ${{A}_{{\rm total}}}={{A}_{{\rm D}2}}+2\left( {{A}_{{\rm D}1}}+{{A}_{{\rm D}3}} \right)$. In a uniform system ($\Delta {{V}_{{\rm gg}}}=\Delta {{V}_{{\rm cg}}}$), the guiding effect is absent and ${{Y}_{g}}$ is almost constant at $\sim 0.15$ (black solid line in figure 2(j)). When the spatial modulation is induced, on the other hand, ${{Y}_{{\rm g}}}$ monotonically increases with decreasing $|\Delta {{V}_{{\rm gg}}}|$ and reaches the maximum value of 0.87 at $\Delta {{V}_{{\rm gg}}}=0$ V (red solid line in figure 2(j)). It is important to note that, although the guiding effect is partially due to the modulation of R through n in the guiding-gate region, the value ${{Y}_{{\rm g}}}=0.87$ is larger than the guiding yield for Dc current [22]. The dashed line in figure 2(j) represents the ratio of the injected and detected currents ${{Y}_{{\rm dc}}}={{I}_{{\rm in}}}/{{I}_{{\rm det} }}$ obtained by a Dc measurement using a sample with the same gate structure (section 1 of the supplementary information). ${{Y}_{{\rm dc}}}$ is smaller than ${{Y}_{{\rm g}}}$, indicating that the modulation of Z plays an essential role in the plasmon guiding.

We carried out similar measurements for several values of $\Delta {{V}_{{\rm cg}}}$ (figure 3). ${{Y}_{{\rm g}}}$ is maximized by setting the guiding-gate region at the CNP and the maximum value increases with $|\Delta {{V}_{{\rm cg}}}|$. The data are almost symmetric with respect to $\Delta {{V}_{{\rm gg}}}=0$ V and $\Delta {{V}_{{\rm cg}}}=0$ V, indicating that the guiding effect does not depend (or at least depends only weakly) on the carrier type. The velocity of guided plasmons, that is, ${{v}_{{\rm D}2}}$ at $\Delta {{V}_{{\rm gg}}}=0$ V, increases with $|\Delta {{V}_{{\rm cg}}}|$. The variation of ${{v}_{{\rm D}2}}$ at $\Delta {{V}_{{\rm gg}}}=0$ V indicates that the velocity of guided plasmons is controllable. Note that, in principle, it is possible to guide plasmons by setting n in the channel smaller than that in the guiding-gate region. However, in such density distribution, R in the channel is large and thus plasmon damping is strong.

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An important advantage of the gate-tunable guiding is that it is possible to change the route of the guiding channel simply by changing local gate biases. We demonstrate the plasmon routing using a sample consisting of a Y-shaped channel defined by etching, a channel gate, and two routing gates covering the branches of the channel (figure 4(a)). The routing gates serve to select the branch for the plasmon transport. In figure 4(b), ${{A}_{{\rm D}1}}$ and ${{A}_{{\rm D}2}}$ are plotted as a function of the D2 routing gate bias $\Delta {{V}_{{\rm rg}2}}$. During the measurement, biases of the channel gate and the other routing gate are fixed at $\Delta {{V}_{{\rm cg}}}=\Delta {{V}_{{\rm rg}1}}=71\;{\rm V}$. When $\Delta {{V}_{{\rm rg}2}}=71\;{\rm V}$ too, ${{A}_{{\rm D}1}}$ and ${{A}_{{\rm D}2}}$ are almost the same. As $|\Delta {{V}_{{\rm rg}2}}|$ is decreased, ${{A}_{{\rm D}2}}$ decreases and, at the same time, ${{A}_{{\rm D}1}}$ increases. This demonstrates the plasmon routing to detector D1. The routing to detector D2 is also possible by changing $\Delta {{V}_{{\rm rg}1}}$ (figure 4(c)). The routing yield, defined as

$$\begin{eqnarray}&&{{Y}_{{\rm r}}}=\frac{\left| {{A}_{{\rm D}1}}-{{A}_{{\rm D}2}} \right|}{{{A}_{{\rm total}}}},\end{eqnarray} \tag{ 4 }$$

reaches 0.94 (figure 4(d)). The plasmon routing is also possible in a square-shaped device using guiding gates (figure 4(e)). By setting $\Delta {{V}_{{\rm gg}}}=0$ V, the Y-shaped channel is defined (figure 4(f)). Then, by sweeping one of the routing gates, the routing is achieved (figures 4(g) and (h)). These results indicate that plasmons can be guided in a bent channel (section 6 of the supplementary information). As the ratio of ${{A}_{{\rm D}1}}$ to ${{A}_{{\rm D}2}}$ can be tuned, the system can also work as a plasmonic switch, splitter, and intensity modulator.