Bilayer-induced asymmetric quantum Hall effect in epitaxial graphene

Raman spectroscopy is a powerful tool to differentiate between mono- and multilayer graphene [35]. In the case of SiC substrates, most studies concentrate on the so-called 2D peak at around 2700 cm⁻¹ caused by a double resonant scattering process involving two phonons. It is the signal of choice since its frequency lies far away from the excitation structures of the underlying SiC substrate that make the detection of other graphene signatures more difficult.

The intensity, width, and frequency of the 2D peak are sensitive to the number of layers [35]. However, the 2D peak shows also a great sensitivity to strain and charge inhomogeneity: this makes the assessment of layer number not conclusive. The ultimate signature of monolayer graphene is considered to be the line-shape of the 2D peak [36, 37]. In particular, there is general agreement that the peak of monolayer graphene can be fitted with a single Lorentzian, while a decomposition into four Lorentzians is necessary for bilayer graphene [38].

As a first step, we produced a fine Raman-scattering map to determine the layer topography of the entire device. Figure 2(a) shows our result for a step size of 0.5 μm with integration times as long as 10 s for low-noise spectrum detection. As discussed in the following, for our device we were able to detect the number of graphene layers directly by integrating the scattering intensity in suitable spectral ranges. A systematic analysis of the spectra acquired at several points across the device gives us confidence that we can identify two integration ranges corresponding to monolayer and bilayer graphene.

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In order to obtain a Raman map, we first acquired the whole spectrum (1300–2800 cm⁻¹) at each point and corrected it for the SiC contribution by subtracting a reference spectrum of the bare substrate. The resulting spectral data were then integrated over the ranges indicated in figures 2(b–d), where three typical spectra acquired at different positions are shown, each one exhibiting different and peculiar characteristics. At point (b), the shape of the 2D peak is perfectly fitted by a single Lorentzian, thus demonstrating that monolayer graphene is present at this position. At point (c) the peak has a complex shape, and it is significantly red-shifted. Most importantly, the peak can be satisfactorily fitted only by using four Lorentzians, a clear signature of bilayer graphene. Point (d) shows the intermediate situation, where the laser spot covers an area where mostly monolayer graphene is present, but the bilayer contribution cannot be neglected. By exploiting the red-shift occurring for bilayer graphene, we can identify two separate integration ranges 2680–2720 cm⁻¹ (range I) and 2720–2760 cm⁻¹ (range II) for the detection of the number of graphene layers. We carefully checked that red-shifted peaks could be fitted by four Lorentzians only by analyzing the spectra at several locations, and we could assign range I and range II to monolayer and bilayer graphene, respectively. As an example, the points where the spectra in figures 2(b–d) were acquired are shown in figure 2(e), which is a low noise map of integrated intensity in range II, where white is high intensity and dark red is low intensity.

The Raman topography of the whole device was obtained first by calculating the two integrated intensity maps in ranges I and II, and then by combining them into a composite image using RGB channels. The final map is shown in figure 2(a). In the figure, light green (violet) indicates a large integral value in range I (range II) and thus the presence of monolayer (bilayer) graphene. The Hall-bar is intersected by tens of narrow bilayer inclusions. These bilayer inclusions are located at the SiC step edges where they form due to the growth mechanism of graphene on the Si-face via step-edge nucleation and lateral layer-by-layer growth on the terrace [25]. Some of these inclusions are isolated islands, whereas others form long continuous stripes. In particular, some bilayer stripes connect one side of the device to the other.

Magnetoresistance traces of our device are shown in figure 3. The traces of longitudinal (figure 3(a)) and transverse (figure 3(b)) resistance were measured using different contact pairs. For $|B|< 5\;{\rm T}$, the traces of the longitudinal resistance are similar, and both show the typical behavior expected for clean graphene monolayer Hall-bars, comprising a weak localization peak around zero-field, and the developing of magneto-oscillations precursory to the Shubnikov–de Haas oscillations [39]. In the same field range, both traces of transverse resistance R_xy show a monotonic dependence and display kinks at $|B|\approx 3\ {\rm T}$. These kinks coincide with the most pronounced minima in R_xx and correspond to a filling factor $\nu =6$.

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For $|B| > 5\ {\rm T}$, the curves measured using different contact pairs are strongly asymmetric, and deviate significantly from the quantized values expected for homogeneous monolayer graphene in the QH regime. Considering the longitudinal resistance, both ${{R}_{1-3}}$ and ${{R}_{4-6}}$ saturate at a value $\approx 10\ {\rm k\ \Omega\ }$ for $B\gt 0$. For $B\lt 0$, ${{R}_{1-3}}$ saturates at the same value $\approx 10\ {\rm k\ \Omega\ }$, whereas ${{R}_{4-6}}$ displays a vanishing value. Correspondingly, the transverse resistance ${{R}_{3-6}}$ displays a $0.5\times h/{{e}^{2}}$-plateau for both field signs, whereas ${{R}_{1-4}}$ reaches the $0.5\times h/{{e}^{2}}$-plateau only for $B\gt 0$, and ${{R}_{1-4}}\approx -1.6\ {\rm k\ \Omega\ }$ for $B\lt 0$.

A magnetic-field dependence similar to ${{R}_{1-3}}$ was already observed for Hall-bars intersected by bilayer inclusions which are believed to shunt the transport channels at opposite sides of the bar [31, 33, 40]. In these structures, R_xx was insensitive to an inversion of the direction of circulation of the channels, and it thus exhibited the same saturation values regardless of the sign of magnetic field. In contrast to these previous findings, however, here we show that ${{R}_{4-6}}$ has a remarkably asymmetric field-dependence, and is not invariant under inversion of the magnetic field. Analogously, in the transverse direction, a symmetric behavior is displayed only by ${{R}_{3-6}}$, with clear plateaus corresponding to filling factor $\nu =2$, as expected for transverse contact pairs connected by a continuous monolayer region. On the other hand, ${{R}_{1-4}}$ is markedly asymmetric, thus implying the presence of an effect which depends on the chirality of the edge-channels. We note that for $|B| > 5\ {\rm T}$, all magnetoresistance curves are flat, suggesting that their values are pinned to quantized resistance plateaus.

In order to analyze our results, in the following we develop a model based on the information provided by the Raman data shown in figure 2. In particular, we start with the presence of bilayer stripes which connect one side of the device to the other. In agreement with previous publications [31, 33, 40] we assume that additional edge channels are present in the bilayer regions that cause a shunting of the edge states circulating in the monolayer graphene. Our quantitative analysis exploits the Büttiker–Landauer formalism [41], in an approach similar to the one applied to monolayer–bilayer planar junctions on exfoliated graphene [42–45]. For the sake of simplicity we model our device as consisting of monolayer graphene intersected by one bilayer region. Of course in the real device more than one stripe might contribute to the transport, but as we will show, this simple model already explains the main features of our data. The direction of the bilayer stripe is chosen so as to be consistent with the Raman map of figure 2(a). Furthermore, as motivated in the previous paragraph, we assume that both monolayer and bilayer graphene are in a quantized QH state, possibly hosting different numbers of edge states. If we assume that the bilayer stripe connects the two sides of the Hall-bar without being in direct contact with any of the Ohmic contacts, we obtain deviations from the quantized values of monolayer graphene, but the curves are symmetric upon inversion of the direction of the magnetic field. In order to obtain asymmetric curves it turns out to be necessary that the stripe connects an Ohmic contact with a region between two contacts on the other side of the device. Finally, in order to reproduce the observed peculiar asymmetry of the curves in figure 3, it is necessary to impose that the bilayer connection runs between contact 4 and the other side of the device between contacts 1 and 2, as shown in figure 4. Any other configuration fails to reproduce the observed symmetries while simultaneously remaining consistent with the Raman data. The fact that in our device the Ohmic contacts stick out into the Hall-bar makes this configuration more likely than for a Hall-bar with recessed Ohmic contacts and is probably the reason why this asymmetry was not observed before.

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The arrangement considered is depicted in figure 4 for clockwise (CW) channel chirality ($B\gt 0$) and filling factors in the monolayer (${{\nu }_{M}}$) and in the bilayer (${{\nu }_{B}}$) such that ${{\nu }_{B}}\gt {{\nu }_{M}}$. In such a case, the channels exiting contact 1 and the channels from contact 4 undergo a full equilibration while co-propagating along the upper device edge, and emerge at point A at the same potential. By making use of current conservation relations, it is possible to write down the equation at each node, obtaining

$$\begin{eqnarray}&&{{V}_{4-6}}={{V}_{1-3}}=\frac{{{\nu }_{B}}-{{\nu }_{M}}}{2{{\nu }_{B}}-{{\nu }_{M}}}{{V}_{{\rm SD}}}\end{eqnarray} \tag{ 1 }$$

for the longitudinal direction, and

$$\begin{eqnarray}&&{{V}_{1-4}}={{V}_{3-6}}=\frac{{{\nu }_{B}}}{2{{\nu }_{B}}-{{\nu }_{M}}}{{V}_{{\rm SD}}}\end{eqnarray} \tag{ 2 }$$

for the transverse direction. We note that for positive sign of the magnetic field, longitudinal voltages have identical values for opposite sides of the devices, as in the conventional quantum Hall regime. However, their value is considerably larger due to the presence of shunting channels in the bilayer.

For $B\lt 0$, a counter-clockwise (CCW) propagation is expected and the corresponding values for the longitudinal voltage drops are

$$\begin{eqnarray}&&{{V}_{4-6}}=0,\;\;\;{{V}_{1-3}}=\frac{{{\nu }_{B}}-{{\nu }_{M}}}{2{{\nu }_{B}}-{{\nu }_{M}}}{{V}_{{\rm SD}}},\end{eqnarray} \tag{ 3 }$$

while in the transverse direction we obtain

$$\begin{eqnarray}&&{{V}_{1-4}}=-\frac{{{\nu }_{M}}}{2{{\nu }_{B}}-{{\nu }_{M}}}{{V}_{{\rm SD}}},\ \ \ {{V}_{3-6}}=-\frac{{{\nu }_{B}}}{2{{\nu }_{B}}-{{\nu }_{M}}}{{V}_{{\rm SD}}}.\end{eqnarray} \tag{ 4 }$$

The values of the resistance, calculated using the source–drain current ${{I}_{{\rm SD}}}={{\nu }_{M}}{{\nu }_{B}}{{V}_{{\rm SD}}}/(2{{\nu }_{B}}-{{\nu }_{M}})$, are summarized in table 1 in units of $h/{{e}^{2}}$. In the following we shall focus on the case $|B| > 5\ {\rm T}$, in which ${{\nu }_{M}}=2$ channels propagate along the device, as inferred from the high-field values of ${{R}_{3-6}}$. By setting ${{\nu }_{M}}=2$, and using the measured value ${{R}_{1-3}}{{|}_{-7{\rm T}}}=0.393\times h/{{e}^{2}}$, we obtain the filling factor in the bilayer region ${{\nu }_{B}}=9.33$. The nearest Hall plateau for bilayer graphene is at ${{\nu }_{B}}=8$ [46]. We can estimate the variation in our data by considering the difference in ${{R}_{4-6}}$ and ${{R}_{1-3}}$ at $B=\;+7{\rm T}$, which should be identical according to our model (see table 1). We obtain ${{R}_{1-3}}{{|}_{+7{\rm T}}}=0.397\times h/{{e}^{2}}$ and ${{R}_{4-6}}{{|}_{+7{\rm T}}}=0.346\times h/{{e}^{2}}$, i.e., a variation in R of $0.051\times h/{{e}^{2}}$. This slight difference in the traces measured with different contacts is presumably caused by an inhomogeneity of the charge density and a mixing between resistance components due to geometrical effects. Using the filling factor values ${{\nu }_{M}}=2$ and ${{\nu }_{B}}=8$, we calculate a transverse resistance value ${{R}_{1-4}}=-0.125\times h/{{e}^{2}}$, which is roughly consistent with the measured value ${{R}_{1-4}}{{|}_{-7{\rm T}}}=-0.064\times h/{{e}^{2}}$ within the variation. Using the formulae given in table 1, we have calculated all quantum Hall resistance values both for ${{\nu }_{M}}=2$ and ${{\nu }_{B}}=8$ as well as for ${{\nu }_{M}}=6$ and ${{\nu }_{B}}=24$ and included them as thin horizontal lines in figure 3. The excellent agreement between experiment and model further strengthens our interpretation. In summary, this simple model explains the main features of our data. In particular, it fully accounts for the peculiar asymmetry of the magnetoresistance curves. Our results imply the coexistence of QH states on both monolayer and bilayer graphene grown on the same substrate.

Table 1.  Calculated quantum Hall resistances (in units of $h/{{e}^{2}}$) obtained from the model in figure 4.

${{R}_{1-3}}$${{R}_{4-6}}$${{R}_{1-4}}$${{R}_{3-6}}$

B $\gt$ 0 (CW)	$\frac{{{\nu }_{B}}-{{\nu }_{M}}}{{{\nu }_{B}}{{\nu }_{M}}}$	$\frac{{{\nu }_{B}}-{{\nu }_{M}}}{{{\nu }_{B}}{{\nu }_{M}}}$	$\frac{1}{{{\nu }_{M}}}$	$\frac{1}{{{\nu }_{M}}}$
B $\lt$ 0 (CCW)	$\frac{{{\nu }_{B}}-{{\nu }_{M}}}{{{\nu }_{B}}{{\nu }_{M}}}$	0	$-\frac{1}{{{\nu }_{B}}}$	$-\frac{1}{{{\nu }_{M}}}$

This topic was addressed in a very recent work [40], where an electrostatic model defining the domain of coexistence of QH in both monolayer graphene and bilayer inclusions was proposed. It was stated that QH conditions in mono- and bilayer regions at large carrier density are not expected to be simultaneously present, so that the bilayer inclusions act as dissipative shunts when the monolayer graphene is pinned at a certain filling factor. However, we argue that if monolayer and bilayer regions have different carrier densities, it is possible to have the conditions for dissipationless transport in both graphene domains. A density of $3.4\times {{10}^{11}}$ cm⁻² in monolayer graphene and ${{\nu }_{M}}=2$ would require a 4 times higher carrier density in bilayer graphene to reach ${{\nu }_{B}}=8$ at the same magnetic field B, i.e, a carrier density of $1.4\times {{10}^{12}}$ cm⁻². These values are in excellent agreement with values reported in literature for similar mono- and bilayer samples [47].