Probing Chirality of the Quantum Hall Effect via the Landauer-Büttiker Formalism with Two Current Sources

1. Introduction

The quantum Hall effect (QHE) is characterized by the precise quantization of Hall resistance and the vanishing of longitudinal resistance in the presence of a strong perpendicular magnetic field [1]. Depending on the system, the quantized values can correspond to integer [2], half-integer [3], or fractional multiples of the fundamental resistance quantum [4]. This quantization arises from the formation of topologically non-trivial insulating bulk states, which give rise to robust and dissipationless edge channels [5,6,7,8]. These topological features are believed to ensure the robustness and precision of quantized transport [9,10] and serve as the foundation for related phenomena such as the quantum anomalous Hall effect [11] and the quantum spin Hall effect [12], which can occur even in the absence of an external magnetic field. The exactness of quantized Hall resistance has enabled its use in applications such as quantum resistance standards and electrical metrology [9,10].

One of the defining features of QHE is the presence of chiral edge currents coexisting with an insulating bulk [1]. Classically, this behavior can be understood as arising from skipping cyclotron orbits along the sample boundaries [13]. Quantum mechanically, it originates from the formation of discrete Landau levels and their bending near the sample edge [13]. From a topological perspective, edge states arise from the bulk–edge correspondence [5,6,7,8]. The existence of chiral edge states and an insulating interior has been investigated through local probe techniques [14,15] and noise measurements [16,17]. This unique transport phenomenon has motivated the exploration of the QHE in a variety of geometries, including three-dimensional structures [18], series and parallel connections [19], and anti-Hall bar or Corbino type configurations [20,21,22,23,24,25,26]. The QHE remains at research frontier, with topics including the incompressible nature of the bulk [27], the appearance of hot spots [28], and the intriguing behavior of snake states [29].

In this work, we investigate the chirality of edge states in QHE using two current source (TCS) excitations in a standard Hall bar geometry. The use of TCS enables a variety of excitation and measurement configurations, offering new approaches to probing the chirality of edge currents generated by each source. All measurements across different configurations are analyzed using the Landauer-Büttiker formalism (LBF) [30,31]. The results are consistent with the presence of chiral edge currents and a bulk that remains incompressible: the total current contributed by each source flows in the same direction regardless of the configuration. This behavior contrasts sharply with that of conventional diffusive conductors, where oppositely directed currents would be expected to partially or fully cancel. The TCS-based method thus provides compelling additional evidence for the chiral nature of edge states in the QHE.

2. Materials and Methods

For this study, we used monolayer epitaxial graphene (epigraphene) grown on silicon carbide substrates as the two-dimensional channel [9,10]. The epigraphene was patterned into a Hall bar geometry (Figure 1(a), width = 30 µm, length = 180 µm) using electron-beam lithography and was doped near the charge neutrality point (~5 × 10¹⁰ electrons/cm²) through a molecular doping method (See Figure 1(f)) [32]. This low carrier density enables high mobility, resulting in a robust QHE. After doping, the sample exhibited a carrier mobility of 32,000 cm²/V·s and a sheet resistance of 4 kΩ/□ at a temperature of 2 K. A perpendicular magnetic field was applied to the sample surface to induce a QH state with a filling factor of ν = 2, corresponding to the first plateau in the half-integer QHE of epigraphene [3]. All measurements were performed at T = 2 K using two Keithley current sources and nanovoltmeters.

Before performing the main measurements, we evaluated the contact resistances of all eight terminals on the sample using the three-terminal method under quantum Hall conditions [33]. In the three-terminal method, contact resistance is estimated by measuring the longitudinal resistance with one of the voltage probes placed on the drain contact. Then the measured resistance is the sum of the contact resistance of the drain contact and the lead resistance. All contact resistances were found to be below 10 Ω, except for contact 2 (Rc = 60 Ω), contact 7 (Rc = 60 Ω), and contact 8 (Rc = 40 kΩ) (see Figure 1 for contact numbering). The carrier density extracted from the Hall voltage measured between terminals 2 and 8 was 3.5 × 10¹⁰ electrons/cm², which deviates from the values obtained using other voltage probes (e.g., terminals 3–7 and 4–6), where the carrier density was approximately 5 × 10¹⁰ electrons/cm². This discrepancy could be related to the high contact resistance of terminal 8. For the carrier density measurements, current was sourced through contacts 1 and 5. The robustness of the QHE in our sample is further confirmed by the observation of a critical current as high as 5 µA at a magnetic field of 2 T and a temperature of 2 K.

3. Results

Figure 1(a) shows a schematic of the TCS configuration, where the current source A and B supply currents $I_a$ and $I_b$, which are directed horizontally and vertically, respectively. The voltage measured between terminals 8 and 6 when both current sources are active ($I_a=I_{15}$ , $I_b=I_{37}$) is denoted by $V_{\mathrm{15,37,86}}$. Due to the chirality of the edge currents in the QH regime, we expect that the principle of superposition to hold in the TCS configuration: the voltage when both current sources are on should equal the sum of the voltages measured when each source is applied individually. To test this, we compare $V_{\mathrm{15,37,86}}$ with $V_{\mathrm{15,86}}=I_{15}{R}_{\mathrm{15,86}}$ and $V_{\mathrm{37,86}}=I_{37}{R}_{\mathrm{37,86}}$, where each is measured with the other current source turned off. As shown in Figure 1(b) we find that $V_{\mathrm{15,37,86}}=V_{\mathrm{15,86}}+V_{\mathrm{37,86}}$, consistent with the quantized Hall resistance $R=h/{2e}^2$ or zero in the QH state. We further verify this superposition behavior by fixing $I_{15}$ = 100 nA and varying $I_{37}$ as 100 nA, 200 nA, 300 nA. The resulting voltage $V_{\mathrm{15,37,86}}$ changes accordingly, with $V_{\mathrm{15,86}}$ remaining constant and $V_{\mathrm{37,86}}$ varying linearly, as expected (Figure 1(c,d)). The superposition principle is also confirmed at another voltage terminal pair: $V_{\mathrm{15,37,28}}=V_{\mathrm{15,28}}+V_{\mathrm{37,28}}$, where $V_{\mathrm{15,28}}=I_{15}\bullet (h/2e^2)$ in the QH regime with $\nu =2$ as shown in Figure 1(e).

We explain the linear superposition behavior observed in the QHE with TCS as shown in Figure 1 using the LBF. In the QH regime at filling factor $\nu =2$, the LBF accounts for chiral edge channels that contribute a quantum conductance of $2e^2/h$, while the bulk remains incompressible. This quantization $2e^2/h$ arises from the half integer QH effect in monolayer graphene, where the Hall conductivity is given by ${\sigma }_{xy}=g(N+{\displaystyle \frac{1}{2}})e^2/h$ with degeneracy $g=4$ reflecting the spin and valley degeneracy, and the zeroth Landau level ($N=0$) [3]. Within the LBF framework, the current and voltage at each terminal can be expressed as:

$I_i=(2e^2/h)\sum _j\left(T_{ji}{V}_i-{T_{ij}V}_j\right)=(2e^2/h)\left(V_i-V_{i-1}\right)$ (Eq. 1)

Here, $I_i$denotes the current flowing out of $i$th terminal into the sample, $V_i$ is the voltage at the $i$th terminal, and $T_{ji}$ is the transmission probability for an electron injected from terminal $i$ to reach terminal $j$. Charge conservation requires $\sum _i{I}_i=0$. For dissipationless chiral transport in the QH regime, the transmission probabilities are defined as $T_{ji}=1$ if $j=i+1$ and $T_{ji}=0$ otherwise [30,31]. When both current sources on, LBF allows linear superposition:

${I_i}^{\left(a,b\right)}={I_i}^{\left(a\right)}+{I_i}^{\left(b\right)}$, ${V_i}^{\left(a,b\right)}={V_i}^{\left(a\right)}+{V_i}^{\left(b\right)}$, (Eq. 2)

where ${I_i}^{\left(a,b\right)}$ and ${V_i}^{\left(a,b\right)}$are the total current and voltage at terminal $i$ with both sources are on, and ${I_i}^{\left(a\right)}$, ${V_i}^{\left(a\right)}$ (respectively ${I_i}^{\left(b\right)}$, ${V_i}^{\left(b\right)}$) are those when only source $a$ (respectively $b$) is on. In the specific configuration shown in Figure 1, where $I_a=I_{15}$ and ${I_b=I}_{37}$, the LBF gives:

${I_1}^{\left(a\right)}=I_a=-{I_5}^{\left(a\right)}$, ${I_3}^{\left(b\right)}=I_b=-{I_7}^{\left(b\right)}$,

with zero currents at all other terminals. For the voltages, we find:

${V_i}^{\left(a\right)}={V_1}^{\left(a\right)}=V_a$ for $i=\mathrm{1,2},\mathrm{3,4}$, ${V_i}^{\left(b\right)}={V_3}^{\left(b\right)}=V_b$ for $i=\mathrm{3,4},\mathrm{5,6}$

and zero for other terminals. From this, we obtain:

${V_8}^{\left(a,b\right)}-{V_6}^{\left(a,b\right)}=-{V_3}^{\left(b\right)}=-V_b=-(h/2e^2)I_b$, ${V_2}^{\left(a,b\right)}-{V_8}^{\left(a,b\right)}={V_1}^{\left(a\right)}=V_a=(h/2e^2)I_a$. Each voltage drop is determined solely by the corresponding current source. These results are in excellent agreement with the experimental observations in Figure 1(b-e).

We investigate additional configurations and conclude that QHE with TCS consistently follows the additive nature of the LBF, regardless of the configuration. In Figure 2, the positions of the current sources are fixed as in Figure 1, but the voltage is measured between terminal 2 and 6. In this set up, both $V_{\mathrm{15,26}}$ and $V_{\mathrm{37,26}}$ contain contributions from both longitudinal and transverse conductance arising from current $I_a$ and $I_b$, respectively. According to the LBF, the superposition of contributions from both sources predicts:

${V_2}^{\left(a,b\right)}-{V_6}^{\left(a,b\right)}=V_a-V_b=(h/2e^2)(I_a-I_b)$,

indicating that both currents contribute to the Hall voltage. This prediction is verified in Figure 2. In Figure 2(b), the Hall voltage is completely canceled when $I_a=I_b$, consistent with the expected result. Figure 2(c) and 2(d) demonstrate that the Hall voltages can be tuned and made asymmetric by varying the relative magnitudes of the two current sources.

In Figure 3, we explore configurations where the two current sources $I_a$ and $I_b$ are applied parallel (Figure 3(a)) and antiparallel (Figure 3(d)) directions, such that the current either add or cancel. According to the LBF, for the parallel configuration in Figure 3(a), the currents are distributed as:

${I_1}^{\left(a\right)}=I_a=-{I_5}^{\left(a\right)}$, ${I_2}^{\left(b\right)}=I_b=-{I_4}^{\left(b\right)}$,

with zero currents at all other terminals. The corresponding voltages are:

${V_i}^{\left(a\right)}={V_1}^{\left(a\right)}=V_a$ for $i=\mathrm{1,2},\mathrm{3,4}$, and ${V_i}^{\left(b\right)}={V_2}^{\left(b\right)}=V_b$ for $i=\mathrm{2,3}$

; the other terminals have zero voltage. In this configuration, the expected voltage differences are:

${V_8}^{\left(a,b\right)}-{V_6}^{\left(a,b\right)}=0$, ${V_3}^{\left(a,b\right)}-{V_7}^{\left(a,b\right)}={V_a+V}_b=(h/2e^2)(I_a+I_b)$.

The experimental results in Figure 3(b) and (c) confirm these predictions.

In the antiparallel configuration shown in Figure 3d, only the direction of current for source B is reversed.

${I_4}^{\left(b\right)}=I_b=-{I_2}^{\left(b\right)}$ and ${V_i}^{\left(b\right)}={V_2}^{\left(b\right)}=V_b$ for $i=\mathrm{4,5},\mathrm{6,7},\mathrm{8,1}$

; voltages at the other terminals remain zero. The resulting voltage differences are:

${V_8}^{\left(a,b\right)}-{V_6}^{\left(a,b\right)}=V_b-V_b=0$ and ${V_3}^{\left(a,b\right)}-{V_7}^{\left(a,b\right)}={V_a-V}_b=(h/2e^2)(I_a-I_b)$

, which are in excellent agreement with the experimental data shown in Figure 3(e) and (f). These results clearly demonstrate that even when $I_b$ is applied from terminal 4 to terminal 2 in the antiparallel configuration, the current does not flow directly from the short path 4→3→2. Instead, due to the chirality of the edge states, it follows a longer clockwise path: 4→5→6→7→8→1→2.

We further examine a configuration in which the two current sources share a common drain contact, as shown in Figure 4. According to the LBF, the current distribution in this setup is ${I_1}^{\left(a\right)}=I_a=-{I_4}^{\left(a\right)}$, ${I_8}^{\left(b\right)}=I_b=-{I_4}^{\left(b\right)}$, with zero currents at all other terminals. The corresponding voltages are ${V_i}^{\left(a\right)}={V_1}^{\left(a\right)}=V_a$ for terminals $i=\mathrm{1,2},3$ and ${V_i}^{\left(b\right)}={V_8}^{\left(b\right)}=V_b$ for $i=\mathrm{8,1},\mathrm{2,3}$. All other terminals have zero voltages. From this, we find the following voltage differences:

${V_2}^{\left(a,b\right)}-{V_3}^{\left(a,b\right)}=(V_a+V_b)-(V_a+V_b)=0$,

${V_2}^{\left(a,b\right)}-{V_7}^{\left(a,b\right)}=V_a+V_b=(h/2e^2)(I_a+I_b)$.

These predictions are in excellent agreement with the experimental results shown in Figure 4(b) and Figure 4(c).

4. Discussion

We explored additional current source configurations and confirmed that the LBF consistently describes transport behavior in all cases. The chiral edge currents along the sample boundary are schematically illustrated in Figure 5 for a configuration where ${\mathit{I}}_{\mathit{a}}$ and ${\mathit{I}}_{\mathit{b}}$ are nominally applied in opposite directions. Unlike in conventional conductors, where opposing currents may cancel each other, the zero quantum Hall voltages observed in Figure 3(e) and 3(f) do not result from vanishing net current. Instead, they arise from the specific voltage distribution dictated by chiral edge transport, as described by Eq. (1).

5. Conclusions

We have experimentally demonstrated the linear superposition principle of QH transport under TCS excitations in a conventional Hall bar geometry. Our device is based on epitaxial monolayer graphene, doped near the charge neutrality point to enhance mobility and stabilize the QH state. The observed superposition reveals a combinatorial transport behavior in which both longitudinal and transverse QH resistances coexist in a tunable way. This TCS-based approach provides a powerful and flexible method for probing chirality and edge transport in QH systems.