Parity effect of bipolar quantum Hall edge transport around graphene antidots

Parity effect, which means that even-odd property of an integer physical parameter results in an essential difference, ubiquitously appears and enables us to grasp its physical essence as the microscopic mechanism is less significant in coarse graining. Here we report a new parity effect of quantum Hall edge transport in graphene antidot devices with pn junctions (PNJs). We found and experimentally verified that the bipolar quantum Hall edge transport is drastically affected by the parity of the number of PNJs. This parity effect is universal in bipolar quantum Hall edge transport of not only graphene but also massless Dirac electron systems. These results offer a promising way to design electron interferometers in graphene.

(M, N ) = (1, 0) case : For calculation of the conductance, we need to know the transmission probability of the current. We define I l0 and I l1 (I r0 and I r1 ) as the current flowing in the region underneath top gate electrode (region that is not covered by the electrode) as shown in Fig. S1. The sign of the current is defined as the direction of the chirality in the region underneath the top gate electrode. Since we assume the uniform mixing, the relationship between these currents can be written as From these equations, we derive the following relations, They mean that the current difference between two edge states in one region, (I l0 − I l1 ), is equal to the difference between the other region, (I r0 − I r1 ). This property is available to calculate the conductance of the device with several antidots and the junctions. as discussed above, we define the current difference, ∆I 0 , as ∆I 0 ≡ I l0 − I l1 = I r0 − I r1 and we have Since both sides are connected to the source or drain electrodes, 0 = I r0 − ∆I 0 holds. From these two relations, we obtain ∆I 0 described by I l0 , We define I li , I ri (i = 0, 1, 2, 3) shown in this picture.
The transmission probability of the device is given by I r0 /I l0 ; therefore the conductance, G, can be calculated from the Landauer-Büttiker formula as We note that this representation is consistent with the formula reported before [3].
In this case, we can describe From this relation and Eq. 2 with converting i into i − 1, we obtain We define this current difference as ∆I.
). Since Eq. 2 with i = 0 gives the transmission probability, the conductance can be represented as It is easy to show that this representation is valid for both bipolar and unipolar cases; namely the conductance in the bipolar regime is written as and the conductance in the unipolar regime is obtained as case as shown in Fig S4. The relationship between I li and I ri using Eq. 2 is written as Then, using we obtained the recurrence relation of ∆I i as therefore, we have ∆I N = ( m−1 m ) N ∆I 0 . In the bipolar regime, namely, sgn(ν tg ) ̸ = sgn(ν bg ), the chirality of the edge states is opposite between I li and I ri . In this regime, I r0 = 0 holds. I r(N +1) satisfies Additionally, we get From these relations, the transmission probability I r(N +1) /I l0 and hence, the conductance is obtained as This is expressed as In the unipolar regime, sgn(ν tg ) = sgn(ν bg ), we should calculate I r0 as transmitted current into the drain electrode. I r(N +1) equals to 0, which is different from the bipolar regime. We , we can obtain the transmission probability, I r0 /I l0 . Consequently, the conductance is represented as The above calculation using the Landauer-Büttiker formula gives the conductance formulas in all the combinations of the filling factors. We complie these representations in  Table I.

THE UNIPOLAR REGIME
We obtained the conductance representation in the unipolar regime as well. The calculated resistance from the representations complied in Table I  First, we consider the series FPI, which consists of N + 1 mirrors. The reflectivity and transmittance are represented as r and t (≡ 1 − r). The N = 2 case is shown in Fig. S5(a).
Then we define r N and t N as the reflectivity, "Output2" and transmittance "Output1" in The inverse of the transmittance is given as which is calculated from Eq. 4. From Eq. 4 and t 2 = 1−r 1+r , the reflectivity is Next, we discuss the series MZI. This system consists of N + 1 of beam splitters, which split the beam into two paths. The N = 2 case is shown in Fig. S5(b). We consider r N and t N , which are the detected probability in the two ports, for example, the "Output1" and "Output2" shown in Fig.1(d) of the main text. The transmittance of the MZIs consisting of N + 1 mirrors is written as Herein, solving this recurrence relation gives

THE ANTIDOT
We measured the graphene device without an antidot to confirm the quality of the graphene devices and uniform mixing of the QH edge state along the PNJ. The device without an antidot was fabricated by the same method as the device with an antidot. We also experimentally tested the (M, N ) = (2, 0) case. This case, the resistance as a function of V tg and V bg is shown in Fig. S7(a) and its crosssection is also shown in Fig. S7(b).
The observed resistance is consistent with the calculated results shown in Fig. S7(c) as previously reported [3][4][5].
Additionally, we check the (M, N ) = (3, 0) case. Fig. S7(d) represents the resistance as a function of V tg and V bg and its crosssection is shown in Fig. S7(e). The resistance derived from our calculation are in Fig. S7   mixing along the PNJ occurs in our device. This also supports that our graphene devices have enough quality to discuss the chirality control around the graphene antidots.