Abstract
Parity effect, which means that evenodd 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.
Introduction
Various parity effects, which mean that evenodd property of an integer physical parameter results in an essential difference, has been discovered in many physical systems, especially in mesoscopic physics. For example, the significance of the parity effect regarding the number of electrons is seen in the micronsized superconducting island^{1}, in the quantum dot in the Kondo regime^{2,3} and so on^{4,5,6}. In case of quantum Hall (QH) edge transport, the electron spins are polarized in the edge state for odd filling factors, while they are unpolarized with even filling factors. This parity effect leads to polarizing nuclear spins^{7,8} and reading spin state in quantum dot^{9,10}. In this way, the parity effects have played important roles on developing the research area.
Since graphene, which is monolayer film of graphite, was first isolated^{11,12}, this wonder material has been studied extensively^{13}. Especially, as electrons in graphene follow massless Dirac equation, the graphene is a nice platform to address universal features of the massless Dirac electron system, as is the case in the surface state of topological insulators^{14,15}. In such systems, we can study the QH edge transport in bipolar regime, where the PNJs are formed as reported previously^{16,17}. The QH edge states copropagate along the PNJ and are uniformly mixed, reflected by a massless Dirac electron nature that a Landau level is formed at charge neutrality point. This bipolar QH edge transport is not realized on conventional twodimensional electron gas in GaAs/AlGaAs heterostructure and there are only several experimental reports about the bipolar QH edge transport in graphene^{17,18,19,20,21,22,23} despite great interest.
Here we report a new parity effect of QH edge transport in graphene antidot devices with several pn junctions (PNJs). We study a graphene device with two types of regions in which the carrier types are n (electrons) and p (holes) which are arranged in series. The number of such regions is M + 1; therefore, the device has M PNJs. Furthermore, we place N antidots on all of the PNJs. The (M, N) = (2, 1) (even case) and (3, 1) (odd case) are schematically illustrated in Fig. 1(a,b), respectively. We calculated the conductance of graphene devices with such structures and discovered a significant parity effect of the resistance as a function of N, depending on parity of M. We implemented the proofofconcept experiments and established the parity effect using a graphene device, whose optical picture is shown in Fig. 1(c). Our finding is a universal feature of bipolar QH edge transport in massless Dirac electron systems.
We first present parity effect found in the calculated results of the twoterminal resistance. We define ν_{tg} and ν_{bg} as the Landau level filling factors of the two types of regions tuned by both the top and backgate voltages and by only the backgate voltage, respectively. When we calculate the resistance, we obtain the twoterminal conductance using the LandauerBüttiker formula^{24} and then obtain the resistance as the inverse of the conductance. In this paper, we only consider the bipolar regime, sgn(ν_{tg}) ≠ sgn(ν_{bg}), which is unique to graphene (see the Supplemental Material for the unipolar regime). Our calculation treated here does not include possible interference effects.
We discuss the conductance in the even M case. We consider the filling factor in the regions connected to the source and drain electrodes, denoted by ν_{tg}. We calculate the twoterminal conductance G as a function of M, N and when the filling factors ν_{tg} and ν_{bg} are ±2, ±6, . The result is
The resistance is obtained as R = G^{−1}. As observed from this formula, the conductance increases as N increases. Interestingly, when N increases to infinity, the conductance becomes ν_{tg}e^{2}/h, namely, the conductance with no PNJs. Note that this representation is not symmetric with regard to the exchange of ν_{bg} and ν_{tg}. Consequently, this means that the chirality of the QH edge states at the left and right sides are the same (see Fig. 1(a) for the case (M, N) = (2, 1).).
Next, we discuss the odd M case. In this case, we obtain the conductance as
The resistance is obtained as R = G^{−1}. The conductance increases as N increases, as found in the even M case. However, the conductance is quantitatively different from that in the even M case (Eq. (1)) in several aspects. First, the conductance dependence on N in the odd M case in Eq. (2) is expressed as a power of N, which is very different from the even M case in Eq. (1). In addition, we can obtain a peculiar result from Eq. (2), ν_{bg}ν_{tg}/(ν_{bg} + ν_{tg}) · e^{2}/h, as N approaches infinity. This conductance is the same as that observed in graphene with (M, N) = (1, 0). Furthermore, Eq. (2) is symmetric with regard to the exchange of ν_{bg} and ν_{tg}. The physical laws possess charge, parity and time (CPT) reversal symmetry. From parity (mirror) symmetry, the conductance in odd M case is demanded to be symmetric about the exchange of ν_{bg} and ν_{tg}, different from even M case. These unique properties in the bipolar regime originate from the difference in the chirality at the left and right (see the case (M, N) = (3, 1) in Fig. 1(b)). In particular, we note that Eq. (2) has a powerlaw dependence on the number of antidots N, which is only realised in bipolar QH edge transport.
These calculated results mean that the resistance formulas as a function of the number of antidots is strongly dependent on the parity of M, the number of PNJs. Such fundamental difference, the parity effect, should be useful for not only graphene but also QH devices of massless Dirac electron systems, such as the surface state of topological insulators^{14,15} and Dirac electrons in HgTe/CdTe heterostructure^{25}. This parity effect is because of the chirality difference of the QH edge states around the antidots, which will be discussed later using an analogy to optical interferometers.
We implemented the following proofofconcept experiments. While there are several reports regarding the transport in graphene PNJs in the QH regime^{18,19,20,21,22,23,26,27}, there are no reports on devices with antidots and PNJs in the QH regime. We fabricated two types of graphene devices; one has an antidot and the other has no antidot. In this article, we mainly discuss the results of the device with the antidot. We indicate the results of the device with no antidots in Supplemental Material. These devices were cleaved from a highly ordered pyrolytic graphite crystal via Scotch tape and transferred onto a Si substrate covered with 285nmthick SiO_{2}. We estimated the thickness of graphene from an analysis of contrast in the optical pictures. Then, we etched the graphene samples using an oxygen plasma and created an antidot in graphene, as observed in the inset of Fig. 1(a). Subsequently, we pattered the electrodes using electronbeam lithography (EBL). Then, we deposited 5 nm of palladium and 30 nm of gold as the source and drain electrodes. Crosslinked PMMA was patterned by EBL on graphene as the insulating layer following the same method reported previously^{28,29}. Finally, we placed two topgate electrodes (3 nm of chromium and 30 nm of gold) on the insulating layer. We measured the twoterminal resistance of this graphene device at 2 K and 7 T. The graphene device, whose optical picture is shown in Fig. 1(c), has an antidot and two topgate electrodes marked α and β in the inset of Fig. 1(c). We discuss the device for (M, N) = (1, 1),(2, 1) and (3, 1) by controlling the number of PNJs.
We begin with the (M, N) = (1, 1) case. We applied a topgate voltage to only the electrode marked as α in Fig. 1(c) and measured the twoterminal resistance. Changing the top and backgate voltages corresponds to tuning ν_{bg} and ν_{tg}. Figure 1(d) shows the device resistance as a function of the topgate voltage V_{tg} and backgate voltage V_{bg} at 2 K and 7 T. Cross sections of the image plot at V_{bg} = 15, 6, −4 and −13 V are shown in Fig. 1(e). The bipolar regime corresponds to the area in Fig. 1(d) surrounded by the green line marked as ‘pn’ and ‘np’. Several plateaus at which the resistance is almost constant exist in Fig. 1(d,e). We can decide the filling factors, ν_{tg} and ν_{bg}, in bipolar regime from the resistance in unipolar regime. As shown in Fig. 1(d,e), for example, the device exhibits a resistance plateau of h/e^{2} in (ν_{tg}, ν_{bg}) = (±2, ), which is consistent with the calculated resistance from Eq. (2) with (M, N) = (1, 1) shown in Fig. 1(f). The resistances on the other plateaus of (ν_{tg}, ν_{bg}) are also consistent with our calculated results. This calculated result for (M, N) = (1, 1) is consistent with the resistance for the (M, N) = (1, 0) case as reported before^{17}. The QH edge transport is not affected by the existence of the antidot in M = 1 case, so that we increase the PNJs to confirm the validity of Eq. (2).
We now discuss the device for the (M, N) = (2, 1) case (even case), where pnp and npn systems are realised for the condition of sgn(ν_{tg}) ≠ sgn(ν_{bg}). Experimentally, we applied the topgate voltage β in Fig. 1(c). In Fig. 2(a), we plotted the resistance as a function of V_{tg} and V_{bg} in units of h/e^{2}. The image plot is not symmetric with regard to the exchange of the vertical and horizontal axes. This property means that the conductance should be nonsymmetric about the exchange of ν_{tg} and ν_{bg}. Furthermore, we present the cross sections at V_{bg} = 15, 6, −4 and −13 V of Fig. 2(a) in Fig. 2(b). The resistance is different from the results for graphene pnp junctions without the antidot^{18,19,26} (see the Supplemental Material). For example, the largest resistance in our device is close to h/e^{2} in (ν_{tg}, ν_{bg}) = (±2, ) case, which is different from the predicted value of 3h/2e^{2} for graphene without the antidot. The calculated results for (M, N) = (2, 1) case in Fig. 2(c) not only in (ν_{tg}, ν_{bg}) = (±2, ) case but also the other plateaus are in good agreement with these experimental results in Fig. 2(a). This agreement means that Eq. (1) can well explain the QH edge transport with even M case on graphene antidots. Additionally, these experimental results in the unipolar regime are also consistent with the calculated results shown in the Supplemental Material.
Finally, we show the resistance for the (M, N) = (3, 1) case (odd case). We applied the same topgate voltage to the two topgate electrodes, α and β, in Fig. 1(a). Figure 3(a) shows the observed resistance dependence on V_{bg} and V_{tg}. The image plot is symmetric with regard to the exchange of the vertical and horizontal axes; therefore, the conductance should be symmetric with regard to the exchange of ν_{tg} and ν_{bg}. Figure 3(b) shows the cross sections at V_{tg} = 15, 6, −4 and −13 V of Fig. 3(a). These results also indicate a different resistance compared to the results of the device without an antidot (see Supplemental Material). For example, the resistance in (ν_{tg}, ν_{bg}) = (±2, ) case is equal to 4h/3e^{2}, which is different from the expected resistance, 2h/e^{2} without the antidot. Our calculated results in Fig. 3(c) demonstrate good consistency and provide an explanation of the experimental results. This experimental consistency validates the formula derived in Eq. (2) which explains the edge transport of graphene with an antidot and the three PNJs. Herein, our experimental study reveals the existence of the parity effect in the QH edge transport in graphene antidot devices in bipolar regime.
The confirmed parity effect in graphene antidot devices has a good analogy to optical systems, although the interference effect is not explicitly taken into account here. The conductance in the even case can be readily understood by comparing the FabryPerot interferometers (FPIs), which are constructed with two parallel beam splitters with reflectivity r > 0. The conductance dependence on N is consistent with the reflectivity, (N + 1)/((N + 1) + (1–r)/r), of FPIs arranged in series (not including the interference effect), which are constructed with N + 1 parallel beam splitters arranged in series (see the Supplemental Material). A schematic picture for the N = 1 case is shown in Fig. 4(b). Utilizing this analogy, we can regard PNJs between antidots as a beam splitter with r = ν_{bg}/((M/2 + 1)ν_{bg} + M/2ν_{tg}) = G(M, 0)h/ν_{tg}e^{2} and able to calculate the conductance representation without explicitly using the Landauer–Büttiker formula. In the graphene device, the edge states at the left (or right) side marked as ‘Input’, ‘Output1’ and ‘Output2’ in Fig. 4(a) correspond to the passing (or reflected) beam between the beam splitters marked as ‘Input’, ‘Output1’ and ‘Output2’ in Fig. 4(b).
In the odd case, we adopted MachZender interferometers (MZIs) arranged in series, where two beams passing through a beam splitter of the MZI enter into the next beam splitter from different sides. This dependence with the condition of ν_{tg} = ν_{bg} can be viewed as the probability obtained at one port, (1–(1–2t)^{N + 1})/2, of N (see the Supplemental Material). N + 1 and t indicate the number of the beam splitters in the MZIs and the transmittance of the beam splitter, respectively. In condition with ν_{tg} = ν_{bg}, t = 1/(M + 1) in (1–(1–2t)^{N + 1})/2 gives G(M, N)h/ν_{tg}e^{2}. A schematic picture of the N = 1 case is shown in Fig. 4(d). In this case, we can regard the edge states at the left and right, ‘Input’, ‘Output1’ and ‘Output2’ in Fig. 4(c) as the beams passing through the beam splitter, ‘Input’, ‘Output1’ and ‘Output2’ in Fig. 4(d).
As we discuss above, there are strong analogy between bipolar QH edge transport in graphene antidot device and optical interferometer. Thus, we believe that this system has a possibility to be a device for investigating interference effect of QH edge state in graphene. We ignored the interference effect between the edge states in our calculation and did not detect any interference pattern experimentally. This is because our device has not enough mobility so that the coherence length in the QH edge state is short. The higher quality graphene devices, such as graphene on hBN substrate or suspended graphene will enable to measure the interference pattern.
In conclusion, we have studied the bipolar QH edge transport phenomena around graphene antidots with several number of PNJs. We derived the formulas to calculate the conductance of graphene with the antidots and PNJs in the QH regime. We experimentally confirmed these formulas with devices having one, two and three PNJs. Consequently, we have established the parity effect that the conductance dependence on the number of antidots relies on the evenodd parity of the number of the PNJs. In particular, the odd M case is a unique transport regime realised on graphene antidot devices or other QH devices of massless Dirac electron systems. Furthermore, we suggest that this difference characterised by the parity can be understood by considering the analogy between the QH edge transport and the optical interferometers. Our results may be available to construct electron interferometer in graphene.
Additional Information
How to cite this article: Matsuo, S. et al. Parity effect of bipolar quantum Hall edge transport around graphene antidots. Sci. Rep. 5, 11723; doi: 10.1038/srep11723 (2015).
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03 August 2015
Parity effect, which means that evenodd 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.
References
Lafarge, P., Joyez, P., Esteve, D., Urbina, C. & Devoret, M. H. Nature 365, 422 (1993).
GoldhaberGordon, D. et al. Nature 391, 156 (1998).
Cronenwett, S. M., Oosterkamp, T. H. & Kouwenhoven, L. P. Science 281, 540 (1998).
Van Dam, J. A., Nazarov, Y. V., Bakkers, E. P. A. M., De Franceschi, S. & Kouwen hoven, L. P. Nature 442, 667 (2006).
Matveev, K. A. & Larkin, A. I. Phys. Rev. Lett. 78, 3749 (1997).
Guo, Y. et al. Science 306, 1915 (2004).
Wald, K. R., Kouwenhoven, L. P., McEuen, P. L., van der Vaart, N. C. & Foxon, C. T. Phys. Rev. Lett. 73, 1011 (1994).
Kawamura, M. et al. Appl. Phys. Lett. 90, 022102 (2007).
Kiyama, H., Fujita, T., Teraoka, S., Oiwa, A. & Tarucha, S. Appl. Phys. Lett. 104, 263101 (2014).
Otsuka, T., Abe, E., Iye, Y. & Katsumoto, S. Phys. Rev. B 79, 195313 (2009).
Novoselov, K. S. et al. Nature 438, 197 (2005).
Zhang, Y., Tan, Y.W., Stormer, H. L. & Kim, P. Nature 438, 201 (2005).
Geim, A. K. & Novoselov, K. S. Nat. Mater. 6, 183 (2007).
Fu, L., Kane, C. L. & Mele, E. J. Phys. Rev. Lett. 98, 106803 (2007).
Hasan, M. Z. & Kane, C. L. Rev. Mod. Phys. 82, 3045 (2010).
Abanin, D. A. & Levitov, L. S. Science 317, 641 (2007).
Williams, J. R., DiCarlo, L. & Marcus, C. M. Science 317, 638 (2007).
Özyilmaz, B. et al. Phys. Rev. Lett. 99, 166804 (2007).
Velasco Jr, J., Liu, G., Bao, W. & Lau, C. N. New J. Phys. 11, 095008 (2009).
Woszczyna, M., Friedemann, M., Dziomba, T., Weimann T. & Ahlers, F. J. Appl. Phys. Lett. 99, 022112 (2011).
Ki, D.K., Nam, S.G., Lee, H.J. & Özyilmaz, B. Phys. Rev. B 81, 033301 (2010).
Schmidt, H., Rode, J. C., Belke, C., Smirnov, D. & Haug, R. J. Phys. Rev. B 88, 075418 (2013).
Nakaharai, S., Williams, J. R. & Marcus, C. M. Phys. Rev. Lett. 107, 036602 (2011).
Beenakker, C. & van Houten, H. Solid State Physics 44, 1 (1991).
Buttner, B. et al. Nat. Phys. 7, 418 (2011).
Velasco Jr, J. et al. Solid State Commun. 152, 1301 (2012).
Ki, D.K. & Lee, H.J., Phys. Rev. B 79, 195327 (2009).
Stander, N., Huard, B. & GoldhaberGordon, D. Phys. Rev. Lett. 102, 026807 (2009).
Choi, J.H. et al. Nat. Commun. 4, 2525 (2013).
Acknowledgements
This work is partially supported by GrantinAid for Scientific Research (S) (No. 26220711) from JSPS and GrantinAid for Scientific Research on Innovative Areas “Science of Atomic Layers” (No. 25107004) from MEXT.
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S.M. planned this research, performed the calculation and the experiments, analysed the data and wrote the manuscript. K.Kobayashi. supervised the research. S.N., K.Komatsu and K.T. supported to fabricate devices. T.M. and T.O. supported to fabricate and measure devices. All authors discussed the results and commented on the manuscript.
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Matsuo, S., Nakaharai, S., Komatsu, K. et al. Parity effect of bipolar quantum Hall edge transport around graphene antidots. Sci Rep 5, 11723 (2015). https://doi.org/10.1038/srep11723
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DOI: https://doi.org/10.1038/srep11723
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