Supplementary material
Magnetoresistance oscillationsarising from edge-localized electrons inlow-defect graphene antidot-lattices
T.Shimizu, J. Nakamura, K. Tada,Y. Yagi, J. Haruyama*
Faculty of Science and Engineering, Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara, Kanagawa 252-5258, JAPAN Corresponding authors:
(1)Fabrication of nanoporous alumina template (NPAT) masks and three advantages of the present nonlithographic method
The NPAT (Fig.1), which consists of a honeycomb array of hexagonal-shaped antidots (AD),was fabricated by the anodicoxidation of a pure aluminum (Al) substrate (Al = 99.99%) using electrochemical methods witha carbon electrode as thecathode13. Due to self-organization, a NPAT provides structure parameters (e.g., anda)with exceptionally high regularity and high reproducibility. After the formation, the NPAT with an area of 1 cm2was detached from the Al substrate by alternating the polarity of the two electrodes. The detached NPATs were then placed onto the graphenes on Si(SiO2) substrateas etching masks (supplementary material(SM)(3) below).SQUID measurements proved that the NPATs do not include any magnetic impurities.
(2)The advantages of using the NPAT mask in conjunction with low-power gas etching (SM(4) below) and high-temperature annealing (SM(5) below) can be explained as follows:
①Because this processuses the nonlithographic fabrication method, it ensures minimal damage to the nanopore edges.
②The honeycomb-like hexagonal ADL can result in the formation of a large number of GNRs with sufficient lengths (e.g., 40 nm in the present case) because of the presence of six boundaries among the neighboring six ADs compared with that of lattices of the square- and round-shaped ADs. In the actual GADL, it is speculated that zigzag and armchair edges exist with mixed states in one GNR (one boundary), as confirmed by the STM observation. Even in this case, a large number of GNRs in the present GADLs can yield a large area of assembled zigzag-edge GNRs.
③When the edge structure of one boundary of a hexagonal AD is aligned with the hexagonal carbon lattice of graphenes, the other five AD boundaries can have the same edge atomic structure. This also strongly contributes to the advantage mentioned in (2).
(3) Fabrication of bulk graphenes
The thin-multilayer (< 10 layers) and monolayer graphene samples as the base for formation of GADLs were extracted from bulk Kish graphite (Toshiba Ceramics) onto degenerately doped Si wafers with a 250-nm-thick SiO2 surface layer by means of mechanical exfoliation. Interference-induced color shifts in an optical microscope (3D-CCD), Raman spectroscopy, and cross-correlation with an AFM profile allowed us to identify the number of deposited graphene layers of all graphene flakes, which ranged from 1 to 10. Moreover, ferromagnetism and spin polarization have been recently confirmed also in GADLs fabricated on a few-layer graphenes synthesized on SiC substrates by CVD as shown in Fig.1(c).
(4) Formation of ADLon graphenes using NPAT masks
Using the NPAT explained inSM(1) as a mask, assembled graphenesexplained in SM(2) were etched by a carefully optimized low-power Ar gas (e.g., 200–600 V for 10 –40 min) to avoid damage. We carried out the low-power etching step by step. Each after etching in 10 minutes, we performed FESEM(or AFM) observation and checked formation of nano-pores on Si-substrate under the NPAT mask. Until confirmation of formation of the ADs on Si-substrate, we repeated the slight etching. This is very important process to avoid giving damages to the AD edges. The boundaries of ADs are not intentionally aligned along the hexagonal carbon lattice of graphene in this process.
After formation of the ADL on graphenes, the NPAT mask was entirely dissolved by a H3PO4 solution (or detached mechanically from the GADLs in some cases). It left no contamination.
Then, all the thicker multi-layer GADL flakes (i.e. those with 10 or more graphene layers), under the NPAT, were individually removed using plastic tweezers. Their complete removal was confirmed following the approach shown in SM (3).
(5) Observation of AD-edge atomic structures and electronic states
There are three different edge structures: zigzag, armchair, and a mixture of the two. Because the possibility of the formation of each structure is basically equal (33%), the[XJ1] value of 50% for formation of a zigzag edge in the inset of Fig.1(e) (samples 1-4) indicates that almost half of the edgesfor the case of the mixtures (i.e., 33%/2 = 17%), which are close to the behavior of a zigzag edge, can be reconstructed to a zigzag edge by annealing (i.e., 33% + 17% = 50%). The remaining 50% of the samples, as shown in the inset of Fig.1(e), will have armchair edges (sample 5) or large-volume defects (samples 6–8).
Observation of edge atomic structures is indispensable for the present experiments. However, it is extremely difficult at thecurrent stage to do this, because for HRTEM observation, GADLs cannot be fabricated on a TEM grid, and for STM observation, the GADLs do not have a conductivity enough large for such measurements. The GADLs formed on Cu substrates may solve this problem for STM observations, because such a substrate provides a high conductivity, although the edge atomic structures may be affected by the Cu substrate.
(6) Rich physics in ADL systems in conventional 2DEG12
In large-/a sample, cyclotron orbit is mainly pinned around the ADs owing to the narrow space between ADs, which does not allow cyclotron motion8. Consequently,a commensurability MR peak and AB-type oscillations appear around a value of B that satisfies 2Rc = a, where the electrons undergoing cyclotron motion are localized around the AD edges and quantized electronic levels En are formed within the AD unit cell (Fig.2(a))8. En is given by h2/2mL2(n - /0), where L is the length of the orbital around the AD, /0 is the ratio of magnetic flux to the unit flux, and n is the number of orbitals.Each when the Fermi level crosses one quantized orbital, an MR oscillation for AB-type effect appears. The oscillation period corresponds to the addition of one flux quantum per unit cell, i.e., ΔBABT = (h/e)/S, where S is the area of the unit cell.
Applying a larger B eliminates the AB-type oscillations and yields Shbunikov–de Haas (SDH) oscillations for (spin-split) Landau levels (LLs) as well as QHEs associated with edge current channels, particularly in the limit of low /a10,11. In samples with a small /a, electrons can travel through the regions between the widely spaced ADs and are scattered by the ADs and sample edges in the manner of pin-balls. These introduce characteristic electron trajectories at B values satisfying 2Rc = a. (Fig. 2a)), e.g., (1) trajectories that encircle the ADs and (2) chaotic trajectories involving the periodic scattering by the ADL (runaway orbitals). In (1), electrons localize around the ADs, resulting in commensurability MR peaks, whereas in (2), electrons can be flee from the center of sample, resulting in also MR peaks. In the latter of Ref.[12], Ando suggested the strong contribution of case (2). When ADL act as scattering center for electrons, anomalous FQHEs were observed with filling factors = 1/2 and 3/211.
(7) Correlation of Fig. 1 (monolayer GADLs) with Fig. 2 (10-layer GADLs)
Fig.1 is for monolayer GADLs, while Fig.2isthe results in10-layer GADLs. However, the results are consistent because the zigzag edge sates of the surface layer in the 10-layer GADL at least are similar to those in monolayer GADL as reported by Otani et al. (as below). Indeed, percentages (50%) that the H-annealed samples have exhibited features of Figs.1(e) and 2 are mostly common between mono and 10-layer GADLs(i.e., numbers of 5/11 and 11/21, respectively. The different values of /a are also not a dominant factor because the ferromagnetism results from the regions between the ADs (i.e., GNRs; Fig.1(a)) with H-terminated zigzag edges in the monolayer GADLs. On the other hand, AB-type oscillations are produced by AD edges with H-terminated zigzag edges in the 10-layer GADL s.
M. Otani, M. Koshino et al., Phys. Rev.B81,161403 (R)(2010).
(8) Residual weak-3D nature in 10-layer GADLs
In Fig. 2(b), xx does not drop to 0 even for high B. This indicates the absence of QHEs. Neither are conventional SDH oscillations observed in Fig. 2(b). The absence of SDH oscillations and QHEs is attributed to the low mobility caused by formation of ADs and the residual weak-3D nature of the specimens caused by multi-layer structures even without ADL (pristine graphene).
In contrast, the quantized electron orbits for AB-type oscillations can be confirmed in text. This suggests that cyclotron electrons strongly localize around antidots, resulting in extremely high EDOSs even in the present weak-2D system. This also might be contribution of AD edge states within the unit cell.
(9) Difference between AB-type effect and AB effect
It should be noted that this AB-type effect is different from the AB effect with an oscillation period ΔBAB=(h/e)/(r2) (where r is the radius of the AB ring (AD radius)) for a single AB ring. The vector potential existing inside the single AB ring is modulated by the magnetic flux penetrating the ring, resulting in modulation of phase interference of electron waves in the AB ring and, then, appearance of subsequent AB oscillation. Also in an ADL system, the electron waves encircling an AD can produce a similar effect in an applied B. However, this effect is smeared out by statistical averaging over a large ensemble of ADs in the ADL systems like the present GADLs.
(10) AB-type effect for large unit cells
In a conventional 2DEG ADL, AB-type oscillations appear at a rate of one flux quantum per unit cell. For larger unit cells, the AB-type oscillations become quickly smeared out by averaging, because the edge states in the 2DEG system are not strong and the electrons localize at the AD edges only for B values satisfying 2Rc =a. Only for small /a values which provide the AD spacing large enough to allow electron cyclotron motion, commensurability peaks for large unit cells (e.g., including 4, 9, and 12 ADs) were observed for very low B (< 0.4 T).
(11) Smaller-period MR oscillation
Furthermore, much smaller MR oscillations with a period of△B270 mT are found in a manner superposed ontothe MR oscillation below B1T, as shown in Fig. 2(a). Figure 2(b) shows the Fourier power spectrum of the data shown in Fig.2(a) over the range 0.6 T < B 1T. There is a strong peak around 1/△B = 13 T-1. The corresponding oscillation period B 75 mTis also consistent with AB-type oscillations in a larger unit cell with size 2a (labeled as the “second unit cell” in Fig. 2(a) of text), because this cell gives ΔBABT =(h/e)/S =50 mT. This result provides supporting evidence for the strong correlation between the edge states of the six ADs located along the boundary of thissecond unit cell and the electron cyclotron orbital along the boundary. It can be also the manifestation of the edge states of the present honeycomb GADLs, because for larger unit cells, the AB-type oscillations become quickly smeared out by averaging in conventional 2DEGs (SM (9)).
Fig.2 (a) Higher B resolution for MR (ρxx) below B=1Tin sample used forFig.2(b) of text. (b) Fourier power spectrum of (a) for 0.6T < B < 1T.
(12) Indirect evidence for zigzag-structured pore edges
Evidence for this is: (1) Edge chirality can reportedly be distinguished by the observed I(D) of graphene edges being stronger (weaker) at armchair (zigzag) edges14. This is because the double resonance process, which induces the D peak, can be only fulfilled at an armchair edge (as a result of the one-dimensional character of the edges).Indeed, Ref. 14found an I(D)/I(G) value < 0.1for observation of the zigzag edge of graphene flakes using angle-dependent Raman spectroscopy with a polarized laser beam. This is qualitatively consistent with Fig.1(e). (2) The low I(D)/I(G) values are also qualitativelyconsistent with those reported forGADLs in Ref.6, where the hexagonal-pore boundaries were intentionally aligned along the carbon hexagonal lattice, resulting in formation of thepure zigzag pore edges.
(13) Expectation for spin-related phenomena and spintronics
In a view of recent reports on the possibility of spin-rectification effect (shown as ref.1 as below) and quantum spin Hall effect (QSHE) (refs.2 and 3 as below) using edge spin current of graphene, our observations contribute towards creation of novel spintronic devices as well as realization of spin quantum bits (ref.4 as below).
- Y-W.Son et al.,Nature444,347 (2006).
- C. L. Kaneand E. J. Mele, Phys. Rev. Lett. 95,226801 (2005).
- M. J. Schmidt and D. Loss, Phys. Rev. B81, 165439 (2010).
- T. G. Pedersen et al.,Phys. Rev. Lett.100, 136804 (2008).
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[XJ1]Millie says: Not sure if this is obvious – at least not obvious to me.