Signatures of pseudo diffusive transport in ballistic suspended graphene superconductor junctions. Piranavan Kumaravadivel 1 and Xu Du 1

Transcription

1 Signatures of pseudo diffusive transport in ballistic suspended graphene superconductor junctions Piranavan Kumaravadivel 1 and Xu Du 1 1 Department of Physics and Astronomy, Stony Brook University Abstract We report transport measurements on short ballistic suspended graphene-niobium Josephson weak links where we observe a strong gate dependence of the multiple Andreev reflection associated sub-harmonic gap structures at low carrier densities. As the Fermi energy is tuned closer to the neutrality point (n<10 10 cm -2 ), these sub-harmonic gap structures become more pronounced whereas the normalized excess current decreases sharply. These observations differ from the density-independent behavior previously reported in disordered graphene superconductor junctions. Our results are in qualitative agreement with theoretical predictions for the emergence of pseudo-diffusive behavior in ballistic graphene due to charge transmission via evanescent modes. Since its isolation, graphene has been a subject of intense research leading to new theoretical insights and experimental demonstrations of unique electronic transport properties that stem from its 2D, Dirac-like energy spectrum. One such property emerges when approaching the Dirac point where the conducting channels in ballistic graphene transition from propagating to evanescent mode. The result is an onset of novel transport signatures that are theoretically identified as pseudo-diffusive [1]. For example, in an undoped ballistic graphene strip, when the width (W) is sufficiently larger than the length (L), the presence of a finite number of highly transmitting evanescent channels (N) leads to a length-dependent conductance, similar to that of a diffusive metal. At the Dirac point, even though the carrier density is zero, the conductivity remains finite and reaches the quantum limited value [2]. Besides conductivity, pseudo-diffusive transport of Dirac electrons also manifests as fluctuations in electrical current (shot

2 noise), when the transmission distribution among the channels is tuned by Fermi energy in the vicinity of the Dirac point. This leaves a density dependent signature in the Fano factor (F) in ballistic graphene devices. At Dirac point F = 1/3, the universal, density independent value for disordered metals. This finite noise at zero energy and density is related to the jittery motion of relativistic quantum particles ( Zitterbewegung ) [2]. Pseudo-diffusive transport signatures can also be observed in ballistic graphene (G)- superconductor (S) hybrid devices: both in S-G and S-G-S junctions [3-5]. In particular, relevant to the work presented here, it has been shown that in ballistic S-G-S, both the sub-harmonic gap structures (SHGS) that form due to multiple Andreev reflections (MARs) and the excess current across the junction due to the superconducting proximity effect should be gate tunable (i.e., Fermi-energy dependent) near Dirac point, reflecting the crossover from ballistic to pseudo-diffusive charge carrier transmission in graphene. While these long-standing theoretical proposals are of conceptual simplicity, the experimental realizations have been impeded by a few technical challenges. First of all, to reach the energy scale required for evanescent transmission, the Fermi wavelength should satisfy. Here m/s is the energy independent Fermi velocity and E F is the Fermi energy. Therefore the potential fluctuations (δe F ) near the neutrality point (NP) should be small, a few mevs, even for a micrometer long channel. Secondly, charge carrier scattering should be largely eliminated, so that the transmission of the carriers reflects the intrinsic nature of the transverse modes. This requires the devices to be ballistic. In previous work on shot noise and superconductivity induced SHGS in graphene [6-9], the devices used were fabricated directly on SiO 2 substrate. Due to the strong substrate-associated disorder, the observations were marred by the presence of large potential fluctuations (for graphene on SiO 2, δe F typically ranges from 25 to 100meV) and short mean free path (usually <<100 nm). More recently, graphene/h-bn hetero-structures have demonstrated ballistic transport [10]. Josephson current has also been observed in these structures when coupled with superconductors [11,12].However, methods for achieving both very low carrier density and highly transparent S-G interfaces

3 are still under development. To the best of our knowledge, an unambiguous experimental study of pseudo-diffusive transport in graphene has not been reported. In this letter, we present our study on charge transport in ballistic suspended graphene-niobium (Nb) superconducting weak links. When approaching the NP (n<10 10 cm -2 ), where the evanescent mode transport starts to dominate over the conventional propagating modes, the SGHS become more pronounced. We also find that the normalized excess current, which remains constant at high carrier densities, becomes suppressed rapidly at low carrier densities near NP. Both observations are in contrast with previous experimental observations in disordered graphene superconductor junctions where both SGHS and I exc R N show no significant gate dependence [9,13,14]. Our results are in qualitative agreement with the theoretical predictions and provide strong evidence for pseudo-diffusive transport in ballistic graphene. The graphene device used in this study was fabricated using mechanically exfoliated highly oriented pyrolytic graphite (HOPG). The graphene channel was designed to have a large aspect ratio (W/L) ~9 with width W =5.5µm and length L = 0.6µm. Such geometry minimizes any effect from the edges of graphene and complies with the theoretical prescription [1]. Moreover, a larger W/L improves experimentally observation by allowing more evanescent modes to contribute to the conductance. To suspend graphene we use an etch-free method as outlined in previous work[15]. The contacts were formed by electron beam evaporation of Ti and Pd to form a thin buffer layer ~2nm followed by DC magnetron sputtering of ~60 nm Niobium (T c ~9K, Hc 2 ~3.5 T). The Ti/Pd buffer layer helps in achieving a low contact resistance. The sputter conditions and device geometry were determined to minimize the stress on graphene. This is essential for a transparent SNS junction and also improves the chances of getting a ballistic graphene channel by current annealing. All the measurements were performed in an Oxford Instruments VTI refrigerator, with room temperature ferrite and π-filters, and cryogenic 2-stage RC filters.

4 The gating curves of the device after current annealing measured for two different temperatures, 9K (~ T c ) and 1.5K, are presented in figure 1b. At T~9K the device shows a maximum mobility of >250,000 cm 2 /Vs and the mean free path is limited the sample length. From the smear of the resistance near the NP, we estimate a minimum carrier density of cm -2. This corresponds to a potential fluctuation at NP, the smallest value observed so far in such devices. To characterize the Fermi wavelength, similar to Ref.[3], we use a dimensionless parameter. Therefore the maximum δe F ~ 4.4meV in our sample corresponds to κ~4.0. The resistivity at NP is. The small discrepancy from the theoretical value of may be attributed to the presence of electron hole puddles and finite Coulomb scattering which in practice cannot be completely avoided. The excellent quality of the device is also evident from the quantum Hall (QH) measurements. As shown in figure 1c, at T=1.5K, and in a low magnetic field of B= 300mT, pronounced magneto-oscillations are already observed. At B=500mT, the sample displays fully developed anomalous quantum hall plateaus at where. From these QH plateaus, we find the carrier density (n)-gate voltage (V g ) relation: where is the gate voltage at NP. This is consistent with the estimation using the geometrical capacitance considering 285nm SiO 2 in series with 220nm (thickness of the PMMA spacer) of vacuum. On the higher mobility electron side, we also observe additional oscillatory features in resistance, R (Vg), at, etc., which may be attributed to the onset of broken symmetry. The resistance at the NP displays diverging behavior with increasing field, starting at a low B~0.3T. All the evidences observed here demonstrate that the sample is of extremely high quality, with long mean free path and minimal potential fluctuations.

5 FIG. 1. (color online) Device characteristics: (a) Main panel: Device schematics. Inset: SEM Image of the device. Scale bar is 2µm. The graphene channel is highlighted by the open rectangle (b) Resistivity in units of (πh/4e2) as a function of gate voltage (Vg) at T=9K (blue) and T (<TC) =1.5K (red). (c) Quantum Hall measurements: Conductance versus filling factor for two different magnetic fields 300mT (blue) and 500mT (red). d. Differential resistance as a function of bias voltage (Vbias) at T=1.5K. Gate voltage is 3.5V away from the NP gate voltage (VNP). The corresponding κ ~43(see text) and the induced gap (Δ) =0.34meV. Next, we study the superconducting proximity effect at T=1.5K in absence of magnetic field. Compared to T>TC, the junction resistance is significantly reduced. For, the resistance vanishes reflecting the presence of a finite supercurrent. Figure 1d shows the differential resistance at as a function of bias voltage (Vbias) taken (κ~43). From the curve we obtain the proximity induced gap on graphene (Δ) ~0.34meV. This value is significantly smaller than the BCS gap of Nb

6 ( ) and varies slightly from sample to sample. Similar reduction has also been observed in superconductor-nanowire weak links [16,17] and can be attributed to the Fermi velocity mismatch and the presence of a finite-thickness of Ti/Pd buffer layer. At finite, the resistance drops down to ~ 40% of the normal resistance (R N ) due to superconducting proximity effect. Evaluating the value of the normalized excess current and using the OTBK model [18] we estimate the dimensionless barrier strength of the interface, Z ~0.5. The MAR-associated SGHS are very weak, consistent with the theoretical prediction for ballistic channels with high transmission [19]. Other factors that may affect the weak SGHS include the small induced superconducting gap and the relatively high measurement temperature. Now we focus on the behavior of the differential resistance as we approach the NP. Figure 2a shows the normalized differential resistance as a function of V bias obtained at various gate voltages. When ( and ), the differential resistance curve starts to develop a pronounced dip at. As the V g is ramped further towards the NP, i.e., for ( and ), while the dip at continues to be deeper, other SHGS start to emerge at low V bias. All the observed features appear at where n=1, 2, 3 as expected for MAR processes and their positions in V bias are independent of V g. The higher order features are within noise. The observed features are sharpest at NP where the SHGS at n=1, 2 and 3 are all easily resolvable. But once V g is ramped to the hole side, slightly away from the NP (ĸ~ -1.9), the SHGS begin to weaken. The dip at n=1 shows the most prominent response to the gate voltage. In conjunction with the appearance of the pronounced SGHS, the overall shape of the normalized differential resistance curve transforms from a V -shape to a shallow profile. We note that the observed gate-dependence of the vs. curves is specific to

7 superconductivity. Above T c the background vs., within the bias voltage range studied here, shows roughly no curvature. FIG.2. (color online) Normalized differential resistance versus bias Voltage (V bias ) for different. Individual curves are shifted for clarity. Dotted lines indicate SHGS. a. Ballistic S-G-S: Ti/Pd/Nb contacts at T=1.5K. Evolving SHGS at V bias =±2Δ/ne for n=1, 2 and 3 where 2Δ=0.68meV. b. Diffusive S-G-S: (Upper panel) Ti/Pd/Nb contacts at T=1.5K. Unchanged Gap feature at V bias =±2Δ/e=±0.3mV. (Lower Panel) Pd/Al contacts at T~0.27K. Unchanged SHGS at V bias =±2Δ/ne for n=1 and 2 where 2Δ=0.13meV. For comparison, we present a similar set of data for diffusive samples with Ti/Pd/Nb (Fig.2b, upper panel) and Pd/Al (Fig.2b, lower panel) contacts. In both cases the graphene channel sits on SiO 2 and has a mean free path and. In contrast to the ballistic sample, there is no significant gate voltage dependence of the SHGS for both disordered devices. We note that the higher order MAR features are unresolvable in the Ti/Pd/Nb device, which may be attributed to the smaller induced superconducting gap and the relatively high measurement temperature.

8 To understand the observed gate-dependent MAR features, we consider Fermi energy modulation of charge transmission in a ballistic graphene device. In general, the transmission of Dirac electrons in graphene can be described by a summation of contributions from the boundary-defined transverse modes, each satisfying the Dirac- Wyle equation. Depending on the Fermi energy, the lowest modes are propagating with real wave vectors, while the higher modes are evanescent with imaginary wave vectors[5]. At high densities, the channels in the ballistic graphene strip are propagating with high transmission. Approaching the NP, however, the propagating modes become suppressed and the evanescent modes contribute increasingly to the conduction with more channels having a lower transmission. This change in the transmission distribution is the underlying reason for the emergence of pseudo-diffusive transport signatures. With superconducting contacts, it has been shown that the oscillatory amplitude of the MAR-associated SGHS is due to the contribution from the low-transmitting channels [3,20]. As a result, close to the NP where evanescent modes with low transmitting probability dominate, the SGHS are more pronounced than at large gate voltages. The stronger gate-dependence of lower order SGHS, especially n=1, is also consistent with the theory. This is because compared to the higher order SHGS, the lower order ones are formed by Andreev quasi-particles that traverse the graphene channel fewer times and hence involves more contribution from the low transmitting channels.

9 FIG.3. (color online) Normalized Excess current (I exc R N ) as a function of a. Experimental data: I exc R N is calculated from the IV characteristics of the device for different V g at 1.5K.The (blue) line is drawn as a guide to the eye. b. Theoretical calculation: for short ballistic S-G-S junction with W=5.5µm and L=0.6µm at T=0K. A more quantitative assessment for comparing with the theoretical predictions for pseudo-diffusive transport can be made by characterizing the excess current through the S-G-S device. Generally in a SNS junction when ev bias >>2Δ, the current though the sample consists of a normal Ohmic current (I N ) and an excess current (I exc ) due to the superconducting proximity effect. Compared to the Josephson current, excess current is much more robust against the influence of the electromagnetic environment, and hence provides a reliable parameter for characterizing the proximity effect. In S-G-S junctions the abundance of nearly ballistic modes at large Fermi energies leads to a large excess current. But, in the vicinity of the NP the number of highly transmitting propagating modes decreases and the charge transport becomes increasingly evanescent thereby decreasing the excess current. Figure 3a shows the normalized excess current I exc R N in our device (extracted from IV curves at various gate voltages) as a function of κ. For κ >9, i.e., in the ballistic transport regime I exc R N ~0.4mV~1.2Δ/e whereas for κ <9, there is a clear gate dependence. The excess current sharply reduces when approaching the NP. This reduction coincides with the onset of the enhanced SHGS as shown in figure 2a. For short ( ) ballistic graphene Josephson junctions, the excess current been

10 theoretically studied in reference [3]. With L=0.6µm, and since the induced gap our device marginally satisfies the short junction limit. The observed gate modulation of normalized excess current is in qualitative agreement with the zero temperature theoretical calculation as shown in Figure 3b. Here, we use W=5.5µm and L=0.6µm of our sample and following [3], the total excess current in a short Josephson junction is calculated as a sum of its individual contribution from all the transverse modes, each determined by the corresponding transmission coefficient. The discrepancy between the theory and our observation, especially in the values for normalized excess current, may be attributed to a few factors. First, the theory assumes an ideal SN interface (Z=0), whereas in our device Z~0.5. Secondly, our measurements were carried out at a base temperature of ~ 1.5K while the theory does not consider a finite temperature. Both these factors contribute to the reduction of the normalized excess current. In addition, for lowest values of κ, the theory does not consider the presence of the electron hole puddles that exists in an actual device. Together with the smearing from finite temperature, we observe a slight broadening in the excess current dip with an onset at κ~9 compared to the theory (κ~4). We also consider the possible impact of imperfect S-G interface that may also result in Fermi energy modification of transmission coefficients without any direct consequence of the Dirac fermionic nature of graphene. Indeed it has been suggested that in graphene-metal junctions, the doping of the metal contacts can extend into graphene, forming a p-n junction that imposes the charge carrier reflections [21,22]. A direct evidence of the presence of such interfacial p-n junction is the electron-hole asymmetry in the R vs. V g dependence. As the gate voltage is swept across the NP, the S-G interface changes from p-n to n-n and the asymmetry in the R vs. V g dependence can be associated with the transmission probably across the S-G junction. While the presence of such junctions may affect the MARs, we expect its gate voltage dependence to be gradual with no particular energy scale. In addition, such contact-doping associated reflection should give rise to asymmetric MAR characteristics which persists up to large gate voltages on both the electron and hole sides[11]. These are apparently not consistent with the

11 observation of a sharp dip on the I exc R N vs. E F dependence for E F <8meV, and the qualitatively symmetric behavior with respect to the NP. In summary, we carried out transport measurements on ballistic suspended graphene-niobium superconducting weak links. The high quality of our device enables us to study the gate voltage dependence of the SGHS and normalized excess current in the low carrier density regime close to the NP. Our results are consistent with theoretical predictions for evanescent transport mediated pseudo-diffusive behavior in short ballistic Josephson weak links. We note that similar devices of high quality, with the ability of achieving both low-field quantum Hall effect and superconducting proximity effect, can also be used to study magneto-transport at lower temperatures that may lead to interesting insights of superconducting correlations in the quantum hall regime. Acknowledgements The authors thank Dmitri Averin for insightful discussions, Laszlo Mihaly for support with cryogenic facilities, and Peter Stephens for providing single crystal HOPG. This work was supported by AFOSR under grant FA References: [1] J. Tworzydlo, B. Trauzettel, M. Titov, A. Rycerz, and C. W. J. Beenakker, Phys Rev Lett 96 (2006). [2] M. I. Katsnelson, Eur Phys J B 51, 157 (2006). [3] J. C. Cuevas and A. L. Yeyati, Physical Review B 74 (2006). [4] A. R. Akhmerov and C. W. J. Beenakker, Physical Review B 75 (2007). [5] M. Titov and C. W. J. Beenakker, Physical Review B 74 (2006). [6] L. DiCarlo, J. R. Williams, Y. M. Zhang, D. T. McClure, and C. M. Marcus, Phys Rev Lett 100 (2008). [7] R. Danneau, F. Wu, M. F. Craciun, S. Russo, M. Y. Tomi, J. Salmilehto, A. F. Morpurgo, and P. J. Hakonen, Phys Rev Lett 100 (2008). [8] H. B. Heersche, P. Jarillo- Herrero, J. B. Oostinga, L. M. K. Vandersypen, and A. F. Morpurgo, Nature 446, 56 (2007). [9] X. Du, I. Skachko, and E. Y. Andrei, Physical Review B 77 (2008). [10] L. Wang et al., Science (New York, N.Y.) 342, 614 (2013). [11] M. Ben Shalom et al., in arxiv: [12] V. E. Calado, S. Goswami, G. Nanda, M. Diez, A. R. Akhmerov, K. Watanabe, T. Taniguchi, T. M. Klapwijk, and L. M. K. Vandersypen, in arxiv:

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