Electronic Transport and Possible Superconductivity at Van Hove

Oct 27, 2015 - Department of Physics, CNAM, and JQI , University of Maryland, College Park, Maryland 20742, United States. Nano Lett. , 2015, 15 (12),...
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Electronic Transport and Possible Superconductivity at Van Hove Singularities in Carbon Nanotubes Y. Yang,† G. Fedorov,† S. E. Shafranjuk,‡ T. M. Klapwijk,§,∥ B. K. Cooper,§ R. M. Lewis,⊥ C. J. Lobb,⊥ and P. Barbara*,† †

Department of Physics, Georgetown University, Washington, District of Columbia 20057, United States Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, United States § Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands ∥ Laboratory for Quantum Limited Devices, Physics Department, Moscow State Pedagogical University, 29 Malaya Pirogovskaya Street, Moscow, 119992, Russia ⊥ Department of Physics, CNAM, and JQI , University of Maryland, College Park, Maryland 20742, United States ‡

S Supporting Information *

ABSTRACT: Van Hove singularities (VHSs) are a hallmark of reduced dimensionality, leading to a divergent density of states in one and two dimensions and predictions of new electronic properties when the Fermi energy is close to these divergences. In carbon nanotubes, VHSs mark the onset of new subbands. They are elusive in standard electronic transport characterization measurements because they do not typically appear as notable features and therefore their effect on the nanotube conductance is largely unexplored. Here we report conductance measurements of carbon nanotubes where VHSs are clearly revealed by interference patterns of the electronic wave functions, showing both a sharp increase of quantum capacitance, and a sharp reduction of energy level spacing, consistent with an upsurge of density of states. At VHSs, we also measure an anomalous increase of conductance below a temperature of about 30 K. We argue that this transport feature is consistent with the formation of Cooper pairs in the nanotube. KEYWORDS: carbon nanotubes, tunable superconductivity, van Hove singularities

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Notwithstanding the rich physics that arises from the strong enhancement of electronic interactions when the Fermi energy is close to VHSs, experimental studies of their effect on electronic transport properties of materials are lacking. Different factors contribute to this lack. In graphene, the carrier densities required to approach the VHSs are quite high and hard to achieve either by electrical or chemical gating.9 In carbon nanotubes, VHSs can be reached by gating and have also been clearly identified with Raman spectroscopy17 and scanning tunneling microscopy.18−21 In electrical transport measurements, it has been difficult to determine the region of gate voltage corresponding to a VHS because characteristic features in the conductance versus gate voltage curve have not been identified. In this article, we report two clear signatures of VHSs in the conductance versus source−drain voltage and gate voltage of carbon nanotube field-effect transistors. First, the VHSs are found to strongly affect the Fabry−Perot interference patterns

he van Hove singularities (VHSs) in the density of states have profound effects on the structural and electrical properties of solids. Electronic instabilities at VHSs have been related to the occurrence of magnetic ordering1 and charge density waves.2 A van Hove scenario has also been proposed as a route to superconductivity since the discovery of cuprate superconductors.3−8 More recent studies of both conventional and unconventional pairing mechanisms at VHSs in graphene and carbon nanotubes predict superconductivity with critical temperatures above 10 K.9−12 As an example, Figure 1a shows the density of states (DOS) of a semiconducting carbon nanotube with sharp VHSs at the onset of each subband. The energy spacing between VHSs, indicated as ΔE1−2 VHS for the spacing between subband 1 (red) and subband 2 (black) in Figure 1a, is determined by the nanotube diameter.13−16 Within each subband, the electronic energy level spacing is determined by the length of the nanotube and it is almost constant in the linear part of a subband, but it gets much smaller at the onset of a subband, as shown by ΔE1 and ΔE2 for subbands 1 and 2, respectively, in Figure 1a. © 2015 American Chemical Society

Received: June 29, 2015 Revised: September 25, 2015 Published: October 27, 2015 7859

DOI: 10.1021/acs.nanolett.5b02564 Nano Lett. 2015, 15, 7859−7866

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Figure 1. (a) Schematic of the electronic band structure (right) and the density of states (left) of a semiconducting carbon nanotube. The energy level spacing near the onset of a new subband, ΔE2 (black vertical bar) is smaller than the energy level spacing ΔE1 in the linear part of the dispersion relation (red vertical bar). (b) Sketch of a gated carbon nanotube. Cq is the quantum capacitance and Ces is the electrostatic capacitance. (c) SEM image of the sample showing the short (L = 400 nm) device B on the right and the long (L = 1200 nm) device A on the left. (d) Zero-bias differential conductance as a function of back gate voltage for device B at different temperatures. At the threshold gate voltage, the Fermi energy is aligned with the onset of the first (red) subband (right vertical dashed line). The left vertical dashed line corresponds to the peak in quantum capacitance in Figure 2b.

of the electronic wave functions22 in the nanotube by changing their characteristic energy. Second, the “gate efficiency”, the factor of proportionality between the shift in Fermi energy and total energy provided by the applied gate voltage, changes at the VHS, due to the increased nanotube quantum capacitance.26 With the VHS clearly identified by these two signatures, we find that when the gate voltage is fixed in proximity of a VHS, the conductance shows an anomalous increase at low temperature and a zero bias conductance peak. We explain this as a possible signature of intrinsic superconductivity in the carbon nanotube and use it to extract the temperature dependence of the superconducting gap. Our carbon nanotubes are synthesized by catalytic chemical vapor deposition on highly doped Si substrates covered by 400 nm of thermally grown SiO2, which serves as the back gate.27 Figure 1c shows a sample made of a 35 μm long semiconducting nanotube. Two pairs of Pd electrodes are patterned by e-beam lithography and sputtered on top of the nanotube with spacing of 400 nm for device B (right) and 1200 nm for device A (left), respectively. The contact resistance between a nanotube section and the source−drain contacts is about 10 kΩ (the contact resistance of a carbon nanotube with ideal source− drain contacts cannot be lower than 6.5 kΩ). This is extracted by measuring the resistance of devices with different lengths

(device A, B, and the device formed by the section between them) at room temperature and in vacuum. The zero-bias differential conductance of device B as a function of gate voltage at temperatures from 50 mK to 28 K is shown in Figure 1d (curves obtained at temperature higher than 50 mK are shifted upward for clarity). This device is a ptype FET with a threshold voltage Vth ≈ 2 V, which corresponds to the onset of the valence band. When the voltage is decreased below the threshold value, the conductance increases up to about 3G0 at VG ≈ − 8 V (here G0 = e2/h) and flattens out when the gate voltage is lowered further. Steps in the conductance as a function of gate voltage have been previously observed in InAs nanowires and attributed to the onset of one-dimensional sub-bands,28,29 suggesting that the step at VG ≈ − 8 V may be due to the onset of the second subband in the nanotube (see Figure 1a). The flattening of conductance for voltages VG < −8 V is noticeable at all temperatures, even though it is less clear at lower temperature due to the appearance of large conductance oscillations. As we will argue in detail below, we identify these oscillations as due to Fabry−Perot interference.22 The characteristic diamond patterns in the plots of conductance as a function of source−drain and gate voltages are shown in Figure 2a. If these are Fabry−Perot patterns, the value of source−drain voltage VL where the two diagonal lines cross (indicated by the green dots) can be used to infer the spacing 7860

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Figure 2. (a) The 2D plot of the differential conductance of device B as a function of gate voltage VG and source−drain voltage VSD at 50 mK. The gate voltage scale is the same for (a) and (b). The green dots mark the characteristic voltage VL discussed in the text and the black arrow marks a sharp drop of VL at VG ≈ −7.5 V. (b) Quantum capacitance in units of electrostatic capacitance. The data are extracted from the 2D plots of differential conductance as a function of source− drain and gate voltage measured at 50 mK (black squares) and 5K (red squares) by measuring the gate efficiency (see Figures 3 and 4).

of electronic energy levels in the carbon nanotube.22 Fabry− Perot patterns have been widely observed in both metallic and semiconducting CNTs with very good contacts.22−24 In our measurements, the features are smeared out at higher temperature (see Figure 1d) but are still distinguishable in the region of gate voltage where the device conductance is high. For values of VG > − 4 V, the conductance quickly decreases below 2G0 and no Fabry−Perot patterns can be distinguished. This is because in semiconducting carbon nanotubes the shape of the Schottky barriers at the contacts and the contact transparency are tuned by the gate voltage.25 The pattern in Figure 2a shows a remarkable feature: There is a change of the size of the diamonds at VG ≈ − 7.5 V indicated by the vertical black arrow with a corresponding sharp drop of VL. To understand the cause of this drop, we analyzed the curves of conductance versus source−drain voltage and conductance versus gate voltage underlying the 2D plot. The gate efficiency α is one important parameter that we can extract from each diamond in the 2D plot, by measuring the ratio between the voltage VL and the width of the diamond along gate voltage axis, α = VL/ΔVG. The energy required to transfer charge δq to the carbon nanotube corresponding to a change of gate voltage δVG is equal to the charging energy of the nanotube electrostatic capacitance Ces plus the change in the nanotube chemical potential δμc, which depends on the density of states D(E). This δμc is expressed as a charging energy through the quantum capacitance, Cq ≡ eδq/δμc = e2D(E)L where L is the length of the nanotube between the contact electrodes in series with the geometrical capacitance, as shown in Figure 1b. Therefore, the gate efficiency α = δμc/eδVG depends on the quantum capacitance and the electrostatic capacitance, α = Ces/ (Cq + Ces).26 Figure 3a,b shows the 2D plot measured at 50 mK from Figure 2 and the differential conductance at zero source−drain bias as a function of gate voltage extracted from this plot with

Figure 3. (a) Differential conductance of the short section device, device B, as a function of source−drain and gate voltage at 50 mK. (b) Zero-bias differential conductance extracted from (a). The vertical lines mark local minima of this plot at the center of Fabry−Perot diamonds. (c) Conductance versus source−drain voltage corresponding to the gate voltage values marked by the vertical lines in (b). The local maxima of conductance versus source−drain and gate voltage are extracted from (b) and (c) and marked with red dots in (a). (d) Table showing the measured values of average VL and ΔVG and the corresponding gate efficiency α.

several maxima and minima corresponding to the conductance oscillations. The gate voltage corresponding to each minimum marks the center of a conductance oscillation diamond. The values of the minima are listed in the table in Figure 3d in the dip column Vdip G . For each VG value, the corresponding curve of conductance as a function of source drain voltage is plotted in Figure 3c, vertically shifted for clarity. They show a minimum at zero bias and two maxima at finite source−drain voltage. The patterns often show a slight asymmetry. The maxima appear at slightly different absolute values of negative and positive source−drain voltage. Such asymmetry might originate from a slight asymmetry of the Schottky barriers. We mark these values with vertical red lines on the curves and list them in the table as VL1 and VL2. For the calculation of the gate efficiency α, we use VL = (|VL1| + |VL2|)/2, also listed in the table. We measure ΔVG, the gate voltage difference between the 7861

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capacitance identifies a peak in the density of states, a van Hove singularity. The sharp drop in the size of the diamonds for VG ≈ − 7.5 V can now be understood by considering the dispersion relations of the 1D subbands sketched in Figure 1a. In the energy range well below the valence band edge, where the carbon nanotube dispersion relation is approximately linear, the spacing of the energy levels is constant, as shown by the energy level spacing for the red subband, with ΔE1 = ℏvFπ/L, where vF = 8 × 105 m/s is the nanotube Fermi velocity and L is the length of the nanotube between the contact electrodes. For device A, which is 400 nm long, ΔE1 is about 4.5 meV. This is consistent with the larger values of VL measured at VG > − 7.5 V (see Figures 3 and 4). The sudden drop of the energy level spacing at VG ≈ −7.5 V is thus explained with the onset of a new subband. At the edge of a new subband, the dispersion relation is nonlinear and the spacing of the energy levels is smaller, reflecting an increased density of states, as shown by the energy level spacing ΔE2 for the black curve in Figure 1a. The level spacing gets larger as the gate voltage is lowered further toward the linear region of the new subband. We thus identify the conductance oscillations as Fabry−Perot oscillations with the unusual feature around VG ≈ −7.5 V identifying the onset of a new subband. The energy spacing between the VHSs at the onsets of the first and second valence subbands depends on the nanotube diameter d15,16 via

two maxima of zero-bias differential conductance versus gate voltage adjacent to Vdip G . The maxima extracted from the conductance versus source drain voltage and gate voltage curves are marked with red dots in the 2D plot. We then calculate the gate efficiency α(Vdip G ) = VL/ΔVG, also listed on the table. The conductance oscillations at 50 mK are not very clear for gate voltages VG > −5 V. We could better distinguish conductance oscillations for VG > −5 V at higher temperatures when the measured curves are smeared out. Figure 4a shows 2D patterns obtained at 5K for section B. We used the same procedure outlined above to analyze the conductance curves to obtain additional values for the gate efficiency.

1−2 ΔE VHS =

|Vppπ |s d

(1)

where Vppπ ≈ 2.5 eV is the nearest-neighbor ppπ interaction and s = 0.142 nm is the spacing of the carbon atom bonds. For our nanotube with diameter measured by AFM of 3.0 ± 0.5 nm, ΔE1−2 VHS = 0.12 ± 0.02 eV. The average gate efficiency extracted from the Fabry−Perot 2D plot measured at gate voltages VG > −7.5 V (corresponding to the first valence subband) is α ≈ 1%. This yields ΔE = eα |VVHS − Vth| ≈ 0.1 eV between the threshold and the quantum capacitance peak, which is in reasonable agreement with the expected spacing between the first and the second VHS. The second VHS also marks the onset of an anomalous transport feature: A large zero-bias conductance peak that is strongly dependent on temperature develops in proximity to the second VHS, just beyond the onset of the second subband. This zero-bias anomaly (ZBA) occurs within a narrow range of gate voltage, about 0.5 V to 1 V wide,31 as shown in the Supporting Information. Figure 5a shows the temperature dependence of this peak, measured at VG = −8.6 V, very close to the gate voltage corresponding to the VHS measured at 50 mK. This peak in source−drain conductance decreases with increasing temperature but persists up to about 30 K. This feature is quite different from typical Fabry−Perot oscillations because it spans a wider range of source−drain voltages and persists up to much higher temperatures (Fabry−Perot oscillations for this sample are completely smeared out at temperatures above 12 K). A similar zero-bias anomaly (ZBA) also occurs in the longer device, device A, in a similar range of gate voltage, as shown in Figure 5c. The gate voltage dependence of the ZBA for both sections is included in Supporting Information. At temperatures below liquid helium temperature, Fabry− Perot oscillations are superimposed on the ZBA. Figure 6a shows the ZBA at 5K at VG = −8.6 V along with the curves

Figure 4. (a) Differential conductance of the short section device, device B, as a function of source−drain and gate voltage at 5 K. (b) Zero-bias differential conductance extracted from (a). The vertical lines mark local minima of this plot at the center of Fabry−Perot diamonds. (c) Conductance versus source−drain voltage corresponding to the gate voltage values marked by the vertical lines in (b). The local maxima of conductance versus source−drain and gate voltage are extracted from (b,c) and marked with red dots in (a). (d) Table showing the measured values of average VL and ΔVG and the corresponding gate efficiency α.

We next extract Cq/Ces = (1 − α)/α from the measured gate efficiency for every value of Vdip G . We estimated Ces ≃ 35 aF/μm by calculating the capacitance of a metallic wire separated from a metallic plane by the 400 nm dielectric layer. (Although stray capacitances from the leads should also be included to obtain an accurate measurement of Cq, here we are focusing on its qualitative gate voltage dependence.) Figure 2b clearly shows a sharp peak in Cq at VG ≈ −7.5 V. The peak of quantum 7862

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extracted from the 2D plot in the region close to the VHS. The ZBA starts developing as a background to the Fabry−Perot oscillations and grows wider for lower values of gate voltage in high bias region, while the zero-bias region is dominated by the Fabry−Perot dip. The growing Fabry−Perot oscillations superimposed to the ZBA are more clear in the temperature dependence of the ZBA in the long section, shown in Figure 6b. We previously reported ZBAs at large negative values of gate voltage in several devices with length longer than 1 μm, similar to device A.31,32 We considered different mechanisms as possible causes of these ZBAs, including resonant tunneling, Kondo effect, and Fano resonance, but none of these could explain the data.32 Because the values of gate voltage where the ZBAs occurred matched the location of the VHS estimated from AFM measurements of the nanotube diameter and reasonable values of gate efficiency, we proposed that the ZBAs may be a signature of intrinsic superconductivity in carbon nanotubes, occurring when the gate voltage shifts the Fermi energy into VHSs of the density of states. However, in our previous work where the lowest measurement temperature was 4.2 K and most nanotube devices were longer than one micrometer we could not detect any evidence of VHS in our transport measurements. This is because the steps of conductance versus gate voltage that may be an indication of the onset of new sub-bands are often not clearly distinguishable in carbon nanotubes, especially at low temperature (see for example Figure S2 in Supporting Information). The analysis of Fabry−Perot patterns presented in Figures 2, 3, and 4 leading to the peak in quantum capacitance provides a clear signature of a VHS, but it cannot be easily observed in long nanotubes due to the smaller characteristic energies and smaller diamonds in the Fabry−Perot patterns. Figure 2 represents the central result of this work and is the first transport measurement clearly showing the location in gate voltage of the VHS because we are using shorter samples and much lower temperatures. There are two clear signs of the VHS: (1) the reduction in size of the Fabry−Perot pattern that is expected at the onset of a new band due to the decreased spacing in electronic energy levels and (2) a peak of quantum capacitance extracted from the gate efficiency that is measured from the Fabry−Perot patterns. Measurements from the short device show that the VHS and ZBA occur in the same gate voltage region. While it is not possible to measure similar evidence of VHS in the long section (device A), we do measure a ZBA at the same gate voltage spacing from the threshold, as expected for VHSs in sections of nanotubes with the same density of states. We expect the occurrence of similar ZBAs at gate voltages 1−2 corresponding to the next VHS with spacing ΔE2−3 VHS = 2ΔEVHS 15,16 from the VHS in Figure 2. While we observed features similar to the first ZBA, we could not obtain repeatable measurements for VG < −20 V due to charge trapping in the dielectric layer causing hysteresis in the conductance versus gate voltage curve (see Supporting Information). If Cooper pairs form in the carbon nanotube, Andreev reflection will occur at the interface between the carbon nanotube and the normal Pd electrode, when electrons in the Pd are converted into Cooper pairs in the nanotube. Even though classic signatures of superconductivity, such as zero resistance and the Meissner effect, are suppressed in one dimension, Andreev reflection remains an important indication

Figure 5. (a) Differential conductance as a function of source−drain bias showing the ZBA for device B at VG = −8.6 V at different temperatures (5, 8, 12, 21, and 31 K) and (b) with an external magnetic field applied perpendicular to the nanotube axis. The normal conductance GN in (b) is obtained by fitting the data for magnitude of source drain bias larger than 10 mV. (c) ZBA as a function of temperature for device A. (d) Ratio of the ZBA curves measured for device A (GS) to the normal conductance (GN) obtained by fitting each curve as shown in (b) at three different temperatures. The experimental GS/GN(VSD) curves are used to fit the model described in ref 31 and 43.

Figure 6. (a) Differential conductance as a function of source−drain bias for device B at 5 K and 50 mK for different values of gate voltage. (b) Differential conductance as a function of source−drain bias for device A at −9.1 V at different temperatures (50 mK, 390 mK, 700 mK, and 5K). At low temperature, strong Fabry−Perot oscillations appear superimposed to the ZBA.

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experiments with multiple nanotubes no electrostatic doping was applied to control the shift of Fermi energy into VHSs.44−49 In contrast to the work on ropes and arrays, we use isolated carbon nanotubes and electrostatic doping via gating. Our measurements show that these zero-bias anomalies occur in proximity of VHSs, where the increased density of states favors superconductivity. Another factor possibly enhancing intrinsic superconductivity is interband pairing via Fano−Feshbach resonances, since the VHS marks the onset of an additional subband.50−53 In conclusion, we have shown that VHSs strongly affect interference patterns of the electronic wave functions in carbon nanotubes and thus can be clearly identified in transport measurements. We also showed that the occurrence of zerobias conductance peaks when the gate voltage shifts the Fermi energy into VHS can consistently be explained as due to a gatetunable intrinsic superconductivity with critical temperatures in the range 30−40 K and critical magnetic field substantially higher than 7 T. We find that sections of the same carbon nanotube with different length show qualitatively the same features, however the conductance increase and the critical temperature are larger in the shorter section. Further studies of these phenomena will focus on understanding the role played by the carbon nanotube length for isolated nanotubes as well as measurements in higher magnetic field and samples made with ropes of nanotubes with similar diameter.

of superconductivity and can explain the zero-bias anomaly that we observe.30,32 Because electronic transport is ballistic at low temperature for such short nanotubes, the device resistance is dominated by the contacts. It would be a challenge to clearly distinguish superconductivity (including the thermal and quantum fluctuations that are expected in a one-dimensional systems) from ballistic transport with traditional four-probe measurements. Moreover, the additional probes that are needed for the four-point configuration create new scattering sites in the nanotube and additional quantum interference effects in some cases even leading to negative four-contact resistance.33 For these reasons, two-point measurements are a better probe of superconductivity in this system. If Cooper pairs are present, they will cause a change in the contact resistance due to Andreev reflection that can be readily measured in our twopoint measurement configuration. We note that the ZBA is only slightly suppressed by a magnetic field as high as 7 T (see Figure 5b). This is consistent with previous work showing that orbital and spin pair-breaking effects from applied magnetic field are weakened in superconducting nanowires with diameter as small as 10 nm.34,35 Similar effects are presumably the cause of the weak magnetic field dependence in Figure 5b. Andreev reflection explains the data quite well, as we have shown in previous work.31,43 It assumes an s-wave superconducting state, but Andreev reflection also occurs with other pairing states. The reflection causes an increase in the conductance. We calculated this increase by combining the Blonder−Tinkham−Klapwijk approach36 with the scatteringmatrix technique.37 We included interference of evanescent waves in the nanotube section between the electrodes38 and a prolonged dwell time of an electron in the Pd close to the interface, leading to reflectionless tunneling.39,40 Reflectionless tunneling further increases the conductance by including coherent diffusive scattering in the normal metal. (Those electrons that are not Andreev reflected as holes and instead undergo normal reflection still have some probability to be coherently scattered back toward the interface by collisions with defects. When this happens, they can be Andreev reflected as holes.) In this case, Andreev reflection can increase the conductance by a factor higher than two. We included inelastic scattering in the theory by adding an imaginary part to the electron energy, ECNT = E + iΓCNT(T).41,42 By fitting the conductance calculated from the model to the experimental curves of normalized conductance versus bias at different temperatures (see Figure 5d), we extract the temperature dependence of the superconducting gap, ΔCNT(T) (details of the procedure are described in refs 31 and 43). The energy gap magnitude ΔCNT and the critical temperature are bigger for the short CNT section with ΔShort TCNT c CNT (T = 0) ≃ 37 K. For the longer section ΔLong ≃ 5 mV and TCNT,Short c CNT (T = 0) ≃ 3.8 mV and TCNT,Long ≃ 29 K. From these values, we can c estimate the BCS coherence length ξ0 = ℏvF/(πΔ) ≃ 40 nm for the short section while it is 55 nm for the long section. Intrinsic superconductivity has been reported in ropes and arrays of carbon nanotubes44−49 but those measurements showed a wide variation of critical temperatures from 0.5 to 18 K, and in many cases the results could not be systematically reproduced on multiples samples. This is expected due to the difficulty to control the nanotube uniformity in samples made of multiple nanotubes with different chirality and diameters yielding very different densities of states. Furthermore, in

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ASSOCIATED CONTENT

* Supporting Information S

This material is available free of charge via the Internet at http://pubs.acs.org/. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.5b02564. Details on the gate voltage dependence of the ZBA and a discussion on the effect of VHSs on the nanotube conductance. (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Present Addresses

(G.F.) Moscow Institute of Physics and Technology (State University), Dolgoprundy, Moscow Region, 141700, Russia. (R.M.L.) Sandia National Laboratories, MS 1423, P.O. Box 5800, Albuquerque, New Mexico 87185-1423, United States (Y. Y.) Joint Quantum Institute, NIST and the University of Maryland, 100 Bureau Drive Gaithersburg, MD 20899, United States Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS ́ The authors thank Antonio Bianconi and Antonio GarciaGarciá and for fruitful discussions. This work was supported by the NSF (DMR-0907220, DMR-1008242, and DMR0521170). S.E.S. acknowledges support by the AFOSR Grant FA9550-11-1-0311. G.F. acknowledges support of the Russian Foundation for Basic research (Grant 15-02-07841). T.M.K. acknowledges financial support from the European Research Council Advanced Grant 339306 (METIQUM) and from the Ministry of Education and Science of the Russian Federation under Contract No. 14.B25.31.0007. 7864

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