Abstract

In electronic transport, umklapp processes play a fundamental role as the only intrinsic mechanism that allows electrons to transfer momentum to the crystal lattice and, therefore, provide a finite electrical resistance in pure metals1,2. However, umklapp scattering is difficult to demonstrate in experiment, as it is easily obscured by other dissipation mechanisms1,2,3,4,5,6. Here we show that electron–electron umklapp scattering dominates the transport properties of graphene-on-boron-nitride superlattices over a wide range of temperature and carrier density. The umklapp processes cause giant excess resistivity that rapidly increases with increasing superlattice period and are responsible for deterioration of the room-temperature mobility by more than an order of magnitude as compared to standard, non-superlattice graphene devices. The umklapp scattering exhibits a quadratic temperature dependence accompanied by a pronounced electron–hole asymmetry with the effect being much stronger for holes than electrons. In addition to being of fundamental interest, our results have direct implications for design of possible electronic devices based on heterostructures featuring superlattices.

Main

In umklapp electron–electron (Uee) scattering, a crystal lattice gives a pair of interacting electrons a momentum kick such that

$${\bf{k}}_1 + {\bf{k}}_2 = {\bf{k}}_3 + {\bf{k}}_4 + {\bf{g}}$$
(1)

where \(\hbar {\bf{k}}_{1,2}\) and \(\hbar {\bf{k}}_{3,4}\) are the initial and final momenta of the two electrons near the Fermi level, respectively, and \({\bf{g}}\) is a non-zero reciprocal lattice vector of the crystal. In clean metals, normal electron–electron scattering, such that \({\bf{g}} = 0\), does not lead to a finite resistance because electron–electron collisions do not relax the momentum imparted to the electron system by the electric field (unless the charge carriers involved have the opposite polarity so that, for example, electrons scatter at thermally excited holes7,8,9). This can be understood by considering the case of head-on collisions along the direction of the electric field: if one electron is scattered backwards, the other must be scattered in the forward direction to conserve momentum, as illustrated in Fig. 1a (left) for Dirac electrons in one of the graphene valleys. In contrast, in umklapp processes (Fig. 1a, right), both electrons near the Fermi level can be scattered in the backward direction with the Bragg momentum, \(\hbar{\bf{g}}\), transferred to the lattice. This behaviour originates from the peculiar nature of electrons in periodic potentials, whose momentum is conserved only up to one reciprocal lattice vector (\(\hbar{\bf{g}}\)).

Fig. 1: Umklapp scattering and excess resistivity in graphene superlattices.
Fig. 1

a, Normal electron–electron scattering for, for example, holes in graphene does not lead to resistivity (left), in contrast to the umklapp scattering for holes in a graphene superlattice (right). Here we also illustrate the superlattice Brillouin zone (purple hexagon) and superlattice minibands in the valence band of graphene. b, Longitudinal resistivity for non-aligned (orange) and aligned (green) graphene/hBN devices. Solid curves: low T = 10 K. Dashed: 200 K. Inset: optical image of the superlattice device in b. Scale bar, 5 μm. c, T-dependent resistivity, Δρ, at a fixed n= −1 × 1012 cm−2 for four superlattice devices and the non-aligned device (orange symbols). Error bars are smaller than the data points. Inset: device schematic and measurement scheme. The top illustration is a moiré pattern arising from 1.8% lattice mismatch in aligned graphene (blue) and hBN (grey) crystals.

Although recent theories predict a dominant role of Uee scattering in some classes of conductors10,11, experimental evidence has so far been reported only for a few ultraclean metals1,2 and in laterally modulated two-dimensional (2D) electron gases in GaAs/AlGaAs quantum wells3,4,5,6. In both cases, the umklapp contribution was relatively small and noticeable only at T< 15 K, being dwarfed by other thermal processes at higher T. In this report, we show both theoretically and experimentally that, in graphene moiré superlattices, Uee scattering dominates T-dependent resistivity over a wide range of carrier densities, n (a representative miniband12,13,14 for Dirac electrons in graphene superlattices is shown in Fig. 1a, right).

The devices we studied (inset of Fig. 1b) were fabricated using standard methods for assembling encapsulated graphene/hexagonal boron nitride (hBN) heterostructures (see Methods) where a superlattice was engineered by aligning graphene with an hBN substrate. This produces a moiré pattern15 due to the small lattice mismatch (δ ≈ 1.8%) between the two crystals (inset of Fig. 1c), which in turn creates a superlattice potential with a period of around 15 nm for perfect alignment. The superlattice potential acts on charge carriers in the graphene and causes significant reconstruction of the electronic spectrum. In particular, a mini-Brillouin zone is created around the Dirac points of graphene13,14, whose size is determined by the misalignment angle, θ, and resulting moiré period, λ. As the Brillouin zone is small compared to normal metals, Uee scattering becomes dominant in graphene/hBN superlattices. We present our results referring to 6 superlattice devices and, for comparison, a reference device in which the graphene and hBN axes were intentionally misaligned (θ > 15°; λ < 3 nm). For aligned samples, λ was determined from the frequency of Brown–Zak oscillations using magnetotransport measurements16 (Supplementary Section 1). At low T and small n, all of our devices exhibited high mobilities of up to 500,000 cm2 Vs−1.

Figure 1b plots the resistivity ρxx as a function of doping (n) for 2 of our graphene devices at 10 K (solid lines) and 200 K (dashed). One of them is the reference, non-aligned sample (orange curves) whilst the other has a misalignment angle close to 0° and λ ≈ 15 nm (green). At low T, both devices exhibit comparable values of ρxx for small n, with sharp peaks at zero doping and the resistivity that drops off rapidly with increasing n for both electrons and holes (positive and negative n, respectively). The measured ρxx are rather similar except for additional satellite peaks that occur in the superlattice at \(n = \pm n_0 = 8/\sqrt 3 {\lambda }^2\) because of secondary Dirac points located at the edges of the superlattice Brillouin zone. At 200 K, however, the two devices exhibit remarkably different behaviour even for |n| < |n0|. In non-aligned graphene, the resistivity at 200 K is only marginally larger than that at 10 K. This weak T dependence stems from the low electron–phonon coupling intrinsic to the stiff atomic lattice of the graphene. In stark contrast, the superlattice device exhibits a huge increase in ρxx, which is accompanied by a pronounced electron–hole asymmetry. Such a behaviour cannot be attributed to electron scattering on thermally activated holes7,8,9 because the effect is much stronger for doping away from the main Dirac point where the Fermi energy εF>kBT and the system behaves as a metal rather than a gapless semiconductor. To compare devices with different electronic quality, we analysed the T-dependent part of resistivity, Δρ, by subtracting ρxx at the base T of 10 K from the measured data: Δρ = ρxx(T) – ρxx (10 K). We have chosen 10 K to avoid an obscuring contribution from mesoscopic fluctuations at lower T. Figure 1c plots Δρ(T) for the studied devices at a fixed density of holes. There is a huge excess resistivity in the graphene superlattice that grows rapidly with the moiré period. As shown below, this behaviour can accurately be described by a dominant contribution from Uee scattering.

Figure 2a details our observations by plotting ρxx as a function of n (normalized by n0) for four superlattice devices, focusing on the doping level 0.2n0 < |n| < 0.7n0, away from the Dirac points, miniband edges and van Hove singularities. In this range of n, the reconstruction of the Dirac spectrum is weak and thermal excitations of carriers with the opposite sign of effective mass can be neglected9. The solid and dashed lines in Fig. 2a represent ρxx at 10 K and 100 K, respectively, whereas the coloured shaded areas emphasize changes in resistivity. Notably, the electron–hole asymmetry increases with n and, also, becomes more pronounced with increasing λ. Such asymmetry is absent in our reference graphene at any T. To emphasize this observation, Fig. 2b plots Δρ(100 K) for these superlattice devices.

Fig. 2: Electron–electron scattering and its electron–hole asymmetry in graphene superlattices.
Fig. 2

a, Resistivity for different λ (colour-coded) as a function of density n. Their n0 were between 2 and 3.7 × 1012 cm−2. Solid curves: 10 K. Dashed: 100 K. The curves for λ = 13.6 and 15.1 nm are offset for clarity by 200 and 400 ohm, respectively. The coloured shaded areas emphasize the T-dependent parts of ρxx for different λ. Data close to the neutrality points n = 0 and ±n0 are omitted (grey shading) because they exhibit activated behaviour of little interest for this study. b, Open circles show experimental Δρ(T) (same colour coding as in a). The error bars are smaller than the symbols. Solid curves: calculated Uee contribution. Note that a small density-independent offset of ρBG = 10 has been added to the theoretical curves to account for the resistivity generated by scattering at acoustic phonons in the Bloch–Grüneisen regime19,24,25 . The inset depicts an umklapp process for the threshold density nc such that kF = g/4 where the momentum transferred to the superlattice corresponds to the exact backscattering of a pair of electrons (orange and green balls).

To explain this behaviour, we model Uee scattering for Dirac electrons in either the conduction (s = +) or valence (s = −) band of graphene using perturbation theory in both the electron–electron Coulomb interaction and moiré superlattice potential. The use of this approach is justified by considering that, in the range of densities 0.2n0 < |n| < 0.6n0, the superlattice inflicts only a weak change on the Dirac spectrum and the Dirac velocity13 as, for example, shown previously in angle-resolved photoemission studies of graphene/hBN heterostructures17,18. In the perturbative approach, one can envisage an Uee process as a scattering event in which one of the electrons scatters into a state on the opposite side of the Fermi circle, whereas the other goes into an intermediate state with a much larger momentum (and therefore off the energy shell) and then is returned back to the Fermi line by Bragg scattering off the superlattice. The overall amplitude of such a process is accounted for by four Feynman diagrams, in which Bragg scattering can occur either before or after an electron–electron collision and involves either the first or second electron,

Here stands for the screened Coulomb interaction, ■ stands for the superlattice perturbation leading to Bragg scattering with momentum transfer \(\hbar \bf{g}\) and = describes the electronic propagator of the intermediate virtual state, where v is the Dirac velocity in graphene and s′ = ± refers to the conduction and valence band, respectively. To account for the superlattice scattering, we employ the previously developed model13,14 to describe electron scattering with the shortest six moiré superlattice reciprocal lattice vectors,

\({\bf{g}}_{m = 0, \cdots 5} = \left( { - {\mathrm{sin}}\left[ {\phi + \frac{{{\rm \uppi} m}}{3}} \right],{\mathrm{cos}}\left[ {\phi + \frac{{{\rm \uppi} m}}{3}} \right]} \right)g\), where \(g = \frac{{4{\rm \uppi }}}{{\sqrt 3 \lambda }}\) and \(\phi = {\mathrm{arctan}}\left[ {\frac{{\sin }}{{{\mathrm{\delta }} + 1 - {\mathrm{cos}}}}} \right]\).

Hence, for the first diagram in equation (2), the intermediate state has a wavevector \({\bf{p}}^\prime = {\bf{k}}_1 + {\bf{g}}_m\), and the matrix element for superlattice scattering is

$$\blacksquare \equiv W\left({{\bf{g}}_m} \right) = \frac{1}{2}\left[ {U_0h_ + + i\left( { - 1} \right)^mU_3h_ - + \left( { - 1} \right)^mU_1h_1} \right]$$
(3)

with \(h_ \pm = 1 \pm ss^\prime {\rm{e}}^{i\left( {\vartheta _{{\bf{k}}_1} - \vartheta _{{\bf{p}}^\prime }} \right)}\) and \(h_1 = s{\rm{e}}^{{{i}}\left( {\vartheta _{{\bf{k}}_1} - \frac{{\uppi m}}{3}} \right)} + s^\prime e^{{{i}}\left( {\frac{{{\rm \uppi} m}}{3} - \vartheta _{{\bf p}^\prime }} \right)}\), determined by the chirality of the electron states and the sublattice structure of the superlattice Hamiltonian13,14 (\(\vartheta _{\bf k}\) is the angle between \({\bf{k}}\) and the x axis) and \(U_i\) are the phenomenological parameters controlling the superlattice potentials. We use U0 = 8.5 meV, U1 = −17 meV and U3 = −14.7 meV, which were determined from the previous independent study of transverse magnetic focusing in graphene/hBN superlattices12.

For the Coulomb interaction in the first diagram of equation (2),

we use19,20 with the Thomas–Fermi wavevector \(q_{{\mathrm{TF}}} = \frac{{4e^2k_{\rm F}}}{{v\kappa }}\), and the dielectric constant of hBN, κ ≈ 3.2. Then, the first diagram in equation (2) is given by

$$M_{ss\prime }^{{\bf{g}}_m}\left( {{\bf{k}}_1,{\bf{k}}_2,{\bf{k}}_3,{\bf{k}}_4} \right) = \frac{{W\left( {{\bf{g}}_m} \right)V\left({{\bf{g}}_m} \right)}}{{sv\left| {{\bf{k}}_1} \right| - s^\prime v\left| {{\bf{p}}^\prime } \right|}}$$

To determine the Uee contribution to resistivity, ρUee, we use the Boltzmann transport theory21 assuming the thermal energy \(k_{\mathrm{B}}T < {\it{\epsilon }}_{\mathrm{F}}\) (Supplementary Section 2), which yields the tensor with \(\alpha ,\beta = x,y\).

$$\rho _{{\mathrm{Uee}}}^{\alpha \beta } = \frac{h}{{e^2}}\frac{{\left( {k_{\rm B}T} \right)^2}}{{24{\rm \uppi }^2v^4k_{\rm F}^2}}\mathop {\sum }\limits_{m = 0, \cdots ,5} g_m^\alpha g_m^\beta {\int} {\left| {\mathop {\sum }\limits_{s^\prime = \pm } M_{ss^\prime }^{{\bf{g}}_m}} \right|^2} \frac{{d\vartheta _{{\bf{k}}_1}d\vartheta _{{\bf{k}}_3}}}{{\left| {{\mathrm{sin}}\vartheta _{24}} \right|}}$$
(5)

This expression was derived using the approximation \({\bf{k}}_i \approx k_{\rm{F}}( {{\mathrm{cos}}[ {\vartheta _{{\bf{k}}_i}} ],{\mathrm{sin}}[ {\vartheta _{{\bf{k}}_i}} ]} )\), where ki are related by equation (1), and the scattering angle ϑ24 is such that

$${\mathrm{cos}}\left( {\vartheta _{24}} \right) = \cos \left( {\vartheta _{{\bf{k}}_1} - \vartheta _{{\bf{k}}_3}} \right) - \frac{g}{{k_{\rm F}}}\left( {\frac{g}{{2k_{\rm F}}} + \sin \left( {\vartheta _{{\bf{k}}_1} - \phi - \frac{{{\rm \uppi} m}}{3}} \right) - {\mathrm{sin}}\left( {\vartheta _{{\bf{k}}_3} - \phi - \frac{{{\rm \uppi} m}}{3}} \right)} \right)$$

As a result of the three-fold rotational symmetry of graphene superlattices, the resistivity tensor is isotropic, that is, \({\mathrm{\rho }}_{{\mathrm{Uee}}}^{\alpha \beta } = {\mathrm{\rho }}_{{\mathrm{Uee}}}\delta ^{\alpha \beta }\). We note that Uee scattering can occur only above the threshold kF > g/4 (inset of Fig. 2b), which arises from the fact that all scattering states must be in the vicinity of the Fermi level, ki ≈ kF in equation (1) and yields the critical density \(n_{\rm c} = n_0{\rm \uppi} /8\sqrt 3 \approx 0.227n_0\) below which Uee scattering is not allowed.

Using the superlattice parameters Ui stated above, we calculated ρUee for the specific experimental parameters in Fig. 2a. The results (solid curves in Fig. 2b) are in good agreement with the experiment, which is particularly impressive considering that no additional fitting parameters were used. Note that the deviations between the experiment and theory for electron doping in Fig. 2b are mostly due to a limited accuracy of our analytical method as the full numerical analysis shows (see Supplementary Fig. 5a). Furthermore, the analytical theory, equation (5), suggests that close to the threshold density \({\mathrm{\rho }}_{{\mathrm{Uee}}}\left( {\left| n \right| - n_{\rm{c}}} \right)^{3/2}\), which stems from the interplay between the size of the phase space available for Uee scattering and the ‘chirality factor’ (for example, \(\left( {1 + {\rm{e}}^{i\left( {\vartheta _{{\bf{k}}_2} - \vartheta _{{\bf{k}}_4}} \right)}} \right)/2\) in equation (4)) which suppresses the amplitude of backscattering22. The large asymmetry between ρUee for electrons and holes arises from the fact that kinematic constraints dictate that the electron Bragg scattering by the superlattice must be almost backscattered (inset of Fig. 2b). The probability for such backscattering,

$$P \sim \left| {U_1 - sU_3} \right|^2$$
(6)

is much higher in the valence band (s = −1) than the conduction band (s = 1) for the given U1 and U3 used in equation (3) so that the Uee process is much more effective for hole rather than electron doping. Note that this feature of Uee distinguishes itself from other scattering mechanisms including the potential disorder in the moiré superlattice23, which results in almost electron–hole symmetric ρxx within the density range −0.7n0 < n < 0.7n0.

Our analysis predicts two further signatures of Uee scattering. First, for a given |n| > nc, the efficiency of umklapp processes depends on λ such that ρUeeλ4. This behaviour is governed entirely by the Uee matrix element (\(M_{ss^\prime }^{{\bf{g}}_m}\)) and can be understood by counting the number of Bragg vector factors (\(g \propto \lambda ^{ - 1}\)) that appear in equations (3)–(5), recalling that momenta kF, \(\left|{{\bf{k}}_i} \right|\) and \(\left| {{\bf{p}}^\prime } \right|\) scale linearly with g for a fixed n/n0. Figure 3 shows that the λ4 dependence describes well the experimentally observed behaviour of Δρ. This unusually strong dependence is one of the reasons why misaligned devices with small superlattice periods do not exhibit any discernible umklapp resistivity. Second, equation (5) yields a quadratic temperature dependence typical for electron–electron scattering in the Fermi liquid theory, ρUee T2, in agreement with the experimental behaviour plotted in the inset of Fig. 3. The T2 behaviour holds over a wide T range for all of our superlattice devices, showing the dominance of Uee scattering. However, at high T > 150 K, one can see some deviations from the T2 dependence. We attribute those to the thermal excitation of carriers with the opposite sign of effective mass, resulting in deviations of resistivity from the values described by equation (5). Indeed, these deviations become stronger as we approach either the main Dirac point (|n| < 0.3n0) or van Hove singularities13 (|n| > 0.6n0) where scattering at thermally excited carriers of the opposite polarity starts playing a role.

Fig. 3: Characteristics of umklapp electron–electron scattering.
Fig. 3

The temperature-dependent part of the resistivity as a function of the moiré period for n = −0.5n0 in all of our six superlattice devices. The circles are experimental data; the dashed line is the best fit of a λ4 dependence to the data; and the solid line is the calculated Uee contribution to the resistivity (also proportional to λ4). Inset: symbols are experimental data for the superlattice devices (colour-coded). The dashed lines are T2 fits to the experimental data. Logarithmic scales are used in both the main plot and the inset. Standard deviations in our measurements are smaller than all of the symbols.

Finally, we analysed the normal and umklapp (due to the moiré superlattice) scattering of electrons at acoustic phonons in graphene. The normal electron–phonon scattering, studied in detail previously19,24,25, can result in approximately a 10 Ω contribution for the relevant n at 100 K (the value used as an offset in Fig. 2b) and up to 30 Ω at 300 K. An additional scattering from phonons in the hBN may also contribute to the deviations. This is discussed in Supplementary Section 2C–E, where we consider a possibility that electrons scatter off acoustic phonons in graphene and hBN by transferring additional momentum \(\hbar \bf{g}\) to the moiré superlattice (Supplementary Fig. 5c). When analysing such processes, we took into account the intrinsic electron–phonon coupling (deformation potential) in graphene, piezoelectric coupling with deformations in hBN and dynamical variations of the moiré potential due to a mutual displacement of graphene and hBN, which are caused by vibrations of the two crystals. We find that the calculated phonon-induced umklapp resistivity is much smaller and has different dependences on T and λ, as compared to those caused by Uee scattering and observed experimentally (Supplementary Fig. 5).

To conclude, Uee scattering in long-period moiré superlattices degrades the intrinsic high-T mobility of graphene’s charge carriers. This limits the potential applications of epitaxial grown graphene/hBN heterostructures26, which are inherently aligned, for room-temperature high-mobility devices. To achieve high carrier mobility at room temperature, the 2D crystals that form the heterostructure should be misaligned. We expect that Uee scattering should strongly influence electron transport in twisted graphene bilayers, where superlattice effects have also been predicted27,28 and recently observed29,30,31,32,33.

Methods

Device fabrication

The graphene/hBN devices presented in the main text were fabricated following similar methods reported previously34. First, we used a dry-transfer method35 for assembling the heterostructures. We obtained monolayer graphene and few-layer hBN by mechanical exfoliation of graphite and bulk hBN crystals onto a silicon/silicon dioxide (Si/SiO2) wafer. After the appropriate flakes were identified, we used a polymer membrane attached to the tip of a micromanipulator to assemble the heterostructure. This was performed on a rotating stage that allowed us to control the relative angle between two crystal lattices to a precision of about 0.5°. We first assembled monolayer graphene on the hBN substrate. During this step, we used the rotating stage to try to align their crystallographic axes to produce a moiré superlattice15. As the flakes cleave preferentially along their crystallographic directions, the straight edges of the few-layer crystals tell us their relative orientation. However, alignment of straight edges does not guarantee alignment of the crystal axes because of the two types of edge that exist (armchair or zigzag). Therefore, we performed atomic force microscopy measurements of the graphene/hBN stack to check for an underlying moiré superlattice36. If a superlattice was obtained, we then placed a second hBN flake on top of the stack to fully encapsulate the graphene flake and preserve its intrinsic electronic quality34. We then used standard methods in electron beam lithography to fabricate the Hall bar geometry and define quasi-one-dimensional contacts to the graphene edge18,37. After fabrication, the structural properties of the moiré superlattice were then confirmed in transport experiments by measuring the frequency of Brown–Zak oscillations16 and the position of secondary Dirac points38,39,40 (see Supplementary Section 1 for details).

Transport measurements

Our electrical measurements were carried out by standard low-frequency (10–30 Hz) a.c. techniques using a lock-in amplifier. The measured resistance (Rxx) was determined by driving a small excitation current (Ia.c. = 0.1–1 μA) down the length of the device channel while simultaneously measuring the 4-probe voltage drop (Va.c.) between a different pair of contacts; R = Va.c./Ia.c. Electrostatic gating of our samples was achieved by applying a d.c. voltage between the silicon substrate and the device channel, which are electrically isolated by a thin (90/290 nm) SiO2 dielectric and the bottom hBN (30–50 nm) layer of our graphene heterostructure. For temperature control, we performed experiments in the variable temperature insert of a helium-flow cryostat.

Data availability

The data that support plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Additional information

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Acknowledgements

We would like to thank C. Woods, S. Slizovskiy and F. Guinea for useful discussions. This work was supported by the European Research Council Synergy Grant and Advanced Investigator Grant, Lloyd’s Register Foundation Nanotechnology Grant, EC European Graphene Flagship Project, the Royal Society and EPSRC (including the EPSRC CDT NOWNANO).

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Affiliations

  1. National Graphene Institute, University of Manchester, Manchester, UK

    • J. R. Wallbank
    • , R. Krishna Kumar
    • , M. Holwill
    • , G. H. Auton
    • , J. Birkbeck
    • , A. Mishchenko
    • , K. S. Novoselov
    • , A. K. Geim
    •  & V. I. Fal’ko
  2. School of Physics & Astronomy, University of Manchester, Manchester, UK

    • R. Krishna Kumar
    • , M. Holwill
    • , Z. Wang
    • , J. Birkbeck
    • , A. Mishchenko
    • , K. S. Novoselov
    • , A. K. Geim
    •  & V. I. Fal’ko
  3. Department of Physics, Lancaster University, Lancaster, UK

    • L. A. Ponomarenko
  4. National Institute for Materials Science, Tsukuba, Japan

    • K. Watanabe
    •  & T. Taniguchi
  5. Physics Department, Columbia University, New York, NY, USA

    • I. L. Aleiner

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Contributions

This study was consummated by J.R.W., V.I.F. and A.K.G.; J.R.W, I.L.A and V.I.F. have developed theory for the studied effect. hBN was provided by T.T. and K.W. The devices were fabricated by M.H., G.H.A. and J.B. Transport measurements were performed by R.K.K., Z.W. and A.M. under the supervision of K.S.N., L.A.P. and A.K.G. All authors have contributed to the discussions of results. The manuscript was written by J.R.W, R.K.K., V.I.F. and A.K.G.

Competing interests

The authors declare no competing interests.

Corresponding author

Correspondence to V. I. Fal’ko.

Supplementary information

  1. Supplementary Information

    Supplementary Figures 1–5; Supplementary References 1–18; Additional mathematical derivations

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DOI

https://doi.org/10.1038/s41567-018-0278-6