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Imaging resonant dissipation from individual atomic defects in graphene

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Science  08 Dec 2017:
Vol. 358, Issue 6368, pp. 1303-1306
DOI: 10.1126/science.aan0877

Watching electrons lose steam in graphene

Although graphene can be fabricated to be extremely clean, it still has a nonzero electrical resistance. Resistance is associated with turning electrons' energy into heat, but how exactly does this happen? Halbertal et al. used a tiny scanning temperature probe based on a superconducting quantum interference device to investigate this problem. As the current flowed through a square-shaped sample of graphene, electrons lost energy predominantly in the vicinity of atomic-scale defects, which were few and far between in the bulk but much more common on the edges of the sample.

Science, this issue p. 1303

Abstract

Conversion of electric current into heat involves microscopic processes that operate on nanometer length scales and release minute amounts of power. Although central to our understanding of the electrical properties of materials, individual mediators of energy dissipation have so far eluded direct observation. Using scanning nanothermometry with submicrokelvin sensitivity, we visualized and controlled phonon emission from individual atomic-scale defects in graphene. The inferred electron-phonon “cooling power spectrum” exhibits sharp peaks when the Fermi level comes into resonance with electronic quasi-bound states at such defects. Rare in the bulk but abundant at graphene’s edges, switchable atomic-scale phonon emitters provide the dominant dissipation mechanism. Our work offers insights for addressing key materials challenges in modern electronics and enables control of dissipation at the nanoscale.

Understanding the microscopic mechanisms of momentum and energy dissipation is a central problem in fields ranging from condensed matter to particle physics. It is also of keen interest to researchers designing new approaches to handle, convert, and use energy. Dissipation pathways are particularly intriguing in ultrapure materials, such as graphene (1), because of tight restrictions on the phase space for electron-phonon scattering (24). Furthermore, in condensed matter physics, we are interested in processes concealed within materials. This lack of access, along with the fact that the power released in such processes is very small, poses a key challenge for experimentally probing dissipation at the nanoscale. We used a recently developed ultrasensitive scanning nanothermometer with submicrokelvin sensitivity (5), achieved with a superconducting quantum interference device (SQUID) placed on an extremely sharp tip (SQUID-on-tip, or SOT), to probe these subtle effects in high-mobility graphene. Owing to its exceptional cleanliness and the two-dimensional nature of its electrons and phonons (1), graphene is an excellent platform to study electron-phonon relaxation.

Our measurements were performed on exfoliated graphene encapsulated between hexagonal boron nitride (hBN). Charge carriers were injected into a micron-scale “electron chamber” by applying electrical current through narrow constrictions [Fig. 1, A and B, and section 1 of (6)]. Transport measurements in such samples routinely show ballistic signatures over a wide range of temperatures and carrier densities (79). Our SQUID-on-tip (10) acts as an extremely sensitive thermometer (tSOT) (5) with an effective diameter of 33 nm and thermal sensitivity of 510 nK/Hz1/2 at 4.2 K, and it provides a map of the local temperature variations δT(x, y) upon scanning at a height h of 10 to 40 nm above the sample surface (as specified for each measurement). The tip is mounted on a quartz tuning fork (11), which allows the tSOT to vibrate parallel to the sample surface with a controlled amplitude xac = 2.7 nm at a frequency of ~37 kHz [sections 2 and 3 of (6)]. The resulting ac signal, Tac(x, y) = xac∂δT(x, y)/∂x, renders higher-sensitivity imaging (Fig. 1C) by avoiding the low-frequency 1/f noise of the tSOT. We control the carrier density in graphene globally by a back-gate bias Vbg to the Si/SiO2 substrate and locally by applying a tip potential Vtg to the tSOT (Fig. 1A). Description of sample fabrication and thermal imaging is provided in (6).

Fig. 1 Observing individual dissipation sources in a graphene heterostructure.

(A) Schematic side view of the measurement setup with the hBN-graphene-hBN heterostructure and SQUID-on-tip nanothermometer (tSOT). (B) Optical image of the device patterned into a square chamber (bright; 4 μm × 4 μm). A fixed current Idc = 3 μA is driven through the connecting constrictions (arrows). (C) Scanning ac nanothermometry Embedded Image of the area outlined in (B) at Vbg = –2 V, Vtg = 9 V, and 4.2 K. The tSOT scans the sample while oscillating with amplitude xac = 2.7 nm at 12° to the x axis. Scan area, 5.5 μm × 5 μm; pixel size, 18 nm; scan speed, 20 ms per pixel; height h = 20 nm; Idc = 3 μA. The sharp rings (marked A, B, and C) uncover three isolated sources of dissipation in the bulk of graphene, in addition to a dense array of resonances along the graphene edges. (D) Zoomed-in view of defect C at Vtg = 5 V. Scan area, 140 nm × 150 nm; pixel size, 1.9 nm; scan speed, 20 ms per pixel; h = 20 nm; Idc = 3 μA. (E) Illustration of an atomic defect in graphene that creates a localized resonant dissipative state at the center of the ring (D), which in turn mediates inelastic scattering of an impinging electron (red) into a phonon (orange) and a lower-energy electron (blue). The defect, forming an sp3 orbital (green), can arise from a carbon vacancy, adatom, or admolecule.

Figure 1C shows the thermal signal Tac(x, y) measured while applying a fixed current Idc = 3 μA through two of the constrictions, as shown in Fig. 1B. The image reveals a complex array of fine rings along the edges of the heterostructure (5). In addition, three isolated rings are observed in the bulk of graphene, labeled A, B, and C. The bulk rings are rare and have comparable diameters, in sharp contrast to the rings at the edges, which are dense and display widely varying sizes (movies S1 and S2). We show below that the rings mark dissipation from single atomic defects positioned at their centers.

Electron-phonon cooling pathways in graphene are particularly interesting for several reasons. Owing to the exceptional stiffness of the carbon bonds, scattering by optical phonons in graphene is inefficient below room temperature. Moreover, the small size of the electron’s Fermi surface restricts the phase volume for scattering by acoustic phonons, blocking the intrinsic electron-lattice relaxation pathway (24). However, theory predicts that disorder can substantially ease the electron-phonon scattering (4). Our key finding is that hot electrons, generated by the applied current, dissipate their energy through a very specific disorder-induced mechanism: resonant inelastic scattering by local electronic resonances caused by individual defects. Each such localized state (LS) mediates cooling through resonant electron scattering, creating an atomic-scale thermal link between the electronic bath, at an effective hot-electron temperature Te, and the phonon bath, at a base temperature Tp [section 4 of (6)]. To characterize this thermal link, we define the electron-phonon heat conductivity of a defect, κep(ε), which describes the power transferred between the baths, Pep(ε) = κep(ε)ΔT (where ΔT = TeTp and ε is the energy relative to the Dirac point). The resulting “cooling power spectrum” Pep(ε) (CPS)—the fundamental quantity accessed in our experiment through local temperature increase δT(ε) ∝ Pep(ε)—is found to peak sharply when the Fermi level EF is aligned with the quasi-bound LS resonant energy ELS.

Our analysis [section 4 of (6)] of the observed resonances suggests that they originate from quasi-bound states arising at a carbon vacancy or adatom bonded to a single C atom when its sp2 orbital transforms to an sp3 state (1215). Such defects are known to produce sharp electronic resonances at energies near the Dirac point (16, 17). Although these defects have been extensively investigated by means of ab initio calculations and scanning tunneling microscopy (STM) (1219), their prominent role in dissipation has not been anticipated by previous work. The defect-induced CPS originates from the part of the local density of electronic states that mediates electron-phonon coupling [EP-LDOS, Dep(ε)]—a hitherto inaccessible quantity—convoluted with the electron and phonon Fermi and Bose energy distribution functions, respectively [section 4 of (6)]. Our measurements and analysis indicate that the spectral width of Dep(ε) is much greater than the thermal broadening, in which case the CPS can be approximated as Pep(ε) ∝ Dep(ε) [section 4 of (6)]. This quantity can be probed experimentally, as illustrated schematically in Fig. 2, A to C. By parking the tSOT above the defect and varying Vtg, we can induce local band bending that shifts Dep(ε) with respect to EF [Fig. 3A and section 6 of (6)]. The resulting variation in the measured temperature, δT(Vtg), provides a spectroscopic measurement of Pep(ε) ∝ Dep(ε) (Fig. 2B). Additional information can be obtained by tuning EF by the back-gate Vbg, yielding resonant peaks in Dep(ε) aligned as diagonal lines in the δT(Vtg, Vbg) map (Fig. 2C). Importantly, in this configuration, phonon emission can be turned on and off by applying a potential on the tSOT tip, demonstrating control of nonequilibrium dynamics and probing them at the nanoscale.

Fig. 2 Thermal nanospectroscopy of dissipative localized states.

(A) Schematic example of a spectrum of EP-LDOS [Dep(ε), blue] and the graphene Dirac dispersion relation (green). a.u., arbitrary units; k, wave number. (B) Temperature variation δT(Vtg) as would be measured above the defect versus tip potential Vtg, providing nanoscale spectroscopy of the cooling power spectrum Pep(ε) by tip-induced band bending. (C) The expected δT(Vbg, Vtg), showing diagonal resonance lines that map the peaks in Dep(ε) [calculation details are given in section 6 of (6)]. (D) The experimental δT(Vbg, Vtg), measured by the tSOT at h = 10 nm above the center of defect C in the presence of Idc = 3 μA, revealing a single resonance dissipation line (data were not taken for the white portion of the plane in this measurement). The signal was obtained through integration of Embedded Image over Vtg [section 2 of (6)]. (E) A zoomed-in line cut of δT(Vtg) along the dashed line in (D) at Embedded Image V. (F) The cooling power spectrum Pep(ε) ∝ Dep(ε), derived from the measurement of the resonant dissipation ring [section 9 of (6)], showing a single sharp peak near the Dirac point at εLS = –22 meV.

Fig. 3 Origin of the resonant ring structures.

(A) Schematic description of the system’s energy levels corresponding to tip position marked by the black point in (C): two localized states with a peak in EP-LDOS at ELS pinned to Dirac energy ED. The tip potential Vtg induces band bending, resulting in the calculated position-dependent ED(x) (blue). The left LS is off-resonance, whereas the tSOT positioned at x = –R brings the right LS into resonance for inelastic scattering of electrons injected at energy Ee (red arrows). The result is a point source of phonon emission (red arcs). The Fermi energy EF (gray dashed line) is determined by the back-gate voltage Vbg. (B) Red dashed lines, schematic temperature profiles resulting from ballistic two-dimensional phonon emission from LS at resonant conditions, Embedded Image, with Embedded Image nm [section 5.2 of (6)]. Red solid line, calculated temperature variation δT(x) measured by the scanning tSOT that brings the LS into resonance at a distance ±R from each defect. Blue solid line, calculated Tac(x) = xacδT(x)/∂x, as would be measured by the tSOT vibrating with amplitude xac parallel to the surface [section 6 of (6)]. (C) Tac(x, y) measured in the central region of Fig. 1C, showing dissipation rings around defects B and C. The dashed line describes the scan direction depicted in (A). Scan area, 3.5 μm × 0.4 μm; pixel size, 8 nm; scan speed, 20 ms per pixel; h = 20 nm; Vbg = –0.3 V; Vtg = 5.5 V; Idc = 6 μA; xac = 2.7 nm directed at 18° to the x axis.

The experimental δT(Vtg, Vbg) map of defect C (Fig. 2D) displays, in agreement with the discussion above, a sharp resonance (Fig. 2E) along a single diagonal line, which passes close to the origin in the Vtg-Vbg plane. This resonance is caused by the presence of a LS with a narrow single peak in Dep(ε) (Fig. 2F) close to the Dirac energy [section 8 of (6)]. Such LSs give rise to the sharp thermal rings observed in Fig. 1C as follows. For the LS to cause inelastic electron scattering, its energy level ELS has to be aligned with the energy of the impinging electrons, EeEF (the global Fermi energy). For a given tip potential Vtg, this resonant condition occurs when the tip is located at a distance R from a LS [Fig. 3A, movie S3, and section 6 of (6)]. As a result, each LS displays a sharp peak in δT(x, y) and Tac(x, y) along a ring of radius R (Fig. 3B), describing the ring formations in Figs. 1C and 3C.

The variable tip and back-gate potentials also provide the spectroscopic means to extract the energy level and the CPS of the LS. By repeatedly scanning the tSOT along the line crossing the defect and incrementing Vbg, a bell-shaped resonance trace is obtained (Fig. 4A). The shape and polarity of the trace confirm the electrostatic picture described above [Fig. 4, A to C; section 7 of (6); and movies S5 and S6], and its asymptotic value Embedded Image describes the energy level ELS of the LS. Similar information can be obtained by sweeping Vtg at various values of Vbg [section 7 of (6) and movies S4 and S7].

Fig. 4 Thermal spectroscopy of individual bulk and edge localized states.

(A) Map of Tac(x) line scans through defect C while varying Vbg at Vtg = –5 V, showing the bell-shaped resonance trace. (B) Resonance traces for various values of Vtg that switch their polarity at Embedded Image. (C) Resonance trace as in (A) at Vtg = 4.2 V. In (A) to (C), h = 20 nm; pixel width, 5 nm; pixel height, 30 mV; scan speed, 20 ms per pixel; xac = 2.7 nm; Idc = 3 μA; a linear fit was subtracted from each line to emphasize the resonance traces (fig. S11 and movie S5 show raw images). (D and E) Maps of Tac(x) line scans along the bottom edge of the square graphene sample in Fig. 1C at Vtg = –10 V (D) and 10 V (E). Each bell-shaped trace originates from a single dissipative atomic defect. In (D) and (E), h = 20 nm; pixel width, 4 nm; pixel height, 100 mV; scan speed, 60 ms per pixel; xac = 2.7 nm; Idc = 3 μA. (F) Optical image of sample 2, with a patterned hBN/graphene/hBN heterostructure (bright) on a SiO2 substrate (dark) and dc current Idc chopped at 35.5 Hz with a root mean square value of 1 μA, applied as indicated by the arrows. (G) Thermal image of δT(x, y) revealing dissipation along the graphene edges. Scan area, 4.7 μm × 3.7 μm; pixel size, 50 nm; scan speed, 200 ms per pixel; h = 20 nm; Vbg = 8 V; Vtg = 0.85 V, corresponding to Embedded Image (flat-band condition) at this Vbg.

Our spectroscopic analysis of bulk LSs leads to the following conclusions: (i) The LS energy resides slightly below the Dirac point (εLS = ELSED ≅ –22 meV), as derived from the analysis of the resonance lines [section 8 of (6)]. (ii) The spectral width of the CPS [Pep(ε)] is about 13 meV [Fig. 2F and section 9 of (6)]. (iii) No additional LS energy levels are observed at least 180 meV above and below ELS. If present, additional concentric rings and bell-shaped traces would have formed; however, these are absent for the entire Vbg and Vtg applied range of ±10 V [section 7 of (6)]. (iv) This level spacing puts an upper bound on the spatial extent of the LS of less than 2 nm, based on a charging energy calculation [section 7 of (6)]. (v) The sharp energy level and nanometer spatial extent of the LS closely resemble the characteristic features of atomic defects in graphene derived from ab initio calculations (14) and observed using STM (16, 17), in contrast to more extended nonresonant “puddles” originating from disordered substrate potential (20, 21). (vi) The resonant character of the defects and the energy level close to the Dirac point are consistent with the sp3 band vacancy model. In particular, vacancies and monovalent adatoms are known to form LSs at energies comparable to our measured values (14, 16) [section 4 of (6)] and thus are strong candidates for the observed bulk defects. (vii) The bulk defects are extremely rare (we found only seven such defects in total in 13 samples with a total area of 3500 μm2), corresponding to an estimated average areal density of 2 × 105 cm−2 or volume concentration of 5 × 10−5 parts per million if originating from the parent graphite. Their spectral properties appear to be the same within our experimental resolution [section 8 of (6)], pointing to a common chemical or structural origin.

These conclusions are reinforced by examining the LSs along the edges of graphene, which display spectroscopic features very similar to those of the bulk defects, but with some notable differences. Shown in Fig. 4, D and E, are tSOT line scans along the bottom edge of the sample that cross through numerous LSs, taken while increasing Vbg (movies S6 and S7). Each LS is visible as a bell-shaped trace, similar to the ones in Fig. 4, A to C, indicating the same microscopic origin of different LSs. In contrast to LSs in the bulk, the edge LSs are extremely dense, with some adjacent states only about 1 nm apart (fig. S15E). This puts an additional bound on their spatial extent [section 10 of (6)]. We ascribe their origin to the carbon dangling bonds near the edge with high affinity to adatoms and molecules, giving rise to resonance states formed by the resulting sp3 vacancies (1215). Notably, unlike the LSs in the bulk, the edge LSs display large variations in ELS values, manifested in Fig. 4, D and E, by the vertical spread of the bell-shaped traces over the entire Vbg range of 20 V. This translates into a 260-meV spread in ELS, limited by our bias range. We are able to resolve states that are less than 2 nm apart but differ in their ELS by as much as 160 meV [section 10 of (6)]. Such large energy-level variation may arise from the atomic defects at graphene edges having more diverse chemical origins than the native bulk defects. An alternative explanation is Coulomb interaction between the charged defects. Charging a LS by one electron would shift the energy of a neighboring LS at a distance of 2 nm by ~240 meV [section 7 of (6)], consistent with our observations.

The above results suggest that hot electrons lose most of their energy to phonons at the edges of graphene. To verify this conjecture, we measured the bare δT(x, y) in the absence of tSOT electrostatic influence [Embedded Image at a flat-band condition; section 5 of (6)]. Measurement of sample 2 (Fig. 4, F and G) revealed a higher temperature along the sample edges relative to the temperature in the sample bulk. Together with numerical simulations [section 5 of (6)], this finding implies that the LSs along the graphene edges are the dominant source of dissipation at all doping levels reachable in our experiment. Therefore, the excess phonons, corresponding to the overall temperature rise, are not generated locally in the graphene bulk. Instead, the phonons are predominantly emitted through inelastic electron scattering by those LSs at graphene edges that are at resonance at a given Vbg value [section 5 of (6)]. The observed atomic-scale resonant LSs thus emerge as the leading mechanism of dissipation in clean graphene, each acting as a pointlike source emitting phonons that then propagate ballistically throughout the entire sample.

The observation of sharply localized resonant states and their role in dissipation should have considerable implications for understanding the thermal (22), magnetic (12, 16, 23), chemical (24), and transport (2528) properties of graphene. These states are distinct from the extended edge states anticipated for crystalline graphene edges (23). Further, resonant dissipation is completely different from the conventional nonresonant picture of electron-phonon coupling (24), posing interesting questions for future experimental and theoretical work. This dissipation mechanism may affect the edge transport characteristics (2931) and explain previous observations of the mean free path being limited by the device size in state-of-the-art encapsulated graphene (79). The resonance states, localized at the edge and in the bulk, thus emerge as a key factor governing the dissipation and possibly limiting the carrier mobility in pure graphene.

Supplementary Materials

www.sciencemag.org/content/358/6368/1303/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S15

Movies S1 to S7

References and Notes

  1. Materials and methods are available as supplementary materials.
Acknowledgments: We thank M. E. Huber for the SOT readout setup and M. Solodky for assistance with data analysis. This work was supported by the Minerva Foundation with funding from the German Federal Ministry of Education and Research; NSF/DMR-BSF Binational Science Foundation (BSF) grant no. 2015653 and NSF grant no. 1609519; the Weizmann–UK Making Connections Programme; and a Rosa and Emilio Segré Research Award. A.K.G. and M.B.S. acknowledge support from the Lloyd's Register Foundation and the European Research Council ARTIMATTER project (ERC-2012-ADG). L.S.L. and E.Z. acknowledge the support of the MISTI (MIT International Science and Technology Initiatives) MIT–Israel Seed Fund. Data presented in this paper can be obtained from the corresponding authors upon reasonable request. A patent describing the dissipation imaging technique has been submitted (WO 2016/142945).
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