Abstract
Symmetrybroken electronic phases support neutral collective excitations. For example, monolayer graphene in the quantum Hall regime hosts a nearly ideal ferromagnetic phase at specific filling factors that spontaneously breaks the spinrotation symmetry^{1,2,3}. This ferromagnet has been shown to support spinwave excitations known as magnons that can be electrically generated and detected^{4,5}. Although longdistance magnon propagation has been demonstrated via transport measurements, important thermodynamic properties of such magnon populations—including the magnon chemical potential and density—have not been measured. Here we present local measurements of electron compressibility under the influence of magnons, which reveal a reduction in the gap associated with the ν = 1 quantum Hall state by up to 20%. Combining these measurements with the estimates of temperature, our analysis reveals that the injected magnons bind to electrons and holes to form skyrmions, and it enables the extraction of free magnon density, magnon chemical potential and average skyrmion spin. Our methods provide a means of probing the thermodynamic properties of chargeneutral excitations that are applicable to other symmetrybroken electronic phases.
Main
The interplay between electron–electron interactions and isospin degeneracy in Landau levels can give rise to symmetrybroken states such as quantum Hall ferromagnets (QHFMs)^{1,2}. The spinpolarized ν = 1 state in monolayer graphene stands out due to its remarkable magnetic properties^{3} and represents a unique platform for exploring chargeneutral spin excitations that may be useful for spintronics applications. Recent transport experiments have shown that voltage biases exceeding the Zeeman energy scale E_{Z} = gµ_{B}B, where g is the electron gfactor and µ_{B} is the Bohr magneton, provide enough energy for electrons in the ν = 1 edge channel to flip their spin and scatter out of the edge channel. This transition launches a magnonic excitation that may propagate through the bulk and produce a nonlocal voltage that can be detected micrometres away^{4,5}. However, measurements of the thermodynamic properties of this magnon system—critical for harnessing these potentially useful chargeneutral excitations—remain outside the reach of both transport studies and conventional direct magnetic sensing owing to the dilute magnetization of the system. Here we perform local electronic compressibility measurements of the ν = 1 QHFM with a scanning singleelectron transistor (SET) and examine its response to the presence of magnons. We find that pumping magnons into the system results in a marked reduction in the charge gap, typically of about 15–20%. We argue that this gap reduction is a result of magnons binding with electrons or holes to form skyrmions, which, together with estimates of the temperature, allows us to determine the local magnon chemical potential and free magnon density in the system. The method of extracting the thermodynamic properties of magnons introduced in our experiments suggests novel routes towards realizing and probing Bose–Einstein condensation in QHFMs^{6}, and is more broadly applicable to other flatband systems with spontaneously symmetrybroken states.
The device and measurement setup are shown in Fig. 1a,b. Figure 1c shows the twoterminal conductance G_{2T} between contacts 2 and 3 as a function of backgate voltage V_{BG} and d.c. bias V_{d.c.} at a magnetic field B = 11 T. Consistent with previous studies^{4,5}, for –E_{Z} < V_{d.c.} < E_{Z}, G_{2T} exhibits a plateau as a result of the quantization of the Hall conductance. However, the quantized Hall plateau disappears as soon as V_{d.c.} reaches E_{Z}, signalling the onset of magnon generation and absorption processes.
To study the dependence of the ν = 1 gap on the magnon population, we measure the electron chemical potential µ(ν) as a function of filling factor ν at each value of V_{d.c.} (Methods). Figure 1d shows two representative measurements of the electron chemical potential µ(ν) near ν = 1, with topgate voltage V_{TG} = 0. The trace at V_{d.c.} = 10 mV (Fig. 1d, red curve) clearly exhibits a reduced gap compared with that at 0 mV (Fig. 1d, blue curve). Similar to the transport behaviour, the gap begins to principally change when V_{d.c.} exceeds E_{Z} (Fig. 1e), initially dropping sharply and reaching suppression of about 20% at the highest biases investigated. The gap reduction shown in Fig. 1d,e, observed at many different locations (Extended Data Fig. 1), demonstrates the remarkable sensitivity of the ν = 1 gap to the presence of magnons.
An important piece of evidence that the ν = 1 gap suppression observed for d.c. biases V_{d.c.} > E_{Z} results from magnon generation and absorption is its dependence on the local filling factor under the top gate, ν_{TG}. As a consequence of the spin order present in the region under the top gate, magnons freely propagate across when ν_{TG} = ±1, but only weakly for ν_{TG} = 0 and not at all for ν_{TG} = ±2. Figure 2a shows the a.c. nonlocal voltage V_{NL} measured across contacts 5 and 6, normalized by the a.c. bias V_{a.c.} applied between contacts 2 and 3. In addition to the vanishing nonlocal voltage for V_{d.c.} < E_{Z}, we find that for V_{d.c.} > E_{Z}, no appreciable signal is detected for ν_{TG} > 2 or for ν_{TG} = 0; on the other hand, a strong nonlocal voltage is observed for 0 < ν_{TG} < 2, in accordance with the expected transport characteristics and energysplitting hierarchies shown in previous studies^{4,5}. Next, we perform gap measurements near contact 5 as a function of ν_{TG} and V_{d.c.}, using the same contacts for magnon generation. Figure 2b shows the reduction in gap at each V_{d.c.,} determined by subtracting—from each point—the average of the three traces with V_{d.c.} < E_{Z} at each ν_{TG}, which is similar to the transport measurement (Fig. 2a); deviations in the ν = 1 gap are only observed for V_{d.c.} > E_{Z} and 0 < ν_{TG} < 2. Intriguingly, the bias dependence of the gap and behaviour of V_{NL} appear to define three regimes. First, biases V_{d.c.} < E_{Z} result in no magnon generation and thus leave the gap intact. Second, for biases E_{Z} < V_{d.c.} ≲ 4E_{Z}, the gap is rapidly suppressed and the magnitude of V_{NL} is large. Finally, for larger biases V_{d.c.} ≳ 4E_{Z}, the suppression is more gradual and the magnitude of V_{NL} is vanishingly small. These observations unambiguously establish that gap suppression results from magnon propagation into the bulk.
The first step toward understanding gap suppression is to identify the nature of charge excitation associated with the ν = 1 gap in the absence of magnons. Theoretical studies^{2,7,8,9,10,11,12,13,14} have proposed that the lowestlying charged excitations at ν = 1 are finitesized skyrmions, consisting of a single charge ±e ‘dressed’ by one or more extra overturned spins or a valley texture. For skyrmions comprising flipped spins (referred to as ‘spin skyrmions’), the excitation energy is determined by the competition between E_{Z} and exchange energy. We, therefore, consider a phenomenological model of spin skyrmions with s flipped spins^{15}, whose occupation follows a Boltzmann distribution (Methods and Supplementary Information). Extended Data Fig. 2 shows the best fits to the data using this phenomenological model. The satisfactory agreement between the fit and data at many different locations validates our model and allows us to determine the Coulomb energy E_{C}, which sets the overall scale for the skyrmion and magnon energies (Methods), which is around 21.4 meV, consistent with previous local compressibility measurements of the ν = −1 gap^{11}. Most notably, we find that <s>, the mean number of extra spins carried by a charge excitation, is less than 6% of an electron spin in the absence of injected magnons, establishing that the lowestlying charge excitation consists of bare electrons and holes.
The observed gap suppression can be naturally captured by extending the phenomenological spin skyrmion model to incorporate the presence of magnons^{16} (Methods), where we describe the magnons by an effective Bose–Einstein distribution with chemical potential µ_{m} (refs. ^{17,18}) and electron temperature T. The magnon chemical potential µ_{m} defines an equilibrium between free magnons and those bound as flipped spins in skyrmions, and may be nonzero due to S_{Z} conservation (that is, magnon number conservation) in the bulk, which results from the weak spin–orbit coupling and small number of nuclear spins present in graphene. Since each magnon represents one flipped spin and therefore one unit of E_{Z}, pumping magnons into the system amounts to externally supplying some of the work needed to flip spins. Assuming there is equilibration between the charge excitations and free magnons (Methods), this results in a reduction in the Zeeman freeenergy cost by µ_{m} per flipped spin for the spin skyrmion, favouring the formation of skyrmions over bare electrons or holes and thus suppressing the overall charge gap. To compare the predictions of this model with our experiments, we compute the ν = 1 gap as a function of µ_{m} and T with the parameters obtained by fitting the measured zerobias µ(ν) curves (Fig. 3a and Methods). The results of these calculations indicate that considerable enhancements of µ_{m} and T are required to achieve the measured gap suppression at large biases (as shown by the constantgap contours in Fig. 3a).
To use our model to extract the magnon chemical potential µ_{m}, an independent estimate of electron temperature T as a function of d.c. bias is required. Such an estimate is furnished by a measurement of longitudinal resistance R_{xx} in the presence of magnon pumping using a circuit configuration that is insensitive to magnon absorption at the contacts (Methods). Strikingly, we find that the measured R_{xx} displays a sudden increase for V_{d.c.} > E_{Z}, indicative of its magnon origin. The comparison of the biasdependent R_{xx} measurement with the measurement of R_{xx} at zero bias as a function of temperature (Fig. 3c) suggests that injecting magnons into the system results in the electron temperature heating up to approximately 3 K. By finding the bestfit temperature for each V_{d.c.}, we extract the quantitative values of electron temperature T (Fig. 3d), which spans three distinct regimes. In the lowbias regime of V_{d.c.} < E_{Z}, no magnons are generated and T remains at the base temperature. Between E_{Z} and approximately 4E_{Z}, T rapidly increases as a function of bias. Finally, above approximately 4E_{Z}, T saturates and once again remains constant to the highest biases investigated. We have performed similar estimates using a variety of circuit configurations, both two and fourterminal configurations (Extended Data Figs. 3–5), which point to a similar range of temperatures.
Estimates of T(V_{d.c.}) and the results of our model calculations allow us to relate the measured gap values to µ_{m}. Specifically, we determine µ_{m}(V_{d.c.}) (Fig. 3e) by matching our measured gap values and T to the simulation results (Fig. 3a,d; Extended Data Fig. 6 provides the analysis at another location). As in the case of T, the measured gap and V_{NL}, we find that µ_{m}(V_{d.c.}) exhibits three separate regimes. At low bias, that is, V_{d.c.} < E_{Z}, we have µ_{m} = 0 in accordance with the general properties of the Bose–Einstein distribution. At intermediate bias, namely, E_{Z} < V_{d.c.} ≲ 4E_{Z}, we observe no increase in µ_{m}, despite the presence of magnon transport signatures in V_{NL} (Extended Data Fig. 7). Thus, the behaviour of the measured gap in the intermediatebias regime can be explained as a result of heating due to the injected magnons without invoking the possibility of skyrmion formation. At high bias, that is, 4E_{Z} ≲ V_{d.c.}, where T is approximately 3 K, we extract the values of µ_{m} in excess of zero, as expected in the presence of magnon pumping. We emphasize that gap suppression observed in this regime cannot be explained by heating alone, as this would require the temperature to continue to linearly increase beyond V_{d.c.} = ±5 mV and reach as high as 6 K at V_{d.c.} = ±10 mV, in direct contradiction to the temperature estimated from our zerobias R_{xx} measurements (Fig. 3b,c and Extended Data Fig. 3).
A further insight can be gained by examining the density n_{m} of the equilibrated free magnons obtained from our calculations and the mean number of overturned spins per skyrmion <s> as a function of V_{d.c.}. Figure 3f shows the extracted n_{m}(V_{d.c.}) in units of magnons per flux quantum ϕ_{0}. In the range of E_{Z} < V_{d.c.} ≲ 4E_{Z}, the finite V_{NL} and gap suppression measurements demonstrate that magnons are at work (Extended Data Fig. 7). However, we find that µ_{m} does not increase in this range and n_{m}, therefore, remains negligibly small. We speculate that two possible scenarios may explain this apparent contradiction. One hypothesis is that for E_{Z} < V_{d.c.} ≲ 4E_{Z}, there is an additional population of magnons, possibly of a very long wavelength, which is not in thermal equilibrium with the electrons and thus is not captured in the computed n_{m} values despite contributing to G_{2T} and V_{NL}. A comparison of the lower bound on the magnon lifetime obtained from our transport measurements and device geometry with theoretical estimates of the magnon–electron scattering time shows that magnons with k less than approximately 0.01l_{B}^{–1}, where l_{B} is the magnetic length, may scatter with skyrmions slowly enough to fail to equilibrate. A second, more exotic possibility is that, in fact, only a very small number of magnons are present in this bias regime, which would imply that a highly efficient mechanism of transport is responsible for the changes in G_{2T} and V_{NL}. On the other hand, for 4E_{Z} ≲ V_{d.c.}, a finite population of equilibrated free magnons emerges, which appears to linearly scale with V_{d.c.}. We note that for a 100 µm^{2} sample at 11 T, the highest equilibrium magnon density of about 3 × 10^{–3} per ϕ_{0} corresponds to a total number of equilibrated magnons only of the order of 300. It is possible that a population of nonequilibrated magnons also persists in this regime. In any case, these observations suggest that the absorption rate of magnons at the contacts may be outpaced by the finite population of free magnons, causing V_{NL} to weaken and G_{2T} to level off at high biases (Extended Data Fig. 7). Finally, the corresponding <s> (Fig. 3g) displays a similar trend as n_{m} and is three excess overturned spins at the highest biases, consistent with our overall mechanism of gap suppression. The change in behaviour as V_{d.c.} exceeds 4E_{Z} may be related to the large increase in specific heat as µ_{m} increases (Supplementary Information) and/or to higherenergy valley–spin excitations hosted by a valleypolarized chargedensitywave ground state (Methods)^{19}.
In a lowdensity electron system, correlation effects induced by the Coulomb repulsion between carriers can result in negative (inverse) electronic compressibility dµ/dn (refs. ^{20,21}), which can be observed at ν = 0 + ε, 1 ± ε and 2 ± ε (Fig. 4a,b). The associated correlation energy scale, approximated in our model to the leading order by E_{WC} ~ \(\sqrt \varepsilon\), governs the magnitude of negative compressibility. Intriguingly, we find that the negativecompressibility features at ν = 1 ± ε respond differently to V_{d.c.} than those at ν = 0 + ε and 2 ± ε, with those at ν = 1 ± ε being greatly diminished at high bias voltages. The pronounced reduction with increasing bias for E_{Z} < V_{d.c.} ≲ 4E_{Z} is presumably due to heating, but the reduction with increasing bias beyond this point—where the electron temperature is found to be constant—signals that the strength of correlations is suppressed by the presence of magnons. A possible explanation is that the formation of skyrmions may decrease the magnitude of correlation energy, because the electric charge of a skyrmion is more spread out than a bare electron or hole in the lowest Landau level. A comparison of the average negative compressibility for ν = 0 + ε, 1 ± ε and 2 ± ε (Fig. 4c,d) shows that it is sensitive to V_{d.c.} only near ν = 1, providing additional evidence for the magnon origin. Further study is required to fully establish the microscopic mechanism of these effects.
Looking ahead, the methods of measuring µ_{m} demonstrated here can be used to map out this important quantity over extended spatial regions. As the gradient of µ_{m} is the driving force of magnon currents, such studies may provide further new insights into the nature of magnon transport in the system^{22}, including the ν = 0 state in monolayer graphene, which may support spin superfluidity^{6,23}. The ability to tune µ_{m} in situ raises the possibility of dynamical control of quantum phases analogous to recent pump–probe experiments^{24}, but using magnetic excitations instead of terahertz frequencies. Finally, our combined ability of manipulating and probing the magnon chemical potential is immediately applicable to intriguing correlated insulating states recently reported in moiré superlattice systems, which are expected to support electrically addressable neutral excitations similar to the ν = 1 QHFM^{25,26,27}.
Methods
Sample preparation
The device consists of monolayer graphene encapsulated by two layers of hexagonal boron nitride (hBN) on a pdoped Si substrate with a 285 nm layer of SiO_{2}, and was fabricated using a dry transfer technique. A gold top gate was defined using electronbeam lithography and thermally evaporated Cr/Au. The final device geometry was defined by electronbeam lithography and reactiveion etching. Edge contacts were made by thermally evaporating Cr/Au while rotating the sample using a tilted rotation stage.
Measurements
All the measurements were carried out in a ^{3}He cryostat with a base temperature of approximately 500 mK. The transport measurements were performed using standard lockin techniques with a 100 µV excitation with frequencies ranging from 17 to 40 Hz. The temperaturedependent measurements were recorded by applying current to a resistive heater located at the ^{3}He stage. The SET tips were fabricated using the procedure described elsewhere^{28}. The diameter of the SET is approximately 100 nm, and it was held about 300 nm above the encapsulated graphene. Compressibility measurements were performed using d.c. and a.c. techniques similar to those described elsewhere^{28}. The SET serves as a sensitive detector of changes in electrostatic potential δφ, which is related to the chemical potential of the graphene flake by δµ = –eδφ when the system is in equilibrium. In the a.c. scheme used to measure dµ/dn, an a.c. voltage is applied to the Si back gate to weakly modulate the carrier density of graphene, and the corresponding changes in SET current are converted to chemical potential by normalizing the signal by that from a small a.c. bias directly applied to the sample. For d.c. measurements, an analogue proportional–integral–derivative controller is used to maintain the SET current at a fixed value by changing the tip–sample bias. The corresponding change in sample voltage provides a direct measure of µ(n).
Spin skyrmion model
We briefly summarize the skyrmion model used for estimating the magnon chemical potential µ_{m}. Assuming that both density of overturned spins and deviation from ν = 1 is small, the densities of electronlike and holelike skyrmions with s overturned spins, denoted as \(n_s^\mathrm{e}\) and \(n_s^\mathrm{h}\), respectively, follow the Boltzmann distributions given by
where \(E_s^\mathrm{e}\) and \(E_s^\mathrm{h}\) are the energy of elementary charged electronlike and holelike excitations at ν = 1, respectively; µ is the electron chemical potential; T is the temperature; δµ_{WC}(ν) is a Wigner crystallike energy functional; a_{e} and a_{h} are fit parameters; and µ_{m} is the magnon chemical potential (Supplementary Information provides complete details of the parameters used). The energies \(E_s^\mathrm{e}\) and \(E_s^\mathrm{h}\) as well as δµ_{WC}(ν) are parametrized by an overall phenomenological Coulomb energy scale E_{C} that is treated as a fit parameter to be obtained by fitting the zerobias µ traces. Extended Data Fig. 2 shows examples of the zerobias fit results, which are in excellent agreement with the experimental traces, along with the fit parameters E_{C}, a_{e}, a_{h} and a Gaussian densitybroadening parameter Δν. These fit parameters, along with our independent measurements of the ν = 1 gap and T, can then be combined with the distributions \(n_s^\mathrm{e}\) and \(n_s^\mathrm{h}\) to determine μ_{m} and other thermodynamic properties (Fig. 3e–g, Extended Data Fig. 6b–d and Supplementary Information).
R _{xx} and G _{2T} thermometry
To obtain an estimate of the electron temperature T independent of our compressibility measurements, we perform R_{xx} measurements in the presence of magnon generation using the circuit shown in Extended Data Fig. 3. Keeping contact 2 grounded, we apply an a.c. bias between contacts 1 and 4 and measure the longitudinal a.c. voltage V_{xx} across contacts 5 and 6. We emphasize that this measurement of R_{xx} is different than the nonlocal voltage and is not expected to be directly sensitive to contributions from magnon generation and absorption, and we expect the phonon contribution to R_{xx} to be small in the temperature range of interest (Supplementary Information provides further details). To generate magnons, a d.c. bias is applied to contact 3; in this case, no a.c. modulation is applied to the magnon generation contacts. Strikingly, the measured R_{xx} as a function of d.c. bias (Extended Data Fig. 3) displays an abrupt change when the applied d.c. bias exceeds the Zeeman energy, reminiscent of the response observed in the magnon transport experiments with a.c. modulation applied to contact 3 (Fig. 1). However, we emphasize that the change in R_{xx} is not caused by magnon absorption events as in the case of the V_{NL} signal discussed earlier or by other hotcarrier effects, because the a.c. modulation used for monitoring R_{xx} is not applied to the d.c.biased contacts used for magnon generation. Extended Data Fig. 3d,e shows R_{xx} measured using the same circuit with no d.c. bias applied to contact 3 as a function of temperature. Under these conditions, the electron temperature is expected to be well equilibrated with the lattice temperature^{29}. Remarkably, we find good agreement between an R_{xx} trace measured at a given d.c. bias and that at a given temperature (Fig. 3e, where the error bar is estimated by matching the R_{xx} value with a d.c. bias to the temperaturedependent R_{xx} with up to 5% error), suggesting that the change in R_{xx} at a d.c. bias greater than the Zeeman energy is equivalent to an increase in the temperature of the system. We note that although the phonon temperature is expected to remain near the base temperature in the biased case, we do not expect phonons to play an important role in our measurements (Supplementary Information). A comparison of these two R_{xx} measurements, therefore, allows us to determine the temperature of the system when magnons are pumped into the system and uniquely determine μ_{m}.
Alternatively, the twoterminal conductance G_{2T} may be used as a proxy for the temperature instead of the fourterminal R_{xx}. Extended Data Fig. 4a,b shows the bias and temperaturedependent twoterminal conductance G_{2T}, respectively, measured with an a.c. voltage between contacts 1 and 4 using the circuit shown in Extended Data Fig. 3a. As in the case of R_{xx} measurements, once the system has heated beyond about 5 K, the principal signatures of the quantum Hall effect vanish (in this case, the plateau), thus placing an overestimated but crucial upper bound for the temperature of our system. Extended Data Fig. 4c shows G_{2T} at a d.c. bias of –10 mV compared with a selection of zerobias traces taken at various temperatures, which points to a temperature of about 3 K—in good agreement with the results obtained by analysing R_{xx}. We have verified this behaviour in numerous circuit configurations—both twoterminal and fourterminal configurations—which consistently point to the same range of temperatures (Extended Data Fig. 5, except for the positive biases shown in Extended Data Fig. 5h, which is likely due to a bad contact). We regard the R_{xx} measurements as a more reliable indicator of temperature, as G_{2T} is more susceptible to effects stemming from the contact resistance. Nevertheless, our observation that both R_{xx} and G_{2T} thermometry techniques yield approximately the same electron temperature leads us to conclude that reliable estimates can be derived from either technique.
Thermalization of magnons and electrons
To understand the degree of equilibration between the skyrmions and free magnons and why there could be an additional population of magnons that are not in equilibrium with the electrons, we note that the degree of thermalization between a magnon and the skyrmion population depends on its momentum k. Moderatetoshortwavelength magnons with \(k \gtrsim l_\mathrm{B}^{  1}\) are equivalent to wellseparated electron–hole pairs and therefore may be expected to thermalize with the skyrmion population very quickly—at a rate nearly equal to that of a single free electron or hole. On the other hand, longwavelength magnons with momenta \(k \ll l_\mathrm{B}^{  1}\) are equivalent to tightly bound electron–hole pairs, which only carry a small electric dipole moment and therefore are expected to couple more weakly to skyrmions and thermalize more slowly. In general, for a given magnon to equilibrate with the skyrmion population, its lifetime must exceed the mean magnon–electron scattering time τ_{me}(k). The magnon–electron scattering rate \({{\varGamma }}_k = \tau _{\mathrm{me}}^{  1}\) has been calculated within the context of bilayer QHFMs to be^{30}
where δν is the difference in the Landaulevel filling factor from the nearest integer (taken as 0.01 here) and v(k) is the magnon velocity. To obtain a lower bound for the magnon lifetime, we note that in the absence of magnetic impurities, we expect magnon absorption events at the contacts^{4} to be the dominant mechanism by which magnons are removed from the bulk. The requirement that magnons survive long enough to travel the approximately 10 μm distance between the contacts in the device, therefore, allows us to estimate a lower bound on the lifetime τ_{min} ≈ d/v(k), where d = 10 μm. Extended Data Fig. 8 plots the two timescales τ_{me}(k) and τ_{min}(k) as a function of k, and shows that only verylongwavelength magnons with \(k \lesssim 0.01l_\mathrm{B}^{  1}\) are expected to scatter slowly enough with skyrmions to fail to come into equilibrium.
Role of valley skyrmions
A number of theoretical studies have considered the nature of the lowestlying charged excitations in the ν = 1 QHFM^{10,11,12,13,14}. Although valley skyrmions may be favoured under ideal conditions, the presence of a boron nitride substrate in encapsulated devices may result in the breaking of sublattice symmetry and therefore disfavour the formation of valley skyrmions. Although we do not find direct evidence for a gap at the chargeneutrality point (CNP) in our device, we observe a robust incompressible state at ν = 5/3, with an incompressible peak comparable in magnitude to those occurring at ν = 1/3 and 2/3 (Extended Data Fig. 9). The conspicuous absence of this state in previous local compressibility measurements on suspended devices was attributed to lowlying valleyskyrmionic excitations with energy less than that of a Laughlin quasiparticle^{11,28}. Thus, the observation of robust incompressible states at ν = 5/3 strongly suggests that valley skyrmions are disfavoured in our sample. Furthermore, within the spin skyrmion model of gap suppression, we do not expect the presence of magnons to alter the energy cost of adding a valley skyrmion. These conclusions also hold for valleycoherent ground states, in which the spin excitation spectrum is expected to be the same as the valleypolarized case considered above^{19}. Hence, we conclude that valley skyrmions are unlikely to play an important role in the observed ν = 1 gap reduction.
Discussion of a possible gap at the CNP
To search for evidence of sublattice symmetry breaking, we performed highresolution local compressibility measurements near the CNP at zero magnetic field, which was compared with a model that considers the sublatticegapped Dirac form \(\mu \left( n \right) = \sqrt {\frac{{{\varDelta}_0^2}}{4} + \frac{{\uppi v_\mathrm{F}^2n}}{{{\hbar}^2}}}\), where Δ_{0} and ν_{F} are the sublattice gap and Fermi velocity, respectively, and ℏ is the reduced Planck constant. Extended Data Fig. 10 shows two fits of the measured inverse compressibility at zero magnetic field to the sublatticegapped Dirac model, one with disorder broadening and the other one without broadening. The unbroadened fit favours a scenario in which the sublattice gap is zero. The broadened fit, however, yields a mean squared error approximately onehalf that of the unbroadened fit, as well as favours a scenario in which the sublattice gap is approximately 12.3 meV with a broadening of 7 × 10^{9} cm^{–2}, consistent with that extracted from our fit to the ν = 1 gap at a high magnetic field. These considerations suggest that sublattice symmetry is likely broken by the boron nitride substrate, disfavouring the formation of valley skyrmions, despite the compressibility signature of the gap being obscured by disorder broadening at zero magnetic field.
Data availability
Source data are provided with this paper. All other data that support the findings of this paper are available from the corresponding author upon reasonable request.
Code availability
The code that supports the findings of this study is available from the corresponding author upon reasonable request.
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Acknowledgements
We acknowledge discussion with P. JarilloHerrero and P. Wang. This work was primarily supported by the US Department of Energy, Basic Energy Sciences Office, Division of Materials Sciences and Engineering, under award DESC0001819. Fabrication of samples was supported by the US Department of Energy, Basic Energy Sciences Office, Division of Materials Sciences and Engineering, under award DESC0019300. A.Y. also acknowledges the Gordon and Betty Moore Foundations EPiQS Initiative through grant no. GBMF9468; ARO grant no. W911NF1410247; and the STC Center for Integrated Quantum Materials, National Science Foundation (NSF), grant no. DMR1231319. A.T.P. acknowledges support from the Department of Defense through the National Defense Science and Engineering Graduate Fellowship (NDSEG) program. Y.X. acknowledges partial support from the Harvard Quantum Initiative in Science and Engineering. A.T.P., Y.X. and A.Y. acknowledge support from the Harvard Quantum Initiative Seed Fund. P.R.F. acknowledges support from the NSF Graduate Research Fellowship under grant no. DGE 1745303. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, grant no. JPMXP0112101001; JSPS KAKENHI grant no. JP20H00354; and the CREST (JPMJCR15F3), JST. This work was performed, in part, at the Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Infrastructure Network, which is supported by the NSF under award no. ECS0335765. CNS is part of Harvard University.
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A.T.P., Y.X. and A.Y. designed the experiment. A.T.P. and Y.X. performed the scanning SET experiment and temperaturedependent transport measurements and they analysed the data with input from A.Y. S.H.L. fabricated the device and performed the transport measurements using the dilution refrigerator. B.I.H., A.T.P. and Y.X. performed the theoretical analysis and carried out the numerical calculations. K.W. and T.T. provided the hBN crystals. All the authors participated in discussions and in writing of the manuscript.
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Extended data
Extended Data Fig. 1 Additional examples of the ν=1 gap suppression by the presence of magnons.
a, Optical image of the device indicating the circuit used for magnon generation and the locations where the gap measurements were taken. b, Biasdependent energy gap measured at location 2. The grey dotted lines mark ±E_{Z}. While the origin of the small asymmetry for V_{DC}< E_{Z} is unclear, its magnitude is much smaller than overall suppression observed at higher bias, and the top gate dependence shows that the onset consistently occurs near E_{Z} (see Fig. 2). cg, Chemical potential µ near ν = 1 measured with V_{DC} = 0 mV (blue) and V_{DC} = 10 mV (red) at 6 different locations. Although the local value of the ν = 1 gap varies, its reduction by the presence of magnons is clearly reproduced in all the data sets.
Extended Data Fig. 2 Fitting of the chemical potential in the absence of magnons.
ag, Chemical potential µ near ν = 1 measured with 0 mV (blue circles) DC bias applied to contact 3 and the best fit (red curves) using the skyrmion model by setting μ_{m} = 0 meV (see Methods) at 7 different locations. The values of E_{C} obtained at these positions correspond to effective dielectric constants ε ranging from 10.0 to 12.4.
Extended Data Fig. 3 R_{xx} thermometry.
a, Circuit used for R_{xx} measurements. Contacts 2 and 3 are used to generate magnons. Contacts 1, 6, 5, and 4 are used to measure R_{xx}. The white arrows indicate the chirality of the current flow. b, R_{xx} as a function of V_{DC} at V_{BG} = 3.8 V. The value of R_{xx} tends to saturate for V_{DC} greater than ~4E_{Z}, suggesting that the temperature saturates. c, Individual R_{xx} traces measured at base temperature with various values of V_{DC} applied to contact 3 near ν=1 using the circuit shown in a. The center of the ν=1 plateau is around a back gate voltage of 3.5 V. d, Individual R_{xx} traces measured at various temperatures with zero DC bias applied to contact 3 near ν=1 using the circuit shown in a. e, Individual R_{xx} traces measured at base temperature with 10 mV applied to contact 3 (blue dots) and at various temperatures with 0 mV applied to contact 3 (orange, yellow and purple lines). The close agreement between the blue dotted line and the red line suggests that the effect of magnon generation on the R_{xx} measurement is primarily due to heating. These measurements also demonstrate that the increase in temperature due to magnon generation does not exceed 4.5 K at V_{DC} = 10 mV.
Extended Data Fig. 4 Alternative derivation of electron temperature using twoterminal conductance G_{2T}.
a, Biasdependent twoterminal conductance G_{2T} measured in the vicinity of the ν = 1 plateau. b, Temperaturedependent G_{2T} measured at zero bias in the same range of electron densities. c, G_{2T} measured at 10 mV DC bias compared with selected zerobias traces at elevated temperature.
Extended Data Fig. 5 Temperature extraction from additional circuit configurations.
ad, circuit configurations in which additional biasdependent (eh) and temperature dependent (il) twoterminal transport measurements were carried out. mp, comparison of traces taken at V_{DC} = 10 mV and at base temperature with zerobias traces taken at various temperatures. In each panel the middle value of temperature is that found to agree best with the V_{DC} = 10 mV trace in the leastsquares sense. The good agreement between the 10 mV trace and the bestfit zerobias trace indicates that the main effect of the bias in this circuit configuration is to elevate the temperature. qs, additional R_{xx} data acquired simultaneously with G_{2T} using the circuit configuration shown in d. Estimation from R_{xx} gives a slightly lower temperature than G_{2T}.
Extended Data Fig. 6 Thermodynamics of free and bound magnons extracted from location 2.
a, ν=1 gap as a function of magnon chemical potential \(\mu _m/E_z\) and temperature T computed using the skyrmion model. bd, Magnon chemical potential \(\mu _m/E_z\) (b), free magnon density per flux n_{m} (c) and the mean number <s > of extra flipped spins carried by a charge (d), extracted from the skyrmion model (see Methods). The shaded region corresponds to a medium bias regime where heating due to magnon injected plays a key role.
Extended Data Fig. 7 Three regimes in magnon transport characteristics.
a, G_{2T} averaged over values of V_{BG} on the ν=1 plateau as a function of DC bias. b, V_{NL}/V_{AC} averaged over values of ν_{TG} for 0 < ν_{TG} < 2. On each plot, the low, medium and highbias regimes are indicated by shading in the same manner as Fig. 3. c, zerobias measurement of G_{2T} showing a welldeveloped ν = 0 plateau, implying an insulating ν = 0 ground state as observed in previous transport studies of magnon generation.
Extended Data Fig. 8 Magnon lifetime and electronmagnon scattering time.
Magnon lifetime and magnonelectron scattering time as a function of momentum, showing that only for very small momenta \(k \lesssim 0.01l_B^{  1}\) is τ_{min}(k) expected to exceed τ_{me}(k). Thus, all magnons with \(k \gtrsim 0.01l_B^{  1}\) are expected to be wellthermalized with the skyrmion population at temperature T.
Extended Data Fig. 9 Robust ν = 5/3 state.
Local inverse compressibility \(d\mu /dn\) measured for 0< ν < 2. In contrast to local compressibility studies performed on suspended monolayer graphene devices, a prominent peak at ν = 5/3—comparable in strength to those at 1/3 and 2/3, and stronger than that at 4/3—is evident, suggesting that valley skyrmion formation in the device is disfavored.
Extended Data Fig. 10 Zerofield fits to the Dirac point.
a, Fit comparing measured data to a model with no disorder broadening. The fit favors zero sublattice gap. b, Fit comparing measured data to a model with disorder broadening. The fit favors a scenario with a 12.3 meV sublattice gap with a disorderbroadening parameter of approximately 7 × 10^{9} cm^{2}, similar in magnitude to the broadening inferred from highfield compressibility measurements. The mean squared error (MSE) of the broadened fit is improved compared to that of the unbroadened fit by more than a factor of two.
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Supplementary Fig. 1 and Discussion.
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Source Data Fig. 1
Source data for Fig. 1c–e.
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Source data for Fig. 2.
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Source data for Fig. 3.
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Source data for Fig. 4.
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Pierce, A.T., Xie, Y., Lee, S.H. et al. Thermodynamics of free and bound magnons in graphene. Nat. Phys. 18, 37–41 (2022). https://doi.org/10.1038/s4156702101421x
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DOI: https://doi.org/10.1038/s4156702101421x
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To measure a magnon population
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