Thermodynamics of free and bound magnons in graphene

Symmetry-broken electronic phases support neutral collective excitations. For example, monolayer graphene in the quantum Hall regime hosts a nearly ideal ferromagnetic phase at filling factor $\nu=1$ that spontaneously breaks spin rotation symmetry. This ferromagnet has been shown to support spin-wave excitations known as magnons which can be generated and detected electrically. While long-distance magnon propagation has been demonstrated via transport measurements, important thermodynamic properties of such magnon populations--including the magnon chemical potential and density--have thus far proven out of reach of experiments. Here, we present local measurements of the electron compressibility under the influence of magnons, which reveal a reduction of the $\nu=1$ gap by up to 20%. Combining these measurements with estimates of the temperature, our analysis reveals that the injected magnons bind to electrons and holes to form skyrmions, and it enables extraction of the free magnon density, magnon chemical potential, and average skyrmion spin. Our methods furnish a novel means of probing the thermodynamic properties of charge-neutral excitations that is applicable to other symmetry-broken electronic phases.


novel means of probing the thermodynamic properties of charge-neutral excitations that is applicable to other symmetry-broken electronic phases.
The interplay between electron-electron interactions and isospin degeneracy in Landau levels can give rise to symmetry-broken states such as quantum Hall ferromagnets (QHFMs) 1,2 .
The spin polarized ν=1 state in monolayer graphene stands out due to its remarkable magnetic properties 3 and represents a unique platform for exploring charge-neutral spin excitations that may be useful for spintronics applications. Recent transport experiments have shown that voltage biases exceeding the Zeeman energy scale EZ=gµBB 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 non-local voltage that can be detected microns away 4,5 . However, measurements of the thermodynamic properties of this magnon system-critical for harnessing these novel charge-neutral 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 single electron transistor (SET) and examine its response to the presence of magnons. We find that pumping magnons into the system results in a marked reduction of the charge gap, typically of about 15-20%. We argue that the 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 thermodynamic properties of magnons introduced in our experiments suggests novel routes toward realizing and probing Bose-Einstein condensation in quantum Hall ferromagnets 6 , and is more broadly applicable to other flat-band systems with spontaneously symmetry-broken states.
The device and measurement setup are shown in Fig. 1a-b. The hBN-encapsulated graphene device rests on a standard conducting Si/SiO2 substrate, and a narrow top gate (TG) covers part of the device to enable independent tuning of the local filling factor νTG. Fig. 1c shows the two-terminal conductance G2T between contacts 2 and 3 as a function of back-gate filling factor ν and DC bias VDC at a magnetic field B=11 T. Consistent with previous studies 4,5 , for -EZ< VDC <EZ, we see that G2T exhibits a plateau as a result of the quantization of σxy.
However, the quantized Hall plateau disappears as soon as |VDC| reaches EZ, signaling 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 as a function of filling factor µ(ν) at each value of VDC. Fig. 1d shows two representative measurements of the electron chemical potential µ(ν) near ν=1, with top-gate voltage VTG = 0. Here, µ(ν) jumps sharply as ν passes through 1 due to the ν=1 gap, with the latter taken to be the maximal excursion of µ. The trace at VDC =10 mV (red curve in Fig. 1d) clearly exhibits a reduced gap compared to that at 0 mV (blue curve in Fig. 1d). A detailed VDC dependence of the gap values (summarized in Fig. 1e) exhibits a striking resemblance to that of the transport behavior. Specifically, the gap begins to change principally when VDC exceeds EZ, initially dropping sharply and reaching a ~20% suppression at the highest biases investigated.
The gap reduction shown in Fig. 1d-e, observed at many different locations (see Extended Data An important piece of evidence that the ν=1 gap suppression observed for DC biases |VDC|>EZ results from magnon generation and absorption is its dependence on νTG. As a consequence of the spin order present in the region under the TG, magnons propagate freely across when νTG = ±1, but only weakly for νTG = 0 and not at all for νTG = ±2. Fig. 2a shows the AC nonlocal voltage VNL measured across contacts 5 and 6, normalized by the AC bias VAC applied between contacts 2 and 3. In addition to the vanishing nonlocal voltage for |VDC|<EZ, we find that for |VDC|>EZ, no significant signal is detected for |νTG|>2 nor for νTG=0; on the other hand, a strong non local-voltage is seen for 0<|νTG|<2, in accordance with the expected transport characteristics shown in previous studies 4,5 . Next, we perform gap measurements near contact 5 as a function of νTG and VDC, using the same contacts for magnon generation. Fig. 2b shows the reduction in gap at each VDC, determined by subtracting from each point the average of the three traces with |VDC|<EZ at each νTG. As in the case of the transport measurement (Fig. 2a), deviations in the ν=1 gap are only observed for |VDC|>EZ and 0<|νTG|<2. Intriguingly, the bias dependence of the gap and the behavior of VNL appear to define three regimes. First, biases |VDC|<EZ result in no magnon generation and thus leave the gap intact. Second, for biases EZ<|VDC|≲4EZ the gap is suppressed rapidly and the magnitude of the VNL is large. Finally, for larger biases |VDC|≳4EZ the suppression is more gradual and the magnitude of VNL is vanishingly small. Overall, the observed similarities between the νTG dependence of VNL and the gap unambiguously establish that the gap suppression results from magnon propagation into the bulk.
The first step toward understanding the gap suppression is to identify the nature of the charge excitation associated with the ν=1 gap in the absence of magnons. Theoretical studies 2,[7][8][9][10][11][12] have proposed that the lowest-lying charged excitations at ν=1 are finite-size skyrmions, consisting of a single charge ±e "dressed" by one or more extra overturned spins or a valley texture. While valley skyrmions are believed to set the ν=1 gap under certain idealized conditions 10-12 , we find that they are unlikely to play a role in our observations (see Methods).
For skyrmions comprised of flipped spins (referred to as "spin skyrmions"), the excitation energy is determined by the competition between EZ and the exchange energy: larger skyrmions are favored by the exchange interaction, at the expense of EZ per flipped spin, resulting in an optimal number of flipped spins of order unity. To illustrate this point, we consider a model of spin skyrmions with s flipped spins 13 , whose occupation follows a Boltzmann distribution (see Methods). We also include electron-hole asymmetric Wigner crystal (WC)-like terms in the total energy to account for the regions of negative slope in µ stemming from the effects of correlation ( Fig. 1d) 14 , along with an overall Gaussian broadening of µ(ν) to account for disorder. Extended Data Fig. 2 shows the best fits to the data using this phenomenological model. The satisfactory agreement between the fit and the data at many different locations validates our model and allows us to determine the Coulomb energy EC, which sets the overall scale for the skyrmion and magnon energies (see Methods), to be around 21.4 meV. 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 lowest-lying 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 15 (see Methods), where we describe the magnons by an effective Bose-Einstein distribution with chemical potential µm and electron temperature T. Since each magnon represents one flipped spin and therefore one unit of EZ, 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 the free magnons, this results in a reduction of the Zeeman free-energy cost by µm per flipped spin for the spin skyrmion, favoring 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 µm and T with the parameters obtained by fitting the measured zero-bias µ(ν) curves (see Fig. 3a and Methods).
The results of these calculations indicate that considerable enhancements of µm and T are required to achieve the measured gap suppression at large biases (see constant gap contours in Fig. 3a).
In order to use our model to extract the magnon chemical potential µm, an independent estimate of the electron temperature T as a function of DC bias is required. Such an estimate is furnished by a measurement of the longitudinal resistance Rxx in the presence of magnon pumping. Keeping contact 2 grounded, we apply an AC bias between contacts 1 and 4 and measure the longitudinal AC voltage Vxx across contacts 5 and 6. We emphasize that this measurement of Rxx is different than the non-local voltage and is not expected to be directly sensitive to contributions from magnon generation and absorption (see Methods). Strikingly, we find that the measured Rxx displays a sudden increase for |VDC|>EZ, indicative of its magnon origin. The comparison of the bias-dependent Rxx measurement with a measurement of Rxx 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 best-fit temperature for each VDC, we extract quantitative values of the electron temperature T (Fig. 3d), which spans three distinct regimes. In the low-bias regime |VDC|<EZ, no magnons are generated and T remains at base temperature. Between EZ and approximately 4EZ, T increases rapidly as a function of bias. Finally, above ~4EZ, T saturates and once again remains constant to the highest biases investigated. We have performed similar estimates using a variety of circuit  Fig. 7). However, we find that µm does not increase in this range and nm therefore remains negligibly small. We speculate that two possible scenarios may explain this apparent contradiction. One hypothesis is that for EZ<|VDC|≲4EZ there is an additional population of magnons, possibly of very long wavelength, which are not in thermal equilibrium with the electrons, and thus are not captured in the computed nm despite contributing to G2T and VNL. A second, more exotic possibility is that in fact only a very small number of magnons is present in this bias regime, which would imply that a highly efficient mechanism of transport is responsible for the changes in G2T and VNL. On the other hand, for 4EZ≲|VDC|, a finite population of equilibrated free magnons emerges that appears to scale linearly with VDC. We note that for a ~100 µm 2 sample at 11 T, the highest equilibrium magnon density of ~3×10 -3 per ϕ0 corresponds to a total number of equilibrated magnons only of order 300. It is possible that a population of non-equilibrated 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 VNL to weaken and G2T to level off at high biases (Extended Data Fig. 7). Finally, the corresponding <s> (Fig. 3g) displays a similar trend as nm and reaches 3 excess overturned spins at the highest biases, consistent with our overall mechanism of gap suppression. The reason for the change in behavior as VDC exceeds 4EZ is not known.
In a low-density electron system, correlation effects induced by the Coulomb repulsion between carriers can result in negative (inverse) electronic compressibility dµ/dn 16,17 , which we observe at ν=0+ε, ν=1±ε and 2±ε (Figs. 4a-b). The associated correlation energy scale, approximated in our model to leading order by the energy of a classical Wigner crystal !"~√ , governs the magnitude of the negative compressibility. Intriguingly, we find that the negativecompressibility features at ν=1±ε respond differently to VDC 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 EZ<|VDC|≲4EZ 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 formation of skyrmions may decrease the magnitude of the correlation energy, because the electric charge of a skyrmion is more spread out than for a bare electron or hole in the lowest Landau level. Comparison of the averaged negative compressibility for ν=0+ε, ν=1±ε and 2±ε ( Fig. 4c and d) shows that only near ν=1 is it sensitive to VDC, 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 µm demonstrated here can be used to map out this important quantity over extended spatial regions. As the gradient of µm is the driving force of magnon currents, such studies may provide further new insights into the nature of magnon transport in the system 18 . One can also envision applying our technique to electronic states with exotic magnetic order. In particular, the ν=0 state in monolayer graphene has been predicted to support spin superfluidity, in which magnons can propagate without dissipation 6,19 . Our experiments also suggest a novel strategy to effectively reduce EZ, or equivalently the spin-gfactor, by increasing the magnon chemical potential µm, which can be used to drive spin transitions in complex systems like fractional quantum Hall states 11,[20][21][22] . This raises the possibility of dynamical control of quantum phases analogous to recent pump probe experiments 23 , but using magnetic excitations instead of THz. Finally, our combined ability of manipulating and probing magnon chemical potential is immediately applicable to intriguing , and the total charge density, or equivalently the filling factor ν, is These formulae show that, for fixed total density, both the population of charge carriers with total spin s and the electron chemical potential-and thus the gap to charged excitationsdepend on the temperature and the magnon chemical potential.
The above formulation determines the filling factor n as a function of , < , and under the assumption that the charged excitations do not interact. However, the experimental $ ( ) curves exhibit strong negative compressibility near n=1, indicating that substantial correlation effects are present. At =0, for sufficiently low carrier densities, in the absence of impurities, electrons or holes are expected to form a Wigner crystal, whose energy per carrier has been where $,& are parameters we fit to experiment, which we allow to be different in order to reflect the observed asymmetries between electrons and holes in our system. We then calculate the densities of skyrmions at finite temperatures using The total density of free magnons < is given by , with < being the density of magnons at wavevector k given by the Bose-Einstein distribution Equating these quantities, we obtain the result for NBL stated above. This relation is consistent with the observation that for a system of linear size , with just a single magnon present, the requirement < , with a constant of order unity, gives NBL of order / * ( .

Rxx and G2T thermometry.
To obtain an estimate of the electron temperature T independent of our compressibility measurements, we perform Rxx measurements in the presence of magnon generation using the circuit shown in Extended Data Fig. 3. To generate magnons, a DC bias is applied to contact 3, while contact 2 is grounded; in this case, no AC modulation is applied to the magnon generation contacts. Strikingly, the measured Rxx as a function of DC bias (Extended Data Fig. 3) displays an abrupt change when the applied DC bias exceeds Zeeman energy, reminiscent of the response observed in the magnon transport experiments with AC modulation applied to contact 3 (Fig. 1). However, we emphasize that the change in Rxx is not caused by magnon absorption events as in the case of the VNL signal discussed in the main text, because the AC modulation used for monitoring Rxx is not applied to the contacts used for magnon generation. Extended Data Fig. 3c and d show Rxx measured using the same circuit with no DC bias applied to contact 3 as a function of temperature. Remarkably, we find good agreement between an Rxx trace measured at a given DC bias and that at a given temperature (Fig. 3d, where the error bar is estimated by matching the Rxx with DC bias to the temperature Rxx up to 5% error), suggesting that the change in Rxx at a DC bias greater than Zeeman is equivalent to raising the temperature of the system. Comparing these two Rxx measurements therefore allows us to determine the temperature of the system when magnons are pumped into the system and determine < uniquely.
Alternatively, the two-terminal conductance G2T may be used as a proxy for the temperature instead of the four-terminal Rxx. Extended Data Fig. 4a and b respectively show the bias-and temperature-dependent two-terminal conductance G2T measured with an AC voltage between contacts 1 and 4 using the circuit shown in Extended Data Fig. 3a. As in the case of the Rxx measurements, once the system has heated beyond T~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 G2T at -10 mV DC bias compared with a selection of zero-bias traces taken at various temperatures, which points to a temperature in the range of ~3 K, in good agreement with the results obtained by analyzing Rxx.
We have verified this behavior in numerous circuit configurations, both in two-terminal and four-terminal configurations, which consistently point to the same range of temperatures (Extended Data Fig. 5, except for positive biases in Extended Data Fig. 5h, which is likely due to a bad contact). We regard the Rxx measurements as a more reliable indicator of temperature, as G2T is more susceptible to effects stemming from contact resistance. Nevertheless, our observation that both the Rxx and G2T thermometry techniques yield approximately the same electron temperature leads us to conclude that reliable estimates can be derived from either technique.
Role of valley skyrmions. The n=1 quantum Hall state in suspended graphene was originally proposed to support valley skyrmions as its lowest-lying charged excitations 10,11,30 . In encapsulated devices, however, the presence of the boron nitride substrate may break sublattice symmetry and therefore disfavor the formation of valley skyrmions. Although we do not find direct evidence for a gap at the charge neutrality point in our device, we observe a robust incompressible state at ν=5/3, with an incompressible peak comparable in magnitude to that occurring at ν=1/3 and 2/3 (Extended Data Fig. 8). The conspicuous absence of this state in previous local compressibility measurements on suspended devices was attributed to low-lying valley-skyrmionic excitations with energy less than that of a Laughlin quasiparticle 11,27 .         While the origin of the small asymmetry for |VDC|<EZ 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 EZ (see Fig. 2). c-g, Chemical potential µ near ν=1 measured with VDC=0 mV (blue) and VDC=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. Fig.2 | Fitting of the chemical potential in the absence of magnons. a-g, 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 < =0 mV (see Methods) at 7 different locations. The values of EC obtained at these positions correspond to effective dielectric constants ranging from 10.0 to 12.4. Fig. 3 | Rxx thermometry. a, Circuit used for Rxx measurements. Contacts 2 and 3 are used to generate magnons. Contacts 1, 6, 5, and 4 are used to measure Rxx. The white arrows indicate the chirality of the current flow. b, Individual Rxx traces measured at base temperature with various values of VDC applied to contact 3 near n=1 using the circuit shown in a. The center of the n=1 plateau is around a back gate voltage of 3.5 V. c, Individual Rxx traces measured at various temperature with no bias applied to contact 3 near n=1 using the circuit shown in a. d, Individual Rxx 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 Rxx 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 VDC=-10 mV. Fig. 4 | Alternative derivation of electron temperature using two-terminal conductance G2T. a, Bias-dependent two-terminal conductance G2T measured in the vicinity of the ν=1 plateau. b, Temperature-dependent G2T measured at zero bias in the same range of electron densities. c, G2T measured at -10 mV DC bias compared with selected zero-bias traces at elevated temperature. Fig. 5 | Temperature extraction from additional circuit configurations. a-d, circuit configurations in which additional bias-dependent (e-h) and temperature dependent (i-l) two-terminal transport measurements were carried out. m-p, comparison of traces taken at VDC=-10 mV and at base temperature with zero-bias traces taken at various temperatures. In each panel the middle value of temperature is that found to agree best with the VDC=-10 mV trace in the least-squares sense. The good agreement between the -10 mV trace and the best-fit zero-bias trace indicates that the main effect of the bias in this circuit configuration is to elevate the temperature. q-s, additional Rxx data acquired simultaneously with G2T using the circuit configuration shown in d. Estimation from Rxx gives a slightly lower temperature than G2T. Fig. 6 | Thermodynamics of free and bound magnons extracted from location 2. a, n=1 gap as a function of magnon chemical potential < / P and temperature T computed using the skyrmion model. b-d, Magnon chemical potential < / P (b), free magnon density per flux < (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. Fig. 7 | Three regimes in magnon transport characteristics. a, G2T averaged over values of VBG on the n=1 plateau as a function of DC bias. b, VNL/VAC averaged over values of nTG for 0<nTG<2. On each plot, the low-, medium-and high-bias regimes are indicated by shading in the same manner as Fig. 3. Fig. 8 | Robust ν=5/3 state. Local inverse compressibility / 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. Fig. 9 | Zero-field 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 disorder-broadening parameter of approximately 7×10 9 cm -2 , similar in magnitude to the broadening inferred from high-field compressibility measurements. The MSE of the broadened fit is improved compared to that of the unbroadened fit by more than a factor of two.