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Ravi Kumar, [Saurabh Kumar Srivastav](https://orcid.org/0000-0002-0498-4217), Ujjal Roy, [Jinhong Park](https://orcid.org/0000-0003-3570-6356), [Christian Spånslätt](https://orcid.org/0000-0001-6746-5433), [K. Watanabe](https://orcid.org/0000-0003-3701-8119), [T. Taniguchi](https://orcid.org/0000-0002-1467-3105), [Yuval Gefen](https://orcid.org/0009-0002-4553-6039), Alexander D. Mirlin, [Anindya Das](https://orcid.org/0000-0002-6310-1576)

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[Electrical noise spectroscopy of magnons in a quantum Hall ferromagnet](https://mdr.nims.go.jp/datasets/bb8bd92a-b9af-4948-bd1e-c4751995c2f4)

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Electrical noise spectroscopy of magnons in a quantum Hall ferromagnetArticle https://doi.org/10.1038/s41467-024-49446-zElectrical noise spectroscopy of magnons ina quantum Hall ferromagnetRavi Kumar1,7, Saurabh Kumar Srivastav 1,7, Ujjal Roy1,7, Jinhong Park 2,3,7,Christian Spånslätt 4, K. Watanabe 5, T. Taniguchi 5, Yuval Gefen 6,Alexander D. Mirlin2,3 & Anindya Das 1Collective spin-wave excitations, magnons, are promising quasi-particles fornext-generation spintronics devices, including platforms for informationtransfer. In a quantum Hall ferromagnets, detection of these charge-neutralexcitations relies on the conversion of magnons into electrical signals in theform of excess electrons and holes, but if the excess electron and holes areequal, detecting an electrical signal is challenging. In this work, we overcomethis shortcoming bymeasuring the electrical noise generated bymagnons.Weuse the symmetry-broken quantum Hall ferromagnet of the zeroth Landaulevel in graphene to launch magnons. Absorption of these magnons createsexcess noise above the Zeeman energy and remains finite even when theaverage electrical signal is zero.Moreover, we formulate a theoreticalmodel inwhich the noise is produced by equilibration between edge channels andpropagating magnons. Our model also allows us to pinpoint the regime ofballistic magnon transport in our device.The emergence of charge-neutral collective excitations presents apowerful platform for developing data processing as well as informa-tion transfer with small power consumption. Among these excitations,spin-wave excitations, or their quanta ‘magnons’, inmagneticmaterialsare promising. An obviously important task is to develop new techni-ques for the detection of these charge-neutral quasi-particles. So far,various experimental tools, such as inelastic neutron scattering1,2,inelastic tunneling spectroscopy3,4, terahertz spectroscopy5,6, micro-wave Brillouin light scattering7,8, nitrogen-vacancy center9,10, andsuperconducting qubits11 have been used to detect magnons in bulkmagnetic materials. However, their detection in device geometries,which is necessary for information processing applications, hasremained challenging until very recently. In particular, it was demon-strated by ref. 12 thatmagnons can be converted into electrical signalsin a quantum Hall ferromagnet (QHF) in graphene.Graphene offers a very versatile platform for new kinds of elec-tronic devices. When subjected to a perpendicular magnetic field,graphene shows several unique quantum Hall (QH) phases, related toits peculiar sequence of Landau levels (LL), manifesting both spin andvalley degrees of freedom13–15. In particular, the particle-hole sym-metric zeroth LL (ZLL) has a rich variety of QHF phases16–22: When theZLL is partially filled, Coulomb interactions break spin and valleysymmetries, and for a quarter (ν = − 1) or three-quarters (ν = 1) filling,the QH phases comprise ferromagnetic insulator bulks with spin-polarized edge states23–26. While the charge excitations in the bulk ofthese QHF insulators have a gap determined by the exchange energy(EX ∼ e2ϵ‘B, where e, ϵ, and ℓB are the elementary charge, dielectric con-stant, and the magnetic length), the spin-waves (magnons) haveinstead a gap determined by the Zeeman energy (EZ = gμBB, where g isthe Landé g-factor, and μB is the Bohr magneton)27 and are in fact thelowest energy excitations of the system. However, magnons do notcarry electrical charge, and therefore do not have a large impact onelectrical transport, which in turn makes it a difficult task to detectthem. There are a few reported attempts of generating and detectingReceived: 30 September 2023Accepted: 5 June 2024Check for updates1Department of Physics, Indian Institute of Science, Bangalore 560012, India. 2Institute for Quantum Materials and Technologies, Karlsruhe Institute ofTechnology, 76021 Karlsruhe, Germany. 3Institut für Theorie der Kondensierten Materie, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany.4Department of Microtechnology and Nanoscience (MC2), Chalmers University of Technology, S-412 96 Göteborg, Sweden. 5National Institute of MaterialScience, 1-1 Namiki, Tsukuba 305-0044, Japan. 6Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel. 7Theseauthors contributed equally: Ravi Kumar, Saurabh Kumar Srivastav, Ujjal Roy, Jinhong Park. e-mail: anindya@iisc.ac.inNature Communications |         (2024) 15:4998 11234567890():,;1234567890():,;http://orcid.org/0000-0002-0498-4217http://orcid.org/0000-0002-0498-4217http://orcid.org/0000-0002-0498-4217http://orcid.org/0000-0002-0498-4217http://orcid.org/0000-0002-0498-4217http://orcid.org/0000-0003-3570-6356http://orcid.org/0000-0003-3570-6356http://orcid.org/0000-0003-3570-6356http://orcid.org/0000-0003-3570-6356http://orcid.org/0000-0003-3570-6356http://orcid.org/0000-0001-6746-5433http://orcid.org/0000-0001-6746-5433http://orcid.org/0000-0001-6746-5433http://orcid.org/0000-0001-6746-5433http://orcid.org/0000-0001-6746-5433http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0009-0002-4553-6039http://orcid.org/0009-0002-4553-6039http://orcid.org/0009-0002-4553-6039http://orcid.org/0009-0002-4553-6039http://orcid.org/0009-0002-4553-6039http://orcid.org/0000-0002-6310-1576http://orcid.org/0000-0002-6310-1576http://orcid.org/0000-0002-6310-1576http://orcid.org/0000-0002-6310-1576http://orcid.org/0000-0002-6310-1576http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-49446-z&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-49446-z&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-49446-z&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-49446-z&domain=pdfmailto:anindya@iisc.ac.inspin-wave excitations or magnons in graphene-based QHFdevices12,28–32. While magnon generation in these phases is based on anout-of-equilibrium occupation of edge channels with opposite spin,the detection of the magnons relies on the absorption of magnons byedge modes in the vicinity of ohmic contacts. The absorption ofmagnons by the edge modes creates excess electrons or holes in dif-ferent corners of the graphene devices, and the measured electricalsignal depends on the relative difference between the electron andhole signal magnitudes, which, in turn, critically depend on the devicegeometry. One may, therefore, not be able to detect any electricalsignal if both the excited electrons and hole signals are equal. Thus, analternative technique, which does not rely on the difference betweenexcess electron and hole signals, is necessary for sensitive detection ofmagnons.In this work, we demonstrate that electrical noise spectroscopy ofmagnons is a powerful method that satisfies the detection sensitivityrequirement.We first establish that our device hosts symmetry-brokenrobust QH phases and study the magnon transport when the bulkfilling is kept at ν = 1. In order to generate magnons, we inject an edgecurrent through an ohmic contact. While the injected current onlyflows in the downstream direction (as dictated by the electron motionsubject to an external magnetic field), we measure the non-localelectrochemical potential of a floating ohmic contact placed upstreamfrom the source contact. Whenever the bias voltage of the injectioncontact corresponds to an energy smaller than the Zeeman energy EZ,no non-local signal is detectable. As the bias energy exceeds EZ, wemeasure afinite non-local signal for negative bias voltages. By contrast,the non-local signal remains zero for the entire positive bias voltages,whichmay naively suggest that magnons are not generated in this biasregime. Next, we switch to measuring the electrical noise and showthat, as expected, no noise is detected below EZ. On the other hand, assoon as the bias energy exceeds EZ, the noise increases for both signsof the bias voltage. We show that the noise contributions created dueto magnon absorption at different corners in our devices are additive,even when the average electron and hole currents mutually cancel(which happens for positive bias voltages). This renders noise spec-troscopy a highly sensitive tool for magnon detection. Finally, ourtheoretically calculated noise captures well the experimental data andfurther suggests that the detected noise is a result of an increase in theeffective temperature of the system as a result of equilibrationbetween edge channels and magnons.ResultsDevice and experimental principleFigure 1a shows the schematics of our device andmeasurement setup.The device consists of hBN encapsulated graphite-gated high-mobilitysingle-layer graphene, fabricated by the standard dry transfertechnique33,34. Device fabrication and characterization are detailed inthe Supplementary Information (SI-S1). The QH response of the deviceat a magnetic field (B) of 1 T is shown in Fig. 1c, indicating robust QHplateaus and the inset depicts the activation gap at ν = 1, which isestimated to be ∼4K (see SI-S1). As seen in Fig. 1a, the device has leftand right ground contacts, while the upper transverse contact is uti-lized to inject current for magnon generation. The lower transversecontact is used to detect the change in the chemical potential of thefloating contact (FC) due to magnon absorption. The device’s bulk istuned to the ν = 1 QHF state, allowing it to host magnons. Importantly,the local doping due to the attached metallic contacts increases thefilling factor to ν = 2 near these contacts; this is represented (shownonly for the right side of the FC) by additional loop-shaped edgemodes at each contact, and are referred to as the “inner edge”. Incontrast, the outer edge propagates between contacts, as shown inFig. 1a. A noiseless current, Idc + dI, comprising a dc and an ac com-ponent, is injected into the red-colored source contact in Fig. 1a. Theinjected current flows along the outer edge with up-spin polarization.This current exits the sample at the right-most grounded contact. Thecurrent along the inner edge, which flows around the source contact,has a down-spin polarization, does not contribute to the electricalconductance in the circuit. The dc voltage drop at the source contact,VS = Idc ×he2, is shown as the electrochemical potential μ in Fig. 1a. Thecorresponding ac voltage that drops at the source contactis dVL =dI ×he2.Whenever μ exceeds the Zeeman energy EZ, i.e., ∣μ∣ ≥ EZ, the elec-trons flowing along the circulating inner edge can tunnel into the outeredge by flipping their spin via magnon emission near point ‘A’, asshown in Fig. 1a. This process does not directly alter the electricalconductance since the tunneling current flows back and is absorbedbythe same injection contact. The emitted magnons propagate throughthe bulk of the device and can be absorbed at the device corners (‘B’,‘C’, ‘D’, ‘E’ and ‘F’) via tunneling of electrons from theouter edge to theinner edge through the reverse spin flipping process. However, onlyparts of the currents generated at the two corners ‘B’ and ‘D’ arrive atthe FC and contribute to the fluctuations of the electrochemicalpotential δμFC of the FC. This is so since generated electron and holeexcitations are separated at points ‘B’ and ‘D’ into two respectivecurrents, only one of whichflows towards the FC, as shown in Fig. 1a, b.The fluctuations δμFC or noise are measured in the lower transversecontact placed to the left of the FC on the lower edge, by using an LCRresonance circuit at a frequency of ∼740 kHz, followed by an amplifierchain and a spectrum analyzer35,36. At zero bias, the measured noisepredominantly arises from the equilibrium thermal noise,SV(I =0) = 4kBTR. At finite bias above the Zeeman energy, due tomagnon absorption, excess voltage noise will be generated andquantified as δSV = SV(I) − SV(I =0). The δSV is converted to excess cur-rent noise by δSI = δSV/R2, where, R= hνe2 is the resistance of the con-sideredQH edge. Further details about noise detection are specified inthe Method section and in SI-S8.We also measure the average chemical potential of the FC (dVNL)via the same transverse contacts with standard lock-in measurements.It should be noted that the magnon generation in Fig. 1a is shown onlyfor negative bias voltage; for positive bias voltage, magnons areinstead generated near point ‘E’, as shown in Fig. 2b. We carried outmeasurements in two devices, where for the second device (bilayergraphene), the filling near the contacts was tuned by local gating,showing similar results (see SI-S6, S7).Magnon detection using non-local resistance and noisespectroscopyFigure 1d shows a 2D color map of the differential resistance RL = dVL/dI (with L denoting “local”) measured in the injection contact as afunction of the bias voltage (VS) and gate voltage (VBG) around thecenter of the ν = 1 plateau. It can be seen that within EZ [white verticaldashed lines in Fig. 1d], RL remains constant at he2 ∼ 25:8kΩ anddecreases on both sides above EZ, as shown by the solid magentaline in Fig. 1(d). This feature is similar to that in ref. 12, and can beunderstood as follows: For negative bias voltages, magnons aregenerated at ‘A’. Absorption at ‘B’ and ‘F’ reduce (via holes) andincrease (via electrons) the chemical potential (dVL) of the sourcecontact, respectively, and thus affect RL. However, since theabsorption at ‘B’ dominates over that at ‘F’, RL decreases. Note that tobe absorbed at ‘F’, the magnons have to bend around the injectedcontact in contrast to their straight propagation when reaching ‘B’.Similarly, for positive bias voltage, the generated magnons from ‘E’[see Fig. 2b] are absorbed dominantly at ‘F’ in comparison to ‘B’ andthus RL decreases.A more powerful approach to magnon detection, which permitsto explicitly demonstrate and to explore magnon transport throughthe system, is providedbynon-localmeasurements12,31. Figure 1e showsa 2D colormapof the non-local differential resistance,RNL = dVNL/dI, vsbias and gate voltages, where dVNL is the chemical potential of the FC.Article https://doi.org/10.1038/s41467-024-49446-zNature Communications |         (2024) 15:4998 2As seen from the line cut in Fig. 1f (top panel), RNL remains zero withinEZ (vertical, dashed lines), and increases for negative bias voltageabove EZ. However, RNL is almost zero for the entire positive bias vol-tage range. When the bulk filling was set to ν = 2, no detectable non-local signal (Fig. 1f, lower panel) was observed as the ground state isthen non-magnetic. The RNL in Fig. 1e, f can be understood as follows:As schematically shown in Fig. 1b, themagnon absorption at ‘B’ and ‘D’contributes to the non-local signal of the FC via excess electrons andholes, respectively. For negative bias voltage, the magnons are gen-erated at ‘A’, but the absorption at ‘B’ dominates over ‘D’ due toshorter distance [Fig. 1a], and thusRNL takes afinite value. However, forpositive bias voltage, the magnons are generated at ‘E’ [Fig. 2b], andthe absorption at ‘B’ and ‘D’ are almost equal due to their similardistance from ‘E’. Thus, RNL becomes almost zero.Figure 2c shows a 2D color map of the measured excess noise (SI)in the FC as a function of bias and gate voltages. The correspondingline cuts are shown in Fig. 2d (upper panel).We see that SI remains zeroas long as ∣eVS∣ ≤ EZ, and keeps increasing for larger values of eitherpositive or negative bias voltage. This feature stands in stark contrastto Fig. 1e, f. We have repeated this measurement at different magneticfields (see SI, S2). For example, Fig. 2e showsRNL and SI atB = 2 T, whichdisplay features very similar to the data at B = 1 T. The noise generationmechanism can be understood as follows: The absorption of magnonsresults in a change in the electrochemical potential of the FC either viaexcess electrons or holes, which are created at different absorbingcorners. This process of magnon absorption at different cornersoccurs randomly, rendering the absorption events uncorrelated.Whenan equal number of excess electrons and holes reach the FC, themean22 26 30V BG(x10-1 V)0.680.700.720.740.760.780.800.660.700.740.780.820 1 20481201dVNL/dI (k Ω)dVNL/dI (kΩ)dVL/dI (kΩ)ν = 2VS   (mV)51015202530R (kΩ)VBG (V)c) d) e) f)4567ln R (Ω)1/T (K-1)∆ ~ 4 KDataLinear fitν = 2ν = 1ν = 6ν = 3B = 1Ta) + dI b)FC1620VBG= 0.065 V0.072 VVBG=0.160 V0.1 0.40.2 0.3 0.40.2-0.2-0.4 0V BG(x10-1 V)0.40.2-0.2-0.4 0VS   (mV)0.5 1.0 1.5 2.0VS   (mV)0.40.2-0.2-0.4 0dVNL/dI (k Ω)ν = 1Fig. 1 | Device schematic, magnon generation, and detection in quantum Hallferromagnet. a The device has a left, right, transverse, and floating contact. Thedevice is set to ν = 1, whereas regions adjacent to the contacts are tuned to ν = 2, asshown by the additional circulating inner edges near the contacts. The spinpolarization of the outer and inner edges are orthogonal, denoted by up and downarrows, respectively. A dc plus ac current (Idc+ dI) is injected through the upper redtransverse contact, and when the electrochemical potential (μ) exceeds Zeemanenergy (EZ), magnons are generated near point “A” via a spin-flip process. Thesemagnons propagate through theQH bulk and are absorbed at other corners via thereverse spin-flip process. The bottom transverse contact is used to measure thevoltage (dV) and noise (SV) of the floating contact using standard lock-in (∼ 13Hz)and LCR resonance circuit (∼740 kHz), respectively. b Magnon absorption at thedifferent corners creates electron-hole excitations, but only points “B” and “D”contribute excess electrons and holes to the floating contact, respectively. c QHresponse at B = 1 T. The inset shows the activation gap of ν = 1, which is ∼4 K. d 2Dcolor map of the differential resistance (dVL/dI) measured at the source contact vsthe dc bias voltage (VS = Idc ×he2) and the gate voltage around the center of the ν = 1plateau. A line cut at VBG = 0.079 is shown in solid magenta. e Non-local dVNL/dI ofthe floating contact vs source and gate voltages. f (upper panel) Line cuts from (e).Each plot is shifted vertically for clarity. (bottom panel) Non-local dVNL/dI for bulkν = 2. The vertical lines in d–f represent the Zeeman energy at B = 1 T.Article https://doi.org/10.1038/s41467-024-49446-zNature Communications |         (2024) 15:4998 3electrochemical potential change of the FC is zero, resulting in a van-ishing signal for the non-local resistance. In contrast, the variance ofthe electrochemical potential is independent of the signs of theimpinging charges and thus remains nonzero because of the randomarrival of excess electrons and holes, leading to fluctuations in theelectrochemical potential of the FC.Note that, in general, thefinite noisemeasured in Fig. 2dmay arisefrom hot phonons excited from Joule heating at the hot spot corners(‘A’ and ‘E’) in Fig. 2a and thus increase the effective temperature of theFC. In order to distinguish this noise from noise generated bymagnonabsorption, we study the non-magnetic state ν = 2 (which supportsphonons but not magnons). No significant noise was detected for thisnon-magnetic state at B = 1 T, as shown in Fig. 2d (bottom panel), andfurther shown in SI-Fig. 5 for higher magnetic fields. These resultsestablish that the phonon contribution is negligible at magneticfields B ≤ 2 T.The threshold voltage, Vth for magnon detection, extracted fromSI at different B, is plotted in Fig. 2f (solid red circles with error bars). Ata given B, we calculate the root mean square (rms) value of the data,and a sudden change in its magnitude is marked as the thresholdvoltage. The threshold voltage was extracted for several back gatevoltage points across theHall plateau, and itsmean value and standarddeviationas an error bar are shown in Fig. 2f. The detailedprocedureofour Vth extraction is discussed in SI–S3 and shown in SI-Fig 3. The solidblack line in Fig. 2f represents the Zeeman energy EZ = gμBB and is seento closely follow the Vth extracted from the noise data. We also showthe threshold voltage extracted from the non-local resistance (fornegative bias voltage) as solid blue circles with error bars. It can beseen from Fig. 2f that Vth extracted from the non-local resistanceexhibits a non-monotonic behavior with increasing B. This featurehighlights the noise as a universal and robust probe for detectingmagnons in contrast to the non-local resistance.Note that the threshold voltage, Vth, above which the non-localresistance arises is significantly higher than EZ [see Fig. 2e, f] forB < 3 T.This behavior has been observed in previous works as well12,32. Incontrast to the resistance data, however, the noise starts to increase atbias voltage ∣eVS∣ ∼ EZ [see Fig. 2d–f]. The difference in threshold vol-tages for the non-local resistance and the noise can be understood if-0.3 -0.15 0 0.15 0.30.650.700.750.800.855 10 15001020304050-0.4 -0.2 0 0.2  0.40510SI (10-29  A2 /Hz)-0.8 -0.4 0 0.4 0.8010200510SI (10-29  A2 /Hz)a)e)f)ν = 2ν = 110-29A2/Hzν = 1b)c) d)err =err =err =1 2 3 4 5B (T)0.10.20.30.40.50.60.7V th(meV)NoiseConductancegμBBVBG= 0.065 V0.072 VVBG = 0.16 VVBG =      0.118VV BG(x10-1 V)SI (10-29  A2 /Hz)B = 1 TB = 2 TVS   (mV)VS   (mV)VS   (mV)B = 1 TdVNL/dI (k Ω)Fig. 2 | Noise spectroscopy of magnons.Magnon generation for negative (a) andpositive (b) bias voltages, where magnons are generated at point 'A' and 'E',respectively. The generated magnons propagate through the QH bulk and areabsorbed at different corners. Only magnon absorption at points 'B' and 'D' gen-erates noise at the floating contact. c 2D colormapof excess noise generated at thefloating contact for different bias and gate voltages. d (top panel) Line cuts ofexcess noise from c) for different VBG around the center of the ν = 1 plateau. Eachplot is shifted vertically for clarity. (bottom panel) Noise spectra for bulk ν = 2. Asexpected, no excess noise is visible. eNoise spectra and dVNL/dI for bulk filling ν = 1atB = 2 T. The vertically dashed lines in c–e depict the Zeeman energy EZ. Each dataset in (d, e) is the five-point average of the raw data (shown in SI-Fig. 3), and “err”represents the standard deviation of the raw data. The “err” remains almost similarin magnitude for each data set in (d), and is shown for one of them. f Thresholds ofthe bias voltage (with error bars) vs magnetic field. The thresholds are extractedboth from noise spectroscopy (solid red circles) and non-local differential resis-tance measurements (solid blue circles). Plotted is also EZ = gμBB (solid black line).Article https://doi.org/10.1038/s41467-024-49446-zNature Communications |         (2024) 15:4998 4magnons are absorbed in ‘B’ and ‘D’with equal probabilities within thebias voltage window EZ < ∣eVS∣ < eVth. Hence, this absorption process isinvisible in the non-local resistance data while strikingly visible in thenoise data. Such an equal magnon absorption at ‘B’ and ‘D’ (geome-trically located at asymmetric distances from the magnon generationpoint) for negative bias voltage may arise from ballistic magnontransport in the bias voltage window EZ < ∣eVS∣ < eVth, where generatedmagnons propagate with a long wavelength λ≫ ℓB. Such magnonsexperience little scattering from other degrees of freedom, particu-larly phonons or skyrmions29, and reach all absorption corners withalmost equal probabilities.However, the ballisticmotion ofmagnonsmaynot bepossible at ahighermagnetic field,B > 2T. At higherB, a larger current is required togeneratemagnons, and thus also,more phononsmay be excited at thehot spots near ‘A’ and ‘E’ [see Fig. 2a] due to increased Joule heating.Indeed, the proliferation of phonons is observed while measuring afinite noise for the non-magnetic state, ν = 2, at B > 2T; see SI-Fig. 5.These excited phonons at higher B can play an important role inscattering the magnons. As a result, the magnon transport may notremain ballistic, and hence the threshold voltage for the non-localresistance at B > 2T is reduced to the vicinity of EZ, and in fact, is evenslightly lower than EZ as seen in Fig. 2f. The reductionbelow EZ could bedue to the temperature-broadening effect as the excited phononselevate the temperature of the entire system and thus soften theZeeman gap EZ. In order to validate the claim about the temperature-broadening effect, we havemeasured the non-local resistance atB = 1T(where there are no phonons generated at the hot spot) at increasingbath temperature. We see the evolution of Vth from higher than EZ atlower bath temperature to lower than EZ at higher temperature. Theseresults are summarized in SI–S4. It is worth noting that while hotphonons contribute to the measured excess noise at higher magneticfields (B > 2T), a distinct sudden increase in noise magnitude due tomagnons occurs around VS∼ EZ, as shown in SI-Fig. 2.Theoretical model and comparison to experimentIn this section, we theoretically model the noise spectroscopyobserved at a lower magnetic field, such that the effects of hot pho-nons can be neglected. We model the edge segments where themagnon generation and absorption take place as line junctions of co-propagating edgeswith length L, where electrons tunnel between edgechannels (with spin-↑ and spin-↓), see Fig. 3a. Each such tunnelingevent is associated with the generation or absorption of magnons. Weidentify two distinct transport regimes depending on a degree ofequilibration, characterized by the equilibration length ℓeq; a short-junction regime (L < ℓeq) with partial equilibrationof the edge channelsand the magnons, and a long-junction regime (L > ℓeq) with strongequilibration, see Methods and SI-S9 for details. In the strong equili-bration regime, equilibration in the magnon-generation region ‘A’takes place until the chemical potential difference between the edgechannels equals EZ. At this saturation point, further magnon genera-tion is strongly suppressed. All generated magnons propagate in thebulk of the QH state and are eventually absorbed in one of theabsorption regions (‘B’, ‘C’, ‘D’, ‘E’ and ‘F’). Each absorption eventcreates an electron-hole pair (an electron in the spin-↓ channel and ahole in the spin-↑ channel). These pairs produce the measured excessnoise. In each absorption line junction, the excess noise is dominantlygenerated near x = L [yellow circle in Fig. 3a] while remaining con-tributions are exponentially suppressed, see refs. 36–40 for a similarnoise-generating mechanism. The excess noise SI reflects an increasedtemperature of the edge channels during the magnon absorption,given bySI =12e2hðT0 +TÞ � 2e2hT0� �=12e2hðT � T0Þ� �, ð1Þwhere T0 is the bath temperature andT =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT20 +3ðjeVSj � EZ Þð2EZ +3jeVSjÞð5πÞ2θðjeVSj � EZ Þsð2Þis the effective temperature of the system as a result of equilibration.Furthermore, θ(∣eVS∣ − EZ) is the step function, which reflects the factthat no magnons can be absorbed for bias energies below EZ. Thefactor 1/2 inEq. (1) originates from thenoise-measurement scheme, seeMethods. In Fig. 3b, we compare our theoretically calculated excessnoise (solid red line),SI, with the experimentallymeasurednoise versusthe bias energy eVS (for simplicity, only the negative bias side isdisplayed), at fixed T0 = 20mK. Figure 3c shows the measured noise atdifferent bath temperatures (T0), and the corresponding theoreticalplots are shown in Fig. 3d. A comparison between the experiment(orange circles) and theory (blue circles) for SI at VS = −0.3mV as afunction of T0 is shown in Fig. 3e. Our theoretical model captures wellthe characteristic features of the noise. Note that as seen in Fig. 3f, noexcess noise was detected even at higher temperatures (600mK) forν = 2 at B = 1 T.The bias voltage dependence of the excess noise defines threeregimes in Fig. 3b; (i) Biases ∣eVS∣ < EZ result in no magnon generationand thus no excess noise. (ii) In a narrow region 0< jeVSj � EZ<1γL, theequilibration in magnon absorption and generation regions is onlypartial, L< ‘eq � 1γðjeVS j�EZ Þ. Here, γ is a parameter proportional to thetunneling strength in every tunnel junction comprising the line junc-tion. This lack of equilibration allows us to model the magnon-generation and absorption regions as single tunnel junctions in regime(ii), see Methods and SI-S9 for further details of the model. In thismodel, the noise generation is of a non-equilibriumnature, resulting inSI = e2CðjeVSj � EZ Þ2=h with the parameter C = γL. The single para-meter of the model, C, is obtained by fitting to the experimental data,as shown in Fig. 3b by the solid, blue line. (iii) For larger biasesjeVSj> EZ +1γL and hence ℓeq < L, the edge channels and magnonsachieve full equilibration in the magnon absorption and generationregions. We find that our theoretical model is in good agreement withthe experimental data. In particular, at sufficiently large biases [regime(iii)], our equilibrated line junction model correctly describes severalexperimental observations: the sudden increase followed by(approximate) saturation of the non-local conductance as a function ofthe bias voltage [see Fig. 1f], the linear behavior of the noise as afunction of the bias voltage [Fig. 2d], and the temperature dependenceof the excess noise [Fig. 3d, e]. In addition, our single tunnel junctionmodel [partial equilibration regime (ii)] properly describes the cross-over region of bias voltages close to EZ. Note that our theory assumesthatmagnons are absorbed in all the absorption regions with the sameprobability, but in reality, there may be deviations. These can explainsome variations between experimental data curves.DiscussionAs we show in Eq. (1), the excess noise generated in the line junction[regime (iii)] reflects the increase in temperature T − T0 of the edgedue to heating. The temperature behavior extracted from the mea-sured excess noise data in Fig. 3b (right y-axis) is similar to thetemperature behavior in Fig. 3d of ref. 32, a result which wasobtained from Rxx thermometry measurements. However, in thestudy by ref. 32, the equilibration between magnons with electronsor holes to form skyrmions is manifest in two distinct regimes. Forshorter wavelengths (λ≪ lB), equilibration occurs more easily thanfor longer wavelengths (λ≫ ℓB). Consequently, a higher bias energy(VS > 4EZ) was necessary to induce the formation of skyrmions. Theauthors argued that a linear increment of the temperature withincreasing VS originates from free magnons, whereas the saturationof temperatures at higher VS is attributed to the formation of sky-rmions. Contrary to these findings, we have not encountered anyArticle https://doi.org/10.1038/s41467-024-49446-zNature Communications |         (2024) 15:4998 5temperature saturation at higher bias voltages in our experimentalobservations, indicating a lack of skyrmion formation. Therefore, weexpect our experiment predominantly to examine free magnons.However, as previously described, their transport behavior can behindered by hot phonons proliferated at higher B. Note that otherpossibilities, such as impurities or contact transparency, may affectthe magnon transport at higher B.Furthermore, the noise behavior as a function of the bias voltageappears to be correlated with that of the visibility of the Mach-Zenderinterferometry measured in ref. 30. It would be interesting to make adetailed connection between those twodifferent quantities. Finally, weemphasize that ourmeasurements were performed for relatively smallmagnetic fields and lower ambient temperatures than in previousworks12,30,32. These small quantities allowus to fully neglect the effect ofphonons. We have observed a sizeable effect of phonons only formagnetic fields B > 2T (see SI-S5). Finally, we expect that edge recon-struction does not happen in our graphite-gated devices, as theoreti-cally studied in ref. 41 and experimentally established inrefs. 39,40,42,43. Even if edge reconstruction produces additionalpairs of counter-propagating edge modes, it is possible that eachindividual pair localizes over a short-length scale. Thus, suchmodes donot contribute to the low-energy transport, yielding no quantitativechanges in RNL and SI.In summary, we have demonstrated the utility of electrical noisespectroscopy as a highly sensitive tool for detecting and studyingmagnons in aquantumHall ferromagnet.Our newprotocol overcomesnon-universal (e.g., device geometry dependent) features that screenout the presence of magnons, when other detection tools areemployed, most prominently non-local conductance measurements.This robustness paves the way for utilizing magnons as low-powerinformation carriers in future quantum technologies. Intriguing gen-eralizations of our approach, with a promise of novel physics, includebulk phases of the fractional quantumHall regime as well as of integerand fractional Chern insulator phases of twisted bilayer graphene44–46.Further implementations of our approach may include other ferro-magnetic materials and vdW magnets47,48.000.2 0.42.0-51015204.0-00.20.40.60.8SI (10-29  A2 /Hz)VS (mV)err bar =(i)(ii)(iii)0510152020 4060100150200400T (mK) err barVBG = 0.072 V0.40.2-0.4 -0.2 0SI (10-29  A2 /Hz)VS (mV)e)TheoryExperimentVS  = - 0.3 mV510150.40.60.2T (K)ΔTo (K)SI (10-29  A2 /Hz)0 0.2 0.4 0.6 0.805101520 20 ( mK)60100200400600d)SI (10-29  A2 /Hz)-0.4 -0.1 0-0.2-0.30102020 600f)T(mK) err barVBG = 0.156V0.30 0.20.1-0.1-0.3 -0.2SI (10-29  A2 /Hz)b)c)a)T (K)ΔVS (mV)VS (mV)Fig. 3 | Theoretical model and temperature dependence of excess noise. a Themagnon absorption (wiggly green lines with arrow) at any corner is modeled as aline segment of co-propagating edges, where tunneling of electrons occur fromouter to inner edge [Figs. 1a, 2a, b]. The noise from the total tunneling current isdominantly generated in the vicinity of x = L (yellow circle), where the local equi-librium noise dominates over the shot noise. b Comparison between the experi-mentally measured excess noise at 20mK (black solid circles) and the theoreticalresults in a tunneling junction model; While the red solid line is the prediction forthe strongly equilibrated regime, Eq. (1), the blue solid line is for the partiallyequilibrated regimewhich is obtained by fitting the formula SI = e2γLðeVS � EZ Þ2=hwith the experimental data. For this plot, we used the parameter choice γL =0.8/EZ.These regimes of nomagnons, partial and strong equilibration are further indicatedby the horizontal arrows at the topof the axis. The right-hand side axis indicates theexcess temperature, see Eq. (1). c Measured noise at different bath temperatures.(d) Noise calculated from Eqs. (1, 2) at different bath temperatures T0.e Comparison between experiment and theory for the excess noise at VS = −0.3mVas a function of bath temperature. The right side of the axis indicates the excesstemperature. f Excess noise for bulk filling ν = 2 at 20 and 600mK, both at B = 1T. Inb, c, f, 'err bar' represents the standard deviation of the raw data shown in SI-Fig. 3.Article https://doi.org/10.1038/s41467-024-49446-zNature Communications |         (2024) 15:4998 6MethodsDevice and measurements schemeUtilizing the dry transfer pick-up approach, we fabricated encapsu-lated devices consisting of a heterostructure involving hBN (hexagonalboron nitride), single-layer graphene (SLG), and graphite layers. Theprocedure for creating this heterostructure comprised themechanicalexfoliation of hBN and graphite crystals onto an oxidized silicon waferthrough the widely employed scotch tape method. Initially, a layer ofhBN,with a thickness of∼25–30 nm,was picked up at a temperature of90 °C. This was achieved using a poly-bisphenol-A-carbonate (PC)coated polydimethylsiloxane (PDMS) stamp on a glass slide attachedto a home-built micromanipulator. The hBN flake was aligned over thepreviously exfoliated SLG layer picked up at 90 °C. The subsequentstep involved picking up the bottom hBN layer of similar thickness.Following the same process, this bottom hBN was picked up utilizingthe previously acquired hBN/SLG assembly. After this, the hBN/SLG/hBN heterostructure was employed to pick up the graphite flake.Ultimately, this resulting heterostructure (hBN/SLG/hBN/graphite)was placed on top of a 285-nm thick oxidized silicon wafer at a tem-perature of 180 °C. To remove the residues of PC, this final stack wascleaned in chloroform (CHCl3) overnight, followed by cleaning inacetone and isopropyl alcohol (IPA). After this, poly-methyl-methacrylate (PMMA) photoresist was coated on this hetero-structure to define the contact regions using electron beam litho-graphy (EBL). Apart from the conventional contacts, we defined aregion of ∼6μm2 area in the middle of the SLG flake, which acts as afloating metallic reservoir upon edge contact metallization. After EBL,reactive ion etching (mixture of CHF3 and O2 gas with a flow rate of40 sccm and 4 sccm, respectively, at 25 °C with RF power of 60W) wasused to define the edge contact. The etching time was optimized suchthat the bottom hBN did not etch completely to isolate the contactsfrom the bottom graphite flake, which was used as the back gate.Finally, the thermal deposition of Cr/Pd/Au (3/12/60 nm) was done inan evaporator chamber with a base pressure of ∼1 × 10−7 mbar. Afterdeposition, a lift-off procedure was performed in hot acetone and IPA.The device’s schematics and measurement setup are shown in Fig. 1a.The distance from the floating contact to the ground contacts was∼5μm, whereas the transverse contacts were placed at a distanceof ∼2.5μm.All measurements were done in a cryo-free dilution refrigeratorwith a ∼20mK base temperature. The electrical conductance wasmeasured using the standard lock-in technique, whereas the noise wasmeasured using an LCR resonant circuit at resonance frequency∼740 kHz. The signal was amplified by a homemade preamplifier at4 K, followedby a room temperature amplifier, andfinallymeasuredbya spectrum analyzer. At zero bias, the equilibrium voltage noise mea-sured at the amplifier contact is given bySV = g2ð4kBTR+V 2n + i2nR2ÞBW , ð3Þwhere kB is the Boltzmann constant, T is the temperature, R is theresistance of the QH state, g is the gain of the amplifier chain, andBW is the bandwidth. The first term, 4kBTR, corresponds to thethermal noise, and V 2n and i2n are the intrinsic voltage and currentnoise of the amplifier. At finite bias above the Zeeman energy, due tomagnon absorption at points ‘B’ and ‘D’, chemical potential fluc-tuations of FC create excess voltage noise at the amplifier contact. Atthe same time, the intrinsic noise of the amplifier remains unchan-ged. Due to the white nature of the thermal noise and the excessnoise, we could operate at a higher frequency (∼740 kHz), whicheliminates the contribution from flicker noise (1/f) which usuallybecomes negligible for frequencies above a few tens of Hz. Theexcess noise (δSV) due to bias current is obtained by subtracting thenoise value at zero bias from the noise at finite bias, i.e.,δSV = SV(I) − SV(I = 0). The excess voltage noise δSV is converted toexcess current noise SI according to SI =δSVR2 , where R= hνe2 is theresistance of the considered QH edge.Theoretical calculation of the non-local resistance and noiseTo compute the tunneling current, non-local resistance, and noisegenerated in the magnon absorption regions, we model the magnongeneration and absorption regions as line junctions of length L. Theseline junctions are modeled as extended segments with two co-propagating edge channels in which electrons tunnel along a seriesof tunnel junctions, see Fig. 3a. We identify two distinct transportregimes: those of a short (partially equilibrated; L < ℓeq) and long(equilibrated; L > ℓeq) junctions, where ℓeq is the equilibration length.The short-junction regime can be equivalently modeled as a singletunnel junction. Details of the theoretical analysis are presentedin SI, Sec. S9.We first consider the partial-equilibration regime, treating themagnon generation or absorption regions as a single tunnel junction(at position x =0). The Hamiltonian describing this junction readsH = � ivXs =",#Zdxψys ðxÞ∂xψsðxÞ+XqðEZ + _ωqÞbyqbq+Wψy"ðx =0Þψ#ðx =0Þbyðx =0Þ+h:c::ð4ÞEmploying the Keldysh non-equilibrium formalism, we derive zero-temperature expressions for the tunneling current Iab, non-localresistance dVab/dI, and noise Sab generated in an absorption region,respectively:Iab = γ0 e2hðjeVSj � EZ Þ2θðjeVSj � EZ Þ , ð5ÞdV abdI��������= γ0 he2 ðjeVSj � EZ ÞθðjeVSj � EZ Þ , ð6ÞSab =e2hγ0ðjeVSj � EZ Þ2θðjeVSj � EZ Þ : ð7ÞHere, γ0 is a parameter associated with the tunneling strength in thetunnel junction. While the non-local resistance increases linearly withincreasing bias voltage eVS, the noise increases instead quadratically.For finite temperature, we first numerically determine the eVS-dependence of the magnon chemical potential μm, and thereby weobtain the eVS-dependence of the non-local resistance and noise. Thisfinite temperature result is used to fit the experimental data for regime(ii) in Fig. 3b.In the limit of a long line junction, the last term in Eq. (4) ismodified to describe tunneling in the spatial region 0 ≤ x ≤ L. In theequilibrated regime, L > ℓeq, this model yields non-local resistance andnoise characteristics distinct from those in the single tunnel-junctionmodel. Specifically, the equilibrated line-junction model predicts thefollowing tunneling current, non-local resistance, and the excess noisein each individual absorption region,Iab =e2hMðjeVSj � EZ ÞθðjeVSj � EZ Þ , ð8ÞdV abdI��������= h2Me2θðjeVSj � EZ Þ , ð9ÞSab =e2hðT � T0Þ , ð10Þwith the increased temperature T of the system, Eq. (2). Here, M = 5 isthe number of absorption regions. Notably, the non-local resistanceArticle https://doi.org/10.1038/s41467-024-49446-zNature Communications |         (2024) 15:4998 7(9) is constant in eVS, whereas the noise [Eqs. (1)-(2)] instead increaseslinearly in eVS at sufficiently large bias voltage eVS. In the calculation ofEqs. (2), (8), and (9), we have assumed for simplicity that the magnonsare absorbed in each individual absorption region with equalprobabilities. Note that the measured excess noise SI in Eq. (1) hasthe additional factor 1/2 compared with the excess noise generated inan absorption region, i.e., SI =12 Sab = 2 ×14 Sab. The factor of 2 reflectscontributions from two noise spots (‘B’ and ‘D’) and the factor 1/4 = (1/2)2 originates from that only one channel out of the two emanatingfrom the FC is measured at the bottom transverse contact, see Fig. 1a.We also calculate the dependence of the equilibration length ℓeqon the bias voltage eVS. We do this by using the results for the partial-equilibration regime (short L) and inspecting at what L the equilibra-tion becomes strong. The result reads‘eq =1γðjeVSj � EZ Þ, forjeVSj> EZ : ð11ÞThis equation implies a partial-equilibration regime for ∣eVS∣ slightlyexceeding EZ and a strong-equilibration regime for larger ∣eVS∣, asdiscussed in the “Discussion” section above, and also illustrated inFig. 3b. Equation (11) shows that ℓeq increases significantly as the biasenergy approaches EZ, indicating that the equilibration process takesplace very slowly near ∣eVS∣ ∼ EZ. This happens because the absorptionrate per unit length is proportional to ðjeVSj � EZ Þ2 [Eq. (5)] whereasthe total tunneling current in the equilibrated regime scales as(∣eVS∣ − EZ) [Eq. (8)].Reporting summaryFurther information on research design is available in the NaturePortfolio Reporting Summary linked to this article.Data availabilityThe data presented in the manuscript are available from the corre-sponding author upon request.References1. Yuan, B. et al. Dirac magnons in a honeycomb lattice quantum XYmagnet CoTiO3. Phys. Rev. X 10, 011062 (2020).2. Chen, L. et al. Magnetic field effect on topological spin excitationsin CrI3. Phys. Rev. X 11, 031047 (2021).3. Ganguli, S. C. et al. Visualization of moiré magnons in monolayerferromagnet. Nano Lett. 23, 3412–3417 (2023).4. Spinelli, A., Bryant, B., Delgado, F., Fernández-Rossier, J. & Otte, A.F. Imaging of spin waves in atomically designed nanomagnets.Nat.Mater. 13, 782–785 (2014).5. Kampfrath, T. et al. Coherent terahertz control of antiferromagneticspin waves. Nat. Photonics 5, 31–34 (2011).6. Zhuang, S. & Hu, J.-M. 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Layer-dependent ferromagnetism in a vanderWaalscrystal down to the monolayer limit. Nature 546, 270–273 (2017).AcknowledgementsA.D. thanks the Department of Science and Technology (DST) and Sci-ence andEngineeringResearchBoard (SERB), India, forfinancial support(SP/SERB-22-0387) and acknowledges the Swarnajayanti Fellowship ofthe DST/SJF/PSA-03/2018-19. A.D. also thanks financial support fromCEFIPRA: SP/IFCP-22-0005. S.K.S. and R.K. acknowledge the PrimeMinister’s Research Fellowship (PMRF), Ministry of Education (MOE), andInspire fellowship, DST for financial support, respectively. A.D.M., J.P.,and Y.G. acknowledge support by the DFGGrantMI 658/10-2 and by theGerman-Israeli Foundation Grant I-1505-303.10/2019. Y.G. acknowl-edges support from the Helmholtz International Fellow Award, the DFGGrant RO 2247/11-1, CRC 183 (project C01), and the Minerva Foundation.Y.G. acknowledges the Infosys Chair professorship at IISc for this col-laboration. C.S. acknowledges funding from the 2D TECH VINNOVACompetence Center (Ref. 2019-00068). This project has receivedfunding from the European Union’s Horizon 2020 research andinnovation program under grant agreement No 101031655 (TEAPOT).K.W. and T.T. acknowledge support from the Elemental Strategy Initia-tive conducted by the MEXT, Japan, and the CREST (JPMJCR15F3), JST.Author contributionsR.K., S.K.S., and U.R. contributed to device fabrication, data acquisition,and analysis. A.D. contributed to conceiving the idea and designing theexperiment, data interpretation, and analysis. K.W. and T.T. synthesizedthe hBN single crystals. J.P., C.S., Y.G., and A.D.M. contributed to thedevelopment of theory and data interpretation, and all the authorscontributed to writing the manuscript.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-49446-z.Correspondence and requests for materials should be addressed toAnindya Das.Peer review informationNature Communications thanks Petr Stepanov,and the other, anonymous, reviewer(s) for their contribution to the peerreview of this work. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2024Article https://doi.org/10.1038/s41467-024-49446-zNature Communications |         (2024) 15:4998 9https://doi.org/10.1038/s41467-024-49446-zhttp://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Electrical noise spectroscopy of magnons in a quantum Hall ferromagnet Results Device and experimental principle Magnon detection using non-local resistance and noise spectroscopy Theoretical model and comparison to experiment Discussion Methods Device and measurements�scheme Theoretical calculation of the non-local resistance and�noise Reporting summary Data availability References Acknowledgements Author contributions Competing interests Additional information