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Cheng-Li Chiu, Taige Wang, Ruihua Fan, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Xiaomeng Liu, Michael P. Zaletel, Ali Yazdani

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[High spatial resolution charge sensing of quantum Hall states](https://mdr.nims.go.jp/datasets/0af38485-da45-4491-83a4-d396c89cd0df)

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High spatial resolution charge sensing of quantum Hall statesPNAS  2025  Vol. 122  No. 8 e2424781122� https://doi.org/10.1073/pnas.2424781122 1 of 6RESEARCH ARTICLE | Significance Our method integrates scanning tunneling microscopy (STM) spectroscopy with a monolayer graphene detector in the integer quantum Hall regime, achieving sub-meV energy resolution  (0.3 meV at 1.4 K) and spatial resolution at the magnetic  length scale (10 nm at 6 T).  By combining high energy and spatial precision while minimizing sample perturbation from the STM tip, this technique enables nonperturbative and highly sensitive electrostatic potential mapping. We demonstrate its potential by resolving the charge response of composite fermions to a single impurity and observing charge distribution consistent with Friedel oscillation of composite Fermi liquid. This approach paves the way for studying quantum Hall liquids and exotic quasiparticles, such as anyons, offering a transformative tool for condensed matter physics research.Author contributions: C.-L.C., X.L., and A.Y. designed research; C.-L.C., T.W., R.F., X.L., M.P.Z., and A.Y. performed research; K.W. and T.T. contributed new reagents/analytic tools; C.-L.C., T.W., and R.F. analyzed data; and C.-L.C., T.W., R.F., M.P.Z., and A.Y. wrote the paper.Reviewers: J.K.J., The Pennsylvania State University; and A.Y., Harvard University.Competing interest statement: Author A. Yazdani and referee A. Yacoby co-published a review together in 2023. Additionally, K.W. and T.T., as contributors to materials in this field, have co-publications with referee A. Yacoby, this is common for many collaborations in the field of two-dimensional materials.Copyright © 2025 the Author(s). Published by PNAS. This article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).1To whom correspondence may be addressed. Email: yazdani@princeton.edu.This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.​2424781122/-/DCSupplemental.Published February 18, 2025.APPLIED PHYSICAL SCIENCESHigh spatial resolution charge sensing of quantum Hall statesCheng-Li Chiua , Taige Wangb,c, Ruihua Fanb, Kenji Watanabed , Takashi Taniguchie, Xiaomeng Liuf, Michael P. Zaletelb,c, and Ali Yazdania,1Affiliations are included on p. 6.Contributed by Ali Yazdani; received November 29, 2024; accepted January 14, 2025; reviewed by Jainendra K. Jain and Amir YacobyCharge distribution offers a unique fingerprint of important properties of electronic systems, including dielectric response, charge ordering, and charge fractionalization. We develop an architecture for charge sensing in two-dimensional electronic systems in a strong magnetic field. We probe local change of the chemical potential in a prox-imitized detector layer using scanning tunneling microscopy, allowing us to infer the chemical potential and the charge profile in the sample. Our technique has both high energy (<0.3 meV) and spatial (<10 nm) resolution exceeding that of previous studies by an order of magnitude. We apply our technique to study the chemical potential of quantum Hall liquids in monolayer graphene under high magnetic fields and their responses to charge impurities. The chemical potential measurement provides a local probe of the thermodynamic gap of quantum Hall ferromagnets and fractional quan-tum Hall states. The screening charge profile reveals spatially oscillatory response of the quantum Hall liquids to charge impurities and is consistent with the composite Fermi liquid picture close to the half-filling. Our technique also paves the way to map moiré potentials, probe Wigner crystals, and investigate fractional charges in quantum Hall and Chern insulators.quantum Hall effect | charge sensing | scanning tunneling microscopyMultiple techniques have been established to spatially resolve electrostatic and chemical potential in two-dimensional electronic systems, including Kelvin probe force microscopy (1, 2) (KPFM), electrostatic force microscopy (3, 4), scanning quantum dot microscopy (5) (SQDM), and the scanning single electron transistor (6, 7) (SET). We have witnessed their recent success in understanding strongly correlated electronic systems, such as Wigner crystals (8), generalized Wigner crystal (9, 10), topological phases in moiré heterostructures (11–13), and many more. However, high spatial resolution and high energy resolution have never been achieved simultaneously. For example, scanning SET, though offers remarkable energy resolution (<50 μeV), has limited spatial resolution (>100 nm) due to detector size. KPFM and SQDM can achieve a higher spatial resolution (<40 nm), but at the cost of limited energy resolution (>5 meV). Facing these challenges, we develop a chemical potential and charge sensing technique with both high spatial and high energy resolution based on scanning tunneling spec-troscopy (STS). We take advantage of the high spatial resolution of the conventional STS and convert it into a chemical potential probe by adding a detector layer whose spectroscopy reflects the chemical potential of the sample. Specifically, we choose the detector to be monolayer graphene (MLG), which has proven useful for imprinting the charge density variation of the sample layer ( 9 ,  14 ,  15 ). Here, we fix the MLG to be a particular incompressible state such that its chemical potential is locked to that of the sample, which promotes the detector to a full-fledged quantitative chemical potential probe. The high spatial and energy resolution is achieved by preparing the detector MLG into an integer quantum Hall (IQH) state. The Landau levels (LL) can be used as a sharp spectroscopic feature for chemical potential readout, offering high energy resolution; the incompressible nature of the IQH state allows the detector to be put close to the sample while remaining noninvasive, making high spatial resolution possible. We use STS to track the energy of the LLs in the detector MLG, which can be converted to the chemical potential and charge distribution of the sample. Although using a detector layer in the IQH state requires a magnetic field, we found our technique viable at fields as low as 0.2 T (SI Appendix, section 9 ). Interestingly, while our experimental setup is similar to that of refs.  9 ,  10 , and  14 , the mechanisms for contrast differ: refs.  9 ,  10 , and  14  utilize local tip-gating effects to infer the charge profile of the sample layer from discharge events, whereas we rely on the band-bending effect in the detector layer to directly read out the chemical potential of the sample layer. Both effects are present in our setup, but we configure the STM tip and the electronic state of the detector layer Downloaded from https://www.pnas.org by Kenji Watanabe on February 18, 2025 from IP address 144.213.253.16.https://creativecommons.org/licenses/by-nc-nd/4.0/https://creativecommons.org/licenses/by-nc-nd/4.0/https://creativecommons.org/licenses/by-nc-nd/4.0/mailto:yazdani@princeton.eduhttps://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2424781122/-/DCSupplementalhttps://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2424781122/-/DCSupplementalhttps://orcid.org/0000-0001-8396-3408https://orcid.org/0000-0003-3701-8119mailto:https://orcid.org/0000-0003-4996-8904http://www.pnas.org/lookup/doi/10.1073/pnas.2424781122#supplementary-materialshttp://crossmark.crossref.org/dialog/?doi=10.1073/pnas.2424781122&domain=pdf&date_stamp=2025-2-172 of 6   https://doi.org/10.1073/pnas.2424781122� pnas.orgsuch that the band-bending effect dominates, unlike refs.  9 ,  10 , and  14 , which operate in a regime where the local tip-gating effect is more prominent (see SI Appendix, section 10  for a full comparison).  *   Device Structure and Control Parameters. As shown in Fig. 1A, our device consists of two MLG layers: one serving as the detector and the other below as the sample, separated by a thin hexagonal Boron Nitride (hBN) layer (2.5 nm). A graphite back gate is 70 nm beneath the sample, separated by a thicker hBN layer. We fabricate our devices using state-of-the-art Van der Waals stacking techniques (16) and cleaning techniques which leave minimal residue on the exposed top layer MLG. The stack is placed onto a SiO2/Si substrate with prepatterned electrical contacts. The filling of the detector and sample can be controlled individually by the gate voltages Vg and Vm. We use STM to measure the zero-bias tunneling conductance of the detector layer as a function of  Vg  and  Vm  , in the presence of 6 T magnetic field at 1.4 K. As shown in  Fig. 1B  , we can see compressible regimes (white) and incompressible quantum Hall regimes (red) of the detector layer. When we change the gate voltages, the sample and detector layers vary across different IQH plateaus independently, giving rise to the zigzag pattern. These measurements match both the penetration capacitance measurements ( 17 ) (filling factor difference < 2%) of the same device and simulation with an electrostatic charging model (SI Appendix, sections 2 and 4 ). The excellent match highlights the noninvasive nature of our technique and, combined with the pristine quality of our device, enables the first-time STM observation of quantum Hall excitonic insulating states (for STS and penetration capacitance measurement of excitonic insulat-ing states, see SI Appendix, sections 3 and 4 ). †     ResultsChemical Potential Sensing. The basis of our chemical potential sensing technique is to measure the energy shift of the detector’s LLs as we tune the back gate Vg  (SI Appendix, section 6). Following the dashed line in Fig. 1B, we tune the detector MLG into an incompressible IQH state at � = − 2  to minimize the tip-sample chemical potential mismatch (18) (see SI Appendix, section 5 for details), and charge the sample MLG to partial fillings of the zeroth Landau level (zLL). As shown in Fig. 1C, the energy of the detector MLG’s LL does not increase linearly with the back gate voltage as one would expect from a conventional MLG device (18, 19). In fact, the LL energy tracks the chemical potential �d of the detector MLG, which is locked to the chemical potential �m of the sample MLG. In a uniform sample, it can be shown from an electrostatic charging model that (SI Appendix, section 6)�d = �m + Vm −ndcd,where nd is the charge density of the detector layer and cd is the geometric capacitance cd = �∕dd  per unit area between the detector and the sample. Both Vm and nd remain the same when we charge the sample layer, so �d can be easily converted to �m. This locking behavior can also be understood from a multistep process shown in  Fig. 1D  . When we apply a positive  ΔVg    to the back gate (step 1), an electrochemical potential shift (step 2) will induce charge transfer between the back gate and the sample detector: graphenespacer hBN (2.5nm)sample: graphenebottom hBN (70nm)graphite gateVmVbiasVbias~ ~Vg-2 -1.5 -1 -0.5Vg (V)-130-120-110-100-90VBias (mV)0 5dI/dV(nS )Vm=-20mV-2 -1.5 -1 -0.5Vg (V)-505101520µ (meV)-5/3-4/3-2/3-1/31/32/3ν=-1 ν=0 ν=1Vg+∆Vg Vmh+ h+ h+∆µg=0∆V=∆Vg∆µm∆µs=∆µm1234h+5 6graphitegatemiddle layergraphenesensor layergraphene1LLzLL-1LL2LL1LLzLL-1LL2LL 3LLh+ h+ h+e--5 0 5Vg (V)-100-50050100Vm (mV)0.1 0.5dI/dV |v=0(nS)6T 1.4KA B CD FE-0.8 -0.6 -0.4 -0.2ν-4-2024µ (meV)2.2 2.4 2.6 2.8ν-4-2024µ (meV)zLL 1LLFig. 1.   Charge sensing technique and local chemical potential probing. (A) Schematic of charge-sensing measurement. Vbias (DC and AC) is applied to the detector layer to control the tunneling while Vm and Vg are applied to the sample layer and the bottom gate to control the layer filling factor. (B) Zero bias conductance phase diagram showing the compressible and incompressible regimes. STM tip height is set at I = 2.2 nA, Vbias = 0.4 V before turning off the feedback and measure the zero bias conductance with Vac = 2 mV. (C) Density-tuned bias spectroscopy of the −1 LL. The Landau level traces the chemical potential of the sample layer. (D) Illustration of the charge sensing process. (E) Chemical potential extracted from (C) showing symmetry-breaking gaps and fractional quantum Hall (FQH) gaps. (F) High-resolution measurements of the chemical potential of partially filled zLL and 1 LL show many FQH states.*  Another imaging technique with a double-layer structure and scanning probe was reported ( 44 ), but since the probe doesn’t directly measure potential, detailed comparisons are omitted.†  The 40 mV shift in the double charge neutrality point (charge neutral point of both layers), indicated by a dot in  Fig. 1B  , arises from the built-in electric field due to the device’s asym-metry, where the top layer is single-side encapsulated by hBN.Downloaded from https://www.pnas.org by Kenji Watanabe on February 18, 2025 from IP address 144.213.253.16.http://www.pnas.org/lookup/doi/10.1073/pnas.2424781122#supplementary-materialshttp://www.pnas.org/lookup/doi/10.1073/pnas.2424781122#supplementary-materialshttp://www.pnas.org/lookup/doi/10.1073/pnas.2424781122#supplementary-materialshttp://www.pnas.org/lookup/doi/10.1073/pnas.2424781122#supplementary-materialshttp://www.pnas.org/lookup/doi/10.1073/pnas.2424781122#supplementary-materialshttp://www.pnas.org/lookup/doi/10.1073/pnas.2424781122#supplementary-materialsPNAS  2025  Vol. 122  No. 8 e2424781122� https://doi.org/10.1073/pnas.2424781122 3 of 6layer, increasing the sample layer’s filling factor (step 3) and modifying its internal chemical potential  �m    (step 4). Meanwhile, the electrochemical potential difference between the detector and the sample layer is held constant by  Vm    (step 5). Since the detector is in an incompressible state, no charge transfer will occur between the sample and the detector. Consequently, the internal chemical potential of the detector layer will be pinned to that of the sample layer (step 6), tracing the internal chemical potential of the sample layer as we change the sample layer’s filling factor. Since all the charge transfer occurs between the back gate and the sample layer, the change in filling factor will be proportional to  ΔVg   . The resulting chemical potential  �m  of the sample MLG is presented in  Fig. 1E   as a function of filling factor. The com-pressibility is negative within each LL due to electron–electron interactions ( 17 ,  20 ,  21 ). We have also observed charge gaps associated with the symmetry-breaking quantum Hall ferro-magnetism (QHFM) in the zLL and the FQH states (at � = −13, −23, −25, −35  ) within each isospin flavor of the zLL (see  Fig. 1F   for a closer look at FQH in both  N = 0  and  N = 1  LL). The QHFM and FQH gaps, measured by the jumps in the chemical potential, are comparable with previous measurements using capacitance techniques ( 15 ,  22 ) and STM ( 18 ).  Charge Sensing and Spatially Resolved Response to Single Impurities. Now, we turn our attention to the proximity of a local impurity. The electrostatic charging model suggests that local changes in the LL energy Δ�d (r) can be viewed as a probe of local changes in electrostatic potential ΔΦd (r) (SI Appendix, section 7), Δ�d (r) = − ΔΦd (r),where Δ�d (r) = �d (r) − �0d , and �0d is the chemical potential far away from the impurity 1. Physically, ΔΦd (r) directly characterizes the screening property of the sample MLG. It has two contributions, one from the impurity itself Φimpd(r) and one from the sample MLG’s response Φmd(r) , only the latter of which contains information about the sample. We show a two-dimensional map of ΔΦd (r) at � = −12 in the Inset of Fig. 2A (Materials and Methods). The positive charge impurities show up as dark disks in ΔΦd (r) , lowering the electrostatic potential for electrons.‡ A closer look at an individual charge impurity ( Fig. 2A  , impu-rity A) reveals oscillations of the underscreening (dark) and over-screening (bright) regions. The oscillatory charge response in a partially filled LL can have multiple origins, including charge order instabilities like melted Wigner crystals ( 8 ), magnetoroton excita-tions ( 23 ), or Friedel oscillations of the composite Fermi liquid ( 24 ,  25 ), depending on the filling factor. At  � = −12  , we show the Fourier transform of  ΔΦd (r)  in  Fig. 2F  , which reveals a peak at wave vector close to  2∕ �B  . The Inset  of  Fig. 2C   shows a scaling collapse of  ΔΦd (r)  at various magnetic field by a scaling function F (x) = ΔΦd(x �B)∕Ec  , where the magnetic length  �B  is the only length scale and the Coulomb energy  Ec =e2� �B  is the only energy scale. Our finding is consistent with the composite fermion liquid [1]-100 -50 0 50 100X (nm)-100-50050100Y (nm)0.5-0.8Impurity Aν = -1/2, 6T100 200 3000100200300ABCDFG0 20 40 60 80 100 120r (nm)-1-0.500.5Real space0 50 100r (nm)-15-10-50Gateimpurity+2DEGDetectorTotalImpurity-2DEGE-40 -20 0 20 40r (nm)-30-20-100energy (meV)30dI/dV(nS)intensityExperiment Simulation0 1 2 3 4-20-15-10-505 Momentum space-10 100 4 8 12-.08-.040c3T4T5T6T1 2 3 4 5-2.5-2-1.5-1-0.50dimp = 4.5 nmdimp = 0 nmRPA 1.4K0K0 50 100r (nm)024610-400.20.4charge density (e/nm2 )Trapped charge = -0.97eFig. 2.   Spatial resolved potential response to a charge impurity at � = − 1∕2 . (A) Potential near a positive charge impurity taken at � = − 1∕2 under 6 T. Inset: a wide field of view potential including multiple charge impurities taken with the same filling and magnetic field. (B) Left: Degeneracy-lifted state of zLL taken near the impurity when the sample and detector are at � = + 2 . The zero energy is shifted to the energy of the Landau level at a far distance. Right: Simulation using first-order perturbation theory with zLL wavefunction. The spectrum is broadened by 1.5 meV for clarity. (C) Azimuthal average of (A). The error bar represents the SEM. Inset: Azimuthally averaged potential taken at different magnetic fields. The data are scaled with Ec and �B . (D) Potential profile of the impurity, 2DEG, and total. The screening potential from 2DEG is inverted for clarity. Inset: Illustration showing that the potential on the detector is the sum of impurity potential and screening potential. (E) Calculated charge density of the 2DEG screening the impurity potential. (F) Fourier transform of the potential. The result is azimuthally averaged from the 2D Fourier transform of (A). Inset: 2D Fourier transform of (A). (G) The composite fermion RPA calculation of the total potential at � = − 1∕2 for strong and weak potential. The dashed line and solid line represent the zero temperature and finite temperature (1.4 K) calculations.‡  We find the density of such impurities in our sample to be around 3 × 109  cm−2 , which is comparable to that of the double-encapsulated devices.Downloaded from https://www.pnas.org by Kenji Watanabe on February 18, 2025 from IP address 144.213.253.16.http://www.pnas.org/lookup/doi/10.1073/pnas.2424781122#supplementary-materials4 of 6   https://doi.org/10.1073/pnas.2424781122� pnas.org(CFL) theory ( 24 ,  26         – 31 ) of the half-filled LL, which predicts a peak at  2kF    in the charge response function, with  kF = 1∕ �B    being the composite fermion Fermi wavevector assuming full spin and valley polarization. We further compare our experimental results to the random phase approximation (RPA) calculation based on CFL the-ory ( Fig. 2G  ; see details in SI Appendix, sections 14 and 15 ) and exact diagonalization (ED) calculations (SI Appendix, section 13 ). The experimental results agree qualitatively with the RPA calculation, both showing a peak at wavevector slightly smaller than  2∕ �B    due to finite temperature effect. The ED calculation shows that the oscillatory behavior of the charge response survives even with a very strong impurity like impurity A. To extract the actual charge response of the sample MLG, we need to isolate the electrostatic potential  Φmd(r)  from the screening charges in the sample MLG. To do so, we first tune the sample MLG to the incompressible  � = − 2  IQH as well so that it cannot have any response  Φmd(r) = 0 . §   Then, we can measure  Φimpd(r)  from  ΔΦd (r)  directly. In this case, the spectroscopy at the impurity ( Fig. 2 B   Left ) shows a set of discretized energy levels correspond-ing to different angular momentum states in the zLL of the detec-tor. We can then estimate the depth of the impurity by matching these energy levels to numerical simulation. Assuming the impu-rity has unit charge +e, we find that the impurity is directly above the sample layer, probably introduced during the stacking process ( Fig. 2 B   Right , and see SI Appendix, section 11  for details). We can then isolate the potential  Φmd(r)    arising from the sample MLG’s response at arbitrary filling by subtracting  Φimpd(r)    from ΔΦd (r)    (yellow line in  Fig. 2 B  and D  ). It becomes clear that the first peak in  ΔΦd (r)    closest to impurity is associated with an overscreening effect as the screening charge is forced to spread out over  �B    and  therefore  −Φmd(r)    exceeds  Φimpd(r)    at scale  �B    ( Fig. 2C  ). Using the electrostatic Green’s function between sample and detec-tor layers, we can further convert  Φmd(r)    into the charge distribu-tion in the sample layer as shown in  Fig. 2E  . The screening charge density sums up to  −0.97 ± 0.2e    (see SI Appendix, section 12  for details) and decays from the center of the impurity with oscilla-tions over 60 nm. From this analysis, we estimate that our tech-nique has a sensitivity of 0.3 meV to changes in the potential landscape (at 1.4 K) with a spatial resolution of about one  �B    (SI Appendix, section 8 ).  Filling Dependence and Impurity Dependence. The screening property of the sample MLG exhibits a rich dependence on the filling factor and the impurity strength. Fig. 3A shows the result for the impurity A across the range � ∈ (−1, 0) as a concrete example. The total charge density is locally bounded by 1∕2� �2B in a single LL, which constrains the mobility of the screening charge density at higher filling. Consequently, the central underscreening region increases monotonically in diameter with the filling factor. Notably, the oscillation is much weaker at rational fillings � = −23, −13 , as expected the incompressible nature of FQH states. The azimutal anisotropy at these fillings might be related to an anisotropic arrangement of fractionalized quasi-particles at the impurity. The amplitude of the oscillation is relative weak close to � = −12 , yet survives farthest away from the impurity and shows more oscillation periods. We therefore interpret the oscillations in the charge response close to � = −12 as a universal feature of underlying electronic system. The behaviors of the oscillation period fall into two distinct cat-egories depending on the strength of the impurity. For a strong impurity, e.g., the impurity A and B (shown in  Figs. 3B   and  4A   respectively), the oscillation period becomes shorter with increasing ν=-5/6 ν=-19/24 ν=-3/4 ν=-17/24 ν=-2/3 ν=-3/5ν=-7/12 ν=-13/24 ν=-1/2 ν=-11/24 ν=-5/12ν=-2/5 ν=-1/3 ν=-7/24 ν=-1/4 ν=-5/24 ν=-1/6-0.50.4(meV)-0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2ν050100r (nm)-0.80.4(meV)ABFig. 3.   Potential response to a charge impurity at various filling factors. (A) Potential taken at different filling factors. Each image is 252 × 252 nm with 100 × 100 pixels. Overscreening regions appear as bright color rings. The first row is � = − 5∕6 ∼ − 3∕5 . The second row is near half filling � = − 7∕12 ∼ − 5∕12 . The third row is � = − 2∕5 ∼ − 1∕6 . (B) Azimuthal average of potential as a function of filling factor. The dispersion of the potential shows the change of ring size and periodicity, which evolves with respect to �  . The separation of the bright stripe, representing the periodicity of the oscillation ring, decreases before � = − 2∕3 and increase after � = − 3∕5.§  More precisely, the sample MLG will exhibit only a dielectric response with 0 total charge, which we implicitly account for as a renormalization of  Φimpd(r)   .Downloaded from https://www.pnas.org by Kenji Watanabe on February 18, 2025 from IP address 144.213.253.16.http://www.pnas.org/lookup/doi/10.1073/pnas.2424781122#supplementary-materialshttp://www.pnas.org/lookup/doi/10.1073/pnas.2424781122#supplementary-materialshttp://www.pnas.org/lookup/doi/10.1073/pnas.2424781122#supplementary-materialshttp://www.pnas.org/lookup/doi/10.1073/pnas.2424781122#supplementary-materialshttp://www.pnas.org/lookup/doi/10.1073/pnas.2424781122#supplementary-materialsPNAS  2025  Vol. 122  No. 8 e2424781122� https://doi.org/10.1073/pnas.2424781122 5 of 6filling factor for  𝜈 < −23    while longer for  𝜈 > −23 , which is clearly not particle–hole symmetric about  � = −12 . In contrast, a weak impurity (C), which is 4.5 nm below the sample, exhibits a more symmetric behavior about  � = −12    ( Fig. 4B  ). Fourier transform of the data for  − 0.6 < 𝜈 < − 0.4    also shows distinctive behaviors: the strong impurity B features a peak at a wave vector which decreases with filling ( Fig. 4C  ), while the weak impurity C features a dip at almost constant wave vector  2∕ �B    ( Fig. 4D  ). These obser-vations are consistent with the ED calculation shown in  Fig. 4 E  and F  . Particle–hole symmetry of the predicted CFL at � = −12    is an important question both experimentally and theoretically  ( 32     – 35 ). Our results suggest that the particle–hole symmetry, unless explicitly broken by a strong impurity, is preserved in the zLL of MLG close to  � = −12 . Nevertheless, inferring particle–hole sym-metry breaking from  2kF     scattering should be with caution since the residual magnetic field composite fermions experience away from  � = −12    can smear out the  2kF    peak in the charge response ( 26 ).        In conclusion, we have developed a technique to detect the collective density response of quantum hall liquid that achieves high spatial and energy resolution simultaneously. Our technique can also be applied beyond quantum Hall effect in graphene and even be adapted to zero magnetic field if the sensor is replaced with an isolated flat band material, such as twisted-bilayer graphene ( 36 ,  37 ). This opens up a way to probe more exotic two-dimensional materials, including detecting fractional excita-tion in fractional Chern insulator ( 11 ,  38 ,  39 ), Hall crystals in pentalayer graphene ( 40 ,  41 ) and so on. After submission, we have become aware of a potential imaging study using atomic SET ( 42 ).   Materials and MethodsSample Preparation. The sample in this work was fabricated using a mechan-ical dry transfer technique. Our pickup stamp is made of polyvinyl alcohol (PVA)-coated transparent tape. We use the stamp to pick up the layer, begin-ning from the top layer (detector). After picking up all the flakes, we align and press the stack on PVA against Au/Ti patterned Si/SiO2 chips. After heating up to 110 °C, we are able to detach the PVA from the tape, leaving the sample covered underneath. Afterward, we wash away the PVA to expose the sample with water, acetone, IPA, and N-methyl-2-pyrrolidone (NMP). The sample is then loaded into the STM’s UHV chamber and baked at 350 °C overnight before being loaded into the microscope of our STM. The optical image of our device is in SI Appendix.STM Measurements. In this study, we use a home-built UHV STM with a max-imum 6 T out-of-plane magnetic field operating at T = 1.4 K to perform multi-ple types of measurements, including point spectroscopy measurement, grid measurement (2D potential mapping), line cut measurement, and penetration capacitance measurement. The tip is made of tungsten wire and prepared on a Cu(111) surface as described in our previous work (16). When performing the point spectroscopy measurement (Fig. 1C), we set our lock-in frequency to  4 kHz, using 1.5 mV oscillation, and the setpoint is 1 nA at Bias = +0.5 V. When performing the grid measurement (Figs. 2A and 3A) and line cut (Figs. 2B and 4), we set the lock-in frequency to 2,022 Hz and read the dI/dV and d2I/dV2 using two synchronized lock-ins, with the setpoint being 1 to 1.5 nA at Bias = +0.5 V. Details on using d2I/dV2 to perform 2D potential mapping are in SI Appendix, section 7.  When doing the penetration capacitance measurement, we use 27.186 kHz  and 2 mV oscillation for interlayer compressibility and 18.648 kHz and 4 mV oscillation for total compressibility. The results of the penetration capacitance measurements are in SI Appendix, section 4.Data, Materials, and Software Availability. All study data are available at the Harvard Dataverse (43). Detailed descriptions can be found in the article and/or SI Appendix.ACKNOWLEDGMENTS. This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences (Grant DE-FG02-07ER46419), Gordon and Betty Moore Foundation as part of the Emergent Phenomena in Quantum Systems initiative (Grants GBMF9469), NSF, Materials Research Science and Engineering Centers programs through Princeton Center for Complex Materials (Grants DMR-2011750 and NSF DMR-2312311), Army Research Office Multidisciplinary University Research Initiative 134396-5117989 (AROW911NF2120147), Office of Naval Research (Grants N00014-21-1-2592 and N00014-24-1-2471), and NSF, Office of Multidisciplinary Activities (OMA-2326767). T.W. and M.Z. are supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division under Contract No. DE-AC02-05-CH11231 (Theory of Materials program KC2301). T.W. is also supported by the Heising-Simons Foundation, the Simons Foundation, and NSF AB D FC EFig. 4.   Potential response to different depths of charge impurities. (A) Total potential near impurity B (2 nm below the detector, or 0.5 nm above the sample). The response shows similarity to impurity A. (B) Total potential near impurity C (7 nm below the detector, or 4.5 nm below the sample). The impurity potential is shallower since the impurity is further from the sample layer. This causes the response to be weaker compared to impurities A and B. (C) The Fourier transform of the total potential in (A). The response is asymmetric about � = − 1∕2 . (D) The Fourier transform of the total potential in (B). The response is symmetric across � = − 1∕2 . (E) ED calculation of Φmd(k) responding to a strong potential. The peak response disperses asymmetrically about � = − 1∕2 similar to (C). (F) ED calculation of Φmd(k) responding to a weak impurity exhibits a more symmetric response about � = − 1∕2 , consistent with the experimental observation in (D). The apparent “noise” is an artifact coming from the interplay of the Jain sequence and the discrete sequence of � sampled by finite-site numerics ( N� = 30 ) at T = 0.Downloaded from https://www.pnas.org by Kenji Watanabe on February 18, 2025 from IP address 144.213.253.16.http://www.pnas.org/lookup/doi/10.1073/pnas.2424781122#supplementary-materialshttp://www.pnas.org/lookup/doi/10.1073/pnas.2424781122#supplementary-materialshttp://www.pnas.org/lookup/doi/10.1073/pnas.2424781122#supplementary-materialshttp://www.pnas.org/lookup/doi/10.1073/pnas.2424781122#supplementary-materials6 of 6   https://doi.org/10.1073/pnas.2424781122� pnas.orggrant No. PHY-2309135 to the Kavli Institute for Theoretical Physics (KITP). R.F. is supported by the Gordon and Betty Moore Foundation (Grant GBMF8688). 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Harvard Dataverse. https://doi.org/10.7910/DVN/LDYCEM. Deposited 30 January 2025.44.  M. Pelliccione et al., Local imaging of high mobility two-dimensional electron systems with virtual scanning tunneling microscopy. Appl. Phys. Lett. 105, 181603 (2014).Downloaded from https://www.pnas.org by Kenji Watanabe on February 18, 2025 from IP address 144.213.253.16.https://doi.org/10.48550/arXiv.2311.05568https://doi.org/10.48550/arXiv.2410.22277https://doi.org/10.7910/DVN/LDYCEM High spatial resolution charge sensing of quantum Hall states Significance Anchor 4 Device Structure and Control Parameters. Results Chemical Potential Sensing. Charge Sensing and Spatially Resolved Response to Single Impurities. Filling Dependence and Impurity Dependence. Materials and Methods Sample Preparation. STM Measurements. Data, Materials, and Software Availability ACKNOWLEDGMENTS Supporting Information Anchor 22