1 Introduction

Sensing and imaging electric signals such as electric field, electric current, and electric potential from a microscopic view is crucial in condensed matter physics and material science. Recently, materials with exotic electric phenomena such as 2D materials [1], multiferroic materials [2], and flexoelectric materials [3] have raised popular attention because they are ideal platforms for both researching fundamental physics and developing new electronic devices. However, the electric signals targeted to the novel phenomena are always subtle and elusive, and there potentially exist complex interactions among multiple degrees of freedom including orbital, charge, spin, and phonon [4]. Therefore, to understand and reveal the correlation between the macroscopic performance and the microscopic physical properties of such materials, a sensitive technology capable of detection in a non-invasive and quantitative way at nanoscales should be desirable.

One of the feasible strategies to address this problem is using scanning probe microscopy (SPM) [5], where a sharp tip can be used to detect both the nanostructure and properties of samples. Owing to the highly localized tip-sample interactions such as tunneling current [6] used in scanning tunneling microscope (STM) and Pauli repulsion [7] used in atomic force microscope (AFM), SPM is able to obtain the surface topography with atomic resolution. Besides, a large family of variations of SPM are developed to capture mechanical, electric, magnetic, optical, and thermal properties at the nanoscale [5]. To image electric quantities such as local charge distribution, electric polarization, and local potential at nanoscales, various advanced SPM technologies such as scanning capacitance microscope (SCM) [8], piezoresponse force microscope (PFM) [9, 10], and Kelvin probe force microscope (KPFM) [11] have been developed. However, the detection scheme of the mentioned SPM techniques lies either on classical devices with finite size or interactions with perturbative probes, leading to limited sensitivity, unavoidable tip disturbances, or nonquantitative interpretations of data during applications [5]. This will be challenging for conventional SPM techniques to visualize the subtle signals from complex glassy systems [12] and hidden quantum phases [13] or fail to disentangle the interactions among intertwined orders in strongly correlated systems such as high-temperature superconductors [14, 15].

Recently, quantum sensing has been proven as a powerful tool for detecting weak signals [16, 17]. Based on the definite quantum energy level with a well-defined Hamiltonian and harnessing the quantum coherence, quantum sensing can realize quantitative measurements of signals with a sensitivity that surpasses the classical methods [17, 18]. Furthermore, because of the ultra-high sensitivity, the quantum sensor can work in a non-perturbative regime, which helps to preserve the intrinsic properties of the targeted elements. Quantum sensors such as superconducting quantum interference devices (SQUID) [19], color centers [16, 20], and adsorbed magnetic atoms [21] have already been demonstrated to image magnetic fields with spatial resolution from sub-micrometers down to atomic scales, and with a sensitivity up to single electrons spins. Therefore, it’s promising that the integration of atomic/nanometer-sized quantum sensors that are sensitive to weak electric signals inside a high-performance SPM system will provide a promising platform for both visualizing the current and charge distribution in nanoelectronics and revealing the mechanism of the complicated quantum orders in novel quantum materials with high spatial resolution.

Since quantum sensing has already been well-reviewed by many excellent works [17, 18], in this perspective, we will focus on the new capability of atomic/nanometer-sized quantum sensors in electric sensing and imaging at nanoscales. First, we briefly introduce the working principle of quantum sensors. Then we review the recent efforts through quantum sensors-based SPM for studying the charge/polarization distribution and electron transport of materials at nanoscales, where a discussion of their privileges over conventional SPM techniques is also included. This part is structured according to the physical parameters such as electric field, electric current, and electric potential. Finally, we propose some promising methods for the future developments and optimizations of quantum sensors in nanoscale electric sensing and imaging.

2 Working principle of quantum sensing

Quantum sensing harnesses the fundamental properties and phenomena of quantum mechanics to precisely measure the physical quantities. Historical examples of quantum sensors include magnetometers such as SQUID [22, 23], atomic vapors [24], and atomic clocks [25]. In recent years, quantum sensing has rapidly evolved within a broad field of quantum science and quantum technology. More candidates such as spin qubits [26–28], cold atomic clouds [29, 30], quantum dot (QD) [31, 32], single electron transistors (SET) [33], and molecules [34, 35] have also been introduced to this field as quantum sensors. In this section, we aim to provide a concise introduction to the working principles of quantum sensing.

Generally, quantum sensing works under three fundamental approaches [17]: (I) Utilizing a quantum object to measure a physical quantity. (II) Employing quantum coherence to measure a physical quantity. (III) Leveraging quantum entanglement to enhance the measurement sensitivity or precision beyond classical limits. Meanwhile, an ideal quantum sensor should obey the DiVincenzo criterion [36] that is, they should have discrete energy levels that can be initialized, coherently manipulated, interacted with physical quantities, and read out (Fig. 1(a)). However, it’s worth noting that the quantum sensor needn’t always involve a two-level system (TLS). And in some cases, coherent modulation is not required, instead, a measurement of relaxation rates [37, 38] or charging events [39] is applied. Since the quantum states and coherence of quantum sensors are very vulnerable to external disturbances from the environment, it can be taken as an important advantage over classical devices in case of sensing tiny interactions from the interested elements, provided that such signals can be selectively extracted from the broadband background noises and the quantum states can be efficiently read out either through electrical [31, 32] or optical [40, 41] methods. In this perspective, we pay particular attention to the atomic or nanometer-sized quantum sensors with ultrahigh sensitivity that can be feasibly integrated into a high-resolution microscope such as SPM.

Figure 1
figure 1

Principle of quantum sensing. (a) Quantum sensing experiments typically involve three main processes: initialization (light yellow box), interaction with the signal (pink box), and readout (blue box). First, the quantum sensor is initialized into a known basis state, e.g., \(|0\rangle \). Then, the energy levels of the TLS experience small shifts caused by the interaction with the external field. Finally, the resulting quantum state is read out through either electrical or optical methods. The green box shows three kinds of quantum sensors of atomic/nanometer size, including color centers, adsorbed molecules or atoms, and QD such as shallow dopants, from left to right. To improve the sensitivity, the quantum sensor can also be coherently controlled (gray box) with advanced sequences such as dynamical decoupling. (b) The lattice structure of a single NV center is highlighted by the region in shaded blue, which consists of a substitutional nitrogen atom and its adjacent carbon vacancy. (c) The energy level structure of the NV center. The triplet states (\(S = 1\)) 3E and 3A2 are excited and ground states, while 1A1 and 1E are singlet states (\(S = 0\)). A zero-field splitting of 2.87 GHz separates the \(m_{s} = 0\) and \(m_{s} = \pm 1\) in the triplet ground states. The green and red lines indicate the optical transition and radiative decay between excited and ground states through spin-conserving processes. The gray dashed lines describe the non-fluorescent processes through ISC. The blue circle marks the spin transition between \(m_{s} = \pm 1\) and \(m_{s} = 0\) with microwave. The magnetic field will lead to Zeeman splitting of \(m_{s} = \pm 1\), which can be measured using ODMR

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Here, we cite one kind of solid-state qubit such as nitrogen-vacancy center (NV) in diamond as an example to explain the experimentations of quantum sensing. NV center is an atomic-sized spin defect consisting of a substitutional nitrogen impurity and its adjacent carbon vacancy (Fig. 1(b)). Two unpaired electrons trapped on NV form a spin triplet (\(S = 1\)). The energy level structure is shown in Fig. 1(c). The triplet states 3E and 3A2 are excited and ground states, while 1A1 and 1E are singlet states (\(S = 0\)). A zero-field splitting of 2.87 GHz induced by the spin-spin interaction separates the \(m_{s} = 0\) and \(m_{s} = \pm 1\) in the triplet ground states. The green and red lines in Fig. 1(c) indicate the optical transition and radiative decay between excited and ground states through spin-conserving processes. However, because of the inter-system crossing (ISC) (dashed lines in Fig. 1(c)), the \(m_{s} = \pm 1\) change into the long-lived singlet states after optical excitations, and finally relax into \(m_{s} = 0\) in a nonradiative way. This will lead to a spin-dependent fluorescence for NV with a contrast of ∼30% between \(m_{s} = \pm 1\) to \(m_{s} = 0\) and provide a method to initialize the quantum states of NV under optical excitations. When sweeping the frequency of the microwave and simultaneously monitoring the photon counts of NV, the changed fluorescence indicates the electron spin resonance of NV and this approach is called optical detectable magnetic resonance (ODMR) [42].

Quantum sensors like single NV centers have found a wide range of applications including magnetometers [20], electrometers [43], pressure sensors [44], and thermometers [45, 46], due to their well-defined spin levels and definite response to external signals through Zeeman splitting, Stark effect, and temperature-induced energy shifts. Fortunately, combined with a long coherence time of up to milliseconds [47] and advanced coherent modulation techniques [16, 48, 49], NV has demonstrated the detection of nanometer-sized volumes of nuclear spin [50–52] and electron spins [53, 54] positioned on the diamond surface with the sensitivity far beyond the conventional instruments for electron paramagnetic resonance and nuclear magnetic resonance measurements. Besides, enhanced sensitivity beating the standard quantum limit can also be realized through the entanglement of NVs [55] and their neighborhood solid-state qubits [56]. Most importantly, because of the atomic size of a single NV center, the integration of such point defect-like qubits into the SPM system is also feasible, making it suitable for nanoscale sensing and imaging for weak magnetic and electric signals.

3 Nanoscale electric field sensing and imaging

The precise imaging of electric fields is important for elucidating the local distribution of electric dipoles and the accumulation of charges at both surfaces and interface, which is important to gain insights for nanostructures and ferroelectric or multiferroic materials. Quantitative electric field sensing requires a sensor with a definite quantity of electric dipole and polarizability factor. To this end, promising candidates include QD [57] and NV center [43]. Due to its atomic size and controllable placement in proximity to the substrate surface, NV is more promising to realize nanoscale electric field imaging than QD. In this section, we will focus on the progress in electric field sensing and imaging using single NV centers.

Different from the linear response of the external magnetic field, the coupling of the electric field with the unpaired electron spins of NV depends on the secondary Stark effect (Fig. 2(a)) [58], which imposes an inherent technical challenge using NV for electrometry [43, 59]. In practice, NV detects the component of the electric field that is perpendicular (ie, \(E_{\bot} \)) to NV’s anisotropic axis connecting the nitrogen and vacancy sites (Fig. 2(b)) [58], because the axial component of NV’s electric dipole moment is very small [43]. The spin-level shifts induced by the electric field can be measured through the ODMR spectrum (Fig. 2(c)), and the magnitude of such shifts is determined by both the direction (Fig. 2(d)) and strength (Fig. 2(e)) of the applied bias magnetic field. Thus, to optimize the sensitivity of NV to \(E_{\bot} \) while concurrently suppressing its response to the longitudinal component of the external magnetic and intrinsic strain fields, a bias field \(B_{\bot} \) of several tens of Gauss should be carefully aligned perpendicular to the NV’s anisotropic axis. Recent experimental work has demonstrated that the electric field sensitivity of NV centers can easily reach the level of mV/nm [60], a notable improvement over the sensitivity of ∼V/nm achieved by PFM [61] and KPFM [62]. Consequently, NV centers-based scanning electrometry holds immense promise for achieving both high spatial resolution and remarkable sensitivity in investigating materials [63] and nanoelectronics [64].

Figure 2
figure 2

Principle of NV-based electrometry. (a) Left: Spin level diagram of NV− changes with different transverse magnetic field \(B_{\bot}\). The hyperfine interaction with the associated 15N isotope is represented as \(A_{\parallel} \). The states \(|{-}1{\downarrow} \rangle \) and \(|{-}1{\uparrow} \rangle \) are equally mixed into the \(|{+}\rangle \) and \(|{-}\rangle \) states. Right: Energy level diagram of the \(|{\pm} \rangle \) states under varying electric field strength. Arrows indicate spin-resonant transitions induced by microwaves (\(f_{+}\) and \(f_{-}\)) [60]. (b) Schematic illustration of the NV center, including coordinate axes, and the magnetic and electric fields defined in the text. Nitrogen atom and vacancy are represented by the red and transparent balls respectively, while the violet balls represent three nearest carbon atoms. The anisotropic axis of NV is marked by the z-axis [60]. (c) The observed shift in the ODMR resonance lines under different electric fields demonstrates the Stark effect [43]. (d) Polar plot illustrating the measured frequency shift of NV’s spin levels Δω under the same electric field strength as a function of the magnetic field orientation \(\varphi _{B}\) in the non-axial plane [43]. (e) The measured frequency shift of NV’s spin levels Δω under the same electric field strength as a function of magnetic field strength along the anisotropic axis [43]

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Typically, for scanning electrometry based on the dynamical AFM, the oscillation of cantilevers at several tens of nanometers will introduce unwanted electric noise, especially under small tip-surface distance, leading to reduced coherence time and sensitivity of NV centers [65]. This challenge can be partially overcome through the implementation of a so-called qPlus-based atomic force sensor (Fig. 3(a)) [66]. Benefiting from the asymmetric tuning-fork design, this technique enables the precise positioning of an AFM tip to the surface within 1 nm while maintaining extremely small oscillation amplitudes down to 100 pm. Through a pulsed-ODMR scheme [67], Bian et al. achieved the quantitative imaging of the electric field gradients generated by a sharp conductive tip with a resolution of ∼17 nm (Fig. 3(b)) [60].

Figure 3
figure 3

Nanoscale imaging of electric field by single NV centers. (a) Schematic of the AFM system integrated with NV technology. A metal tip is precisely positioned near a specific NV center using a qPlus sensor. A piezo actuator (gray arrow) is employed to drive the oscillations of the qPlus sensor. MW represents the microwave pulses fed through the on-chip coplanar waveguide. The laser is used to initialize and read out the spin state of a single NV center, while the microwave is used for driving the spin transition of NV during measurements. The transverse magnetic field \(B_{\bot} \) is marked by a white arrow, with a magnitude of 9.6 Gauss [60]. (b) Left: Scanning field-gradient image at a bias voltage of −16 V with a fixed microwave frequency of 700 kHz as the ODMR setpoint, showing field gradient directions (dashed arrows). Right: Simulated scanning-field gradient image at −10 V bias with a 700 kHz ODMR setpoint. The non-spherical feature is due to the triangular shape of the tip. Scale bars in (b) are 100 nm [60]. (c) ODMR spectra exhibit no observable frequency shift below the applied DC electric field threshold because of the screen effect of the charges near the diamond surface. The double peaks are caused by the hyperfine interaction of the 15N nuclear spin [72]. (d) Frequency-dependent signal attenuation (top) and phase delay (bottom), were obtained by a lock-in. Signals are normalized relative to the expected phase without screening. The red traces indicate a cut-off of 35.4 kHz. PDD represents the periodic dynamical decoupling sequence with multiple π pulses for sensing signals at high frequencies [64]. (e) Scanning a single NV inside a diamond tip, which oscillates horizontally at a frequency of 32 kHz through a shear-mode tuning fork. The blue and red balls represent the near-surface charges both on the diamond tip and sample surface [63]. (f) Ferroelectric domain pattern in hexagonal YMnO3 observed through NV-based scanning electrometry, where the domain boundaries can be resolved by the scheme of scanning gradiometry. Bright features correspond to patched charges at topographic defects [63]

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Another notable challenge arises from the unavoidable electrical shielding effect caused by the residual charges near the diamond surface [68–71], where the DC electric field below the threshold value fails to produce observable Stark shifts in NV’s spin level (Fig. 3(c)) [72]. This is also confirmed in Fig. 3(d). Using a lock-in scheme, the measured signal is attenuated and the demodulated phase is retarded at low frequency, indicating that surface charges are reorganized and screen the electric field. In contrast, NV can readily detect Stark shifts induced by AC with high frequency because of the limited mobility of the surface charges which cannot instantaneously respond to the fast changes of the electric field [64]. Therefore, to surmount the challenge of screening effect and improve the performance of NV for imaging DC electric fields, Huxter et al. developed the technique of scanning gradiometry (Fig. 3(e)) [73]. In this case, an AFM cantilever with a diamond probe oscillates at 32 kHz, where the sequences for quantum sensing such as Hahn echo or dynamical decoupling are synchronized with the harmonic motions of the diamond tip. As a result, the static local electric field gradient is effectively transformed into an AC signal. In this way the sensitivity is limited by \(T_{2}\) of NV instead of the free induction decay \(T_{2}^{*}\), which yields a remarkable sensitivity improvement for detecting DC electric field. As an exemplary demonstration, NV-based scanning electrometry has been employed to visualize the domain patterns in materials like piezoelectric (\(\mathrm{Pb} [ \mathrm{Zr}_{0.2} \mathrm{Ti}_{0.8} ] \mathrm{O}_{3}\)) and ferroelectric (YMnO3) substances with spatial resolution of sub-100 nm (Fig. 3(f)) [63]. Further improvement of spatial resolution can be realized by moving the tip closer to the surface. In this work, Huxter et al. achieved a sensitivity up to \(2.4\times 10^{-2}\text{ mV/nm}\), which is on par with the capabilities demonstrated for AC field detection by bulk NVs [43]. The NV-based scanning electrometry reproduces the major ferroelectric features of YMnO3 imaged by PFM but in a non-perturbative and quantitative manner. It’s also worth noting that, since NV can be sensitive to either electric or magnetic field through a simple re-orientation of the bias magnetic field, it’s suitable to investigate both the ferroelectric and ferromagnetic orders in multiferroic materials in situ without breaking the experimental conditions.

4 Nanoscale electric current sensing and imaging

Traditional electrical measurements follow Ohm’s law, but emerging theories accounting for the electron-electron interactions suggest novel transport phenomena like electron fluid dynamics in quantum materials, which may challenge this convention (Fig. 4(a)) [74, 75]. To investigate such intriguing transport properties, techniques for current measurement with both high sensitivity and high spatial resolution are desired. Nowadays, state-of-the-art micro-fabrications allow the production of miniaturized current sensors such as SQUID [19, 76] and SET [77, 78] with sub-micrometer resolution. However, to realize nanoscale electric current imaging, a nanometer-sized diamond structure housing an NV center is attempted to integrate into the AFM tip [79, 80]. NV-based scanning magnetometry has demonstrated applications in nanoscale magnetic field imaging for a wide range of materials, including multiferroic non-colinear orders [81], intricate skyrmion formations [82], and two-dimensional magnetic materials [83, 84]. As for the current imaging, electron flow produces a quasi-static magnetic stray field aligned with their transport direction, governed by Ampere’s law. The strength and orientation of this stray field depend both on the magnitude and direction of the current, which can be sensed by NV centers through the Zeeman effects. Since the scheme of quantum sensors allows quantitative measurements, by scanning NV and recording the projection of induced magnetic stray field onto the NV axis (\(B_{\mathrm{NV}}\)), one can quantitatively reconstruct the distribution of the electric currents [85].

Figure 4
figure 4

Nanoscale sensing and imaging of the hydrodynamics of electron fluids by single NV centers. (a) Top: Local current density in a 2D conductor oriented along the x direction, illustrating the conventional ohmic transport (left) and viscous Poiseuille flow (right). Bottom: The corresponding magnetic stray field profiles \(B_{x} ( x )\) (solid line) and \(B_{z} ( x )\) (dashed line) generated by an ohmic current flow (left) and viscous Poiseuille flow (right) in the y direction [86]. (b) The schematic of measuring current flow with a near-surface NV center in a diamond tip above an hBN-encapsulated graphene device (hBN–G–hBN) on a SiO2/doped Si substrate with an NV-graphene distance ≤50 nm [86]. (c) Reconstructed current density perpendicular to the current direction acquired at different devices measured under room temperature (red points: graphene at the CNP, grey points: palladium, orange points: low-mobility graphene). The intensity of current density is normalized by the average charge carrier flux and the spatial coordinate x is normalized by the channel width [86]. (d) Flow profiles at \(T = 298\text{ K}\) (top) and \(T = 128\text{ K}\) (bottom), where red and blue lines represent the experimental and calculated data. At 298 K, the data shows the expected Ohmic current profile at CNP with a double-peak feature caused by momentum-relaxing scattering. At 128 K with higher carrier density than CNP, a single-peak profile appears, indicating the reduced momentum-relaxing scattering which breaks the ohmic transport [89]. (e) Left: Contour plot of \(B_{\text{x}}\) induced by the current flow across the WTe2 device (inset) at different temperatures. The gray lines mark the \(B_{\text{x}}\) profile of the narrowest width between 10 K and 20 K. Right: \(B_{\text{x}}\) profile at 90 K, 20 K, and 5 K (as indicated by arrows), with the purple one at 20 K showing the narrowest width [91]. (f) Calculated contour plot of \(B_{\mathrm{x}}\) through ab initio calculations accounting for the scattering rates of electron-electron and electron-phonon interaction. The narrowest width and maximum intensity of \(B_{\text{x}}\) profile appear near 15 K [91]

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As a groundbreaking demonstration, Ku et al. applied a single NV center to directly image the flow of viscous Dirac fluid in high-mobility graphene at room temperature (Fig. 4(b)) [86]. Compared to the non-interacting particles, hydrodynamic electron flow such as Dirac fluid [87, 88] possesses finite viscosity and exhibits notable phenomena such as vortices [76] near the edge of the device or Poiseuille flow that appears in the vicinity of the charge-neutrality point (CNP) (Fig. 4(c)). Ku et al. observed a parabolic flow profile along the axis perpendicular to the direction of current in graphene, and the derived viscosity and scattering rates are closely conformed to the universal values anticipated at quantum criticality, which further confirms the presence of an almost ideal electron fluid in this device. Subsequently, employing the same technique, Jenkins et al. visualized the local deviations from Ohm’s law within a narrow constriction formed in a high-mobility graphene monolayer [89]. They found a transition in electron transport from diffusive to viscous behavior as the temperature decreased below 200 K (Fig. 4(d)). Furthermore, Fatemi et al. extended the application of NV to semimetals such as tungsten ditelluride [90], where they brought new insights into electron interactions mediated by phonons (Fig. 4(e) and (f)) [91]. It’s worth noting that, besides the quasi-static magnetic stray field induced by the viscous or ballistic electrons in a mesoscopic current flow, NV is also applicable for detecting high-frequency magnetic fluctuations caused by the diffusive electrons, which can be used to image the local electric conductivity of nanodevices [92] and analyze the mean free path of electrons of materials [37] with nanometer resolution.

In addition to the current flux closely related to the transport properties of novel electronic devices, current loops such as vortex states of superconductors have also received much attention in condensed matter physics. Over the years, many approaches such as scanning SQUID [93] and magnetic force microscopy [94] have been employed to image the local current distribution in orbital ferromagnetism [95] and superconductor vortex [96]. These methods offer valuable insights into quantum materials but have technical limitations, including restricted magnetic field ranges, limited spatial resolution, and disturbance to the intrinsic properties of samples. NV stands out as a promising solution to defeat all of the mentioned limitations and facilitate nanoscale investigations at low temperatures. For example, Pelliccione et al. and Thiel et al. applied quantitative nanoscale imaging of vortices structures within the superconductor \(\mathrm{BaFe}_{2} ( \mathrm{As}_{0.7} \mathrm{P}_{0.3} )_{2}\) (Fig. 5(a)) [97] and \(\mathrm{YB} \mathrm{a}_{2} \mathrm{C} \mathrm{u}_{3} \mathrm{O}_{7-\delta} \) (Fig. 5(b)) [98]. In this method, a 3D imaging of the stray field of the vortex is also applicable by varying the distance between the diamond tip and the surface (Fig. 5(c)). At a tip-to-sample distance of approximately 10 nm, the observations revealed remarkable deviations from the prevalent model of monopole for vortex [96], while exhibiting excellent quantitative agreement with Pearl’s analytic model (Fig. 5(d) and (e)) [99]. Notably, these experiments provided non-invasive and unambiguous insights into the local penetration depth [100], and NV can be readily extended to explore superconductivity in the regime of higher temperature and magnetic field.

Figure 5
figure 5

Nanoscale imaging of current distributions of the vortex by single NV centers. (a) Imaging of vortex current distributions in BaFe2(As0.7P0.3)2 at \(T = 6\text{ K}\), obtained by recording the fluorescence of NV with the fixed microwave frequency as the ODMR setpoint. Experimental images (top) and simulations (bottom) show an iso-magnetic stray field of 5.9 Gauss (left) and 8.8 Gauss (right) respectively, with dark rings indicating the resonance of the NV center. Scale bars: 200 nm [97]. (b) Imaging of the vortex current distributions in \(\text{YB} \text{a}_{2} \text{C} \text{u}_{3} \text{O}_{7-\delta} \) at \(T = 4.2\text{ K}\), obtained by recording the full ODMR spectrum at each pixel (120 × 120 pixels here). In this way, the projection of the induced magnetic stray field onto the NV axis (\(B_{\text{NV}}\)) can be derived from the measured resonance in each ODMR. The asymmetric shape in the image is caused by the fact that the NV axis is not perpendicular to the surface of the diamond tip and the superconductor [98]. (c) Top: A series of images of \(B_{\text{NV}}\) acquired at \(z_{\text{tip}}\) of 0, 50, and 100 nm. Bottom: Images of \(B_{\text{NV}}\) in the x-z plane [98]. (d) and (e) Line profile of the vortex stray field along the x (left) and y (right) axes as marked in the insets. Blue and green dashed lines represent the fit with a Pearl vortex model and magnetic monopole, respectively. It’s shown that the bulk London penetration depth can be determined (\(\lambda _{L} = 251 \pm 14\text{ nm}\)) only with the Pearl vortex model [98]

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5 Nanoscale electric potential sensing and imaging

Imaging and measuring electric potential from a microscopic view is very important to investigate the local electrostatic interactions and the related fundamental phenomena such as charge distribution, charge transfer, and charge diffusion in materials and electronic devices. A miniaturized quantum device such as SET has already been successfully integrated on SPM [77, 101, 102] and demonstrated ultrahigh sensitivity up to \(5 \times 10^{-2}\) e [77]. However, the spatial resolution of an SET-based SPM is only several hundred nanometers, limited by the size of the device. STM and AFM are powerful tools to measure the electronic properties even with atomic resolution [5, 103], but fail to directly study the electric potentials. KPFM is widely used to detect the local contact potential difference between the tip and surface, which reflects the local electric potential at the surface [11, 104]. However, due to the long-range electrostatic interactions, the measured signals from KPFM always involve lateral averaging effects and limit the spatial resolution. Functionalized tips with a single CO molecule demonstrated sub-molecular resolution using KPFM [62] but require a short tip-surface distance lower than 1 nm, where the chemical interaction between tip and surface starts to contribute to the signals and impedes the quantitative elucidations of intrinsic electric potentials of the surface.

Recently, a technique called scanning quantum dot microscope (SQDM) was developed for electric potential imaging at the level of sub-nanometer [39, 105, 106]. In this method, a single molecule such as the perylene tetracarboxylic dianhydride (PTCDA) [107] can be picked up by an AFM tip as a point-like quantum dot because of the tiny hybridization between the tip and the molecule. Similar to the working principle of SET [101], by sweeping the bias voltage \(V_{\mathrm{b}}\) between tip and surface, a discharging/charging event with a single electron (\(\Delta q_{\mathrm{QD}}= \pm e\)) happens across the quantum dot at the critical bias voltage \(V_{\mathrm{b}} = V^{\pm}\) (Fig. 6(a)) when the electron can tunnel through the PTCDA. Meanwhile, the discharging/charging of PTCDA is sensitively monitored by the qPlus force sensor [66] as sharp dips in the frequency shift Δf and the electric potential imaging can be further realized just by recording \(V^{\pm}\) during the scanning (Fig. 6(b)). Using PTCDA as a quantum sensor for electric potential brings several advantages compared to the conventional KPFM. First, the measured signals of SQDM originate from the electric potential distribution on the point-like QD instead of the entire tip structure, which allows quantitative analyses of the local electric potential [39]. Second, the QD with the size of a single molecule under the screening effects of tip and surface shows increased spatial resolution than KPFM [39] (Fig. 6(c)). Third, due to the high sensitivity of such quantum sensors, SQDM can image at a relatively large working distance (up to 4 nm) [105], which is suitable for surfaces with large roughness and complex nanostructures.

Figure 6
figure 6

Nanoscale imaging of electric potential by single molecular quantum sensors. (a) Principle of measuring electric potential with a molecular QD. Above a bare surface, the potential at the PTCDA (or QD) is \(\Phi _{\text{s}} = \alpha _{0} V_{\text{b}}\) at the bias of \(V_{\text{b}}\), where \(\alpha _{0}\) is the local gating efficiency (top). PTCDA will experience a discharging/charging with e at the critical bias \(V_{\text{b}} = V_{0}^{\pm} \), which causes sharp dips in the frequency shift of a qPlus force sensor (bottom). The local surface electric potential \(\Phi _{\text{s}}\) of a nanostructure such as single adatoms changes \(\alpha _{0}\) and \(V_{0}^{\pm} \) to α and \(V^{\pm} \), which can be furtherly imaged by monitoring \(V^{\pm}\) during the scanning [39]. (b) Electric potential image by SQDM recorded at a tip height of 2 nm. The modulation on the edge of the island reveals the rows of inequivalent PTCDAs. Image size: \(60 \times 60 \text{ nm} ^{2}\) [39]. (c) Zoomed-in electric potential images of several nanostructures in (b), showing the electric multipole moment of different species. Scale bar:1 nm [39]. (d) Top: Schematic of a quantum sensor such as a single H2 molecule trapped under an STM tip coupled with the THz pump-probe pulses with a delay time of τ. Bottom: Asymmetric double-well potential in the STM junction for the trapped H2, effectively leading to a TLS of such quantum sensor. The \(| a \rangle \) and \(| b \rangle \) are basis quantum states represent two adsorption configurations with an energy difference Δε caused by the local electric potential [35]. (e) Time-elapsed rectification tunneling current with different tip positions (marked in the inset) showing the coherent oscillations of the TLS after prepared H2 into the higher energy state \(| b \rangle \) (top). The different FFT peaks indicate the varied local potentials and the corresponding site-dependent energy splitting of H2’s quantum states (bottom) [35]. (f) Temporal snapshots of the rectification tunneling current at different delay times of the pump-probe show the local chemical potential of the Cu2N lattice with atomic resolutions. Image size: 53.2 Å × 44.9 Å [35]

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Since the spatial resolution is still limited by the nanometer size of PTCDA, to further increase the spatial resolution of electric potential imaging, a molecule of atomic size like H2 trapped inside the tunneling junction of STM [108] is used as a quantum sensor [35, 109]. In this case, two low-lying quantum states of H2 inside the asymmetric double-well induced by the tip and surface effectively build up a TLS [110] (Fig. 6(d)), whose energy splitting is closely related to its immediate electrostatic environment. To measure the coherent oscillations of this TLS, a pump-probe scheme is coupled to the STM tunneling junction, where a THz electromagnetic pump pulse excites this TLS into the higher energy state and another probe pulse detects the population distribution of the TLS after a delay time. Decayed oscillations persisting several tens of picoseconds are recorded and the frequencies of such oscillations are obtained through fast Fourier transform (FFT). In principle, the oscillation frequency is determined by the energy splitting of TLS and changed with the local electric potential at different tip positions (Fig. 6(e)). Using H2 as a quantum sensor, Wang et al. can probe the surface chemical environment of Cu2N at the sub-angstrom scale (Fig. 6(f)), which largely surpasses the limitation of KPFM. It’s worth noting that applying ultrashort THz pulses instead of microwave sequences widely used in solid-state qubits [111, 112], Wang et al. opens up the door for quantum sensing in the regime of ultrafast physics, which is inaccessible for NV-based measurements since the temporal resolution is limited by the pulse width and bandwidth of electronics [113].

6 Challenges and new possibilities in quantum sensor

Combining quantum sensing with SPM offers nanoscale electric sensing with high sensitivity, but it still comes with challenges. First, the quantum sensors are very sensitive to the unwanted and random changes of the immediate electromagnetic environments, which might overwhelm the interested signals. This is especially more detrimental for quantum sensors in extreme proximity to surfaces [114, 115] and conductive tips [116], which might suffer from broadband magnetic/charge noise that cannot be efficiently suppressed using Hahn echo and dynamical decoupling [112, 117]. Recently, a more efficient method has been developed which is based on charge manipulation by an SPM tip to eliminate the near-surface paramagnetic noise of shallow NV centers and improve its coherence up to 20-fold [118]. This indicates that making use of the local electric field control through SPM is promising to diagnose and mitigate the decoherence of such atomic/nanometer-sized quantum sensors either hosted in the insulating lattices [28] or adsorbed on the surfaces [109]. Furthermore, the precise tip manipulation and elaborate surface treatments in a high-resolution SPM system can be used to produce quantum sensors through a more controllable manner [119], where the paramagnetic or charge defects inevitably produced during ions implantations [51, 114] can be efficiently decreased, yielding quantum sensors with high coherence.

Besides, as for the color center-based scanning electrometry and current imaging, the spatial resolution is always limited by both the uncontrollable shape of the tip apex and the limited precision in positing a single color center inside the photonics structures. Moreover, the sensitivity is also always degraded due to the complex fabrication processes during the production of diamond tips [79, 120]. To optimize the performance of such quantum sensors, using parabolically shaped diamond tips instead of truncated ones [121] should make it easier to control the tip-surface distance and improve the spatial resolution of imaging. Meanwhile, a high-resolution implanting technique with an accurately designed mask is also developed [122] to enhance the photon collection efficiency and sensitivity of shallow NVs. In addition, finding new kinds of quantum sensors that are more compatible with the SPM tip will also be a tendency for future applications in nanoscale electric sensing and imaging. One promising way is to functionalize the tip with molecular-like quantum sensors with both tunable rotational/vibrational quantum states [35] and permanent electric dipoles [123]. In principle, the larger electric dipole indicates higher sensitivity for measuring electric field and electric potential. It’s also possible to make use of single color centers [124] or confined excitons [125] in two-dimensional (2D) materials as quantum sensors. In this case, the few-layer 2D materials with extremely shallow color centers can be transferred onto a photonics structure as the SPM tip, where a quantum sensor can be positioned close to the tip’s apex with high precisions [126].

7 Summary

In this perspective, we introduce the working principle of quantum sensing in electric sensing and imaging. Specifically, we focus on the atomic/nanometer-sized quantum sensors that can be feasibly integrated into the SPM system. Compared to conventional electric imaging methods such as SCM, PFM, and KPFM, quantum sensors are based on quantum coherence and thus have much higher sensitivity. We also review the recent pioneering applications of quantum sensors in nanoscale sensing and imaging of electric field, electric current, and electric potential. A discussion of their privileges over conventional SPM techniques is also involved. Exemplary applications of atomic/nanometer-sized quantum sensors include the scanning electrometry on ferroelectric materials, the local detection of electron fluid dynamics on quantum materials, the visualization of the current distribution of a vortex, and the electrostatic potential imaging with spatial resolution down to atomic scale. These results pave the way for investigating charge/polarization distribution, electron transport, and electrostatic interactions in materials and electronic devices at nanoscales. Furthermore, we discussed some current technical challenges using quantum sensors in SPM systems. We also propose some future directions for developing advanced scanning probes with high-performance quantum sensors, which would be powerful toolkits for future applications in visualizing current and conductivity distribution in nanoelectronics, detecting the subtle signals from complex glassy systems [12] and hidden quantum phases [13], or disentangling the interactions among intertwined orders in strongly correlated systems [14, 15].