Resonant-Impurity STM/STS Spectroscopy

• Resonant-impurity STM/STS is a technique that uses scanning tunneling microscopy and spectroscopy to detect localized modifications in tunneling conductance—such as near-zero-bias peaks, in-gap bound states, and Fano line shapes—caused by impurities.
• It employs theoretical frameworks like the T-matrix, Anderson impurity models, and rate equations to interpret complex phenomena including YSR resonances, interference effects, and spatial LDOS mapping across various electronic systems.
• Benchmark studies, such as the Zn resonance in Bi2212, demonstrate that impurity responses are highly sensitive to local superconducting gap and carrier density variations, offering insight into the host material's electronic properties.

Resonant-impurity STM/STS is the use of scanning tunneling microscopy and spectroscopy to resolve the local spectral response created when a defect, adatom, donor, charge trap, or impurity state perturbs a host electronic structure. In the literature surveyed here, the relevant observables are localized modifications of the tunneling conductance $dI/dV$, which appear as near-zero-bias resonances, in-gap bound states, Fano line shapes, split Landau-level features, or anisotropic real-space conductance textures. These phenomena have been examined in cuprates, iron pnictides, topological insulators, graphene, Dirac and nodal-line semimetals, heavy-fermion superconductors, low-carrier-density superconductors, and quantum Hall fluids, and they are used both to characterize impurity physics itself and to diagnose properties of the host state such as gap structure, local carrier density, topological surface bands, orbital order, or confined anyonic configurations (Machida et al., 2010, Alpichshev et al., 2011, Park et al., 23 Jun 2026).

1. Core observables and spectroscopic signatures

The common experimental quantity is the local tunneling conductance. In Zn-doped Bi2212, the conductance is written as

$\frac{dI}{dV}=g(\mathbf r,V),$

and the impurity resonance is identified from a sharp low-bias peak near the Zn site, centered close to $-1.5\ \mathrm{mV}$ or $-1.5\ \mathrm{meV}$ (Machida et al., 2010). In theoretical treatments of nodal-line semimetals and related hosts, STM is cast more explicitly as a probe of the local density of states (LDOS),

$$\rho(\mathbf r,\omega)=-\frac{1}{\pi}\mathrm{Im}\,[G_{11}(\mathbf r,\omega)+G_{22}(\mathbf r,\omega)],$$

with $dI/dV\propto \rho(\mathbf r,eV)$ (Zhou et al., 2019). In the 3D Dirac-semimetal two-impurity problem, the experimentally relevant quantity is likewise

$$\mathrm{LDOS}(\varepsilon,\mathbf R_m)=-\frac{1}{\pi}\sum_\sigma \mathrm{Im}\,G^R_\sigma(\varepsilon,\mathbf R_m),$$

and the impurity correction is decomposed into direct and indirect scattering terms $\delta\rho_{jj'}^\sigma$ (Marques et al., 2018).

The phenomenology is correspondingly diverse. Resonant impurities may generate sharp near-zero-bias peaks, as in Zn-substituted Bi2212 and in the unitary-impurity calculations for CeCoIn$_5$ (Machida et al., 2010, Zhang et al., 2015). In superconductors they may also generate in-gap bound states, including Yu–Shiba–Rusinov (YSR) resonances from magnetic impurities and particle-hole-symmetric Andreev-induced features produced by a single localized electronic state inside a gap (Odobesko et al., 2023, Zhong et al., 2024). In systems with competing direct and impurity-mediated tunneling paths, the spectral line shape is not purely Lorentzian; interference generates asymmetric Fano profiles and, in semiconductor-surface models, spatial modulation through factors such as $\cos(2k_x(\omega)x)$ (Mantsevich et al., 2010). Inelastic channels add another layer: in the Dirac-semimetal Anderson-Holstein problem, phonon-assisted sidebands enter the LDOS, while in STM-based IETS the experimentally emphasized quantity is

$\frac{dI}{dV}=g(\mathbf r,V),$0

(Marques et al., 2018).

Spatial mapping is integral rather than auxiliary. The Bi2212 Zn study uses normalized conductance maps $\frac{dI}{dV}=g(\mathbf r,V),$1 to localize bright resonance sites (Machida et al., 2010). YSR work on Fe dimers on Nb(110) images even and odd hybridized states in real space, and strong-field quantum Hall work uses local spectra near charged impurities to infer bound-state structure that changes qualitatively between integer and fractional regimes (Odobesko et al., 2023, Park et al., 23 Jun 2026). A recurring implication is that impurity STM/STS is fundamentally a spatially resolved spectroscopy of the impurity-modified Green’s function, not merely a point spectrum measured above a defect.

2. Theoretical formalisms and model Hamiltonians

A large fraction of resonant-impurity STM/STS is formulated through Green’s functions and $\frac{dI}{dV}=g(\mathbf r,V),$2-matrices. For nodal-line semimetals, the impurity scattering is summed in the $\frac{dI}{dV}=g(\mathbf r,V),$3-matrix approximation,

$\frac{dI}{dV}=g(\mathbf r,V),$4

and the full Green’s function is

$\frac{dI}{dV}=g(\mathbf r,V),$5

The pole condition is monitored through

$\frac{dI}{dV}=g(\mathbf r,V),$6

with resonances occurring when $\frac{dI}{dV}=g(\mathbf r,V),$7 (Zhou et al., 2019). The same broad logic underlies treatments of CeCoIn$\frac{dI}{dV}=g(\mathbf r,V),$8, where the impurity-dressed Green’s function is written in real space as $\frac{dI}{dV}=g(\mathbf r,V),$9, and of iron-pnictide dimer resonances, where the LDOS is evaluated from a standard impurity $-1.5\ \mathrm{mV}$0-matrix built on a reconstructed multiorbital band structure (Zhang et al., 2015, Kang et al., 2012).

Anderson-type descriptions are equally central. In graphene on SiC$-1.5\ \mathrm{mV}$1, the Si-adatom dangling bond is treated as a single impurity orbital with Hamiltonian

$-1.5\ \mathrm{mV}$2

and impurity Green’s function

$-1.5\ \mathrm{mV}$3

The resonance energy and linewidth are then controlled by $-1.5\ \mathrm{mV}$4 and $-1.5\ \mathrm{mV}$5, respectively, with the hollow and top stackings differing because of interference among graphene-coupling paths (Hiebel et al., 2012). In the 3D Dirac-semimetal problem with two localized impurities and local vibrational modes, the model is a two-impurity Anderson-Holstein Hamiltonian treated by a Lang–Firsov transformation, equation-of-motion methods, and the Hubbard I approximation; phonons renormalize $-1.5\ \mathrm{mV}$6 and $-1.5\ \mathrm{mV}$7, and the impurity Green’s function becomes an infinite phonon ladder (Marques et al., 2018).

Interference and Coulomb effects are treated especially explicitly in semiconductor-surface models. There the Hamiltonian

$-1.5\ \mathrm{mV}$8

includes both direct tip–surface tunneling and resonant tunneling through the impurity. For a noninteracting resonant level,

$-1.5\ \mathrm{mV}$9

while shallow impurities are handled by a mean-field shift $-1.5\ \mathrm{meV}$0 and deep impurities by a Hubbard-I self-energy that splits the spectrum into features at $-1.5\ \mathrm{meV}$1 and $-1.5\ \mathrm{meV}$2 (Mantsevich et al., 2010).

Other platforms require different many-body closures. Single donors in silicon are described by resonant transport through two tunnel barriers and a rate-equation model for ground- and excited-state occupations (Voisin et al., 2015). Quantum Hall charge traps are modeled by exact diagonalization, with the STM LDOS written as a sum over many-body addition and removal matrix elements,

$-1.5\ \mathrm{meV}$3

(Park et al., 23 Jun 2026). The field as a whole is therefore methodologically plural: $-1.5\ \mathrm{meV}$4-matrix, Anderson impurity, Keldysh, Hubbard I, Lang–Firsov, rate equations, and exact diagonalization each appear where the host band structure and impurity physics demand them.

3. Zn resonance in Bi2212 as a benchmark problem

A central benchmark for resonant-impurity STM/STS is the systematic Zn study on Bi$-1.5\ \mathrm{meV}$5Sr$-1.5\ \mathrm{meV}$6Ca$-1.5\ \mathrm{meV}$7O$-1.5\ \mathrm{meV}$8, which addresses whether the near-zero-bias Zn resonance is universally produced by every Zn impurity or instead depends on the local electronic environment. The samples were single crystals with Zn content determined by ICP optical emission spectroscopy to be $-1.5\ \mathrm{meV}$9 per Cu atom, and three carrier concentrations were prepared by oxygen depletion: OD87K ($$\rho(\mathbf r,\omega)=-\frac{1}{\pi}\mathrm{Im}\,[G_{11}(\mathbf r,\omega)+G_{22}(\mathbf r,\omega)],$$0 K), UD83K ($$\rho(\mathbf r,\omega)=-\frac{1}{\pi}\mathrm{Im}\,[G_{11}(\mathbf r,\omega)+G_{22}(\mathbf r,\omega)],$$1 K), and UD76K ($$\rho(\mathbf r,\omega)=-\frac{1}{\pi}\mathrm{Im}\,[G_{11}(\mathbf r,\omega)+G_{22}(\mathbf r,\omega)],$$2 K). STS was performed with a home-built low-temperature STM in helium gas at $$\rho(\mathbf r,\omega)=-\frac{1}{\pi}\mathrm{Im}\,[G_{11}(\mathbf r,\omega)+G_{22}(\mathbf r,\omega)],$$3 K on freshly cleaved BiO surfaces. The Zn resonance was mapped by bright spots in $$\rho(\mathbf r,\omega)=-\frac{1}{\pi}\mathrm{Im}\,[G_{11}(\mathbf r,\omega)+G_{22}(\mathbf r,\omega)],$$4, and higher-resolution imaging showed the characteristic fourfold-symmetric structure with maxima at the Zn site and at second- and third-nearest-neighbor sites in the CuO$$\rho(\mathbf r,\omega)=-\frac{1}{\pi}\mathrm{Im}\,[G_{11}(\mathbf r,\omega)+G_{22}(\mathbf r,\omega)],$$5 plane (Machida et al., 2010).

The decisive observation is the correlation with the local superconducting gap $$\rho(\mathbf r,\omega)=-\frac{1}{\pi}\mathrm{Im}\,[G_{11}(\mathbf r,\omega)+G_{22}(\mathbf r,\omega)],$$6. The gap maps show nanoscale inhomogeneity, with $$\rho(\mathbf r,\omega)=-\frac{1}{\pi}\mathrm{Im}\,[G_{11}(\mathbf r,\omega)+G_{22}(\mathbf r,\omega)],$$7 ranging approximately from $$\rho(\mathbf r,\omega)=-\frac{1}{\pi}\mathrm{Im}\,[G_{11}(\mathbf r,\omega)+G_{22}(\mathbf r,\omega)],$$8 to $$\rho(\mathbf r,\omega)=-\frac{1}{\pi}\mathrm{Im}\,[G_{11}(\mathbf r,\omega)+G_{22}(\mathbf r,\omega)],$$9 in OD87K, $dI/dV\propto \rho(\mathbf r,eV)$0 to $dI/dV\propto \rho(\mathbf r,eV)$1 in UD83K, and $dI/dV\propto \rho(\mathbf r,eV)$2 to $dI/dV\propto \rho(\mathbf r,eV)$3 in UD76K). Resonance sites occur only in the smaller-gap regions, and no Zn resonance is observed for $dI/dV\propto \rho(\mathbf r,eV)$4 larger than about $dI/dV\propto \rho(\mathbf r,eV)$5. In the authors’ summary form,

$dI/dV\propto \rho(\mathbf r,eV)$6

This conclusion is reinforced statistically by histograms of $dI/dV\propto \rho(\mathbf r,eV)$7 over total scanned areas of $dI/dV\propto \rho(\mathbf r,eV)$8, $dI/dV\propto \rho(\mathbf r,eV)$9, and $$\mathrm{LDOS}(\varepsilon,\mathbf R_m)=-\frac{1}{\pi}\sum_\sigma \mathrm{Im}\,G^R_\sigma(\varepsilon,\mathbf R_m),$$0 for OD87K, UD83K, and UD76K, respectively. The counted resonance-site densities are $$\mathrm{LDOS}(\varepsilon,\mathbf R_m)=-\frac{1}{\pi}\sum_\sigma \mathrm{Im}\,G^R_\sigma(\varepsilon,\mathbf R_m),$$1 per Cu atom (95 sites), $$\mathrm{LDOS}(\varepsilon,\mathbf R_m)=-\frac{1}{\pi}\sum_\sigma \mathrm{Im}\,G^R_\sigma(\varepsilon,\mathbf R_m),$$2 (60 sites), and $$\mathrm{LDOS}(\varepsilon,\mathbf R_m)=-\frac{1}{\pi}\sum_\sigma \mathrm{Im}\,G^R_\sigma(\varepsilon,\mathbf R_m),$$3 (36 sites), showing that the number of observable Zn resonances decreases strongly with decreasing hole concentration $$\mathrm{LDOS}(\varepsilon,\mathbf R_m)=-\frac{1}{\pi}\sum_\sigma \mathrm{Im}\,G^R_\sigma(\varepsilon,\mathbf R_m),$$4, despite fixed Zn concentration.

The interpretive importance of this result lies in what it excludes. The data support the conclusion that Zn atoms residing in large-gap regions do not generate an observable resonance, rather than the simpler claim that Zn impurities are absent from those regions. The paper discusses two correlated possibilities. One is that large-gap regions correspond to locally suppressed superconductivity, which would matter in theories linking impurity resonances to the superconducting state. The other, emphasized strongly, is that gap inhomogeneity reflects local hole concentration, so the absence of resonance in large-gap areas is fundamentally an absence in low-$$\mathrm{LDOS}(\varepsilon,\mathbf R_m)=-\frac{1}{\pi}\sum_\sigma \mathrm{Im}\,G^R_\sigma(\varepsilon,\mathbf R_m),$$5 regions. The authors state that separating gap size from carrier density is difficult because the two are strongly correlated, but they argue that both are likely central parameters and that carrier density may be especially important. They further note that three theoretical scenarios—a screened Coulomb impurity model, a Kondo impurity model, and a phase-impurity model—all predict suppression of the resonance as the local gap increases, but that future theory should account explicitly for local hole density, not only gap magnitude (Machida et al., 2010).

4. Superconducting impurity resonances and in-gap states

In superconductors, resonant-impurity STM/STS often centers on subgap structure. The YSR case is the canonical magnetic-impurity example: exchange scattering from a magnetic impurity produces a pair of in-gap resonances whose wavefunction reflects the orbital shape of the scattering channel, oscillates with $$\mathrm{LDOS}(\varepsilon,\mathbf R_m)=-\frac{1}{\pi}\sum_\sigma \mathrm{Im}\,G^R_\sigma(\varepsilon,\mathbf R_m),$$6, and decays with factors proportional to $$\mathrm{LDOS}(\varepsilon,\mathbf R_m)=-\frac{1}{\pi}\sum_\sigma \mathrm{Im}\,G^R_\sigma(\varepsilon,\mathbf R_m),$$7 and $$\mathrm{LDOS}(\varepsilon,\mathbf R_m)=-\frac{1}{\pi}\sum_\sigma \mathrm{Im}\,G^R_\sigma(\varepsilon,\mathbf R_m),$$8. A double-functionalized probe combining a superconducting cluster with a CO molecule was used to image hybridized YSR states around Fe dimers on Nb(110), revealing two pairs of YSR resonances at approximately $$\mathrm{LDOS}(\varepsilon,\mathbf R_m)=-\frac{1}{\pi}\sum_\sigma \mathrm{Im}\,G^R_\sigma(\varepsilon,\mathbf R_m),$$9 and $\delta\rho_{jj'}^\sigma$0, as well as rich real-space interference patterns inaccessible with conventional tips. In the soft-contact regime, the CO-induced tip orbital symmetry introduces derivative-like contrast through Chen’s derivative rules, so that $\delta\rho_{jj'}^\sigma$1-type tunneling yields signals proportional to $\delta\rho_{jj'}^\sigma$2 in the $\delta\rho_{jj'}^\sigma$3-$\delta\rho_{jj'}^\sigma$4 plane (Odobesko et al., 2023).

A different superconducting mechanism is the Andreev-tunneling process through a single impurity state in NaAlSi. There the impurity is not a magnetic scatterer producing a conventional YSR state, but a localized electronic level that, when shifted into the superconducting gap, generates energy-symmetric in-gap peaks through Cooper-pair formation or breakup. The linecut maps show an X-shaped spatial evolution: as the tip approaches the ring-like feature associated with a subsurface impurity state, the impurity level moves toward the Fermi level, the in-gap branches move inward, cross at zero bias, and then split again. The paper interprets this as tip-induced gating in a low-carrier-density superconductor with weak screening. Under strong magnetic fields, the impurity state exhibits Zeeman splitting when it is near $\delta\rho_{jj'}^\sigma$5, with extracted Landé factor $\delta\rho_{jj'}^\sigma$6 (Zhong et al., 2024).

The unitary-impurity problem in CeCoIn$\delta\rho_{jj'}^\sigma$7 illustrates a third use of resonant impurity spectroscopy: phase-sensitive diagnosis of unconventional gap symmetry. In a two-band heavy-fermion model treated by the $\delta\rho_{jj'}^\sigma$8-matrix approach, a strong nonmagnetic impurity produces a sharp nearly zero-energy resonance state (ZERS). In the unitary limit $\delta\rho_{jj'}^\sigma$9, the resonance approaches $_5$0, is strongly suppressed on the impurity site itself, and is strongest on nearest-neighbor sites. The real-space pattern depends on the nodal structure of the gap. For $_5$1, the resonance lobes extend mainly along the crystal axes, whereas for $_5$2 the pattern is rotated by $_5$3 and emphasizes the diagonals. The proposed STM criterion is therefore not merely the existence of a near-zero-bias impurity peak but its fourfold spatial orientation (Zhang et al., 2015).

Natural defects in LiFeAs show that impurity spectroscopy in superconductors is not universal even within a single material. Low-temperature STM/STS resolved seven distinct defect types. Fe-$_5$4-1 defects exhibit pronounced positive-bias impurity-induced states with peaks near $_5$5 and $_5$6, while the $_5$7-symmetric As defect enhances the LDOS at both positive and negative bias over a broader spatial range. The paper is explicit that these data alone do not permit a definitive conclusion about pairing symmetry, because the impurity types cannot yet be cleanly classified as magnetic or nonmagnetic; theoretical modeling of real-space impurity potentials, induced magnetism, spin-polarized STM, and higher-resolution ultra-low-temperature STS are identified as necessary for a more complete interpretation (Schlegel et al., 2016).

5. Dirac, topological, and Landau-quantized hosts

In topological and Dirac materials, resonant-impurity STM/STS is frequently used to distinguish local resonances from quasiparticle interference. On Bi$_5$8Se$_5$9, triangular defects associated with subsurface Se vacancies produce broad LDOS resonances near the Dirac point, but the study reports no measurable QPI. The local spectrum is strongly perturbed near the impurity, yet the Dirac point is not locally destroyed: the resonance redistributes spectral weight without eliminating the underlying Dirac crossing. Because the defects are extended rather than point-like, the data are fitted by generalizing the Biswas and Balatsky point-impurity theory to a finite-radius step potential characterized by height $\cos(2k_x(\omega)x)$0 and radius $\cos(2k_x(\omega)x)$1; representative fits give $\cos(2k_x(\omega)x)$2 for a triangular defect and $\cos(2k_x(\omega)x)$3 for an extended Sb-related defect (Alpichshev et al., 2011).

Nodal-line semimetals offer a related but more overtly topological use of impurity resonances. Within a lattice $\cos(2k_x(\omega)x)$4-matrix calculation, a surface impurity produces robust resonant impurity states over a broad range of potentials because the weakly dispersive topological surface states concentrate spectral weight near zero energy and make the pole condition easy to satisfy. By contrast, bulk impurities can induce in-gap-like features, but sharp resonances appear only for certain impurity strengths and are much less robust. The proposed STM signature is therefore not simply any impurity peak, but specifically a localized surface resonance that persists over a wide range of surface impurity potentials, distinguishing the nodal-line surface state from generic metallic bands and from the nearly constant-LDOS surface states of Weyl semimetals (Zhou et al., 2019).

Graphene-based systems show how local geometry and electrostatics can control impurity resonances. In graphene on SiC$\cos(2k_x(\omega)x)$5, the Si-adatom dangling-bond resonance is visible through the graphene overlayer at high bias because graphene is electronically transparent in STM under those conditions. STS shows a stacking-dependent resonance at about $\cos(2k_x(\omega)x)$6 for hollow stacking, $\cos(2k_x(\omega)x)$7 for top graphene/adatom plus hollow graphene/restatom, and $\cos(2k_x(\omega)x)$8 for top graphene/adatom plus top graphene/restatom. The resonance is lower in energy and narrower in hollow stacking, a trend interpreted through an Anderson-impurity model in which destructive interference among multiple hopping paths suppresses low-energy hybridization in the hollow geometry (Hiebel et al., 2012). In backgated graphene with Co adatoms, the central phenomenon is reversible ionization by either the global backgate voltage or the local electric field from the STM tip. Ring-shaped conductance features mark the lateral locus where the impurity level crosses $\cos(2k_x(\omega)x)$9, and the associated screening clouds can exceed $\frac{dI}{dV}=g(\mathbf r,V),$00 in diameter (Brar et al., 2010).

Strong magnetic fields introduce another class of resonant impurity spectroscopy. In monolayer graphene quantum Hall states, local STS near charged defects shows lifting of orbital degeneracy in integer states, while in fractional states such as $\frac{dI}{dV}=g(\mathbf r,V),$01 and $\frac{dI}{dV}=g(\mathbf r,V),$02 the lowest-energy spectral feature acquires an additional splitting that appears only when the chemical potential lies within a fractional gap. Exact-diagonalization calculations attribute this to competing many-body configurations of anyons trapped by the impurity potential, and the splitting vanishes for a rotationally symmetric trap. The anisotropic confining potential

$\frac{dI}{dV}=g(\mathbf r,V),$03

mixes states with $\frac{dI}{dV}=g(\mathbf r,V),$04, producing avoided crossings and a multi-peak LDOS response (Park et al., 23 Jun 2026).

6. Spatial textures, tuning knobs, and diagnostic uses

One of the most consequential developments in resonant-impurity STM/STS is the realization that impurity-induced patterns can diagnose broken symmetry or hidden structure in the host. In iron pnictides near the structural transition temperature $\frac{dI}{dV}=g(\mathbf r,V),$05, a single nonmagnetic impurity can induce a non-local orbital texture through strong quadrupole fluctuations. Real-space mean-field calculations in a 10-orbital model find a progression from no orbital order to diagonal $\frac{dI}{dV}=g(\mathbf r,V),$06 order and then to $\frac{dI}{dV}=g(\mathbf r,V),$07-symmetric order; for representative impurity strength $\frac{dI}{dV}=g(\mathbf r,V),$08, the thresholds are $\frac{dI}{dV}=g(\mathbf r,V),$09 and $\frac{dI}{dV}=g(\mathbf r,V),$10, respectively. The resulting LDOS nanostructure extends roughly $\frac{dI}{dV}=g(\mathbf r,V),$11 along $\frac{dI}{dV}=g(\mathbf r,V),$12 and $\frac{dI}{dV}=g(\mathbf r,V),$13 along $\frac{dI}{dV}=g(\mathbf r,V),$14, and is compared directly with the elongated impurity structures reported by Chuang et al. and Song et al. in STM/STS (Inoue et al., 2011). A related iron-pnictide theory interprets Co-induced dimer resonances aligned with the antiferromagnetic $\frac{dI}{dV}=g(\mathbf r,V),$15 axis as a manifestation of reconstructed nesting at $\frac{dI}{dV}=g(\mathbf r,V),$16; the calculated LDOS maxima at $\frac{dI}{dV}=g(\mathbf r,V),$17 produce a $\frac{dI}{dV}=g(\mathbf r,V),$18 dimer, close to the $\frac{dI}{dV}=g(\mathbf r,V),$19 structures seen experimentally (Kang et al., 2012).

Quantum Hall ferroelectrics and nematics provide a different form of spatial diagnosis. A delta-function impurity in a Dirac Landau level binds two impurity states for $\frac{dI}{dV}=g(\mathbf r,V),$20, and in the ferroelectric case the centroid of the impurity-bound LDOS carries a measurable dipole moment

$\frac{dI}{dV}=g(\mathbf r,V),$21

This establishes a direct relation between a local STM observable and the ideal bulk dipole moment from the modern theory of polarization. Exact diagonalization further shows that a strong impurity can drive the local ground state into a quasihole configuration and create an inter-valley excitonic state whose spatial conductance map differs from the ordinary impurity resonance (Tam et al., 2020).

Tuning parameters are often as important as the impurity itself. Tip position and tip height modulate $\frac{dI}{dV}=g(\mathbf r,V),$22 for single donors in silicon, permitting resonant transport through ground and excited donor states and enabling extraction of charging energies on an absolute scale from thermal broadening (Voisin et al., 2015). In NaAlSi and in backgated graphene, the STM tip acts as a local gate rather than a passive detector, shifting impurity levels through $\frac{dI}{dV}=g(\mathbf r,V),$23 and qualitatively changing the spectrum (Zhong et al., 2024, Brar et al., 2010). Probe-tip orbital symmetry is similarly nontrivial: with CO-functionalized superconducting tips, the measured signal may emphasize derivatives of the sample wavefunction rather than $\frac{dI}{dV}=g(\mathbf r,V),$24 itself (Odobesko et al., 2023).

A final extension is spin-resolved resonant-impurity STM/STS in altermagnets. In a 2D $\frac{dI}{dV}=g(\mathbf r,V),$25-wave altermagnetic substrate, the interplay of direct tip–impurity tunneling and substrate-mediated tunneling produces a generalized Fano form for the spin-resolved local spectral function. In zero field, the impurity correction oscillates with anisotropic spin-dependent period

$\frac{dI}{dV}=g(\mathbf r,V),$26

and the altermagnetic splitting strength $\frac{dI}{dV}=g(\mathbf r,V),$27 can be inferred from the ratio of spin-resolved oscillation periods. In the Landau-quantized regime, the dominant real-space signature becomes a spin-dependent nodal pattern; mismatch between the nodal structures of the two spin channels yields a large local spin contrast, with representative calculations giving $\frac{dI}{dV}=g(\mathbf r,V),$28. The paper identifies resonant-impurity STM/STS in this setting as a phase-sensitive local probe of altermagnetic band anisotropy (Hong et al., 8 Jul 2026).

Taken together, these studies show that resonant-impurity STM/STS is not a single spectroscopy of “impurity peaks,” but a family of local probes whose meaning depends on host topology, symmetry, interaction scale, and electrostatics. The recurrent caution is that the mere presence or absence of a defect-centered spectral feature is rarely sufficient for interpretation. The decisive information often lies in where the resonance occurs in parameter space, how it evolves with tip coupling or carrier density, whether it is accompanied by spatial interference, and how its real-space symmetry tracks the structure of the host state (Machida et al., 2010, Alpichshev et al., 2011, Zhong et al., 2024).