Setpoint-Dependent Spectroscopy

• Setpoint-dependent spectroscopy is a technique in STM/STS where feedback parameters adjust the tip–sample distance and affect the measured signal.
• It identifies and mitigates closed-loop feedback artifacts using constant-height and off-band setpoint protocols to recover true local density of states.
• The method enables advanced parameter extraction, including machine-learning-driven Hamiltonian reconstruction from STM-IETS data.

Setpoint-dependent spectroscopy refers to spectroscopic measurements in scanning tunneling microscopy (STM) and scanning tunneling spectroscopy (STS) where the feedback parameters—specifically the bias voltage and tunneling current setpoints—critically influence not only the tip–sample distance but also the measured spectroscopic signal. Unlike conventional spectroscopy performed at constant tip height, setpoint-dependent protocols impose a closed-loop feedback system that dynamically adjusts the tip position to maintain the pre-set tunneling current at a specified bias. This introduces nontrivial dependencies and possible artifacts in the extracted differential conductance (dI/dV) maps, and, when exploited correctly, enables advanced parameter extraction schemes such as machine-learning-driven Hamiltonian reconstruction from STM inelastic electron tunneling spectroscopy (STM-IETS) datasets.

1. Fundamental Principles of Setpoint-Dependent Spectroscopy

In STM/STS, the tunneling current $I(V, z)$ is described by the Tersoff–Hamann formalism as

$$I(V, z) \propto \int_{0}^{eV} \rho_s(E)\,\rho_t(E-eV)\,T(E, V, z)\,dE$$

where $\rho_s$ and $\rho_t$ are the sample and tip density of states (DOS), and $T(E, V, z) \simeq \exp[-2\kappa(E, V)z]$ is the transmission probability, with $\kappa$ determined by the effective vacuum barrier. Under idealized conditions (constant $\rho_t$, slowly varying $T$, fixed $z$), the measured differential conductance is proportional to the sample LDOS at energy $eV$:

$\frac{dI}{dV} \propto \rho_s(eV)$

Setpoint-dependent operation modifies this paradigm by continuously varying $z$ via closed-loop feedback to maintain $I_s$ at $V_s$, coupling $dI/dV$ to both the local LDOS and the spatial variation of $z$, potentially obscuring spectroscopic information unless specific measurement protocols are followed (Tresca et al., 2022).

2. Feedback-Induced Artifacts and Error Channels

In closed-loop (constant-current) STM/STS, the feedback condition $I(V_s, z) = I_s$ forces $z(x, y; V_s)$ to track local variations in $\rho_s$ at each $(x, y)$. During the measurement of $dI/dV$ at $V \neq V_s$, the total derivative splits as:

$$\left. \frac{dI}{dV} \right|_{\text{meas}} = \left. \frac{\partial I}{\partial V} \right|_{z} + \left( \frac{\partial I}{\partial z} \right) \left( \frac{dz}{dV} \right)$$

Here, $$\left. \frac{\partial I}{\partial V}\right|_{z} \simeq \rho_s(eV)$$ is the true LDOS signal; $$\left(\frac{\partial I}{\partial z}\right)\left(\frac{dz}{dV}\right)$$ is an artifact term coupling feedback dynamics and site-dependent electronic structure. In atomically inhomogeneous phases, such as the $3\times 3$ charge-ordered Pb/Si(111) system, these artifacts can invert apparent site contrast, mask energy-dependent charge order, and result in fundamentally incorrect spectroscopic interpretations: for example, false identification of charge ordering motifs (Tresca et al., 2022).

3. Mitigation Protocols for Setpoint Artifacts

There exist two protocols that rigorously eliminate feedback-induced artifacts:

• Constant-Height STS: Disabling the feedback loop fixes $z$, thereby annihilating the artifact term; $dI/dV$ now directly reflects $\rho_s(eV)$. Thermal drift and surface corrugation limit practicality, but this approach is error-free for true spectroscopic imaging.
• Full Grid STS with Off-Band Setpoint: In closed-loop operation, choosing $V_s$ outside the sample bandwidth (where $\rho_s(eV_s)$ is spatially homogeneous) ensures $z(x, y; V_s)$ is constant over the field of view. Recording an $N\times N$ grid of $I(V, x, y)$ spectra provides a dataset from which true $dI/dV$ maps can be numerically differentiated and, if necessary, normalized at $V_s$. This protocol provides both stability and artifact suppression; with appropriate parameter selection (e.g., $V_s = -1$ V for Pb/Si(111)), the resulting LDOS maps quantitatively agree with first-principles calculations (Tresca et al., 2022).

Practical Guidelines for Measurement

Parameter	Recommended Choice	Rationale
Setpoint bias $V_s$	Outside bandwidth	Ensures spatially homogeneous LDOS
Tunneling current $I_s$	10–200 pA	Stable gap, minimal tip crash risk
Voltage resolution $\Delta V$	$\lesssim$5 meV	Captures fine spectral features
Spatial sampling	$\geq$10 pixels/unit cell	Accurate mapping of charge order

4. Quantitative Applications: Hamiltonian Learning from STM-IETS

Setpoint-dependent spectroscopy can provide access to parameterized multiorbital Hamiltonians via machine learning. For STM-IETS measurements on surface-adsorbed quantum magnets (e.g., FePc on SnTe), the observed evolution of spectral features with setpoint (tip–sample distance) maps the response of excitation energies and tunneling matrix elements to the local electrostatic environment. The model Hamiltonian reads:

$H = H_{CF}(z) + H_{Coulomb} + H_{SOC}$

where $H_{CF}(z)$ encapsulates both static crystal field and tip-induced perturbations $\gamma_{ij}(z)$ (Stark shifts and interorbital mixing), $H_{Coulomb}$ comprises all two-body interactions ($V_{ijkl}$), and $H_{SOC}$ includes spin–orbit coupling ($\lambda_{SOC}, \ell^\alpha_{ij}$).

Machine learning regression, based on convolutional neural networks trained on synthetic spectra from exact diagonalization, can invert experimentally measured $(dI/dV(\omega; z))$ maps to the underlying Hamiltonian parameters ($$\tau, \lambda_{SOC}, \epsilon_{min}, \epsilon_{max}, \ldots$$), with fidelity above 0.9 in the relevant parameter ranges even under moderate noise. This approach enables quantitative atomic-scale characterization of quantum materials (Lupi et al., 27 Jan 2026).

5. Comparison with First-Principles Theory and Experimental Validation

In Pb/Si(111), grid-based STS with off-band setpoint reveals energy-dependent LDOS maps matching ab initio DFT+U+SOC calculations for the one-up/two-down charge ordering. The map topology transitions from triangular (Pb–up dominated) at $E\approx-0.3$ eV, through a balanced point (homogeneous LDOS) at $E\approx-0.115$ eV, to honeycomb (Pb–down dominated) at $E\approx+0.2$ eV. Alternative models, such as two-up/one-down ordering, are ruled out by both the energy location of the balanced point and the predicted contrast reversal dynamics (Tresca et al., 2022).

In molecular STM-IETS, the systematic change of step positions and heights with setpoint encodes nonlinear signatures of $\lambda_{SOC}$, orbital splitting $\tau$, and Stark window $\epsilon_{min}, \epsilon_{max}$. Neural networks trained solely on theoretical spectra correctly recover Hamiltonian parameters from experimental data, transforming setpoint dependence from a source of artifact to a quantitative asset (Lupi et al., 27 Jan 2026).

6. Limitations, Assumptions, and Implications

Setpoint-dependent spectroscopy relies on several key approximations: minimal electronic models (five-orbital for FePc), neglect of explicit tip and substrate band structure beyond one s-orbital, and exclusion of vibrational/Kondo anomalies near zero bias. Pure inelastic (second-order) tunneling is assumed; direct hybridization, non-equilibrium effects, and molecular distortion at extreme setpoints fall outside the validated regime. Within standard operating conditions (setpoint currents 1–100 pA, $z$ such that $\Gamma_{mn}\lesssim k_BT$), these methods remain quantitatively accurate (Lupi et al., 27 Jan 2026).

A plausible implication is that atomic-scale studies of energy-dependent charge order, magnetic excitations, and quasiparticle interference in correlated materials or molecular adsorbates must avoid closed-loop lock-in STS protocols with setpoints inside the electronic bandwidth of interest. Otherwise, artifacts will unavoidably contaminate the extracted electronic/magnetic structure, leading to erroneous conclusions. Setpoint-dependent protocols, when employed with rigorous error suppression and advanced computational modeling, enable robust, quantitative, and even automated reconstruction of local Hamiltonians from single-molecule or single-domain data (Tresca et al., 2022, Lupi et al., 27 Jan 2026).

7. Significance and Broader Impact

Setpoint-dependent spectroscopy establishes both stringent pitfalls and powerful new methodologies for STM/STS-based material characterization. The detailed mapping of artifacts, mitigation strategies, and the integration of machine learning defines a new measurement paradigm: the transition from setpoint-induced error to systematic parameter extraction. This framework is indispensable for the atomic-scale identification of charge order, electronic reconstruction, and many-body quantum states in materials that are inaccessible to bulk probes such as ARPES or x-ray diffraction. The approach alters the interpretation and practice of STM/STS, setting the foundation for future quantitative studies of correlated electrons, magnetism, and quantum molecular phenomena at the atomic scale (Tresca et al., 2022, Lupi et al., 27 Jan 2026).