A scanning tunneling spectroscopy mode introduced by Castenmiller et al. (AIP Advances 8, 075013, 2018) records the tunneling current I(V) and its distance derivative dI(V)/dz simultaneously by adding a small high-frequency sinusoidal modulation to the z-piezo during an open-feedback bias ramp and detecting the response with a lock-in amplifier; within the Simmons barrier picture and the weak-modulation limit the normalized lock-in output satisfies OLIA/⟨I⟩∝κz^2O_{\mathrm{LIA}}/\langle I\rangle \propto \kappa\hat{z}^2, yielding the bias-dependent inverse decay length κ(V)\kappa(V). The unresolved problem is to establish the quantitative validity domain of this method — its accuracy for stronger modulation (κz^≳1\kappa\hat{z} \gtrsim 1), for general tip and sample work functions, and for tunneling situations beyond the simple barrier picture (image-charge rounding, tip-induced band bending, multiple or resonant contributing states) — and to determine under what conditions the measured κ(V)\kappa(V), which is a weighted average over all tunneling states, can be inverted into quantitative k-space band-structure information for general materials rather than favorable single-state cases. Follow-up work has extended the technique case-by-case (dual-modulation mapping on Au(111); Fermi-velocity extraction for graphene on MoS2; group-IV 2D Dirac materials), but no systematic accuracy analysis or general inversion theory exists.

Background

In scanning tunneling microscopy (STM), a sharp tip is held a few angstroms above a surface and the tunneling current II is measured as a function of sample bias VV and tip–sample separation zz. In scanning tunneling spectroscopy (STS), the differential conductance dI/dVdI/dV, usually obtained by modulating the bias and detecting the response with a lock-in amplifier (an instrument that isolates the component of a signal at a reference frequency), is approximately proportional to the local density of states (LDOS) of the sample. A second mode, current–distance or I(z)I(z) spectroscopy, ramps zz at fixed bias: because the current decays roughly as e−2κze^{-2\kappa z}, it yields the inverse decay length κ=2mϕ/ℏ\kappa=\sqrt{2m\phi}/\hbar, where ϕ\phi is the effective tunnel barrier height, approximately the average of the tip and sample work functions (the Simmons model, a simple rectangular-barrier description of a tunnel junction). The apparent barrier height and its bias dependence κ(V)\kappa(V) carry physical information: κ\kappa grows with the parallel momentum k∥k_\parallel of the tunneling states, κ=(2m/ℏ2)(ϕˉ−E+eV/2)+k∥2\kappa = \sqrt{(2m/\hbar^2)(\bar{\phi} - E + eV/2) + k_\parallel^2} with ϕˉ\bar{\phi} the average of tip and sample work functions, so the measured decay length is sensitive to which electronic states — and hence which momenta in k-space — dominate the current.

Classically, κ(V)\kappa(V) is obtained from a series of I(V)I(V) curves recorded at several tip–sample separations, an approach introduced by Stroscio, Feenstra and Fein in 1986 that demands exceptional tip stability. State-resolved decay-constant measurements have been used to extract critical-point energies of band structures, for example for monolayer WSe2 (Zhang et al. 2015). Castenmiller et al. (2018) introduced a dual mode that avoids the series of set points: during an open-feedback bias ramp (the feedback loop — the servo that normally holds the tunneling current constant by adjusting zz — being disabled), a small sinusoidal modulation z(t)=z0+z^sin⁡(ωt)z(t)=z_0+\hat{z}\sin(\omega t) (amplitude about 0.28 Å, frequency 1831 Hz, above the feedback-loop cutoff) is applied to the z-piezo (the piezoelectric actuator that sets the tip–sample separation), and the response at ω\omega is detected with a lock-in amplifier, giving dI(V)/dzdI(V)/dz simultaneously with I(V)I(V). Within the Simmons picture I(V,z)=CVe−2κzI(V,z)=CVe^{-2\kappa z}, the exact time-averaged current and lock-in output are ⟨I⟩=IrefI0(2κz^)\langle I\rangle = I_{\mathrm{ref}} I_0(2\kappa\hat{z}) and OLIA=z^IrefI1(−2κz^)O_{\mathrm{LIA}}=\hat{z} I_{\mathrm{ref}} I_1(-2\kappa\hat{z}), where I0I_0 and I1I_1 are modified Bessel functions (special functions arising when exponentials of sinusoids are averaged over a period) and IrefI_{\mathrm{ref}} is the unmodulated reference current. These relations use the source's lock-in convention, in which the reference signal is the z-piezo displacement itself, Sref=z^sin⁡(ωt)S_{\mathrm{ref}}=\hat{z}\sin(\omega t), so that OLIAO_{\mathrm{LIA}} carries the explicit z^\hat{z} prefactor; under a unit-amplitude reference sine the same Simmons model would instead give OLIA/⟨I⟩∝κz^O_{\mathrm{LIA}}/\langle I\rangle\propto\kappa\hat{z}. For weak modulation (κz^<1\kappa\hat{z}<1), OLIA/⟨I⟩O_{\mathrm{LIA}}/\langle I\rangle is proportional to κz^2\kappa\hat{z}^2 in this displacement-referenced convention, which is how κ(V)\kappa(V) is extracted. The method was demonstrated on Ge(001) at 77 K, where the extracted κ(V)\kappa(V) increases outside the band gap, interpreted as tunneling to states with nonzero k∥k_\parallel. Follow-up work by the same group extended the approach to simultaneous dI/dVdI/dV and dI/dzdI/dz mapping on Au(111), with a treatment in the Tersoff–Hamann framework (the standard s-wave-tip approximation expressing the current as an integral over the sample LDOS) showing that the measured κ\kappa is a weighted average over all states in the bias window, and to quantitative extraction of the Fermi velocity of graphene on MoS2 — the favorable case where a single Dirac band dominates tunneling within the substrate gap. What remains unestablished is the quantitative validity domain of the method — its accuracy for stronger modulation, general work functions, and realistic junctions (image-charge-rounded and bent barriers, multiple or resonant channels) — and the conditions under which κ(V)\kappa(V) can be inverted into state-resolved k-space structure for general materials.

Problem Statement

Establish the quantitative validity domain of simultaneous I(V)I(V) and dI(V)/dzdI(V)/dz scanning tunneling spectroscopy — the dual mode introduced by Castenmiller et al. (2018) in which a small high-frequency z-piezo modulation and lock-in detection yield dI(V)/dzdI(V)/dz during an open-feedback bias ramp — and determine what the simultaneously extracted inverse decay length κ(V)\kappa(V) can quantitatively tell us about k-space electronic structure. Throughout, z^\hat{z} denotes the amplitude of the sinusoidal z-piezo modulation, OLIAO_{\mathrm{LIA}} the output of the lock-in amplifier at the modulation frequency (recorded with the source's convention in which the lock-in reference signal is the displacement itself, Sref=z^sin⁡(ωt)S_{\mathrm{ref}}=\hat{z}\sin(\omega t)), ⟨I⟩\langle I\rangle the time-averaged tunneling current, and IrefI_{\mathrm{ref}} the unmodulated reference current. The requested results are jointly linked by this single objective: knowing when the extracted κ(V)\kappa(V) is quantitatively trustworthy, and what can be inverted from it.

  1. Finite-modulation accuracy: extend the lock-in analysis beyond the weak-modulation limit κz^<1\kappa\hat{z}<1 in which OLIA/⟨I⟩∝κz^2O_{\mathrm{LIA}}/\langle I\rangle\propto\kappa\hat{z}^2; quantify the accuracy and systematic errors of the extracted κ(V)\kappa(V) as a function of modulation amplitude, including how the exact Bessel-function relations and the calibration via ⟨I⟩/Iref\langle I\rangle/I_{\mathrm{ref}} behave for κz^≳1\kappa\hat{z}\gtrsim 1 when the junction is not described by a purely exponential Simmons barrier.

  2. Barrier generality: quantify the accuracy of the extracted κ(V)\kappa(V) for general tip and sample work functions and for realistic tunnel barriers beyond the rectangular Simmons form — trapezoidal bias-dependent barriers with image-charge rounding (smoothing of the rectangular barrier by the electrostatic image charge of the tunneling electron), and junctions with tip-induced band bending (bending of the sample's electronic bands near the surface by the local electric field of the tip) — identifying the regimes in which the simple barrier picture fails quantitatively.

  3. Complex tunneling situations: characterize the behavior of the method when multiple electronic states with different k∥k_\parallel, or resonant or molecular channels, contribute to the current, given that the measured κ\kappa is a weighted average over the contributing states.

  4. k-space inversion: determine the conditions (identifiability and uniqueness) under which the measured κ(V)\kappa(V) — possibly combined with the simultaneously recorded I(V)I(V) or dI/dVdI/dV — can be inverted into quantitative state-resolved band-structure (k-space) information for general materials and spectral regimes, and provide the inversion scheme, or demonstrate that no such inversion is possible without information beyond what the favorable single-dominant-state cases (Ge(001), graphene on MoS2, group-IV Dirac materials) already exploit.

A complete answer is a systematic account — theory, simulation, and/or validated experiment with independent benchmarks — of when the method is quantitatively reliable and when it fails, together with the conditions (identifiability and uniqueness) under which the measured κ(V)\kappa(V) admits a state-resolved k-space inversion, and the inversion scheme itself or a demonstration that no such inversion is possible beyond the favorable single-dominant-state cases already exploited. Items 1–4 above are all mandatory and jointly define this single objective; each must be answered within its own explicitly stated regime — the modulation-amplitude regime for item 1, the barrier and work-function class for item 2, the contributing-state class for item 3, and the material and spectral regime for item 4.

References

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