IETS: Inelastic Tunneling Spectroscopy

• IETS is a quantum transport method that identifies inelastic tunneling events by measuring current changes at specific bias voltages linked to vibrational, spin, or magnetic excitations.
• It utilizes differential conductance (dI/dV) and its second derivative to pinpoint threshold energies where electrons interact with localized excitations in molecular or atomic-scale junctions.
• Advanced computational methods like NEGF and DFT, coupled with lock-in detection, enable precise mapping of nano- and mesoscopic excitations, bolstering research in quantum magnetism and molecular electronics.

Inelastic Tunneling Spectroscopy (IETS) is a quantum transport technique for probing elementary excitations—especially molecular vibrations, phonons, and spin transitions—by resolving small inelastic features in the tunneling current as a function of applied bias. When tunneling electrons lose (or gain) energy to local excitations in the junction (molecule, nanostructure, defect), additional conductance channels are opened at sharply defined threshold voltages, which manifest as steps in the differential conductance (dI/dV) and peaks (or dips) in its second derivative (d²I/dV²). IETS provides direct, high-sensitivity access to vibrational, spin, and magnetic spectra in atomic- and molecular-scale systems, and underpins a broad array of fundamental and applied studies in scanning tunneling microscopy/spectroscopy (STM/STS), molecular electronics, solid-state ionics, quantum magnetism, and topological matter.

1. Physical Principles and Basic Formalism

The basic mechanism of IETS is the inelastic excitation of local quantum modes by electrons tunneling between electrodes. For planar junctions (e.g., metal–insulator–metal), an electron with energy $E$ incident from electrode 1 can tunnel via a barrier to electrode 2. If the applied bias $V$ exceeds the energy of a local excitation ($eV \geq \hbar\omega_\nu$), an inelastic process becomes energetically allowed:

$$I(V)=I_{\rm elastic}(V) + \sum_\nu g_\nu S(V, \hbar\omega_\nu)$$

where $g_\nu$ is the electron–vibration coupling and $S(V, \hbar\omega_\nu)$ is a lineshape function centered at $V = \hbar\omega_\nu/e$ (Krane et al., 2024). The step-onset in conductance arises from the sudden opening of inelastic channels at threshold.

For planar junctions, as well as STM/STS configurations, the total current can be expressed as

$$I(V) = \frac{4\pi e}{\hbar}\int_{-\infty}^\infty T(E)\, [f_1(E) - f_2(E + eV)]\,dE$$

where $T(E)$ is the (in general, bias-dependent) transmission probability, and $f_{1,2}$ are Fermi functions. Differentiation yields

$$\frac{dI}{dV} \propto \int T(E)\left[ -\frac{\partial f(E + eV)}{\partial E} \right]\,dE \quad , \quad \frac{d^2I}{dV^2} \propto \int T(E)\left[ -\frac{\partial^2 f(E + eV)}{\partial E^2} \right]\,dE$$

Thermal broadening imposes a fundamental resolution limit, classically $\Delta E_{\rm min}\approx 5.4 k_BT$ (FWHM at 400 K $\sim$ 186 meV), though tailored resonant tunneling can surpass this (Ngabonziza et al., 2020).

In STM-IETS, lock-in detection of $d^2I/dV^2$ is realized via small bias modulations and phase-sensitive amplifiers (Krane et al., 2024), providing sensitivity to minute inelastic features and enabling spatial mapping.

2. Theoretical Descriptions and Computational Methodologies

Modern IETS modeling describes electron–vibration coupling and inelastic scattering using non-equilibrium Green's function (NEGF) approaches, lowest-order expansion (LOE) formalisms, density-functional theory (DFT)-based dynamics, and many-body perturbation theory.

a) Model Hamiltonians and NEGF

The basic molecular junction model includes an electronic level coupled to a vibrational mode:

$$H_C = \varepsilon_0 d^\dagger d + \hbar\omega_0 a^\dagger a + \gamma_0 (a^\dagger + a) d^\dagger d$$

NEGF formalism yields the retarded Green's function:

$$G^r(\omega) = \big[\omega - \varepsilon_0 - \Sigma_L^r(\omega) - \Sigma_R^r(\omega) - \Sigma_{e\textrm{–}vib}^r(\omega)\big]^{-1}$$

with electron–vibration self-energy $\Sigma_{e\textrm{–}vib}^r(\omega)$ treated in the self-consistent first Born (Hartree–Fock) approximation (Dash et al., 2012).

The Meir–Wingreen formula is used to compute the current, with elastic and inelastic parts:

$$I = (e/\hbar) \int (d\omega/2\pi) \textrm{Tr}\left\{ [\Sigma_L^<(\omega)G^>(\omega) - \Sigma_L^>(\omega)G^<(\omega)] \right\}$$

IETS signatures (peaks in $d^2I/dV^2$) arise due to the opening of inelastic channels at voltages $|eV| \geq \hbar\omega_0$.

b) LOE and Beyond Wide-Band Approximation

For weak electron–vibration coupling, the LOE yields for each mode a contribution proportional to

$$\gamma_\lambda = \mathrm{Tr}\{ M_\lambda \tilde{A}_L(\mu_L) M_\lambda A_R(\mu_R) \}$$

where $M_\lambda$ is the mode-resolved coupling matrix and $A_{L,R}$ are spectral densities (Lü et al., 2013, Palsgaard et al., 2014). Including energy-dependence beyond the wide-band approximation is essential for capturing resonance and band-edge effects and for quantitative matching with experiment.

c) First-Principles Protocols

Workflow involves:

• DFT optimization of device geometries, vibrational analysis (phonons/eigenmodes).
• NEGF evaluation of transmission, dynamical matrices, and electron–phonon matrices.
• Explicit summation over vibrational modes and convolution with thermal and instrumental broadening.
• Practical use of codes such as SIESTA/TranSIESTA with Inelastica (Palsgaard et al., 2014).

3. Vibrational, Magnetic, and Electronic Excitations: Spectral Assignments

IETS identifies a range of elementary excitations:

• Molecular vibrations: Threshold steps at $eV = \hbar\omega_\nu$ signal excitation of molecular stretches, bends, and torsions. For thiol–Au junctions, for instance, the highest IETS feature corresponds to the S–Au stretch ($\sim$45 meV), while the presence/absence of S–H torsion mode ($\sim$55 meV) indicates intact or dissociated thiols (Demir et al., 2012).
• Phonons in low-dimensional materials: In graphene, the dominant out-of-plane acoustic phonon (ZA, K point, $\sim$67 meV) sets a pronounced step in $dI/dV$; optical phonons yield additional peaks at higher energy. Defective graphene exhibits mode-specific inelastic signatures characteristic of local structural rearrangements or adsorbates (Palsgaard et al., 2014).
• Magnetic and spin excitations: IETS can resolve spin-flip transitions, multiplet splitting due to magnetic anisotropy, or Kondo-like zero-bias features. Studies on adatoms and molecular nanomagnets demonstrate that selection rules govern accessible $\Delta J_z$ transitions, which extend to $2\ell+1$ for atoms with strong orbital moments (Kyvala et al., 6 Aug 2025). Many-body effects (Kondo, singlet-triplet transitions) produce threshold peaks with energies and widths set by $T_K^*$, a distinct coherence scale (Korytár et al., 2011).
• Spin-phonon coupling and topological matter: In topological insulators, IETS detects both spin-orbit–locked Dirac quasiparticles and bosonic excitations. Varying coupling strength induces qualitative changes in the local density-of-states and produces spin-polarized inelastic Friedel oscillations (She et al., 2012).
• Fractional quantum excitations: In van der Waals heterostructures (e.g., graphene/α-RuCl₃), IETS detects both spinon continua and Majorana bound states. Features scale as $B^3$ in field, evidencing fractionalization in quantum magnets (Carrega et al., 2020).

4. Experimental Methodologies and Signal Optimization

Signal detection in IETS hinges on energy resolution, junction characterization, and control of electrode/molecule geometries.

• Lock-in detection: Small AC bias modulation is applied; $dI/dV$ and $d^2I/dV^2$ are extracted at the first and second harmonic of modulation frequency, respectively (Krane et al., 2024).
• Tip and contact structure: The fraction of current passing through the molecule (Imol/Itot) and symmetry matching of tip and sample orbitals strongly modulate signal amplitude. Single-atom tips maximize vibrational IETS in STM, while multi-atom tips reduce observed peaks by bypassing the molecule. Positioning the adsorbate atop an adatom compels the current through the molecule, restoring intensity and suppressing tip dependence (Okabayashi et al., 2015, Garcia-Lekue et al., 2011).
• Functionalized and chemically engineered junctions: Using molecule-terminated, multi-molecule or symmetry-matched tips enhances select vibrational modes (via propensity rules) (Garcia-Lekue et al., 2011). Lateral displacement and symmetry mixing can “unlock” otherwise forbidden transitions.
• Spatially resolved IETS: Spatial mapping across the nanostructure reveals the distribution of excitation intensity and aids in distinguishing vibrational from magnetic modes by selection rule analysis and correlation with electronic wavefunction mapping (Krane et al., 2024).
• High-temperature operation: By engineering resonant states within the barrier, it is feasible to surpass the textbook $5.4k_BT$ thermal resolution limit and achieve high-resolution IETS spectra up to 400 K ($\sim$20 meV FWHM), enabling operando studies on solid-state ionic conductors (Ngabonziza et al., 2020).

5. Advanced Topics: Robustness, Control, and Functionality

• Contact independence: In the off-resonant transport regime, the bias threshold and amplitude of vibrational IETS peaks are robust against variations in molecule–lead contact geometry and local potential profile. Peak positions remain fixed at $eV = \hbar\omega_0$, while elastic resonances shift with contact properties (Dash et al., 2012).
• Electrostatic control and device applications: Application of a gate (uniform electric field) can tune vibronic couplings $\gamma$ and frequencies $\omega$, achieving non-linear amplification and switching of the IETS signal at sub-volt gate swings—suggesting a pathway for molecular-scale transistors and amplifiers relying on vibrational rather than electronic-level alignments (Dash et al., 2012).
• In-situ bond tracking: Real-time tracking of chemical bond formation/dissociation (e.g., S–H to Au–S in thiol junctions) is accessible by monitoring shifts and disappearance of vibrational modes (Demir et al., 2012).
• Many-body and correlation effects: In strongly correlated or Kondo systems, inelastic features ride atop narrow many-body resonances, and replica peaks appear shifted by excitation energies. Accurate treatment requires a hybrid approach: DFT/GW for spectral input, nonperturbative solutions (NRG) for Kondo/coherence physics, and inclusion of both “intrinsic” (molecule–substrate) and “extrinsic” (tip–sample) electron–phonon couplings (Eickhoff et al., 2019).

6. Future Directions and Application Domains

• Nanoionic devices and high-temperature solid-state spectroscopy: Spectral analysis of mobile protons, deuterons, and other ions in oxides is now feasible at device-operating temperatures, with direct application in battery, fuel-cell, and memristor diagnostics (Ngabonziza et al., 2020).
• Quantum magnetism and topological quantum matter: IETS underpins the detection of exotic excitation spectra in engineered quantum materials, giving access to Majorana bound states, spinon continua, and the local fingerprints of fractionalization. Field-dependence and symmetry-resolved IETS complement established STM/STS modalities for identifying novel quantum phases (Carrega et al., 2020).
• Single-nuclear-spin readout: With sub–100 mK energy resolution, IETS can resolve hyperfine–split transitions of individual nuclear spins, providing a spectroscopic route to quantum state populations and the possibility of single-nuclear-spin readout (Delgado et al., 2011).
• Chemically resolved imaging: By mapping the local curvature of the tip–sample potential via IETS of a functionalized probe (e.g., CO-decorated tip), ultrahigh-resolution images of intramolecular structure and charge transfer can be achieved, leveraging the mechanical response of the probe particle to surface Hartree potentials and Pauli repulsion (Hapala et al., 2014).

In summary, IETS provides a highly sensitive, mode- and symmetry-resolved spectroscopic tool for quantifying local excitations in atomic-scale systems. Through synergy of advanced experimental techniques, first-principles and many-body theory, and control of nano- and mesoscopic architectures, IETS is central to modern research in molecular electronics, nanomagnetism, materials chemistry, and quantum condensed matter.