| Tersoff–Hamann approximation | |
|---|---|
| Name | Tersoff–Hamann approximation |
| Caption | Conceptual scheme of a scanning tunneling microscope tip and sample |
| Introduced | 1983 |
| Authors | J. Tersoff; D. R. Hamann |
| Field | Condensed matter physics; Surface science |
| Related | Scanning tunneling microscopy; Density functional theory |
Tersoff–Hamann approximation
The Tersoff–Hamann approximation is a theoretical model that simplifies calculation of tunneling currents in scanning tunneling microscopy (STM) by relating the measured current to the local density of states of a sample at the position of the probe tip. It provides a practical bridge between quantum mechanics-based electronic structure calculations and experimental STM images, enabling interpretation of surface electronic features and atomic-scale contrast. The approximation is widely used in conjunction with density functional theory (DFT) and has influenced both computational modeling and experimental analysis in surface physics.
The Tersoff–Hamann approximation was proposed by J. Tersoff and D. R. Hamann to model the tunneling process between an STM tip and a conductive sample under low-bias conditions. Physically, STM tunneling is a quantum-mechanical tunneling phenomenon described by the overlap of electronic wavefunctions across a vacuum barrier; full treatment invokes scattering theory and non-equilibrium approaches such as the Bardeen tunneling formalism and Landauer–Büttiker formalism. The Tersoff–Hamann model reduces complexity by representing the STM tip as an idealized pointlike probe with an s-wave symmetry, allowing the tunneling current to be expressed in terms of the sample's local density of states (LDOS) evaluated at the probe position. This approximation is central to interpreting topographic and spectroscopic images in studies of surface reconstruction, adsorption, and atomic-scale defects.
In the original formulation, the tunneling current I at bias V and tip position r0 is approximated as proportional to the integral of the sample LDOS ρ_s(r0, E) over the energy window set by the bias: I(V, r0) ∝ ∫_{E_F}^{E_F+eV} ρ_s(r0, E) dE, where E_F is the Fermi energy. The model assumes a structureless, spherical tip described by an s-wave state and weak tip–sample coupling so that perturbative approaches apply. Its derivation starts from the Bardeen tunneling matrix element and replaces tip wavefunctions by a decaying radial function, yielding a simplified tunneling matrix element proportional to the sample wavefunction evaluated at r0. Energy- and bias-dependent corrections, finite-temperature smearing, and tip electronic structure can be included perturbatively. The key mathematical simplification is that spatial variation of the current reflects the sample LDOS rather than detailed tip states, making direct comparison with Kohn–Sham eigenstates from DFT feasible.
The approximation underpins simulation of STM images for a wide range of systems: clean metal surfaces (e.g., Cu(111), Au(111)), semiconductor surfaces (e.g., Si(111), Si(100)), molecular adsorbates (e.g., benzene on metal substrates), and two-dimensional materials such as graphene and transition metal dichalcogenides. It is extensively used in combination with DFT codes like VASP, Quantum ESPRESSO, SIESTA, and CASTEP to generate simulated topographs and scanning tunneling spectroscopy (STS) signals. Experimental groups at institutions such as IBM Research, Max Planck Institute for Solid State Research, and university surface-science laboratories routinely employ Tersoff–Hamann-based simulation workflows to assign contrast to atomic species, identify defect states, and interpret quasiparticle interference patterns observed in STM/STS.
Extensions of the original model address finite tip structure, non-s-wave tip states (p- or d-wave symmetry), higher bias regimes, and spin-polarized tunneling. Methods incorporating explicit tip atoms or cluster models relax the point-tip assumption and can reproduce tip-dependent contrast. Time-dependent and non-equilibrium generalizations connect the approximation to nonequilibrium Green's function approaches. Limitations include failure when tip electronic structure dominates the signal, when chemical forces or strong tip–sample interactions perturb sample states (e.g., contact regime), and for inelastic tunneling processes involving vibrations or magnetic excitations. For magnetic systems, spin-polarized STM requires explicit treatment beyond the spin-averaged Tersoff–Hamann form.
The Tersoff–Hamann approximation is commonly paired with density functional theory because DFT yields the single-particle Kohn–Sham wavefunctions and LDOS required by the model. Kohn–Sham eigenstates are used to compute ρ_s(r, E) and thus simulated STM images; exchange–correlation functionals (e.g., LDA, GGA) and pseudopotential or projector-augmented wave representations influence the computed LDOS. Corrections for quasiparticle effects (e.g., via GW approximation) or strong correlations (e.g., DFT+U) may be necessary when DFT inaccurately locates electronic states. Practical workflows often combine structural relaxation with DFT and subsequent Tersoff–Hamann-based postprocessing to compare theory with STM/STS experiments.
Alternative tunneling models include the full Bardeen approach that retains explicit tip wavefunctions, atomistic tip models that compute tunneling matrix elements numerically, and non-equilibrium Green's function methods that treat transport beyond perturbation theory. Compared to these, Tersoff–Hamann is computationally inexpensive and yields transparent interpretation in many weak-coupling cases, but atomistic tip models (used by groups at institutions like CERN and national nanoscience centers) capture tip-dependent contrast and chemical identity more accurately. For inelastic and spin-dependent phenomena, methods developed in the scanning probe microscopy community, including combined DFT+NEGF and many-body treatments, are preferred when the Tersoff–Hamann assumptions break down.
Category:Scanning probe microscopy Category:Density functional theory Category:Surface science