Tersoffhamann ModelEdit

The Tersoff–HAMANN model, commonly called the Tersoff–Hamann approximation, is a foundational framework in the interpretation of scanning tunneling microscopy (STM) data. At its core, it connects the tunneling current between a metallic or semiconductor tip and a sample to the electronic structure of the sample itself, in particular to the local density of states at the position of the tip. By simplifying the complex quantum problem of tunneling to a tractable form, the model provides a practical bridge between experimental images and the underlying surface chemistry and physics. This bridge has made the approach indispensable in surface science, nanoscience, and materials physics, where understanding how electrons are arranged and how they respond at surfaces guides both fundamental research and technological development.

The model was introduced by James Tersoff and David R. Hamann in the mid-1980s as an efficient way to interpret STM signals without requiring a full, computationally intensive description of the tip–surface interaction. It builds on the Bardeen tunneling framework, which treats tunneling as a matrix element problem between the electronic states of the tip and the sample. The key insight of Tersoff and Hamann was to replace the complex, detailed structure of the tip with a simple, idealized representation: an s-wave (spherically symmetric) tip at the apex. With this simplification, the tunneling current becomes proportional to the local density of states of the sample at the tip location and near the Fermi energy. In practice, this means that STM images—whether acquired in constant-current or constant-height modes—can be interpreted as maps of the sample’s LDOS, up to a known proportionality and the geometry of the tip.

Theoretical foundations

Tunneling framework and LDOS interpretation

The Tersoff–Hamann approach rests on the idea that the STM current I at low temperature and small bias is governed by the overlap between the tip’s wavefunction and the sample’s electronic states. In the approximation, the current can be linked to the sample’s local density of states, ρ_s(r0, E), evaluated at the position r0 of the tip (and at energies near the Fermi level). This leads to the widely used practical rule: the STM image reflects the LDOS structure of the surface.
In many treatments, the bias window is incorporated, yielding I(V) that is connected to an energy integral of the LDOS from E_F to E_F + eV. The precise dependence depends on temperature, tip orbitals, and the rate at which electrons tunnel through the vacuum barrier, but the central message remains: the image encodes the spatial variation of electronic states at the surface.

Tip model and its consequences

The apex of the tip is assumed to host a single, dominant orbital with s-character. This simplifies the math and has a powerful practical consequence: the measured current becomes proportional to the LDOS of the sample at the tip position, with the proportionality independent of most details of the rest of the tip. This is why the model is so widely used: it provides a clear, testable link between experiment and electronic structure.
In real experiments, the tip can have p- or d-character contributions, and the geometry can deviate from an ideal sphere. The Tersoff–Hamann framework accommodates this by treating deviations as corrections to the simple s-wave assumption; more elaborate variants exist to handle non-spherical tips and spin polarization.

Practical implementation and connection to theory

The model is typically embedded within a broader workflow that includes first-principles calculations, most often density functional theory (DFT), to compute the surface LDOS. By pairing simulated LDOS maps with experimentally obtained STM images, researchers extract quantitative and qualitative information about surface reconstructions, adsorbates, vacancies, and electronic inhomogeneities.
Because it emphasizes LDOS, the model naturally dovetails with other electronic structure tools and concepts, such as density functional theory and local density of states. When used in conjunction with these tools, it becomes a practical algorithm for predicting and interpreting a wide range of surface phenomena.

Practical considerations and limitations

Bias and energy resolution: The basic picture assumes small bias and low temperature, so the mapping between image contrast and LDOS is most faithful near the Fermi level. Higher biases can involve more complex energy windows and may require extensions to the model.
Tip geometry and orbital character: While an s-wave tip is the standard starting point, real tips can deviate. For materials with strongly directional bonding or with surface states that couple preferentially to specific orbitals, the simple model may miss some features, and more sophisticated tip models may be needed.
Tip–sample interactions: At very close separations or with reactive surfaces, the vacuum barrier picture can break down, introducing significant tip–surface interactions that go beyond the original Bardeen-based derivation. In such cases, explicit tunneling calculations or nonperturbative methods may be warranted.
Spin polarization and magnetic surfaces: For magnetic samples or spin-polarized STM experiments, extensions of the Tersoff–Hamann framework exist to incorporate spin effects, but the basic s-wave, spin-averaged picture is incomplete.
Surface complexity: Complex surfaces, such as those with strong electron correlation, heavy adsorbates, or dynamical fluctuations, can present challenges that push the model beyond its simplest regime. In those cases, combining STM interpretation with complementary techniques (e.g., spectroscopy or neutron scattering) often yields a more complete picture.

Applications and impact

The Tersoff–Hamann model has become a staple in the interpretation of STM images of a wide range of surfaces: clean metal faces, semiconductor surfaces, graphene and other two-dimensional materials, oxide surfaces, and molecular adsorbates. Its simplicity makes it a useful first-pass tool for understanding electronic structure at atomic scales.
It supports a productive loop between theory and experiment: experimental STM topographs guide refinements in surface models, while computed LDOS maps help explain features such as vacancy states, adatoms, and surface reconstructions. This synergy accelerates progress in surface science, catalysis, and nanostructure design.
The framework has influenced the broader philosophy of how we connect measured tunneling signals to electronic structure. As with many effective simplifications, it invites a practical, hierarchical approach: start with a simple, robust model, identify where it works well, and augment with more detailed physics where necessary.

Controversies and debates

Domain of validity versus competing models: Critics sometimes argue that simplified models, while elegant, can obscure important physics in complex materials. Proponents counter that the strength of the Tersoff–Hamann approach lies in its transparency and predictive power within its regime of validity. For many common surface systems, the LDOS-based interpretation yields reliable, experimentally testable insights, and when it does not, researchers have a clear path to more elaborate modeling that incorporates additional orbital character, spin, or many-body effects.
The role of simplifications in engineering practice: Some debates focus on whether the reduction to an s-like tip and a single-orbital picture risks oversimplifying engineering problems in nanodevice design. Advocates of the model reply that simplifications are deliberate and productive: they provide a clear, quantitative link between measurable signals and electronic structure, enabling robust device concepts and materials discovery. When greater accuracy is required, the model can be extended or replaced with more detailed calculations, rather than discarded outright.
Woke critiques of science culture and model-building: In broader discourse, there are critiques that science should reflect social considerations in its priorities and methods. From a practical, outcomes-focused vantage point, the core value of the Tersoff–Hamann model is its utility: it yields reproducible, interpretable predictions that guide experiments and technology. Critics who argue that theoretical physics should foreground social or ideological concerns over technical rigor may misunderstand how scientific progress actually occurs: robust models that accurately predict and explain observations tend to be the most durable foundation for subsequent innovation. Proponents of this view emphasize that a strong, well-validated framework—like the Tersoff–Hamann approach—serves as a reliable platform for applied science, industry collaboration, and national competitiveness, while acknowledging the importance of inclusive and fair scientific communities. In short, the best critique is not to discard useful models, but to ensure the scientific enterprise remains open, rigorous, and effective at solving real-world problems.