The Tersoff-Hamann Approximation

The Scanning Tunneling Microscope (STM) provides astonishingly detailed images of the atomic world, but it is not a conventional camera. It operates on the strange principles of quantum mechanics, "seeing" with a flow of electrons that tunnel across a vacuum gap. Understanding this process is key to deciphering the rich information contained in STM images. While a complete theory of tunneling exists, it is mathematically complex and doesn't offer an immediate, intuitive interpretation of the resulting images. This is the knowledge gap that the Tersoff-Hamann approximation elegantly fills, providing a simple yet powerful framework for understanding what an STM truly measures.

This article explores this foundational model. The first section, ​​"Principles and Mechanisms"​​, will unpack the core idea of the approximation—the direct link between tunneling current and the Local Density of States (LDOS)—and explore its use in both topographic imaging and spectroscopy. We will also examine its limitations and the fascinating imaging rules that emerge when using more complex tips. Following that, the section on ​​"Applications and Interdisciplinary Connections"​​ will showcase how this model is used in practice, from visualizing atomic orbitals and solving materials science puzzles to monitoring chemical reactions and bridging the gap between computational theory and experimental reality. Let us begin by exploring the elegant physics that underpins this powerful approximation.

So, we have this marvelous machine, a Scanning Tunneling Microscope, that gives us postcard pictures of atoms. But how? It’s not a camera in any conventional sense. It doesn't use light. The "seeing" is done by a delicate quantum mechanical conversation between a sharp metal tip and the surface it hovers over. This conversation is carried by a tiny trickle of electrons, a ​​tunneling current​​, that 'jumps' across the vacuum gap—a region that classical physics would call absolutely forbidden territory.

Imagine you're trying to throw a ball over a very high wall. Classically, if you don't throw it high enough, it will never get to the other side. But in the strange world of quantum mechanics, the ball—our electron—has a fuzzy, wave-like nature. Its presence doesn't just stop at the wall; it subsides, decaying exponentially into the forbidden zone. If the wall is thin enough (in our case, the vacuum gap is just a few angstroms wide), there’s a non-zero chance the electron's wave will emerge on the other side, and poof—the electron appears, as if it tunneled right through.

The rate of this tunneling, which determines the current we measure, depends on two crucial factors: the number of available "launching pads" (occupied electronic states) on one side, and the number of available "landing zones" (unoccupied electronic states) on the other. The central task of any theory of STM is to connect this abstract quantum process to the beautiful images we see.

The full, rigorous theory of this process, first laid out by John Bardeen, is a masterpiece of quantum mechanics. It involves calculating a ​​tunneling matrix element​​ by integrating the wavefunctions of the tip and sample over a surface in the vacuum gap. It’s powerful and general, but it's also wonderfully complex and doesn't immediately tell you what a picture of a carbon atom will look like.

This is where the genius of J. Tersoff and D. R. Hamann came in. They made a brilliant simplification. They asked: what is the simplest, most perfect tip we can possibly imagine? Let's model it as a single atom at the apex, with a perfectly symmetric, spherical ​​$s$-wave orbital​​. It's the quantum equivalent of a perfectly round ballpoint pen. Furthermore, they looked at the situation in the limit of a very small voltage difference between the tip and sample.

With these assumptions, the formidable mathematics of Bardeen's formalism collapses into something astoundingly simple and intuitive. The tunneling current, they found, is directly proportional to a quantity called the ​​sample's Local Density of States (LDOS)​​, evaluated right at the position of the tip.

What on earth is the Local Density of States? Let's unpack the term. "Density of States" tells you how many electron states are available at a particular energy. But the Local part is the key. The LDOS, denoted $\rho_s(\mathbf{r}, E)$, tells you not just how many states there are at energy $E$, but where they are in space. Formally, we define it as:

This equation might look a bit scary, but the idea is simple. For every electron eigenstate $\psi_n$ in the sample, we check its energy $E_n$. If its energy is the one we're interested in ($E$), we take its probability density $|\psi_n(\mathbf{r})|^2$ at the location $\mathbf{r}$ and add it to our total. The LDOS is therefore a map of electron probability at a specific energy. The ​​Tersoff-Hamann approximation​​, in its essence, states that an STM with a simple $s$-wave tip is a machine that directly measures this map. The image contrast we see is a direct picture of the electron clouds of the surface atoms.

This simple proportionality is the Rosetta Stone for interpreting STM data. It gives us two powerful ways to explore the atomic world.

Topographic Imaging

This is the most common way we use an STM. The microscope scans the tip laterally across the surface, while a feedback loop constantly adjusts the tip's height ($z$) to keep the tunneling current fixed. What does this mean in the Tersoff-Hamann picture? Since the current is proportional to the LDOS, a contour of constant current is a contour of constant LDOS. The motion of the tip literally traces out the shape of the electron clouds.

For a very small bias voltage, the tunneling electrons come from just below the Fermi level and go to just above it. The resulting image is essentially a map of the LDOS at the Fermi energy, $\rho_s(\mathbf{r}, E_F)$. Imagine scanning an atom with a $d_{z^2}$ orbital, which looks like a dumbbell with a donut around its middle. According to the model, where the orbital's wavefunction magnitude $|\psi(\mathbf{r}_{\text{tip}})|$ is large, the current will be high, and the tip will retract. Where it's small, the tip will move closer. The resulting topographic image will look just like the shape of that orbital. This direct link between the measured topography and the shape of atomic orbitals is the 'miracle' of the Tersoff-Hamann approximation.

Scanning Tunneling Spectroscopy (STS)

But what if we're more curious? What if we want to know about the electronic states at energies other than the Fermi level? We can do that, too. Instead of scanning, we hold the tip fixed at one interesting spot $(\mathbf{r}_0)$ and slowly ramp the bias voltage $V$. By measuring how the current $I$ changes with voltage, we can map out the electronic spectrum of that specific point on the surface.

The key quantity we measure is the ​​differential conductance​​, $dI/dV$. And here lies the second miracle of the Tersoff-Hamann model. Under the right conditions, this quantity is directly proportional to the sample's LDOS at an energy $E_F + eV$:

This means by sweeping the voltage, we are sweeping through the energy levels of the sample, and our $dI/dV$ measurement tells us exactly how many states are available at each energy.

How does this magic work? The trick lies in the properties of the electrons' energy distribution, the ​​Fermi-Dirac distribution​​, at low temperatures. A full derivation shows the current is an integral involving the LDOS and the difference in the Fermi functions of the tip and sample. When we take the derivative with respect to voltage, a term involving the derivative of the Fermi function, $-\frac{\partial f}{\partial E}$, emerges. At low temperature, this derivative behaves like a very narrow peak, acting as a precise "energy filter" or sampling window. It picks out the value of the LDOS at precisely the energy determined by the bias voltage.

Of course, such a simple and beautiful result doesn't come for free. It relies on the same strict assumptions:

1. ​​Low Temperature:​​ To ensure our "energy filter" is sharp. Thermal energy would smear it out, blurring our energy resolution.
2. ​​A "Boring" Tip:​​ We need our idealized $s$-wave tip, which must also have a flat, featureless density of states of its own, so it doesn't imprint its own structure on the measurement.
3. ​​Small Bias:​​ The approximation assumes the tunneling probability doesn't change much with energy, which holds best for small bias voltages.

When these conditions are met, STS gives us an incredibly powerful tool to perform spectroscopy on a single atom or molecule.

The Tersoff-Hamann model is beautiful, but reality is often more interesting. What happens when our tip is not a perfect, featureless sphere? What if the apex atom has a directional orbital, like a $p_z$ or $d_{xz}$ state? This can happen, for instance, if the tip accidentally picks up a molecule like carbon monoxide from the surface.

When the tip's orbital has a more complex shape, the simple proportionality to the LDOS breaks down. A new, richer set of rules emerges, often called the ​​"derivative rule"​​. The idea is that the tunneling matrix element becomes sensitive not just to the value of the sample's wavefunction, but to its spatial derivatives.

• An ​​$s$-wave tip​​ measures $\propto |\psi_s|^2$.
• A ​​$p_z$-wave tip​​ (with its lobe pointing down) measures $\propto |\frac{\partial \psi_s}{\partial z}|^2$.
• A ​​$p_y$-wave tip​​ (lobe pointing along y) measures $\propto |\frac{\partial \psi_s}{\partial y}|^2$.

This has a profound and fascinating consequence. Imagine a molecular orbital that has a nodal plane, a surface where the wavefunction is exactly zero. An $s$-wave tip scanning over this node would measure zero current, creating a dark line in the image. But right at a node, where the function passes through zero, its derivative can be at a maximum! A $p$-wave tip, being sensitive to the derivative, would therefore measure a maximum current at the node. It makes the invisible visible.

Consider imaging a simple, spherically symmetric $s$-orbital adatom with a $p_z$ tip. The sample wavefunction $\psi_s$ is maximum at the center. But its vertical derivative, $\frac{\partial\psi_s}{\partial z}$, is zero at the center (by symmetry) and peaks in a ring around it. The resulting STM image would show a donut shape, a "tip-induced artifact" that actually reveals the $p_z$ nature of our probe!

This establishes a kind of ​​selection rule​​ for STM imaging. The symmetry of the tip orbital interacts with the symmetry of the sample orbital to determine the final image. A $p_x$ tip might be completely blind to a feature that a $p_y$ tip sees brightly. This complexity, once a puzzle, is now a powerful tool for chemists and physicists to identify the precise symmetry and character of orbitals on a surface.

This journey from a simple proportionality to a complex set of derivative rules shows the depth of physics hidden in an STM image. And the story doesn't even end there.

When we apply a ​​finite bias​​ of, say, 1 Volt, we are no longer probing a single energy level. We are collecting electrons from a 1 eV-wide energy window. The resulting image is a superposition of all the electronic orbitals within that window. This is why STM images can change so dramatically with the magnitude and polarity of the voltage: a positive bias probes unoccupied states (like the LUMO), while a negative bias probes occupied states (like the HOMO), and these orbitals almost never have the same shape.

And if the tip gets too close to the sample, the ​​weak coupling​​ assumption fails. The tip and sample orbitals hybridize, forming a single quantum entity. The image then reflects the properties of this new, combined system, with complex phenomena like resonant tunneling taking over.

What began as a simple approximation—the idea that an STM image is a direct photo of electron clouds—unfolds into a much richer story. By understanding the principles of how the tip and sample "talk" to each other, we learn that the STM is not just a camera, but a deeply interactive quantum instrument, capable of revealing not just where atoms are, but the intricate shape, symmetry, and energy of the very orbitals that bind them together.

Now that we have grappled with the machinery of the Tersoff-Hamann approximation, you might be asking a perfectly reasonable question: What is it for? Is it just a clever piece of theoretical physics, an elegant formula destined to live only on blackboards? The answer, I am delighted to tell you, is a resounding no. This approximation is not an end in itself; it is a key. It is the miraculous key that has unlocked a hidden universe, transforming the Scanning Tunneling Microscope (STM) from a mere surface profiler into a true quantum-world explorer. It gives us the power to translate the abstract language of wavefunctions and energy levels into astonishingly direct pictures of reality at the atomic scale. It’s our Rosetta Stone for the nanoscale, and with it, we will now embark on a journey to see what wonders it has revealed.

The first and most famous promise of STM is that it lets us "see" atoms. But what are we truly seeing? If you imagine atoms as tiny billiard balls, the Tersoff-Hamann model immediately disabuses you of this notion. The model tells us that the tunneling current, and thus the apparent height in an STM image, is proportional to the local density of states (LDOS)—essentially, a measure of how much "room" there is for an electron at that particular point in space and energy. An STM image is not a photograph of solid objects; it is a map of electron probability.

This leads to a profound and sometimes counterintuitive consequence: the height you measure in an STM image is not necessarily the true geometric height. Imagine an adatom sitting on a flat surface. You might expect it to always appear as a bump of a certain height. However, the apparent height depends critically on the electronic character of that adatom relative to the substrate underneath. If the adatom has a high density of electronic states available for tunneling at the bias voltage you've chosen, it will appear as a prominent peak. But if its electronic states are sparse in that energy window, it might appear as only a small bump, or even a depression, regardless of its physical size! The STM measures electronic hills and valleys, not just geometric ones. This principle allows us to distinguish between different types of atoms on a surface not just by their position, but by their unique electronic signatures, which manifest as different apparent heights.

The story gets even better. Since the LDOS is built from the quantum mechanical wavefunctions of the electrons, we can, in the right circumstances, see the distinct shapes of the orbitals themselves! We learn in chemistry class that electrons in atoms occupy orbitals with beautiful, characteristic shapes: spherical $s$-orbitals, dumbbell-shaped $p$-orbitals, and so on. These are not just mathematical abstractions. With an STM, you can literally image them. For example, if you were to scan over a surface state described by a $p_x$'-like orbital, the Tersoff-Hamann model predicts that the STM image would not be a single bump centered on the atom. Instead, it would reveal two distinct lobes of high current, aligned along the x-axis, with a line of near-zero current in between—a direct visualization of the nodal plane of the $p$'-orbital. For the first time, we could see the ghostly shapes that dictate the rules of chemical bonding.

This ability to see electronic structure is an incredibly powerful tool for solving real-world materials science puzzles. Consider the famous case of the silicon (100) surface, the workhorse of the semiconductor industry. To minimize its energy, the atoms on its surface reconstruct, forming pairs called dimers. For years, there was a debate about their exact structure: were they symmetric, or were they "buckled," with one atom of the pair pushed up and the other pulled down? STM, interpreted through the lens of Tersoff-Hamann, settled the debate beautifully. By applying a negative bias, experimenters could image the filled electronic states—which, on a buckled dimer, are localized on the "up" atom. They saw one bright spot per dimer. Then, by switching to a positive bias, they imaged the empty states—which are localized on the "down" atom. The bright spot on each dimer moved! This bias-dependent contrast reversal was the smoking gun for the buckled structure and a triumph for the idea that STM images are maps of the LDOS.

So far, we have used the STM as a camera, taking snapshots at a fixed energy. But what if we could use it as a kind of stethoscope, to listen to the electronic "heartbeat" of the surface? This is the technique of Scanning Tunneling Spectroscopy (STS). Instead of just keeping the current constant, we can hold the tip at one spot, sweep the bias voltage $V$, and measure how the current changes. The derivative, $dI/dV$, according to the Tersoff-Hamann model, is directly proportional to the LDOS at the energy $E = eV$. By doing this, we can map out the entire spectrum of available electron states at any point on the surface.

This technique opens up a new world of information. When an atom is adsorbed on a surface, it doesn't just sit there; it electronically couples to the substrate, and its sharp energy levels can broaden into resonances. STS can measure the shape of these resonances with exquisite precision. The exact shape of the peak—whether it's a symmetric Lorentzian or an asymmetric Fano lineshape—tells a detailed story about the quantum mechanical interaction between the adatom and the sea of electrons in the substrate.

With this spectroscopic ability, we can move beyond single atoms and start to probe the collective behavior of electrons. On the surface of a metal, electrons behave like a "gas." If you introduce a single defect—a tiny impurity atom, for example—it acts like a stone thrown into a pond. The electron gas ripples. These ripples are not waves of water, but standing waves of electron probability, known as Friedel oscillations. STM can image these beautiful, concentric rings of high and low electron density emanating from the defect. The Tersoff-Hamann model shows us how to connect the wavelength of these ripples directly to a fundamental property of the metal: the Fermi wavevector, $k_F$. We are, in effect, seeing a quantum interference pattern written in the very fabric of the material.

Sometimes, the electrons on a surface decide to organize themselves in a much more dramatic fashion. In certain materials, below a critical temperature, the electron gas itself can spontaneously form a static, periodic wave of charge density—a Charge-Density Wave (CDW). It's as if the electronic fluid has frozen into a crystalline state. The Tersoff-Hamann framework predicts that both topographic and spectroscopic STM measurements should reveal this modulation directly. A topographic image shows a periodic corrugation, and a $dI/dV$ map shows a periodic modulation in the LDOS. Even more remarkably, STM can see defects in this electronic crystal. A "phase slip," where the wave pattern shifts abruptly, appears as a line across which the crests and troughs of the CDW are interchanged—a topological defect in a purely electronic order, made visible for us to see.

The power of our magic key is not limited to just visualising electron charge. What about spin, the intrinsic magnetic moment of the electron? By replacing the standard metallic tip with a magnetic one, we can create a Spin-Polarized STM (SP-STM). Now, the tunneling current doesn't just depend on the LDOS, but also on the relative alignment between the tip's magnetization and the local spin polarization of the sample's electrons. The Tersoff-Hamann model is readily extended to include this spin-dependent term. Suddenly, we can map magnetism atom-by-atom. We can visualize the swirling structure of a magnetic domain wall or the alternating spin arrangement in an antiferromagnet, revealing the rich textures of the magnetic world with nanoscale resolution.

The interdisciplinary reach of this simple model is astonishing. Let's turn to chemistry. Can we watch a chemical reaction? In a way, yes. Imagine individual molecules sitting on a surface, undergoing a reaction that transforms them from a reactant R into a product P. This transformation inevitably changes the molecule's frontier orbitals—its highest occupied and lowest unoccupied molecular orbitals (HOMO and LUMO). This change in electronic structure means a change in the LDOS. By the Tersoff-Hamann model, this translates directly into a change in the apparent height in an STM image. By setting the bias voltage just right, we can make the product appear significantly "taller" than the reactant, allowing us to pick them out by eye and monitor the progress of a surface-catalyzed reaction one molecule at a time.

Our understanding has even become sophisticated enough to engineer the imaging process itself. The "simple" $s$-wave tip of the original model is an idealization. Real tips have structure. In a stunning display of experimental ingenuity, scientists can pick up a single carbon monoxide (CO) molecule and place it on the apex of their STM tip. This CO-functionalized tip has an effective frontier orbital with $p$'-like symmetry. A $p$'-orbital has a node and odd parity. This completely changes the rules of imaging! Instead of being sensitive to the sample wavefunction $\psi_S$ itself, the tunneling current becomes proportional to the square of its lateral gradient, $|\nabla \psi_S|^2$. This means the tip will measure zero current where the wavefunction is at a maximum (like over an atom) and maximum current where the wavefunction is changing most rapidly (like over a chemical bond). The resulting image is a "contrast-inverted" picture of the molecule, highlighting the bonds instead of the atoms and providing breathtaking sub-molecular resolution.

Finally, the Tersoff-Hamann approximation serves as the crucial bridge between the worlds of computational materials science and experimental reality. Using powerful techniques like Density Functional Theory (DFT), a theorist can sit at a computer and calculate, from first principles, the electronic structure of a hypothetical surface. This calculation yields the energy- and position-dependent LDOS, $\rho(\mathbf{r}, E)$. This is exactly the input needed for the Tersoff-Hamann model. The theorist can then use the model to simulate what the STM image of that surface should look like. This allows for a direct, quantitative comparison between theory and experiment. If the simulated and measured images match, it provides powerful confirmation of our understanding of the surface's atomic and electronic structure. This synergistic loop—predict with DFT, simulate with Tersoff-Hamann, and verify with STM—is at the very heart of modern surface science.

From a conceptually simple starting point—that the tunneling current is proportional to the local density of states—we have unlocked a universe. We have seen the shape of orbitals, solved atomic structures, listened to quantum resonances, mapped the ripples in an electron sea, visualized electronic and magnetic crystals, watched molecules react, and created a powerful synergy with computational prediction. The Tersoff-Hamann approximation is a beautiful example of how a simple, intuitive physical model can provide a profoundly deep and versatile window into the workings of nature.