Fowler-Nordheim Tunneling

In the microscopic world of materials, electrons are confined by an energy barrier at the surface known as the work function. Classically, overcoming this barrier requires a brute-force approach, such as heating the material to induce thermionic emission. This method, however, is energy-intensive and often impractical. This raises a fundamental question: can electrons escape a material without being given enough energy to leap over the barrier? The answer lies not in classical physics, but in the strange and powerful realm of quantum mechanics. This article explores Fowler-Nordheim tunneling, a remarkable quantum effect that provides this alternate pathway. First, in "Principles and Mechanisms," we will examine the quantum theory behind this phenomenon, explaining how a strong electric field reshapes the potential barrier and deriving the core equations that govern the process. Subsequently, in "Applications and Interdisciplinary Connections," we will see how this seemingly esoteric effect has become a cornerstone of modern technology, from digital memory to atomic-scale imaging.

Imagine you are trying to get a small ball over a very high wall. In the world of our everyday experience—the world of classical physics—there are only two ways to succeed. You can either give the ball enough initial speed to launch it clear over the top, or you can somehow lower the height of the wall. There is no third option. If the ball’s energy is less than the potential energy required to be at the top of the wall, it will never, ever get to the other side.

An electron inside a piece of metal finds itself in a similar predicament. It is free to roam within the metal, but it is confined by an energy barrier at the surface. To pull an electron out of the metal and into the vacuum requires a certain minimum amount of energy, a property of the material known as the ​​work function​​, denoted by $\Phi$. This work function acts like a wall.

The classical way to get electrons over this wall is simply to give them more energy. We can do this by heating the metal. As the temperature rises, the electrons inside jiggle around more and more violently. A few of the most energetic electrons will gain enough thermal energy to leap over the work function barrier and escape. This process is called ​​thermionic emission​​, and it’s the principle behind the glowing filaments in old vacuum tubes. It’s a battle of brute force: high temperatures are needed to overcome the high barrier.

But what if we could find another way? A more subtle, more mysterious way that doesn’t require scorching temperatures? This is where the strange and beautiful rules of quantum mechanics enter the picture.

In the quantum world, an electron is not just a tiny ball; it also behaves like a wave. And waves do something that particles cannot: they can have a presence in places they are "not supposed to be." If an electron wave encounters an energy barrier, its amplitude doesn't just drop to zero at the boundary. Instead, it decays exponentially inside the barrier. If the barrier is thin enough, the wave will have a small but non-zero amplitude on the other side. This means there is a finite probability that the electron can simply appear on the far side of the barrier, without ever having had enough energy to go "over the top." This phenomenal process is called ​​quantum tunneling​​. It is as if the electron has ghosted right through the wall.

For an electron at a metal surface with no external forces, the work function barrier is like a rectangular wall of constant height $\Phi$. While tunneling is possible in principle, the probability of an electron burrowing through this thick, tall barrier is infinitesimally small. To make tunneling a practical reality, we need to change the shape of the wall.

This is the crucial insight of Ralph Fowler and Lothar Nordheim. What happens if we apply a very strong external ​​electric field​​, $E$, pulling electrons away from the surface? The field creates an additional potential that slopes downwards, away from the metal. The total potential energy barrier that an electron sees is now a combination of the flat work function and this new downward slope.

The potential energy barrier $U(x)$ at a distance $x$ from the surface is no longer just $U(x) = \Phi$. Instead, it becomes:

where $e$ is the elementary charge. The barrier is no longer a rectangle. It has become a ​​triangle​​.

This change in shape is transformative. The barrier is now thinner at every energy level below its peak. An electron tunneling at its initial energy level doesn't have to cross the full original width; it only needs to penetrate a finite, triangular sliver of potential energy.

It's important to realize that this triangular shape is an idealization that holds true in very strong fields. At more moderate fields, where the applied voltage $V$ is less than the work function (i.e., $eV \lt \Phi$), the barrier is more accurately described as a trapezoid. The process of tunneling through this shape is called ​​direct tunneling​​. As we crank up the voltage and the field becomes stronger, we eventually reach a point where the barrier becomes triangular. This transition marks the onset of the ​​Fowler-Nordheim tunneling​​ regime, a distinct physical process that dominates at high fields.

So, how do we calculate the probability of an electron tunneling through this triangular barrier? Here, physicists employ a powerful tool from quantum mechanics known as the ​​Wentzel-Kramers-Brillouin (WKB) approximation​​. The WKB method provides a way to find approximate solutions to the Schrödinger equation for slowly varying potentials.

Intuitively, the WKB approximation tells us that the probability of tunneling, $T$, decreases exponentially with the "size" of the barrier—a quantity that depends on both its height and its width. The core of the calculation involves an integral across the classically forbidden region (the part of the barrier the electron tunnels through). For our triangular barrier, the calculation yields a beautifully simple and profound result. The tunneling probability is dominated by an exponential factor:

where $m$ is the electron's mass and $\hbar$ is the reduced Planck constant.

Look closely at this formula. It is the very soul of the Fowler-Nordheim mechanism. The most important feature is the electric field, $E$, in the denominator of the exponent. This means that the tunneling probability is extraordinarily sensitive to the electric field. Doubling the electric field doesn't just double the current; it increases it by many orders of magnitude because it dramatically shrinks the exponent. The term $\Phi^{3/2}$ also tells us that a higher work function makes tunneling much harder, which makes perfect sense.

To get the full expression for the tunneling current density, $J$, we must also account for the number of electrons available to tunnel and other quantum factors at the surface. This contributes a pre-exponential factor of $E^2$. Combining these parts gives us the celebrated ​​Fowler-Nordheim equation​​:

Here, $A$ and $B$ are constants that depend on the work function $\Phi$ and other fundamental constants. The constant $B$ is essentially the collection of terms multiplying $1/E$ in our tunneling probability exponent.

A beautiful theory is one thing, but how do we know it's actually happening in a real device? We must look for its unique signature, its "fingerprint" in the experimental data. The Fowler-Nordheim equation provides just such a fingerprint.

If we take the natural logarithm of the equation and rearrange it, we get:

This has the form of a straight line, $y = c + mx$. If we measure the current density $J$ at various electric fields $E$ and plot $\ln(J/E^2)$ on the vertical axis against $1/E$ on the horizontal axis, we should get a straight line with a negative slope equal to $-B$. This specific graphical representation is known as a ​​Fowler-Nordheim plot​​, and its linearity is the smoking gun for FN tunneling.

This ability to identify a mechanism by its characteristic plot is a powerful technique in physics. It allows us to distinguish Fowler-Nordheim tunneling from a whole host of other ways charge can move through a material.

• ​​Direct Tunneling (DT)​​, which occurs through thinner, trapezoidal barriers, does not produce a linear FN plot.
• ​​Schottky Emission​​ and ​​Poole-Frenkel Emission​​ are both thermally activated processes, meaning their currents depend strongly on temperature. In contrast, FN tunneling is largely athermal. These thermal processes also have their own distinct signature plots, typically involving $\sqrt{E}$, which are different from the $1/E$ dependence of FN tunneling.

By comparing the experimental data to these different theoretical models, we can confidently identify the underlying physics at play.

This ghostly quantum effect is not just a physicist's curiosity; it is a workhorse of modern technology. Every time you save a file on a USB flash drive or a solid-state drive (SSD), you are using Fowler-Nordheim tunneling. These devices store information in memory cells that contain a "floating gate," a tiny island of conductive material completely surrounded by a thin insulating layer of oxide.

To write a "1" or a "0," a high voltage (e.g., 12 volts) is applied across the thin oxide layer (e.g., 8 nanometers thick). This creates a colossal electric field—on the order of $10^9$ V/m! Under this intense field, the oxide's potential barrier becomes triangular, and electrons from the semiconductor are compelled by FN tunneling to pass through the supposedly impenetrable insulator and get trapped on the floating gate. This process can happen in a matter of microseconds. The absence or presence of this trapped charge is then read as a digital bit. The entire digital world, in a very real sense, is built upon a foundation of quantum tunneling.

Of course, the real world is always a little messier—and more interesting—than our simplest models. The perfect triangular barrier is an excellent approximation, but it neglects a subtle effect. As an electron begins to leave the conductive metal surface, it induces an "image charge" behind it within the metal, much like your face induces an image in a mirror. This positive image charge pulls the electron back, slightly modifying the potential energy.

This ​​image-force barrier lowering​​ has the effect of rounding off the sharp corner of the triangular barrier and slightly lowering its peak height. The consequence of this is that the Fowler-Nordheim plot is not a perfectly straight line. It exhibits a slight upward curvature, especially at very high fields.

Does this mean the theory is wrong? Not at all! It means our first model was a brilliant simplification, and now we can refine it. Physicists have developed more advanced "Schottky-Nordheim" models that account for the image force. By using these more sophisticated equations, we can correct for the curvature and extract even more accurate information about a material's properties from the tunneling current. This process of starting with a simple, powerful idea and gradually adding layers of refinement to match reality more closely is the very essence of scientific progress.

We have spent our time understanding the strange, almost ghostly, passage of an electron through a solid wall that, by all classical reasoning, it should never be able to penetrate. This quantum mechanical phenomenon, Fowler-Nordheim tunneling, might at first seem like a mere curiosity, a subtle bit of weirdness confined to the blackboard. But what is it good for? It turns out this phantom process is not just a curiosity; it is the silent, tireless engine behind some of our most advanced technologies and a crucial character in the unfolding drama of the nanoworld. Its principles reach far beyond the idealized physics problem, forming a bridge to electrical engineering, materials science, chemistry, and beyond.

Perhaps the most ubiquitous application of Fowler-Nordheim tunneling is resting in your pocket or on your desk right now. Every flash drive, solid-state drive (SSD), and even the memory in your smartphone relies on this quantum effect to store the bits and bytes that constitute our digital lives. The core of this technology is a remarkable device known as a floating-gate transistor. Imagine a tiny, electrically isolated island of conducting material—the floating gate—buried within an insulator, like a ship in a bottle. This island is separated from the main transistor channel by a very thin layer of oxide. To store a '1', we need to place electrons on this island; to store a '0', we leave it empty (or remove them).

But how do you get electrons onto an island surrounded by a seemingly impenetrable insulating wall? You can't just open a door. Instead, you must do something much more clever: you apply a strong voltage to a nearby "control gate." This high voltage creates an immense electric field across the thin oxide layer, on the order of megavolts per centimeter. The field is so strong that it drastically warps the energy landscape of the insulating barrier, transforming it from a wide, rectangular wall into a steep, triangular slide. Under these conditions, electrons in the transistor channel don't need to climb over the barrier; they simply tunnel through it, via the Fowler-Nordheim mechanism, and become trapped on the floating gate. This process of forcing charge through the barrier is how the memory cell is "programmed." The amount of charge that tunnels onto the gate can be precisely controlled by the magnitude and duration of the voltage pulse, which in turn sets the transistor's threshold voltage. This shift in threshold voltage is what the device's circuitry reads as a stored piece of information. The same principle is now being explored for next-generation devices, from implantable medical recorders to the analog memory elements of "neuromorphic" computers that mimic the human brain.

What if, instead of using this tunneling current to trap electrons, we used it to see? This is the revolutionary idea behind the Scanning Tunneling Microscope (STM), an instrument so powerful it can image individual atoms on a surface. In its most common mode of operation, an atomically sharp metal tip is brought incredibly close to a conducting surface, and a small voltage is applied. Electrons "direct tunnel" across the vacuum gap, creating a tiny current that is exquisitely sensitive to the tip-to-sample distance.

But what happens if we crank up the voltage? As the bias increases such that the applied voltage becomes comparable to the work function of the material—the energy required to pluck an electron out of the metal—the shape of the vacuum barrier warps. It thins and tilts, becoming triangular. The electrons no longer just hop across a narrow gap; they are ripped out of the tip by the intense field in a process of field emission, and we enter the Fowler-Nordheim regime.

Here, another fantastic piece of physics comes into play: ​​field enhancement​​. As any student of electromagnetism knows, electric field lines bunch up at sharp points. The local field at the apex of the STM tip can be hundreds or even thousands of times stronger than the average field you would calculate by simply dividing the voltage by the distance. This is the same reason a lightning rod works so well. This geometric effect means that even a modest applied voltage can create the enormous local fields needed for FN tunneling.

This transition is not just a curiosity; it's a powerful analytical tool. By carefully measuring the tunneling current as a function of high voltage and plotting the data in a specific way—a graph of $\ln(I/V^2)$ versus $1/V$, known as a Fowler-Nordheim plot—we can learn a great deal. The slope of this plot is directly related to the material's work function $\phi$ and the field enhancement factor $\beta$. If we know the work function of our tip material, we can use the slope of the FN plot to calculate the effective radius of our tip, giving us a measure of its sharpness!. It is a beautiful example of how a deep understanding of a physical model allows us to turn electrical measurements into geometric information about an object far too small to see.

So far, we've seen Fowler-Nordheim tunneling as a tool we can harness. But like a mischievous gremlin in the machine, this quantum effect can also cause trouble, setting fundamental limits on how small we can build our electronic devices.

Consider the humble capacitor. As we shrink our circuits, the insulating layers that separate conducting elements must also become thinner. A nanoscale capacitor with plates separated by just a few nanometers can have an immense electric field between them even with a small voltage. This field can be strong enough to induce FN tunneling directly through the "insulator," causing the capacitor to slowly leak its charge and discharge itself over time. This quantum leakage is a fundamental source of power drain and a major headache for chip designers, representing a physical wall against infinite miniaturization. The very effect that makes our memory work can undermine the integrity of other components. Understanding the crossover from low-field direct tunneling to high-field FN tunneling is therefore critical for predicting the reliability and lifetime of modern dielectrics.

Yet, in a wonderful twist, this "nuisance" effect can sometimes play the role of a hero. In the high-tech world of semiconductor manufacturing, plasma etching is used to carve microscopic trenches into silicon wafers. A strange problem called "electron shading" can occur in deep, narrow trenches: because positive ions in the plasma are directional and electrons are not, the bottom of the trench can get bombarded with more positive charge than negative charge. This positive charge accumulates on the insulating floor of the trench, creating a large, potentially damaging voltage. But as this voltage builds, so does the electric field across the thin insulator. Eventually, the field becomes strong enough to trigger a Fowler-Nordheim leakage current, allowing electrons from the silicon below to tunnel up and neutralize the accumulating positive charge. The system reaches a steady state where the ion current is perfectly balanced by the FN tunneling current. Tunneling acts as a natural, self-regulating pressure-release valve, preventing catastrophic damage to the device during its own fabrication.

The influence of Fowler-Nordheim tunneling does not stop at the boundaries of physics and electrical engineering. Its extreme sensitivity to the electronic properties of a surface makes it a powerful probe for chemistry and materials science.

The FN tunneling current depends very strongly on the work function $\phi$, which we've described as the energy price to extract an electron. This "price" can be dramatically altered by just a single layer of molecules adsorbed onto the emitting surface. These molecules can form a sheet of tiny electric dipoles that creates an additional electric field at the surface, either helping or hindering the escape of electrons. This change in work function, in turn, causes an exponential change in the field emission current. This principle is the basis for highly sensitive gas detectors and provides a way to study surface chemistry, catalysis, and contamination in vacuum systems.

Finally, let's return to the concept of field enhancement, but in the cutting-edge context of memristors—often touted as a building block for brain-like computers. In some of these devices, the switching mechanism involves the formation and retraction of a tiny metallic filament. The end of this filament can be atomically sharp, acting like a nanoscale lightning rod. When a voltage is applied, it is much easier to trigger FN tunneling from the sharp filament tip than it is to trigger it from the opposing flat electrode. This purely geometric effect creates a profound asymmetry in the current flow; the device acts as a rectifier, allowing current to flow easily in one direction but not the other. This inherent, quantum-mechanical asymmetry, born from geometry, is a key feature governing the behavior of these next-generation computing elements.

From the bits in our computers to the eyes that see atoms, from a parasitic leak to a manufacturing savior, Fowler-Nordheim tunneling is a testament to the profound and practical consequences of fundamental quantum mechanics. It is a single, beautiful thread that weaves together the disparate fields of modern science and technology.