Case Study 3.1: Quantum Tunneling in Technology — From the STM to Flash Memory

Introduction

Quantum tunneling is not a theoretical curiosity confined to textbook problems. It is the operating principle behind technologies that generate hundreds of billions of dollars in economic value every year. In this case study, we trace the tunneling effect from the physics of Chapter 3 through three technologies that depend on it: the scanning tunneling microscope (STM), flash memory, and tunnel diodes. Each technology exploits a different aspect of the tunneling formula $T \approx e^{-2\kappa d}$, and each represents a different way that "genuinely weird" quantum mechanics powers real-world engineering.

The Scanning Tunneling Microscope: Seeing Atoms

The Problem

In the early 1980s, surface science was stuck. X-ray diffraction could reveal crystal structure, and electron microscopy could image features down to a few nanometers, but nobody could see individual atoms on a surface in real space. The diffraction limit of light ($\lambda \sim 500\,\text{nm}$) is a thousand times larger than an atom ($\sim 0.1\,\text{nm}$). How do you see something far smaller than your probe?

The Tunneling Solution

Gerd Binnig and Heinrich Rohrer at IBM Zurich realized in 1981 that the exponential sensitivity of quantum tunneling could be turned into an imaging tool. Their insight: bring a sharp metal tip to within about 1 nm of a conducting surface and apply a small voltage. Electrons tunnel across the vacuum gap, producing a measurable current.

The tunneling current depends on the tip-surface distance $d$ as:

$$I \propto e^{-2\kappa d}, \quad \kappa = \frac{\sqrt{2m\phi}}{\hbar},$$

where $\phi$ is the effective work function (typically 4–5 eV for metals). For $\phi = 4\,\text{eV}$:

$$\kappa = \frac{\sqrt{2(9.109 \times 10^{-31})(4 \times 1.602 \times 10^{-19})}}{1.055 \times 10^{-34}} = 1.025 \times 10^{10}\,\text{m}^{-1}.$$

The decay length is $1/(2\kappa) = 0.049\,\text{nm}$. This means that a change in distance of just $0.05\,\text{nm}$ — about one-third the diameter of an atom — changes the tunneling current by a factor of $e \approx 2.72$.

This extraordinary sensitivity is the key: by keeping the tunneling current constant (using a feedback loop that adjusts the tip height as it scans across the surface), the tip traces a path that follows the atomic-scale topography. The result is an image of the surface with sub-angstrom resolution.

The Physics in Action

The first STM images of the silicon 7×7 reconstructed surface (1983) were a sensation. For the first time, individual atoms were visible as bumps in a real-space image. Binnig and Rohrer received the 1986 Nobel Prize in Physics.

Modern STMs routinely achieve: - Vertical resolution: 0.01 nm (10 pm) — one-tenth the radius of a hydrogen atom. - Lateral resolution: 0.1 nm — sufficient to see individual atoms. - Spectroscopic capability: By varying the bias voltage, the STM maps the local density of states (the energy-resolved electronic structure) at each point on the surface.

The STM does not "see" atoms with light. It feels them with tunneling electrons. The image is a map of the local electronic structure, not a photograph. This distinction matters: what the STM images as "bumps" are regions of high tunneling probability, which correlate with (but are not identical to) the atomic positions.

Quantitative Verification

Consider the specific numbers for a typical STM experiment:

  • Work function: $\phi = 4.5\,\text{eV}$
  • Tip-surface distance: $d = 0.6\,\text{nm}$
  • Bias voltage: $V_{\text{bias}} = 100\,\text{mV}$

The tunneling probability per electron is:

$$T = e^{-2\kappa d} = e^{-2(1.085 \times 10^{10})(6 \times 10^{-10})} = e^{-13.0} \approx 2.3 \times 10^{-6}.$$

With a tunneling current of $\sim 1\,\text{nA}$ and electron charge $e = 1.6 \times 10^{-19}\,\text{C}$, that is about $6.25 \times 10^9$ electrons per second. Each electron that arrives at the gap has only a 1-in-430,000 chance of tunneling through, but with the enormous number of attempts per second, a measurable current flows.

Flash Memory: Storing Data with Tunneling

The Architecture

Flash memory — the storage technology in USB drives, SSDs, and smartphones — stores data using trapped electrons on an electrically isolated "floating gate." Writing and erasing data requires moving electrons through a thin oxide barrier (~8–10 nm of SiO$_2$), and this transport occurs by quantum tunneling.

The basic unit is a floating-gate transistor:

  1. A thin tunnel oxide (SiO$_2$, ~8 nm) separates the silicon channel from the floating gate.
  2. The floating gate is a conductor surrounded by insulator — electrons placed there are trapped indefinitely (retention time > 10 years).
  3. A control gate above the floating gate applies voltage to facilitate tunneling.

Writing: Fowler-Nordheim Tunneling

To write data (store electrons on the floating gate), a high voltage (~20 V) is applied to the control gate. This creates a strong electric field across the tunnel oxide, effectively reducing the barrier height and width. The tunneling mechanism is Fowler-Nordheim tunneling — tunneling through a triangular barrier.

For a triangular barrier created by electric field $\mathcal{E}$:

$$T \propto \exp\left(-\frac{4\sqrt{2m}\phi^{3/2}}{3e\hbar\mathcal{E}}\right),$$

where $\phi$ is the oxide barrier height ($\sim 3.1\,\text{eV}$ for Si/SiO$_2$). The field $\mathcal{E} \sim 10\,\text{MV/cm}$ makes tunneling probable enough to charge the floating gate in microseconds.

Reading: Threshold Voltage Shift

Stored electrons on the floating gate shift the threshold voltage of the transistor. A read operation simply checks whether the transistor turns on at a reference voltage:

  • Electrons present (charged floating gate): higher threshold → transistor OFF → bit = 0.
  • No electrons (empty floating gate): lower threshold → transistor ON → bit = 1.

The read operation does not involve tunneling — it is a classical electrical measurement. Only writing and erasing require tunneling.

The Engineering Challenge

Flash memory represents a remarkable balancing act: - The tunnel oxide must be thin enough for tunneling to occur during write/erase ($\sim 10^{-6}\,\text{s}$). - The tunnel oxide must be thick enough that stored electrons do not leak out ($> 10$ years). - The ratio of these timescales is $\sim 10^{14}$, which is achievable because $T \propto e^{-2\kappa d}$ and the electric field during writing dramatically changes the effective barrier.

Modern flash memory packs 100+ billion floating gates on a single chip. Each one is a quantum tunneling device.

Tunnel Diodes: Negative Resistance from Tunneling

The Esaki Diode

In 1957, Leo Esaki (then at Sony) discovered that heavily doped p-n junctions exhibited a peak in current at low forward bias, followed by a decrease — negative differential resistance. This was the first solid-state tunneling device, and Esaki received the 1973 Nobel Prize for it.

In a tunnel diode, the p-n junction is so narrow ($\sim 10\,\text{nm}$) and the doping so heavy that the conduction band on the n-side overlaps energetically with the valence band on the p-side. At low forward bias, electrons can tunnel from filled states on the n-side to empty states on the p-side. As the bias increases further, the overlap decreases and the tunneling current drops — giving negative resistance.

Applications

Tunnel diodes are used in: - Ultra-high-frequency oscillators (up to hundreds of GHz) — the negative resistance enables oscillation. - Low-noise amplifiers — tunneling is inherently fast and low-noise. - Multi-junction solar cells — tunnel junctions connect sub-cells in series.

The tunnel diode demonstrates that tunneling is not just about passing through barriers — it is about the quantum-mechanical overlap of electronic states across thin barriers, and this overlap can be controlled by engineering the material structure.

Discussion Questions

  1. The STM relies on the exponential sensitivity $I \propto e^{-2\kappa d}$. If tunneling current depended on distance as a power law ($I \propto 1/d^n$), would atomic-resolution imaging still be possible? Why or why not?

  2. Flash memory endurance is limited to about $10^3$–$10^5$ write/erase cycles before the tunnel oxide degrades. Each tunneling event can create defects (trap states) in the oxide. How does this connect to the idea that tunneling, while gentle, still involves real physical processes?

  3. The tunneling formula $T \approx e^{-2\kappa d}$ was derived for a rectangular barrier. Real barriers in the STM, flash memory, and tunnel diodes are not rectangular. How does the WKB approximation (Chapter 20) generalize the tunneling formula to arbitrary barrier shapes?

  4. Could you build a "tunneling switch" — a device that toggles between tunneling ON and tunneling OFF by changing the barrier width by a fraction of a nanometer? In fact, this is essentially what MEMS-based tunnel sensors do. Research MEMS tunneling accelerometers and describe their operating principle.

  5. All three technologies in this case study rely on electron tunneling. Under what circumstances might proton tunneling be technologically relevant? (Hint: consider enzyme catalysis and hydrogen storage.)

Key Takeaways

  • The tunneling formula $T \approx e^{-2\kappa d}$ is not just a textbook result — it is the design equation for billion-dollar technologies.
  • The exponential sensitivity of tunneling to distance enables the STM's atomic resolution.
  • Flash memory exploits the distinction between "tunneling is easy" (with applied field) and "tunneling is impossible" (without field) for the same barrier.
  • Tunnel diodes demonstrate that tunneling current can be controlled by energy-band engineering.
  • Quantum mechanics is not just the physics of atoms — it is the physics of the information age.