Quantum Tunneling in Transistors and Flash Memory, Explained
Quantum tunneling is when an electron gets through a thin "wall" even though it lacks the energy to climb over it. It happens because electrons behave like waves. The thinner the wall, the higher the chance of getting through, by orders of magnitude. Inside chips, tunneling makes current leak through thin insulating films. It is also how flash memory writes and erases data.
What you'll learn
- Why an electron can get through a wall it should not be able to cross
- How the wall's thickness and height, and the particle's mass, set the chance of getting through
- An estimate of how much leakage grows when an insulating film is made 0.3 nm thinner
- Why chipmakers switched to films called high-k
- How flash memory writes data, and devices built around tunneling
Prerequisites: the rough shape of a gate and its thin insulating film, as explained in "What Is a Transistor?". Each equation is explained in words before it appears.
In everyday life
The photos you take on your phone are stored in flash memory. Each time a photo is saved, electrons inside the chip pass through a film that does not conduct electricity. Your photos survive when the power is off because, normally, electrons can hardly get through that film.
The same effect can also cause trouble. In chips that do calculations, such as CPUs, electrons pass through the thin insulating film in each transistor, and current leaks. In 2007, Intel replaced the material of this film, which had been used for more than 40 years. The goal was to cut this leakage.
Both are examples of quantum tunneling, an effect of quantum mechanics. This article first shows how electrons get through a wall. Then it looks at where this matters inside chips.
How it works 1: the "wall" an electron sees
Under the gate of a transistor is a thin film that does not conduct electricity. For a long time, this film was silicon dioxide (SiO₂), made by oxidizing silicon. To an electron in the silicon, this film is a tall "energy wall."
The wall's height is the energy an electron is missing to enter the film. It is measured in eV (electronvolts). One eV is the energy an electron gains when a voltage of 1 V pushes it. At the boundary between silicon and silicon dioxide, the wall is about 3.1 eV high.
At room temperature, the typical energy that heat gives an electron is about 0.026 eV. The wall is more than 100 times higher. The fraction of electrons that heat can lift over the wall is less than 1 in 10⁵⁰. In practice, that is zero.
Yet when the film is thin, current really does flow. The electrons are not climbing over the wall. They are passing through it. It is as if they reach the other side of a mountain through a tunnel instead of over the top. That is why the effect is called tunneling.
How it works 2: electrons behave as waves
An electron is a tiny particle, but it also behaves like a wave. When electrons hit a crystal, their waves add up in some places and cancel in others, just like light or sound. The discovery of this behavior won the 1937 Nobel Prize in Physics.
To keep things simple, think of an electron's wave as showing how likely you are to find the electron at each place. Where the wave is larger, the electron is more likely to be found. Strictly, the square of the wave's size is the guide to that probability.
When this wave hits a wall it lacks the energy to climb, it does not stop dead at the surface. It seeps a short way into the wall. Inside, the wave no longer oscillates; it just gets weaker the deeper it goes. It shrinks exponentially, like multiplying by the same fraction again and again.
If the wall is thick, the wave has all but vanished partway through. But if the wall is thin, a little of the wave is still left at the far side. Beyond the wall, this leftover wave travels on as an oscillating wave again. This is what "the electron appears beyond the wall with some probability" means. Most of the rest bounces back.
This idea comes from quantum mechanics, which took shape in the 1920s. In 1928, it explained why electrons fly out of a metal under a strong electric field. The two physicists behind that explanation, Fowler and Nordheim, will come up again in the section on flash memory.
How it works 3: three things set the chance of getting through
How fast the wave fades inside the wall depends on the missing energy (the wall's height) and on the particle's mass. The larger either one is, the faster the wave fades. More precisely, the fading rate is proportional to the square root of "missing energy × mass."
Roughly, the chance of getting through can be written in the form below. ħ ("h-bar") is a very small constant from quantum mechanics. exp(−3) means "about 1/2.7, multiplied together 3 times." The number in the brackets is how many times you multiply.
chance of getting through ≈ exp(−2 × thickness × √(2 × mass × missing energy) ÷ ħ)
The number of multiplications is proportional to the thickness. So a small change in thickness changes the chance by several or even dozens of times. Height and mass sit inside the square root, so making them 4 times larger only doubles the number of multiplications. Even so, extra multiplications cut the chance by orders of magnitude, so the effect is still large. This formula is a rough guide for a rectangular wall when the chance of getting through is small. Minor factors are left out.
By the numbers: making the insulating film 0.3 nm thinner
Let's use the formula from How it works 3 to estimate the chance that an electron gets through a silicon dioxide film. Take the wall height as about 3.1 eV. Inside the film, an electron behaves as if it were lighter than a free electron. This apparent mass is called the effective mass. Values of 0.4 to 0.5 times the normal electron mass are often used, because they match measurements.
- The chance of getting through drops to about 1/3 to 1/4 for every 0.1 nm of added thickness
- Making the film 0.2 nm thinner raises the chance about 10 to 13 times
- Making it 0.3 nm thinner, from 1.5 nm to 1.2 nm, raises the chance about 30 to 50 times
0.3 nm is roughly one layer of the atoms that make up the film. Removing just one atomic layer multiplies the leakage by tens of times. For a 1.2 nm film, the chance that an electron gets through on a single hit is about one in a few million. That looks small. But a huge number of electrons hit the film all the time, so the resulting current is too large to ignore.
This is a rough estimate of the probability part only, for a low voltage and a wall that stays rectangular. Real leakage current also depends on the applied voltage, the film quality and the number of electrons hitting the wall. Even so, measurements confirm that leakage changes exponentially with thickness (for example, IBM's 1997 study).
Where it matters 1: the gate insulator and high-k
As transistors shrank, the insulating film under the gate was made thinner too. A thinner film lets the gate voltage pull harder on the electrons below. That way, even a small transistor can switch cleanly between on and off. In Intel's 90 nm generation, announced in 2002, the film was 1.2 nm thick, about five atomic layers. In the next 65 nm generation, announced in 2004, it was still 1.2 nm.
Thinning stopped there because, as the numbers showed, a thinner film makes leakage rise sharply. Leakage current wastes power even when the chip is idle, and it heats the chip.
The answer was to change the material. How strongly the gate pulls on electrons depends on the film's thickness. It also depends on how well the film stores electric charge, a property called the dielectric constant. Materials with a high dielectric constant are called high-k materials. Oxides containing hafnium have a dielectric constant more than 5 times that of silicon dioxide.
With 5 times the dielectric constant, a film about 5 times thicker gives the same pull (ignoring a thin layer placed underneath). Most high-k materials present a somewhat lower wall to electrons than silicon dioxide does. Even so, making the film several times thicker matters far more. In the formula from How it works 3, thickness counts directly, while height counts only through a square root.
In January 2007, Intel announced that its 45 nm generation would use a hafnium-based high-k film with a metal gate. Products came out in November the same year. According to Intel, leakage was less than one tenth of that with silicon dioxide. Leading-edge transistors in the 2020s still use this combination.
Where it matters 2: writing and erasing flash memory
A flash memory cell stores a 0 or a 1 by trapping electrons in a tiny "room" surrounded by insulating film. "Memory vs Storage: What's the Difference?" explains this in more detail. The path electrons flow along is called the channel. The film between the channel and the room is called the tunnel oxide.
This film must hold electrons for years without power, so it is made much thicker than the gate insulator. As an example, take a thickness of 8 nm. With no voltage applied, the formula from How it works 3 gives a chance of getting through below 10⁻⁴⁰. In an ideal film, electrons practically never escape.
In high-capacity NAND flash, writing applies a high voltage, around 20 V, to the gate above (the value varies by product). This creates a strong electric field (the slope of the voltage) in the film, and the wall tilts. The part of the wall that is higher than the electron's energy shrinks to only about 3 nm, near the entrance. This value assumes an example field of about 1 V per nm (10 MV/cm).
Current that tunnels through only part of a tilted wall like this is called Fowler–Nordheim tunneling (FN tunneling). It differs from tunneling through the full thickness of a rectangular wall, as in gate leakage, which is called direct tunneling. In FN tunneling, the field sets the distance to tunnel, so the current is very sensitive to the field. Under the same assumptions, weakening the field by just 20% cuts the chance of getting through by a factor of a few hundred. Erasing reverses the voltage to remove electrons from the room.
Reading uses a much lower voltage than writing, so the electrons barely move. But each time electrons cross the film under a strong field, the film is damaged a little. That is why flash memory can be rewritten only a limited number of times.
Where it matters 3: transistors shrunk too far
Electrons enter a transistor at the source and leave at the drain. When the transistor is off, there is an energy wall between the two. In today's transistors, electrons that heat lifts over this wall are one of the main causes of leakage. But once the wall's width (roughly the gate length) shrinks to a few nm, electrons start to tunnel through it. Then the gate struggles to stop the current, even when it tries to turn the transistor off.
A 2019 simulation studied silicon transistors with the gate wrapped around all four sides. With a 3 nm gate length, most of the current came from tunneling. At 6 nm, the effect was small. This is one example calculation, and the results change with shape and material. Even IBM's "2 nm" test chip, announced in 2021, had a 12 nm gate length.
Meanwhile, another form of tunneling already causes leakage in today's chips. One example is gate-induced drain leakage (GIDL), which comes from the strong field where the gate overlaps the drain. It causes problems such as shortening how long DRAM holds its data.
Where it matters 4: devices built around tunneling
Esaki diode
In 1957, Leo Esaki was studying the junction between two types of semiconductor (a PN junction) at Tokyo Tsushin Kogyo (now Sony). His junction was germanium with a very large amount of impurities added. He found that the wall at the boundary became very thin, and electrons tunneled through it. He published the result in 1958. Over a certain range, raising the voltage makes the current fall, the opposite of normal behavior. For this discovery, which led to the Esaki diode, he shared the 1973 Nobel Prize in Physics with Giaever and Josephson.
Tunnel FET (TFET)
An ordinary transistor relies on electrons that heat lifts over a wall. So at room temperature, raising the current tenfold takes at least about 60 mV of gate voltage. "Fermi-Dirac vs Boltzmann Distribution in Semiconductors" explains why. A tunnel FET uses the gate to switch tunneling on and off at a junction boundary like the one in the Esaki diode. This is called band-to-band tunneling. Since electrons are not lifted over the wall by heat, the device is not bound by this limit in principle. But it can carry only a small current, which is a major problem. In the 2020s it is still at the research stage.
Scanning tunneling microscope (STM)
In 1981, Binnig and Rohrer at IBM's Zurich research lab got a new kind of microscope working. It images a surface using the tunneling current across the gap between a sharp tip and the sample. When the gap changes by 0.1 nm, the current changes nearly 10 times. So bumps the size of a single atom show up. The two received the 1986 Nobel Prize in Physics. MRAM, a memory that stores data magnetically, also relies on tunneling. Its memory element, the magnetic tunnel junction, passes current through a wall about 1 nm thick.
Common misconceptions
Myth: The electron borrows energy for a moment to get over the wall
In fact, in the standard calculation of tunneling, the electron has the same energy before and after the wall. It does not go over the top. Instead, the wave fades inside the wall, and part of it is left on the far side. "Borrowing" is a way of telling the story to fit everyday intuition. It does not appear in the steps used to calculate the probability.
Myth: Inside the tunnel, electrons travel faster than light
In fact, tunneling cannot send signals or information faster than light. In some calculations and experiments, the peak of the wave that got through seems to come out sooner than light would. But the position of the wave's peak is not the speed at which a signal arrives. How to define the time an electron spends inside the wall is still debated.
Myth: Tunneling only matters in special devices
In fact, it happens all over ordinary chips. Examples include leakage through the gate insulator and GIDL, and the writing and erasing of NAND flash in phones and SSDs. An ohmic contact connects metal wiring to a semiconductor with low resistance. It, too, works by tunneling: heavy doping makes the wall thin so current can tunnel through. The word "quantum" makes tunneling sound like something from a distant world, which is why this myth spreads.
Learn more
- Gate insulator terms: high-k / metal gate / EOT (equivalent oxide thickness) (a high-k film expressed as an equivalent silicon dioxide thickness)
- Flash and device terms: Fowler–Nordheim tunneling / floating gate / tunnel diode (Esaki diode) / subthreshold swing
- Compare in tables: planar MOSFET vs FinFET / NAND vs NOR flash / MRAM vs ReRAM vs PCM vs FeRAM / static vs dynamic power
- See it in figures: cross-section of an n-channel MOSFET (try changing the gate voltage)
- Related explainers: "Fermi-Dirac vs Boltzmann Distribution in Semiconductors" (electrons lifted over a wall by heat, and 60 mV/decade) / "Memory vs Storage: What's the Difference?"