Quantum tunnelling

A simulation of a wave packet incident on a potential barrier. In relative units, the barrier energy is 20, greater than the mean wave packet energy of 14. A portion of the wave packet passes through the barrier.

The wave function of a particle summarizes everything that can be known about a physical system.[18] Therefore, problems in quantum mechanics analyze the system's wave function. Using mathematical formulations, such as the Schrödinger equation, the wave function can be deduced. The square of the absolute value of this wavefunction is directly related to the probability distribution of the particle's position, which describes the probability that the particle is at any given place. The wider the barrier and the higher the barrier energy, the lower the probability of tunneling.

A simple model of a tunneling barrier, such as the rectangular barrier, can be analysed and solved algebraically. In canonical field theory, the tunneling is described by a wave function which has a non-zero amplitude inside the tunnel; but the current is zero there because the relative phase of the amplitude of the conjugate wave function (the time derivative) is orthogonal to it.

The simulation shows one such system.

An electron wavepacket directed at a potential barrier. Note the dim spot on the right that represents tunneling electrons.

The 2nd illustration shows the uncertainty principle at work. A wave impinges on the barrier; the barrier forces it to become taller and narrower. The wave becomes much more de-localized–it is now on both sides of the barrier, it is wider on each side and lower in maximum amplitude but equal in total amplitude. In both illustrations, the localization of the wave in space causes the localization of the action of the barrier in time, thus scattering the energy/momentum of the wave.

Problems in real life often do not have one, so "semiclassical" or "quasiclassical" methods have been developed to offer approximate solutions, such as the WKB approximation. Probabilities may be derived with arbitrary precision, as constrained by computational resources, via Feynman's path integral method. Such precision is seldom required in engineering practice.

The concept of dynamical tunneling is particularly suited to address the problem of quantum tunneling in high dimensions (d>1). In the case of an integrable system, where bounded classical trajectories are confined onto tori in phase space, tunneling can be understood as the quantum transport between semi-classical states built on two distinct but symmetric tori.[21]

Chaos-assisted tunneling oscillations between two regular tori embedded in a chaotic sea, seen in phase space

In real life, most system are not integrable and display various degrees of chaos. Classical dynamics is then said to be mixed and the system phase space is typically composed of islands of regular orbits surrounded by a large sea of chaotic orbits. The existence of the chaotic sea, where transport is classically allowed, between the two symmetric tori then assists the quantum tunneling between them. This phenomenon is referred as chaos-assisted tunneling.[22] and is characterized by sharp resonances of the tunneling rate when varying any system parameter.

When ${\displaystyle \hbar }$ is small in front of the size of the regular islands, the fine structure of the classical phase space plays a key role in tunneling. In particular the two symmetric tori are coupled "via a succession of classically forbidden transitions across nonlinear resonances" surrounding the two islands.[23]

Tunneling is a source of current leakage in very-large-scale integration (VLSI) electronics and results in the substantial power drain and heating effects that plague such devices. It is considered the lower limit on how microelectronic device elements can be made.[26] Tunneling is a fundamental technique used to program the floating gates of flash memory.

Cold emission of electrons is relevant to semiconductors and superconductor physics. It is similar to thermionic emission, where electrons randomly jump from the surface of a metal to follow a voltage bias because they statistically end up with more energy than the barrier, through random collisions with other particles. When the electric field is very large, the barrier becomes thin enough for electrons to tunnel out of the atomic state, leading to a current that varies approximately exponentially with the electric field.[27] These materials are important for flash memory, vacuum tubes, as well as some electron microscopes.

A simple barrier can be created by separating two conductors with a very thin insulator. These are tunnel junctions, the study of which requires understanding quantum tunneling.[28] Josephson junctions take advantage of quantum tunneling and superconductivity to create the Josephson effect. This has applications in precision measurements of voltages and magnetic fields,[27] as well as the multijunction solar cell.

QCA is a molecular binary logic synthesis technology that operates by the inter-island electron tunneling system. This is a very low power and fast device that can operate at a maximum frequency of 15 PHz.[29]

A working mechanism of a resonant tunneling diode device, based on the phenomenon of quantum tunneling through the potential barriers.

Diodes are electrical semiconductor devices that allow electric current flow in one direction more than the other. The device depends on a depletion layer between N-type and P-type semiconductors to serve its purpose. When these are heavily doped the depletion layer can be thin enough for tunneling. When a small forward bias is applied, the current due to tunneling is significant. This has a maximum at the point where the voltage bias is such that the energy level of the p and n conduction bands are the same. As the voltage bias is increased, the two conduction bands no longer line up and the diode acts typically.[30]

Because the tunneling current drops off rapidly, tunnel diodes can be created that have a range of voltages for which current decreases as voltage increases. This peculiar property is used in some applications, such as high speed devices where the characteristic tunneling probability changes as rapidly as the bias voltage.[30]

The resonant tunneling diode makes use of quantum tunneling in a very different manner to achieve a similar result. This diode has a resonant voltage for which a lot of current favors a particular voltage, achieved by placing two thin layers with a high energy conductance band near each other. This creates a quantum potential well that has a discrete lowest energy level. When this energy level is higher than that of the electrons, no tunneling occurs and the diode is in reverse bias. Once the two voltage energies align, the electrons flow like an open wire. As the voltage further increases, tunneling becomes improbable and the diode acts like a normal diode again before a second energy level becomes noticeable.[31]

A European research project demonstrated field effect transistors in which the gate (channel) is controlled via quantum tunneling rather than by thermal injection, reducing gate voltage from ≈1 volt to 0.2 volts and reducing power consumption by up to 100×. If these transistors can be scaled up into VLSI chips, they would improve the performance per power of integrated circuits.[32][33]

Quantum tunneling is an essential phenomenon for nuclear fusion. The temperature in stellar cores is generally insufficient to allow atomic nuclei to overcome the Coulomb barrier and achieve thermonuclear fusion. Quantum tunneling increases the probability of penetrating this barrier. Though this probability is still low, the extremely large number of nuclei in the core of a star is sufficient to sustain a steady fusion reaction.[34]

Radioactive decay is the process of emission of particles and energy from the unstable nucleus of an atom to form a stable product. This is done via the tunneling of a particle out of the nucleus (an electron tunneling into the nucleus is electron capture). This was the first application of quantum tunneling. Radioactive decay is a relevant issue for astrobiology as this consequence of quantum tunneling creates a constant energy source over a large time interval for environments outside the circumstellar habitable zone where insolation would not be possible (subsurface oceans) or effective.[34]

Quantum tunnelling may be one of the mechanisms of hypothetical proton decay.[35][36][25]

By including quantum tunneling, the astrochemical syntheses of various molecules in interstellar clouds can be explained, such as the synthesis of molecular hydrogen, water (ice) and the prebiotic important formaldehyde.[34]

Quantum tunneling is among the central non-trivial quantum effects in quantum biology. Here it is important both as electron tunneling and proton tunneling.[37] Electron tunneling is a key factor in many biochemical redox reactions (photosynthesis, cellular respiration) as well as enzymatic catalysis. Proton tunneling is a key factor in spontaneous DNA mutation.[34]

Spontaneous mutation occurs when normal DNA replication takes place after a particularly significant proton has tunnelled.[38] A hydrogen bond joins DNA base pairs. A double well potential along a hydrogen bond separates a potential energy barrier. It is believed that the double well potential is asymmetric, with one well deeper than the other such that the proton normally rests in the deeper well. For a mutation to occur, the proton must have tunnelled into the shallower well. The proton's movement from its regular position is called a tautomeric transition. If DNA replication takes place in this state, the base pairing rule for DNA may be jeopardised, causing a mutation.[39] Per-Olov Lowdin was the first to develop this theory of spontaneous mutation within the double helix. Other instances of quantum tunneling-induced mutations in biology are believed to be a cause of ageing and cancer.[40]

While the Drude-Lorentz model of electrical conductivity makes excellent predictions about the nature of electrons conducting in metals, it can be furthered by using quantum tunneling to explain the nature of the electron's collisions.[27] When a free electron wave packet encounters a long array of uniformly spaced barriers, the reflected part of the wave packet interferes uniformly with the transmitted one between all barriers so that 100% transmission becomes possible. The theory predicts that if positively charged nuclei form a perfectly rectangular array, electrons will tunnel through the metal as free electrons, leading to extremely high conductance, and that impurities in the metal will disrupt it significantly.[27]

The scanning tunneling microscope (STM), invented by Gerd Binnig and Heinrich Rohrer, may allow imaging of individual atoms on the surface of a material.[27] It operates by taking advantage of the relationship between quantum tunneling with distance. When the tip of the STM's needle is brought close to a conduction surface that has a voltage bias, measuring the current of electrons that are tunneling between the needle and the surface reveals the distance between the needle and the surface. By using piezoelectric rods that change in size when voltage is applied, the height of the tip can be adjusted to keep the tunneling current constant. The time-varying voltages that are applied to these rods can be recorded and used to image the surface of the conductor.[27] STMs are accurate to 0.001 nm, or about 1% of atomic diameter.[31]

In chemical kinetics, the substitution of a light isotope of an element with a heavier one typically results in a slower reaction rate. This is generally attributed to differences in the zero-point vibrational energies for chemical bonds containing the lighter and heavier isotopes and is generally modeled using transition state theory. However, in certain cases, large isotope effects are observed that cannot be accounted for by a semi-classical treatment, and quantum tunneling is required. R. P. Bell developed a modified treatment of Arrhenius kinetics that is commonly used to model this phenomenon.[41]

The time-independent Schrödinger equation for one particle in one dimension can be written as

$${\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\Psi (x)+V(x)\Psi (x)=E\Psi (x)}$$

or

$${\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)={\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)\Psi (x)\equiv {\frac {2m}{\hbar ^{2}}}M(x)\Psi (x),}$$

where

• ${\displaystyle \hbar }$ is the reduced Planck's constant,
• m is the particle mass,
• x represents distance measured in the direction of motion of the particle,
• Ψ is the Schrödinger wave function,
• V is the potential energy of the particle (measured relative to any convenient reference level),
• E is the energy of the particle that is associated with motion in the x-axis (measured relative to V),
• M(x) is a quantity defined by V(x) − E which has no accepted name in physics.

The solutions of the Schrödinger equation take different forms for different values of x, depending on whether M(x) is positive or negative. When M(x) is constant and negative, then the Schrödinger equation can be written in the form

${\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)={\frac {2m}{\hbar ^{2}}}M(x)\Psi (x)=-k^{2}\Psi (x),\qquad {\text{where}}\quad k^{2}=-{\frac {2m}{\hbar ^{2}}}M.}$

The solutions of this equation represent travelling waves, with phase-constant +k or -k. Alternatively, if M(x) is constant and positive, then the Schrödinger equation can be written in the form

${\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)={\frac {2m}{\hbar ^{2}}}M(x)\Psi (x)={\kappa }^{2}\Psi (x),\qquad {\text{where}}\quad {\kappa }^{2}={\frac {2m}{\hbar ^{2}}}M.}$

The solutions of this equation are rising and falling exponentials in the form of evanescent waves. When M(x) varies with position, the same difference in behaviour occurs, depending on whether M(x) is negative or positive. It follows that the sign of M(x) determines the nature of the medium, with negative M(x) corresponding to medium A and positive M(x) corresponding to medium B. It thus follows that evanescent wave coupling can occur if a region of positive M(x) is sandwiched between two regions of negative M(x), hence creating a potential barrier.

The mathematics of dealing with the situation where M(x) varies with x is difficult, except in special cases that usually do not correspond to physical reality. A full mathematical treatment appears in the 1965 monograph by Fröman and Fröman. Their ideas have not been incorporated into physics textbooks, but their corrections have little quantitative effect.

The wave function is expressed as the exponential of a function:

$${\displaystyle \Psi (x)=e^{\Phi (x)},}$$

where

$${\displaystyle \Phi ''(x)+\Phi '(x)^{2}={\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right).}$$

${\displaystyle \Phi '(x)}$ is then separated into real and imaginary parts:

$${\displaystyle \Phi '(x)=A(x)+iB(x),}$$

where A(x) and B(x) are real-valued functions.

Substituting the second equation into the first and using the fact that the imaginary part needs to be 0 results in:

$${\displaystyle A'(x)+A(x)^{2}-B(x)^{2}={\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right).}$$

Quantum tunneling in the phase space formulation of quantum mechanics. Wigner function for tunneling through the potential barrier ${\displaystyle U(x)=8e^{-0.25x^{2}}}$ in atomic units (a.u.). The solid lines represent the level set of the Hamiltonian ${\displaystyle H(x,p)=p^{2}/2+U(x)}$.

To solve this equation using the semiclassical approximation, each function must be expanded as a power series in ${\displaystyle \hbar }$. From the equations, the power series must start with at least an order of ${\displaystyle \hbar ^{-1}}$ to satisfy the real part of the equation; for a good classical limit starting with the highest power of Planck's constant possible is preferable, which leads to

$${\displaystyle A(x)={\frac {1}{\hbar }}\sum _{k=0}^{\infty }\hbar ^{k}A_{k}(x)}$$

and

$${\displaystyle B(x)={\frac {1}{\hbar }}\sum _{k=0}^{\infty }\hbar ^{k}B_{k}(x),}$$

with the following constraints on the lowest order terms,

$${\displaystyle A_{0}(x)^{2}-B_{0}(x)^{2}=2m\left(V(x)-E\right)}$$

and

$${\displaystyle A_{0}(x)B_{0}(x)=0.}$$

At this point two extreme cases can be considered.

Case 1

If the amplitude varies slowly as compared to the phase ${\displaystyle A_{0}(x)=0}$ and

$${\displaystyle B_{0}(x)=\pm {\sqrt {2m\left(E-V(x)\right)}}}$$

which corresponds to classical motion. Resolving the next order of expansion yields

$${\displaystyle \Psi (x)\approx C{\frac {e^{i\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(E-V(x)\right)}}+\theta }}{\sqrt[{4}]{{\frac {2m}{\hbar ^{2}}}\left(E-V(x)\right)}}}}$$

Case 2

If the phase varies slowly as compared to the amplitude, ${\displaystyle B_{0}(x)=0}$ and

$${\displaystyle A_{0}(x)=\pm {\sqrt {2m\left(V(x)-E\right)}}}$$

which corresponds to tunneling. Resolving the next order of the expansion yields

$${\displaystyle \Psi (x)\approx {\frac {C_{+}e^{+\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)}}}+C_{-}e^{-\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)}}}}{\sqrt[{4}]{{\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)}}}}$$

In both cases it is apparent from the denominator that both these approximate solutions are bad near the classical turning points ${\displaystyle E=V(x)}$. Away from the potential hill, the particle acts similar to a free and oscillating wave; beneath the potential hill, the particle undergoes exponential changes in amplitude. By considering the behaviour at these limits and classical turning points a global solution can be made.

To start, a classical turning point, ${\displaystyle x_{1}}$ is chosen and ${\displaystyle {\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)}$ is expanded in a power series about ${\displaystyle x_{1}}$:

$${\displaystyle {\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)=v_{1}(x-x_{1})+v_{2}(x-x_{1})^{2}+\cdots }$$

Keeping only the first order term ensures linearity:

$${\displaystyle {\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)=v_{1}(x-x_{1}).}$$

Using this approximation, the equation near ${\displaystyle x_{1}}$ becomes a differential equation:

$${\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)=v_{1}(x-x_{1})\Psi (x).}$$

This can be solved using Airy functions as solutions.

$${\displaystyle \Psi (x)=C_{A}Ai\left({\sqrt[{3}]{v_{1}}}(x-x_{1})\right)+C_{B}Bi\left({\sqrt[{3}]{v_{1}}}(x-x_{1})\right)}$$

Taking these solutions for all classical turning points, a global solution can be formed that links the limiting solutions. Given the two coefficients on one side of a classical turning point, the two coefficients on the other side of a classical turning point can be determined by using this local solution to connect them.

Hence, the Airy function solutions will asymptote into sine, cosine and exponential functions in the proper limits. The relationships between ${\displaystyle C,\theta }$ and ${\displaystyle C_{+},C_{-}}$ are

$${\displaystyle C_{+}={\frac {1}{2}}C\cos {\left(\theta -{\frac {\pi }{4}}\right)}}$$

and

Quantum tunneling through a barrier. At the origin (x = 0), there is a very high, but narrow potential barrier. A significant tunneling effect can be seen.

${\displaystyle C_{-}=-C\sin {\left(\theta -{\frac {\pi }{4}}\right)}}$

With the coefficients found, the global solution can be found. Therefore, the transmission coefficient for a particle tunneling through a single potential barrier is

$${\displaystyle T(E)=e^{-2\int _{x_{1}}^{x_{2}}dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left[V(x)-E\right]}}},}$$

where ${\displaystyle x_{1},x_{2}}$ are the two classical turning points for the potential barrier.

For a rectangular barrier, this expression simplifies to:

$${\displaystyle T(E)=e^{-2{\sqrt {{\frac {2m}{\hbar ^{2}}}(V_{0}-E)}}(x_{2}-x_{1})}.}$$