Quantum tunnelling: Wikis

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Quantum mechanics
\Delta x\, \Delta p \ge \frac{\hbar}{2}
Uncertainty principle
Introduction · Mathematical formulation
Fundamental concepts
Quantum state · Wave function
Superposition · Entanglement
Measurement · Uncertainty
Exclusion · Duality
Decoherence · Ehrenfest theorem · Tunnelling

Quantum tunneling refers to the phenomena of a particle's ability to penetrate energy barriers within electronic structures. The scientific terms for this are Wave-mechanical tunneling, Quantum-mechanical tunneling and the Tunnel effect. The Tunnel Effect is an evanescent wave coupling effect that occurs in the context of quantum mechanics. Particles behave in a manner calculated with Schrödinger's wave-equations. All waves die away, but according to the laws of physics, the energy in these waves pass on. Wave coupling effects, mathematically equivalent to quantum tunnelling mechanics, can occur with Maxwell's wave-equation (both with light and with microwaves), and with the common non-dispersive wave-equation often applied (for example) to waves on strings and to acoustics.

For these effects to occur there must be a situation where a thin region of "medium type 2" is sandwiched between two regions of "medium type 1", and the properties of these media have to be such that the wave equation has "traveling-wave" solutions in medium type 1, but "real exponential solutions" (rising and falling) in medium type 2. In optics, medium type 1 might be glass, medium type 2 might be a vacuum. In quantum mechanics, in connection with motion of a particle, medium type 1 is a region of space where the particle's total energy is greater than its potential energy, medium type 2 is a region of space (known as the "barrier") where the particle's total energy is less than its potential energy - for further explanation see the section on "Schrödinger equation - tunnelling basics" below.

If conditions are right, amplitude from a traveling wave, incident on medium type 2 from medium type 1, can "leak through" medium type 2 and emerge as a traveling wave in the second region of medium type 1 on the far side. If the second region of medium type 1 is not present, then the traveling wave incident on medium type 2 is totally reflected, although it does penetrate into medium type 2 to some extent. Depending on the wave equation being used, the leaked amplitude is interpreted physically as traveling energy or as a traveling particle, and, numerically, the ratio of the square of the leaked amplitude to the square of the incident amplitude gives the proportion of incident energy transmitted out the far side, or (in the case of the Schrödinger equation) the probability that the particle "tunnels" through the barrier.

Contents

Introduction

Schematic representation of quantum tunnelling through a barrier. The energy of the tunneled particle is the same, only the quantum amplitude (and hence the probability of the process) is decreased.

The scale on which these "tunnelling-like phenomena" occur depends on the wavelength of the traveling wave. For electrons, the thickness of "medium type 2" (called in this context "the tunnelling barrier") is typically a few nanometres; for alpha-particles tunnelling out of a nucleus, the thickness is much less; for the analogous phenomenon involving light, the thickness is much greater.

With the Schrödinger's wave-equation, the characteristic that defines the two media discussed above is the kinetic energy of the particle if it is considered as an object that could be located at a point. In medium type 1 the kinetic energy would be positive, in medium type 2 the kinetic energy would be negative. There is some inconsistency in this, because particles cannot physically be located at a point: they are always spread out ("delocalised") to some extent, and the kinetic energy of the delocalised object is always positive.

What is true is that it is sometimes mathematically convenient to treat particles as behaving like points, particular in the context of Newton's Second Law and classical mechanics generally. In the past, people thought that the success of classical mechanics meant that particles could always and in all circumstances be treated as if they were located at points. But there never was any convincing experimental evidence that this was true when very small objects and very small distances are involved, and we now know that this viewpoint was mistaken. However, because it is still traditional to teach students early in their careers that particles behave like points, it sometimes comes as a big surprise for people to discover that it is well established that traveling physical particles always physically obey a wave-equation (even when it is convenient to use the mathematics of moving points). Clearly, a hypothetical classical point particle analysed according to Newton's Laws could not enter a region where its kinetic energy would be negative. But, a real delocalised object, that obeys a wave-equation and always has positive kinetic energy, can leak through such a region if conditions are right. An approach to tunnelling that avoids mention of the concept of "negative kinetic energy" is set out below in the section on "Schrödinger equation tunnelling basics".

An electron approaching a barrier has to be represented as a wave-train. This wave-train can sometimes be quite long – electrons in some materials can be 10 to 20 nm long. This makes animations difficult. If it were legitimate to represent the electron by a short wave-train, then tunnelling could be represented as in the animation alongside.

Reflection and tunnelling of an electron wavepacket directed at a potential barrier. The bright spot moving to the left is the reflected part of the wavepacket. A very dim spot can be seen moving to the right of the barrier. This is the small fraction of the wavepacket that tunnels through the classically forbidden barrier. Also notice the interference fringes between the incoming and reflected waves.

It is sometimes said that tunnelling occurs only in quantum mechanics. Unfortunately, this statement is a bit of linguistic conjuring trick. As indicated above, "tunnelling-type" evanescent-wave phenomena occur in other contexts too. But, until recently, it has only been in quantum mechanics that evanescent wave coupling has been called "tunnelling". (However, there is an increasing tendency to use the label "tunnelling" in other contexts too, and the names "photon tunnelling" and "acoustic tunnelling" are now used in the research literature.)

With regards to the mathematics of tunnelling, a special problem arises. For simple tunnelling-barrier models, such as the rectangular barrier, the Schrödinger equation can be solved exactly to give the value of the tunnelling probability (sometimes called the "transmission coefficient"). Calculations of this kind make the general physical nature of tunnelling clear. One would also like to be able to calculate exact tunnelling probabilities for barrier models that are physically more realistic. However, when appropriate mathematical descriptions of barriers are put into the Schrödinger equation, then the result is an awkward non-linear differential equation. Usually, the equation is of a type where it is known to be mathematically impossible in principle to solve the equation exactly in terms of the usual functions of mathematical physics, or in any other simple way. Mathematicians and mathematical physicists have been working on this problem since at least 1813, and have been able to develop special methods for solving equations of this kind approximately. In physics these are known as "semiclassical" or "quasiclassical" methods. A common semiclassical method is the so-called WKB approximation (also known as the "JWKB approximation"). The first known attempt to use such methods to solve a tunnelling problem in physics was made in 1928, in the context of field electron emission. It is sometimes considered that the first people to get the mathematics of applying this kind of approximation to tunnelling fully correct (and to give reasonable mathematical proof that they had done so) were N. Fröman and P.O. Fröman, in 1965. Their complex ideas have not yet made it into theoretical-physics textbooks, which tend to give simpler (but slightly more approximate) versions of the theory. An outline of one particular semiclassical method is given below.

A working mechanism of a Resonant Tunneling Diode device, based on the phenomenon of quantum tunneling through the potential barriers.

Three notes may be helpful. In general, students taking physics courses in quantum mechanics are presented with problems (such as the quantum mechanics of the hydrogen atom) for which exact mathematical solutions to the Schrödinger equation exist. Tunnelling through a realistic barrier is a reasonably basic physical phenomenon. So it is sometimes the first problem that students encounter where it is mathematically impossible in principle to solve the Schrödinger equation exactly in any simple way. Thus, it may also be the first occasion on which they encounter the "semiclassical-method" mathematics needed to solve the Schrödinger equation approximately for such problems. Not surprisingly, this mathematics is likely to be unfamiliar, and may feel "odd". Unfortunately, it also comes in several different variants, which doesn't help.

Also, some accounts of tunnelling seem to be written from a philosophical viewpoint that a particle is "really" point-like, and just has wave-like behaviour. There is very little experimental evidence to support this viewpoint. A preferable philosophical viewpoint is that the particle is "really" delocalised and wave-like, and always exhibits wave-like behaviour, but that in some circumstances it is convenient to use the mathematics of moving points to describe its motion. This second viewpoint is used in this section. The precise nature of this wave-like behaviour is, however, a much deeper matter, beyond the scope of this article on tunnelling.

Although the phenomenon under discussion here is usually called "quantum tunnelling" or "quantum-mechanical tunnelling", it is the wave-like aspects of particle behaviour that are important in tunnelling theory, rather than effects relating to the quantization of the particle's energy states. For this reason, some writers prefer to call the phenomenon "wave-mechanical tunnelling".

History

By 1928, George Gamow had solved the theory of the alpha decay of a nucleus via tunnelling. Classically, the particle is confined to the nucleus because of the high energy requirement to escape the very strong potential. Under this system, it takes an enormous amount of energy to pull apart the nucleus. In quantum mechanics, however, there is a probability the particle can tunnel through the potential and escape. Gamow solved a model potential for the nucleus and derived a relationship between the half-life of the particle and the energy of the emission.

Alpha decay via tunnelling was also solved concurrently by Ronald Gurney and Edward Condon. Shortly thereafter, both groups considered whether particles could also tunnel into the nucleus.

After attending a seminar by Gamow, Max Born recognized the generality of quantum-mechanical tunnelling. He realized that the tunnelling phenomenon was not restricted to nuclear physics, but was a general result of quantum mechanics that applies to many different systems. Today the theory of tunnelling is even applied to the early cosmology of the universe.[1]

Quantum tunnelling was later applied to other situations, such as the cold emission of electrons, and perhaps most importantly semiconductor and superconductor physics. Phenomena such as field emission, important to flash memory, are explained by quantum tunnelling. Tunnelling is a source of major current leakage in Very-large-scale integration (VLSI) electronics, and results in the substantial power drain and heating effects that plague high-speed and mobile technology.

Another major application is in electron-tunnelling microscopes (see scanning tunnelling microscope) which can resolve objects that are too small to see using conventional microscopes. Electron tunnelling microscopes overcome the limiting effects of conventional microscopes (optical aberrations, wavelength limitations) by scanning the surface of an object with tunnelling electrons.

Quantum tunnelling has been shown to be a mechanism used by enzymes to enhance reaction rates. It has been demonstrated that enzymes use tunnelling to transfer both electrons and nuclei such as hydrogen and deuterium. It has even been shown, in the enzyme glucose oxidase, that oxygen nuclei can tunnel under physiological conditions.[2]

Schrödinger equation - tunnelling basics

Consider the time-independent Schrödinger equation for one particle, in one dimension. This can be written in the forms

-\frac{\hbar^2}{2m} \frac{d^2}{dx^2} \Psi(x) + V(x) \Psi(x) = E \Psi(x)
\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 \hbar is Planck's constant divided by  2 \pi, \; m is the particle mass, x represents distance measured in the direction of motion of the particle, Ψ(x) is the Schrödinger wave function, V(x) is the potential energy of the particle (measured relative to any convenient reference level), E is that part of the total energy of the particle that is associated with motion in the x-direction (measured relative to the same reference level as V(x)), and M(x) is a quantity defined by this equation. Explicitly, M(x) is given by

M(x) = V(x) − E.

The quantity M(x) has no accepted name in physics generally; the name "motive energy" is used in the article on field electron emission.

The solutions of the Schrödinger equation take different forms for different values of x, depending on whether M(x) is positive or negative. This is easiest to understand if we consider a situation in which we have regions of space in which M(x) is (a) constant and negative and (b) constant and positive. When M(x) is constant and negative, then the Schrödinger equation can be written in the form

\frac{d^2}{dx^2} \Psi(x) = \frac{2m}{\hbar^2} M(x) \Psi(x) = -k^2 \Psi(x),\;\;\;\;\;\; \mathrm{where} \;\;\; 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

\frac{d^2}{dx^2} \Psi(x) = \frac{2m}{\hbar^2} M(x) \Psi(x) = {\kappa}^2 \Psi(x), \;\;\;\;\;\; \mathrm{where} \;\;\; {\kappa}^2= \frac{2m}{\hbar^2} M.

The solutions of this equation are rising and falling exponentials, which take the form exp(+κx) for rising exponentials, or the form exp(-κx) for decaying exponentials (also called "evanescent waves"). When M(x) varies with position, the same difference in behaviour occurs, depending on whether M(x) is negative or positive, but the parameters k and κ become functions of position. It follows that the sign of M(x) determines the "nature of the medium", with negative M corresponding to the "medium of type 1" discussed above, and positive M corresponding to the "medium of type 2". It thus follows from well-established mathematical principles of classical wave-physics - but applied to the Schrödinger equation - that evanescent wave coupling can occur if a region of positive M is sandwiched between two regions of negative M. This occurs if V(x) has a "hill-type" shape.

A problem is that the mathematics of dealing with the situation where M(x) varies with x is intensely difficult, except in certain mathematical special cases that usually do not correspond quantitatively well to physical reality. A discussion of the simple (but quantitatively unrealistic) case of the rectangular potential barrier appears elsewhere. A discussion of the "semi-classical" approximate method, as sometimes found in physics textbooks, is given in the next section. A full (but very complicated) complete mathematical treatment appears in the 1965 monograph by Fröman and Fröman noted below. Their ideas have not yet made it into physics textbooks, but probably in most cases their corrections have little quantitative effect. A brief statement of the outcome of the Fröman and Fröman treatment appears in the article on field electron emission (which was the first major physical effect to be identified as due to electron tunnelling, in 1928), in the section on escape probability.

Note that, in the hypothetical physical picture of "particle" motion used in the 1800s and earlier, in which a "particle" is assumed to have the behaviour of a moving point mass, positive values of M(x) correspond to negative values of the kinetic energy of a point mass located at position "x". There is, however, no logical need to introduce the concept of "negative kinetic energy at a point in space" into discussion of evanescent wave coupling (i.e., there is no logical need to introduce this concept into discussions of "tunnelling" based on the Schrödinger equation.)

A semiclassical method for determining a formula for tunnelling probability

Now let us recast the wave function Ψ(x) as the exponential of a function.

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

Now we separate Φ'(x) into real and imaginary parts using real valued functions A and B.

\Phi'(x) = A(x) + i B(x) \,
A'(x) + A(x)^2 - B(x)^2 = \frac{2m}{\hbar^2} \left( V(x) - E \right),

because the pure imaginary part needs to vanish due to the real-valued right-hand side:

i\left(B'(x) - 2 A(x) B(x)\right) = 0.

Next we want to take the semiclassical approximation to solve this. That means we expand each function as a power series in \hbar. From the equations we can see that the power series must start with at least an order of \hbar^{-1} to satisfy the real part of the equation. But as we want a good classical limit, we also want to start with as high a power of Planck's constant as possible.

A(x) = \frac{1}{\hbar} \sum_{k=0}^\infty \hbar^k A_k(x)
B(x) = \frac{1}{\hbar} \sum_{k=0}^\infty \hbar^k B_k(x).

The constraints on the lowest order terms are as follows.

A_0(x)^2 - B_0(x)^2 = 2m \left( V(x) - E \right)
A0(x)B0(x) = 0

If the amplitude varies slowly as compared to the phase, we set A0(x) = 0 and get

B_0(x) = \pm \sqrt{ 2m \left( E - V(x) \right) }

which is only valid when you have more energy than potential - classical motion. After the same procedure on the next order of the expansion we get

\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)}}

On the other hand, if the phase varies slowly as compared to the amplitude, we set B0(x) = 0 and get

A_0(x) = \pm \sqrt{ 2m \left( V(x) - E \right) }

which is only valid when you have more potential than energy - tunnelling motion. Resolving the next order of the expansion yields

\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)}}

It is apparent from the denominator, that both these approximate solutions are bad near the classical turning point E = V(x). What we have are the approximate solutions away from the potential hill and beneath the potential hill. Away from the potential hill, the particle acts similarly to a free wave - the phase is oscillating. Beneath the potential hill, the particle undergoes exponential changes in amplitude.

In a specific tunnelling problem, we might suspect that the transition amplitude is proportional to e^{-\int dx \sqrt{\frac{2m}{\hbar^2} \left( V(x) - E \right)}} and thus the tunnelling is exponentially dampened by large deviations from classically allowable motion.

But to be complete we must find the approximate solutions everywhere and match coefficients to make a global approximate solution. We have yet to approximate the solution near the classical turning points E = V(x).

Let us label a classical turning point x1. Now because we are near E = V(x1), we can expand \frac{2m}{\hbar^2}\left(V(x)-E\right) in a power series.

\frac{2m}{\hbar^2}\left(V(x)-E\right) = v_1 (x - x_1) + v_2 (x - x_1)^2 + \cdots

Let us only approximate to linear order \frac{2m}{\hbar^2}\left(V(x)-E\right) = v_1 (x - x_1)

\frac{d^2}{dx^2} \Psi(x) = v_1 (x - x_1) \Psi(x)

This differential equation looks deceptively simple. Its solutions are Airy functions.

\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)

Hopefully this solution should connect the far away and beneath solutions. Given the 2 coefficients on one side of the classical turning point, we should be able to determine the 2 coefficients on the other side of the classical turning point by using this local solution to connect them. We are able to find a relationship between C and C + ,C .

Fortunately the Airy function solutions will asymptote into sine, cosine and exponential functions in the proper limits. The relationship can be found as follows:

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

Now we can construct global solutions and solve tunnelling problems.

The transmission coefficient, \left| \frac{C_{\mbox{outgoing}}}{C_{\mbox{incoming}}} \right|^2, for a particle tunnelling through a single potential barrier is found to be

T = \frac{e^{-2\int_{x_1}^{x_2} dx \sqrt{\frac{2m}{\hbar^2} \left( V(x) - E \right)}}}{ \left( 1 + \frac{1}{4} e^{-2\int_{x_1}^{x_2} dx \sqrt{\frac{2m}{\hbar^2} \left( V(x) - E \right)}} \right)^2}

Where x1,x2 are the 2 classical turning points for the potential barrier. If we take the classical limit of all other physical parameters much larger than Planck's constant, abbreviated as \hbar \rightarrow 0, we see that the transmission coefficient correctly goes to zero. This classical limit would have failed in the unphysical, but much simpler to solve, situation of a square potential. A related subject is above barrier reflection: in classical physics a particle will not reflect if its energy is above the potential barrier, but in the quantum case it is possible. In this case, the reflection coefficient is exponentially small in Planck constant. The semiclassical technique of calculation of the reflection coefficient is similar to the calculation of the tunnelling described above.

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