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Home , Quantum tunnelling

Tunneling Applications

Can the effects of quantum tunneling be seen on the macroscopic scale?

Maxence Caron

Philippe Chatigny

Emile Demers

Contemporary Physics 203-BNL-LW

Gwendoline Simon

Champlain St-Lawrence College

April 15th 2016 Table of Content 1 Research Question ...... 2 2 Introduction to Quantum Tunneling ...... 2 2.1 The -Particle Duality of Matter ...... 2 2.2 Heisenberg's ...... 3 2.3 Schrödinger’s Equation and 1-dimensional tunneling ...... 5 3 Applications ...... 7 3.1 ...... 7 3.1.1 ...... 7 3.1.2 ...... 10 3.2 Technology ...... 11 3.2.1 Scanning Tunneling ...... 11 3.2.2 Tunnel ...... 13 3.3 Biology ...... 15 3.3.1 ...... 15 3.3.2 Vibration Theory of Olfaction ...... 17 4 Conclusion ...... 18 Bibliography ...... 19

1 1 Research Question

In contemporary physics, there still exists a clash between the classical and quantum approach to the prediction of the behavior of the universe. In the macroscopic realm, classical physics seems to prevail as it can model our universe. However, at the atomic scale, it is quantum physics which appears to accurately describe the behavior of particles. If a process which is impossible according to classical physics can have repercussions in our everyday life, the importance of adapting the current model to incorporate such predictions would be proven. An example of such classically impossible occurrence is quantum tunneling. The question is whether or not the effects of quantum tunneling manifest themselves in the macroscopic scale. If they do, it would prove the need for a reconciliation between the two models. In order to answer this question, the process and applications of quantum tunneling must be investigated.

2 Introduction to Quantum Tunneling

2.1 The Wave-Particle Duality of Matter

Fundamentally, every elementary particle and entity that can be found in its minimal amount

() exhibits both particle and wave properties. This was demonstrated in 1801 by

Thomas Young in his “Double-slit experiment”, in which is displayed as simultaneously behaving as a classical wave and particle. His experiment also displayed the fundamentally probabilistic of quantum (Encyclopaedia Britannica). However, this duality does not simply apply to quantum entities. Indeed, as proposed by , all matter constituted from these particles and entities also behave according to this wave-particle duality.

The “de Broglie wavelength” of a macroscopic object can be calculated using the following formula, where λ is the wavelength of the object, p its and h the

(Cohen-Tannoudji et al.).

2 Equation 1

ℎ � = �

Thus, for a macroscopic particle, the mass is so large that the momentum is always sufficiently large to make the de Broglie wavelength smaller than the range of experimental detection (R.

Eisberg and R. Resnick). This explains why, in the domain of the macroscopic, the intuitive seem to describe the universe accurately. There are, of course, exceptions to this rule as can indeed interfere with the larger scale, such as in the quantum tunnelling effect (Stanford Encyclopedia of Philosophy).

2.2 Heisenberg's uncertainty principle

In the realm of subatomic particles, the size of the de Broglie wavelength is comparatively large since the particle’s momentum is very small. Because of the small size and momentum of these particles, they more blatantly exhibit another property of quantum mechanics which is

Heisenberg's uncertainty principle. This principle, introduced in 1927 by physicist Werner

Heisenberg, states that for any particle in the quantum realm, both its position and momentum cannot be known simultaneously (D., Sen). More formally, this relation can be described by the following inequality where � is the standard deviation of the position, �is the standard deviation of the momentum and ħ is the reduced Planck constant (h/2π).

Equation 2

ħ � � ≥ 2

3 This inequality illustrates a fundamental principle of quantum mechanics. There is minimum uncertainty in nature that makes it impossible to know the exact position and momentum of a particle.

Heisenberg’s uncertainty can be illustrated by Heisenberg’s famous thought experiment called

“Heisenberg’s microscope”. Imagine you want to know the position and the momentum of a certain quantum particle using a microscope. In order to precisely know the position of this quantum particle a must be used. If a photon hits and is then reflected by a quantum object onto the microscope, the position of the object can be known. The precision of this position would then depend on the wavelength of the oscillating photon. For instance, if we take as our quantum particle an , we can then use as our medium photon a gamma ray since its wavelength is smaller than the size of the photon. Furthermore, we know, from Einstein, the following equation, where E is the of the photon, h is Planck’s constant, c the and � the wavelength:

Equation 3

ℎ� � = �

Thus, if our medium photon has a very small wavelength, which is required in order to have an uncertainty smaller than the size of the observed particle, it also must have a very large energy.

Because this additional energy must be converted into kinetic energy, the medium photon will significantly affect the momentum of the observed particle. This will result in a larger uncertainty for the momentum of the quantum particle. Hence, Heisenberg’s uncertainty is coherent with de Broglie’s wavelength since the instruments used to measure the position and the momentum of a macroscopic object are not precise enough for their uncertainty to be observed.

4 The uncertainty of the instruments, in this case, would be greater than the standard deviation of both the position and the momentum of the object.

2.3 Schrödinger’s Equation and 1-dimensional tunneling

Schrodinger’s equation is quantum physics’ analogue to Newton’s second law of as they both are used to observe the effect of time on a system. Where Schrodinger’s differs is that, as previously illustrated, quantum physics can only predict via possibilities, that is to say, only to probabilistically predict the effect of time on a system. Indeed, as illustrated by Heisenberg’s uncertainty, there is a fundamental uncertainty attributed to the momentum of a quantum object before it is observed. Momentum is only one of the possible quantum state vectors. Thus,

Schrodinger’s equation must describe many potential outcomes in the form of wave functions derived from a single quantum state (Schrödinger, E).

Schrodinger’s equations can be used to demonstrate the effect of 1-dimensional quantum tunneling through a potential barrier, as shown in Graph 1.

Graph 1

(Graph 1 was drawn using paint.net)

5 Indeed, from a classical standpoint, to overcome the potential barrier V0 of length a, a particle must have enough kinetic energy, E. This means that if � < �, the particle will not be able to cross the potential barrier. On the other hand, if � > �, the particle will assuredly cross the barrier.

Yet quantum mechanics portray the situation quite differently. From Schrodinger’s equation, we can obtain the equation transmission probability, T, which represents the probability that the particle will cross a certain potential barrier. This relation is represented by Equation 4, where V0 is the potential barrier, E the energy of the particle, m the mass of the object, a the length of the barrier and ħ the reduced plank’s constant (Sharma, B).

Equation 4

1 � = � = ∗ ∗ � ∗ sinh ħ 1 + 4 ∗ � � − �

Equation 4 is compatible with Heisenberg’s approach since it demonstrates a reduced tunneling probability for macroscopic objects. Indeed, the magnitude of T decreases exponentially with mass, which correlates with the de Broglie wavelength and with the length of the potential barrier, meaning tunneling is less likely on potential barrier of a larger size. More importantly, the inequality in Equation 5 can be derived from equation 4:

Equation 5

0 < � < 1

Equation 5 means that even if � ≪ �, there will always be a small chance that the object will tunnel through the potential barrier, as T must be greater than 0. Conversely, even if � ≫ �,

6 there is a small chance that the particle will be reflected by potential barrier. This is represented by Graph 2.

Graph 2

(Graph 2 was drawn using paint.net)

In graph 2, a particle, represented by the red , has a total energy E inferior to the required energy to pass the potential barrier Vo. Yet, as its transmission probability is superior to

0, there is a probability that the particle will cross the classically forbidden barrier. If the particle does cross, its will have exponentially decayed in the process, but its frequency will be conserved, meaning the “de Broglie” wavelength will be conserved (Mohsen, Razavy)

Thus, the Schrödinger equation manifests the intuitiveness of tunneling and its contradiction with classical approaches.

3 Applications

3.1 Nuclear Physics

3.1.1 Alpha Decay

Alpha decay is a physical phenomenon which requires quantum tunneling to be carried out. This type of nuclear reaction occurs when an unstable element emits an to become a smaller, more stable, element. An alpha particle consists of two protons and two neutrons bound

7 together into a particle identical to a helium nucleus. For an alpha particle to be emitted, it must be bound by the residual strong force to a larger, parent nucleus. Inside of its bounds, this particle, which is in fact a helium particle, may move about in its restricted space. Because of the residual strong force of the surrounding neutrons and protons, the energy required for the alpha particle to escape this potential barrier is greater than even its maximum kinetic energy. For instance, an alpha particle in a Polonium-212 carries an average kinetic energy of approximately 9MeV while the energy required for it to leave the nucleus classically is of 26.4

MeV (Rohlf, James William). Graph 3 illustrates this .

Graph 3

(Source : http://abyss.uoregon.edu/~js/glossary/quantum_tunneling.html)

In graph 3, the x-axis is the radial distance between the center of the parent particle and the alpha particle and the y-axis represents the energy requirement for alpha particle to increase its radial distance. E represent the average kinetic energy of the alpha particle, which is well below the

8 required energy needed for it to exit the boundaries of the nucleus. Indeed, the alpha particle must overcome the residual strong force, represented by the central potential well. After the peak is crossed, the electromagnetic attraction between the protons and of the alpha particle and those in the initial nuclei decrease with distance, following Coulomb’s inverse square law.

Thus, from a classical perspective, alpha decays would not be possible, since the alpha particle does not have the required energy to escape the parent nuclei.

This is where quantum tunneling comes in and allows a small probability for the formation of a

“tunnel” to be formed across the central peak in the graph 3, allowing the alpha particle to escape

(Serot. O). However, for quantum tunneling to have a significant chance of occurring, specific condition must be met, or else stable would continuously emit alpha particles. As previously demonstrated in equation 4, a particle always has a slight change to tunnel through a potential barrier. The amplitude of this statistical chance depends on the mass and size of the particle and, more importantly, on the absolute value of the difference between the particle’s kinetic energy and the required of the barrier. Thus, a particle has an exponentially greater chance of tunneling if the said particle has a small mass, if the distance that must be tunneled is small and if |V0-E| is small. In a large nucleus that is about to undergo alpha decay, all of these conditions are met because the alpha particle has a small mass, the distance is of about 5-10 femtometers, depending on the atomic mass (Eisberg, Robert), and the farthest particles from the center of mass have a high kinetic energy because the atom is unstable and the residual strong force is weaker at the edge of the parent nucleus. Thus, only in large, unstable atoms will alpha particles have a significant chance of tunneling through the nucleic potential barrier.

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Because of quantum tunneling, this phenomenon has noticeable impacts in our daily lives, particularly in technology. Alpha tunneling is used to create a constant flow of current in chambers in ionization smoke detectors through the decay mechanism of Americium-

241 (Fleming, Joseph M.) and to power artificial heart pacemakers (LANL).

3.1.2 Stellar Nucleosynthesis

Nuclear fusion is defined as a nuclear reaction in which two atoms collide at very high speeds and form another, bigger atom. Such reaction may only occur when both atoms go at very high velocities because they must be close enough such that the residual strong force becomes greater than the coulomb force (Yoon, Jin-Hee) in the region before the potential well in graph 3, which corresponds to a distance of about 10 femtometers in (Eisberg, R.). Such nuclear reactions are known to occur in stars and supernovas, which are responsible for the creation of the heavy elements we can find on Earth (Chaisson, Eric). In space, the only available energy the atoms of a star have to overcome Coulomb’s force is the converted gravitational potential energy. This energy is converted, via friction, to kinetic energy in the form of thermal motion

(Böhm-Vitense, E). If we assume that the stellar gas at the core of massive stars, with temperatures in the range of ≈107K (Hathaway, David H.), is a perfect gas (Schneider, Stephen), we can convert it into an average energy using the following equation, where Ek is the average kinetic energy, k is Boltzmann’s constant of 8.617385 x 10-5 eV/K and T is the temperature in kelvin.

Equation 6

3 � = ∗ � ∗ � 2

10 This yields an average kinetic energy of about ≈1keV which 1000 times lower than the required

≈1MeV (Trixler, Frank) needed to overcome coulomb’s barrier. Yet, the kinetic energy in the particles can get them significantly close for quantum tunneling to close the gap. Such occurrence of tunneling is statistically significant because atoms in a star’s core are light and are found in astronomical quantity, meaning atoms bump into each other quite frequently. Without tunneling, fusion could only occur in stars whose core temperature is in the realm of 1010 Kelvins

(107Kelvin*103) which might only occur in supermassive stars if possible at all. Hence, quantum tunneling played a key role in the development of our solar system, by making the sun’s possible.

3.2 Technology

3.2.1 Scanning Tunneling Microscope

Scanning tunneling use quantum tunneling to electronically map objects that are very small with a resolution of 0.1nm by 0.1nm laterally and 0.01nm in depth (Bai, Chunli). This technique, developed in 1986 by Gerd Binning and , can be used to observe atoms of a sample in a vacuum, as their size ranges from 0.1 to 0.5 nanometers (Robert H.

March). It is done by approaching an electrically charged tip fixed to a piezoelectric tube, which allows 3-dimensional control of the tip, to the sample to be observed. The tip of the microscope, as it is supplied an , becomes an that attracts electrons towards the surface of the sample, without directly touching the atoms.

From a classical perspective, the probability that an electron would be transferred from the atom to the tip is supposed to linearly increase as the tip approaches because the classical electron

11 radius assumes that electrons are distributed uniformly around the nucleus. However, it can be observed that the probability of such electron transfer actually increases exponentially as the tip approaches the atom due to quantum tunneling. This exponential increase in probability is proportional to the current detected by the microscope since a greater probability of electron tunneling to the tip is synonymous to a greater flow of electrons to the tip. The tunneling current is proportional to a power of the tunneling voltage, which is the difference of potential between the tip and the atom. This relation is described according to equation 7, where JT is the tunneling current, VT is the tunneling voltage, ∅ is the average barrier height () in eV, s is the distance of the gap between the two interacting together, and A is a constant.

Equation 7

∅/ � ∝ �

This equation illustrate the two methods that can be used to map a surface down to the atomic scale. The “constant height mode” method is used by applying a constant voltage � and keeping the tip at a constant height, while moving in the other 2 such that the distance � can be obtained for any coordinate by knowing the current JT. Alternatively, the “constant current mode” is used if the microscope is programmed to adjust the distance � via the piezotube so that the current � is constant, which allows the distance � to be obtained for any set of x,y,z coordinate by simply knowing the voltage � (Binning, G.). The “constant height” method is mainly used to rapidly map a surface at the cost of precision, as it does not require computer tip adjustment. Conversely, the “constant current” method is used to precisely map a surface, at the cost of a longer mapping time. Thus, quantum tunneling can be successfully used to accurately map surfaces down to very high resolutions, which was previously unheard of via photon magnification methods.

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3.2.2 Tunnel Diodes

Diodes are components of an electrical circuit which conduct electricity mainly in one direction.

A p-n junction is a type of consisting of two pure semi-conductors in which a voltage is applied, creating both a positively and negatively charged , named p-site and n- site respectively. At the atomic scale, the “n” site contains most of the excited electrons while the

“p” site contain most of the valence sites to which they can bound, called the holes. Between the two parts is placed an electric potential barrier, like an isolating material for instance (D.

Walsh.). From a classical perspective, the system would have to amass the required voltage to overcome the potential barrier, such that electrons can be exchanged between the two diodes, creating a current. Thus, there should be no current between the n and p sections until electrons start to have enough energy to overcome the potential barrier.

Tunnel diodes are a special type of p-n junctions in which quantum tunneling occurs. Such diodes are made of 2 highly doped semi-conductors, meaning that they contain a high number of impurities, which alters their electrical properties. One particular property of a doped semi- conductor is that its number of excited electrons in the “n” semi-conductor is not the same as the number of available holes for the electrons in the “p” semi-conductor. Since � ≠ �, there is a local coulomb force created by the residual charge between the n and p. This local charge exists even though the system is at thermodynamic equilibrium, meaning there is no net exchange of electrons between the 2 doped semi-conductors (Weller, Paul). Another requirement for tunnel diodes is that a part of the potential barrier between p and n is made very thin, increasing the tunneling statistical chances of occurring.

13 Graph 4 below describes, in blue, the behavior of the when a conventional forward bias is applied, meaning there is an applied voltage such that the electrons are going from the “n” part to the “p” part. Simultaneously, in red, a conventional p-n diode is showed to contrast the two.

Graph 4

(Graph 4 was drawn using paint.net)

In graph 4, the current output of the tunnel diode when no voltage is applied is of 0 as the system is in thermodynamic equilibrium. Then, in region 1, in which a voltage is applied, the system is no longer in thermodynamic equilibrium. Since p is a doped , it has some additional free electron holes which can be filled with the electron excited by the external voltage. Since the potential barrier between the two semiconductors is very small and because the electrons have a very small mass, electrons have a chance of tunnelling, creating a current. At

Vp, the first current peak in the graph, there are exactly as many tunneling electrons as there are holes. In region 2, where even more voltage is applied, there are now more excited electrons as there are holes. These extra electrons, which cannot find a valence hole to fill, create a local coulomb force which slows down the electron exchange between n and p, decreasing the current.

14 This region corresponds to a “negative resistance” since increasing the voltage decreased the current, a rare property amongst diodes, which is partly why tunnel diodes are unique. Vv corresponds to the maximum current that can be applied on the diode in which electrons can tunnel without being able to regularly overcome the potential barrier. Thus, the large quantity of excited electrons without a hole to fill decreases the current to a local minimum. In region 3, some of the electrons have enough energy to regularly overcome the potential barrier, by being conducted in the isolating material (Kebeer).

The most significant property of a tunnel diode is that they can generate high variations in current from a constant increase of voltage because of negative resistance. This allows them to generate power oscillations of high frequencies quite easily. Overall, other properties include stability with respect to temperature changes, modest power supply requirement and small size

(Hebb, Malcom H). Their capacity to generate high frequencies from the rapid changes in current allows them to be used in oscillators which, in turn, allows circuits with tunnel diodes to reach frequencies often the microwave range and above (Chattopadhyay, D.). Also, their good stability and critical environment survivability added to their high frequency capabilities means that they are used in radio communications. More precisely, they are used in satellites, allowing for telephones, radios, and the internet and cellphone data.

3.3 Biology

3.3.1 Photosynthesis

Photosynthesis is the process in which an organism converts energy from a light source into chemical potential energy, which can later be used by it. As one of the organism’s pigment

15 molecules captures an incident photon, the pigment shifts into an in which electrons are produced. For the electrons to be used, they must be transported from outside the mitochondria to its interior, in the cytoplasm by traversing a 4nm wide mitochondrial cell membrane. Assuming that the electrons vibrate at the frequency of blue light, which is color the chlorophyll pigment absorbs the most, it would take approximately 1016 years for the electron to tunnel through the membrane (Jones, Lauren V). Thus, for the tunneling process to be efficient, the electrons must tunnel through 4 distinct dye units imbedded in the membrane between which the electrons may jump, effectively reducing the tunneling distance. The arrangement of these pigments enables an exponentially more efficient excitation transfer, allowing the electrons to achieve the process in 4x10−5 seconds (Arndta, Markus).

Additionally, it is important that the electrons do not back flow, that is, they must not tunnel from the mitochondria’s cytoplasm back to the pigment molecule. This is prevented by the fact that each jump corresponds to a lower energy well (Jones, Lauren V), thus increasing the difference between the particle’s energy and barrier’s potential energy requirement, � − � , which was shown to decrease the transmission probability in Equation 4. In other words, the electrons have exponentially more probability to flow from the chlorophyll to the mitochondria than the other way around, which means that electrons will flow, on average, in one direction.

The presence of quantum tunneling in such an important biological process demonstrates its omnipresence in our lives.

16 3.3.2 Vibration Theory of Olfaction

The quantum tunneling effect might also be present in the , through the olfactory system. The way we smell different molecules may partially be explained by the lock and key mechanism in which olfactory receptors of certain shapes and sizes can only recognize a specific molecule. The majority of smells can be explained this way, where an odor molecule, the key, binds to its appropriate, specific receptor combination, the lock, which depends on the size, shape and chemical group of the molecule (Arndta, Markus). However, this mechanism cannot explain other aspects of smell like the fact that drosophila have only 62 olfactory sites (Franco,

Maria Isabel), yet they can differentiate between more than 62 types of smell. Moreover, drosophilae can differentiate some by smell, like hydrogen and deuterium. For these reasons, the lock and key model cannot hold.

An alternative model to the “lock and key” theory is the vibration theory of olfaction in which smell is detected partly by the shape of the molecule and partly from the vibration of its electrons. This principle relies on the fact that electrons from an odorant molecule could tunnel onto a specific, electrically-charged amino acid residue within the olfactory receptor (Turin,

Luca.). In other words, the olfactory system could detect the molecular vibration of the odorant molecule via its tunneling electrons, a sort of “non-optical form of vibrational spectroscopy”.

This could complete the lock and key theory because it will allow for a theoretically infinite amount of distinct smells from the different combinations of electron vibrations and olfactory receptors. This would also explain how isotopes of hydrogen can be differentiated by smell, since deuterium does not have the same molecular vibration as hydrogen, for instance. Thus,

17 quantum tunneling might allow insects, and even animals, to smell a larger range of odor molecules.

4 Conclusion

From its various applications in the domain of nuclear physics, technology and biology, quantum tunneling was shown to have macroscopic impacts in our lives. Indeed, it can be argued that quantum tunneling was key factor in the development of life on earth, and arguably a requirement for life to exist in the universe, as it is required in nuclear fusion for instance.

Therefore, reconciling classical physics and quantum physics is primordial in our “theory of everything”, because overlooking seemingly negligible effect such as tunneling would lead to major macroscopic paradoxes, such as the impossibility of life in such theory.

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