Tag / Quantum tunneling

by David D. Nolte

The Ubiquitous George Uhlenbeck

There are sometimes individuals who seem always to find themselves at the focal points of their times.  The physicist George Uhlenbeck was one of these individuals, showing up at all the right times in all the right places at the dawn of modern physics in the 1920’s and 1930’s. He studied under Ehrenfest and Bohr and Born, and he was friends with Fermi and Oppenheimer and Oskar Klein.  He taught physics at the universities at Leiden, Michigan, Utrecht, Columbia, MIT and Rockefeller.  He was a wide-ranging theoretical physicist who worked on Brownian motion, early string theory, quantum tunneling, and the master equation.  Yet he is most famous for the very first thing he did as a graduate student—the discovery of the quantum spin of the electron.

Electron Spin

G. E. Uhlenbeck, and S. Goudsmit, “Spinning electrons and the structure of spectra,” Nature 117, 264-265 (1926).

George Uhlenbeck (1900 – 1988) was born in the Dutch East Indies, the son of a family with a long history in the Dutch military [1].  After the father retired to The Hague, George was expected to follow the family tradition into the military, but he stumbled onto a copy of H. Lorentz’ introductory physics textbook and was hooked.  Unfortunately, to attend university in the Netherlands at that time required knowledge of Greek and Latin, which he lacked, so he entered the Institute of Technology in Delft to study chemical engineering.  He found the courses dreary. 

Fortunately, he was only a few months into his first semester when the language requirement was dropped, and he immediately transferred to the University of Leiden to study physics.  He tried to read Boltzmann, but found him opaque, but then read the famous encyclopedia article by the husband and wife team of Paul and Tatiana Ehrenfest on statistical mechanics (see my Physics Today article [2]), which became his lifelong focus.

After graduating, he continued into graduate school, taking classes from Ehrenfest, but lacking funds, he supported himself by teaching classes at a girls high school, until he heard of a job tutoring the son of the Dutch ambassador to Italy.  He was off to Rome for three years, where he met Enrico Fermi and took classes from Tullio Bevi-Cevita and Vito Volterra.

However, he nearly lost his way.  Surrounded by the rich cultural treasures of Rome, he became deeply interested in art and was seriously considering giving up physics and pursuing a degree in art history.  When Ehrenfest got wind of this change in heart, he recalled Uhlenbeck in 1925 to the Netherlands and shrewdly paired him up with another graduate student, Samuel Goudsmit, to work on a new idea proposed by Wolfgang Pauli a few months earlier on the exclusion principle.

Pauli had explained the filling of the energy levels of atoms by introducing a new quantum number that had two values.  Once an energy level was filled by two electrons, each carrying one of the two quantum numbers, this energy level “excluded” any further filling by other electrons. 

To Uhlenbeck, these two quantum numbers seemed as if they must arise from some internal degree of freedom, and in a flash of insight he imagined that it might be caused if the electron were spinning.  Since spin was a form of angular momentum, the spin degree of freedom would combine with orbital angular momentum to produce a composite angular momentum for the quantum levels of atoms.

The idea of electron spin was not immediately embraced by the broader community, and Bohr and Heisenberg and Pauli had their reservations.  Fortunately, they all were traveling together to attend the 50th anniversary of Lorentz’ doctoral examination and were met at the train station in Leiden by Ehrenfest and Einstein.  As usual, Einstein had grasped the essence of the new physics and explained how the moving electron feels an induced magnetic field which would act on the magnetic moment of the electron to produce spin-orbit coupling.  With that, Bohr was convinced.

Uhlenbeck and Goudsmit wrote up their theory in a short article in Nature, followed by a short note by Bohr.  A few months later, L. H. Thomas, while visiting Bohr in Copenhagen, explained the factor of two that appears in (what later came to be called) Thomas precession of the electron, cementing the theory of electron spin in the new quantum mechanics.

5-Dimensional Quantum Mechanics

P. Ehrenfest, and G. E. Uhlenbeck, “Graphical illustration of De Broglie’s phase waves in the five-dimensional world of O Klein,” Zeitschrift Fur Physik 39, 495-498 (1926).

Around this time, the Swedish physicist Oskar Klein visited Leiden after returning from three years at the University of Michigan where he had taken advantage of the isolation to develop a quantum theory of 5-dimensional spacetime.  This was one of the first steps towards a grand unification of the forces of nature since there was initial hope that gravity and electromagnetism might both be expressed in terms of the five-dimensional space.

An unusual feature of Klein’s 5-dimensional relativity theory was the compactness of the fifth dimension, in which it was “rolled up” into a kind of high-dimensional string with a tiny radius.  If the 4-dimensional theory of spacetime was sometimes hard to visualize, here was an even tougher problem.

Uhlenbeck and Ehrenfest met often with Klein during his stay in Leiden, discussing the geometry and consequences of the 5-dimensional theory.  Ehrenfest was always trying to get at the essence of physical phenomena in the simplest terms.  His famous refrain was “Was ist der Witz?” (What is the point?) [1].  These discussions led to a simple paper in Zeitschrift für Physik published later that year in 1926 by Ehrenfest and Uhlenbeck with the compelling title “Graphical Illustration of De Broglie’s Phase Waves in the Five-Dimensional World of O Klein”.  The paper provided the first visualization of the 5-dimensional spacetime with the compact dimension.  The string-like character of the spacetime was one of the first forays into modern day “string theory” whose dimensions have now expanded to 11 from 5.

During his visit, Klein also told Uhlenbeck about the relativistic Schrödinger equation that he was working on, which would later become the Klein-Gordon equation.  This was a near miss, because what the Klein-Gordon equation was missing was electron spin—which Uhlenbeck himself had introduced into quantum theory—but it would take a few more years before Dirac showed how to incorporate spin into the theory.

Brownian Motion

G. E. Uhlenbeck and L. S. Ornstein, “On the theory of the Brownian motion,” Physical Review 36, 0823-0841 (1930).

After spending time with Bohr in Copenhagen while finishing his PhD, Uhlenbeck visited Max Born at Göttingen where he met J. Robert Oppenheimer who was also visiting Born at that time.  When Uhlenbeck traveled to the United States in late summer of 1927 to take a position at the University of Michigan, he was met at the dock in New York by Oppenheimer.

Uhlenbeck was a professor of physics at Michigan for eight years from 1927 to 1935, and he instituted a series of Summer Schools [3] in theoretical physics that attracted international participants and introduced a new generation of American physicists to the rigors of theory that they previously had to go to Europe to find. 

In this way, Uhlenbeck was part of a great shift that occurred in the teaching of graduate-level physics of the 1930’s that brought European expertise to the United States.  Just a decade earlier, Oppenheimer had to go to Göttingen to find the kind of education that he needed for graduate studies in physics.  Oppenheimer brought the new methods back with him to Berkeley, where he established a strong theory department to match the strong experimental activities of E. O. Lawrence.  Now, European physicists too were coming to America, an exodus accelerated by the increasing anti-Semitism in Europe under the rise of fascism. 

During this time, one of Uhlenbeck’s collaborators was L. S. Ornstein, the director of the Physical Laboratory at the University of Utrecht and a founding member of the Dutch Physical Society.  Uhlenbeck and Ornstein were both interested in the physics of Brownian motion, but wished to establish the phenomenon on a more sound physical basis.  Einstein’s famous paper of 1905 on Brownian motion had made several Einstein-style simplifications that stripped the complicated theory to its bare essentials, but had lost some of the details in the process, such as the role of inertia at the microscale.

Uhlenbeck and Ornstein published a paper in 1930 that developed the stochastic theory of Brownian motion, including the effects of particle inertia. The stochastic differential equation (SDE) for velocity is

where γ is viscosity, Γ is a fluctuation coefficient, and dw is a “Wiener process”. The Wiener differential dw has unusual properties such that

Uhlenbeck and Ornstein solived this SDE to yield an average velocity

which decays to zero at long times, and a variance

that asymptotes to a finite value at long times. The fluctuation coefficient is thus given by

for a process with characteristic speed v0. An estimate for the fluctuation coefficient can be obtained by considering the force F on an object of size a

For instance, for intracellular transport [4], the fluctuation coefficient has a rough value of Γ = 2 Hz μm2/sec2.

Quantum Tunneling

D. M. Dennison and G. E. Uhlenbeck, “The two-minima problem and the ammonia molecule,” Physical Review 41, 313-321 (1932).

By the early 1930’s, quantum tunnelling of the electron through classically forbidden regions of potential energy was well established, but electrons did not have a monopoly on quantum effects.  Entire atoms—electrons plus nucleus—also have quantum wave functions and can experience regions of classically forbidden potential.

Uhlenbeck, with David Dennison, a fellow physicist at Ann Arbor, Michigan, developed the first quantum theory of molecular tunneling for the molecular configuration of ammonia NH3 that can tunnel between the two equivalent configurations. Their use of the WKB approximation in the paper set the standard for subsequent WKB approaches that would play an important role in the calculation of nuclear decay rates.

Master Equation

A. Nordsieck, W. E. Lamb, and G. E. Uhlenbeck, “On the theory of cosmic-ray showers I. The furry model and the fluctuation problem,” Physica 7, 344-360 (1940)

In 1935, Uhlenbeck left Michigan to take up the physics chair recently vacated by Kramers at Utrecht.  However, watching the rising Nazism in Europe, he decided to return to the United States, beginning as a visiting professor at Columbia University in New York in 1940.  During his visit, he worked with W. E. Lamb and A. Nordsieck on the problem of cosmic ray showers. 

Their publication on the topic included a rate equation that is encountered in a wide range of physical phenomena. They called it the “Master Equation” for ease of reference in later parts of the paper, but this phrase stuck, and the “Master Equation” is now a standard tool used by physicists when considering the balances among multiples transitions.

Uhlenbeck never returned to Europe, moving among Michigan, MIT, Princeton and finally settling at Rockefeller University in New York from where he retired in 1971.

Selected Works by George Uhlenbeck:

G. E. Uhlenbeck, and S. Goudsmit, “Spinning electrons and the structure of spectra,” Nature 117, 264-265 (1926).

P. Ehrenfest, and G. E. Uhlenbeck, “On the connection of different methods of solution of the wave equation in multi dimensional spaces,” Proceedings of the Koninklijke Akademie Van Wetenschappen Te Amsterdam 29, 1280-1285 (1926).

P. Ehrenfest, and G. E. Uhlenbeck, “Graphical illustration of De Broglie’s phase waves in the five-dimensional world of O Klein,” Zeitschrift Fur Physik 39, 495-498 (1926).

G. E. Uhlenbeck, and L. S. Ornstein, “On the theory of the Brownian motion,” Physical Review 36, 0823-0841 (1930).

D. M. Dennison, and G. E. Uhlenbeck, “The two-minima problem and the ammonia molecule,” Physical Review 41, 313-321 (1932).

E. Fermi, and G. E. Uhlenbeck, “On the recombination of electrons and positrons,” Physical Review 44, 0510-0511 (1933).

A. Nordsieck, W. E. Lamb, and G. E. Uhlenbeck, “On the theory of cosmic-ray showers I The furry model and the fluctuation problem,” Physica 7, 344-360 (1940).

M. C. Wang, and G. E. Uhlenbeck, “On the Theory of the Brownian Motion-II,” Reviews of Modern Physics 17, 323-342 (1945).

G. E. Uhlenbeck, “50 Years of Spin – Personal Reminiscences,” Physics Today 29, 43-48 (1976).

Notes:

[1] George Eugene Uhlenbeck: A Biographical Memoire by George Ford (National Academy of Sciences, 2009). https://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/uhlenbeck-george.pdf

[2] D. D. Nolte, “The tangled tale of phase space,” Physics Today 63, 33-38 (2010).

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[3] One of these was the famous 1948 Summer School session where Freeman Dyson met Julian Schwinger after spending days on a cross-country road trip with Richard Feynman. Schwinger and Feynman had developed two different approaches to quantum electrodynamics (QED), which Dyson subsequently reconciled when he took up his position later that year at Princeton’s Institute for Advanced Study, helping to launch the wave of QED that spread out over the theoretical physics community.

[4] D. D. Nolte, “Coherent light scattering from cellular dynamics in living tissues,” Reports on Progress in Physics 87 (2024).

by David D. Nolte

A Short History of Quantum Tunneling

Quantum physics is often called “weird” because it does things that are not allowed in classical physics and hence is viewed as non-intuitive or strange.  Perhaps the two “weirdest” aspects of quantum physics are quantum entanglement and quantum tunneling.  Entanglement allows a particle state to extend across wide expanses of space, while tunneling allows a particle to have negative kinetic energy.  Neither of these effects has a classical analog.

Quantum entanglement arose out of the Bohr-Einstein debates at the Solvay Conferences in the 1920’s and 30’s, and it was the subject of a recent Nobel Prize in Physics (2022).  The quantum tunneling story is just as old, but it was recognized much earlier by the Nobel Prize in 1972 when it was awarded to Brian Josephson, Ivar Giaever and Leo Esaki—each of whom was a graduate student when they discovered their respective effects and two of whom got their big idea while attending a lecture class. 

Always go to class, you never know what you might miss, and the payoff is sometimes BIG

Ivar Giaever

Of the two effects, tunneling is the more common and the more useful in modern electronic devices (although entanglement is coming up fast with the advent of quantum information science). Here is a short history of quantum tunneling, told through a series of publications that advanced theory and experiments.

Double-Well Potential: Friedrich Hund (1927)

The first analysis of quantum tunneling was performed by Friedrich Hund (1896 – 1997), a German physicist who studied early in his career with Born in Göttingen and Bohr in Copenhagen.  He published a series of papers in 1927 in Zeitschrift für Physik [1] that solved the newly-proposed Schrödinger equation for the case of the double well potential.  He was particularly interested in the formation of symmetric and anti-symmetric states of the double well that contributed to the binding energy of atoms in molecules.  He derived the first tunneling-frequency expression for a quantum superposition of the symmetric and anti-symmetric states

where f is the coherent oscillation frequency, V is the height of the potential and hν is the quantum energy of the isolated states when the atoms are far apart.  The exponential dependence on the potential height V made the tunnel effect extremely sensitive to the details of the tunnel barrier.

Fig. 1 Friedrich Hund

Electron Emission: Lothar Nordheim and Ralph Fowler (1927 – 1928)

The first to consider quantum tunneling from a bound state to a continuum state was Lothar Nordheim (1899 – 1985), a German physicist who studied under David Hilbert and Max Born at Göttingen and worked with John von Neumann and Eugene Wigner and later with Hans Bethe. In 1927 he solved the problem of a particle in a well that is separated from continuum states by a thin finite barrier [2]. Using the new Schrödinger theory, he found transmission coefficients that were finite valued, caused by quantum tunneling of the particle through the barrier. Nordheim’s use of square potential wells and barriers are now, literally, textbook examples that every student of quantum mechanics solves. (For a quantum simulation of wavefunction tunneling through a square barrier see the companion Quantum Tunneling YouTube video.) Nordheim later escaped the growing nationalism and anti-semitism in Germany in the mid 1930’s to become a visiting professor of physics at Purdue University in the United States, moving to a permanent position at Duke University.

Fig. 2 Nordheim square tunnel barrier and Fowler-Nordheim triangular tunnel barrier for electron tunneling from bound states into the continuum.

One of the giants of mathematical physics in the UK from the 1920s through the 1930’s was Ralph Fowler (1889 – 1944). Three of his doctoral students went on to win Nobel Prizes (Chandrasekhar, Dirac and Mott) and others came close (Bhabha, Hartree, Lennard-Jones). In 1928 Fowler worked with Nordheim on a more realistic version of Nordheim’s surface electron tunneling that could explain thermionic emission of electrons from metals under strong electric fields. The electric field modified Nordheim’s square potential barrier into a triangular barrier (which they treated using WKB theory) to obtain the tunneling rate [3]. This type of tunnel effect is now known as Fowler-Nordheim tunneling.

Nuclear Alpha Decay: George Gamow (1928)

George Gamov (1904 – 1968) is one of the icons of mid-twentieth-century physics. He was a substantial physicist who also had a solid sense of humor that allowed him to achieve a level of cultural popularity shared by a few of the larger-than-life physicists of his time, like Richard Feynman and Stephen Hawking. His popular books included One Two Three … Infinity as well as a favorite series of books under the rubric of Mr. Tompkins (Mr. Tompkins in Wonderland and Mr. Tompkins Explores the Atom, among others). He also wrote a history of the early years of quantum theory (Thirty Years that Shook Physics).

In 1928 Gamow was in Göttingen (the Mecca of early quantum theory) with Max Born when he realized that the radioactive decay of Uranium by alpha decay might be explained by quantum tunneling. It was known that nucleons were bound together by some unknown force in what would be an effective binding potential, but that charged alpha particles would also feel a strong electrostatic repulsive potential from a nucleus. Gamow combined these two potentials to create a potential landscape that was qualitatively similar to Nordheim’s original system of 1927, but with a potential barrier that was neither square nor triangular (like the Fowler-Nordheim situation).

Fig. 3 George Gamow

Gamow was able to make an accurate approximation that allowed him to express the decay rate in terms of an exponential term

where Zα is the atomic charge of the alpha particle, Z is the nuclear charge of the Uranium decay product and v is the speed of the alpha particle detected in external measurements [4].

The very next day after Gamow submitted his paper, Ronald Gurney and Edward Condon of Princeton University submitted a paper [5] that solved the same problem using virtually the same approach … except missing Gamow’s surprisingly concise analytic expression for the decay rate.

Molecular Tunneling: George Uhlenbeck (1932)

Because tunneling rates depend inversely on the mass of the particle tunneling through the barrier, electrons are more likely to tunnel through potential barriers than atoms. However, hydrogen is a particularly small atom and is therefore the most amenable to experiencing tunneling.

The first example of atom tunneling is associated with hydrogen in the ammonia molecule NH3. The molecule has a pyramidal structure with the Nitrogen hovering above the plane defined by the three hydrogens. However, an equivalent configuration has the Nitrogen hanging below the hydrogen plane. The energies of these two configurations are the same, but the Nitrogen must tunnel from one side of the hydrogen plane to the other through a barrier. The presence of light-weight hydrogen that can “move out of the way” for the nitrogen makes this barrier very small (infrared energies). When the ammonia is excited into its first vibrational excited state, the molecular wavefunction tunnels through the barrier, splitting the excited level by an energy associated with a wavelength of 1.2 cm which is in the microwave. This tunnel splitting was the first microwave transition observed in spectroscopy and is used in ammonia masers.

Fig. 4 Nitrogen inversion in the ammonia molecule is achieved by excitation to a vibrational excited state followed by tunneling through the barrier, proposed by George Uhlenbeck in 1932.

One of the earliest papers [6] written on the tunneling of nitrogen in ammonia was published by George Uhlenbeck in 1932. George Uhlenbeck (1900 – 1988) was a Dutch-American theoretical physicist. He played a critical role, with Samuel Goudsmit, in establishing the spin of the electron in 1925. Both Uhlenbeck and Goudsmit were close associates of Paul Ehrenfest at Leiden in the Netherlands. Uhlenbeck is also famous for the Ornstein-Uhlenbeck process which is a generalization of Einstein’s theory of Brownian motion that can treat active transport such as intracellular transport in living cells.

Solid-State Electron Tunneling: Leo Esaki (1957)

Although the tunneling of electrons in molecular bonds and in the field emission from metals had been established early in the century, direct use of electron tunneling in solid state devices had remained elusive until Leo Esaki (1925 – ) observed electron tunneling in heavily doped Germanium and Silicon semiconductors. Esaki joined an early precursor of Sony electronics in 1956 and was supported to obtain a PhD from the University of Tokyo. In 1957 he was working with heavily-doped p-n junction diodes and discovered a phenomenon known as negative differential resistance where the current through an electronic device actually decreases as the voltage increases.

Because the junction thickness was only about 100 atoms, or about 10 nanometers, he suspected and then proved that the electronic current was tunneling quantum mechanically through the junction. The negative differential resistance was caused by a decrease in available states to the tunneling current as the voltage increased.

Fig. 5 Esaki tunnel diode with heavily doped p- and n-type semiconductors. At small voltages, electrons and holes tunnel through the semiconductor bandgap across a junction that is only about 10 nm wide. Ht higher voltage, the electrons and hole have no accessible states to tunnel into, producing negative differential resistance where the current decreases with increasing voltage.

Esaki tunnel diodes were the fastest semiconductor devices of the time, and the negative differential resistance of the diode in an external circuit produced high-frequency oscillations. They were used in high-frequency communication systems. They were also radiation hard and hence ideal for the early communications satellites. Esaki was awarded the 1973 Nobel Prize in Physics jointly with Ivar Giaever and Brian Josephson.

Superconducting Tunneling: Ivar Giaever (1960)

Ivar Giaever (1929 – ) is a Norwegian-American physicist who had just joined the GE research lab in Schenectady New York in 1958 when he read about Esaki’s tunneling experiments. He was enrolled at that time as a graduate student in physics at Rensselaer Polytechnic Institute (RPI) where he was taking a course in solid state physics and learning about superconductivity. Superconductivity is carried by pairs of electrons known as Cooper pairs that spontaneously bind together with a binding energy that produced an “energy gap” in the electron energies of the metal, but no one had ever found a way to directly measure it. The Esaki experiment made him immediately think of the equivalent experiment in which Cooper pairs might tunnel between two superconductors (through a thin oxide layer) and yield a measurement of the energy gap. The idea actually came to him during the class lecture.

The experiments used a junction between aluminum and lead (Al—Al2O3—Pb). At first, the temperature of the system was adjusted so that Al remained a normal metal and Pb was superconducting, and Giaever observed a tunnel current with a threshold related to the gap in Pb. Then the temperature was lowered so that both Al and Pb were superconducting, and a peak in the tunnel current appeared at the voltage associated with the difference in the energy gaps (predicted by Harrison and Bardeen).

Fig. 6 Diagram from Giaever “The Discovery of Superconducting Tunneling” at https://conferences.illinois.edu/bcs50/pdf/giaever.pdf

The Josephson Effect: Brian Josephson (1962)

In Giaever’s experiments, the external circuits had been designed to pick up “ordinary” tunnel currents in which individual electrons tunneled through the oxide rather than the Cooper pairs themselves. However, in 1962, Brian Josephson (1940 – ), a physics graduate student at Cambridge, was sitting in a lecture (just like Giaever) on solid state physics given by Phil Anderson (who was on sabbatical there from Bell Labs). During lecture he had the idea to calculate whether it was possible for the Cooper pairs themselves to tunnel through the oxide barrier. Building on theoretical work by Leo Falicov who was at the University of Chicago and later at Berkeley (years later I was lucky to have Leo as my PhD thesis advisor at Berkeley), Josephson found a surprising result that even when the voltage was zero, there would be a supercurrent that tunneled through the junction (now known as the DC Josephson Effect). Furthermore, once a voltage was applied, the supercurrent would oscillate (now known as the AC Josephson Effect). These were strange and non-intuitive results, so he showed Anderson his calculations to see what he thought. By this time Anderson had already been extremely impressed by Josephson (who would often come to the board after one of Anderson’s lectures to show where he had made a mistake). Anderson checked over the theory and agreed with Josephson’s conclusions. Bolstered by this reception, Josephson submitted the theoretical prediction for publication [9].

As soon as Anderson returned to Bell Labs after his sabbatical, he connected with John Rowell who was making tunnel junction experiments, and they revised the external circuit configuration to be most sensitive to the tunneling supercurrent, which they observed in short time and submitted a paper for publication. Since then, the Josephson Effect has become a standard element of ultra-sensitive magnetometers, measurement standards for charge and voltage, far-infrared detectors, and have been used to construct rudimentary qubits and quantum computers.

By David D. Nolte: Nov. 6, 2022

YouTube Video

YouTube Video of Quantum Tunneling Systems

References:

[1] F. Hund, Z. Phys. 40, 742 (1927). F. Hund, Z. Phys. 43, 805 (1927).

[2] L. Nordheim, Z. Phys. 46, 833 (1928).

[3] R. H. Fowler, L. Nordheim, Proc. R. Soc. London, Ser. A 119, 173 (1928).

[4] G. Gamow, Z. Phys. 51, 204 (1928).

[5] R. W. Gurney, E. U. Condon, Nature 122, 439 (1928). R. W. Gurney, E. U. Condon, Phys. Rev. 33, 127 (1929).

[6] Dennison, D. M. and G. E. Uhlenbeck. “The two-minima problem and the ammonia molecule.” Physical Review 41(3): 313-321. (1932)

[7] L. Esaki, New Phenomenon in Narrow Germanium Para-Normal-Junctions, Phys. Rev., 109, 603-604 (1958); L. Esaki, (1974). Long journey into tunneling, disintegration, Proc. of the Nature 123, IEEE, 62, 825.

[8] I. Giaever, Energy Gap in Superconductors Measured by Electron Tunneling, Phys. Rev. Letters, 5, 147-148 (1960); I. Giaever, Electron tunneling and superconductivity, Science, 183, 1253 (1974)

[9] B. D. Josephson, Phys. Lett. 1, 251 (1962); B.D. Josephson, The discovery of tunneling supercurrent, Science, 184, 527 (1974).

[10] P. W. Anderson, J. M. Rowell, Phys. Rev. Lett. 10, 230 (1963); Philip W. Anderson, How Josephson discovered his effect, Physics Today 23, 11, 23 (1970)

[11] Eugen Merzbacher, The Early History of Quantum Tunneling, Physics Today 55, 8, 44 (2002)

[12] Razavy, Mohsen. Quantum Theory Of Tunneling, World Scientific Publishing Company, 2003.

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