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by Roberto Lalli

Robert Schrieffer

The Nobel Prize in Physics 1972 together with John Bardeen and Leon N. Cooper
"for their jointly developed theory of superconductivity, usually called the BCS-theory".

John Robert Schrieffer was born in Oak Park, IL, on May 31, 1931, in the difficult aftermath of the Stock Market Crash of 1929. Since an early age, Schrieffer was fascinated by activities such as building amateur radios and constructing homemade rockets. These pastimes sparked his interest in sciences and, especially, electrical engineering. After graduation, Schrieffer was admitted to the Massachusetts Institute of Technology in Cambridge, MA, in 1949. There, he decided to major in electrical engineering, but he switched to physics before earning his bachelor. Eventually, he worked on a bachelor’s thesis on the theory of multiplets in atoms with a large atomic number under the supervision of John C. Slater. Since the 1930s, Slater was among the major scientific figures working on the application of quantum mechanics to many-body problems. Slater had built one of the two American theoretical schools aimed at the training of young scholars in quantum mechanical solid-state physics, the other being the Princeton group under the leadership of Eugene Wigner.
In the late 1940s, the field of solid-state physics had grown enormously both for the number of physicists working on it and for the achievements that had been made, and Slater was considered one of the founding fathers of that field in the United States. Working with Slater, Schrieffer developed a strong interest in solid-state physics and its unsolved theoretical problems. Following this interest, Schrieffer enrolled at the University of Illinois at Urbana-Champaign to study under the American theoretical physicist John Bardeen, who had recently made enormous contributions to the field by putting forward the theoretical investigation underlying the discovery of the first transistor: the point-contact transistor.

A PhD Dissertation Worthy of a Nobel Prize
At Urbana, Schrieffer immediately began to closely collaborate with John Bardeen, who at the time was immersed in the problem of superconductivity. Discovered in 1911, the phenomenon of the abrupt vanishing of resistance in certain metals and alloys when they were cooled down below their critical temperature had proved to be one of the most arduous problems for the extension of quantum mechanics to many-body systems. After the development of quantum mechanics, the application of this theoretical framework to solids seemed to be natural and produced rewarding results. However, while many macroscopic phenomena concerning the behaviour of condensed matter had been successfully explained by means of quantum mechanical laws, superconductivity had resisted all attempts to produce a similar explication.
Progress was only made in the 1930s, when the recently emigrated German-born Jews Fritz and Heinz London put forward a phenomenological theory of superconductivity that was able to provide a useful description of the connection between superconducting currents and externally applied magnetic fields. The London theory successfully described the recently discovered phenomenon of the Meissner effect—the sudden expulsion of weak magnetic fields from a superconductor when it is cooled below the critical temperature. In 1933, the German physicists W. Meissner and R. Ochsenfeld had made the discovery that a material in the superconducting state was a perfect diamagnet. This discovery led some physicists, including the London brothers, to believe that diamagnetism was the basic property of superconducting states and that the vanishing resistance was more a consequence. Bardeen was one of the first physicists to recognize the importance of the London theory as a starting point to build a microscopic theory of superconductivity. Bardeen, indeed, understood that a microscopic theory should be able to derive some basic properties suggested by the London brothers; especially, a certain “rigidity” of the wave function of the superconducting state, which is essentially unaffected by the presence of a sufficiently weak external magnetic field. This conjecture was later extended by Fritz London himself in 1950, when he suggested that in superconductors there was a condensation of the average momentum distribution, i.e., they had the long-range property that the average momentum of each electron was tied to a common value.

In 1950, extending the general theory of second-order (i.e. continuous) phase transitions developed by Lev Landau in 1937, Vitaly Ginzburg and Landau himself put forward their phenomenological theory of superconductivity, which included a spatial dependence of the density of superconducting electrons. The two Soviet theoretical physicists introduced an order term—a square of a wave function—that indicated the density of superconducting electrons. By introducing this term and by putting forward an equation relating the order term, the temperature and the free energy difference between superconducting and normal states, Ginzburg and Landau were able to calculate the surface energies at the boundary between the normal and superconducting states. Moreover, it was the first theoretical approach to suggest that there were two different kinds of superconductors, with different reactions with respect to the applied magnetic fields. As was later recognized, type-I superconductors were those superconductors in which the superconducting phase is abruptly destroyed as soon as the applied magnetic field reach a certain critical value. In type-II superconductors, instead, there is an intermediate phase characterised by the formation of magnetic vortices when the magnitude of the applied magnetic field lies between two different critical values H1 and H2, while the material comes back to the normal state for values of the magnetic field greater than H2.
Around the same period, important experimental discoveries gave a strong impetus to increase the efforts towards a microscopic theory of superconductivity. By exploring the phenomenon of superconductivity arising in bulks of different isotopes of mercury, two independent groups of researchers discovered that the critical temperature depends on the isotopic mass of the ionic lattice: The critical temperature increased when the ionic mass decreased. These observations suggested that the phenomenon of superconductivity strongly depended on the interactions between the conducting electrons and the vibrations of the lattice, which were described as interaction between conducting electrons and phonons—quanta of the collective excitations of the ions forming the lattice.
Bardeen was stimulated by these results to attack again the problem of superconductivity from a quantum mechanical perspective in 1950. From 1951, Bardeen began working with younger collaborators to the resolution of this problem in his new institution—the University of Illinois at Urbana-Champaign. In collaboration with D. Pines, and elaborating on other theoretical proposals put forward by Herbert Fröhlich and A. Brian Pippard, Bardeen was exploring a quantum field theoretical approach to superconductivity that could explain the isotopic mass dependence and the other empirical properties of superconductivity. While he was making progress, the theoretical structure they put forward was unable to address the various properties of superconductivity. Superconductivity still retained the status of one of the most arduous problem of many-body quantum theory.
This was the situation when Schrieffer enrolled as a graduate student at the University of Illinois. In 1954, he visited Bardeen to decide his dissertation topic. Bardeen proposed ten different subjects, the last one of which being superconductivity. Faced with the difficult choice, Schrieffer opted for superconductivity on the ground that he had time to shift in case it would prove to be too hard. When Schrieffer began working on that problem, Bardeen also invited Leon Cooper—a recent PhD with a background in theoretical quantum field theory—to join their group and to substitute Pines, who had recently obtained a teaching position in Princeton.
By September 1956, Cooper made a momentous step towards the resolution of the problem by demonstrating that if the net force between a couple of electrons, whose energy lies just below the Fermi energy, is attractive they form bound pairs, however weak the net attraction might be. This discovery, which came to known as Cooper pairs, suggested that the superconducting state was a set of such bound electrons. In this perspective, the critical temperature was the temperature below which the attractive force become greater that the thermal excitation of conducting electrons; namely, it is the temperature that allows the formation of the Cooper pairs. However, the main problem of finding the ground state wave function that accounted for the formation and the presence of such pairs was still a difficult problem. The major difficulty was that the average separation between the electrons forming a Cooper pair was about 10-4 cm, much greater than the average spacing between these condensed electrons; in other words, the picture offered a strong spatial overlapping of the superconducting electrons suggesting that the pair-pair collisions would interrupt the formation of pairs.

Discouraged by the small progress made after Cooper had put forward the Cooper pair hypothesis, Schrieffer was wondering whether he should change the topic of his dissertation. To that question, Bardeen famously replied: “Give it another month or a month and a half […] keep working. Maybe something will happen […]”
While riding in a underground train in New York City—where he was attending a conference on the many-body problem—Schrieffer wrote down a wave function that turned to be a satisfying response to the problem the trio of researchers was facing. Schrieffer put forward the superconducting ground state as a product of operators (each corresponding to one pair) acting on a vacuum, in which the occupancies of pair states were uncorrelated. After introducing some assumptions in order to insure that the average number of electrons represented by the Schrieffer wave function remained fixed, Schrieffer was able to derive various parameters; especially, the energy gap between the superconducting ground state and the continuum of energies above the Fermi energy.
As soon as Schrieffer discussed his wave function with Bardeen and Cooper, they understood that he was on the right track and worked hard to derive all the possible empirical consequences of the theory. Bardeen assigned Schrieffer the task to derive the thermodynamic properties, while Cooper worked on the Meissner effect and the other electrodynamic properties and Bardeen made calculations on the nonequilibrium properties. In order to secure their priority and fearing that other theoretical physicists—especially R. Feynman—might soon put forward a theory of superconductivity, Bardeen, Cooper and Schrieffer published a short letter in Physical Review with the outline of the theory on February 1957 and announced the discovery at the March 1957 meeting of the American Physical Society.
It took six months, however, to derive the entire set of consequences of their theory. After Bardeen was eventually able to derive the second order transition, they hurried to prepare the final paper with the full description of the theory and the confrontation with the empirical properties. The 30-page paper was submitted in July 1957 and published in Physical Review in December of the same year. By that time, the Bardeen-Cooper-Schrieffer theory—or BCS theory as came to be known—was already considered as one of the best candidates for the microscopic description of superconductivity although some theorists raised some concerns about its physical implications.
On the experimental side the theory met with great success. The calculations made by Bardeen, Cooper, and Schrieffer were able to explain many of the properties of superconductivity. For instance, in 1957, R. W. Morse and H. V. Bohm at Brown University observed a rapid decrease of ultrasonic attenuation in a pure superconductor after it was cooled down before its critical temperature.

Defending the BCS Theory
While experimental physicists showed great enthusiasm towards the BCS theory, some theorists who had been working on the same problem raised several doubts on its physical implications. These doubts were related to the apparent lack of gauge invariance of the theory. In order to simplify the calculation, Bardeen, Cooper and Schrieffer had used some approximations and the calculation of the Meissner effect was done in radiation gauge, in which the scalar potential becomes zero in regions far from the electric charge. But other researchers, most notably P. W. Anderson, L. P. Gor’kov, and Y. Nambu, corroborated the view that the lack of invariance was only apparent. It was later understood in better terms that the degeneracy of the BCS ground state was an example of spontaneous symmetry braking—in this case the breaking of the U(1) electromagnetic gauge invariance of the Hamiltonian of the system in the thermodynamic limit. The concept of spontaneous symmetry breaking would rapidly become relevant in other subfields of theoretical physics, becoming a necessary mechanism for the understanding of particle interactions in the Standard Model—the so-called Higgs mechanism put forward in 1964 by several physicists including P. Higgs and F. Englert who were awarded the Nobel Prize in Physics in 2013.
After completing his dissertation, Schrieffer won a National Science Foundation Fellowship for the academic year 1957-58, which he spent at the University of Birmingham and the Niels Bohr Institute in Copenhagen. He continued to make research on the theory of superconductivity and gave some seminars on that topic. In Copenhagen, the influential Nobel Laureate Niels Bohr attacked the BCS theory on the grounds that the theory was too simple to be a faithful account of the complexity of nature. Encouraged by Bardeen, Schrieffer continued to trust the theory and to make further improvements. In particular he collaborate with D. Pines to solve the problem of the apparent gauge symmetry breaking.
A second criticism that was raised concerned the inability of the BCS theory to make specific predictions concerning which material would show the property of superconductivity under a certain critical temperature. Schrieffer became an expert for the application of the Green’s function formalism to superconductivity, which had been developed by Soviet theoretical physicists. Schrieffer began employing these tools to make computer integrations of the gap equations. This work was aimed at producing a quantitative theory of the critical temperature. Along with his collaborators D. Scalapino and J. Wilkins, and after having discussed the work with P. W. Anderson, Schrieffer published the influential paper “Effective Tunneling Density of States in Superconductors,” in Physical Review Letters in 1963.
Schrieffer continued to work on theory of superconductivity and on other collective phenomena of condensed matter as a Professor at the University of Pennsylvania in Philadelphia. In 1964, he wrote the first textbook on the BCS theory entitled Theory of Superconductivity, which was very influential for younger students and researchers.
In 1972, after BCS had become the standard theory of superconductivity and it had been applied in other fields such as the properties of superfluid helium, nuclear structure and astrophysics, Schrieffer, along with Bardeen and Cooper was awarded the Nobel Prize in Physics “for their jointly developed theory of superconductivity, usually called the BCS-theory.”

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Schrieffer R. (1972) Nobel Lecture: Macroscopic Quantum Phenomena from Pairing in Superconductors. Nobel Media AB 2014. Retrieved 10 December 2014.