Royal Society publishing Informing the sence of the futue Recollections of Early Solid State Physics Author(s):R.E. Peierls Source: Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 371, No. 1744, The Beginnings of Solid State Physics(Jun. 10, 1980), pp 28-38 Published by: The Royal Society StableUrl:http://www.jstor.org/stable/2990272 Accessed:12/03/20100208 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp.JstOr'sTermsandConditionsofUseprovidesinpartthatunless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showpublisher?publishercode=rsl Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission JStOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about STOR, please contact support@jstor. org The Royal Society is collaborating with JSTOR to digitize, preserve and extend access to Proceedings of the Royal Society of London. Series A, Mather atical and Phvsical sciences ittp://www.jstor.org
Recollections of Early Solid State Physics Author(s): R. E. Peierls Source: Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 371, No. 1744, The Beginnings of Solid State Physics (Jun. 10, 1980), pp. 28-38 Published by: The Royal Society Stable URL: http://www.jstor.org/stable/2990272 Accessed: 12/03/2010 02:08 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=rsl. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. The Royal Society is collaborating with JSTOR to digitize, preserve and extend access to Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. http://www.jstor.org
Pro,R.Soo,Lond.A371,2838(1980) Printed in Great britain Recollections of early solid state physics BY R, E. PEIERLs, F R.S. Nuclear Physics Laboratory, University of Oxford, U. K Peierls, Sir Rudolf Ernst. Born Berlin 1907. Studied at Berlin, Munich and Leipzig. From 1932 held research positions in Manchester and Cambridge. Was Professor of mathematical physics in Birmingham and Oxford. Knighted 1968 Author of many papers applying quantum mechanics to problems in solid state and in particle physics; author of Quantum theory of solids, 1955. Author with O. R. Frisch in 1940 of 'Confidential memorandum to British Government on possibility and critical size of nuclear bomb My first contact with the electron theory of metals was in Sommerfeld's department at Munich, where I was a student from October 1926 to Easter 1928. These were my third to fifth semesters as a student, and i was only beginning to acquire an under- standing of physics, and of the new quantum and wave mechanics Sommerfeld was then developing his approach to electrons in metals, with some assistance from two American visitors, w. v. Houston and c. Eckart. i heard him give a series of lectures summarising the problems, and his results, even before the papers were published. () Sommerfeld was very much at home with the classical Lorentz-Drude theory nd familiar with the experimental data, which produced such a mysterious mixture of confirmation(Wiedemann-Franz ratio, 'normal'Hall effect, and many of the thermoelectric and thermomagnetic coefficients) and utter contradiction (para magnetism, specific heat, temperature dependence of resistivity). Pauli had shown(a) that the application of fermi statistics to the conduction electrons had resolved the aradox of paramagnetism. Nobody seemed worried at the time about the existence of diamagnetic metals; it was known that the ion cores would be diamagnetic. Sommerfeld built on Paulis idea,, and showed that Fermi statistics would also account for the absence of an observable electronic specific heat, and this was the greatest success of his approach. He showed that the specific heat of a degenerat Fermi gas would be proportional to T, and should be observable at very low ter peratures. It was gratifying that the constancy of the Wiedemann-Franz ratio could still be accounted for, although on the face of it it had appeared to requir a classical specific heat, and that the numerical value, which differed a little from the classical value fitted the data even better His theory failed to account for the temperature dependence of the resistivity, and for the magnetoresistance. The latter is now known to depend essentially on variations in the electron mobility Sommerfeld thought of resistance in terms of a constant mean free path. This made the collision time, and hence the mobility
Proc. R. Soc. Lond. A 371,28-38 (1980) Printed in Great Britain Recollections of early solid state physics By R. E. PEIERLS, F.R.S. Nuclear Physics Laboratory, University of Oxford, U.K. Peierls, Sir Rudolf Ernst. Born Berlin 1907. Studied at Berlin, Munich and Leipzig. From 1932 held research positions in Manchester and Oambridge. Was Professor of mathematical physics in Birmingham and Oxford. Knighted 1968. A uthor of many papers applying quantum mechanics to problems in solid state and in particle physics; author of Quantum theory of solids, 1955. Author with O. R. Frisch in 1940 of 'Oonfidential memorandum to British Government on possibility and critical size of nuclear bomb'. My first contact with the electron theory of metals was in Sommerfeld's department at Munich, where I was a student from October 1926 to Easter 1928. These were my third to fifth semesters as a student, and I was only beginning to acquire an understanding of physics, and of the new quantum and wave mechanics. Sommerfeld was then developing his approach to electrons in metals, with some assistance from two American visitors, W. V. Houston and C. Eckart. I heard him give a series oflectures summarising the problems, and his results, even before the papers were published.(l) Sommerfeld was very much at home with the classical Lorentz-Drude theory, and familiar with the experimental data, which produced such a mysterious mixture of confirmation (Wiedemann-Franz ratio, 'normal' Hall effect, and many of the thermoelectric and thermomagnetic coefficients) and utter contradiction (paramagnetism, specific heat, temperature dependence of resistivity). Pauli had shown(2) that the application of Fermi statistics to the conduction electrons had resolved the paradox of paramagnetism. Nobody seemed worried at the time about the existence of diamagnetic metals; it was known that the ion cores would be diamagnetic. Sommerfeld built on Pauli's idea, and showed that Fermi statistics would also account for the absence of an observable electronic specific heat, and this was the greatest success of his approach. He showed that the specific heat of a degenerate Fermi gas would be proportional to T, and should be observable at very low temperatures. It was gratifying that the constancy of the Wiedemann-Franz ratio could still be accounted for, although on the face of it it had appeared to require a classical specific heat, and that the numerical value, which differed a little from the classical value, fitted the data even better. His theory failed to account for the temperature dependence of the resistivity, and for the magnetoresistance. The latter is now known to depend essentially on variations in the electron mobility. Sommerfeld thought of resistance in terms of a constant mean free path. This made the collision time, and hence the mobility, [ 28 ]
Recollections of solid state physics inversely proportional to the velocity. Since even in a degenerate Fermi gas there is a small spread of velocities of conducting electrons around the Fermi velocity he could obtain a small spread of collision times, and hence a small magneto- resistance, several orders of magnitude less than the observed effect He followed the classical ideas by thinking of the electrons as free, except for collisions with atoms(therefore the magnitude of the mean free path required to explain the observed conductivity, even larger than in the classical picture because of the greater velocity, was another paradox ). He also ignored the mutualinteraction of the electrons. This had always been done in the classical theory, electron- electron encounters conserve the total electron momentum hence the total current and thus do not contribute to the resistance. I do not recall Sommerfeld mentioning and there was no reason why he should have taken a different pointon reatments this argument explicitly, but it is always clearly stated in the classical As a very junior theoretician I listened to Sommerfeld's exposition and was duly mpressed, but was not yet at the stage of criticizing or questioning the assumptions It was characteristic of Sommerfeld's positive attitude that one learnt more about the successful solution of difficulties than about the mysteries that remained. He was completely fair in listing the contradictions-it was just a matter of emphasis When at Easter 1928 Sommerfeld left for a sabbatical year, I joined Heisenbergs group at Leipzig, where Felix Bloch had just completed his treatment of electrons in periodic potentials(a)and his explanation of the order of magnitude and tempera ture dependence of the resistivity in terms of lattice vibrations. Bloch also did not orry about the electron-electron interaction- I do not know whether he recalled the old arguments that it should not matter, or whether he was simply content to extend the theory by taking in one more factor which had previously been ignored without necessarily including everything My first substantial research assignment in leipzig was to see whether Bloch's starting-point of independent electrons(electron orbitals we would say today was unavoidable, and how far one could get if one started from a Heitler-London model (which Heisenberg was about this time applying to ferromagnetism). Today it would be obvious to any undergraduate that there could not be any conductivity in the Heitler-London model unless it is supplemented by ionized states, in which ome atoms have more, and others fewer, than their normal complement of electrons But at the time this conclusion was not obvious to me, and evidently not to Heisen- berg. It took some struggle with exchange integrals for a many-body system before I concluded that, at least in the linear chain I was using as a model, the only way a current could be obtained was by all electrons making a quantum jump simul taneously, and for a macroscopic dimension this makes therate astronomically small In the summer of 1928 I was fairly sure about this conclusion. I spent the summer vacation in England, mostly as a tourist, but I visited Cambridge and called on Dirac, whom I had met. He introduced me to R. H. Fowler. When Fowler heard I was from Leipzig, he asked me to talk to the Kapitza Club about bloch's work. At that time neither my English nor my command of physics was really adequate for
Recollections of solid state physics 29 inversely proportional to the velocity. Since even in a degenerate Fermi gas there is a small spread of velocities of conducting electrons around the Fermi velocity, he could obtain a small spread of collision times, and hence a small magnetoresistance, several orders of magnitude less than the observed effect. He followed the classical ideas by thinking of the electrons as free, except for collisions with atoms (therefore the magnitude of the mean free path required to explain the observed conductivity, even larger than in the classical picture because of the greater velocity, was another paradox). He also ignored the mutual interaction of the electrons. This had always been done in the classical theory, since electronelectron encounters conserve the total electron momentum, hence the total current, and thus do not contribute to theresistance. I do not recall Sommerfeld mentioning this argument explicitly, but it is always clearly stated in the classical treatments, and there was no reason why he should have taken a different point of view. As a very junior theoretician I listened to Sommerfeld's exposition and was duly impressed, but was not yet at the stage of criticizing or questioning the assumptions. It was characteristic of Sommerfeld's positive attitude that one learnt more about the successful solution of difficulties than about the mysteries that remained. He was completely fair in listing the contradictions - it was just a matter of emphasis. When at Easter 1928 Sommerfeld left for a sabbatical year, I joined Heisenberg's group at Leipzig, where Felix Bloch had just completed his treatment of electrons in periodic potentials(3) and his explanation of the order of magnitude and temperature dependence of the resistivity in terms of lattice vibrations. Bloch also did not worry about the electron-electron interaction - I do not know whether he recalled the old arguments that it should not matter, or whether he was simply content to extend the theory by taking in one more factor which had previously been ignored, without necessarily including everything. My first substantial research assignment in Leipzig was to see whether Bloch's starting-point of independent electrons (' electron orbitals' we would say today) was unavoidable, and how far one could get if one started from a Heitler-London model (which Heisenberg was about this time applying to ferromagnetism). Today it would be obvious to any undergraduate that there could not be any conductivity in the Heitler-London model unless it is supplemented by ionized states, in which some atoms have more, and others fewer, than their normal complement of electrons. But at the time this conclusion was not obvious to me, and evidently not to Heisenberg. It took some struggle with exchange integrals for a many-body system before I concluded that, at least in the linear chain I was using as a model, the only way a current could be obtained was by all electrons making a quantum jump simultaneously, and for a macroscopic dimension this makes the rate astronomically small. In the summer of 1928 I was fairly sure about this conclusion. I spent the summer vacation in England, mostly as a tourist, but I visited Cambridge and called on Dirac, whom I had met. He introduced me to R. H. Fowler. When Fowler heard I was from Leipzig, he asked me to talk to the Kapitza Club about Bloch's work. At that time neither my English nor my command of physics was really adequate for
R. E. Peierls this, but it did not occur to me to refuse, and I did my best. But, more relevant to the present story, he also introduced me to one of his research students, W.H MoCrea(now a distinguished astrophysicist), who was also thinking about con ductivity using what he and Fowler called the Heitler-London model. Actually this 4)was a one-electron tight-binding model with two centres of for On my return to Leipzig the project was abandoned, and Heisenberg suggested I looked at the‘ anomalous’,ic itive, Hall effect. i tackled this on the basis of Bloch' s theory of electrons in periodic fields, and first had to convince myself that the effect of the magnetic field on the wave vector of the electron was the same as for a free electron of the same velocity, but that the mean velocity of the electron same k, if the energy function E(k) different. It was obvious, in particulAr was given by dE/dk, and therefore different from that for a free electron of the that in Bloch' s tight-binding model the energy would flatten off near the band edge, so that the current would there go to zero. Thus for an electron near the band edge an electric field could cause a decrease, rather than an increase, in the velocity in the field direction. One's first shock on seeing this result is the fear that it might lead to a negative conductivity. One soon realizes, however, that for an ensemble of electrons in statistical equilibrium the positive acceleration of the electrons near the bottom of the band outweighs the negative acceleration of those near the top until for a full band the current just vanishes t this point one was close to an explanation of the positive Hall effect, subject only to the proof that the rate of change of the wavevector in the magnetic field is still given by the Lorentz force. At this point I cheated a little by disregarding inter-band terms, which for the purpose in hand were unimportant, but in other problems can cause headaches So the explanation of the positive Hall effect came out without much difficulty. I recall a comment by Heisenberg that this was similar to the situation in atomic spectra(pointed out, I think, by Pauli)where an atom with one, or a few, electrons missing from a closed shell was dynamically similar to one with just one, or a few electrons in that shell, except for some signs. My memory is confused, however, on the question whether this comment was made when Heisenberg suggested the problem to me, or when I showed him the answer. In other words, I am not clear whether Heisenberg had, with his usual powerful intuition, guessed in advance how the solution would come out. I reported previously that he had, but I am now rather doubtful whether this was right In any event it was gratifying to have solved one of the remaining mysteries I wrote a paper on the subject, 5)which was not too clearly written, and also gave a talk to a conference, of which a summary is published. (6)It contains a sketch the Fermi surface for a two-dimensional square lattice for the case of an almost empty, and an almost full band with tight binding. In the latter case the boundar consists of circular quadrants inside the four corners of the square which forms the Brillouin zone for that case. In the longer paper there is also the remark that the conductivity vanishes for a full band
30 R. E. Peierls this, but it did not occur to me to refuse, and I did my best. But, more relevant to the present story, he also introduced me to one of his research students, W. H. McCrea (now a distinguished astrophysicist), who was also thinking about conductivity using what he and Fowler called the Heitler-London model. Actually this(4) was a one-electron tight-binding model with two centres of force. On my return to Leipzig the project was abandoned, and Heisenberg suggested I looked at the' anomalous', i.e. positive, Hall effect. I tackled this on the basis of Bloch's theory of electrons in periodic fields, and first had to convince myself that the effect of the magnetic field on the wave vector of the electron was the same as for a free electron of the same velocity, but that the mean velocity of the electron was given by dE jdk, and therefore different from that for a free electron of the same k, if the energy function E(k) was different. It was obvious, in particular, that in Bloch's tight-binding model the energy would flatten off near the band edge, so that the current would there go to zero. Thus for an electron near the band edge an electric field could cause a decrease, rather than an increase, in the velocity in the field direction. One's first shock on seeing this result is the fear that it might lead to a negative conductivity. One soon realizes, however, that for an ensemble of electrons in statistical equilibrium the positive acceleration of the electrons near the bottom of the band outweighs the negative acceleration of those near the top, until for a full band the current just vanishes. At this point one was close to an explanation of the positive Hall effect, subject only to the proof that the rate of change of the wavevector in the magnetic field is still given by the Lorentz force. At this point I cheated a little by disregarding inter-band terms, which for the purpose in hand were unimportant, but in other problems can cause headaches. So the explanation of the positive Hall effect came out without much difficulty. I recall a comment by Heisenberg that this was similar to the situation in atomic spectra (pointed out, I think, by Pauli) where an atom with one, or a few, electrons missing from a closed shell was dynamically similar to one with just one, or a few, electrons in that shell, except for some signs. My memory is confused, however, on the question whether this comment was made when Heisenberg suggested the problem to me, or when I showed him the answer. In other words, I am not clear whether Heisenberg had, with his usual powerful intuition, guessed in advance how the solution would come out. I reported previously that he had, but I am now rather doubtful whether this was right. In any event it was gratifying to have solved one of the remaining mysteries. I wrote a paper on the subject,(5) which was not too clearly written, and also gave a talk to a conference, of which a summary is published.(6) It contains a sketch of the Fermi surface for a two-dimensional square lattice for the case of an almost empty, and an almost full band with tight binding. In the latter case the boundary consists of circular quadrants inside the four corners of the square which forms the Brillouin zone for that case. In the longer paper there is also the remark that the conductivity vanishes for a full band
Recollections of solid state physics This seems relevant to another question on which my memory fails to serve namely when and how it was first realized that a filled band would give an insulator. In retrospect it seems to be an obvious consequence of the existence of bands, at least in the tight-binding limit, and particularly obvious from the arguments sketched above. It seems almost incredible that this point could have been missed but i have no clear recollection of when I became aware of it, and it is certainly not mentioned in any paper of that time This work was complete by the spring of 1929, and since at that time Heisenberg ent on sabbatical leave, i moved to Zurich to work with Pauli. here i left metals for a while, since Pauli suggested to me the problem of heat conduction in non- metallic crystals, under the influence of the anharmonic forces. This was a problem hich, at least at high temperatures, could be treated classically. Pauli had been interested in this problem and had looked at the related question of the absorption of sound waves because of anharmonicity. The abstract of a talk he gave to a meeting is published, ) and the answer given there is wrong (probably the only error in print under Paulis name) because it gives a finite damping in a linear chain, for hich in fact the three-phonon processes, which he was studying, do not occur He showed me a few pages of notes on this problem, to start me off. Apart from this guidance I looked at the problem from first principles, and this was probably fortunate, because there were a number of different wrong approaches in the literature, and it was less confusing to find the solution first, and then discover where others had gone wrong This led to the concept(and the ugly word) of Umklapp processes, and to the prediction of the exponential rise of the heat conductivity at low temperatures, (8) verified only in 1951 by Berman (9) In many ways my paper did not dispose of the problem. For example, it failed to point out that a pure substance, to show the exponential rise, had to be also isotopically pure. This omission made the experi mental discovery of the effect more difficult. Other, more sophisticated parts of the problem are still not completely sorted out. I submitted a thesis on this topi to leipzig(my one semester in Zurich not being an adequate residence qualification there)and returned to Zurich as Pauli s assistant I then started thinking further about electrons in metals. I felt uncomfortable about having, in my work on the Hall effect, relied on the flattening of the energy surface near the band edge, a result then known only in the tight-binding limit which was not realistic for conduction electrons. It seemed obvious that in the opposite limit of free electrons this effect was absent, and one therefore did not know what was happening in the intermediate case. It suddenly dawned on me that, if a weak potential was added as a small perturbation, there would be band gaps near the Bragg reflexions, and that the energy surface there had zero slope though, for a very weak potential this flattening was confined to a very narrow region near the edge, and the slope returned to its free-electron value more rapidl the weaker the potential(o) Few pieces of work have given me as much pleasure as this discovery, which
Recollections of solid state physics 31 This seems relevant to another question on which my memory fails to serve, namely when and how it was first realized that a filled band would give an insulator. In retrospect it seems to be an obvious consequence of the existence of bands, at least in the tight-binding limit, and particularly obvious from the arguments sketched above. It seems almost incredible that this point could have been missed, but I have no clear recollection of when I became aware of it, and it is certainly not mentioned in any paper of that time. This work was complete by the spring of 1929, and since at that time Heisenberg went on sabbatical leave, I moved to Zurich to work with Pauli. Here I left metals for a while, since Pauli suggested to me the problem of heat conduction in nonmetallic crystals, under the influence of the anharmonic forces. This was a problem which, at least at high temperatures, could be treated classically.Pauli had been interested in this problem and had looked at the related question of the absorption of sound waves because of anharmonicity. The abstract of a talk he gave to a meeting is published,(7) and the answer given there is wrong (probably the only error in print under Pauli's name) because it gives a finite damping in a linear chain, for which in fact the three-phonon processes, which he was studying, do not occur. He showed me a few pages of notes on this problem, to start me off. Apart from this guidance I looked at the problem from first principles, and this was probably fortunate, because there were a number of different wrong approaches in the literature, and it was less confusing to find the solution first, and then discover where others had gone wrong. This led to the concept (and the ugly word) of Umklapp processes, and to the prediction of the exponential rise of the heat conductivity at low temperatures,(8) verified only in 1951 by Berman.(9) In many ways my paper did not dispose of the problem. For example, it failed to point out that a 'pure' substance, to show the exponential rise, had to be also isotopic ally pure. This omission made the experimental discovery of the effect more difficult., Other, more sophisticated parts of the problem are still not completely sorted out. I submitted a thesis on this topic to Leipzig (my one semester in Zurich not being an adequate residence qualification there) and returned to Zurich as Pauli's assistant. I then started thinking further about electrons in metals. I felt uncomfortable about having, in my work on the Hall effect, relied on the flattening of the energy surface near the band edge, a result then known only in the tight-binding limit, which was not realistic for conduction electrons. It seemed obvious that in the opposite limit of free electrons this effect was absent, and one therefore did not know what was happening in the intermediate case. It suddenly dawned on me that, if a weak potential was added as a small perturbation, there would be band gaps near the Bragg reflexions, and that the energy surface there had zero slope, though, for a very weak potential this flattening was confined to a very narrow region near the edge, and the slope returned to its free-electron value more rapidly the weaker the potentiaL<lO) Few pieces of work have given me as much pleasure as this discovery, which