Present Focus: Magnetic Molecules
(41 publications: Nos. 78, 81, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124 of Publication List)
Magnetism at the nanoscale level presents many new questions and challenges that are of great importance to basic science. Also, there is a prospect for a variety of applications in physics, chemistry, material sciences, biomedical sciences, as well as quantum computing. Together with many worthy collaborators world-wide, I am investigating the magnetic properties of “magnetic molecules”. These are a rapidly emerging, diverse class of compounds that have become possible due to remarkable advances in recent years by synthesis chemists. Magnetic molecules are realized as macroscopic crystalline samples composed of identical molecular units each of which hosts a relatively small number (2 to 30 or more) of paramagnetic ions that mutually interact via Heisenberg exchange. These systems are rather unique in that macroscopic measurements directly establish the magnetic properties of an individual magnetic molecule. This is because the intermolecular magnetic interactions (dipole-dipole for the most part) are typically negligible as compared to the intramolecular magnetic interactions. These molecules thus constitute bona fide “zero-dimensional” magnetic systems, and their properties are distinct from those of bulk magnetic systems that typically involve cooperative effects in one, two, and three dimensions.
In one class of magnetic molecules, “ring-type”, the paramagnetic ions occupy the uniformly spaced sites of a planar ring embedded within the molecule. For such cyclic arrays the nearest-neighbor isotropic Heisenberg model for a (finite) ring of N interacting spins s serves as a very useful theoretical platform for the theoretical determination of the magnetic properties. Another specific example, discussed at length in subsequent paragraphs, is the magnetic molecule named {Mo72Fe30}, which includes thirty Fe3+ magnetic ions that occupy the 30 sites of an icosidodecahedron, and their interactions are well described by the nearest-neighbor isotropic Heisenberg Hamiltonian. These are but two examples from among a very large variety of magnetic molecules.
Because their properties can often be varied by the synthesis chemist, magnetic molecules provide a very promising platform for the study of such topics of considerable current interest as spin frustration, metamagnetic transitions, the discrete magnetic excitation spectrum, the transition from quantum to classical characteristics, quantum tunneling of magnetization, and unique dynamical effects at very low temperatures including hysteresis loops and magnetization steps, all at the nanoscale (molecular) level.
For many magnetic molecules the theoretical analysis can proceed via diagonalization of the Heisenberg Hamiltonian followed by the use of standard methods of statistical mechanics. However, this approach is impractical once the dimensionality, D = (2s + 1)N, of the underlying Hilbert space, exceeds about 107, a situation that can occur even for relatively small numbers, N, of identical spins s. Fortunately the determination of the thermodynamic properties can nevertheless be made for many of these challenging magnetic molecules by employing the quantum Monte Carlo method. (During the past few years my Ph.D. student, (now Dr.) Larry Engelhardt, has successfully introduced this technique to the study of magnetic molecules, and has applied it to a variety of systems including one for which D = 10120 ; see Publication #122). Important progress can also be achieved for magnetic molecules when s = 5/2 or larger, by using simulational methods of classical spin dynamics and the classical Monte Carlo method. (My collaborator Christian Schröder, has demonstrated this for a number of models of magnetic molecules; see Publications #94, 101, 108, 114, 116, 122, and 123.) Upon employing both classical and quantum Monte Carlo methods it becomes possible to study finite quantum Heisenberg spin models and their approach to the classical limit (see Publication #122). This allows one to clarify the conditions when classical Monte Carlo methods can provide useful predictions for finite quantum Heisenberg spin systems.
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To pursue a successful research program in magnetic molecules requires a closely coordinated, interdisciplinary effort coupling the skills and facilities of synthesis chemists, and experimental and theoretical physicists using a wide variety of experimental (resonance techniques, inelastic neutron scattering, thermodynamic measurements to milli-kelvin temperatures and large static and pulsed magnetic fields), analytical (classical and quantum statistical mechanics), and computational techniques (matrix diagonalization methods, classical spin dynamics, and both classical and quantum Monte Carlo methods).
The U.S. Department of Energy is funding a comprehensive program in magnetic molecules at Ames Laboratory. The benefits of this support is compounded by the fact that Ames scientists are enmeshed through very active collaborations with chemists and physicists world-wide. To date my own published work in this field spans the 41 articles listed above. Some of these publications deal with a wide variety of successfully synthesized magnetic molecules. Those articles were either written jointly with experimentalists or address issues concerning experimental findings. Other articles address broader theoretical issues concerning finite arrays of interacting classical and quantum spins, or predictions for systems that have not yet been synthesized, methodology issues, etc.
In the following, rather than attempt to summarize the major features of my published articles on the subject of magnetic molecules, I provide summaries of some of our findings for two systems that are of particular interest to me. At the very least, these summaries can provide the reader with a taste of the rich subject of magnetic molecules. Of course, the reader is invited to peruse among all of the articles listed at the outset of this section, and to contact me with comments, questions, etc.
The magnetic molecule {Mo72Fe30}
A spectacular example of a magnetic molecule, one to which I and some of my collaborators have devoted a great deal of attention over the past 6 years goes under the name {Mo72Fe30}, a necessary abbreviation for an extremely lengthy chemical formula. First synthesized by my distinguished collaborator Achim Müller (Bielefeld) in 1999, this neutral, spherical-shaped molecule (see Fig. 1 (a)) has a molecular weight of approx 18,600 and a diameter of 2.5 nm (a bona fide “nanomagnet”!). As a result of many fruitful investigations we now have a rather well developed understanding of its magnetic properties (see below). The 30 iron ions (Fe3+, spin 5/2) are situated on the sites of an icosidodecahedron (a polyhedron, or “polytope” in the language of mathematicians, consisting of 12 regular pentagons and 20 equilateral triangles) embedded within the interior of the molecule, one of the so-called archimedean polytopes. All 30 sites are equivalent, and each iron ion interacts with four nearest-neighbors via isotropic antiferromagnetic exchange (see Fig. 1 (b)).
Fig. 1 (a) Ball and stick representation of the magnetic molecule {Mo72Fe30}. The O-Mo-O bridges (one is highlighted) act as highly efficient superexchange ligands leading to antiferromagnetic exchange interaction between each Fe ion and its four nearest-neighbor Fe ions. (b) The Fe ions (spins s = 5/2) occupy the sites of an icosidodecahedron. In both panels those Fe ions described by the same color have parallel spin vectors. The 30 spins are decomposable into three subgroups of 10 spins, where all the spins of a given subgroup are mutually parallel, and the sublattices are characterized by three coplanar unit vectors with an angular spacing of 120o.
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On a personal note, I cannot contain my awe at the fact that such a complex, aesthetically pleasing structure can be achieved by “pouring and mixing” a quantity of raw chemicals!
From a theoretical point of view {Mo72Fe30} presents a major challenge. The Heisenberg matrix for this system is defined on a Hilbert space of dimensionality 630, which is approximately one-third of Avogadro’s number. So we can be assured that a frontal attack based on matrix diagonalization is out of the question! Despite this huge technical hurdle one can make major progress.
Two important facts were discovered early on by a collaborator at the time, a topologist, Maria Axenovich of Iowa State University’s Mathematics Department, and these facts were of key importance for further progress. (See our Publication #92.) We dispensed with the fact that the Fe3+ ion is a spin 5/2 entity and instead chose it to be a classical spin (i.e., a unit vector free to rotate continuously in 3-space) and adopted the classical Heisenberg model as our starting point. Axenovich’s first result was to note that the icosidodecahedron is “three-colorable”: One can paint each of the 30 sites with one of three different colors while its four nearest-neighbor sites are to be painted with the other two colors. Her second result was a rigorous derivation of the exact value of the minimal (“ground state”) energy for the classical Heisenberg icosidodecahedron (with nearest-neighbor antiferromagnetic exchange) in zero magnetic field.
Combining these two results we arrived at two distinct statements of physical importance: First, for zero magnetic field, in the ground state the configurational ordering of the spins is “frustrated”, illustrated in Fig. 1 (b), namely the 30 spins consist of three distinct “sublattices” each of 10 co-parallel spins, whereby nearest-neighbor spins differ in angular orientation by 120o. Second, introducing an external magnetic field, the three sublattice vectors uniformly cant (a useful mental picture to consider is that of a uniformly closing umbrella) so as to give rise at 0 K to a linear increase of magnetic moment with field, achieving “saturation” (all 30 spins parallel) at a well defined magnetic field, proportional to the nearest-neighbor exchange constant. The latter crucial quantity had been determined (as 1.57 K) slightly earlier by an analysis of the measured temperature dependence of the weak-field susceptibility using the only feasible, available theoretical tool, namely a classical spin dynamics approach performed by my collaborator Christian Schröder. For {Mo72Fe30}, our formula for the saturation magnetic field gave Hsat = 17.7 Tesla. To our great satisfaction, a subsequent measurement at 0.46 K by my collaborator Robert Modler, confirmed the linear increase of the magnetization with magnetic field and its saturation, apart from some expected slight rounding due to thermal effects, at approximately 17.7 Tesla. It is satisfying, but nevertheless somewhat surprising, that this experimental result for quantum spins 5/2 was provided first by our theory using a classical Heisenberg Hamiltonian, when attacked with topology arguments and the techniques of classical spin dynamics.
Somewhat later it was the turn of experiment to provide an important new and surprising result concerning magnetization versus magnetic field, a result that puzzled us initially. For 0 K our rigorous classical result is that dM/dH is strictly constant with H below Hsat and then it is zero above Hsat. It would be reasonable to expect that the effect of finite temperatures is merely to round out this sharp kink in the 0 K magnetization that occurs at Hsat. Instead, in a measurement at 0.42 K, our collaborator Hiroyuki Nojiri (U. Tohoku) found that the differential susceptibility dM/dH exhibits a sharp narrow dip at Hsat/3. Apparently this dip at Hsat/3 emerges as T is increased from zero.
Stimulated by Nojiri’s experiment, we subsequently found by theoretical methods that a pronounced minimum in dM/dH at Hsat/3 does indeed emerge as the temperature is raised from 0 K for antiferromagnetically coupled spins on the vertices of an icosidodecahedron, as well as an equilateral triangle, an octahedron, and a cuboctahedron. This can be traced to the fact that each of these “zero-dimensional” structures is based on corner-sharing triangles. [It is not surprising then that a similar enhancement at Hsat/3 occurs for the classical Heisenberg model of spins (with antiferromagnetic exchange) on the infinite planar kagome lattice, which is also based on corner-sharing triangles.] As to the underlying mechanism, we have found that for low temperatures, when H is in the vicinity of Hsat/3, two competing families of spin configurations exist, where one behaves “magnetically stiff” leading to a reduction of the differential susceptibility. [See our Publication #114, Phys. Rev. Lett. 94, 017205 (2005), for details.]
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The rigorous classical 3-sublattice picture described above also provided us (my collaborator Jürgen Schnack and me) with a useful starting point for constructing a solvable, quantum model of {Mo72Fe30} based on a replacement of the nearest-neighbor Heisenberg Hamiltonian. In this “quantum rotational band model”, described in Publication #98, the Hamiltonian consists of the sum of three terms that describe symmetric, isotropic exchange interaction between the total spin operators for each of the three sublattices. The set of all eigenenergies and eigenvectors of this replacement Hamiltonian can be obtained in a straightforward manner. In particular, the ground state eigenvector can be pictured in terms of the parallel alignment of the 10 spins within each sublattice, precisely the quantum analogue of the exact classical ground state configuration found by Axenovich and me. (This provides the rationale for daring to replace the Heisenberg Hamiltonian by the rotational band model Hamiltonian.) The low energy portion of the spectrum of energy eigenvalues can be described in terms of several parallel “rotational bands” (quadratic dependence of the energies in each band on the total spin quantum number.) The energy gap (in units of Boltzmann’s constant) between the two lowest rotational bands works out to be approximately 8 K, corresponding to 0.69 meV. This result begs confirmation by experiment! Indeed, repeated low temperature inelastic neutron scattering studies have been performed on {Mo72Fe30}, and in our Publication #119 the experimental results are summarized and compared to our (M. Exler, J. Schnack, and me) theoretical predictions based on the quantum rotational band model. (In particular see Fig. 3 of article #119.) Quoting from the abstract of that publication, “The overall energy scale of the excitation and the gross features of the temperature dependence of the observed neutron scattering are explained by the (sic) quantum rotational band model”. Despite this favorable summary, the final sentence of the abstract reads: “However, no satisfactory theoretical explanation is yet available for the observed magnetic field dependence.” {Mo72Fe30} still provides us with fascinating challenges!
A “baby brother” of {Mo72Fe30}, abbreviated as {Mo72V30} since it is based on spin ½ vanadyl ions, has also been synthesized by Achim Müller and his group in 2005 (see our Publication #117). Using the quantum Monte Carlo method so as to compare the calculated magnetic susceptibility, based on the Heisenberg model, with experimental data, we estimated that the exchange constant (in units of Boltzmann’s constant) is approximately 245 K, to be compared to 1.57 K for {Mo72Fe30}. Unfortunately this huge value makes it impractical to experimentally observe the saturation field Hsat (many hundreds of Tesla), a dip in the differential susceptibility at Hsat/3, or to observe the energy gap between the lowest two rotational bands. Efforts, however, are afoot to synthesize a perhaps more convenient {Mo72Cu30}, based on spin ½ Cu ions. I am also hopeful that a spin 3/2 “sibling” of {Mo72Fe30} may be forthcoming in the near future. It is indeed satisfying that the methods of synthesis chemistry have reached the point where conscious design, not only the results of serendipity, can be a driver of new science.
The antiferromagnetic Heisenberg icosahedron in a magnetic field
We have found that a Heisenberg system of spins mounted on the vertices of an icosahedron that interact via isotropic antiferromagnetic nearest-neighbor exchange undergoes a first-order metamagnetic transition between two distinct families of ground state configurations at a critical value of the magnetic field. On reducing the field to below the critical value, instead of a transition to the low-field phase, the high-field phase persists as a metastable phase with a lifetime that appears to diverge as 1/T as the temperature T is reduced towards 0 K. This system is of special interest in that the metastable phase occurs as a result of the geometrical properties of the icosahedron, even though the exchange coupling between nearest-neighbors is strictly isotropic. This system might as a prototype for a nanosize electronic switch, where one would exploit the two phases as 0 and 1 bits. Detailed results are given (our Publication #116) in Phys. Rev. Lett. 94, 207203 (2005).
Fig. 2 (a) Spin configuration for the classical icosahedron for a particular magnetic field exceeding the transition field. (b) The magnetic molecule {Fe9}. Nine of the twelve sites of the icosahedron are occupied by magnetic Fe3+ ions (green spheres) and three are occupied by diamagnetic ions (violet spheres).
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A synthesis effort by our collaborator Richard E.P. Winpenny and his group at the University of Manchester has led to a magnetic molecule, named {Fe9}, where 9 of the 12 sites of an embedded icosahedron are occupied by Fe3+ magnetic ions. Unfortunately the remaining three sites are occupied by diamagnetic ions, leading to what is known in the mathematical theory of polytopes as a “tri-diminished icosahedron”. The results of a combined experimental and theoretical study of this magnetic molecule are described in our Publication #123. Efforts are continuing towards achieving the successful synthesis of a magnetic molecule that is a bona fide realization of an antiferromagnetic Heisenberg icosahedron.
Partial list of collaborators on magnetic molecules
Prof. Dr. Klaus Bärwinkel (klaus.baerwinkel@uos.de)
Fachbereich Physik, Universität Osnabrück, Germany
Classical and quantum statitstical mechanics
Prof. Ferdinando Borsa (borsa@fisicavolta.unipv.it)
Department of Physics, University of Pavia, Italy
Experimental nuclear magnetic resonance (NMR)
Dr. Euan Brechin (ebrechin@staffmail.ed.ac.uk)
Department of Chemistry, University of Edinburgh, UK
Chemical synthesis
Prof. Leroy Cronin (l.cronin@chem.gla.ac.uk)
Department of Chemistry, University of Glasgow, UK
Chemical synthesis
Dr. Larry Engelhardt (lengelhardt@fmarion.edu)
Completed PhD under my direction, Iowa State University, June 2006.
Assistant Professor, Dept. of Physics, Francis Marion University,
Florence SC
Quantum Monte Carlo method
Dr. Ovideau Garlea (garleao@ornl.gov)
Oak Ridge National Laboratory (USDOE), Oak ridge, TN
Experimental neutron scattering
Prof. Dr. Paul Kögerler (kogerler@ameslab.gov,
paul.koegerler@ac.rwthaachen.de, p.koegerler@fz-juelich.de)
Ames Laboratory, Universität Aachen, and Forschungszentrum Jülich,
Germany
Chemical synthesis
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Dr. Bella Lake (bella.lake@hmi.de)
Hahn-Meitner Institute, Berlin, Germany
Experimental neutron scattering
Prof. Dr. Achim Müller (a.mueller@uni-bielefeld.de)
Anorganische Chemie I, Universität Bielefeld, Germany
Chemical synthesis
Prof. Janice Musfeldt (musfeldt@utk.edu)
Department of Chemistry, University of Tennessee, Knoxville, TN
Experimental optics
Dr. Stephen E. Nagler (naglerse.ornl.gov)
Oak Ridge National Laboratory (USDOE), Oak Ridge, TN
Experimental neutron scattering
Prof. Hiroyuki Nojiri (nojiri@imr.tohoku.ac.jp)
Department of Physics, Tohoku University, Sendai, Japan
Experimental electron spin resonance (ESR) and pulsed magnetic fields
Dr. Ruslan Prozorov (prozorov@ameslab.gov)
Ames Laboratory & Dept. of Physics and Astronomy, Iowa State University,
Ames, IA
Experimental milli-kelvin magnetization studies
Prof. Dr. Heinz-Jürgen Schmidt (hschmidt@uos.de)
Fachbereich Physik, Universität Osnabrück, Germany
Application of analytical methods to classical and quantum spin systems
Prof. Dr. Jürgen Schnack (jschnack@uos.de)
Fachbereich Physik, Universität Bielefeld, Germany
Numerical methods for quantum Heisenberg models of magnetic
molecules
Prof. Dr. Christian Schröder (christian.schroeder@fh-bielefeld.de)
Applied Sciences University Bielefeld, Germany
Computer simulational methods (classical spin dynamics, classical Monte
Carlo method).
[Schröder is also the founder and Director of Spinhenge@home, a distributed computing facility devoted to calculations on magnetic molecules that harnesses individual PCs world-wide while their processors are otherwise idle. Currently (mid-November 2006) there are over 16,000 PCs subscribed to Spinhenge@home. Web site: http://spin.fh-bielefeld.de]
Dr. Joris van Slageren (slageren@pi1.physik.uni-stuttgart.de)
Physikalisches Institut, Universität Stuttgart, Germany
Experimental electron spin resonance (ESR) and thermodynamic
measurements
Prof. Byoung-Jin Suh (bjsuh@catholic.ca.kr)
Currently: Visiting Scientist, Ames Laboratory; Permanent post: Dept. of
Physics, Catholic University of Korea, Seoul
Experimental nuclear magnetic resonance
Dr. David Vaknin (vaknin@ameslab.gov)
Ames Laboratory, Ames, IA
Experimental X-ray and neutron scattering
Prof. Richard E.P. Winpenny (Richard.Winpenny@man.ac.uk)
Department of Chemistry, University of Mancester, UK
Chemical synthesis
Quantum many-body systems
(including superfluid 4He and Bose-Einstein condensation)
(12 publications: Nos. 1, 2, 3, 4, 5, 6, 7, 10, 11, 19, 20, 27)
These 12 articles written between 1962 and 1975 deal with models of interacting boson systems. The primary motivation of this work was to provide an improved first-principles understanding of various aspects of superfluid 4He as well as the modifications of Bose-Einstein condensation due to the interaction between bosons. The model Hamiltonian adopted in article #1 is essentially the boson version of the BCS pairing model of superconductivity. The main objectives achieved were the determination of (i) the wave vector and temperature dependence of the excitation spectrum, (ii) detailed properties of the Bose-Einstein condensation, and (iii) the thermodynamic properties both above and below the transition temperature. In #2 we derived a hard-sphere pseudopotential which overcomes a major limitation of the pseudopotential developed by K. Huang - C.N. Yang (1957). In #3 we applied our pseudopotential to study the properties of a low-density hard-sphere boson gas at T = 0 K. In #7 we showed that the divergence of the constant-pressure specific heat and the isothermal compressibility as one lowers the temperature of the ideal boson gas towards the transition temperature can be directly related to the onset of off-diagonal long-range order (ODLRO) of the one-particle density matrix. In #19, 20, and 27 aspects of the temperature and momentum dependence of phonon energies in superfluid 4He were treated in detail.
Phase transitions and critical point phenomena
(20 publications: Nos. 9, 12, 13, 15, 16, 17, 22, 23, 26, 29, 30, 33, 34, 35, 36, 40, 42, 43, 44, 45)
These articles were written between 1969 and 1979. In the three most important of these articles (underlined), Richard Hornreich, Shmuel Shtrikman, and I introduced a new type of multicritical point, which we called a Lifshitz point, that separates ordered phases with wave vector k = 0 and non-zero k along the lambda line. In #29 and 30 we calculated the critical exponents, first using the renormalization-group method to provide the first few terms of an expansion in powers of ε =4 – d, where d is the dimensionality, and then (to first order) using the 1/n method. In 33 we proposed and treated an exactly solvable model that exhibits a Lifshitz point: a spherical model describing a hypercubic d-dimensional lattice (arbitrary d) of spins with nearest-neighbor ferromagnetic coupling and next-nearest-neighbor antiferromagnetic coupling along a single axis. Some 30 years later there continues to be considerable activity in the theoretical and experimental aspects of both magnetic and non-magnetic systems that possess a Lifshitz point.
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General statistical mechanics
(15 publications: Nos. 18, 21, 28, 46, 48, 52, 54, 55, 57, 61, 65, 74, 75, 76, 86)
These 15 articles were written between 1972 and 1999. In #21 we showed that the solvable Baker-Essam model of a compressible Ising lattice is equivalent to a set of linear chains each described by the Mattis-Schultz one-dimensional magnetostriction model. In #46 we applied the method of Levin approximants to the virial expansion for the pressure of hard discs and hard spheres and the results suggest that the pressure diverges at the density of closest-packing with the critical exponent γ = 1. In #48 we derived the second, third, and (two of the three terms of the) fourth virial coefficients of a classical gas of hard hyperspheres of arbitrary dimensionality. In #52 it was shown that application of the method of Levin approximants to the first six terms of the (low density) virial series expansion of the equation of state for the classical hard-disk fluid are in excellent agreement with accurate Monte Carlo – molecular dynamics data (extending to within 70% of the density of closest-packing). The finite-size scaling properties of the 1-D Glauber kinetic Ising model of critical dynamics was derived in #74. In #75 a detailed derivation was given for the time and wave vector dependence of the nonequilibrium elastic-scattering structure factor for the Kawasaki spin-conserving kinetic Ising model of a one-dimensional system. In the analysis we assumed that the system was in equilibrium at an initial temperature and then suddenly placed in contact with a heat bath at a much higher temperature. As an application of this work, in #76 we considered the behavior of the non-equilibrium structure factor describing the disordering of adsorbed monolayers following an abrupt temperature increase.
Electron localization in one-dimensional incommensurate potentials
(5 publications: Nos. 49, 56, 58, 60, 62)
These five articles, written between 1984 and 1988, deal with the fascinating yet very difficult question, what are the conditions for the occurrence of spatially localized eigenfunctions for independent particles in a one-dimensional incommensurate potential? The simplest types of incommensurate potentials are compound potentials consisting of two periodic components, but where the ratio of the two spatial periods is an irrational number, giving rise to a compound potential that is only “quasi-periodic”, but not periodic. Retreating to the time-independent Schrödinger equation in the single orbital tight-binding approximation does not really help simplify matters; one faces an intractable three-term difference equation. (The same type of difference equation arises in other contexts: As examples, a linear chain with a non-periodic array of two or more distinct point masses or force constants: a non-periodic stack of dielectric layers.) It became essential to develop and implement a reliable and efficient numerical algorithm, one that suppresses the numerical instability originating from the existence of a growing solution which inevitably accompanies a localized solution. An easily implemented algorithm with this capability was developed by James Luscombe and me and presented (both the underlying theory and applications) in article #60. With this algorithm one calculates a certain auxiliary function of the energy variable; the zeros of the auxiliary function coincide with the energies of localized eigenstates. We demonstrated explicitly that our algorithm provides both ultra-high precision results for localized, normalized energy eigenfunctions and their associated energy eigenvalues. [This algorithm also provides the essential ingredient for our technique given in article #83 (“Simplified recursive algorithm for Wigner 3j and 6j symbols”).] A year later we published a follow-up article, #62, entitled “Incipient Infinite Degeneracy and the Delocalization Transition in a Quasiperiodic Potential” in Physical Review Letters. We used our algorithm to explore the onset of extended eigenstates when the strength parameter of the second periodic component of the compound potential is decreased toward the Aubry-André critical value. I believe that this is a clearly written but very difficult article, in part because of its heavy use of profound theorems from number theory. Understandably this article is hardly ever cited, but that does not detract from its importance. This however was a clear signal for me that it was time to move on to other issues!
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Bloch oscillations and other dynamical phenomena of electrons in semiconductor superlattices
(8 publications: Nos. 50, 56, 70, 72, 73, 77, 79, 80)
These 8 articles were written between 1985 and 1998. The goal of this research was to determine the dynamical properties of independent electrons in a one-dimensional periodic lattice in the presence of a uniform external dc electric field of strength E. The impetus for this work grew with the successful fabrication of semiconductor superlattices in the early 1970s allowing for tunable laboratory structures. However, already in 1928 Felix Bloch, adopting highly simplifying assumptions, showed that the electric field gives rise to time-periodic electron oscillations, commonly referred to as “Bloch oscillations”, with period τ = h/(eEa), where h is Planck’s constant, e is the magnitude of the electron charge, and a is the lattice (or superlattice) spacing. However, for nearly 70 years the debate raged as to whether Bloch oscillations survive as a feature of the exact dynamics, i.e., when the complete Hamiltonian is dealt with consistently, free of all approximations (such as one- and two-band truncations made by Bloch, Zener, and many others). The debate was particularly intense because of potential applications, for if Bloch oscillations can occur the oscillating electrons should radiate electromagnetic radiation in the terahertz regime (for superlattice parameters). This radiation was in fact first observed in 1992. In parallel, at the very same time, my Ph.D. student Ann Bouchard and I succeeded in solving the exact time-dependent Schrödinger equation, using high-accuracy numerical methods, without adopting any truncations or uncontrolled approximations. Our finding was that in general three distinct types of dynamical phenomena can occur: Bloch oscillations; unbounded acceleration of a portion of the electron wave packet; localized (intrawell) high frequency oscillations. The precise mix of these three phenomena depends on the detailed parameters of the superlattice potential, the field strength, and the specific form of the initial electron wave packet. A first summary of our results appeared in article #70, and a lengthy, comprehensive summary of our work was given in #72. In a later article (#77) my Ph.D. student Joseph Reynolds and I showed that as a result of unavoidable interface roughness in superlattices there would be a degradation of the THz radiation signal, consistent with experimental findings. In article #79we, along with James H. Luscombe (Naval Postgraduate School), proposed an experimental strategy for achieving Bloch oscillations with a significantly longer lifetimes than had been observed heretofore. The idea involved selective photoexcitation of electrons by using an aperiodic superlattice having a single modified layer in an otherwise periodic superlattice. Related to this proposal, in #80, Luscombe and I derived the analytical form of the time-dependent wave function for a single-band tight-binding model of independent electrons within this aperiodic superlattice. The mathematical treatment is very elegant!
Semiconductor quantum nanostructures
(7 publications: Nos. 64, 66, 67, 68, 69, 71, 83)
These 7 articles deal with quantum dots, wires, and superlattices and were written between 1989 and 1999, in collaboration with scientists at Texas Instruments, Inc., especially James H. Luscombe. One motivation of industrial and government laboratories for the study of quantum nanostructures is the hope of developing futuristic, ultra-compact and ultra-fast semiconductor devices that specifically exploit the quantum properties of ultra-small semiconductor systems. The associated issues of basic science are both novel and extremely interesting. For example, in article #64, we dealt with an application of basic graduate quantum mechanics. Using an isotropic two-dimensional anharmonic oscillator model we were able to reproduce the energy level spacings that had been observed in resonant tunneling measurements through an axially symmetric double-barrier quantum dot structure fabricated at Texas Instruments. In article #66 we investigated the problem of lateral confinement of electrons in quantum nanostructures using a finite-temperature Thomas-Fermi method. A more sophisticated treatment of this issue, using a self-consistent Poisson-Schrödinger treatment was given in articles #68 and #69. Closely related to articles #79 and #80 that are cited in the section on Bloch oscillations, in article #83 we performed a specialized calculation of the binding energy of an exciton formed when an electron-hole pair is photoexcited in a single layer of an otherwise periodic superlattice.
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Slit height correction in small angle X-ray scattering
(5 publications: Nos. 37, 38, 39, 41, 47, 59)
These five articles were written between 1977 and 1987 specifically to provide a practical method for X-ray scatterers to eliminate, once and for all, the “slit-height correction problem in small angle X-ray scattering.” Such scattering experiments are usually performed with long, narrow slits collimating the incident and scattered beams since considerations of prohibitively low intensity make it impractical to employ pinhole collimation. The question put to me was, is there a way to correct experimental data for “slit height smearing” effects so as to yield the scattered intensity that would be obtained with pinhole collimation? Moshe Deutsch, a graduate student at the time - now a professor at Bar-Ilan University, and I found that this could in fact be done; by employing a few slick mathematical tricks we were able to find the general solution of the Volterra-type integral equation that links the (measurable) “slit transmission function” and the “correction function”. All that remains is the straightforward numerical evaluation of a one-dimensional integral containing the correction function and the measured intensity data. These five articles provided both a comprehensive, complete solution to a long-standing experimental problem and, for me, many delightful hours playing with non-trivial issues of applied mathematics.
Miscellaneous
(12 publications: Nos. 8, 14, 24, 25, 31, 32, 51, 53, 63, 82, 104, 113)
The following articles deal with topics that were of considerable interest to me in the course of the work but were not ultimately part of a sustained effort over a lengthy period or involving a significant number of other publications. In order to help the reader quickly identify the subject matter of these projects I list the title of each of the above publications. The reader is of course invited to peruse the actual article to obtain a full picture of the work.
#8: “Spin relaxation in Mössbauer spectra of magnetically ordered systems”, with L. Levinson.
#14: “Microscopic model for reorientation of the easy axis of magnetization”, with L. Levinson and S. Shtrikman.
#24: “Transition from the cholesteric storage mode to the nematic phase in critical restricted geometries”, with D. Mukamel and S. Shtrikman
#25: “Exact solution for the linear response of atomic hydrogen to an external electromagnetic field”, with B. Nudler and I. Freund
#31: “Director field and lifetime of the cholesteric storage mode”, with J. Isaacson and S. Shtrikman
#32: “Exact solution for the linear response of a hydrogenic atom to an external electromagnetic field. I. Frequencies below photoelectric threshold”, with B. Nudler-Blum.
(Complete results were also obtained by us for frequencies above photoelectric threshold, and are summarized in Ms. Nudler-Blum’s M.Sc. thesis (in Hebrew) approved by Bar-Ilan University in 1977. Various difficulties at the time prevented the preparation of an article for publication.)
#51: “Estimates of the electron g factor: Application of convergence acceleration methods to the QED series”, with H. Chew
#53: “New Schrödinger equations for old: Inequivalence of the Darboux and Abraham-Moses constructions”, with D. Pursey
#63: “An efficient method for generating a uniform distribution of points within a hypersphere”, with L. P. Staunton
#82: “Simplified recursive algorithm for Wigner 3j and 6j symbols”, with J. H. Luscombe
#104: “Almost-periodic wave packets and wave packets of invariant shapes”, with D. Mentrup
#113: “Analytic solution of the microcausality problem in discretized light cone quantization”, with L. Martinovic
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