aa r X i v : . [ phy s i c s . pop - ph ] M a r A Brief history of mangnetism
Navinder Singh ∗ and Arun M. Jayannavar ∗∗∗ Physical Research Laboratory, Ahmedabad,India, Pin: 380009. ∗∗ IOP, Bhubaneswar, India. ∗ †
Abstract
In this article an overview of the historical development of the key ideas in the field ofmagnetism is presented. The presentation is semi-technical in nature.Starting by notingdown important contribution of Greeks, William Gilbert, Coulomb, Poisson, Oersted,Ampere, Faraday, Maxwell, and Pierre Curie, we review early 20th century investigationsby Paul Langevin and Pierre Weiss. The Langevin theory of paramagnetism and the Weisstheory of ferromagnetism were partly successful and real understanding of magnetism camewith the advent of quantum mechanics. Van Vleck was the pioneer in applying quantummechanics to the problem of magnetism and we discuss his main contributions: (1) hisdetailed quantum statistical mechanical study of magnetism of real gases; (2) his pointingout the importance of the crystal fields or ligand fields in the magnetic behavior of irongroup salts (the ligand field theory); and (3) his many contributions to the elucidation ofexchange interactions in d electron metals. Next, the pioneering contributions (but lesserknown) of Dorfman are discussed. Then, in chronological order, the key contributionsof Pauli, Heisenberg, and Landau are presented. Finally, we discuss a modern topic ofquantum spin liquids.
In this presentation, the development of the conceptual structure of the field is highlighted ,and the historical context is somewhat limited in scope. In that way, the presentation is nothistorically very rigorous. However, it may be useful for gaining a ”bird’s-eye view” of the vastfield of magnetism.
The ”magical” properties of magnets have fascinated mankind from antiquity. A naturallyoccurring magnet, lodestone i , was known to ancient Greeks. The famous Greek philosopherThales of Miletus documented loadstone’s magnetic properties in the 6th century BC[2]. Inthe same century the ancient Indian physician Sushruta was aware of the magnetic properties ∗ Email: [email protected] † Email: [email protected] i Lodestone is a naturally magnetized lump of the mineral magnetite (an oxide of iron). Earth’s magneticfield is too weak to magnetize it. It is probably lightning bolts that magnetize it, as lodestone is found near thesurface of earth, not deep underneath.
1f loadstone, and used it to remove metal splinters from bodies of injured soldiers ii . However,according to reference[4], ancient Chinese writings that date back to 4000 BC mention mag-netite and use of magnetic compass in navigation. Apart from the question of the discovery ofloadstone, it is clear that it was known in the BC era, and was used for the benefit of mankindwhether in surgery or in navigation. iii However, before William Gilbert of England in 17th century, study of magnetism was meta-physical. Along with loadstone’s good use many superstitions were attached to it (due to its”magical” magnetic properties). It was believed that loadstone feeds on iron, an iron piece keptnear loadstone would loose its weight while loadstone would gain. But falsey of such meta-physical claims was clear when the experimental method of science started in 16th century, andexperiments to test this ”weight gain-weight loss” effect gave negative results. Another popularsuperstition attached to loadstone was that it can act as a pain killer, and it was used as anamulet to be tied around affected body parts. Today we know that these beliefs have no scientificbasis and magnetism has no known direct effect on biological or psychological processes.
Scientific study of magnetism started in the 16th century with the investigations of WilliamGilbert (1544 - 1603) of England. He studied medicine and started his practice as a physicianin 1573. He lived in good times of the Elizabethan renaissance in music, art, and in Naturalphilosophy, and was private physician of the queen Elizabeth I from 1601 to 1603. Alongwith medicine did his pioneering investigations in natural philosophy (as subject of sciencewas called in those times). iv The most important work of his life is ”De Magnete” in whichhe summarized all his investigations of magnetic and static electricity phenomena. He clearlydifferentiated between static electricity and magnetism. He also studied the effect of temperatureon magnetic properties of iron and discovered that when iron is red hot it ceases to be attractedby a magnet[5].
His most important discovery is the realization that earth itself is a large magnetand North-South pointing property of compass needle can be understood from the attraction ofunlike poles of magnets thereby bringing the mysterious pointing properties of compass needle tocorrect scientific explanation . Then 17th century was sort of long lull regarding the subject of magnetism, not much was donein this field. It is interesting to note that great Issac Newton (1642 - 1726) did not contributemuch to magnetism, although he developed all the other subjects significantly, and created manynew, as is well known. It could be possible that he thought about cause of magnetism and couldnot reach on to experimentally testable concrete conclusions. His account of magnetism in hisgreatest work ”Principia” is very ambiguous[7].18th century saw a major development. Charles-Augustin de Coulomb (1736 - 1806) exper-imentally found the inverse square force law (now famously known as the Coulomb law) usinghis torsion balance set-up. He started his career as an engineer in French army in 1761, and ii His work ”Sushruta-Samhitha” has been described as the most important treatise on medicine and surgeryin ancient times and is the founding work of Ayurveda[3] iii
It is interesting to note that the phenomenon of frictional electricity was also discovered in that era and thesetwo subjects developed largely independently from each other until these were ”wedded” together in a widersubject of electromagnetism in the 19th century, by the investigations of Faraday, Maxwell, and others[5, 6]. iv He supported the philosophy of Nicolus Copernicus (1473 - 1543) that earth is not the centre of the universe,and rejected the old Aristotelian philosophy. v It is interesting to note that in those days the theoretical understandingof electric and magnetic phenomena was a sort of meta-physical. It was believed that thereis an invisible fluid that carriers magnetic and electric effects from one place to another. Thiswas known as the ”effluvia theory”. vi The effluvia theory was discarded much later, in the19th century, when field concept was originated by Faraday. It is also interesting to note thatCoulomb believed in the effluvia theory but performed accurate experiments. And these are hisaccurate experimental findings that stood the test of the time and immortalized his name!
Next major development in the field of magnetism was brought by Simeon Denis Poisson (1781-1840) and Carl Friedrich Gauss (1777 - 1855). These men formulated Coulomb’s experimentalfindings into an elegant mathematical theory. Poisson obtained mathematical expressions forforce law for an arbitrarily shaped magnetic material in terms of surface and volume integrals.Calculus was extensively used to analyze these problems. The concept of ”potential” wasinvented for quantitative analysis of problems related to electric and magnetic phenomena. vii
After the invention of the first electric battery in 1800 (the Voltaic pile) a major discovery hap-pened in the field of electricity and magnetism which would bridge the gap between the two. In1820, Danish physicist Christian Oersted (1777 - 1851) discovered that electric current producesmagnetic effects. In the famous experiment with his lab assistant Hansteen he observed thata current carrying wire has the capacity to deflect a nearby placed magnetic needle. However,this discovery was not totally unexpected as Oersted believed that there might be a connec-tion between electricity and magnetism. His belief of this intimate connection was a result ofinfluence on him of Immanuel Kant’s philosophy which he studied for his dissertation in 1799.Kantian philosophy asserts deep connections in natural phenomena and unity of Nature.
The French scientist Andre - Marie Ampere (1775 -1836) took it further and developed theidea that ”magnetic force” around a wire has circular character and it leads to a law which isnow called the Ampere law (written down in a mathematical form by J C Maxwell). He alsoperformed further experiments on magnetic effects of currents like measuring forces betweencurrent carrying wires. Other French investigators of his time (in this field) were Biot, Savart, v It turns out that John Mitchell (1724-1793) who was contemporary of Coulomb discovered torsion balance,and found inverse square force law between magnetic poles prior to Coulomb[5]. However, Coulomb performedmuch more systematic and detailed studies, and wrote about it extensively. vi This state of affairs is much like that in the Caloric theory of heat in which it was believed that heat is akind of fluid (the Caloric) which flows from hot bodies to cold bodies. Later on, it was experimentally provedwrong by Count Rumford who observed that large amount of heat is created when iron rods are drilled to makegun barrels, this violated the conservation of Caloric, and mechanical theory of heat was advanced. vii
Poisson was a leading opponent of Huygen’s wave theory of light even though in 1800 Thomas Youngconvincingly demonstrated wave nature of light using his two slit experiment. Poisson strongly believed inNewton’s corpuscular theory and also opposed Jean Fresnel’s wave - theoretical explanation of diffraction.
Perhaps his most importantcontribution with respect to magnetism of matter is his explanation that Ferromagnetism is aresult of internal currents in these materials.
This is much close to present day understanding.He was aware of Joule heating effects and ascribed the internal currents to ”molecules” of ironas giving magnetic effects. Notice that notion of molecules and atoms at that time was notwell established. Thus most important contributions of Ampere are his
Ampere’s law and hisexplanation of ferromagnetism from microscopic circulating currents.
One of the most important name in the development of the field of magnetism is MichaelFaraday (1791 – 1867)–the self-trained experimental genius of 19th century. Born in a familywhich was not well to do financially, Faraday had to struggle and had only very basic schooleducation. He studied mostly by himself. At a young age of 14 he became an apprentice to alocal book binder and seller. This gave an opportunity to read, and Faraday read most of thebooks that came to him for binding. The second lucky break came to him when the eminentEnglish chemist Humphry Davy of the Royal institution supported him and offered a job oflab assistant in his lab. This started Faraday’s scientific career. In 1831 Faraday discoveredthe law of electromagnetic induction – arguably the most important discovery of the 19thcentury. Faraday was motivated by his studies, in collaboration with Sir Charles Wheatstoneon vibrational phenomena in iron plates, in which acoustic induction caused one plate to vibratewhen a nearby plate is vibrated by an external means. It seems that this acoustic inductionis behind Faraday’s most famous experiment of the ”electromagnetic” induction. He wounda coil of insulated copper wire around a thick iron ring on one side, and an other coil on theopposite side. He connected one of the coils to a galvanometer, and the other to a voltaic pilethrough a switch. When the switch was closed or opened he observed sudden deflection thegalvanometer. This marks the discovery of the phenomenon of electromagnetic induction. Inthe fall of the same year 1831 Faraday also noticed that a current can be induced in a coil bymoving a permanent magnet close to it. We now know how important these discoveries are inthe development of the filed of electricity and magnetism, and how these discoveries lead to thetechnological revolution.Thus Faraday took the studies of Oersted and Ampere forward and developed the subjectof electromagnetism. Most importantly in the theoretical understanding of the phenomenaFaraday advanced the field of electromagnetism from metaphysical conceptions of magneticfluids (effluvia theory) to the modern concept of magnetic and electric fields . The field conceptand the concept of lines of force are his one of the biggest contributions. With the concepts of”magnetic field” and ”lines of forces” he could explain these phenomena, and latter on Maxwelldeveloped the mathematical theory of electromagnetism based on Faraday’s conceptions. Alongwith his pioneering experiments in electricity and magnetism Faraday also discovered the fieldof electrochemistry. As this brief historical introduction is aimed at the historical developmentof magnetism, we skip these interesting discoveries and interested reader can read about thesein a dedicated biography of Faraday, for example in[9].Before him only strongly magnetic materials were known like iron and cobalt which showsferromagnetism (a strong form of magnetism). Faraday also discovered ”weak magnetism”, i.e.,dia- and para-magnetism in substances like oxygen and bismuth. He observed that paramagneticsubstances are attracted towards stronger magnetic fields while diamagnetic substances arerepelled. 4
Enter Maxwell
A consolidation of Faraday’s experimental findings was achieved by James Clerk Maxwell (1831-1879) who expressed those findings by an elegant mathematical language. The result was thefamous Maxwell equations. With his profound intuition he was able to go beyond Faraday andintroduced his ”displacement term”. His theoretical synthesis was primarily based on Maxwellsinvestigations. The following lines by Maxwell shows how closely he followed Faraday:”.........resolved to read to no mathematics on the subject till I had first read through Fara-day’s Experimental researchers in Electricity. I was aware that there was supposed to be a differ-ence between Faraday’s way of conceiving phenomena and that of the mathematicians.........As Iproceeded with the study of Faraday, I perceived that his method.......capable of being expressedin ordinary mathematical forms. For instance, Faraday, in his mind’s eye, saw centers of forcetraversing all space where the mathematicians saw centers of force attracting at a distance:Faraday saw a medium where they saw nothing but distance.[10]”In a nutshell, Maxwell tied the final remaining lose threads of the unification of electricity andmagnetism and the field of electromagnetism was born. This further enabled Albert Einstein topoint out fundamental problems in classical mechanics of Issac Newton and lead to the advent ofthe special theory of relativity. The technological revolution brought about by Maxwell’s workis well known. Without going too far away from our main topic of the history of magnetism westress here that the findings of Coulomb, Oersted, Ampere, Poisson, Faraday, and others wereexpressed in a beautiful mathematical language by Maxwell which lead to further developmentsin magnetism and other fields.
10 Enter Pierre Curie
Pierre Curie (1859 -1906) advanced further the experimental findings of Michael Faraday onweakly magnetic substances i.e., paramagnetic and diamagnetic materials. He undertook thestudy of these materials for his doctoral thesis with the aim to investigate whether there aretransitions between various kinds of magnetism in a given material. He performed a thoroughstudy of magnetic properties of some twenty substances[11]. These painstaking investigationsleads to three important discoveries: (1) paramagnetism in various salts is temperature de-pendent and magnetic susceptibility (ratio of induced magnetism to applied magnetic field) isinversely proportional to temperature (known as the Curie law ( χ ∝ T )); (2) Ferromagnetismis also a function of temperature and completely vanishes when temperature is raised above acritical temperature called the Curie temperature ( T c ); and (3) diamagnetism is approximatelytemperature independent. To measure the magnetic coefficients he designed and perfected avery sensitive torsional balance that could measure up to 0.01 mg!Pierre Curie, along with his wife madam Marie Currie, discovered radium and polonium anddid the pioneering investigations in the field of radioactivity. In 1903 they, along with HenriBecquerel, were awarded with Nobel prize for their investigations in radioactivity. But withoutany doubt Curie’s experimental investigations of weakly and strongly magnetic substances aremajor contributions to magnetism. However, microscopic understanding of various forms ofmagnetism remained unclear.
11 Enter Langevin
The experimental studies of Faraday and Curie on weakly magnetic substance remained com-pletely unexplained up to the end of 19th century. The first ”microscopic” understanding5f the behavior of diamagnetic and paramagnetic substances came with the investigations ofPaul Langevin (1872 - 1946). Langevin, using then newly-discovered statistical mechanics ofBoltzmann and Gibbs came up with a mathematical theory which showed that paramagneticsusceptibility is inversely proportional to temperature, whereas the diamagnetic one shows tem-perature independent behavior. At the core of the Langevin theory was his phenomenologicalintroduction of the tiny magnetic moment of atoms.At the time when Langevin advanced his theory (in 1905) atomic structure was unknown(Rutherford’s alpha scattering experiment (1911) and Bohr’s model of the atoms came manyyears later (1913)). However, Langevin’s ad-hoc introduction of tiny atomic magnetic momentswas more or less correct. We now know that magnetic moments in atoms originate eitherfrom the orbital motion of unpaired electrons, or from the spin of electrons, or some vectorialcombination of these two. For paired electrons magnetic effects of orbital motions nullify eachother, and two paired electrons in an orbital give zero net spin by Puali’s exclusion principle, thusno magnetic effects. All noble gases in the right most column of the periodic table are examplesof this nullifying effect and they are thus not paramagnetic. They show only diamagnetism.With this ad-hoc assignment of a tiny magnetic moment to each atom of a paramagneticsubstance and by including the thermal agitation of magnetic moments Langevin was able toobtain the correct temperature dependence of magnetization of a sample. At very low temper-atures, when thermal agitation is feeble, the tiny atomic magnetic moments tend to align alongthe direction of an applied magnetic field thus showing paramagnetic effect. At higher temper-atures due to thermal agitation lesser atomic magnetic moments align along the direction ofexternal magnetic field leading to a weaker paramagnetic effect. It turns out that at the averagemagnetic moment per unit volume of the whole sample (called magnetization) is given by M = nµ k B T H,
Where µ is the tiny magnetic moment of each atom, n is the number density of atoms, k B isthe Boltzmann constant, T is the absolute temperature, and H is the applied magnetic field[ ? ].Thus magnetic susceptibility ( χ = MH ) is inversely proportional to temperature in accordancewith observations. This is one of the pioneering achievement of Langevin. On similar linesLangevin was able to show that diamagnetic effects occur in substances in which there are noatomic magnetic moments ( µ = 0). Although in this case atoms appear magnetically neutral,but there could be induced magnetic effects due to the internal motions of electrons within eachatom. Using Faraday’s induction law, he was able to show an opposing magnetic effect i.e., theinduced magnetization in opposite direction to the applied magnetic field, and from statisticalmechanics obtained it’s temperature independence in accordance with experiments. Thus by1905, a partial understanding of weak magnetism (para- and dia- magnetism) was achieved. Theorigin of the tiny magnetic moments was a mystery at that the time, and another importantproblem of ferromagnetism remained unresolved.
12 Enter Pierre Weiss
Pierre Weiss (1865-1940) generalized the Langevin theory by introducing a new and very im-portant concept of ”mean molecular field” and with his ”generalized theory” he could accountfor ferromagnetism (however, only partly).The Weiss theory goes like this: Suppose that a ferromagnetic material (like iron) is placedin an external magnetic field H . The field acting on each atom of iron, according to Weiss, isnot given by the external field alone but an additional field (a molecular field as Weiss called it)6lso acts on each atom. This molecular Feld is proportional to the magnetization of the sample.Thus the field acting on each atoms is H + λM instead of H . Here M is the magnetization of thesample and λ is a material dependent constant. This is the main point of the Weiss theory. Thetemperature dependence of magnetization is much more stronger in ferromagnetic case and canbe qualitatively explained in the following way. When temperature is lowered more and morespins align along H and thereby enhancing M . Increased M reinforces the alignment processthrough the field H + λM until at sufficiently low temperatures saturation sets in, in which allspins point along H . Similarly, increasing temperature leads to more and more de-stabilizationof the spin alignment thereby reducing M . The field H + λM acting on each spin furtherweakens and at sufficiently high temperatures and in vanishingly small external field H it leadsto zero magnetization. So temperature dependence of magnetization is much more stronger inthe case of ferromagnetism, as compared to para- and dia- magnetism. It turns out that atsufficiently low temperatures it is energetically favorable for spins to align in one direction, thusthere can be a molecular field alone. The Weiss theory can be mathematically expressed if wechange H to H + λM in the above Langevin equation: M = nµ k B T ( H + λM ) . Thus leading to M = nµ / k B T − λ ( nµ / k B ) H. This is known as the Curie-Weiss law. If T is much greater than T c = λ nµ k B it gives a strongerparamagnetism. Thus Weiss theory qualitatively explained Curie’s experimental results forFerromagnetism. However, it has a serious problem. When experimentally known quantities( T c , n etc) are inserted in the above expression, it turns out that λ ∼ ! Within thesemiclassical theories it is not possible to explain such a large value of λ . Thus Weiss theorycould only partly shed light on the origin of ferromagnetism. Such large value of λ could onlybe explained with the advent of quantum mechanics, and in 1926 Werner Heisenberg was ableto explain such large values of λ using the concept of quantum mechanical exchange (refer toarticle II in this series and references therein).Also Weiss theory is a classic example of a mean-field theory. When T → T c , susceptibilitydiverges, and the actual scaling law of susceptibility with temperature ( T − T c ) can only beobtained by going beyond the mean-field approximation. In summary, the above Weiss expres-sion could explain Curie’s experimental observations in ferromagnetic substances at T > T c ,and from the Weiss expression, negative magnetization for T < T c means that molecular field isstrong enough to spontaneously magnetize a ferromagnetic sample without an external appliedfield. Pierre Weiss also designed large electromagnets and in 1918 established a Laboratorydedicated to magnetism in Strasbourg, France, where later in 1932 Louis Neel discovered antiferromagnetism during his doctoral study. Neel was awarded with Nobel prize for this funda-mental discovery in 1970.
13 Pre-quantum mechanical era and the problems of theold quantum theory
The success and failure of the old quantum theory of Bohr and others are well known[13].And how the new quantum mechanics developed by Heisenberg, Born, Schroedinger, and Dirac7eplaced the patch-work of old quantum theory by a coherent picture of new quantum mechanics,in early 1920s, is also well known. In 1922, Stern-Gerlach experiment showed that magneticmoment of atoms can orient itself only in specific directions is space with respect to externalmagnetic field. This quantum mechanical phenomenon of spatial quantization was certainlymissing in the Langevin treatment of paramagnetism. In the Langevin theory atomic momentscan take any orientation in space. The required discretization of the spatial orientations wasintroduced, for the first time, by Pauli viii who found that susceptibility expression with respectto the temperature variation is the same as that of Langevin but with different numericalcoefficient C in χ = C Nµ k B T . He found the value 1.54 instead of 1/3 of the Langevin theory.Pauli used integer quantum numbers but analysis of the band spectrum showed the need forhalf-integer values. Linus Pauling revised Pauli’s calculation by using half-integer instead ofinteger values, and it resulted in another value of the coefficient C [14]. The status of thefield was far from satisfactory by 1925. There was another big problem. The calculations ofsusceptibility within the regime of old quantum theory appeared to violate the celebrated Bohr’scorrespondence principle, which states that in the asymptotic limit of high quantum numbersor high temperatures, the quantum expression should go over to the classical one ( as in blackbody radiation problem for ~ ωk B T << In conclusion, the old quantum theory of magnetism was adismal failure.
14 Quantum mechanical and post-quantum mechanicalera, and the development of the quantum theory ofmagnetism
The modern quantum mechanics was in place by 1926. The equivalence of the matrix formula-tion of Heisenberg (1925) and wave-mechanical formulation of Schroedinger (1926) by shown bySchoredinger in 1926. In the same period van Vleck attacked the problem of magnetism with”new” quantum mechanics.
15 Enter van Vleck
One of the pioneer of the quantum theory of magnetism is van Vleck who showed how newquantum mechanics could rectify the problems of the old quantum mechanics, and restored thefactor of 1/3 of the Langevin’s semi-classical theory. In doing so he took space quantization ofmagnetic moment into account (instead of the integral in the partition function, proper sum-mation was performed). In one of the pioneer investigation, van Vleck undertook a detailedquantum mechanical study of the magnetic behavior of gas nitric oxide (
N O ). He showed quan-titative deviations from semi-classical Langevin theory in this case, and his results agreed verywell with experiments[15]. The quantum mechanical method was applied to other gases, andhe could quantitatively account for different susceptibility behavior of gases like O , N O , and N O . ix The differences in magnetic behavior arise from the comparison of energy level spac-ings ( ~ ω if ) with the thermal energy k B T . He showed that the quantum mechanical expression viii Actually Pauli calculated electrical susceptibility. It turns out that same calculation goes through formagnetic susceptibility except one has to replace electric moment by magnetic moment[14]. ix For a detailed account refer to his beautifully written book[14]. | ~ ω ij | << k B T ). In the opposite regime (when forall | ~ ω ij | >> k B T ) χ showed temperature independent behavior. In the intermediate regime( | ~ ω ij | ∼ k B T ) susceptibility showed a complex behavior (the case of nitric oxide). Thus vanVleck re-derived the Langevin theory by properly taking into account the space quantization.Another major contribution of van Vleck is related to magnetism in solid-state. When a freeatom (suppose a free iron atom) becomes a part in a large crystalline lattice (like iron oxide), itsenergy levels change. The change in the electronic structure of an atom is due to two factors (1)outer electrons participate in the chemical bond formation, thus their energy levels change, and(2) in a crystalline lattice, the remaining unpaired electrons in the outer shells of an atom arenot in a free environment, rather they are acted upon by an electrostatic field due to electronson neighboring atoms. This field is called the crystalline field.Van Vleck and his collaborators introduced crystalline field theory (also known as the ligandfield theory in chemical physics departments) to understand magnetic behavior in solid-state.With crystalline field ideas they could understand different magnetic behaviors of rare earthsalts and iron group salts. It turns out that in rare earth salts 4 f electrons are sequestered inthe interior of the atom, and do not experience the crystalline field very strongly. The energylevel splitting due to crystalline electric field is small as compared to thermal energy ( k B T ), andit remains small even at room temperatures. Due to this the magnetic moment of the atomsbehave as if the atom is free and shows the Langevin-Curie behavior χ ∼ T [15, 16, 17]. Iron group: strongly affected by crystal field Rare earth group: weakly affected by crystal field "Expossed" outer d (cid:1) orbitals "Sequestered" f (cid:0) orbitals
Figure 1: A cartoon showing why crystal field effects differently an iron group ion and a rareearth ion.In contrast to this case, in the iron group salts crystalline field is so strong that it quenchesa large part of the orbital magnetic moment, even at room temperatures, leaving mainly thespin part to contribute to magnetism of salts of iron.Magnetism of iron group metals is a different story (as compared to salts). In this case it turnsout that charge carriers are also responsible for magnetism. The magnetism due to itinerantelectrons was developed by Bloch, Slater, and Stoner [1]. The other extreme of localized electronswas investigated by Heisenberg. Van Vleck advanced ideas that can be dubbed as ”middle ofthe way” approach (refer to [1]). For his pioneering contributions van Vleck was awarded withthe Nobel prize in physics in 1977 along with Phil Anderson and Nevill Mott. His articles arebeautifully written and extremely readable and should form an essential element in a course(graduate or undergraduate) on magnetism. One can say that van Vleck is the father of themodern theory of magnetism, and his name will be forever remembered.
16 Enter Dorfman
When quantum mechanical study of magnetism of real gases was started by van Vleck in mid1920s, the quantum mechanical study of magnetism in metals also started in the other continenttransatlantic. 9he discovery of the paramagnetic properties of conduction electrons in metals is generallyattached to Wolfgang Pauli. Pauli’s paper came in 1927. Even before that, in 1923, Russianphysicist Yakov Grigor’evich Dorfman put forward the idea that conduction electrons in metalsposses paramagnetic properties[18]. His proposal was based on a subtle observation: when onecompares susceptibility of a diamagnetic metal with its ion, the susceptibility of the ion is alwaysgreater than its corresponding metal. It implies that there is some positive susceptibility in thecase of the diamagnetic metal that partly cancels out the larger negative diamagnetic suscep-tibility. And this cancellation is prohibited in the case of metal’s ion (due to ionic bonding).It was Dorfman’s intuition that some positive susceptibility is to be attributed to conductionelectrons in the metal i.e., some paramagnetic susceptibility has to be there. x Dorfman’sconclusion is based on his careful examination of the experimental data. After the discovery ofthe electron spin, Pauli gave the theory of paramagnetism in metals due to free electron spin.However, Dorfman was the first to point out paramagnetism in metals[18].One of the other important contributions of Dorfman is his experimental determinationof the nature of Weiss molecular field responsible for ferromagnetism in the Weiss theory. Itwas believed that the Weiss field is of magnetic origin due to which spins align to give a netspontaneous magnetization. To determine whether the Weiss field is of magnetic origin or ofnon-magnetic origin, Dorfman passed beta-rays (a free electron beam) in two samples of nickelfoils, one magnetized and the other unmagnetized. From deflection measurements he determinedthat Weiss field is of non-magnetic origin[19].In conclusion, Dorfman was an early contributor to the quantum theory of magnetism. Buthe is not as well known as he should have been.
17 Enter Pauli
Pauli’s contribution to magnetism is well known. He formulated paramagnetic behavior of con-duction electrons in metals in 1927 and showed that paramagnetic susceptibility is temperatureindependent (in the leading order). The derivation is discussed in almost all books devoted tomagnetism and solid state physics[20]. Pauli’s derivation of the paramagnetic susceptibility canbe described as one of the early application of Fermi-Dirac statistics of electrons in metals. Inthe standard derivation[20] one calculates the thermodynamical potential Ω( H ) of free electrongas in a magnetic field H . Magnetization is obtained by the standard algorithm of statisticalmechanics: M = − ∂ Ω ∂H , and susceptibility χ = ∂M∂H . For illustration purpose there is a simplerargument[21] which goes like this. For metals at ordinary temperatures one has k B T << E F where T is the temperature and E F is the Fermi energy. Thus electrons only in a tiny diffusionzone around the Fermi surface participate in thermodynamical, electrical, and magnetic prop-erties (other electrons are paired thus dead). If N is the total number of electrons, then fractionof electrons in the diffusion zone is N TT F where T F is the Fermi temperature ( k B T F = E F ). Eachelectron in the diffusion zone has magnetic susceptibility roughly given by χ ∼ µ k B T where µ is itsmagnetic moment. Thus total magnetic susceptibility of metal is given by: N TT F × µ k B T = N µ k B T F which is independent of temperature as the more accurate calculation shows. x It is important to note that the notion of the electron spin came in 1925 with a proposal by Uhlenbeck andGoudsmit and paramagnetism due to electron spin was discovered in 1927 by Pauli as mentioned before. ButDorfman’s proposal came in 1923! As mentioned before Dorfman in 1927 pointed out that the Weiss molecular field required in thetheory of ferromagnetism is of non-magnetic origin. The puzzle of the Weiss molecular field wasresolved by Heisenberg in 1928. The central idea is that it is the quantum mechanical exchangeinteraction which is responsible for the ferromagnetic alignment of spins. Quantum mechanicalexchange interaction has no classical analogue, and it results due to the overlapping of orbitalwave functions of two nearby atoms. Symmetry of the hybrid orbital is dictated by the natureof the spin alignment which obeys the Pauli exclusion principle. Thus there is an apparentspin-spin coupling due to orbital symmetry and under specific circumstances the ferromagneticspin alignment significantly lowers the bonding energy thereby leading to a stable configuration. xi The Heisenberg model based on exchange interactions is related to the resonance-energy-lowering model for chemical bonding by Heitler and London[22]. In the Heitler-London theoryof the chemical bond in hydrogen molecule, it is the exchange of electrons on two hydrogenatoms that leads to the resonant lowering of the energy of the molecule. Electrons stay inan antiparallel spin configuration thereby enhancing the overlap of orbital wave functions inthe intermediate region of two hydrogen atoms. This leads to bond formation. This idea ofresonant lowering of energy via exchange of electrons is greatly used by Linus Pauling in hisgeneral theory of the chemical bond[22]. The Heisenberg model is built on similar ideas andgoes like this[16, 23]. Let S i be the total spin at an atomic site i . If exchange interactionbetween nearest neighbors is the only one important, then the interaction energy (under certainapproximationsis given by V ij = − J ij S i .S j .J ij is called the exchange integral xii . For ferromagnetism the sign of J ij has to be positive, andfor anti-ferromagnetism it has to be negative. The question on what parameters the sign of J depends is complicated and vexed one [1]. The above exchange interaction is now known asthe Heisenberg exchange interaction or the direct exchange interaction. There is a variety ofexchange interactions (both in metals and insulators) that are discussed in [1].To compare predictions of the model with experiment, one needs its solution. The veryfirst solution provided by Heisenberg himself is based on some very restrictive assumptions. Sotight agreement with experiments may not be expected, and it leads to some qualitative results.Heisenberg used complicated group theoretical methods and a Gaussian approximation of thedistribution of energy levels to find an approximate solution. xiii From his solution Heisenbergobserved that ferromagnetism is possible only if the number of nearest neighbors are greater xi It is very important to note that energy associated with spin-spin coupling of two electrons via exchange isvery large as compared to the magnetic dipole-dipole interaction energy which is given by V ij = u i .u j r ij − u i .r ij )( u j .r ij ) r ij . This very small magnetic energy cannot lead to ferromagnetic alignment. In other systems, like ferro-electricsit is an important energy. xii J ij = Z dτ Z dτ φ i (1) φ j (2) H c φ j (1) φ i (2) . xiii An alternative and comparatively simpler method was provided by Dirac using the vector model with similarconclusions[12, 13]. z = 8). This conclusion is certainly violated as many alloys show ferro-magnetism with z = 6. The second result which is much more important is that of magnitudeof λ it turns out that λ of the Weiss molecular field takes the form λ = z J N µ B . The large value of λ required for ferromagnetism is not a problem anymore, as the exchangeintegral J can be large, thus resolving the problem of Weiss theory. This is the biggest successof the Heisenberg model. In conclusion, Heisenberg’s model resolved the puzzle of the Weiss molecular field usingthe concept of exchange interaction. This concept turns out to be the key to the modernunderstanding of magnetism in more complex systems. Heisenberg’s solution was based on manydrastic assumptions which were later improved upon. Literature on the Heisenberg model andits various approximate solutions is very vast. Some references are collected here[16, 17, 23, 24].
19 Enter Landau
Metals which are not ferromagnetic show two weak forms of magnetism, namely, paramagnetismand diamagnetism. Paramagnetism we have discussed, diamagnetism due to free conductionelectrons is a subtle phenomenon and was a surprise to the scientific community[12] when Levlandau discovered it in 1930. To appreciate it consider the following example. Consider theclassical model of an atom in which a negatively charged electron circulates around a positivenucleus. A magnetic moment will be associated with the circulating electron (current multipliedby area). Let a uniform magnetic field be applied perpendicular to the electrons orbit. Let themagnitude of the magnetic field be increased from zero to some finite value. Then, it is an easyexercise in electrodynamics to show that an electromotive force will act on the electron in sucha manner that will try to oppose the increase in the external magnetic field (i.e., Lenz’s law).The induced opposing current leads to an induced magnetic moment in the opposite directionto that of the external magnetic field, and the system shows a diamagnetic behavior (inducedmagnetization in the opposite direction to the applied magnetic field).However, when a collection of such classical model-atoms is considered the diamagnetic effectvanishes. The net peripheral current from internal current loops just cancels with the oppositecurrent from the skipping orbits (refer, for example, to [12]). This observation also agrees withthe Bohr-van Leeuwen theorem of no magnetism in a classical setting. Thus in a classical settingit is not possible to explain the diamagnetic effect.However, in 1930, Landau surprised the scientific community by showing that free electronsshow diamagnetism which arises from a quantum mechanical energy spectrum of electrons in amagnetic field. As described in many text books[20] the solution of the Schroedinger equationfor a free electron in a magnetic field is similar to that of the solution of the harmonic oscillatorproblem. There exits equally spaced energy levels - known as Landau levels. Each Landaulevel has macroscopic degeneracy. Statistical mechanical calculation using these Landau levelsshows that there is non-zero diamagnetic susceptibility associated with free electrons which isalso temperature independent as Pauli paramagnetism is. And as is well known Landau levelphysics plays a crucial role in de Haas - van Alphen effect and related oscillatory phenomena,and in quantum Hall effects.Our historical survey ends here. We will now discuss a current topics in magnetism that is the12opic of spin liquids. There are other important developments in the field of magnetism, someof them are discussed in [1].
20 Spin Liquids
We have studied the traditional magnetic orders (such as Ferromagnetism, paramagnetism etc)in which one can visualize and define magnetic order (like in ferromagnetism, all the spins arealigned in a particular direction and in antiferromagnetism one has two sub-lattices with op-positely directed spins). These magnetic orders can be incorporated into the general schemeof Landau’s order parameter theory of second order phase transitions[25]. In Landau’s theory,order parameter is zero at
T > T c (higher symmetry phase) and is non-zero at T ≤ T c . (lessersymmetry phase). This is the standard paradigm. However, there are magnetic transition whichdo not fall under this general scheme. In fact, in these transitions an order parameter cannotbe defined in the Landau sense. These systems with elusive magnetic order are called QuantumSpin Liquids(QSL). We will consider two examples where spin liquids are thought to exist. Oneis the enigmatic case of underdoped high transition temperature cuprate superconductors. Forexample consider the case of the parent compound La CuO . This is a Mott insulator [Mottinsulator are different from band insulator. They have partially filled bands but are insulatorsdue to strong Coulomb repulsion]. It turns out that Cu is in the oxidation state of Cu ++ i.e, ithas one un-paired electron in the hybrid Cu d x − y - O P σ orbitals. This forms a narrow band,and two electrons on the same Cu atom leads to large Coulomb repulsion (the Hubbard U )and it is not energetically favourable to have double occupancy at a Cu site. Thus, the systemforms a 2 D spin- lattice with AFM arrangement. Historically, it was point out by Andersonthat in 2 D , strong quantum fluctuations will disrupt the long range 2 D AFM arrangement andthe state that will be realized is called RVB( Resonating Valence Bound state). In RVB, spinon nearest neighbours from singlet pairs and these pairs ”resonate” between various configura-tions. Anderson’s original proposal was along Pauling’s theory of aromatic molecules in whicha double bond resonates over single bonds, for example in Benzene. This quantum mechanicalresonance lowers the energy of the system. It turns out that Anderson’s original proposal ofRVB in La CuO was not correct, the state realised was in fact AFM(as has been verified byneutron scattering and other probes). But when one dopes the system ( La − x Sr x CuO ), thislong range order is melted away and leads to some sort of dynamical order with short rangeAFM correlations. This state in underdoped cuprates is called the pseudogap state. Exact na-ture of the electronic order in this phase is not known as there is no order parameter in Landausense. However, Anderson’s original insight of resonating singlet pairs may be true in this phaseput it is open to questions. It may be that pairs are making and breaking and are resonatingover a vast number of configurations. How to mathematically formulate it is a difficult currentopen problem in magnetism and in condensed matter physics.Our second example of Quantum Spin Liquid (QSL) is Herbertsmithite with chemical for-mula ZnCu ( OH ) Cl . The magnetically active element is again Cu with spin S = . Inthis case also ordering is frustrated. If forms what is calles Kagome lattice, with edge sharingtriangles [26]. The well defined magnetic order is not stabilized in this case also. And the spinsform single pairs and resonate between various quantum mechanical configurations ( fundamen-tal feature of a spin liquid ). In contrast to the pseudogap state in cuprates, we have concreteevidence of a quantum spin liquid in Herbertsmithite. There are two experiments that pointsto the existence of QSL in this system. One is the neutron scattering and the other is the NMRKnight Shift experiments. Here we first discuss the neutron scattering experiment. If we havea nice magnetic order (like FM order) then neutron scattering leads to sharp Bragg peaks. And13f there are short range magnetic correlations then we have diffuse Bragg peaks. But if we havea diffused neutron scattering intensity over a considerable part of the Brillouin zone, even atvery low temperatures one can then say that no magnetic order is selected by the system and itremains into a spin liquid state. This is what observed in Herbertsmithite. Neutron scatteringintensity remains diffuse even at very low temperatures—clear signature of a QSL.Another clear signature of the presence of QSL in this system comes from NMR relaxationrate. To understand that we need to first understand the nature of excitations in a QSL. Asis well known excitations in a ferromagnetic are magnons. These are the quanta of deforma-tion in magnetisation much like phonons which are quanta of lattice vibrations. It turns outthat magnons are not gapped i.e. these can be excited with vanishingly small energy but thecorresponding magnons will be of very long wavelength. In contrast, excitations in a QSL aregapped. These are called spinons.For example in 1 D AFM chain, if we flip one spin, then it generates two ”unhappy” bondsand these excitations can propagate along the chain. The excitation of a spinon requires a finiteamount of energy thus their excitation spectrum is gapped. When NMR relaxation rate of Cu nuclear spin is measured it shows very weak relaxation at very low temperatures because at verylow temperatures (such that k B T < ∆, where ∆ is spin gap) spinon excitations are not thereand alignment of nuclear spins is not degraded. Whereas at high temperatures there is a plentyof spinon excitations in QSL and nuclear relaxation becomes faster, as nuclear spin alignmentis relaxed by spinon excitations. Thus we observe that QSLs is not an academic topic but theydo exist in nature.
Acknowledgement
Authors dedicate this article to the memory of Prof. N Kumar from whom they learned thistopic. AMJ thanks DST for J.C. Bose national fellowship. NS would like to thank his friendNagaraj (PRL Library) to provide required literature.
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