From Heisenberg, Slater, and Stoner to Van Vleck

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interaction in localized models. We start by discussing issues related to the origin of exchange interaction in Stoner model. Then we discuss the nature of exchange interaction in the Heisenberg model and an important work- ing rule "the Slater curve" for the sign of this interaction. After highlighting its problems we introduce the contributions of Vonsovsky and Zener which introduce the idea of indi- Figure 1: Through this article we pay homage to John rect s-d exchange interactions. Then Paul- Hasbrouck Van Vleck (March 13, 1899 – Oc- ing’s valence bond theory for the iron group tober 27, 1980) who set the foundation of the- metals is presented. Next comes the famous ories of electron correlation with his "middle- road" theory. debate between the itinerant picture (Stoner model) and the localized picture (Heisenberg model). Pros and cons of both approaches are elucidation of exchange interactions in d elec- discussed. The debate was settled in the fa- tron metals. Next, the pioneering contribu- vor of the itinerant model in the 1960s, when tions (but lesser known) of Dorfman are dis- d-band Fermi surface was observed in iron cussed. Then, in chronological order, the key group transition metals. However, the issue contributions of Pauli, Heisenberg, and Lan- of correlation effects in the itinerant model dau are presented. Finally, the advent of the remained open. The debate still appears in ﬁeld of itinerant electron magnetism starting its varied avatars in the current literature on with the investigations of Bloch and Wigner, unconventional superconducting strongly cor- and more successful formulation by Slater and related materials. Next, we brieﬂy discuss Stoner is presented. The physical basis of the the well settled issues of exchange interactions Slater-Stoner theory is discussed and its prob- in insulator compounds (direct exchange; su- lems are summarized. perexchange; and double exchange). Then we

In the second part an overview of the de- review the "middle-road" theory of van Vleck bate between itinerant electron view (Stoner) and Hurwitz (the very ﬁrst theory which and localized electron view (Heisenberg) is takes into account the electron correlation ef- presented. Connected with this debate are the fects in the itinerant model). We then intro- issues of exchange interactions. These can be duce Friedel-Alexander-Anderson-Moriya the- divided into two categories: (1) exchange in- ory of moment formation in pure iron group teraction in itinerant models, and (2) exchange transition metals which is a kind of gener-

2 An overview of magnetism alization of the famous Anderson impurity and highly successful branch of physics called problem and further advances the "middle- statistical mechanics which bridged the gap road" ideas of Hurwitz and van Vleck. Finally between the microscopic dynamical laws that the discussion of the very important issue of govern the motion of atoms and molecules strong electron correlation in the itinerant pic- and the macroscopic laws of thermodynamics. ture is presented. One of the ﬁrst successful application of statistical mechanics is the Langevin theory of i PART A paramagnetism (1905) [refer paper I ]. How- ever, there is one subtlety involved. In 1911, Niels Bohr in his PhD thesis applied the method of statistical mechanics to understand I. Failure of the classical magnetism from atomic point of view. He concluded that within the setting of classical picture: the Bohr-van Leeuwen statistical mechanics it is not possible to ex- theorem plain any form of magnetism of matter! His

The 19th century saw two major advance- method yielded zero magnetization. Thus ments in fundamental physics. One is there is an apparent contradiction between the "wedding" of electricity and magnetism Bohr’s approach and Langevin’s approach, as through investigations of Oersted, Faraday, both came in the pre-quantum era. Maxwell and others. The other major devel- The result of zero magnetism in classical sta- opment occurred in the understanding of ther- tistical mechanics was re-discovered and elab- modynamical phenomena from molecular– orated independently in 1919 by Miss J. H. kinetic point of view. Thermodynamical con- Van Leeuwen. The result is now famous as cepts like temperature, pressure, and thermo- Bohr-van Leeuwen theorem. It can be ex- dynamical laws were understood from the mo- plained in the following way[1]. Consider the tion and interactions of atoms/molecule–the case of a material in which all the degrees-of- building blocks of matter. Maxwell for the freedom are in mutual thermodynamical equi- ﬁrst time used probabilistic or statistical argu- librium including electrons. In statistical me- ments to derive the physical properties like chanics thermodynamical quantities, includ- pressure, viscosity etc of gases starting from ing magnetization, are computed from free en- the molecular–Kinetic point of view. This sta- ergy which can be expressed through partition tistical method was greatly extended by Lud- function which is further expressed as a phase wig Boltzmann (and independently by Gibbs), iHistory of magnetism I: from Greeks to Paul Langevin and they transformed it into a well respected and Pierre Weiss, Navinder Singh, hereafter referred as I.

3 An overview of magnetism integral of the Boltzmann factor (exp(− H )) treated classically by Langevin. He attributed kBT involving the Hamiltonian (H). In an external a permanent magnetic moment to each atom p2 magnetic ﬁeld, the Hamiltonian ( 2m + V(r)) without worrying about its origin. This state 1 e 2 ii must be replaced by ( 2m (p − c A) + V(r)) of affairs is best explained by J. H. Van Vleck where A is the vector potential and p is the "When Langevin assumed that the magnetic canonical momentum. It turns out that the moment of the atom or molecule had a ﬁxed phase integral (the partition function, Z) be- value µ, he was quantizing the system without comes independent of vector potential when realizing it." the integration over momentum in the phase Assignment of a permanent magnetic mo- ′ e space integration is changed to p = p − c A, ment to an atom is actually an introduction i.e., when momentum variable is changed. So of a quantum mechanical ingredient in to the partition function becomes independent the problem which Langevin did not recog- of vector potential, and resulting free energy nize explicitly. Also, Langevin did not take

(F = −kBTlnZ) also becomes independent of into account the space quantization (spin can vector potential and magnetic field. It gives only have discrete quantized values along the zero magnetization when differentiated (M = magnetization direction). In Langevin’s the- ∂F − ∂H ). In conclusion, this theorem raises an ory magnetic moment can point in any di- apparent paradox: how does magnetic effects rection and the phase integral was computed arise in the Langevin theory which also uses for all possible orientations. Thus one can classical statistical mechanics? Quantum me- regard the Langevin theory as semi-classical, chanics was not known when Langevin ad- and the apparent paradox with fully classical vanced his theory (in 1905). Bohr-van Leeuwen theorem is immediately re- moved. As a side remark it is to be noted that II.Reconciling the Langevin when a fixed magnetic moment is assigned to theory with the Bohr-van Leeuwen an atom, one is departing from the principles of classical electrodynamics that an orbiting theorem (i.e., accelerating) electron inside of an atom It turns out that the Langevin theory is not must radiate energy. Permanent magnetic fully classical. It is actually semi-classical or moment implies permanent circling electrons semi-quantum in nature. Langevin did not inside the atom. The Langevin assumption consider all the degrees-of-freedom classically, of fixed magnetic moment directly leads to as considered in the Bohr-van Leeuwen theo- Bohr’s principle of stationery states on which rem. The internal motion of electrons within iiJ. H. Van Vleck in Nobel lectures in physics 1971 - 1980, the atom which gives magnetism was not Landquist (ed), World Scientific, 1992.

4 An overview of magnetism he built the quantum theory of the hydrogen but with different numerical coefficient C in 2 atom. However, Langevin did not explicitly χ = C Nµ . He found the value 1.54 instead of kB T state the stationarity of the circling electrons, 1/3 of the Langevin theory. Pauli used integer and it was Bohr who fully recognized it, and quantum numbers but analysis of the band stated it as an essential principle of the quan- spectrum showed the need for half-integer val- tum theory[2]. ues. Linus Pauling revised Pauli’s calculation by using half-integer instead of integer values, III. Pre-quantum mechanical era and it resulted in another value of the coef- and the problems of the old ficient C[3]. The status of the field was far quantum theory from satisfactory by 1925. There was another big problem. The calculations of susceptibility The success and failure of the old quantum within the regime of old quantum theory ap- theory of Bohr and others are well known[2]. peared to violate the celebrated Bohr’s corre- And how the new quantum mechanics devel- spondence principle, which states that in the oped by Heisenberg, Born, Schroedinger, and asymptotic limit of high quantum numbers Dirac replaced the patch-work of old quantum or high temperatures, the quantum expression theory by a coherent picture of new quantum should go over to the classical one ( as in black mechanics, in early 1920s, is also well known. body radiation problem for h¯ ω << 1). In the kBT In 1922, Stern-Gerlach experiment showed calculations of Pauli and Pauling there was that magnetic moment of atoms can orient it- no asymptotic connection with the Langevin self only in specific directions is space with theory. Then there was issues related to the respect to external magnetic field. This quan- weak and strong spatial quantization in the tum mechanical phenomenon of spatial quan- old quantum theory[3]. Also the origin of tization was certainly missing in the Langevin the Weiss molecular field remained a complete treatment of paramagnetism. In the Langevin mystery. In conclusion, the old quantum theory of theory atomic moments can take any orienta- magnetism was a dismal failure. tion in space. The required discretization of the spatial orientations was introduced, for the IV. Quantum mechanical and first time, by Pauliiii who found that suscepti- post-quantum mechanical era, bility expression with respect to the tempera- and the development of the ture variation is the same as that of Langevin quantum theory of magnetism iiiActually Pauli calculated electrical susceptibility. It turns out that same calculation goes through for magnetic susceptibility except one has to replace electric moment by The modern quantum mechanics was in magnetic moment[3]. place by 1926. The equivalence of the ma-

5 An overview of magnetism trix formulation of Heisenberg (1925) and pression for susceptibility reduces to the semi- wave-mechanical formulation of Schroedinger classical Langevin result when all energy level (1926) by shown by Schoredinger in 1926. In spacings are much less than the thermal en- the same period van Vleck attacked the prob- ergy (|h¯ ωij| << kBT). In the opposite regime lem of magnetism with "new" quantum me- (when for all |h¯ ωij| >> kBT ) χ showed tem- chanics. perature independent behavior. In the inter-

mediate regime (|h¯ ωij| ∼ kBT) susceptibility showed a complex behavior (the case of ni- V. Enter van Vleck tric oxide). Thus van Vleck re-derived the One of the pioneer of the quantum theory of Langevin theory by properly taking into ac- magnetism is van Vleck who showed how new count the space quantization. quantum mechanics could rectify the problems of the old quantum mechanics, and re- stored the factor of 1/3 of the Langevin’s semi-classical theory. In doing so he took space quantization of magnetic moment into account (instead of the integral in the partition function, proper summation was performed). In one of the pioneer investigation, Figure 2: Van Vleck (1899-1980). [Photo: Wikipedia van Vleck undertook a detailed quantum me- Commons] chanical study of the magnetic behavior of gas nitric oxide (NO). He showed quantitative de- viations from semi-classical Langevin theory Another major contribution of van Vleck is in this case, and his results agreed very well related to magnetism in solid-state. When a with experiments[4]. The quantum mechani- free atom (suppose a free iron atom) becomes cal method was applied to other gases, and a part in a large crystalline lattice (like iron he could quantitatively account for different oxide), its energy levels change. The change susceptibility behavior of gases like O2, NO2, in the electronic structure of an atom is due iv and NO. The differences in magnetic behav- to two factors (1) outer electrons participate ior arise from the comparison of energy level in the chemical bond formation, thus their en- spacings (¯hωif ) with the thermal energy kBT. ergy levels change, and (2) in a crystalline lat- He showed that the quantum mechanical ex- tice, the remaining unpaired electrons in the

ivFor a detailed account refer to his beautifully written outer shells of an atom are not in a free en- book[3]. vironment, rather they are acted upon by an

6 An overview of magnetism electrostatic field due to electrons on neighbor- mainly the spin part to contribute to mag- ing atoms. This field is called the crystalline netism of salts of iron (refer to figure 2). field. Magnetism of iron group metals is a differ- Van Vleck and his collaborators introduced ent story (as compared to salts). In this case it crystalline field theory (also known as the lig- turns out that charge carriers are also responsi- and field theory in chemical physics depart- ble for magnetism. The magnetism due to itin- ments) to understand magnetic behavior in erant electrons was developed by Bloch, Slater, solid-state. With crystalline field ideas they and Stoner (refer to part B). The other ex- could understand different magnetic behav- treme of localized electrons was investigated iors of rare earth salts and iron group salts. It by Heisenberg. Van Vleck advanced ideas turns out that in rare earth salts 4 f electrons that can be dubbed as "middle of the way" are sequestered in the interior of the atom, approach (refer to part B). For his pioneer- and do not experience the crystalline field very ing contributions van Vleck was awarded with strongly (figure 2). The energy level splitting the Nobel prize in physics in 1977 along with due to crystalline electric field is small as com- Phil Anderson and Nevill Mott. His articles pared to thermal energy (kBT), and it remains are beautifully written and extremely readable small even at room temperatures. Due to this and should form an essential element in a the magnetic moment of the atoms behave as if course (graduate or undergraduate) on mag- the atom is free and shows the Langevin-Curie netism. One can say that van Vleck is the fa- 1 behavior χ ∼ T [4, 5, 6]. ther of the modern theory of magnetism, and his name will be forever remembered. "Expossed" outer d−orbitals "Sequestered" f−orbitals

VI. Enter Dorfman

When quantum mechanical study of mag- Iron group: strongly affected by crystal field Rare earth group: weakly affected by crystal field netism of real gases was started by van Vleck Figure 3: A cartoon showing why crystal ﬁeld effects in mid 1920s, the quantum mechanical study differently an iron group ion and a rare earth of magnetism in metals also started in the ion. other continent transatlantic. The discovery of the paramagnetic proper- In contrast to this case, in the iron group ties of conduction electrons in metals is gener- salts crystalline ﬁeld is so strong that it ally attached to Wolfgang Pauli. Pauli’s paper quenches a large part of the orbital magnetic came in 1927. Even before that, in 1923, Rus- moment, even at room temperatures, leaving sian physicist Yakov Grigor’evich Dorfman

7 An overview of magnetism

(figure 3) put forward the idea that conduction electrons in metals posses paramagnetic properties[7]. His proposal was based on a subtle observation: when one compares susceptibility of a diamagnetic metal with its ion, the susceptibility of the ion is always greater than its corresponding metal. It implies that there is some positive susceptibility in the case Figure 4: Yakov Grigor’evich Dorfman (1898-1974) of the diamagnetic metal that partly cancels standing on extreme left. The person sitting out the larger negative diamagnetic suscepti- in the center is A .F. Ioffe. [Photo: Wikipedia bility. And this cancellation is prohibited in Commons] the case of metal’s ion (due to ionic bonding). It was Dorfman’s intuition that some positive magnetic origin, Dorfman passed beta-rays (a susceptibility is to be attributed to conduction free electron beam) in two samples of nickel electrons in the metal i.e., some paramagnetic foils, one magnetized and the other unmagne- susceptibility has to be there. v Dorfman’s tized. From deflection measurements he de- conclusion is based on his careful examination termined that Weiss field is of non-magnetic of the experimental data. After the discovery origin[8]. of the electron spin, Pauli gave the theory of In conclusion, Dorfman was an early con- paramagnetism in metals due to free electron tributor to the quantum theory of magnetism. spin. However, Dorfman was the first to point But he is not as well known as he should have out paramagnetism in metals[7]. been. One of the other important contributions of Dorfman is his experimental determination of VII. Enter Pauli the nature of Weiss molecular field responsible for ferromagnetism in the Weiss theory. It was Pauli’s contribution to magnetism is well believed that the Weiss field is of magnetic ori- known. He formulated paramagnetic behav- gin due to which spins align to give a net spon- ior of conduction electrons in metals in 1927 taneous magnetization. To determine whether and showed that paramagnetic susceptibility the Weiss field is of magnetic origin or of non- is temperature independent (in the leading or- v It is important to note that the notion of the electron der). The derivation is discussed in almost spin came in 1925 with a proposal by Uhlenbeck and all books devoted to magnetism and solid Goudsmit and paramagnetism due to electron spin was discovered in 1927 by Pauli as mentioned before. But Dorf- state physics[9]. Pauli’s derivation of the para- man’s proposal came in 1923! magnetic susceptibility can be described as

8 An overview of magnetism

which is independent of temperature as the more accurate calculation shows.

VIII. Enter Heisenberg

Figure 5: Wolfgang Pauli (1900 - 1958). [Photo: Wikipedia Commons]

one of the early application of Fermi-Dirac statistics of electrons in metals. In the stan- Figure 6: Werner Heisenberg (1901 - 1976). [Photo: dard derivation[9] one calculates the thermo- Wikipedia Commons] dynamical potential Ω(H) of free electron gas in a magnetic field H. Magnetization is ob- As mentioned before Dorfman in 1927 tained by the standard algorithm of statisti- pointed out that the Weiss molecular field re- ∂Ω cal mechanics: M = − ∂H , and susceptibility quired in the theory of ferromagnetism is of ∂M χ = ∂H . For illustration purpose there is non-magnetic origin. The puzzle of the Weiss a simpler argument[10] which goes like this. molecular field was resolved by Heisenberg in For metals at ordinary temperatures one has 1928. The central idea is that it is the quan- kBT << EF where T is the temperature and tum mechanical exchange interaction which is re-

EF is the Fermi energy. Thus electrons only sponsible for the ferromagnetic alignment of in a tiny diffusion zone around the Fermi sur- spins. Quantum mechanical exchange interac- face participate in thermodynamical, electrical, tion has no classical analogue, and it results and magnetic properties (other electrons are due to the overlapping of orbital wave func- paired thus dead). If N is the total number tions of two nearby atoms. Symmetry of the of electrons, then fraction of electrons in the hybrid orbital is dictated by the nature of the diffusion zone is N T where T is the Fermi spin alignment which obeys the Pauli exclu- TF F temperature (kBTF = EF). Each electron in sion principle. Thus there is an apparent spin- the diffusion zone has magnetic susceptibility spin coupling due to orbital symmetry and un- 2 roughly given by χ ∼ µ where µ is its mag- der specific circumstances the ferromagnetic kBT netic moment. Thus total magnetic susceptibil- spin alignment significantly lowers the bond- 2 2 ity of metal is given by: N T × µ = N µ ing energy thereby leading to a stable configu- TF kBT kBTF

9 An overview of magnetism ration.vi is given by The Heisenberg model based on exchange interactions is related to the resonance-energy- Vij = −2JijSi.Sj. lowering model for chemical bonding by viii Jij is called the exchange integral . For fer- Heitler and London[11]. In the Heitler- romagnetism the sign of Jij has to be positive, London theory of the chemical bond in hydro- and for anti-ferromagnetism it has to be neg- gen molecule, it is the exchange of electrons ative. The question on what parameters the on two hydrogen atoms that leads to the res- sign of J depends is complicated and vexed onant lowering of the energy of the molecule. one (we will discuss these issues in part B). Electrons stay in an antiparallel spin conﬁgu- The above exchange interaction is now ration thereby enhancing the overlap of orbital known as the Heisenberg exchange interaction wave functions in the intermediate region of or the direct exchange interaction. There is a two hydrogen atoms. This leads to bond for- variety of exchange interactions (both in met- mation. This idea of resonant lowering of en- als and insulators) that will be discussed in ergy via exchange of electrons is greatly used part B. by Linus Pauling in his general theory of the in mth orbital at site i and an electron in nth orbital at site chemical bond[11]. The Heisenberg model is j is given by built on similar ideas and goes like this[5, 12]. Vi,m;j,n = −2Ji,m;j,nSi,m.Sj,n. Let Si be the total spin at an atomic site i. If ex- Total interaction is obtained by summing over all m and n change interaction between nearest neighbors is the only one important, then the interac- Vi,j = −2 ∑ Ji,m;j,nSi,m.Sj,n. m,n vii tion energy (under certain approximations ) The main assumption is that the exchange integral be- m i n j as- viIt is very important to note that energy associated tween th orbital at site and th orbital at site is sumed to be independent of m and n. It is like assuming the with spin-spin coupling of two electrons via exchange is same exchange integral between two s-orbitals or two d-orbitals very large as compared to the magnetic dipole-dipole in- or between s and d orbitals on two different sites i and j. That teraction energy which is given by is ui.uj (ui.rij)(uj.rij) V = − 3 . ij 3 5 Ji,m;j,n ≃ Ji,j ≃ J. rij rij This very small magnetic energy cannot lead to ferromag- Validity of this assumption depends crucially on the na- netic alignment. In other systems, like ferro-electrics it is ture of the system under consideration. Of course, over- an important energy. lap of two S-orbitals is different from that of two d-orbitals. vii Here Si is the total spin at an atomic site ”i”, i.e., it But let us accept this assumption. Under this assumption includes a vector sum over all the spins of unpaired elec- Vi,j = −2JijSi.Sj where Si = ∑n Si,n etc. Hence one obtains trons. In our notation i and j label two nearest sites. Let the Heisenberg model as given in the main text. viii m and n denote orbital numbers on a given site i or j (in J = dτ dτ φ (1)φ (2)H φ (1)φ (2). cases where there are many unpaired spins in different or- ij Z 1 Z 2 i j c j i bitals). Exchange interaction energy between an electron

10 An overview of magnetism

To compare predictions of the model with many drastic assumptions which were later experiment, one needs its solution. The very improved upon. Literature on the Heisenberg first solution provided by Heisenberg himself model and its various approximate solutions is based on some very restrictive assumptions. is very vast. Some references are collected So tight agreement with experiments may not here[5, 6, 12, 31]. be expected, and it leads to some qualitative results. Heisenberg used complicated group IX. Enter Landau theoretical methods and a Gaussian approximation of the distribution of energy levels Metals which are not ferromagnetic show two ix to find an approximate solution. From his weak forms of magnetism, namely, paramag- solution Heisenberg observed that ferromagnetism and diamagnetism. Paramagnetism we netism is possible only if the number of near- have discussed, diamagnetism due to free con- est neighbors are greater than or equal to eight duction electrons is a subtle phenomenon and (z = 8). This conclusion is certainly violated as was a surprise to the scientific community[1] many alloys show ferromagnetism with z = 6. when Lev landau discovered it in 1930. To The second result which is much more impor- appreciate it consider the following example. tant is that of magnitude of λ it turns out that Consider the classical model of an atom in λ of the Weiss molecular field takes the form which a negatively charged electron circulates J around a positive nucleus. A magnetic mo- λ = z 2 . 2NµB ment will be associated with the circulating

The large value of λ required for ferromagnetism electron (current multiplied by area). Let a is not a problem anymore, as the exchange integral uniform magnetic field be applied perpendic- J can be large, thus resolving the problem of Weiss ular to the electrons orbit. Let the magnitude theory. This is the biggest success of the Heisenberg of the magnetic field be increased from zero model. to some finite value. Then, it is an easy exer- In conclusion, Heisenberg’s model resolved cise in electrodynamics to show that an elec- the puzzle of the Weiss molecular field using tromotive force will act on the electron in such the concept of exchange interaction. This con- a manner that will try to oppose the increase cept turns out to be the key to the modern in the external magnetic field (i.e., Lenz’s law). understanding of magnetism in more complex The induced opposing current leads to an in- systems. Heisenberg’s solution was based on duced magnetic moment in the opposite direction to that of the external magnetic field, and ixAn alternative and comparatively simpler method was provided by Dirac using the vector model with simi- the system shows a diamagnetic behavior (in- lar conclusions[1, 2]. duced magnetization in the opposite direction

11 An overview of magnetism to the applied magnetic ﬁeld). X. Enter Bloch and Wigner

Bloch in 1929[14] advanced the idea that mag- However, when a collection of such classical netism in iron group metals might be origi- model-atoms is considered the diamagnetic ef- nating from itinerant electrons (in contrast to fect vanishes. The net peripheral current from Heisenberg’s localized electron model)x. The internal current loops just cancels with the op- basic principle behind Bloch’s theory is as fol- posite current from the skipping orbits (refer, lows. As is well known conduction electrons for example, to [1]). This observation also form a sphere in momentum space known as agrees with the Bohr-van Leeuwen theorem of the Fermi sphere. Each momentum state is no magnetism in a classical setting. Thus in a doubly occupied with one electron of up spin classical setting it is not possible to explain the and the other with down spin. This configura- diamagnetic effect. tion minimizes the total kinetic energy (K.E.) of the system. Now, if there is an exchange interaction between the conduction electrons However, in 1930, Landau surprised the sci- then they tend to align their spins. Pauli’s ex- entific community by showing that free elec- clusion principle then prohibits them to be in trons show diamagnetism which arises from the same momentum state, and electrons must a quantum mechanical energy spectrum of migrate to higher momentum states. This electrons in a magnetic field. As described migration of electrons to higher momentum in many text books[9] the solution of the states leads to increased K.E. of the system. Schroedinger equation for a free electron in Thus there is a competition between exchange a magnetic field is similar to that of the solu- interaction energy which tends to lower the en- tion of the harmonic oscillator problem. There ergy of the system by aligning spins of elec- exits equally spaced energy levels - known as trons and K.E. which tend to pair them up Landau levels. Each Landau level has macro- with two electrons in each momentum state. scopic degeneracy. Statistical mechanical cal- Under "suitable conditions" exchange interac- culation using these Landau levels shows that tion is the winner and system becomes unsta- there is non-zero diamagnetic susceptibility as- ble to ferromagnetism. The "suitable condi- sociated with free electrons which is also temperature independent as Pauli paramagnetism xThe idea that itinerant electrons might be responsible is. And as is well known Landau level physics for ferromagnetism was already there. Frenkel in 1928 discussed the possibility of ferromagnetism due to itin- plays a crucial role in de Haas - van Alphen erant electrons via Hund’s coupling. Experimental in- effect and related oscillatory phenomena, and vestigations were made by Dorfman, Kikoin, and their in quantum Hall effects. colleagues[7].

12 An overview of magnetism tions" according to Bloch are sufficiently low glects Coulomb electrostatic interaction be- electron density or sufficiently large electron tween electrons with anti-parallel spins. This mass (this will be made more precise in the fol- "correlation energy" is not taken into account lowing paragraph). But Bloch’s argument has in Bloch’s calculation. Wigner estimated these problems as was first pointed out by Wigner. correlation effects and showed that the possi- Before we present Wigner’s argument, let us bility of ferromagnetism in Bloch’s picture is discuss exchange in Bloch’s picture in little nil. more detail. Exchange energy basically originates from XI. Enter Slater and Stoner and electrostatic Coulomb repulsion between two the arrival of the itinerant electrons. By having parallel spins two elec- electron magnetism tron are spatially pulled apart due to Pauli’s exclusion principle, and this lowers the elec- Slater in 1936[16] discussed the possibility of trostatic energy of the system. Bloch showed ferromagnetism due to itinerant electrons. He that when argued that the exchange interaction respon- e2m∗ 4.5 > n1/3 sible for the spin alignment of itinerant elec- h2 trons is not the itinerant exchange, as argued the system exhibits ferromagnetism. That by Bloch, but it is of the intra-atomic origin[17]. is when, either, electron density (n) is very It is an extension of the Hund rule of max- low, or, when effective mass of electron is very imum total spin for a less than half filled large. shellxi. Consider the case of itinerant electrons of narrow d-band in iron group metals. An itinerant electron flits from one atom to another thereby changing atom’s polarity (that is atomic ionization states change and it is also known as the polar model in contrast to Heisenberg’s non-polar model where the polarity of an atom remains the same due Figure 7: Eugene Wigner (1902 - 1995). [Photo: to localized electrons). The minimum energy Wikipedia Commons] configuration is that when this itinerant electron has the same spin polarization as that The above condition is too restrictive. of electrons already there in the correspond-

Wigner[15] in 1938 showed that Bloch’s ar- xiSimilar ideas were advanced by Frenkel as mentioned gument is not realistic one in that it ne- before in a previous footnote.

13 An overview of magnetism ing shell of the atom with degenerate orbitals. In contrast to all these complications Stoner When this electron flits from that atom to a in 1936 adopted a completely phenomenolog- nearby one it takes with it its prejudice of be- ical approach[18]. He basically superposed ing in that spin configuration. For example, Weiss molecular field (i.e., the exchange field) if electrons in less than half full degenerate on itinerant electrons without worrying much d-orbitals of an atom have spin polarization about the origin of the exchange interaction along the positive z-direction (say) then this in metals. The stoner theory is computation- flitting electron will have its spin polarized ally successful and its results can be compared along the same direction. In the nearby atom with experiments. The basic mechanism of fer- the very same mechanism works and it leads romagnetism in the Stoner theory is the same to spin alignment. In total, this intra-atomic as that of Bloch’s—-the competition between exchange leads to ferromagnetic state.xii In the exchange energy and the K.E. At zero temper- language used here we are using both "band" ature this leads to the following condition for concept and "orbital" concept at the same time. ferromagnetism: It appears incoherent, but it turns out that va- Iρ(E ) > 1. lence electrons of the iron group elements re- F tain their atomic character to some extent[6]. Here I is the average exchange interaction With the Slater model one can appreciate energy and ρ(EF) is the electronic density of the fact that alkali and alkaline earth metals states (EDOS) at the Fermi level. So, according are not ferromagnetic as in these metals intra- to the Stoner condition, metals having large atomic exchange is not possible in the conduc- value of EDOS at the Fermi level or having tion s-bands as these are non-degenerate. On large value of exchange interaction are tend to the other hand d-band metals can be ferromag- be ferromagnetic. For example d-band metals netic as intra-atomic exchange can provide the have a chance of being ferromagnetic as EDOS required mechanism for spin alignment due to for d-band is large, whereas EDOS for s-band d-band degeneracy. and p-band metals is smaller and they are not But this criterion based on degenerate ver- ferromagnetic. Not all d-bands metals are fer- sus non-degenerate bands leaves open the romagnetic, so Stoner model is definitely not question of no ferromagnetism in p-band met- the complete answer. But it captures the phe- xiii als which are also degenerate. nomenon in a qualitative way.

xiiVan Vleck also developed a model on similar lines Stoner also performed extensive calcula- called the "minimum polarity model" which is discussed tions of temperature dependence of ferromag- in Part B and references to the related literature are given netism. The ﬁnite temperature model can be there. xiiiThe complications due to various exchange interactions are discussed in detail in part B.

14 An overview of magnetism

much greater than Fermi energy then width of the band is negligible as compared to exchange energy scale, and then the results of a localized model are expected to appear.

M χ−1

Figure 8: Edmund C. Stoner (1900 - 1976). [Photo: Wikipedia Commons]

1 easily described in the following way. Let ∆ T/Tc be the energy due to internal exchange ﬁeld Figure 9: Magnetization and inverse susceptibility as a which is given by IM where M is the uniform function of scaled temperature in Stoner the- magnetization (in the literature ∆ is also called ory. the band splitting). Let N be the total number of electrons given by However, the plot of 1 i.e., inverse sus- ∞ ∞ χ N = dǫρ(ǫ + ∆) f (ǫ) + dǫρ(ǫ − ∆) f (ǫ), ceptibility versus temperature shows signiﬁ- Z−∆ Z∆ cantly more curvature as compared to the well where f (ǫ) = 1 and ∆ = IM. And eβ(ǫ−µ)+1 obeyed Curie-Weiss law (which is a straight the total magnetization is given by line for inverse susceptibility versus tempera- 1 ∞ 1 ∞ ture). This is a drawback of the Stoner model. M = dǫρ(ǫ + ∆) f (ǫ) − dǫρ(ǫ − ∆) f (ǫ). Z Z 2 −∆ 2 ∆ The other major drawback of Stoner theory is

These two equations can be easily solved that when Tc is calculated using magnitude numerically in a self-consistent manner if the of the saturation magnetization it results in a

EDOS is given as a function of energy. When very high values of Tc. Sometimes even an magnetization is plotted as a function of tem- order of magnitude larger[19]. Thus, one can perature in the case of parabolic band one ob- say that Stoner model is not quantitatively suc- tains M − T graph which roughly agrees with cessful, but qualitatively it captures the phe- that obtained with the Weiss theory. Actually, nomenon of itinerant electron magnetization. in the limit I >> 1 Stoner’s M − T curve The story how Random Phase Approximation EF exactly matches with that of Weiss localized (RPA) and the Moriya-Kawabata theory im- model with S = 1/2 on each site. This agree- proves upon it is presented in next article of ment is expected. When exchange energy is the current series.

15 An overview of magnetism

XII. Physical basis of the not have full understanding! Slater-Stoner theory PART B As discussed in the previous section the Slater model can provide a basis to understand magnetism of iron group metals using the idea of d-band degeneracy. The intra-atomic ex- XIII. Heisenberg versus Stoner change can provide the required spin alignment via an extension of Hund’s mechanism In Part A, the Heitler-London approach mo- between a flitting electron of d-band and its tivated Heisenberg model and the Bloch- localized companion in other d-orbital. On Wigner-Slater approach motivated Stoner the other hand missing orbital degeneracy in model are discussed. In the Heisenberg model, the valence orbitals in alkali and alkaline earth electrons which are responsible for ferromag- metals blocks this mechanism of Slater, and netism are localized on atomic sites. The local- hence these turn out to be non-ferromagnetic. ized electron picture is true for magnetic insu- But this mechanism of Slater leaves open the lator compounds, but it is not true for metals question of why no ferromagnetism in p-band in which charge carriers are also responsible metals which are also degenerate. for magnetic effects (as in the case of 3d tran- The issues are partly resolved by using sition iron group metals). There are experi- Stoner theory. Stoner condition requires that mental proofs of it, for example, ferromagnetic for having favorable circumstances for ferro- transition metals show large electronic specific magnetism in a metal, there should be large heat, and d-electron Fermi surfaces[6]. exchange interaction energy or large value of In the opposite picture of itinerant electrons, EDOS. EDOS for s-band and p-band metals is contributions of Bloch, Wigner, Slater, and smaller as compared to that in d-band metals. Stoner are discussed in part A. In a nutshell, But as mentioned before not all d-bands met- Stoner superposed Weiss molecular field (or als are ferromagnetic. Thus both approaches exchange field) on the Sommerfeld free elec- have their own problems. Although Stoner tron model of metals. Free electrons undergo theory provides a clue and quantitative results spin polarization under the action of Weiss but a complete answer cannot be given within molecular field or exchange field which leads Slater-Stoner ideas. Many other exchange in- to ferromagnetism. The finite temperature be- teraction ideas were advanced which are dis- havior of the Stoner model can be studied us- cussed in PART B. Ferromagnetism in d-band ing standard method of statistical mechanics metals is a complicated issue, and still we do i.e., by calculating the free energy etc. How-

16 An overview of magnetism ever, the Stoner model alone is not sufficient to of exchange interaction J in the Heisenberg understand magnetic properties of iron group model. Answers to these questions help to un- metals and Heisenberg model alone is not suf- derstand and resolve the debate. ficient to understand magnetic properties of insulating systems, as will be discussed in sub- sequent sections. he issue of the origin of The above picture was not available before XIV. T early 1950s. So it was not clear whether Stoner Weiss field in the Stoner model model is more appropriate or the Heisen- berg model to discuss ferromagnetism of iron At a more fundamental level, the origin of group metals. Central to this dichotomy was the Weiss molecular field in the itinerant the question whether d electrons in iron group model was attributed to intra-atomic exchange metals are localized or itinerant (de Haas - (within an atom) by Slater. In other words, van Alphen Fermi surface studies of transition Slater’s intra-atomic exchange mechanism pro- metals came in the late 1950s). Thus at that vides a microscopic basis to the Hund rules time it was a real confusion whether Heisen- which states that if d shell in a transition metal berg model is more appropriate for ferromag- ion is less than half full then spins tend to netic transition metals as it reproduced the ex- align parallel to each other to give maximum perimental Curie-Weiss law very well, or, the total S. In this way phenomenologically in- itinerant Stoner model as it reproduced frac- troduced Weiss molecular field in the Stoner tional Bohr magneton numbers of saturation model receives its microscopic justification. magnetization. Heisenberg model failed to re- But the intra-atomic exchange mechanism has produce fractional magneton numbers while its own problems as was discussed in part A. itinerant Stoner model failed to reproduce the The other possible explanation was given by Curie-Weiss lawxiv. Before we enter into this Bloch. The exchange interaction between free very interesting debate and list pros and cons electrons is "inherently" ferromagnetic. But, it of both models, we would like to delve into to be effective, electronic density has to be low much more important and deeper questions to make exchange energy dominant over ki- related to the origin of Weiss molecular field netic energy (refer to part A). This condition in the itinerant model and the origin and sign is not valid for transition metals. On the top of it, Wigner pointed out that electronic cor- xivStoner model leads to much more curvature in relation effects completely destroy the effect the graph of inverse susceptibility versus temperature whereas it should be linear according to the Curie-Weiss of exchange interactions. Thus, Slater’s intra- law. atomic exchange is the most likely candidate.

17 An overview of magnetism

XV. The issue of the sign of J in metals in 2d and 3d transition periods of the periodic table have still larger principle quan- the Heisenberg model tum numbers, but they are not ferromagnetic. For occurrence of ferromagnetism in the This situation was made partly clear by Heisenberg model a much more relevant argument due to Slater[12]. He argued that positive sign of J H = −J ∑ Si.Sj in ferromagnetic metals should be attributed

the sign of J must be positive. The Heisen- to larger interatomic distances as compared berg model is motivated by the homopolar to atomic radii involved. And the sign of J bond formation theory of Heitler and Lon- changes from negative to positive when in- don. In the bond formation, say in hydrogen teratomic distance is varied from smaller to molecule, it is the exchange of electrons that larger value with respect to the atomic radius. leads to resonant lowering of energy. Here the The Slater Curve J exchange interaction turns out to be negative, and electrons pair up in the hybrid molecular orbital with anti-parallel spins whereas in Interatomic distance the Heisenberg model for ferromagnetism exchange interaction has to be positive and electrons must have parallel spins to show fer- Figure 10: The Slater curve. romagnetism. Thus it seems difficult to reconcile two opposite pictures: one requiring positive J for ferromagnetism (Heisenberg), So called "the Slater curve" (depicted in the and the other requiring negative J for chemi- above figure) not only explains why ferromag- cal bond formation (Heitler-London) whereas netism does not occur in the second and third both originate from exchange mechanism. row transition elements (as inter-atomic dis- In his original contribution (refer to [20] tance is too small) but also why only last el- page 192) Heisenberg argued that the sign of ements of the first row show ferromagnetism J is positive in ferromagnetic metals because (as interatomic distance is just appropriate). large principle quantum numbers are involved Thus the Slater curve provided a "rule of in this case whereas in chemical bond prob- thumb" when to expect J to be positive. How- lems principle quantum numbers involved are ever, Slater’s idea is also not free from criti- smaller. With this idea one can reconcile cism. There is no single example where it is the opposite pictures. However, Heisenberg’s theoretically proved that J in a given ferromag- guess is wrong as according to his argument netic material is positive. Realistic theoretical

18 An overview of magnetism calculations to compute J are extremely com- were to be discarded. However, these ideas plicated as wave functions deform from free form seeds of very important progress in un- atomic state to something complicated when it derstanding magnetism in f-electron systems is present in a matrix.xv Thus Slater’s guess re- and dilute magnetic alloys in which f-electrons mained unproved (i.e., without theoretical jus- can be treated localized and which further tification, although empirically it seemed pos- lead to the development of the Kondo effect sible) and further investigations were needed. and RKKY (Ruderman-Kittel-Kasuya-Yosida) Keeping roughly the chronological order, interaction. We will not delve into these we next discuss the Zener-Vonsovsky model very interesting topics. There is a vast litera- which leads to positive J through a different ture on these, and interested reader can con- mechanism. sult[]. Here we present the debate regarding the sign of J in the Heisenberg model, and XVI. Enter Vonsovsky and Zener how Vonsovsky-Zener (VZ) model leads to a positive J through an entirely different mecha- We state at the outset that Vonsovsky-Zener nism. model as originally invented for ferromagnetic The VZ mechanism states that there is a d-electron metals is not the correct mecha- Hund’s type coupling of highest multiplicity nism responsible for ferromagnetism in transi- between conduction s-electrons and localized tion metals. Vonsovsky and zener maintained, d-electrons in partly filled d-shells[6]. To imag- when they put forward the ideaxvi, that d- ine this mechanism one can consider this pic- electrons form an isolated systems with local- ture. Consider that a flitting s-electron enters ized electrons while s-electrons form running into a partly filled d shell. It stays there for waves i.e., bands. This was clearly in contra- h¯ a tiny time interval of the order of E where diction to later experimental investigations us- ion Eion is the energy required to remove that elec- ing de Haas van Alphen effect which showed tron from partly filled d shell, i.e., ionization d-electron Fermi surfaces in the early 1960s. energy. During this tiny time interval (which So d-electrons are itinerant rather than local- is of the order of femtoseconds) Hund’s mech- ized. Therefore Vonsovsky-Zener (VZ) model anism works and it tends to align its spin in xvA computation of the same direction as that of already present d J = dτ dτ φ (1)φ (2)Vφ (1)φ (2) Z 1 Z 2 i j j i electrons. Thus there is an effective ferromag- requires a thorough knowledge of wavefunctions which netic coupling between s electron spin and d- are not exactly known in an environment where atom is electron spin, and VZ model postulate that it present in condensed state. xviThe basic idea is due to S. V. Vonsovskii 1946[21], and can be written as −βSdSs where β is a pos- later on developed by Zener in 1951[22]. itive parameter of the model and Sd (Ss) is

19 An overview of magnetism the total spin of d (s) electrons. The internal It turns out that Heisenberg model alone in interactions between two s electrons, and be- its original formulation cannot address the is- tween two d electrons is assumed to be anti- sue of ferromagnetism in iron group metals. xvii ferromagnetic . Thus the total interaction It is applicable to a few systems like CrO2 energy can be written as and CrBr3[23]. And its advanced versions can

1 1 be applied to a variety of magnetic insulator E = JS2 + γS2 − βS S . 2 d 2 s d s compounds. Below we present another "failed

For a given value of Sd, energy is minimized theory" of ferromagnetism in iron group met- β als. This has historical value only, however, it when Ss = γ Sd and one can write that E = 1 2 leads to other important concepts and devel- − 2 Jeff Sd where opments. β2 J = − J. eff γ Thus there is an effective interaction be- XVII. Enter Pauling with his tween d electrons with exchange coupling Jeff . valence bond theory This can be clearly positive if β is sufficiently large ( that is coupling between s electrons Linus Pauling advanced a theory of ferromag- and d electrons is sufficiently large). Zener netism in iron group metals in 1953[?, 24]. did quantitative calculations to prove his point His theory is an application of his resonat- and got partial success[?, 21]. Later on refined ing valence bond ideas which are successful calculations showed that induced interaction in the chemical bond theory. But these ideas between d-electrons via conduction s electrons were not very successful when applied to met- is not exactly ferromagnetic but it has a com- als. Central to his theory is the concept of plex oscillatory character as a function of dis- hybridization. He explained the ferromag- tance between d electrons (i.e., RKKY interac- netism of iron group metals in the following tion). Thus VZ model was an oversimplified way. According to him, minimum energy con- model and had to be abandoned. figuration is obtained when nine wave func- xviiThe internal exchange interaction between two con- tions ((3d)5, (4s)1, (4p)3) are combined to duction s electrons is ferromagnetic in nature as first produce nine hybrid wave functions (spd hy- pointed out by Bloch. However, this to be effective required very low electron density (part A). This is not the bridization). Out of nine, six have conductive case with ferromagnetic metals and kinetic energy wins hybrid orbital character (these form extended over exchange energy and electrons pair up with anti- states from atom to atom running throughout parallel spins in a given momentum state. Also Wigner’s the lattice), and remaining three have localized correlation effects completely destroy parallel alignment in Bloch’s model. The exchange interaction between local- atomic character. Then he postulates Zener ized d electrons is assumed to be anti-ferromagnetic. type mechanism. There is Hund’s coupling

20 An overview of magnetism between electrons in localized (atomic) hybrid i. In favor of itinerant (Stoner) model orbitals and electrons in conductive hybrid or- Stoner model is conceptually elegant and com- bitals. This Hund’s coupling tends to align putationally easy to implement. It qualita- the spins of electrons in the conductive hybrid tively reproduced magnetism versus temper- orbitals and of electrons in localized orbitals, ature graph. The phenomenologically intro- thereby leading to spin polarization and ferro- duced exchange interaction by Stoner finds its magnetism. He neglected the direct or inter- justification in Slater’s intra-atomic exchange atomic exchange interaction between adjacent and Hund’s mechanism. However, this justifi- atoms (i.e., J term in the VZ model is not cation is open to criticism (refer to part A). The there). Pauling’s model have similar problems most important success of Stoner model is that as that of VZ, and the division of d-orbitals it can address fractional Bohr magneton num- as postulated by Pauling is never observed ex- bers found in saturation magnetization (refer perimentally. But, according to Anderson[25], to table 1). these valence bond ideas find their way in high-Tc cuprate superconductors. ii. Against itinerant (Stoner) model

Ferromagnetic metals obey Curie-Weiss (CW) law (linear graph between inverse susceptibility and temperature) to a reasonably good ap- XVIII. Debate and its resolution proximation. The plot of inverse susceptibility versus temperature from Stoner model shows After discussing these developments which appreciable curvature, instead of being linear. started along the approach of localized model, Thus it fails to reproduce CW law. Also the cal- let us return back to the itinerant picture and culated values of Tc for a reasonable value of to the debate between these two extreme pic- exchange parameter extracted from the spec- tures. As mentioned before, in 1950s when it troscopic data is an order of magnitude higher was not clear whether d-electrons in transition than experimental value. Thus Stoner model metals are localized or itinerant it was a real also fails to reproduce the value of the Curie problem to decide whether Heisenberg model temperature. is more appropriate to understand ferromag- Another drawback of the Stoner model is netism of some transition metals, or the itin- that electronic correlation effects are com- erant Stoner model is more appropriate. Both pletely neglected. As discussed in the next approaches have their pros and cons. We dis- section full itineracy requires momentarily cuss them one by one: arbitrary ionization state of a given atom.

21 An overview of magnetism

This costs correlation energy. For example, Table 1: Heisenberg versus Stoner free state of iron atom has the configuration [Ar]3d64s2. s-orbitals have extended wave Heisenberg Model Stoner model functions and in the condensed state they over- Heisenberg model Stoner model is for lap considerably to form wide s-bands. d- treats localized itinerant electron orbitals overlap comparably less, and form electron cases problems and it narrow d-bands. Fully itinerant d-electrons, and it is a gen- is a generaliza- according to Stoner model, can leave a vari- eralization of the tion of the Bloch ety of d-orbital configurations on a given atom: Heitler-London approach. d8, d7, d6, d5, d4 etc. In Stoner model they approach. are all equally probable. But this of course It always give Stoner theory can wrong as states d8 and d4 will have higher en- integral number of explain saturation ergies due to Coulomb repulsion between elec- Bohr magnetons intensity with frac- trons (two extremes of the ionization states). for the saturation tional Bohr magne- Therefore these configurations are very un- intensity. tons. likely to occur, but Stoner model implicitly as- It can address It cannot explain sumes that these occur with equal probability. Curie-Weiss law the Curie-Weiss Roughly speaking Coulomb repulsion acting for magnetic law adequately. between opposite spin electrons lead to corre- susceptibility. And calculated lation energy. This is completely neglected in values of Tc are too the Stoner model. And it is one of the major high drawbacks of the model (refer to section VIII). iii. In favor of localized (Heisenberg) lator compounds (next section). model iv. Against localized (Heisenberg) The most important success of the Heisenberg model model is that it can address CW law in an elegant way. Due to this fact before the resolution The biggest drawback of the Heisenberg of the debate, it was the model of choice to an- model is that it always give integer number of alyze experimental data even in metals. Later Bohr magnetons for saturation magnetization. on it became evident that this most suited for This result of the model is at variance with ob- insulators, and became the seed to further de- served facts. Also, as discussed previously, the velopments in the field of magnetism of insu- sign of the exchange integral J is a vexed issue.

22 An overview of magnetism

The Slater curve provides a rule of thumb but cases of insulating behavior. Insulating behav- no theoretical derivation of it exists. ior can be due to a different mechanism and From the above discussion it is clear that most importantly it leads to magnetic prop- Heisenberg model is not appropriate at all to erties. Consider for concreteness the exam- discuss ferromagnetism of metals. The Stoner ple of NiO. In this compound Ni is in the model seems appropriate but it has serious valence state Ni2+ with only valence 8 elec- drawbacks in that (1) it completely neglects trons partly filling the d band. The partly correlation effects and (2) it is not able to cap- filled d band according to Bloch-Wilson picture thermodynamical properties at finite tem- ture should lead to conduction. However, as peratures. Before we enter into further de- is now well known, there is strong electron velopments along itinerant (stoner) approach correlation in the narrow d bands which en- which rectify the above drawbacks, we would ergetically prohibits the flit (hop) of an elec- like to briefly brush-up the well settled issues tron from one Ni atom to another, thus giv- of magnetism in insulator compounds in the ing insulating behavior. Such insulators are next section. often magnetic as localized electrons carry a magnetic moment. These magnetic-insulator compounds are now called Mott-Anderson or XIX. Exchange interactions in Mott insulators. Magnetism of such systems insulator compounds was elaborated, among other investigators, by

Magnetism and exchange interactions in insu- Anderson[26]. lator compounds are comparably well under- Magnetic behavior in insulating systems stood topics, and are well treated in the litera- originates from a variety of exchange interac- ture (refer for example to[?, 23, 26]). Thus our tions. Below we discuss the main ones: discussion of this topic here is brief. The key 1. Direct exchange point of the issue of magnetism in insulator compounds was clarified by Mott and Ander- 2. Superexchange son in 1950s, and it can be stated in the fol- 3. Double exchange lowing way. According to Bloch-Wilson theory of energy band formation in crystalline We briefly discuss each of them. Detailed ex- materials, materials with completely filled or positions can be found in[?, 26]. completely empty bands are insulators while Direct exchange is basically the Heisenberg materials with partly filled bands are metals. exchange between two nearby spins as dis- It turns out, as first pointed out by Mott[27], cussed in section. There we noticed that it that it is only the half-truth, not true in all the is not the exchange mechanism in metals. In

23 An overview of magnetism insulator compounds also it is not the most tioned degenerate ground states). But this also common one. One can cite the example of removes the degeneracy. An excited state in

MnF2 where direct exchange is thought to be the above example corresponds to transfer of operative. The exchange integral J in this case an electron from ligand atom to its neighbor ↑↑ ↓ ↓ is negative and leads to anti-ferromagnetism metal atom (M1 − O − M2). If the valence with TN ≃ 10 K. orbital on the metal atom is less than half full, The most common exchange mechanism then the transferred electron will align paral- in insulating magnets is the superexchange lel to already existing spin on M1 (i.e., the mechanism first pointed out by Kramers and Hund’s rule). Similar mechanism happen on elaborated, among others, by Anderson. To M2. Thus it is clear that state1 with antipar- illustrate the basic principle of the mecha- allel spins on metal atoms will have lower en- nism consider the example of manganese ox- ergy, and leads to antiferromagnetism. In the ide (MnO). In the simplest structural unit oxygen p orbitals electrons stay paired due to two Mn cations are bounded together by one Pauli principle. Thus it is the Hund mech- oxygen anion in the center. Denote two met- anism operating in metal atoms and virtual als ions by M1 and M2 and spin states by transfer of electrons from the ligand atom that ↑ ↑↓ ↓ M1 − O − M2 . There are two states state1 = leads to antiferromagnetism. A perturbational ↑ ↑↓ ↓ ↓ ↑↓ ↓ M1 − O − M2 and state2 = M1 − O − M2 , calculation was done by Anderson to calcu- one with antiparallel spins on metal atoms late the magnitude of the effect, and there are and the other with parallel spins. If there is other details. Our discussion here is restricted no coupling between the spin states on two to be semi technical and brief, for more details metal cations, then these states will be degen- readers are advised to refer to[?, 26]. erate (will have the same energy) and form the The double exchange mechanism is thought ground state. An excited state will involve a to be operative in mixed valency systems, and virtual transfer of an electron from central an- was first proposed by Zener[21, 28]. Con- ion to its nearby cation, for example, transfer- sider a system in which two configurations + ++ ring one electron from O to M1. Now accord- (M , and M ) of the metal ion are possi- ing to the quantum mechanical theory of res- ble. For example in crystals of LaMnO3 both onance phenomenon first elaborated by Paul- Mn3+ and Mn4+ co-exist. Then the transi- ing, the excited state wave functions combine tions between two states M++ − O−− − M+ with the ground state wave function to pro- and M+ − O−− − M++ can occur i.e., valency duce a hybrid state with even lower energy of the ligand atom remains the same, but va- thus stabilityxviii (lower than the above men- lency of the metal atoms fluctuates. Now xviiiThis is the most fundamental principle of the chemical bond theory[10].

24 An overview of magnetism watch out the spin states on metal atoms At one instant it may have more electrons than ↑↑ ↑↓ ↑ M1 − O − M2 with Hund’s rule operating the average and at the other instant less elec- on metal atoms (orbitals less than half full and trons than the average number. ferromagnetic spin alignment). Suppose that To make the discussion concrete consider one up spin electron transfers from O to M2 the example of metal nickel. Nickel atom in and simultaneously an up spin electron trans- free state has the configuration [Ar]3d84s2. In ↑ − ↑↓ − ↑↑ fers from M1 to O giving: M1 O M2 i.e., condensed state 4s wave functions of adjacent there is a double exchange (from M1 to O and atoms overlap considerably and form a wide from O to M2). Then by the general theory, res- conduction s band. 3d wave functions overlap ↑↑ − ↑↓ − ↑ onance between states M1 O M2 and comparably weakly and form a narrow d band. ↑ − ↑↓ − ↑↑ M1 O M2 leads to lower energy and If we assume, as van Vleck[29] assumed, that thus more stability. It can be easily checked 70 percent of the wide 4s band is above the 3d that this mechanism will not operate if spins band, then the minimum energy configuration on metal atoms are anti-parallel. In conclu- will be obtained when some of the 4s electrons sion, this mechanism stabilizes the ferromag- migrate to 3d band. Most stable configuration netic spin arrangement[?, 21, 28]. corresponds to the case when on the average After this brief discussion of the exchange 2 × 0.7 = 1.4 electrons per atom migrate to interactions in insulator compounds, we re- 3d orbitals making their average occupancy to turn to the question of the correlation effects 8 + 1.4 = 9.4. The average occupancy of a 4s in the itinerant (Stoner) model. orbital will be 2 − 1.4 = 0.6. Thus the minimum energy configuration symbolically can XX. Van Vleck again and his be written as [Ar]3d1.44s0.6. This could actu- "middle-road" theory ally imply that there are two types of configurations: [Ar]3D94s1 on 60 percent of atoms, As discussed before there are two extreme and [Ar]3d104s0 on 40 percent of atoms, mak- models of ferromagnetism. One is localized ing the average configuration [Ar]3d9.44s0.6. Heisenberg model also called the non-polar Define A ≡ [Ar]3D94s1 and B ≡ [Ar]3d104s0, model in which polarity of atoms stay con- and consider two extreme situations: one is stant in time. Electron migration from one that the configurations A and B on a given atom to another is prohibited. The other ex- set of atoms are permanent in time. That is treme is the itinerant model where electron mi- if an atom is in configuration A it will remain gration from one atom to another is the main in it (no electron migration from the atom or feature. This is also called the polar model, as into the atom takes palce). This "stagnant" polarity of an atom is ever changing with time. situation corresponds to the Heisenberg non-

25 An overview of magnetism polar model. On the other extreme end which figuration B. They performed rough estimates corresponds to the itinerant or polar picture along these lines, but detailed calculations electron migration is freely allowed, and all were not feasible due to the complexity of the the configurations like d10, d9, d8, d7, etc are problem[29], and the success was partial. In possible with equal ease. This situation corre- conclusion, these were the "first steps" of incor- sponds to the Stoner model. Clearly, ioniza- porating correlation effects in the Stoner the- tion energies of the states d10, d9, d8, d7, etc ory. We would like to end our presentation of differ, and this fact is completely neglected the "middle-road" theory with the appropriate in the Stoner theory as mentioned before. In words of Kubo and Nagamiya[6]: other words these "correlation effects" are com- ".....Many investigators agree in that this is pletely neglected in Stoner theory. most desirable. But it is still very difficult to pave this road...... "

XXI. The Friedel-Alexander- Anderson-Moriya theory of moment formation in pure iron group metals

Building on the above ideas of van Vleck on correlation effects in transition metals, Ander- Figure 11: An artistic impression of the "middle-road" son developed the theory of moment forma- theory. tion in a magnetic impurity atom in a metal (as dilute alloy of Mn in Cu). There is an In1940, van Vleck with his student Hurwitz, extensive literature on this very important proposed a "middle-road" theory. As the name and interesting topic of Anderson impurity suggests Vleck-Hurwitz theory avoids both problem[23, 30, 31, 32, 33] and we will not the extreme situations (localization and fully go into this here, rather we will continue our uncorrelated itineracy). In their view the con- discussion of the traditional and complicated figurations A or B on a given atom are not problem of ferromagnetism in iron group met- permanent in time, rather configuration of a als (pure metal not alloys) within the itiner- given atom fluctuates between A and B (that ant picture but including electron correlation is with minimum polarity). And on the whole effects. at a given instant 60 percent of atoms will have The original ideas go back to van Vleck (the configuration A and 40 percent will have con- middle-road theory) and to Friedel (virtual

26 An overview of magnetism bound states formation in dilute magnetic alone can consider the spin as being associated loys). Anderson argued that the idea of vir- with the atom rather than with the individual tual bound states can be applied to local mo- electrons. Here we see the possibility of an ment formation in pure ferromagnetic met- atomic or a Heisenberg model emerging from als. Thus an effective Heisenberg picture be- the effect of correlations in the band model." comes emergent and the problems of the itin- Cyrot’s words are sufﬁciently clear and we erant model in explaining the Curie-Weiss law cannot add to it more. In conclusion, electron can be resolved using an effective Heisenberg correlation leads to quasi-localization. Next model for metals! It appear counter-intuitive comes the question of exchange interactions in to imagine local moments emerging from itin- these "induced" moments. This was addressed erant electrons. There is a very physical pic- by Alexander and Anderson[35] and by Toru ture, due to Cyrot[34], to understand how lo- Moriya[36, ?]. These investigations showed cal moments can form in an otherwise itiner- that instead of the Heisenberg’s J which is ant model. In Cyrot’s words: given by the Slater curve, one obtains an ef-

"The spin of an atom, i.e., the total spin of fective exchange interaction between the in- all the electrons on that atom, ﬂuctuates ran- duced moments which follows a different rule. domly in magnitude and direction. What ef- The effective exchange interaction is of ferro- fect might one expect from the electron inter- magnetic nature when atomic d shell is either action? We recall that Hund’s ﬁrst rule for nearly empty or nearly full, otherwise the in- atoms indicates that the intra-atomic interac- teraction is antiferromagnetic. tions will aline the electron spins on an atom. This line of approach was further extended We might expect a tendency to produce the by the introduction of powerful functional in- same result in a metal, since if an atom has a tegral methods by Schriefer and by Cyrot (re- spin up it will tend to attract electrons with fer to [19]). Although this approach via strong spins up and repel those with spins down. On correlation seemed quite promising but it fails this account one would suppose that the to- to address weakly ferromagnetic systems like tal spin on an atom at any one instant tends ZrZn2, Sc3 In etc. Thus it could not provide to be self-perpetuating, so that the spin value a comprehensive picture. It turns out that a can persist for a period long compared to the completely new mechanism for CW law op- d-electron hopping time. The electrons on the erates in these materials (or may be in iron atoms are always changing about, due to the group metals) in which moments are not lo- band motion, but the magnetic moment of the calized in real space rather they are localized atom persists due to the correlated nature of in momentum space. Such an approach was the electrons’ motion. In these circumstances, advanced by Moriya and kawabata in 1970s,

27 An overview of magnetism and it is know known as the Self-Consistent- itinerant (Stoner) picture, and to resolve its Renormalization (SCR) theory. This theory problems. Several investigators contributed takes into account the correlation effects be- in this important development. The investiga- yond mean field theory and in addition takes tions of Kanamori, Gutzwiller, Hubbard, Cy- into account the renormalization effects of rot, Moriya, Kawabata, and Okabe (among spin fluctuations on the equilibrium state. Our others) are important ones, and will be re- next article in the series is devoted to the SCR viewed in next paper this series. theory and other theories that take correlation In this paper, the difficult topics of exchange effects into account. For a detailed account of and correlation in itinerant and localized mod- the SCR theory consult[19]. To consider the ef- els are explained using semi-technical lan- fect of spin fluctuations on thermodynamical guage. In summary, early attempts were fo- properties, another theory along the lines of cused on obtaining ferromagnetic exchange Landau theory of phase transitions was devel- in the Heisenberg model. The Slater curve oped by Lonzarich and Taillefer[37]. provided a provisional picture or a rule of thumb. The Vonsovsky-Zener model sug- XXII. Conclusion gested a mechanism (s-d exchange) which could in principle provide the required fer- The following lines by Toru Moriya are suffi- romagnetic exchange, however it was found ciently clear to sort out the debate between lo- later on that it could not be applied to calized and itinerant pictures: the iron group metals (due to itinerant na- ".... the magnetic insulator compounds and ture of the d electrons) and was discarded. rare earth magnets are described in terms of But it lead to other very interesting develop- the localized electron model, while the fer- ments in f-electron and magnetic impurity sys- romagnetic d-electron metals should be de- tems (RKKY interaction and the Kondo effect). scribed on the itinerant electron model with the Pauling tried to develop a theory of ferromag- approximation method beyond the mean field level, netism in iron group metals using his valence properly taking account of the effects of electron- bond ideas, but this was also not successful electron correlation.... " as discussed in the text. The debate between —–Toru Moriya[38]. the itinerant model and localized model was When it became clear (from 1960s onwards) resolved and the main conclusion from 1960s that d electrons in iron group metals are to onwards was to properly incorporate the cor- be treated as itinerant electrons, the main relation effects in the itinerant model. Early aim of the ensuing investigations was to in- attempts were made by van Vleck in this di- corporate electron correlation effects into the rection with his "middle-road" theory or min-

28 An overview of magnetism imum polarity model. But to build a quan- [6] R. Kubo and T. Nagamiya, editors, Part titative theory along these lines turned out III–magnetism, in Solid State Physics, to be hard. The Friedel-Alexander-Anderson- McGraw-Hill Book Co. Inc (1969). Moriya theory of moment formation in pure [7] S. V. Vonsovsky, P. L. Kapitza, I. K. Kikoin, iron group metals builds upon these ideas. As M. N. Mikheev, N. N. Semenov, and Ya. S. discussed, the problems of these approaches Shur, Sov. Phys. Usp. 18, 995 (1976). set the stage for the arrival of the SCR theory. In author’s opinion one of the most success- [8] J. Dorfman, Nature, 119, 353 (1927). ful development is the introduction of the SCR [9] R. E. Peierls, Quantum theory of solids, Ox- theory by Moriya and kawabata which goes ford uni. press (2001). beyond the Hartree-Fock and random phase approximations in treating the correlation ef- [10] C. Kittel, Introduction to solid state physics, fects, and in addition, it takes into account John Wiley and sons, Inc. (2003). the effect of thermal excitations of magnetic [11] L. Pauling, The nature of the chemical bond, nature (i.e., thermal spin ﬂuctuations) on the Cornell uni. press (1939). equilibrium state i.e., the renormalization as- pect of the SCR theory. These theories form [12] J. H. Van Vleck, Rev. Mod. Phys. 17, 27 the subject matter of our next article in this (1945). series. [13] R. J. Baxter, Exactly solved models in statistical mechanics, Academic press (1982). References

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29 An overview of magnetism

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