A Survey of Superconductors
I've posted nine web pages about superconductivity to Chemexplore so far (May , 2011) , and my ebook on metallic solids also examines many superconductors (the references are listed at the end of this web page . Underlined blue hyperlinks can be clicked when online to download the PDF or HTML file , which will open in a new window) . Considerable superconductor science is scattered about in these various reports , with much disconnected or partly developed material . I thought it might be useful to provide a sort of overview of the subject at this time , to weave all of these loose threads together .
Superconductor research is done mostly by condensed matter physicists . Their approach to understanding superconductivity and the descriptive terminology they have developed over the past decades in this endeavour are highly specialized and complex . All scientific disciplines tend to accumulate an extensive vocabulary over the years , as does my own profession , that of chemistry . But is superconductivity really all that complicated ? My thesis , expressed in the ebook and in the succeeding Chemexplore web pages on the subject , is that it is not ; the basis of superconductivity is actually quite simple , as I try to show in the following section .
In this web page I'll review many different varieties of superconductors , sorting them into three general classes based on two criteria : first , to what extent antiparallelism is induced in their free electrons ; and second , whether their metallic bond is of the monolayer or bilayer type . Superconductors with the highest transition temperatures (> 100 K) all have a very strong antiparallelism in their free electrons (via antiferromagnetic induction) and a bilayer metallic bond (with the participation of nonmetallic mostly oxygen atoms in it) . Because I'm a chemist , my survey will of necessity be expressed in chemistry terms , and from a chemist's point of view . I hope this chemistry-oriented approach to understanding superconductors will be stimulating and thought-provoking to all the readers of this review .
The Basis of Superconductivity
The most widely held virtually universal view of the origin of superconductivity in metallic solids is that the conduction electrons in them are pushed together by lattice vibrations , the phonons , as described by the BCS (Bardeen-Cooper-Schrieffer) theory , thus forming the Cooper pairs . However , I believe that such phonon mediation is an enabling mechanism assisting in the Cooper pair formation , and is not the cause of superconductivity . There are several identifiable enabling mechanisms that promote the formation of Cooper pairs ; each of these specific enabling mechanisms gives rise to a distinctive class of superconductors .
Cooper pairs are the charge and energy carriers in a superconductor , consisting of somewhat loosely associated pairs of electrons with an antiparallel spin orientation with respect to each other . I have never been satisfied with the BCS description of how the phonons can cause this association between the two electrons , and have sought an alternate explanation of it . My picture of the bonding between the Cooper pair electrons is very simple and straightforward .
The negative charge of the electrostatic field surrounding electrons causes a repulsive electric force , Fe , between two electrons in close proximity . This repulsive force between electrons is commonly observed in static electricity effects , for example in the classroom demonstration of making someone's hair stand on end when he or she receives a substantial charge from an electrostatic generator .
The electrons orbiting atoms are also surrounded by a magnetic field , and they behave like tiny spherical magnets with a north and a south magnetic pole . In common experience two bar or horseshoe magnets will be mutually attracted and stick together if their poles are aligned in an antiparallel manner (northsouth , southnorth) , but will repel one another if their poles are aligned in a parallel manner . What most people seem to forget is that this magnetic field around the electron can similarly exert an attractive force , Fm , with another nearby electron .
Cooper pair electrons have mutually antiparallel orientations , so their magnetic fields must be acting in a reinforcing manner , producing an attractive magnetic force between them . Which is greater , the attractive magnetic force Fm , or the repulsive electrostatic force Fe ? The two forces can be calculated from Coulomb's Law of Electric Force (discussed in most physics textbooks) , and its less well-known magnetic analogue , Coulomb's Law of Magnetic Force (which should be applicable to the two Cooper pair electrons , as they are the smallest and simplest of any system of two magnets ; their magnetic field strength is the quantum of magnetic flux , the fluxoid) . Both forces are inversely proportional to square of the distance the coherence length between the two electrons . The relative magnitudes of the two forces can be compared if the only variable , the coherence length , is mathematically cancelled out by taking a ratio of Fm to Fe :
From this simple calculation we see that the attractive magnetic force Fm between the two Cooper pair electrons is almost 1200 times greater than the repulsive electrostatic force Fe between them . I'm not sure how Coulomb's Law of Magnetic Force is applied to the poles of two magnets ; is the force applicable to one or two sets of poles ?
Whether the Fm / Fe ratio is 1200 or 2400 is unimportant to me ; the key point is that the magnetic force is responsible for the association of the two electrons in a Cooper pair , and is the true cause or basis of superconductivity . The Fm / Fe ratio is a universal constant and applies to Cooper pairs in all physical , chemical , and crystallographic conditions , and of course at all temperatures . We can also use it to rationalize other phenomena involving pairs of electrons , such as a contributing factor in covalent bonding ; to electrons in antibonding molecular orbitals , such as those in the oxygen molecule , which have a parallel spin orientation and so repel one another and thereby cancel out one of oxygen's bonds ; and to the Pauli exclusion principle in the distribution of electrons in their various energy shells (most importantly in the valence shells) of atoms .
What force ultimately causes the antiparallel orientations of the valence shell electrons in magnetic materials ? The protons and neutrons in the atomic nuclei also have magnetic spins , and the packing of these nucleons may result in a shaped magnetic field unique to each nucleus . The magnetic fields of the nuclei undoubtedly induce magnetic fields in their surrounding electrons . The nuclear magnetic fields of certain elements , such as iron , cobalt , and nickel , might have shapes that cause the observed magnetic states (eg. ferromagnetisn , ferrimagnetism , antiferromagnetism) in these Transition metals and their compounds . In most elements the nuclear magnetic fields are probably spherically symmetrical , with no specific sort of spin ordering induction in their respective valence shell electrons . As a result , they are diamagnetic or paramagnetic . In order to make their valence electrons ferromagnetic (FM) or antiferromagnetic (AFM) they must be electronically coupled via electron superexchange to a FM or AFM material . AFM induction is proposed as the enabling mechanism to convert the (Pauli) paramagnetic metals into AFM metals , and so assist their free electrons to condense magnetically stick together more easily into Cooper pairs .
At the same time we should recognize that the absolute Fm strengths are extremely small , and Cooper pairs must be quite fragile , ephemeral entities as a result . The original Drude model (1900) of the electron gas in metals portrayed the conduction electrons as a cloud of hard spheres behaving somewhat like gaseous atoms such as helium . These electrons were in a constant state of motion , richocheting off the lattice atomic kernels as they were pulled downfield by the electric potential . The forceful impacts of the Cooper pairs on the atomic kernels will , if sufficiently strong , break the pairs apart into singlet electrons . While impressively large , the Fm / Fe ratio will in no way unbreakably glue together the Cooper pairs . On the contrary , at a certain temperature they will be uncoupled by increasingly violent collisions with the lattice kernels . The challenge to solid state scientists investigating superconductivity is to discover new enabling mechanisms that will promote the formation of Cooper pairs with stronger and stronger absolute Fm strengths , and with correspondingly higher and higher temperature resistances , that is , critical temperatures .
An antiparallel spin order must be imposed on the free electrons in a metallic solid to enable their condensation into Cooper pairs . This is true of whether the induction comes directly from the nuclear magnetic field (as with the ferromagnetic elements) , or from a neighbouring atom's magnetic field . The stronger the AFM induction , the stronger the magnetic coupling will be , and the higher the transition temperature of the material will be . Tc is also a function of the coherence length between the Cooper pair electrons , which I'll discuss further on in this web page . Three classes of superconductors can be identified , based on the AFM induction strength alone :
* weak induction superconductors , in which the AFM induction from the nucleus or from neighbouring atoms in the solid is very weak . Strong cooling the enabling mechanism in this case is required to induce an antiparallel spin order in the free electrons :
The list of compounds which become antiferromagnetic , however , is very large , as most paramagnetic substances become antiferromagnetic at a sufficiently low temperature (H.M. Rosenberg) .
Weak induction superconductors generally have very low transition temperatures , typically in the 030 K range .
* moderate induction superconductors , in which formation of the Cooper pairs is mediated by a medium strength AFM induction , via electron superexchange from nonmetal linking atoms , eg. arsenic and selenium . They organize the electron spins of adjacent metal atoms whose valence electrons and orbitals form the actual metallic bond in the solid . Their transition temperatures are usually in the 3060 K range .
* strong induction superconductors , which are of two types : internal , in which a metallic bond is chemically created (usually by controlled-valence doping) within an AFM matrix ; and external , in which formation of the Cooper pairs is mediated by a strong induction from AFM layers in a heterostructure [multilayered sandwich compound] into adjacent metallic layers in that composite material . All genuine HTS [high temperature superconductor] compounds are of the strong induction type , and have transition temperatures higher than 77 K , the boiling point of liquid nitrogen . AFM induction layers usually consist of copper(II) oxide and related compounds . Nickel(II) and iron(II) and (III) oxides and their derivatives can also have strong AFM properties .
Let's look more closely at these three classes , which I'll illustrate with various examples .
Weak Induction Superconductors
Superconductivity was discovered in April , 1911 one hundred years ago , as I'm writing this web page ! by H. Kamerlingh Onnes at the University of Leiden in the Netherlands . He was examining the electrical conductivity of a sample of mercury cooled in a bath of liquid helium to about 4 K . To his surprise , the mercury had no detectable electrical resistance . In the following years and decades many more elementary metals (twenty-nine to date at one atmosphere and several more under high pressure) , alloys , intermetallic compounds , synthetic metals , and other chemical compounds were found to be superconducting , but only at extremely low temperatures , usually quite close to Absolute Zero . The one thing all these wildly different materials had in common was that they were metallic solids of one sort or another . Their metallic bonds were the conduit for the superconducting current , just a they were for the materials in their normal state. The following tabulation lists a selection of these superconductors with very low transition temperatures :
A common physical property of all of the above materials is their Pauli paramagnetism , produced by the random orientation of their metallic bond free electrons , resulting in a small net magnetism in the solid . The free electrons are also responsible for the lustrous , shiny appearance of metals , so Pauli paramagnetism and a high reflectivity are often closely associated . Thus , as a rule of thumb , a bright , shiny sample of metal will also exhibit Pauli paramagnetism . Such is the case with most of the weak induction superconductors listed above . It's also the case with many other metallic solids that never become superconducting , like the three coinage metals , copper , silver , and gold . On the other hand , all of the HTS compounds are jet black in appearance and are AFM , not Pauli paramagnetic . So , as another rule of thumb , if a metallic solid is shiny and is Pauli paramagnetic , it will be a very low temperature superconductor , if it is one at all .
The Cooper pairs can form only if this paramagnetism can be converted into an antiparallel spin ordering . With little or no AFM induction this is very difficult to do , and only in a minority of metallic solids can the random spins (paramagnetic) be re-ordered into the correct antiparallelism (diamagnetic) for superconductivity , and only by extreme cooling close to Absolute Zero . In all of the atoms in the materials listed above the AFM induction is very weak , so they barely squeak by as superconductors . An interesting exception might be the selenium atoms in the molecular metal (TMTSF)2 ClO4 . Selenium has a moderate sort of AFM induction capability , as will be discussed in the next section , although it's not very effective in the TMTSF salts . (TMTSF)2 PF6 becomes superconducting at ~ 1 K , but only under 12,000 atmospheres of applied pressure . It has semiconducting properties at normal pressure .
Moderate Induction Superconductors
Nonmetal atoms in a metallic solid can in some cases induce an antiparallel spin order in its free electrons . The induction will be relatively weak , and the resulting superconductors usually have low to medium range transition temperatures . To date , the only nonmetal atoms discovered so far that can induce antiparallelism in the free electrons are from the pnictides (Group VB/15 elements , essentially phosphorus and arsenic) and the chalcogenides (Group VIB/16 , essentially selenium and tellurium) . This is reflected in the types of compounds that can be classified as moderate induction superconductors :
I surveyed the ferropnictides in another Chemexplore web page , Electron Doping of Transition Metal Pnictides and Chalcogenides, so I won't discuss them any further here . The maximum transition temperature for these materials seems to have peaked at ~ 56 K , indicating a certain degree of maturity in them . In the Electron Doping web page I've suggested various chemical techniques that might be applied to the ferropnictides in order to raise their Tc values a little higher , possibly even into the liquid nitrogen range . However , they have direct metalmetal (mostly FeFe) metallic bonds ; these are monolayer metallic bonds , and as will be explained further on in this web page , superconductivity is limited in monolayer metallic bonds by the Fermi-Dirac distribution of the energies of the free electrons . It's therefore unlikely that Tc will ever exceed 90 K for these moderate induction superconductors , in my opinion .
MgB2 , CaC6 , and the Alkali Metal Fullerides
Magnesium diboride , calcium-graphite , and the alkali metal fullerides could also be classified as moderate induction superconductors . The boron and graphite sheets and C60 spheres may be acting as antiparallelism induction agents to a certain extent . The main superconducting phase in these compounds might actually be in their metal atom components . ARPES experiments have shown that there are at least two principal superconduction bands in magnesium diboride :
Though generally believed to be a conventional (phonon-mediated) superconductor , it [MgB2] is a rather unusual one . Its electronic structure is such that there exist two types of electrons at the Fermi level with widely differing behaviours , one of them (sigma-bonding) being much more strongly superconducting than the other (pi-bonding) .This is at odds with usual theories of phonon-mediated superconductivity which assume that all electrons behave in the same manner .Theoretical understanding of the properties of MgB2 has almost been achieved with two energy gaps . In 2001 it was regarded as behaving more like a metallic than a cuprate superconductor.
from the Wikipedia web page , Magnesium Diboride .
The boron sheets in MgB2 are isostructural and isoelectronic with the carbon sheets in graphite , whose metallic bond consists of the carboncarbon 2pz p XO . A similar pi XO in MgB2 is the pi superconducting band (XO = crystal orbital = polymerized molecular orbital = metallic bond = conduction band) . The larger , denser sigma band must be derived from the magnesium intercalated between the boron sheets (not the BB sigma bonds , which are covalent , not metallic) . Since purely ionic Mg2+ cations would be nonmetallic , there must be an electrically conducting mixture of Mg0 and Mg2+ intercalated between the the boron sheets .
Magnesium is a powerful reducing agent (E0ox = 2.372 V) , so it's usually assumed that 100% of the Mg0 3s2 valence electrons are transferred to the boron sheets in MgB2 . Prassides and co-workers state , with respect to the alkali metal fullerides :
Charge transfer is essentially complete and the conduction band of C60 , which arises from its lowest unoccupied molecular orbital (LUMO) of t1u symmetry , is half filled (p. 151) ; In the alkali fulleride phases , the metal ions residing in the interstitial spaces of the C60 structure are electronically [inert] and the electronic properties are entirely determined by those of the C603- sublattice ; ......... in contrast to the alkali metal fullerides where full charge transfer from the metals to C60 occurs (p. 152) .
However , they considered that in the rare earth fullerides they studied , M2.75C60 (M = Sm , Yb , and Eu) , charge transfer was incomplete , and a mixed-valent state was created in the rare earth(II,III) cations in the fulleride compounds :
....... the electronically active fulleride sublattice acts as an electron reservoir that can accept electrons from or donate electrons to the rare-earth 4f/5d bands , thereby sensitively modulating the valence of the rare-earth sublattice (p. 151) .
I would offer the hypothesis that an incomplete transfer of valence electrons , from the alkali metal to the carbon atoms , occurs in the synthesis of the alkali metal fullerides . As the reaction between the fullerene and alkali metal proceeds , the two reagents become mixed-valent , with electron resonance between the partially charged components . The oxidation potential of the alkali metal declines as it becomes mixed-valent , and the reduction potential of the fullerene similarly falls . That is , the mixed-valent alkali metal atoms/cations become weaker and weaker reducers , and the mixed-valent buckyballs become more and more negatively charged , and more and more resistant to accepting any more electrons . Finally a point is reached in the reaction where no further electron transfer takes place . At this point there is a precise electrical balance between the positively charged alkali metal atoms/cations and the negatively charged mixed-valent buckyballs , with respect to the crystallographic structure most energetically stable for that particular set of atoms , cations , and buckyballs (which is mostly the fcc packing) .
KCP is a nice example of this precise electronic balance . KCP is a partially oxidized (by bromine) platinum coordinate covalent compound , K2Pt(CN)4Br0.3 . 3H2O , in which its Transition metal atom components , the platinums , have the somewhat unusual NIOS (non-integral oxidation state) valence of Pt(2.3+) . The Pt(CN)4 units form molecular stacks in the KCP crystals , which are copper-coloured , lustrous , slender needles . The platinum valences consist of about 80% Pt(II) and 20% Pt(IV) , which in KCP at higher temperatures are perfectly blended to Pt(2.3+) . KCP crystallizes from solution with that exact platinum valence , no more , no less . Neutral Pt(CN)4 molecules intercalate in the stacks between the negatively charged Pt(CN)42- anions , diluting their negative charge which would otherwise prevent stacking of the units and their cohesion by the PtPt metallic bond [continuous 5dz2 orbital overlap] along the spines of the stacks . At the 80% Pt(II)20% Pt(IV) point the stabilizing metallic and other chemical bonds in KCP precisely balance the destabilizing coulombic repulsive force between the Pt(CN)42- anions . This mixed-valent Pt(2.3+) produces a partially-filled PtPt XO that results in KCP being metallic . Unoxidized K2Pt(CN)4 [filled Pt(II)] and presumably the neutral Pt(CN)4 molecules [empty Pt(IV)] are electrically insulating .
A similar situation undoubtedly occurs in the alkali metal fullerides . A full valence electron transfer would make the anionic buckyballs too strongly negative and repulsive to each other ; the energetically stable fcc cubic structure can form if some of this repulsive negative charge is reduced , which can occur only if a partial transfer takes place . The electron resonance between the mixed-valent alkali metal atoms/cations and between the mixed-valent buckyballs results in a stabilizing equilibrium in the lattice that permits the close packing of the metal and carbon spheres in the solid . As with KCP , this coulombicversusmetallic bonding balance within the fcc packing structure can be achieved only when the components are mixed-valent , with less than 100% charge transfer . The alkali metal phase of the fulleride compounds will thus be mixed-valent , and indeed is probably the principal conduction pathway in them .
Electrical conductivity measurements in the normal state of the superconducting alkali metal fullerides seem to support this mixed-valent atoms/cations hypothesis . The fullerides have a metallic conductivity, that is , an inverse temperatureelectrical conductivity relationship :
Temperature dependent measurement shows that K3C60 and Rb3C60 solids are metallic (Tanigaki and Zhou , p. 2161) ; a graph of the electrical resistivity of K3C60 over the temperature range 0 to 250 K shows an inverse relationship (Gunnarsson , Fig. 9 , p. 588) ; and , calcium-intercalated graphite , CaC6 , also has such an inverse relationship [Emery and co-workers , Fig. 5(a) , p. 6] . MgB2 exhibits an inverse temperatureelectrical conductivity relationship [Canfield and Crabtree , Figure 2 , p. 36] .
In order to form intermolecular metallic bonds between an assembly of buckyballs , the carbon 2pz pi atomic orbitals must overlap end-to-end ; however , the resulting CC sigma bonds have nodes around the carbon atoms . These nodes would make the buckyball solid behave like a semiconductor , with a direct temperatureelectrical conductivity relationship , which doesn't seem to be the case with the alkali metal fullerides .
On the other hand , mixed-valent alkali metal atoms/cations could form a nodeless sigma XO as the metallic bond in the fullerides . Such a nodeless metallic bond would result in an inverse temperatureelectrical conductivity relationship in the fullerides , which is actually observed in them :
From this electrical conductivity behaviour I conclude that superconductivity in the alkali metal fullerides occurs in their mixed-valent alkali metal atoms/cations , and not in the buckyballs . A similar situation undoubtedly occurs in the calcium-graphite intercalation compound , CaC6 , and in MgB2 , which must have mixed-valent Ca0 / Ca2+ and Mg0 / Mg2+, respectively . In the latter case the Mg0 / Mg2+ 3s sigma XO has been experimentally detected by ARPES . This technique should also demonstrate the presence of similar sigma superconduction bands in the alkali metal fullerides and in CaC6 .
The powerful electron donor molecule , TDAE [tetrakis(dimethylamino) ethylene] , can form a 1 : 1 charge-transfer compound with fullerene buckyballs . TDAE is said to have a reducing strength comparable to that of zinc (E0ox = 0.7618 V , Hoffmann in TDAE) . There are no mixed-valent metal atoms/cations in this material , so there won't be any nodeless sigma XO metallic bond in it . TDAE-C60 may have p orbital s nodal metallic bonds , which would impart semiconductor properties to it :
The conductivity [of TDAE-C60] is of the order of 10-4 S/cm at room temperature and the temperature dependence is clearly nonmetallic(Schilder and co-workers , abstract) . [10-4 S/cm = 10-4 ohm-1cm1 ; this is in the semiconductor range] .
TDAE-C60 is a molecular ferromagnet ; below ~ 16 K the singlet electrons in its crystal lattice assume a parallel ordering pattern . Italian researchers claim to have detected a Meissner signal below 17.3 K in TDAE-C60 , indicating the co-existence of superconductivity with its ferromagnetism . This superconductivity in TDAE-C60 may not be genuine , given the co-existing ferromagnetism in the solid . After all , the prime hallmark of genuine superconductivity is the very strong diamagnetism exhibited by all conventional superconductors , which can be visually observed by the Meissner-Oschenfeld levitation effect (a splendid demonstration of which is provided in this YouTube video) . An ARPES spectrum of TDAE-C60 would be very interesting . I predict it would reveal only a feeble pi band in the compound ; no sigma band would be observed in the analysis .
The electrical conductivity of pure buckyball fullerene is stated in this web page to be 10-16 ohm-1cm-1 at 293 K , making it an electrical insulator [the web page actually gave its electrical resistance as 1 x 1014 ohm-m ; however , they also listed graphite as having a resistance of 1.37 x 10-5 ohm-m , which translates into a conductivity of 730 ohm-1cm1 . The electrical conductivity of a typical sample of virgin (undoped) graphite is usually around 25,000 ohm-1cm1 or so . This same source listed the resistivity of diamond as 1 x 1020 ohm-m , which seems about right] . Foley and co-workers measured the electrical conductivity of fullerene films over a range of temperatures and found them to be semiconducting : their ambient conductivity was ~ 1 ohm-1m-1 [10-2 ohm-1cm-1] , and they had a direct temperatureconductivity relationship . In any case , pure buckyballs are either insulating or semiconducting ; when they're doped with alkali , alkaline earth , or lanthanide metals their M-C60 compounds are indeed highly conducting and even superconducting , but the nodeless XO in the mixed-valent metal atoms/cations is the principal electrical conduit in them . Electron transfer between buckyballs occurs only in nodal s-type bonds , which are a much less favorable conduction pathway , and results in semiconductor behaviour in the carbon solid .
Chinese researchers have recently (January , 2011) reported preparing epitaxial thin films of MgB2 having thicknesses of 7.540 nm , grown on an alumina substrate , by a CVD method . The 7.5 nm film was superconducting , with Tc = 34 K , slightly degraded from the Tc = 39 K for pure bulk magnesium diboride . Alumina has the corundum crystal structure , as does a-Fe2O3 . It would be interesting to compare the superconductivity properties of MgB2 /a-Al2O3 [nonmagnetic alumina] with those of MgB2 /a-Fe2O3 [AFM , TN = 948 K] . Similarly , the superconductivity of magnesium diboride epitaxial thin films on other nonmagnetic vs. AFM substrates could be studied , eg. MgB2 /MgO [rocksalt , nonmagnetic] versus MgB2 /NiO [rocksalt , AFM , TN = 525 K] , and MgB2 /SrTiO3 [cubic perovskite , nonmagnetic] versus MgB2 /LaFeO3 [orthorhombic perovskite , AFM , TN ~ 740 K] . I'm confident that a significant increase in Tc for the MgB2 films would be observed for those on AFM substrates , compared to those on the nonmagnetic control substrates .
Variation of MgB2 films' Tc on several different AFM rocksalt compounds would also be quite interesting . As mentioned , NiO [TN = 525 K] could be one such substrate . Others could be CoO , FeO , and MnO , with TN = 291 , 198 , and 122 K , respectively . CuO , with TN = 230 K , would also be fascinating to examine in this regard . Pure copper(II) oxide has a peculiar crystal structure consisting of layers of three-dimensional interlocking CuO2 ribbons . If the ribbons could be straightened out into planar [CuO2]n sheets , epitaxial CuO would have the potential to be a very powerful AFM induction agent , and might induce superconductivity in MgB2 films at an exceptionally high Tc . [Similarly for the hypothetical compound (ReO3)2CuO] .
On the other hand , MgB2 could be layered with a ferromagnetic induction agent , which would induce a parallel spin order in the metallic layer's free electrons . Then the MgB2 would never become superconducting , even very close to Absolute Zero . Titanium dioxide (rutile crystal structure) could serve as the nonmagnetic control substrate. Chromium dioxide also has the rutile structure and is both metallic and ferromagnetic (TC ~ 391 K , meff = 2.0 BM , ambient) , and could act as the FM substrate layer for MgB2 .
Europium monoxide has the rocksalt crystal structure and is ferromagnetic below its TC (~ 70 K) . But will it also inhibit superconductivity in MgB2 ? Eu2+ is 4f7 electronically . In the Transition metal oxides such as NiO the metal cations' 3d valence shell electrons can successfully superexchange via intervening oxide anions . Can the rare earth f valence electrons also superexchange through the oxygens ? EuO has been used in epitaxial thin films . It would be both interesting and educational to see how it would affect superconductivity in epitaxial thin films of MgB2 .
Strong Induction Superconductors
All genuine high temperature superconductors (HTS , Tc > 77 K) are of the strong induction type . A strongly AFM metal oxide matrix or layers are inducing an efficient antiparallel spin ordering in the metallic bond free electrons , thus assisting them in condensing into Cooper pairs . Typically , oxides of Transition metal divalent cations such as iron , cobalt , nickel , and especially copper , are strong AFM induction agents ; certain iron(III) oxide compounds , such as Fe2O3 , BiFeO3 , and LaFeO3 , are also strongly AFM .
Strong induction superconductors can be either of the internal or external variety . Internal induction occurs when a strongly AFM compound , normally an insulator or poor semiconductor , is converted by some sort of chemical process into a metallic solid . This can often be accomplished by the controlled valence method , thereby creating a mixed-valent compound , as will be reviewed in the next section . The liberated free electrons in the metallic bond immediately assume an antiparallel spin orientation , imposed on them by the internal AFM spin ordering of the host matrix . The famous HTS material YBCO is such an internal strong induction superconductor .
External induction can occur when a multilayer heterostructure is comprised of alternating metallic layers for the electrical conduction and superconduction and nonmetallic , but strongly AFM layers for the induction process . BSCCO-2212 (Bi2Sr2Ca2Cu3O10+x) is an excellent example of an external strong induction superconductor . It's composed of alternating metallic layers of Bi2Sr2O5+x and nonmetallic AFM Ca2Cu3O5 layers . Various strong induction superconductors , including YBCO and BSCCO-2212 , are listed in the following tabulation :
In the above examples , the Cooper pairs are mostly derived from the mixed-valent copper(II,III) component of the material ; these are all internal strong induction superconductors . Only in BSCCO-2212 are the mixed-valent bismuth(III,V) oxide layers the source of the Cooper pair electrons . This is a result of selective oxidation of the metal cations in the preparative process .
For example , the compound Bi2Sr2Ca2Cu3O10 could be prepared rather easily , perhaps by a simple chemie douce procedure . By valence counting we see that it's a homovalent compound , i.e. there are no mixed-valent cations in it : (Bi3+ Bi3+ Sr2+ Sr2+ Ca2+ Ca2+ Cu2+ Cu2+ Cu2+) (O10)-20 . This complex bismuthate compound would undoubtedly be an electrical insulator , or at best a poor semiconductor . If it's heated to say , around 800 ºC in a stream of pure , flowing oxygen , it will be partially oxidized to the metallic and electrically conducting and superconducting BSCCO-2212 . Which metal cations are being oxidized ? Clearly the strontium and calcium cations aren't being affected in any way by the oxygen . Their valences are invariable at +2 each . Either the copper(II) or the bismuth(III) cations are being partially oxidized . To find out which , we'll consult the following tabulation of standard redox potentials for various metal cations :
It's easier to oxidize Bi(III) to Bi(V) at 1.759 V than it is to oxidize Cu(II) to Cu(III) at 2.4 V ; copper(III) is in fact the most powerfully oxidixing metal cation known , so conversely copper(II) is the metal cation most difficult to oxidize to its higher valence state . From this simple High School redox analysis I conclude that the metallic layers in BSCCO-2212 are composed of the mixed-valent ternary bismuth(III,V) oxide , Bi2Sr2O5+x , and the AFM induction layers are composed of the homovalent ternary copper(II) oxide compound Ca2Cu3O5 . To the best of my knowledge , the value for x in Bi2Sr2O5+x has never been specified .
Mixed-Valent HTS Compounds
Mixed-valent compounds are fascinating materials . I've written a lot about them , both with respect to their application in superconductivity , and also about how they might be used in photovoltaic cells . There are four types of mixed-valent compounds : I , II , IIIA , and IIIB . They are now generally referred to as Robin-Day Classes in honour of the two chemists , M.B. Robin and P. Day , who devised their classification and who wrote the definitive review of them in 1967. In Classes I and II the mixed-valent metal cations are separated by anions , while in Classes IIIA and IIIB there is direct metal-metal bonding . Only the electronically-active Classes II and IIIB are of interest with respect to superconductivity . The Class I compounds are insulators or poor semiconductors , and the Class IIIA compounds have electronically-isolated metal clusters .
The primary , perhaps unique role of mixed-valency with respect to superconductors is to unpin valence electrons that would otherwise be frozen in place around their respective kernels . Homovalent insulators or at best , poor semiconductors can be converted into respectable metallic solids by the chemical technique of controlled valence doping , which was first studied by the Dutch solid state physicist E.J.W. Verwey (19051981) in the late 1940s . We can understand this unpinning ability of mixed valency in Robin-Day Class II and IIIB compounds by comparing the redox chemistry of homovalent copper(II) and mixed-valent copper(II,III) . To save myself some nuisance in re-writing , I'll copy and paste from my Antiferro web page the following section :
Consider the redox situation in a sample of a homovalent copper(II) oxide compound . Suppose an electron is transferred by resonance from copper(II) cation A to copper(II) cation B . Cu2+A becomes Cu3+A , while Cu2+B becomes Cu1+B . The redox equations for this electron transfer are as follows :
Cu2+A e- -------------> Cu3+A ........... E0ox = 2.4 V
Cu2+B + e- -------------> Cu1+B ........... E0red = 0.153 V
Net reaction : Cu2+A + Cu2+B -------------> Cu3+A + Cu1+B ........... E0T = 2.247 V
The very large negative cell potential E0T indicates that this reaction the disproportionation of copper(II) to copper(I) and copper(III) is highly unfavourable thermodynamically at STP and is essentially impossible . The copper(II) 3d9 valence electrons are strongly pinned on their respective kernels , and thus homovalent copper(II) oxide compounds are insulators (or poor semiconductors) .
Now consider what happens in a typical Robin-Day Class II mixed-valent copper compound , for example in YBCO , with its Cu2+ Cu3+ Cu2+ . The copper(II) 3d9 valence electron will resonate with the 3d8 copper(III) . When that happens , the Cu2+A will become Cu3+A as before . However , now the Cu3+B becomes Cu2+B , as represented in the following redox equations :
Cu2+A e- -------------> Cu3+A ........... E0ox = 2.4 V
Cu3+B + e- -------------> Cu2+B ........... E0red = + 2.4 V
Net reaction : Cu2+A + Cu3+B -------------> Cu3+A + Cu2+B ........... E0T = 0 V
We see from this simple redox analysis that there is no thermodynamic barrier to the electron resonance in Robin-Day Class II [and IIIB] mixed-valent compounds . That's why they have such extraordinary optical and electrical properties , compared to their corresponding homovalent parent compounds , in which the valence electrons are pinned to their atomic kernels . This extremely fast (~ 108 cycles/s , 10-1 teraherz) electron resonance permits the free electrons in the metallic bond to closely approach other nearby free electrons ; and , if the two electrons have an antiparallel orientation with respect to each other , they can magnetically couple together , as discussed above .
Reproportionation is a chemical technique to produce Robin-Day Class II mixed-valent compounds via the controlled valence process , and as such is an invaluable method to synthesize metallic solids and superconductors . Two different valence states of an element can be reproportionated to form an intermediate valence state , often NIOS (non-integral oxidation state) , whose resulting mixed-valent compound is metallic and sometimes even superconducting . Such was the case with the moderate induction superconductor Ba0.6K0.4BiO3 (Tc = 29.8 K , onset) . Bismuth(III) 6s2 was blended with bismuth(V) 6s0 to produce bismuth(4.4+) : (Ba2+)0.6(K1+)0.4Bi4.4+(O3)6- . The bismuth(III) 6s2 electrons are an example of an inert pair of electrons on a heavy element atom . The inert pairs from the covalent bonds provide the free electrons that condense into Cooper pairs in the metallic bond . Other heavy elements with inert pairs might similarly be reproportionated with their corresponding higher valence states to obtain intermediate NIOS valences in metallic compounds .
The idea of using the inert pairs of electrons in heavier elements as the Cooper pair source in superconductors isn't new . Aleksandrov and co-workers (1989) comment :
One might suggest that other ions with an unshared pair effect would be capable of forming superconducting oxide compounds of copper (p. 756) ; and , The existence of superconductivity in a tin-based system raises the hope that cations other than the familiar bismuth , thallium , and lead cations which have an unshared electron pair can form superconducting compounds (p. 758) .
With all due respect to these perceptive researchers , the superconductivity in their optimum tin compound SnBaSrCu3Ox wasn't derived from the tin's inert pairs . That's because there couldn't have been any inert pairs in their product ! Their tin precursor was tin(IV) oxide , SnO2 , which has empty tin, i.e. 5s0, and they used oxidizing conditions in their preparation of SnBaSrCu3Ox ; therefore , it has no inert pairs . The minor superconductivity detected in the material (Meissner component ~ 2% by volume) undoubtedly came from its slightly oxidized , mixed-valent , Cu3Ox part .
While there are many research papers from the late 1980s and early 1990s describing various superconductors supposedly utilizing the inert pairs of heavy metal elements as the Cooper pair source , upon careful examination we discover that either the heavy metal components have been over-oxidized , and there aren't any inert pairs like the tin compound above or the inert pairs are still intact in the metaloxide layers [as with Cava's lead cuprate , Pb2Sr2YCu3O8 = Pb2+ Pb2+ Sr2+ Sr2+ Y3+ Cu2+ Cu1+ Cu2+ O816- , which actually contains copper(I) , indicating considerable under-oxidation] . In the following paragraphs I'll propose various new HTS candidates in which a careful control of the chemistry should result in composites having metallic Robin-Day Class II MO layers alternating with homovalent ternary copper(II) and nickel(II) oxides as the AFM induction layers . The inert pairs of electrons will have been properly dispersed in the metallic layers , their lower valences having been blended with their corresponding higher valences .
SrCuO2 has parallel (CuO2)2- layers , between which the Sr2+ cations are nested :
Thallium(I) 6s2 could be reproportionated with thallium(III) 6s0 to provide metallic thallium(II) 6s1 compounds . TlCuO2 , i.e. Tl1+0.5Tl3+0.5CuO2 , should be isostructural with SrCuO2 . It might be prepared by reproportionating thallium metal with Tl2O3 to produce TlO, which would then react with CuO :
1/3 Tl0 + 1/3 Tl2O3 + CuO ----- [heat , sealed container] -----> TlCuO2 .
Covalent Tl2O3 melts at 717 ºC , but it actually starts to sublime at a considerably lower temperature when heated . Its highly toxic fumes can spread into the laboratory , polluting it with poison , which is gradually absorbed by anyone working there . Some researchers avoid using thallium compounds in their work ; however , if suitable safety precautions are taken by experienced chemists (i.e. they should have an excellent laboratory technique and observe careful housekeeping) , the very noxious thallium and its compounds can be safely used in research programs . After all , considerable thallium chemistry has been reported in the scientific literature ! Since no gases are evolved in this reproportionation of thallium(0) with thallium(III) oxide , the reaction could be carried out in a sealed container to prevent the very toxic , volatile Tl2O3 fumes from escaping into the laboratory , and incidentally maintain the proper stoichiometry in the desired compound .
Unlike the HTS thallium cuprates such as Tl2Ba2Ca2Cu3O10+x (Tc = 122 K) , TlCuO2 would have metallic , mixed-valent Tl2+ and nonmetallic, homovalent Cu(II) . The metallic bond in TlCuO2 would be a Tl 6s1O 2s2 sigma XO . There should be a reasonably strong AFM induction in TlCuO2 from the copper oxide layers into the TlO metallic bond free electrons . The following sketch shows the TlO layers in TlCuO2 alternating with the CuO layers :
TlCuO2 is predicted to be a genuine HTS compound , with Tc > 77 K , possibly ~ 120 K or so .
The thallium triad is Tl3+Tl1+Tl3+ , with 6s06s26s0 = 6s06s06s0 (e22-) . The (Tl37+) can be combined with the (CuO2)2- layers in the compound (Tl37+)2[(CuO2)2-]7 , i.e. Tl6(CuO2)7 . This HTS candidate might be synthesized by a reproportionation of Tl0 and Tl(III) , as with TlCuO2 :
4/3 Tl0 + 7/3 Tl2O3 + 7 CuO -------- [heat , sealed container] --------> Tl6(CuO2)7 .
Lead's inert pair is in Pb(II) , with 6s2 ; the corresponding empty lead is Pb(IV) , with 6s0 . The mixed-valent lead(II,IV) cuprate will be Pb2(CuO2)3 , i.e. Pb(II)Pb(IV)[(CuO2)3]6- :
2 PbO + 3 CuO + ½ O2 (g) -------- [heat , air or O2 atmosphere] --------> Pb2(CuO2)3 .
We hope that the crystal structure of this mixed-valent compound will resemble that of SrCuO2 and TlCuO2 shown above . Sr2+ is strictly ionic , as is Tl1+, while Tl(III) is mostly covalent when bonded to oxygen atoms . In the case of TlCuO2 this shouldn't be a problem , as Tl(III) is empty, i.e. 6s0 . Both the Tl1+ and the Tl(III) should nest in between the CuO2 sheets , as shown in the sketch of TlCuO2 above . However , the 6s2 inert pairs in Pb(II) can be difficult to disperse . For example , in the compound Pb3O4 (minium , red lead oxide) they remain intact on the Pb(II) , and as a result minium is a Robin-Day Class I mixed-valent compound , with very distinct Pb(II) [tetrahedral] and Pb(IV) [octahedral] crystallographic sites :
Minium is an insulator (or very poor semiconductor) . It's unfortunately possible that Pb2(CuO2)3 could have this same sort of minium structure , with the same electronic behaviour . Verwey's Rule says that in order for a mixed-valent compound to be electronically active , the metal cations with the two different valences must be in identical crystallographic sites . The lead(IV) in Pb2(CuO2)3 would undoubtedly be octahedrally coordinated by the oxygens , as in minium . Can its Pb(II) also be octahedrally coordinated , and thereby disperse its 6s2 inert pairs into frontier orbitals ? Lead(II) sulfide (galena) has a cubic rocksalt crystal structure in which the 6s2 inert pairs are stereochemically invisible , but PbS is nevertheless a poor semiconductor . It would require some sort of doping to become metallic . Nevertheless , the octahedral coordination of the sulfides about the Pb(II) atoms does indeed force their 6s2 inert pairs into higher energy level frontier orbitals .
Sufficient BaO could be added to the reaction mixture to convert the CuO2 to CuO3 , which would form a box-like lattice in which both Pb(II) and Pb(IV) would be octahedrally coordinated :
3 BaCO3 + 2 PbO + 3 CuO + ½ O2 (g) -------- [heat , flowing air or O2 atmosphere]
--------> Ba3Pb2(CuO3)3 + 3 CO2 (g) .
Ba3Pb2(CuO3)3 should have a perovskite-like structure ; its planar CuO2 layers would be joined together by bridging oxygen atoms , and the axial CuO bonds would be elongated (Jahn-Teller effect) compared to those in the CuO2 layers . The Pb(II,IV) would form alternating planar layers with them , with the barium cations nesting in between the CuO3 and PbO3 layers :
The lead triad is Pb(IV)Pb(II)Pb(IV) , with 6s06s26s0 = 6s06s06s0 (e22-) . It might be incorporated into a layered cuprate as follows : (Ba2+)x(Pb310+)[(CuO3)4-]x ; (2x + 10)+ = (4x)- ; x = 5 , so the lead triad compound would be Ba5Pb3(CuO3)5 . Let's have a valence-counting check on that : (Ba2+)5 (Pb4+Pb2+Pb4+) (Cu2+)5 O15 30- . Correct ! A suggested synthesis :
5 BaCO3 + 3 PbO + 5 CuO + O2 (g) -------- [heat , flowing air or O2 atmosphere]
--------> Ba5Pb3(CuO3)5 + 5 CO2 (g) .
The barium cations act as controllers, as in Verwey's controlled valence process for synthesizing mixed-valent compounds . They electrically balance the system , and should prevent any excess oxidation of the lead(II) atoms . The Ba2+ cations are ionically bonded between the layers , in which there is covalent PbO and CuO bonding . Although the oxygen atoms in the above formula are written as combined with the copper (CuO3) , they are actually shared with the lead atoms .
Bismuth(III) is similar to lead(II) in requiring an octahedral coordination to pop its 6s2 inert pairs up into frontier orbitals . This occurs in the cubic perovskite Ba0.6K0.4BiO3 (Tc = 29.8 K) . Therefore , the box-like (CuO3)4- units must be used with it . The bismuth triad is Bi(V)Bi(III)Bi(V) , i.e. Bi(4.33+) , with 6s06s26s0 = 6s06s06s0 (e22-) . It might be incorporated into a layered cuprate as follows : (Ba2+)x(Bi313+)[(CuO3)4-]x ; (2x + 13)+ = (4x)- ; x = 6.5 , so the bismuth triad compound would be Ba6.5Bi3(CuO3)6.5 , or Ba13Bi6(CuO3)13 , which has a correct valence count . A suggested synthesis :
13 BaCO3 + 3 Bi2O3 + 13 CuO + 2 O2 (g) ---------- [heat , flowing oxygen atmosphere] -----------> Ba13Bi6(CuO3)13 + 13 CO2 (g) .
In the simpler compound Ba2Bi(CuO3)2 Bi(IV) is the mixed-valent bismuth component :
2 BaCO3 + ½ Bi2O3 + 2 CuO + ¼ O2 (g) ----------- [heat , flowing oxygen atmosphere] ------------> Ba2Bi(CuO3)2 + 2 CO2 (g) .
The analogous antimony triad , Sb(V)Sb(III)Sb(V) , i.e. Sb(4.33+) , might similarly be incorporated into the compound Ba13Sb6(CuO3)13 . Sb(III) is much easier to oxidize to Sb(V) , at about 0.671 V , than it is to oxidize Bi(III) to Bi(V) , at 1.759 V . Antimony has been used in various superconducting compounds , with modest results . For example , BaPb0.75Sb0.25O3 [with mixed-valent Sb(4.0+)] had a low Tc = 3.5 K , compared to the bismuth analogue , BaPb0.7Bi0.3O3 [Tc = 13 K, with Bi(4.0+)] . The metallic SbO layer could really use some strong AFM induction from an adjacent CuO layer !
Revisiting the thallium triad , Tl3+Tl1+Tl3+ , a mixed-valent thallium cuprate in which the thallium atoms all have an octahedral coordination , i.e. with the box-like (CuO3)4- units , is : (Ba2+)x(Tl37+)[(CuO3)4-]x ; (2x + 7)+ = (4x)- ; x = 3.5 , so the thallium triad compound would be Ba3.5Tl3(CuO3)3.5 , or Ba7Tl6(CuO3)7 , which has a correct valence count . A suggested synthesis :
7 SrO + 4/3 Tl0 + 7/3 Tl2O3 + 7 CuO -------- [heat , sealed container] --------> Sr7Tl6(CuO3)7 .
In this case , reagent grade strontium oxide (99.5% pure) is substituted for the BaCO3 ; no gases are evolved , so the reaction could be carried out in a sealed container to prevent the volatile and highly toxic Tl2O3 fumes from escaping into the laboratory .
In the simpler compound SrTlCuO3 Tl(II) replaces half of the Sr2+ cations in Sr2CuO3 :
SrO+ 1/3 Tl0 + 1/3 Tl2O3 + CuO -------- [heat , sealed container] --------> SrTlCuO3 .
La2CuO4 is known to be strongly AFM (TN ~ 315 K) . Could it be used as the AFM induction layer with these mixed-valent metallic oxide layers ? For the thallium triad we have : (La3+)x(Tl37+)[(CuO4)6-]x ; (3x + 7) + = (6x)- ; x = 7/3 , so La7/3Tl3(CuO4)7/3 , i.e. La7Tl9(CuO4)7 is the target compound :
7/2 La2O3 + 2 Tl0 + 7/2 Tl2O3 + 7 CuO -------- [heat , sealed container] -------->La7Tl9(CuO4)7 .
Similarly for the lead triad : (La3+)x(Pb310+)[(CuO4)6-]x ; (3x + 10) + = (6x)- ; x = 10/3 , so La10/3Pb3(CuO4)10/3 , i.e. La10Pb9(CuO4)10 is the target compound :
5 La2O3 + 9 PbO + 10 CuO + 3 O2 (g) ------ [heat , air or O2 atmosphere] -------> La10Pb9(CuO4)10 .
In the simpler compound LaPbCuO4 Pb(III) replaces half of the La3+ cations in La2CuO4 :
½ La2O3 + PbO + CuO + ¼ O2 (g) -------- [heat , air or O2 atmosphere] --------> LaPbCuO4 .
Similarly for the bismuth and antimony triads : (La3+)x(Bi313+)[(CuO4)6-]x ; (3x + 13) + = (6x)- ; x = 13/3 , so La13/3Bi3(CuO4)13/3 , i.e. La13Bi9(CuO4)13 is the target compound :
13/2 La2O3 + 9/2 Bi2O3 + 13 CuO + 3 O2 (g) -------- [heat , oxygen atmosphere]
--------> La13Bi9(CuO4)13 .
Copper(II) oxide derivatives would be unsuitable as AFM induction agents for chemically reducing triads , for example the tin triad , Sn(IV)Sn(II)Sn(IV) , with 5s05s25s0 = 5s05s05s0 (e22-) . A simple redox analysis indicates that the mildly oxidizing copper(II) will oxidize the mildly reducing tin(II) , but the redox-inert nickel(II) won't :
Sn(II) 2e- ---------------> Sn(IV) ; E0ox = 0.151 V ;
2 Cu2+ + 2e- ---------------> 2 Cu1+ ; E0red = + 0.153 V ;
Net : Sn(II) + 2 Cu2+ ---------------> Sn(IV) + 2 Cu1+ ; E0T = + 0.002 V .
But , Ni2+ + 2e- ---------------> Ni0 ; E0red = 0.257 V ;
Net : Sn(II) + Ni2+ ---------------> Sn(IV) + Ni0 ; E0T = 0.408 V .
Nickel(II) oxide is quite strongly antiferromagnetic , with TN = 525 K . Its ternary oxides are also AFM , so they might be tried as induction agents with the tin triad , which could conveniently be obtained by reproportionating tin metal powder and tin(IV) oxide to Sn(3.33+) :
5 BaCO3 + ½ Sn0 + 2½ SnO2 + 5 NiO -------- [heat , flowing nitrogen or argon atmosphere]
--------> Ba5Sn3(NiO3)5 + 5 CO2 (g) .
A flowing atmosphere of pure , dry nitrogen or argon is recommended to protect the powdered tin from air oxidation , and to sweep out the by-product carbon dioxide , thereby helping the BaCO3 to decompose and drive the reaction to completion (Le Chatelier's Principle) .
It might be possible to protect the air-sensitive tin without nitrogen or argon by heating the reaction mixture in a graphite crucible in air , with a layer of graphite powder tamped down over it . Graphite is highly refractory , and should be chemically inert to the underlying reagents . Since this graphite blanket could inhibit decomposition of the barium carbonate , SrO could be substituted for it as with the strontium thallium cuprates above . The loose graphite powder would be tapped off the top of the solid , sintered reaction mass after it has been cooled to room temperature . Annealing of the mixed-valent tin compound's pellet should also be carried out either under an inert atmosphere or under a protective layer of graphite , to ensure the viability of the tin's 5s2 inert pairs , now dispersed in the SnO metallic bond .
La2NiO4 is strongly antiferromagnetic (TN ~ 320 K) , so it could be tried as the AFM induction layer with the tin triad metallic oxide layer : (La3+)x(Sn310+)[(NiO4)6-]x ; (3x + 10) + = (6x)- ; x = 10/3 , so La10/3Sn3(NiO4)10/3 , i.e. La10Sn9(NiO4)10 is the target compound :
5 La2O3 + 3/2 Sn0 + 15/2 SnO2 + 10 NiO -------- [heat , nitrogen or argon atmosphere , or graphite layer] --------> La10Sn9(NiO4)10 .
LaSnNiO4 would have Sn(III) , which is found in the metallic rocksalt compound SnP (Tc ~ 3 K ) :
½ La2O3 + ¼ Sn0 + ¾ SnO2 + NiO -------- [heat , nitrogen or argon atmosphere , or graphite layer] --------> LaSnNiO4 .
Note : commercial nickel(II) oxide is often black in appearance . This type of NiO has been slightly oxidized , and contains some Ni(III) , which is oxidizing in nature (1.17 V to Ni2+) . Only pure , green , stoichiometric , non-oxidizing NiO should be used in these proposed reactions .
In the above chemical reactions the smaller Sr2+ (from SrCO3 or SrO) could be substituted for the Ba2+ (from BaCO3) as the controller cation, probably with a beneficial effect on Tc . The medium sized strontium cations (1.44 Å , 12-coordinated , per Shannon-Prewitt) should shrink the crystal lattice relative to the larger (1.61 Å) barium cations . This in turn should slightly reduce the coherence length of the free electrons and hopefully raise Tc a little .
The Bilayer Metallic Bond in HTS Compounds
Anions can play a crucial role in the electronic structure of metallic compounds . In elementary metals , their alloys , most intermetallic compounds , and the Robin-Day Class IIIB synthetic metals (such as Alchemist's Gold and silver subfluoride) there is a direct metalmetal bonding , without any anion participation . Electrons in such metallic bonds are subject to the Fermi-Dirac distribution , in which most of their contributing valence electrons are located in lower energy levels below EF as spin-paired couples . Only a few of the valence electrons remain as singlet electrons above EF . For example , in sodium metal , about 99% of the sodium atoms' 3s1 valence electrons are paired off below EF , while the remaining 1% are above EF as singlets . While all the 3s1 electrons contribute to the electrical conductivity in sodium , only those above EF are responsible for its other properties such as high reflectivity , thermal conduction , and Pauli paramagnetism . It's these upper level singlet electrons that must be condensed into Cooper pairs in order for the metal to become superconducting (which apparently sodium never does ; its Pauli paramagnetism is too strong) .
In metallic bonds with participating anions the electronic situation is dramatically different . The anions provide some of their valence shell orbitals and electron pairs to the XO , but as electron pairs they are by definition in the energy levels below EF . The metal atoms' singlet electrons then occupy the higher energy levels above EF . This bilayer metallic bond is illustrated for rhenium trioxide , a homovalent compound with an amazingly high electrical conductivity :
I use a simple picture version of the classic Valence Bond (VB) theory to portray the covalent bonds in molecules and infinite atomic lattices . I also use a simple picture version of the Molecular Orbital (MO) theory to visualize their metallic bonds . First , picture VB is applied to determine the lower energy covalent bond framework or skeleton of the solid ; then any leftover singlet electrons are located in suitable higher energy frontier (LUMO) orbitals , which are then polymerized into the metallic bond XO enveloping or coating the covalent structure .
The metallic bond in ReO3 is predicted by this picture VB analysis to be the the Re 6py,zO 2py,z p XO , which resembles graphite's p XO , for example . Unlike the anisotropic two dimensional graphite layers , however , the pi XO in ReO3 is isotropic and three dimensional ; and of course , the metallic bond in ReO3 is bilayer , while that one in graphite is monolayer .
Because all of the metal atoms' valence electrons will float above EF in a bilayer metallic bond , it will have a very rich population of mobile , free electrons to carry the electrical charge and energy . This is reflected in the high electrical conductivity of ReO3 , which is 2½ times (149,300 ohm-1cm1) that (58,140 ohm-1cm1) of its parent elementary metal , rhenium , with its closely packed atoms . But equally important for superconductivity in such a compound , the coherence length between the free electrons is very short , essentially the distance between the metal atoms :
These new pairs [in the high Tc cuprates] differ from BCS pairs [in the low Tc classical superconductors] in one respect at least : the distance between the charge carriers of each pair in the new superconductors is much shorter , by a factor of around 100 (G. Vidali) .
We can readily understand from the magnetic force association of Cooper pair electrons discussed above why shorter coherence lengths are so desireable in superconducting materials : the magnitude of the magnetic force is inversely proportional to the square of the coherence length , so as the separation distance between the two antiparallel electrons decreases , their bonding magnetic strength greatly increases . BCS theory also predicts that the transition temperature is a function of the coherence length and the density of states (DOS) , which is the population of metallic bond free electrons above EF . The coherence length will be greatly decreased , and the DOS will be greatly increased in compounds with a bilayer metallic bond , so obtaining compounds with such a feature will be very important in the search for new HTS materials .
In metallic solids with a monolayer metallic bond (elementary metals , their alloys , most intermetallic compounds , and synthetic metals) the Fermi-Dirac distribution in essence makes the formation of stable Cooper pairs with strong magnetic coupling very difficult , if not impossible . The DOS is quite low , with the small population of free electrons in sodium metal , as mentioned above , it's only ~ 1% of the 3s1 valence electrons scattered about the crystal lattice . Statistically , the free electrons have very long coherence lengths , making their magnetic coupling forces virtually nil . The phonons are required to push them together , and if their Pauli paramagnetism is particularly strong (as in sodium) they will never be able to become superconducting . ReO3 , with its bilayer metallic bond , rich DOS , and short coherence lengths fails to superconduct (PDF , 258 KB) even extremely close to Absolute Zero ; it apparently can't get rid of the last traces its Pauli paramagnetism . The tungsten bronzes are similar to ReO3 ; the compound Rb0.14WO3 becomes superconducting only at ~ 5.3 K , which incidently is one of the highest Tc values measured for any bronze compound . We again recall that rule of thumb I mentioned above : a bright , shiny , metallic solid (eg. a bronze) with a high electrical conductivity will be Pauli paramagnetic and a dud as a superconductor . A dull , black metallic solid , eg. YBCO , will be a poor electrical conductor but will have an antiparallel ordering of its free electrons in an AFM matrix , and so will be a HTS material .
The following sketch illustrates the Great Divide between those superconductors having a monolayer metallic bond and those with a bilayer metallic bond . It also shows the effect of the AFM induction strength on the transition temperatures , emphasing that Tc values are dependent on both the nature of the metallic bond and the strength of the AFM induction in it :
In the design of new HTS compounds researchers must ensure that the candidates have both a bilayer metallic bond , i.e. with oxygen atom or anion participation in it , and strong AFM layers for induction of antiparallelism at an elevated temperature in the metallic layers' free electrons . An additional design consideration , that of internal or external AFM induction , is discussed in the following final section .
Internal versus External AFM Induction
AFM induction is an enabling mechanism for the condensation of the free electrons into Cooper pairs at a temperature higher often , substantially higher than they otherwise would in the absence of such a spintronic ordering . The stronger the induction , the higher the transition temperature will be . Antiferromagnetism is usually gauged in terms of its Néel temperature (TN) , the temperature at which the material's magnetic susceptibility is at a maximum value . Below that temperature the singlet electron spins are aligning more and more into an antiparallel spin order , so the susceptibility decreases . The temperature range below TN , i.e. from Absolute Zero up to TN , is the induction range of the AFM component in an HTS material . Therefore , the higher the TN of the material , the stronger its AFM induction capability will be .
In a mixed-valent compound the valence shell electrons are resonating extremely rapidly (~ 108 hz) between adjacent cations in the crystal lattice . However , in an AFM compound one or more of these valence shell electrons , per atom , has the antiparallel spin ordering that makes it AFM . When a nonmetallic AFM compound with homovalent metal cations is chemically converted into a mixed-valent compound , usually by partial oxidation or reduction or by controlled valence doping , its valence electrons are churned up by the resonance and much of its antiferromagnetism is degraded . As discussed in the Antiferro web page , the maximum TN values for AFM compounds are observed in pure , stoichiometric samples of them . Their TN values rapidly decline when they are oxidized or reduced even to a relatively minor extent , making them mixed-valent .
On the one hand , we have to convert the nonmetallic precursor AFM compound into a mixed-valent compound in order for it to become metallic ; but on the other hand , when we do so , much of its antiferromagnetism is ruined and lost . This sort of dilemma arises with internal AFM induction superconductors , in which the metallic bond is created within a nonmetallic AFM precursor . For such internal induction materials there must be a delicate balance between just enough mixed-valence resonance to activate the metallic bond , yet not too much resonance that will destroy the AFM induction . By remarkably good luck serendipity ! that fine balance between the degree of mixed-valency and the competing internal AFM induction was achieved for YBCO by its discoverers C.-W. Chu and M. K. Wu . I believe that YBCO owes much of its success to the fact that it contains almost precisely the copper triad , Cu2+ Cu3+ Cu2+ = Cu3+ Cu3+ Cu3+ (e22-) .
Another problem with the internal induction superconductors is that relatively few AFM precursor compounds can be converted into metallic solids . Copper(II) oxide derivatives immediately come to mind , but after that the repertoire of AFM compounds provides few useful candidates for study . In the Iron web page I suggested the possibility of modifying iron(II) oxide , FeO (TN = 198 K = 75 ºC) into an antiperovskite such as Fe3OSe . It would have the iron triad , Fe1+Fe2+Fe1+ = Fe2+Fe2+Fe2+ (e22-) , with all the Fe2+ base cations in a low spin condition .
In an external AFM induction compound the metallic layers alternate with nonmetallic , but strongly AFM layers . Electrical conduction and therefore superconduction can be optimized in the metallic layers , while the antiferromagnetism can be optimized in the pure , stoichiometric layers of the AFM induction agent .
The scope is much wider for the design and synthesis of external induction superconductors than it is for the internal variety . Any sort of metallic layer an elementary metal , a homovalent metallic solid such as the superb rhenium trioxide discussed above , or a mixed-valent compound such as those of Pb , Sn , Tl , Bi , and Sb , also reviewed earlier can theoretically be combined with alternating layers of suitable AFM compounds . Of course , the optimum metallic components will have a bilayer metallic bond , with their rich populations of free electrons and very short coherence lengths . In practice , chemical compatibilities (in particular those of the redox sort : an oxidizer shouldn't be combined with a reducer !) will restrict and reduce the number of metallic and AFM candidate couples . Nevertheless , this remains a very wide field of superconductor chemistry and physics research for future investigation .
Superconductors can be synthesized in either a bulk form (pellets , powders , granules) or as epitaxial thin films . I've discussed the latter variety in the ReO3-NiO web page , so I won't repeat my comments here . Many combinations of metallic and AFM induction layers might in theory be designed and synthesized as thin films , but not in bulk form . This is because they are metastable as very thin layers in which the atomic components are rigorously separated ; but when the same components are combined together in bulk , using for example the shake-'n-bake procedures discussed above , the atoms can intermingle to provide very different and often undesirable chemical products . In recognizing this metastability problem with certain metallic and AFM induction layer combinations , the researcher will have to draw on his ingenuity and wealth of chemical knowledge to re-create that particular combination in a bulk form . Since the practical commercial and industrial uses of superconductors in the future will require mostly bulk forms of these materials [eg. levitation (transportation) , electrical power transmission , and medical MRIs] , any promising new thin film superconductors should be reproduced in bulk form , and that may prove to be quite challenging ! Thin films will dominate superconductor research in the coming decade , and accordingly thin film researchers will be in the vanguard in the continuing quest to explore superconductivity and in the discovery of many marvelous new HTS materials .
References , Notes , and Further Reading
nine web pages : Approaching an Ambient Superconductor ; Prediction of Superconductivity in Transition Metal Chalcogenide Oxides ; Electron Doping of Transition Metal Pnictides and Chalcogenides ; Exploring Some New Chemistry of Layered Compounds ; Electron-Doped Antifluorites as Superconductors ; Antiferromagnetic Induction in High Temperature Superconductors ; New Layered Compounds for High Temperature Superconductivity ; A Rhenium TrioxideCopper Oxide Layer Compound ; Use of the First Derivative Electrical Resistivity Trace for the Accurate Determination of Superconductor Critical Temperatures.
ebook : Exploring the Chemistry of Metallic Solids , including Superconductors [PDF , 6154 KB (6.00 MB)] .
most people seem to forget : Not everybody , though . In 1933 the eclectic Russian physicist Yakov Ilich Frenkel (1894-1952) tried to explain superconductivity in terms of electromagnetic interactions between the free electrons :
. the normal state must be characterized by an opposite orientation of the resulting orbital and spin angular moments . a metal in the superconducting state must behave like a diamagnetic body with a large negative susceptibility ..
J. Frenkel , On a Possible Explanation of Superconductivity, Phys. Rev. 43 (11) , pp. 907-912 (1933) . The quotes were from p. 911 . The J in J. Frenkel stands for Jacob , an anglicization of Frenkel's Russian name , Yakov . Frenkels paper was received by Physical Review on December 27 , 1932 , and was therefore written before the now-famous Meissner-Oschenfeld magnetic levitation experiment in 1933, which confirmed Frenkels hypothesis concerning the strongly diamagnetic nature of superconductors . However , Frenkel didn't seem to bring his mathematical analysis to any firm conclusion .
The modern-day Chinese physicist Xiuqing Huang (Nanjing University) has taken the magnetic fields of the Cooper pair electrons into account , as well as their electrostatic fields , although in a somewhat different way than in my analysis of the Fm / Fe ratio : Xq. Huang , Real Space Coulomb Interaction : A Pairing Glue for FeAs Superconductors, ArXiv.org , 10 July 2008 (PDF , 294 KB) ; idem. , ibid. , A Unified Theory of Superconductivity, 22 September 2008 (PDF , 1201 KB) .
H.M. Rosenberg : H.M. Rosenberg , The Solid State , Clarendon Press , Oxford (UK) , 1975 ; the quotation is from p. 204 .
twenty-nine : C.P. Poole Jr. , H.A. Farach , and R.J. Creswick , Superconductivity , Academic Press , San Diego , CA , 1995 ; Table 3.1 , Properties of the Superconducting Elements, pp. 60-61 . The twenty-nine elements listed are superconducting at ambient pressure (one atmosphere) ; several more elements become superconducting near Absolute Zero under extremely high pressures . A list of superconducting elements is also provided by Wikipedia .
selenium : V. Johnson and A. Wold , Crystal Growth and Magnetic Properties of Compositions in the CoS2 : CoSe2 System, J. Solid State Chem. 2 (2) , pp. 209-217 (1970) . Se substitution [in CoS2] introduces strong antiferromagnetic interactions between cobalt atoms (p. 216) ; K. Adachi , K. Sato , and M. Takeda , Magnetic Properties of Cobalt and Nickel Dichalcogenide Compounds with Pyrite Structure, J. Phys. Soc. Japan 36 (3) , pp. 631-638 (1969) . CoS2 is ferromagnetic , CoSe2 is antiferromagnetic .
ARPES experiments : S. Souma et al. , The Origin of Multiple Superconducting Gaps in MgB2, Nature 423 (6935) , pp. 65-67 (May 1, 2003) .
Prassides and co-workers : K. Prassides , Y. Takabayashi , and T. Nakagawa , Mixed Valency in Rare-Earth Fullerides, Phil. Trans. R. Soc. A 366 (1862) , pp. 151-161 (2008) [PDF , 294 KB] .
mostly the fcc packing : Crystal structures for various alkali metal fullerides are illustrated by : D.W. Murphy et al. , Synthesis and Characterization of Alkali Metal Fullerides : AxC60, J. Phys. Chem. Solids 53 (11) , pp. 1321-1332 (1992) [PDF , 1331KB] ; Fig. 1 , p. 1323 .
metallic and other chemical bonds : S.T. Matsuo , J.S. Miller , E. Gebert , and A.H. Reis , Jr. , One-Dimensional K2Pt(CN)4Br0.3 . 3 H2O , A Structure Containing Five Different Types of Bonding, J. Chem. Educ. 59 (5) , pp. 361-362 (1982) . The five types of chemical bonds are : covalent , ionic (coulombic, electrostatic) , van der Waals dipolar , hydrogen , and metallic .
Tanigaki and Zhou : K. Tanigaki and O. Zhou , Conductivity and Superconductivity in C60 Fullerides, J. Phys. I France 6 (12) , pp. 2159-2173 (1996) [PDF , 972 KB] .
Gunnarsson : O. Gunnarsson , Superconductivity in Fullerides, Rev. Mod. Phys. 69 (2) , pp. 575-606 (1997) [PDF , 545 KB] .
Emery and co-workers : N. Emery et al. , Synthesis and Superconducting Properties of CaC6, Sci. Technol. Adv. Mater. 9(4) , 044102 , 7 pp. (2008) [PDF , 1270 KB] .
Canfield and Crabtree : P.C. Canfield and G.W. Crabtree , Magnesium Diboride : Better Late Than Never, Physics Today 56 (3) , March , 2003 , pp. 34-40 [PDF , 1743 KB] .
TDAE : R.L. Pruett et al. , Reactions of Polyfluoro Olefins . II . Reactions with Primary and Secondary Amines, J. Amer. Chem. Soc. 72 (8) , pp. 3646-3650 (1950) ; R.W. Hoffmann , Reactions of Electron-Rich Olefins, Angew. Chem. Internat. Ed. Engl. 7 (10) , pp. 754-765 (1968) .
1 : 1 charge-transfer compound : P.-M. Allemand et al. , Organic Molecular Soft Ferromagnetism in a Fullerene C60, Science 253 (5017) , pp. 301-302 (1991) . The compound was actually C60TDAE0.86 , with a ferromagnetic Curie transition temperature of TC = 16.1 K.
Schilder and co-workers : A. Schilder et al. , Microwave Conductivity of the Soft Ferromagnet (TDAE)-C60, Phys. Rev. Lett. 73 (9) , pp. 1299-1302 (1994) .
molecular ferromagnet : E. Tosatti , Fullerides in a Squeeze, Science 323 (5921) , pp. 1570-1571 (2009) [PDF , 546 KB] ; D. Mihailovic , Molecular Magnetism with Fullerene Building Blocks : The Race is on for High Tcs , Europhysics News 28 (3) , pp. 78-81 (1997) [PDF , 406 KB] .
Italian researchers : M. Riccò , M. Bisbiglia , R. De Renzi , and F. Bolzoni , Observation of Superconductivity in TDAE-C60, Solid State Commun. 101 (6) , pp. 413-416 (1997) .
Foley and co-workers : M. Foley , C. Ton-That , and L. Kirkup , Electrical Properties of Pure and Oxygen-Intercalated Fullerene Films, Proceedings : 31st Annual Condensed Matter and Materials Meeting (2007) [PDF , 48 KB] ; see Fig. 1 , p. 2 .
Chinese researchers : Y. Zhang et al. , Ultrathin MgB2 Films Fabricated on Al2O3 Substrate by Hybrid PhysicalChemical Vapor Deposition with High Tc and Jc, Supercond. Sci. Tech. 24 (1) , 015013 (2011) .
ferromagnetic induction agent : F. Nolting et al. , Direct Observation of the Alignment of Ferromagnetic Spins by Antiferromagnetic Spins, Nature 405 (6788) , pp. 767-769 (2000) [PDF , 199 KB] ; L. Joly et al. , Laser Control of Spins in a Ferromagnet/ Antiferromagnet System [PDF , 532 KB] .
EuO has been used in epitaxial thin films : J. Holroyd , Y.U. Idzerda , and S. Stadler , Properties of Thin Film Europium Oxide by X-ray Magnetic Circular Dichroism, J. Appl. Phys. 95 (11) (Pt. 2) , pp. 6571-6573 (2004) [PDF , 47 KB] ; S. Thongchant et al. , Preparation and Physical Properties of EuO Nanocrystals Using Eu(II)-Exchanged Zeolite X as a Precursor, Bull. Chem. Soc. Jpn. 77 (4) , pp. 807-812 (2004) [PDF , 24 KB] ; J.M. Honig and L.L. Van Zandt , The Metal-Insulator Transition in Selected Oxides, Ann. Rev. Mater. Sci. 5 , pp. 225-278 ; R.A. Huggins , R.H. Bube , and R.W. Roberts (eds.) , Annual Reviews , Palo Alto , CA (1975) . EuO is discussed on pp. 266-271 .
definitive review : M.B. Robin and P. Day , Mixed Valence Chemistry A Survey and Classification, Adv. Inorg. Chem. Radiochem. 10 , pp. 247-422 , H.J. Emeléus and A.G. Sharpe (eds.) , Academic Press , New York , 1967 ; P. Day , Mixed Valence Chemistry and Metal Chain Compounds, pp. 191-214 in Mixed-Valence Compounds : Theory and Applications in Chemistry , Physics , Geology , and Biology , D.B. Brown (ed.) , NATO Advanced Study Institute , Series C , Mathematical and Physical Sciences Series no. 58 , Reidel-Holland (Kluwer Academic Publications , Hingham , MA) , 1980 ; P. Day , Les Composés à Valence Mixte, La Recherche 12 (120) , pp. 304-311 (mars 1981) ; A.J. Markwell , Mixed-Valency Compounds, Educ. Chem. 25 (1) , pp. 15-17 (January , 1988) .
E.J.W. Verwey : E.J.W. Verwey , Valence Induite, Bull. Soc. Chim. France , mises au point D122 (1949) [Induced Valence, Chem. Abs. 43 , 6015g (1949)] ; E.J.W. Verwey , P.W. Haayman , and F.C. Romeijn , Electrical Properties of Metallic Oxides Containing Other Metals as Impurities, Chem. Abs. 46 , 3424e (1952) ; E.J.W. Verwey et al. , Controlled-Valency Semiconductors, Philips Research Reports 5 , pp. 173-187 (1950) ; E.J.W. Verwey , P.W. Haayman , and F.C. Romeijn , Physical Properties and Cation Arrangement in Oxides with Spinel Structures II. Electronic Conductivity, J. Chem. Phys. 15 (4) , pp. 181-187 (1947) .
extremely fast : ......the magnetic fields at the octahedral sites [on the iron cations in magnetite] are indistinguishable , indicating an oscillation of valence [electrons] more rapid than 108/sec . On the other hand , at 85 K , the Fe(II) and Fe(III) ions in the octahedral holes can be distinguished as expected for a Class II system ; Robin and Day (op. cit . definitive review above , p. 304) .
inert pair : A.R. West , Basic Solid State Chemistry , John Wiley , New York (1988) ; pp. 106-107 ; idem. , Solid State Chemistry and Its Applications , John Wiley , Chichester (UK) , 1984 ; pp. 314-315 . Inert pairs of electrons in inorganic compounds are similar to the lone pairs of electrons in both inorganic and organic compounds (for example , the two lone pairs on the oxygen atom of the water molecule) . The presence of inert pairs in a crystal structure is a reliable diagnostic of covalent bonding in it . The following table lists a selection of inert pairs found in various elements :
Aleksandrov and co-workers: K.S. Aleksandrov et al. , Tin-Based High-Temperature Superconductor, JETP Lett. 49 (12) , pp. 756-758 (1989) [PDF , 127 KB] .
Cava's lead cuprate : R.J. Cava et al. , Neutron Powder Diffraction Study of Pb2Sr2YCu3O8 , the Prototype of a New Family of Superconductors, Physica C : Superconductivity , 157 (2) , pp. 272-278 (1989) .
stereochemically invisible : S.-W. Ng and J.J. Zuckerman , Where are the Lone-Pair Electrons in Subvalent Fourth Group Compounds ?, Adv. Inorg. Chem. Radiochem. 29 , pp. 297-325 , H.J. Emeléus and H.G. Sharpe (eds.) , Academic Press , Orlando , FL , 1985 .
requiring an octahedral coordination : The Bi(III) 6s2 inert pairs may not be displaced into frontier orbitals even in an octahedral coordination environment . LiBiO2 has a rocksalt-like structure in which Bi(III) has three short and three long bonds to neighbouring oxygen atoms : C. Greaves and S.M.A. Katib , LiBiO2 : a Model for Bi3+ Co-ordination in High Temperature Superconductors, J. Chem. Soc. Chem. Commun. 1989 (14) , pp. 902-903 [PDF , 208 KB] . Greaves and Katib comment : The bond angles are similar to those in BiCl3 , and are consistent with a high degree of s-character in the lone pair orbital (p. 903) . The 6s2 inert pair is one of the longer covalent BiO bonds . Apparently electron resonance in a Robin-Day Class II mixed-valent compound is required , in general , to unpin the inert pairs and disperse them into higher energy level frontier orbitals .
Antimony : R.J. Cava et al. , Superconductivity at 3.5 K in BaPb0.75Sb0.25O3 : Why is Tc So Low? , Nature 339 (6222) , pp. 291-293 (1989) ; S.A. Agnihotry et al. , Superconductivity in an Sb-Incorporated Bi-Sr-Ca-Cu-O (BSSCCO) System, Physica C : Superconductivity 212 (3-4) , pp. 381-388 (1993) ; S.A. Agnihotry et al. , Effect of Calcination Temperature on the Sb-Doped Bi-Sr-Ca-Cu-O (BSCCO) System, J. Mater. Sci. Lett. 13 (4) , pp. 241-244 (1994) ; R.Y. Liu et al. , Effects of Sb and Pb Doping on the High-Tc Phase-Formation in Bi-Sr-Ca-Cu-O Superconductors, Supercond. Sci. Technol. 5 (8) , pp. 482-488 (1992) .
ternary oxides are also AFM : M. Arjomand and D. J. Machin , Oxide Chemistry . Part I . Ternary Oxides Containing Nickel in Oxidation States II , III , and IV, J. Chem. Soc. Dalton Trans. 1975 (11) , pp. 1055-1061 ; magnetic properties of various ternary nickel oxides are reported in Table 3 , p. 1057 . Several of these compounds are appreciably AFM .
Fermi-Dirac distribution : A.R. Mackintosh , The Fermi Surface of Metals, Scientific American 209 (1) , pp. 110-120 (July , 1963) . The electron theory of metals is discussed by W.J. Moore , Seven Solid States , An Introduction to the Chemistry and Physics of Solids , W.A. Benjamin , New York , 1967 ; Ch. 2 , Gold, pp. 41-72 ; see Fig. 2.4 , p. 49 for a sketch of a typical Fermi-Dirac distribution curve .
in sodium metal : A.B. Ellis et al. , Teaching General Chemistry , A Materials Science Companion , American Chemical Society , Washington , D.C. , 1993 ; pp. 191-192 .
G. Vidali : G. Vidali , Superconductivity : The Next Revolution ? , Cambridge University Press , Cambridge (UK) , 1993 ; p. 137 .
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