[Springer Series in Solid-State Sciences] High-Temperature Cuprate Superconductors Volume 166 ||...

2

Crystal Structure

It is well known that, in solids, the atomic structure determines the characterof chemical bonding and a number of other related physical properties. Evensmall changes in structure can considerably change electronic properties ofa solid, for example, as in the Peierls metal–insulator phase transitions.Therefore, the investigation of the crystal structure and its dependence ontemperature, pressure, and composition plays an important role in studyinghigh-temperature superconductors. This is important both for understand-ing the mechanisms of high-temperature superconductivity and in predictingpossible ways to synthesize new superconducting compounds.

Many topologically different types of crystal structure of layered copper-oxide superconductors have been studied (for a review, see [444]). Thesevarious structures can be divided into several families depending on the typeof packing of a small number of structure elements, that is, perovskite-likecopper–oxygen CuO2 layers and buffer rock salt or fluorite blocks [1250,1251].It should be pointed out that besides the type of crystal structure, that is,the long-range order, the properties of cuprate superconductors also dependstrongly on the short-range atomic order, which determines the local chargedistribution in the crystal [214,564].

The structure of quaternary copper-oxide compounds with the generalformula (Ln1−xMx)n+1CunO3n+1−m (where Ln is a trivalent rare-earth ionRE or Y, and M is a divalent alkaline ion, Ba, Sr, or Ca) can be character-ized by the packing of CuO6 octahedrons and an ordered system of oxygenvacancies. The number of copper–oxygen planes is defined by the quantityn = 1, 2, . . .. For n = 1, we have the layered perovskite structure of K2NiF4,while for n → ∞ we get a cubic perovskite ABO3 (see Fig. 2.1). The number ofoxygen vacancies is characterized by the quantity m describing the multiplicityof copper coordination. Since copper readily allows the four-fold coordinationCuO4 in a plane and the five-fold coordination CuO5 in a pyramid besides thesix-fold coordination CuO6 in an octahedron, a large number of perovskite-like structures with oxygen deficiency appear. In these compounds, the copper

14 2 Crystal Structure

Ba or K

Bi

Fig. 2.1. The structure of (K–Ba)BiO3 in the cubic phase (reprinted with permis-sion by APS from Pei et al. [970], c© 1990)

is usually in the state Cuv+ with a formal valence 2 ≤ v ≤ 2.4, so that thenumber m turns out to be related to the concentration x of divalent ions M.

Among various copper-oxide structures, only few of them that have regularCuO2 planes with a particular degree of oxidation become superconducting[1254]. The most studied compounds are La1−xMxCuO4 and YBa2Cu3O7−y

and its modification (see, Table 1.1). The structure of the quinary compoundsin Table 1.1, with a general formula AmM2Can−1CunOx, where A is Bi, Tl,or Hg and M is Sr or Ba, can be also represented as a number n copper–oxygen planes: CuO2–(CuO2–Ca)(n−1), coupled by the buffer blocks (MO)–(AO)m–(MO), which define the oxidation level of the planes. We considerthese structures below in detail.

Here, we should also mention the excess-oxygen-doped La2CuO4+δ com-pound (see, e.g., [1347]). While in LMCO compounds the M2+ doping ionsare essentially immobile, the excess oxygen dopants remain mobile down to200K. Therefore, the disorder produced by the randomly substituted M2+ ionsis “quenched,” while the disorder caused by the mobile intercalated oxygen is“annealed.” In the latter case, we should observe a much weaker influence ofthe disorder on the electronic and other properties of cuprates. However, thephase diagram of this oxygen-doped LCO compound appears to be very com-plicated due to phase separation into oxygen-rich and oxygen-poor regionswith the modulation along the c-axis in the form of stages: The stage-ncompound has a periodicity of n CuO2 layers. The highest superconductingTc � 45K was observed for the n = 2 stage compound.

By applying a high-pressure synthesis technique, a number of Sr–Cu–Ophases was produced. Among them are the so-called infinite-layer struc-ture (Sr1−xCax)1−yCuO2 (Tc = 80–110K) [91, 460] and Sr1−xNdxCuO2

(Tc = 40K) [1180], which contain the CuO2 planes with the Sr (Ca) or Ndlayers between them. Hiroi et al. [459] reported also that the Sr2CuO3+δ andSr3Cu2O5+δ compounds show superconductivity below Tc = 70, 100K, respec-tively. However, these results were questioned in later publications. Accordingto Shaked et al. [1137], the infinite-layer phase is not superconducting and thesuperconductivity observed in previous publications is due to other phases,

2.1 The Structure of Ba1−xKxBiO3 15

Srn+1CunO2n+1+δ included as impurity. In a detailed study of the series ofcompounds Sr2Can−1CunOy, Kawashima et al. [592] suggested that the n = 1member of the series, Sr2CuO3+δ is not superconducting. But they observedsuperconductivity for the phases with n = 2, 3, 4 in the series at Tc = 70,109, 83K, respectively.

Another interesting family of cuprates without apical oxygen is cupric oxy-chlorides Sr2CuO2Cl2 (SCOC) and Ca2CuO2Cl2 (CCOC). They have similarcrystal structure to LSCO (also the tetragonal I4/mmm space group) witha replacement of the La2O2 buffer layer by the Sr(Ca)2Cl2 layer. The SCOCcrystal structure and AF ordering of copper spins are shown in Fig. 3.1b (inthe orthorhombic (Bmba) notation). By applying a high-pressure syntheticroute, Hiroi et al. [461] have achieved superconductivity in CCOC materialdoped with sodium, Ca2−xNaxCuO2Cl2 (Na-CCOC). The compound becomessuperconducting at x > 0.08 with a maximum Tc = 28K at optimal doping,x ∼ 0.20.

New ladder compounds with the general chemical formula Srn−1Cun+1O2n

were synthesized by Hiroi et al. [458] by creation lines of oxygen vacanciesordered in the ideal filled copper–oxygen plane. Superconductivity in the lad-der compounds was observed in a more complicated system with the chemicalformula Sr14−xCaxCu24O41. By applying external pressure, it was possibleto increase the metallic conductivity [881] and to discover a superconductingtransition at Tc � 12K at a pressure close to 3 GPa [1287]. We discuss theladder compounds in Sect. 2.2.3.

Observation of the coexistence of ferromagnetic and superconducting prop-erties in the ruthenate–cuprate compound RuSr2GdCu2O8−δ (Ru-1212) byBernhard et al. [130] attracted the interest of many researches. The structureof the Ru-1212 compound can be viewed as a modification of the Y-123 struc-ture in which the CuO chains are replaced by the RuO2−δ sheets. We considerthe ruthenate–cuprate compounds in Sect. 2.4.3.

By applying external pressure, one can change the lattice parameters ofa crystal, which provides useful information for our understanding of thesuperconducting phase transition. The influence of high pressure on crystalstructure and superconducting Tc is discussed in Sect. 2.6.

We begin with the consideration of the crystal structure of the simplestthree-dimensional oxide compound BaBiO3, which has no copper ions andonly a modest Tc ≤ 40K but its classical perovskite CaTiO3 structure sharesmany features with the superconducting cuprates.

2.1 The Structure of Ba1−xKxBiO3

The crystal structures of BaBiO3-based compounds under replacing Ba byK or Bi by Pb can be described as small distortions of the original perovskitecubic phase Pm3m shown in Fig. 2.1 [970]. In this figure, the thermal factorsare shown as ellipsoids whose dimensions characterize the thermal fluctuations


of the ions in the lattice. For the oxygen ions, the strongly anisotropic thermalfactors with large vibration amplitude in the plane of the cube face impliesthat the lattice is predisposed to structural phase transitions related to therotations of the BiO6 octahedrons. In fact, the freezing out of the rotations ofthe octahedrons (in antiphase to neighboring cells) around the cube axes [001]or [110] gives rise to the tetragonal I4/mcm or orthorhombic Ibmm phases,respectively. The lattice constant increases by a factor of

√2 in the basal x−y

plane, while it gets twice larger along the z-axis. An additional condensationof the breathing phonon mode (variation in the lengths of the Bi–O bonds)in the orthorhombic phase Ibmm decreases its symmetry to monoclinic I2/m[970]. In the latter case, the periodic variation in the lengths of the Bi–Obonds gives rise to a similar periodic variation in the charge on the Bi ions,which can be described as a charge density wave (CDW). This monoclinicphase I2/m is just observable in the pure compound BaBiO3, which turnsout to be an insulator with an optical gap of ∼2 eV due to the formation ofa CDW.

A replacement of Ba by K suppresses the CDW. At a potassium concen-tration x > 0.1, nonequivalent positions of Bi are no longer observed. Thelong-range CDW disappears and the symmetry increases to Ibmm. However,metallic conductivity does not occur up to the transition into the cubic phase(at low temperatures for x > 0.37). Then, in the cubic phase, superconductiv-ity with a transition temperature Tc = 30K is observed. As the concentrationincreases up to x = 0.5, which is the solubility limit for potassium ions in asolid solution, the value of the transition temperature decreases. Under dop-ing, the lattice constant of the pseudo-cubic lattice apc smoothly decreases.Its concentration dependence at room temperature is given by the formulaapc = 4.3548− 0.1743x (A). Since the ion radius of K+ and Bi2+ have similarvalues, 1.64 and 1.61 A, respectively, the decrease in the volume of the primi-tive cubic cell can only be related to the decrease in the radius of the bismuthion as the degree of its oxidation increases. The ionic radii of Bi3+ and Bi5+

are 1.03 and 0.76 A, respectively.A similar sequence of structural phase transitions is observed in another

perovskite compound BaBi1−xPbxO3 when Bi is replaced by Pb. At x >0.05, the symmetry of the monoclinic phase I2/m increases to orthorhombic,although the transition to the metallic state takes place only at x > 0.65when the symmetry increases to tetragonal I4/mcm. In the metallic phase,superconductivity is observed with a maximum value Tc = 13K at x = 0.75.Under further increase of the lead concentration, the transition temperatureTc decreases and at x = 1 a nonsuperconducting orthorhombic phase Ibmmis found with typical metallic properties.

Recently, a new family of bismuth-oxide-based superconductors was dis-covered, based on the SrBiO3 compound. A partial substitution of potassiumor rubidium for strontium induces superconductivity with Tc � 12K forSr1−xKxBiO3 (x = 0.45–0.6) and Tc � 13K for Sr1−xRbxBiO3 (x = 0.5)[593]. The crystal structure of the strontium-based compounds also changes

2.2 The Structure of La2−xMxCuO4−y 17

with doping from the less symmetric, monoclinic P21/n for SrBiO3, to thetetragonal I4/ncm in the superconducting phase, as in BaBi1−xPbxO3.

Thus, despite the relative simplicity of the structure of the cubic per-ovskites, the compounds Ba1−xKxBiO3, Sr1−xKxBiO3, and BaBi1−xPbxO3

reveal several phases with various crystal symmetry depending on the concen-tration x of the doping ions and on the temperature. While the semiconductingproperties of the original compound at x = 0 are readily accounted for bythe formation of the CDW in the monoclinic phase, their persistence inorthorhombic phases, where no nonequivalent positions of Bi are observed,cannot be explained so simply. A possible reason why the semiconductinggap in the electronic spectrum should remain in these phases may be theexistence of local (or incommensurate) CDWs with a small coherence lengththat obscure their observation in diffraction experiments. A specific featureof BaBiO3 compounds is the occurrence of superconductivity near to themetal–insulator transition with decrease of the superconducting transitiontemperature Tc, as the number of charge carrier in the region of normalmetallic properties increases. Such a nonmonotonic dependence of Tc on theconcentration of carriers and the suppression of superconductivity in thenormal metallic phase are the features specific to the copper-oxide supercon-ductors also. It should be also pointed out that the highest Tc in bismuth-basedcompounds occurs in the phase with the highest three-dimensional crys-tal symmetry, cubic in the Ba1−xKxBiO3, as the highest Tc in cupratesis observed in compounds with the most regular copper–oxygen planes asdiscussed below.

2.2 The Structure of La2−xMxCuO4−y

Let us consider the first copper-oxide superconductor La2−xMxCuO4−y

(LMCO or 124), where M = Sr, Ba, or Ca. In the high-temperature tetrag-onal (HTT) phase, these compounds have a structure of the K2NiF4 type,that is, a body-centered tetragonal lattice (I4/mmm− D17

4h). The tetragonalunit cell of this lattice, which has two formula units, is shown in Fig. 2.2a[561]. The primitive unit cell that has one formula unit and is determined bythe vectors a1, a2, and a3 is shown in Fig. 2.2b. The corresponding Brillouinzone in the tetragonal body-centered phase and in the folded Brillouin zoneof the face-centered-orthorhombic phase are shown in Fig. 2.3. The structuralparameters of LMCO crystals for different compositions and their tempera-ture dependence are discussed by Hazen [444]. Typical values of the latticeconstants in the tetragonal phase are at = 3.78 A, ct = 13.2 A. The distanceCu–O1 in the plane is given by at/2 = 1.89 A and the distance Cu–O2 tothe apical oxygen O2 is 2.42 A. The large anisotropic thermal factors for theoxygen ions are noteworthy. They indicate a large vibration amplitude ofthese ions in the tilting type mode for the CuO6 octahedron. As the tem-perature decreases, a structural phase transition from the tetragonal to the


a

O2O1

Cu

La/M

ba2 a1

a3

Fig. 2.2. (a) The structure of La2−xMxCuO4 in the tetragonal phase (after [561]).(b) The tetragonal body-centered I4/mmm unit cell of La2CuO4 with the primitiveunit cell defined by the vectors a1,a2, and a3 and the orthorhombic unit cell Cmcawith ao � co � at

√2, bo � ct (see Fig. 3.1)

KZ

Ky

Kx

Z

X

L

X

Fig. 2.3. The Brillouin zone in the tetragonal body-centered phase (solid lines) andthe folded Brillouin zone in the face-centered-orthorhombic phase (dashed lines). TheX point is folded back to Γ , while the X ′ point is folded back to Z (after [981])

low-temperature orthorhombic (LTO) phase (Cmac − D182h) takes place, with

the doubling of the unit cell volume as shown in Fig. 2.2b. The directionsof the orthorhombic axes (ao,bo, co)Cmac are chosen along the directions[110], [001], and [110] of the HTT phase I4/mmm, respectively, and arerelated to the initial tetragonal phase axes as ao � co � at

√2 = 5.35 A

and bo � ct. Another space group, Bmab, with the lattice parameters


(a′o,b

′o, c

′o)Bmab = (ao,−co,bo)Cmac is often used for the characterization

of the orthorhombic phase. In this case, the orthorhombic axes are related tothose of the tetragonal phase as a′

o � b′o � at

√2 and c′o � ct (see Fig. 3.1).

2.2.1 Structural Phase Transitions in La2−xMxCuO4

The study of the phonon spectrum (see [137, 590]) shows that the structuralphase transition HTT → LTO is due to the condensation of a soft tiltingmode at the X-point of the Brillouin zone boundary. In the LTO phase, twodomains that are related to the rotation of the CuO6 octahedron around thetetragonal axes [110] and [110], respectively, appear (see Figs. 2.7 and 3.1a).Under the rotation of an octahedron in the orthorhombic phase, up to 5◦ atlow temperatures [444], the change in the Cu1–O1 bond lengths in the planeis negligible (less than 0.01%). An accompanying orthorhombic deformationincreases the lattice constant along the direction perpendicular to the rotationaxis, that is, co > ao under the rotation around the ao-axis in Fig. 3.1a. Inthe LTO phase, the compressibility of the La2CuO4 lattice turns out to beanisotropic. At room temperature, for the directions ao, bo, and co (Cmacphase, co > ao) it equals 1.8, 1.5, and 4.1(×10−4 kbar−1), respectively [496].

The temperature of the HTT → LTO structural phase transition T0 inLa1−xMxCuO4−y rapidly decreases with increasing concentration x of thedoping divalent M ions. A typical dependence T0(x) for the case M = Sr isshown in Fig. 2.4 [137], where the phases with the long-range antiferromag-netic order (the Neel state), the superconducting state, the boundary betweenthe insulating and metallic phases, and the region of frozen spin states (thespin glass) at low temperatures are also shown. An additional study of thedependence of T0 on the concentration of the oxygen vacancies y shows that T0

decreases as the value x− 2y increases, that is, as the total number of chargecarriers in a CuO2 plane grows up. In fact, the replacement of La3+ by M2+

increases the number of holes by one, while the formation of a vacancy of O2−

increases the number of electrons by two [604, 858]. Under applied pressure,T0(x) goes down and the orthorhombic phase disappears at pressures abovethe critical pc(x) (e.g., pc = 15kbar for x = 0.12 in La2−xSrxCuO4 [604,858]).

More careful studies of the structure of La2−xBaxCuO4 have revealedanother phase transition in the region of low temperatures and concentra-tion x ∼ 0.12 from the orthorhombic phase to the low-temperature tetragonal(LTT) phase, P42/ncm (D16

4h) [90,677,678,1093,1204,1205]. The LTO → LTTphase transition is driven by condensation of the soft optical phonon modeat the Z-point of the Brillouin zone. The temperature of the superconductingtransition in the LTT has a minimum around the concentration x � 0.12. Thesharp decrease of Tc in the LTT phase is correlated with anomalous changes inother electronic properties such as the conductivity, the Hall effect, the ther-moelectric power [677,678,1093,1204,1205]. A freezing of the copper magneticmoments in the LTT phase was also observed by the μSR-method.


600

500

400

300

200

100

00 0.1

Spin-glass

Metal Superconductor

Sr concentration, x

Orthorhombic

Tetragonal

Tem

pera

ture

(K

)

Neelstate

Insula-tor

0.2 0.3 0.4

Fig. 2.4. The temperature–concentration phase diagram of La2−xSrxCuO4 (after[137])

A much weaker decreasing of Tc was observed also in the La2−xSrxCuO4

compound at x ∼ 1/8 = 0.125, which may be explained by incipient LTO–LTTstructural transformation. The latter reveals as the softening of the Z-pointphonon in the LTO phase [137]. In the subsequent inelastic neutron scatter-ing study of the soft Z-point phonon in a single crystal of La2−xSrxCuO4, itwas shown that at concentrations x = 0.15, 0.18 the softening of the Z-pointphonon breaks at Tc, while for lower concentrations, x = 0.10, 0.12, the soften-ing continues below Tc [607]. This observation revealed a certain competitionbetween the LTO–LTT transition and the appearance of superconductivity.Kimura et al. [607] also observed an incommensurate splitting of the centralpeak along the [1, 1, 0] direction in the HTT phase at temperatures muchhigher than the HTT–LTO transition Ts1 = 240 (125)K at x = 0.12 (0.18).The central peak originates from atomic displacements in the HTT phase andit is usually considered as a precursor of short-range order of the LTO phase.Its incommensurate splitting 2ε ∼ 0.24 r.l.u. (reciprocal lattice units) impliesthat incipient lattice modulation appears at very high temperature which maybe coupled with the incommensurate spin modulation (see Sect. 3.2.4).

In a subsequent series of elastic neutron scattering measurements by Fujitaet al. [363] on 1/8-hole doped La1.875Ba0.125−xSrxCuO4 (LBSCO) single crys-tals with x = 0.05, 0.06, 0.075, and 0.085, it has been shown that the CDWorder in the LTT phase is responsible for the suppression of superconductivity.


Fig. 2.5. (a) T–x phase diagram for La1.875Ba0.125−xSrxCuO4 of the superconduct-ing (Tc – closed circles) and the structural phase transitions: LTO–LTT (Td2 – opencircles) and LTT–LTO2 (LTLO). Concentration dependence of the normalized inte-grated peak intensities for (b) the structural phase transition LTO–LTT, (c) CDW,and (d) SDW transitions. The data at x = 0.12 are referred to La1.88Sr0.12CuO4

(after [363])

Figure 2.5a shows the phase diagram of the superconducting (Tc – closedcircles) and structural phase transitions (Td2 – open circles) in the LBSCOcompound. The superlattice peak intensities describing the LTO–LTT, CDWand spin-density-wave (SDW) transitions are shown in Figs. 2.5b–d, respec-tively. In some interval of Sr-ion concentration x > 0.05, an intermediatephase, the second low-temperature orthorhombic phase LTO2 (denoted inFig. 2.5a as LTLO – low-temperature less-orthorhombic), space group Pccn,is observed. It also originates from the oxygen octahedron rotation as we willdescribe below. While in the LTO phase the superconducting Tc does notdepend on the Sr concentration, it abruptly drops in the LTO2 phase belowx < xc ∼ 0.09 and remains low in the LTT phase. This anomalous behaviorin the LTT and LTO2 phases can be explained by appearance of a CDWorder in these phases as shown in Fig. 2.5c. The static CDW (and SDW)order appears at a temperature close to the structural phase transition intothe LTT phase at Td2. The neutron scattering study of a pure LBCO singlecrystal, La1.875Ba0.125CuO4, by Fujita et al. [365] confirmed the simultaneousappearance of static CDW and SDW but at temperature Tcs = 50K, a littlebit lower than the temperature Td2 = 60K of the LTO–LTT structural phasetransition. A substantial oxygen isotope effect on the LTO–LTT transitionwas observed in LSBCO by Wang et al. [1339]. A substitution of 16O by 18Oresulted in an increase of the structural phase transition temperature and


in suppression of the superconducting transition temperature Tc. This showsthat the static CDW stripe phase pinned by the LTT phase competes withthe superconducting transition (see Sect. 6.3).

The incommensurate charge order in LBSCO compounds was investigatedlater by Kimura et al. [608] with much higher precision in the synchrotronX-ray diffraction study. Two single LBSCO crystals were investigated withx = 0.05 in the LTT phase and with x = 0.075 in the LTO2 phase. Thesuperlattice peaks show an incommensurate CDW for both of the crystalswith the propagating wave vectors Qch = ±0.24∓ η (r.l.u.) with η = 0 (0.007)for x = 0.05 (0.075) and two-fold charge periodicity along the c-axis. Here,the reciprocal lattice units (r.l.u.) are defined in the HTT (I4/mmm) symme-try. In the second orthorhombic LTO2 phase, the propagating wave vectoris shifted, which indicates that the pattern of the charge order is pinnedby the crystal structure. The high resolution synchrotron X-ray diffractionenables one to measure the correlation lengths for the CDW. The CDWshows a two-dimensional character with the correlation length in the planeξa � 130 (120) A and ξb = 110, (70) A for the x = 0.05 (0.075), where aand b are the directions along and perpendicular to the propagation vectorQch. The correlation length along the c-axis is much smaller, ξc ∼ 9 A. Thesestudies unambiguously show that the long-standing “1/8 problem” of anoma-lous suppression of Tc in La-124 compounds at this hole concentration can beexplained by appearance of the static CDW, which is pinned by the LTT orLTO2 crystal structure.

This explanation is confirmed by a more detailed study of structural phasetransitions in La1−y−xREySrxCuO4 compounds in which the La3+ ions arereplaced by the rare-earth ions RE3+ = Nd, Sm, Eu, Gd of a smaller radius[183,184]. The phase diagram for RE = Nd compound at y = 0.4 [253,254] andRE = Eu at y = 0.2 [617] are shown in Fig. 2.6. For the RE-doped compounds,the LTT phase exists in a broad region of the Sr concentrations, which enablesone to study this phase at different concentration of doped holes. Figure 2.6ashows the Tc(x) dependence for Nd-compound in the LTO phase, Tc (LTO),(at y = 0) and in the LTT phase, Tc (LTT) (at y = 0.4). Similar to the LBCOcompound, Tc(x) has a minimum at x = 0.12, which is deeper in the LTTphase. For a small Sr concentration, there appears the second low-temperatureorthorhombic phase Pccn, LTO2, as in the LBSCO compound.

A detailed neutron scattering study of CDW and SDW in the LTT phase ofLa1.6−xNd0.4SrxCuO4 compound was performed by Tranquada et al. [1259–1261]. They discovered a cooperative spin and charge order in which dopedholes spatially segregates into stripes that separate antiphase AF domains.The incommensurate spin fluctuations were observed also in the excess-oxygen-doped La2CuO4+y single crystal [1347]. We discuss these results inSect. 3.2.4. in connection with study of spin correlation in La-copper-oxidecompounds.

According to Buchner et al. [182–184], the bulk superconductivity in theLTT phase exists only if the tilting angle Φ < Φc � 3.6◦. At fixed concentration


Fig. 2.6. T–x phase diagrams for La2−y−xREySrxCuO4. (a) For RE = Nd at y = 0.4the phases LTO (D18

2h), LTO2 (Pccn), and LTT (D164h) are shown. The superconduct-

ing temperatures Tc(x) are given in the LTO phase, Tc(LTO), at y = 0 and in theLTT phase, Tc(LTT), at y = 0.4 (after [253]). (b) For RE = Eu at y = 0.4 thephases HTT, LTO, and LTT are shown. Depending on Sr concentration x, long-range (AF LR), short-range (AF SR), and static-stripe (AF SS) antiferromagnetic(AF) phases are found. Bulk superconductivity (SC) in the LTT phase is observedonly for x > 0.18 (after [617])

of holes (Sr2+), the superconducting transition temperature Tc(y) smoothlydecreases in the sequence of phase transitions LTO → LTO2 → LTT. In theintermediate LTO2 phase (Pccn), the oxygen ion displacements can be repre-sented as a sum of the displacements in the two domains in the LTO phase butwith different amplitudes. Starting from one domain of the LTO phase andsmoothly increasing the contribution for the oxygen displacements from thesecond domain, one can get a continuous phase transition from LTO to LTTphase through the Pccn phase, while a direct LTO → LTT phase transition isof the first order. The continuous phase transition LTO → LTO2 → LTT canbe described as a smooth change in the direction of the rotation axis for theCuO6 octahedron from [110] in the LTO phase to [100] in the LTT phase (inthe tetragonal HTT phase notations). Since the tilting angle of rotation doesnot change notably at the LTO → LTT phase transition, Crawford et al. [254]have pointed out that just the change of the rotation direction should explainthe suppression of Tc at fixed concentration of holes (Sr2+) in the consideredsequence of phase transitions.

Strong dependence of Tc on the direction of the CuO6 octahedron rotationaxis was confirmed in studies of the phase diagram for Eu-doped LSCO mate-rial investigated by Klauss et al. [617] with μSR technique shown in Fig. 2.6b.It appears that the temperature Td2 � 130K of the structural phase transitionLTO → LTT at Eu concentration 0.2 only weakly varies with Sr concentra-tion, while the magnetic order changes drastically from the AF long-range


order to a modulated phase (AF static stripes) depending on Sr concentra-tion. Superconductivity in the LTT phase is strongly suppressed and it canbe observed as a bulk property only for large Sr concentration, x > 0.18,when the tilting angle Φ becomes smaller then the critical value, Φc � 3.6◦

[183,184].The electronic property changes in the LTT phase can be related to the

appearance of nonequivalent positions of the O1 oxygen ions in the Cu–Oplane. In the orthorhombic phase, under the rotation of CuO6 octahedronaround the ao-axis (Fig. 3.1) (or co for the other domain) all the four O1oxygen ions move out of the plane, as is shown in Fig. 2.7a,b. In this case,the variation of the crystal field potential, which is proportional to the squareof the displacement of the O1 ion is the same for all the four O1 ions. Inthe LTT phase, which can be represented as a sum of the displacements inthe two domains of the LTO phase shown in Fig. 2.7a,b, the rotation of theoctahedron occurs around the tetragonal axes, that is, the x-axis in one of theCuO2 planes and the y-axis in the neighboring planes. Only two of the four O1ions move out of the plane as shown in Fig. 2.7c. This gives rise to a variationin their crystal field potentials and to a local charge redistribution. In thiscase, the formation of CDW [105] and a gap in the electronic spectrum [983]occur, which should induce considerable changes in the electronic propertiesobserved in the LTT phase.

So, one may conclude that the static CDW formation in the LTT (orLTO2) phase close to the hole doping x = 1/8 is responsible for strongsuppression of superconductivity and appearance of SDW. We consider inSect. 3.2.4 the relevance of the incommensurate static and dynamic SDW

Fig. 2.7. The displacement of the O1 oxygen ions in the two domains of theorthorhombic phase (a, b) and in the low-temperature tetragonal phase (c)


(or stripes) in La-124 compounds to their electronic properties and, in par-ticular, to mechanism of high-temperature superconductivity in cuprates. Itwas suggested that dynamical stripes should exist in other copper-oxide com-pounds at small doping and play an essential role in cuprate superconductors(see, e.g., [615, 1079]).

2.2.2 Theory of Structural Phase Transitions

Studies of the La2CuO4-based copper-oxide superconductors reveal consider-able anomalies in the elastic characteristics of the crystals under structuralphase transitions [828, 1372]. In these compounds, structural phase transi-tions may also influence their electronic and magnetic properties. In thisrespect, we shall consider a phenomenological theory of structural phasetransitions based on a symmetry analysis and the Landau expansion for thefree energy [24, 999, 1001]. The interplay of structural phase transitions andantiferromagnetic ordering in La2CuO4 is discussed in Sect. 3.2.3.

The sequence of structural phase transitions in La2CuO4 from the HTTphase (D17

4h) to the LTO phase (D182h, LTO2 – Pccn) and to the LTT phase

(D164h) can be described as a series of successive condensations of a two-

component order parameter C1, C2. Namely, C1 = 0, C2 = 0 in the LTOphase, C1 > C2 = 0 in the LTO2 phase and C1 = C2 = 0 in the LTT phase.Figure 2.7 shows that the condensation of a soft mode related to the rotationof CuO6 octahedron

C1 ∝ 〈R1(kx(1))〉, C2 ∝ 〈R2(kx(2))〉 (2.1)

corresponds to a two-component order parameter. In (2.1), R1,2 = Rx ∓ Ry

is the rotation of the octahedron around the [110] or [110] axis for the wavevectors kx(1, 2) = (π/a)(±1, 1, 0), respectively. The wave vectors kx(1, 2) forma two-arm star of the wave vector at the X-point of the Brillouin zone of thebody-centered tetragonal lattice (Fig. 2.3).

The primitive cell of the body-centered tetragonal phase, which containsone LMCO formula unit, is given by the translation vectors

a1 = (−τ, τ, τz), a2 = (τ,−τ, τz), a3 = (τ, τ,−τz),

where 2τ = at = a and 2τz = ct are the lattice parameters of the body-centered unit cell shown in Fig. 2.2b. The reciprocal-lattice vectors are deter-mined by the relations

b1 = π

(0,

1τ,

1τz

), b2 = π

(1τ, 0,

1τz

), b3 = π

(1τ

,1τ, 0

).

With this notation,

kx(1) =12b3 =

π

a(1, 1, 0), kx(2) =

12(b1 − b2) =

π

a(−1, 1, 0). (2.2)


The sum of the two vectors, kx(1) + kx(2) = b1 − kz , is equivalent to thevector kz = (π/τz)(0, 0, 1) within the reciprocal-lattice vector b1. Thus, thetwo-component order parameter determines two domains. Each of them isrelated to the irreducible representation X+

3 for the wave vectors kx(1) orkx(2), respectively.

The expansion of the free energy in terms of the order parameter (OP) canbe constructed as a function of the corresponding invariants, I1 = (C2

1 + C22 )

and I2 = C21C2

2 . It is convenient to write the Landau expansion in the form

Fc =12r(C2

1 + C22 ) +

12uC2

1C22 +

14v(C4

1 + C42 ) + · · · , (2.3)

where r = a(T − T0), u, and v are phenomenological constants. The thermo-dynamic potential (2.3) for the two-component order parameter describes awide class of structural transitions (see, e.g., [180]). In the tetragonal phase,the strain contribution to the free energy is

Fε =12C11(ε21 + ε22) + C12ε1ε2 + C13(ε1ε3 + ε2ε3)

+12C33ε

23 +

12C44(ε24 + ε25) +

12C66ε

26, (2.4)

where Voigt’s notations are used for the strain tensor εμ and the elasticcoefficients Cμν of the crystal. The symmetrized square of the irreduciblerepresentation X+

3 contains the invariants (C21 + C2

2 ) and (C21 − C2

2 ). Thisdetermines the interaction of the order parameter with the strains in the form

FCε = [α(ε1 + ε2) + βε3](C21 + C2

2 ) + γε6(C21 − C2

2 ), (2.5)

where higher-order terms are omitted. We point out that the interaction of thetype λε6C1C2 is forbidden since the product C1C2 belongs to the irreduciblerepresentation with wave vector kx(1) + kx(2) = kz = 0.

For second-order phase transitions, the equilibrium values of the orderparameter and the strains εμ are found from the minimum of the Landau freeenergy

∂F

∂Ci= 0,

∂F

∂εμ= 0,

where the full free energy is equal to the sum of the contributions (2.3)–(2.5).An analysis of these equilibrium conditions also enables one to determine thejumps in the elastic coefficients under a structural phase transition. We shallbriefly summarize the results of calculations [999,1001].

The phase transition HTT → LTO occurs when (v−4γ2)/C66 < u. Underthis condition two domains appear, C1 = 0, C2 = 0 or C2 = 0, C1 = 0. In theorthorhombic phase, the elastic coefficients then undergo the jumps

ΔC11 = ΔC12 = −2α2/v,

ΔC13 = −2αβ/v, ΔC33 = −2β2/v (2.6)ΔC66 = −2γ2/v, ΔC44 = 0,


while the equilibrium strains ε1 = ε2, ε3, and ε6 are proportional to the squareof the order parameter, for example, ε6 = −(γ/C66)C2.

The phase transition HTT → LTT occurs when (v−4γ2)/C66 > u. Underthis condition, C1 = C2 = C = 0. The strains ε1 = ε2 and ε3 are propor-tional to the square of the order parameter C2, while the elastic coefficientsexperience the jumps

ΔC11 = ΔC12 = −4α2/(v + u),ΔC13 = −4αβ/(v + u), ΔC33 = −4β2/(v + u), (2.7)ΔC66 = 0, ΔC44 = 0.

By comparing (2.6) and (2.7) with experimental data of the measured veloc-ity of sound in the LMCO crystals and their jumps under structural phasetransitions (see, e.g., [828, 1372]), one can determine the coupling constantsα, β, and γ in (2.5) (see [1003]).

The phase transition LTO → LTT occurs as a first-order transition sincethey are not coupled by space subgroup relationships. The transition tem-perature is determined by the equality of the free energies F in the LTO (atC1 = 0, C2 = 0) and LTT phases (at C1 = C2 = 0). In this case, higherpowers of the invariants in the expansion of the free energy (2.3) start to playa role. For example, Ishibashi [528] proposed a model that contains the orderparameter in sixth power in the form

ΔF = w(I21 − 4I2)I1 = w(C2

1 − C22 )2(C2

1 + C22 ). (2.8)

In the LTO phase, this contribution differs from zero, while in the LTT phaseat C1 = C2, it vanishes. At sufficiently low temperatures when equilibriumvalues of the order parameters become large, the LTT phase can thus be ener-getically more favorable than the LTO phase at w > 0, since the latter containsa large positive contribution (2.8). Ishibashi [528] has successfully describedthe temperature–concentration phase diagram for La2−xBaxCuO4 representedin Fig. 2.5 by assuming a certain dependence of the coefficients u and v on theconcentration x of the doping ions. In other models, this phase diagram wasdescribed by assuming a temperature dependence for the coefficients u and v(see, e.g., [1204,1205]). However, it is difficult to justify such an assumption.To consider the phase transition to the LTO2 (Pccn) phase, one should alsoconsider higher order terms in the free energy (2.3) (see, e.g., [224]).

The phenomenological expansion of the free energy (2.3), (2.5) can beobtained by calculating the free energy on the basis of a microscopic theory.Such a model microscopic theory has been proposed by Flach et al. [340] andPlakida et al. [1003]. In the model, anharmonic vibrations of oxygen ions in asoft tilting mode and their interaction with acoustic phonons are described bythe model Hamiltonian in terms of the local normal modes R1,2 = Rx ∓ Ry.The parameters of the anharmonic model were estimated from the calculationof the ground state energy for the La2CuO4 crystal by the density-functional


–4–6–60

–40

–20

ΔE (

mev

/7 a

tom

s)

0

20

40

–2

LTO

Pccn

–4

–2

0

2

4 4

2

–2

–4–40

–20

0

0

LTT

Q

0 2 4 5

Fig. 2.8. Variation of the ground state energy of the La2CuO4 crystal under a rota-tion of the octahedron in the orthorhombic LTO, LTO2 (Pccn), and low-temperaturetetragonal (LTT) phases as a function of the normal mode amplitude Q (after [983])

method [983]. In the latter calculations, one derives the dependence of theground state energy on the normal mode amplitude Q related to the rotationof the CuO6 octahedron in the LTO and LTT phases as shown in Fig. 2.8.It turned out that the LTT phase has the lowest energy. The sequence ofphase transitions HTT → LTO → LTT can be described as a freezing ofthe soft tilting mode R1,2 = Rx ∓ Ry, (2.1). In the LTO phase, only onecomponent of the soft mode R1 or R2 is condensed, while in the LTT phaseboth the components are condensed, which results in the rotation along x ory tetragonal axis. In the limit of strong anharmonicity of lattice vibrationsin the two-well potential, one can consider the phase transitions as beingthe order–disorder type connected with the ordering of rotations of CuO6

octahedron in the four minima shown in Fig. 2.8. According to Pickett et al.[983], the instability of the tetragonal phase with respect to the rotations ofthe octahedron is due to a competition between repulsive forces in the CuO2

plane and the long-range Coulomb forces determined by the Madelung energy.

2.2.3 Copper-Oxide Ladder Compounds

A new modification of copper-oxide compounds was synthesized by Hiroi et al.[458] under high-pressure by creating oxygen vacancy lines ordered in the ideal


Fig. 2.9. (a) The two-leg ladder SrCu2O3 and (b) the three-leg ladder Sr2Cu3O5

(after [92])

copper–oxygen plane. These materials were called ladder compounds with thegeneral chemical formula Srn−1Cun+1O2n. The first member of the series withn = 3 comprises the two-leg ladder SrCu2O3, the n = 5 compound Sr2Cu3O5

is the three-leg ladder, etc. In Fig. 2.9 [92], we show the two- and three-legladders, where the number of legs is defined by the number of copper chainsstrongly coupled by 180◦ pdσ Cu–O–Cu bonds. The ladders are coupled bythe 90◦ Cu–O–Cu bonds, which are weak due to the orthogonality of the px,py oxygen orbitals at that bonds. This results in some kind of the electronicand magnetic isolation of different ladders, which enables one to study lowdimensional copper–oxygen compounds. Hiroi [462] also synthesized anothermaterial, LaCuO2.5, which contains weakly coupled two-leg ladders.

It turned out that even- and odd-leg ladders have quite different physi-cal properties (for review, see [264, 265]). The ladders with even number oflegs show a large spin gap Δs of the order of the antiferromagnetic (AF)exchange interaction J , namely, Δs/J � 0.5, 0.2, 0.05 for 2-, 4-, and 6-leg lad-ders, respectively. For instance, direct magnetic susceptibility measurementsand 63Cu NMR measurements by Azuma et al. [92] revealed a spin gap ofabout 420K in the two-leg material, SrCu2O3, while for the three-leg lad-der Sr2Cu3O5 a gapless excitation spectrum was observed. The spin gap inthe magnetic excitation spectrum brings about the spin-liquid ground state,where only short range AF spin correlations are observed with the correla-tion length decaying exponentially with the distance. Properties of odd-legladders are much close to that of a single spin-1/2 chain with AF coupling.The spectrum of the magnetic excitation of the odd-leg ladders is gapless andthis results in long-range spin correlations. They can evolve into a true AFlong-range order as for the copper–oxygen plane at zero temperature. Thesequite different magnetic properties were proved by various experiments suchas NMR, neutron scattering, μSR technique.

These simple ladder compounds are insulators and it is difficult to makethem conducting by doping. Hiroi [462] has synthesized another material,


LaCuO2.5, which contains weakly coupled two-leg ladders. This compoundcan be hole doped replacing La by Sr, which bring about metallic conductiv-ity at Sr concentration x � 0.20. The spin gap seems to disappear in metallicphase but superconductivity has not been observed. The theoretical investi-gation predicted quite a different behavior of doped ladder compounds (fora review, see [189]). While even-leg ladders have shown pair correlations fordoped holes and superconductivity, odd-leg ladders should be much closer toa non-Fermi liquid of the Luttinger type. The reasons why, contrary to thetheoretical predictions, superconductivity was not observed in the metallicladder compound La1−xSrxCuO2.5 are unclear (see [265]).

Doping of ladder compounds was successful for the more complicated sys-tem with the chemical formula (Sr,Ca)14(CuO2)10(Cu2O3)7, which show thatit consists of 10 chains (CuO2) and 7 two-leg ladders (Cu2O3). There are sixholes per formula unit: five of which approximately are in the chains and onein the ladders [910]. With increasing of Ca ions concentration (the radii ofwhich are smaller than those of the Sr ions) or under application of externalpressure, it was possible to increase metallic conductivity [881] and to discovera superconducting transition at Tc � 12K at pressure of 3–4GPa [1287]. Itwas observed also that the spin gap, which is quite large at ambient pressure,Δs � 250K, is vanishing at high pressure, P � 3GPa [817]. Experiments witha larger single crystal, however, have shown that an activation-type componentin spin susceptibility that is characteristic for the spin-gap system survives athigh pressure in the superconducting state [366]. They have also observeda superconducting coherence peak (Hebel–Slichter peak), which implies thatthe superconducting gap has no nodes at the Fermi surface.

The large spin gap in the two-leg ladder SrCu2O3 is caused by the strong180◦ pdσ Cu–O–Cu bond along the rung. If the angle of the bond deviatessubstantially from 180◦, the rung coupling decreases and becomes compa-rable with the interladder interaction along the stack in c-direction. As aconsequence, the spin gap can disappear, while the AF long-range orderwill take place. This was observed indeed in the pseudo-ladder compoundCaCu2O3 in which small Ca ions produce bending of the ladder and decreas-ing of the Cu–O–Cu bond angle up to 123◦ (see, e.g., [614]). The crystalstructure of the CaCu2O3 compound can be viewed as corner-shared CuO2

zigzag chains running along the b-axis. They are tilted by nearly 29◦ from astraight Cu–O–Cu bond with the neighboring zigzag chains forming this waypositively and negatively buckled ladders with “kinked” rungs in a-direction.Magnetic susceptibility and neutron-diffraction studies reveal incommensurateAF long-range order below TN = 25K with strong AF exchange interac-tion along the chains, J‖ � 0.17 eV, and much weaker coupling in the rung,J⊥ � 0.09 eV. Experimental (polarized dependent X-ray absorption) and theo-retical (LDA band structure calculation) investigations reveal also a significantcoupling perpendicular to the ladders, in c-direction [606]. This material canbe therefore considered as an anisotropic bilayer system and a candidatefor high-temperature superconductivity at hole doping away from the AF


insulating state investigated so far. Theoretical estimate by Plakida et al.[1016] shows that strong AF interaction along the chains can produce quitehigh Tc � 50K though much smaller, due to large anisotropy of the system, incomparison with ideal copper–oxygen layer materials as the mercury cuprates.As a consequence, this low-dimensional copper–oxygen material is very inter-esting low-dimensional magnetic system where quantum fluctuations play anessential role but strong anisotropy of the material precludes attaining hightemperature of the superconducting transition.

The role of low dimensionality and the interlayer coupling between theCuO2 planes in superconductivity in cuprates were elucidated in studies ofthe artificial YBa2Cu3O7/PrBa2Cu3O7 (YBCO/PrBCO) lattices (see, e.g.,[397, 714, 1266, 1267] and the references therein). With the aid of a speciallayer-by-layer deposition technique, it was possible to obtain films made upby alternating layers of (YBCO)n and (PrBCO)m. In this case, it turnsout that the transition temperature depends both on the thickness of thelayer (YBCO)n and on the distance between the layers, which is definedby the layer (PrBCO)m. Since the PrBCO compound is an insulator, itplays the role of an insulating layer separating the superconducting lay-ers (YBCO)n. As was detected by Triscone et al. [1266], if the thicknessof the Pr-layer exceeds 96 A or m = 8, one can neglect the connectionbetween the superconducting Y-layers. It was revealed that even a YBCOlayer consisting of two unit cells (24 A) resulted in Tc > 50K, while fora one-cell layer (12 A), Tc � 10K. The gradual suppression of Tc as thenumber of unit cells decreases is explained by the growth of fluctuationsof the phase of the superconducting order parameter on crossover to aquasi-two-dimensional system, where a phase transition of the Berezinsky–Kosterlitz–Thouless type should be observed. At the same time, it shouldbe noted that Tc for a superlattice with an equal number of Pr/Y layersproves much higher than for the solid solution Pr0.5Y0.5BCO. For instance,Tc � 50 (60)K for the superlattices with n = m = 1 (2) as compared to Tc ∼20K for the 0.5/0.5 solid solution [1266]. These sublattices, when sub-ject to external magnetic fields, demonstrate rather unconventional stronglyanisotropic properties [1266, 1267]. These investigations again confirm thelocal nature of superconductivity in copper-oxide compounds, which have asmall correlation length of the order parameter. However, a “giant proxim-ity effect” observed in cuprate superconductors, which reveals a supercurrentrunning through very thick barriers in artificial layers, is in conflict withthe standard theoretical picture of small correlation lengths (see, e.g., [174]and the references therein). Interesting results concerning the coexistenceof ferromagnetism and superconductivity were obtained for ferromagnetic-superconducting bilayer structures as in the La2/3Ca1/3MnO3/YBa2Cu3O7−δ

bilayers (see, e.g., [1127, 1181] and references therein), which partly may bedue to a very large magnetic penetration depth in cuprates (see Table 4.6).


2.3 Nd2−xCexCuO4 Compounds

The crystal structure of the Nd2CuO4-based superconducting compoundswith electron conductivity is similar to that of the lanthanum compounds.It is described by the same space group I4/mmm but with the displaced oxy-gen ions O2 from their apex positions to sites on the faces of the tetragonal cell[1248, 1249]. In Fig. 2.10, the tetragonal cell for the compounds Nd(Ce)-124(T′ phase), La(Sr)-124 (T phase), and the mixed compounds Nd(Sr, Ce)-124(T∗ phase) are represented for comparison. In the T∗ phase, the apex oxygenions are maintained only in the layer Nd–Sr, while in the layer Nd–Ce theoxygen ions are shifted to the faces. This reconstruction of the lattice in theT′ phase brings about a corresponding variation in its parameters comparedto the T phase. The length of the tetragonal axis increases by about 4%,at = 3.94 A, while the c-axis decreases by 8%, ct = 12.1 A. The lattice param-eters in the T∗ phase assume intermediate average values between those inthe T and T′ phases. The variation in packing of the O2 ions in the T′ andT∗ phases compared to the T phase can be related to the difference in thesizes of the La and Nd ions. A primitive cell of the T∗ phase contains twoformula unit and coincides with the tetragonal cell in Fig. 2.10 (space groupP4/mmm− D17

4h).The phase diagram of the Nd2−xCexCuO4 and Pr2−xCexCuO4 compounds

are shown in Fig. 2.11. It is qualitatively similar to the phase diagram of theLMCO compounds (Fig. 2.4). At x = 0, the compound displays an AF insu-lating phase (TN � 240K), which is destroyed by doping at much higherconcentration of Ce ions (x � 0.12) than in the LMCO compounds (x � 0.02).

Fig. 2.10. The tetragonal unit cells of the T′, T, and T∗ phases of the electron-doped(Nd-Ce-Sr)2CuO4 compounds (after [1249])

2.4 YBaCuO-Based Compounds 33

500

100

50

AF

TN

a

TC

SC

Met

allic10

T (

K)

5

10 0.1 0.2

x (Ce)0.3

30

10

00 0.10 0.15 0.20

T(K

)

20

b

TC

NdPr

SC Met

allic

Sem

icon

duct

ing

x (Ce)

Fig. 2.11. The temperature–concentration phase diagram for (a) Nd2−xCexCuO4

(after [1282]) and (b) for (Nd-Pr)2−xCexCuO4 (after Takagi et al. [1211])

The superconducting phase (SC) appears in the vicinity of the AF phase. Itexists, however, inside a narrower interval of concentrations than in the LMCOcompounds, 0.13 ≤ x ≤ 0.18 (see Fig. 2.11b), and has a lower transition tem-perature, Tc ≤ 24K. In the T′ and T∗ phases, no structural transitions relatedto displacements of O1 ions in the tilting type modes have been observed. Thiscan be related to the absence of complete CuO6 octahedron in these com-pounds. The absence of the apex oxygen O2 in the T′ phase manifests itselfin a number of the electronic properties of Nd–Ce compounds, for example, ina peculiar dependence of Tc on pressure in T′ and T∗ phases (see Sect. 2.6). Asstudies of mercury compounds show, the role of the apex oxygen in copper-oxide layered compounds is negligible, due to the quite large distance of theapex oxygen to the plane (see Sect. 2.5). This is in strong contrast with theYBa2Cu3O7−y compound in which the electronic properties of the CuO2 planecorrelate with the apex oxygen distance to the plane since the charge transferfrom chains to planes occurs through the apex oxygen ions.

2.4 YBaCuO-Based Compounds

A vast literature is devoted to the study of the compound YBa2Cu3O7−y

(YBCO or Y-123) and its various modifications (see, e.g., [444]). It was thefirst found high-temperature superconductor with a transition temperatureexceeding the boiling point of nitrogen (see Chap. 1). By varying the oxy-gen content, the physical properties of this compound can be changed overa wide range without any significant changes in its structure. A whole classof compounds ABa2Cu3O7−y with A = Y, La, Nd, Sm, Eu, Gd, Ho, Er, andLu was discovered with similar physical properties and Tc ∼ 90K (see, e.g.,[485, 788]). The exception is Pr-123 compound, which shows neither metallicnor superconducting behaviour (see Sects. 3.2.4, 5.1.2, and 5.2.1). As with the


layered copper-oxide compounds, YBCO allows a certain modification of itsstructure through variation of the coupling between copper–oxygen layers. Inparticular, compounds with two chains YBa2Cu4O8 (124) or Y2Ba4Cu7O15

(247), and compounds of type Pb2(Y-Ca) Sr2Cu3O8+y (2123) have been syn-thesized via the modification of Cu–O chain layer into a more complicatedstructure Pb2CuO2. Replacing Cu–O chain layer by RuO2 sheet results inthe magneto-superconductor RuSr2GdCu2O8−δ (see Sect. 2.4.3). All these fea-tures of YBCO have brought about a wide interest in the study of its structureand other properties.

2.4.1 Structure of YBa2Cu3O7−y

The original compound YBa2Cu3O7−y can be synthesized in two structuralmodifications depending on temperature and oxygen content y. The first isthe orthorhombic phase Pmmm(D1

2h). The second is the tetragonal phaseP4/mmm(D7

4h). The elementary cells of these structures with one formulaunit are shown in Figs. 2.12a,b [562]. The main structural parameters in theorthorhombic phase (Fig. 2.12b) (in A) at room temperature and y � 0 arethe following. The lattice constants are a = 3.828, b = 3.888, c = 11.65.The lengths of the bonds are Cu1–O1 �Cu1–O4 = 1.94, Cu2–O2 = 1.92,Cu2–O3 = 1.96, Cu2–O4 = 2.3. The length of the four Cu–O bonds for thefour oxygen ions nearest to the copper both in the plane Cu2–O2, O3 andin the chains Cu1–O1, O4 are approximately the same and correspond tothe lengths of bonds in the CuO2 plane for LMCO compounds. The distancebetween the copper ion in the plane and the apex oxygen, Cu2–O4, (as the

Cu2

Cu1

Y

a

O2

Ba

O1O4

b

Y

O3

Cu2

Cu1

O2

Ba

O1O4

Fig. 2.12. The structure of YBa2Cu3O7−y in (a) the tetragonal and (b) theorthorhombic phases (after [562])


lattice constant c) varies strongly, as the oxygen content decreases under thetransition into the tetragonal phase (see Fig. 2.14). In the tetragonal phase(Fig. 2.12a), oxygen positions O2 and O3 become equivalent with equal bondlength Cu2–O2 and Cu2–O3, while remaining structural parameters are closeto those in the orthorhombic phase.

The orthorhombic phase is observed at low temperatures for an oxygencontent x = 7−y ≥ 6.4. The transition to the tetragonal phase occurs at tem-peratures T ≥ 500◦C when oxygen content starts to decrease together withdisordering of oxygen ions in the Cu1–O1 plane (Fig. 2.12a). It is seen thatYBCO has a typical layered perovskite-like structure with two CuO2 planesseparated by an oxygen free layer of Y ions which are coupled by the bufferlayers Ba–O4–Cu1–O1–Ba–O4. The oxygen O2 and O3 are strongly coupledwith Cu2 in the CuO2 planes, unlike the weakly coupled oxygen O1 in theCu1–O1 chains. Upon heating above 500◦C, the latter oxygen ions diffuseaway from the sample. This enables the oxygen content to be smoothly variedfrom x = 7 (y = 0) to x = 6 (y = 1), when all oxygen ions O1 in chainshave been removed out of the compound. In the latter case, the tetragonalphase persists up to low temperature. At intermediate values of x, the struc-ture of the compound depends on the way in which oxygen is removed [409].Quenching from the HTT phase at x ≤ 6.5 preserves this tetragonal phasewith disordered O1 positions in the Cu1–O1 plane. If the samples are preparedby the lower-temperature Zr-gettered annealing technique, the orthorhombicphase can be maintained up to x � 6.2. In this case, several modifications ofthe orthorhombic phase occur. Besides the OI phase shown in Fig. 2.12b whenx = 7 and all the Cu–O1 chains are filled, an ordered phase OII at x = 6.5 canoccur when alternate chains in the Cu1–O1 plane turn out to be empty. Morecomplicated phases at intermediate values of x are also observed. For exam-ple, at x � 6.35 a phase 2

√2a× 2

√2a occurs when half-filled chains alternate

with chains that are one-quarter filled [1183] Formation of chains of a finitelength was detected in NMR and NQR experiments (see, e.g., [750, 1385]).Theoretical calculations of the x–T phase diagram and studies of the oxygenordering in OII phase were performed within a lattice-gas model described bythe asymmetric next-nearest-neighbor Ising (ASYNNNI) model [272,435] andby band structure calculations [982].

The physical properties of YBa2Cu3O7−y-based compounds depend con-siderably on the oxygen content. The highest value of superconducting tem-perature, Tc = 92K, is attained at the optimal doping in metallic phase atx = 7 − y = 6.92. With decreasing oxygen content, Tc goes down and themetallic phase transforms into the semiconducting phase at y � 0.6. In thelatter phase, long-range AF order appears with a maximum Neel tempera-ture TN � 500K at y = 1 (see Sect. 3.2.1). The way in which Tc depends onx = 7 − y is determined by the type of sample preparation. In Fig. 2.13a, thecurve Tc(x) is plotted. The dots correspond to high-temperature quenchingand the crosses to the lower-temperature Zr-gettering (solid line) [409]. In thelatter case, two plateau are observed at Tc = 90K (for 6.85 < y < 7.0) and


Fig. 2.13. (a) The dependence of Tc on oxygen content x in YBa2Cu3Ox forsamples with oxygen removed by high-temperature quenching (dots) and by lower-temperature Zr-gettering (solid line) (after [409]) and (b) the relation between Tc

and the effective copper valence (after [215])

Tc = 60K (for 6.45 < y < 6.65), which are related to the aforementioned twoorthorhombic phases OI and OII. Thus, the short-range order in the Cu1–O1chain has an essential effect on the electronic properties of the superconduc-tor. This points to a local nature of the doping of the conducting CuO2 planedue to charge (hole) transfer from the Cu1–O1 chains [215,563].

The dependence of Tc and the lattice parameters of the YBCO compoundon the oxygen content has been studied in detail by Cava et al. [215] byapplying the lower-temperature Zr-gettering annealing technique. This tech-nique has permitted a number of low-temperature equilibrium phases to beobtained by varying the oxygen content and has revealed a correlation betweenchanges in electronic properties, in particular, Tc, and structural parameters.Figure 2.14a plots the dependence of the lattice parameters a, b, and c in theorthorhombic phase on the oxygen content x = 7− y [215]. At x � 6.4, underthe transition from the orthorhombic (O) to the tetragonal (T) phase, a con-siderable increase in the c-axis lattice constant is observed. The length of theout-of-plain bond Cu2–O4 (O4 is the apex oxygen in the CuO5 pyramid, seeFig. 2.12, is denoted as O1 in Fig. 2.14b) undergoes equally strong variation,while only a slight variation of the in-plane bond lengths, Cu2–O2, O3, occursas shown in Fig. 2.14b.

In this work, to find the correlation between the change in structuralparameters and electronic properties, a bond valence sum V =

∑i exp[(R0 −

Ri)/Bi] has been calculated for various ions. The sum determines an effective


11.8

11.7

a

a

b

c

c, Å

a,b,

Å

11.6

3.9

3.87

x in YBa2Cu3 Ox6

b

7x in YBa2Cu3 Ox

6

Cu2 - O2

Cu2 - O3

Cu2 - O1

2.5

2.4

2.3

2.0

1.9

1.8

bond

leng

th, Å

Fig. 2.14. The dependence of (a) lattice constants and (b) copper–oxygen bondlengths (Cu2–O2, O3), (Cu2–O4) in YBa2Cu3Ox on oxygen content x (after [215])

valence of a given ion specified by the parameters Ri and Bi. The effectivevalence of copper ions in Cu1 chains has turned out to be linear in x. Itvaries from V = 2.5 at x = 7 to V � 1.3 at x = 6. The effective valence ofCu2 undergoes a downwards jump at the transition from the orthorhombic totetragonal phase, which is accompanied by a correspondingly sharp variationin the length of the Cu2–O4 bond. Figure 2.13b demonstrates a correlationbetween Tc and the effective copper valence. The similar behavior of thesefunctions indicates that Tc is determined by the value of the effective chargein the CuO2 plane. The decrease of Tc from 92 to 60K is due to the trans-fer of (negative) charge of about 0.03 e from the chains to the plane. Thedisappearance of superconductivity at x = 6.45 is connected with a furthertransfer of charge of about 0.05 e to the plane. As the oxygen content furtherdecreases to x = 6, the effective charge of the copper in the plane remainsconstant. The charge transfer from the chains to the plane under oxygendoping was confirmed later in the photoemission experiments (see Sect. 5.2.1,Fig. 5.14).

Thus, detailed structural studies of YBa2Cu3Ox compounds have unam-biguously demonstrated the local nature of the charge transfer from CuOchains to CuO2 planes and revealed drastic changes in the electronic proper-ties of the system, including superconductivity, related to the transfer. Cavaet al. [215] have noted that for samples with the same oxygen content, the tran-sition temperature Tc can vary considerably depending on the oxygen orderingin the Cu1–O1 chains. The positive charge transfer (holes) from chain to planecan occur only when two oxygen positions O1, nearest to a copper site Cu1in the chain O1–Cu1–O1, are filled by oxygen ions. Then, the formal valence


of the Cu1 becomes v > 2, which produces a hole transfer from the planeto oxygen in chains. It explains why the short-range order of oxygen ions inchains is so important for proper hole doping of CuO2 planes.

2.4.2 Modifications of the YBCO Structure

In synthesizing single crystals of YBCO in the orthorhombic phase, poly-domain samples with a twin plane of the (110) type usually appear. In suchtwinned crystals, the anisotropy of physical properties in the (a, b) plane ofthe orthorhombic lattice cannot be studied. In this respect, the synthesis ofuntwinned single crystals YBa2Cu4O8 (or 124) was of a great interest [444]. InFig. 2.15, the structures of three compounds of the Y2Ba4Cu6+nO14+n familyare shown for comparison. The original YBCO-123 structure corresponds ton = 0. The structure YBCO-124 corresponds to n = 2. At n = 1, an inter-mediate structure YBCO-247 occurs. The most important feature of the newmodifications is the appearance of double chains instead of the single Cu1–O1chains in the 123 compound. The oxygen coordination in the chains increasesfrom two (Cu1–O1–Cu1) to three, which stabilizes the entire structure. Dueto this strengthening of the oxygen binding in the chains, the YBCO-124compound can be heated to much higher temperatures (of order 800◦C) with-out any significant loss of oxygen. In the YBCO-124 and -247 compounds,the temperature of the superconducting phase transition attains the valuesTc = 80K and Tc = 87K, respectively [406]. The lower Tc observed in thesemodifications compared to Tc = 92K in YBCO-123 is usually attributed tothe decrease of the effective charge of copper in the CuO2 plane [406] so thesecompounds correspond to the underdoped YBCO-123 material. The synthesisof untwinned YBCO-124 and -247 crystals has enabled a number of interestingstudies of their physical properties to be carried out [181,579].

YBaCu

Fig. 2.15. The crystal structure of the Y2Ba4Cu6+nO14+n compounds at n = 0(YBCO-123), n = 1 (YBCO-247), and n = 2 (YBCO-124) (after Kaldis et al. [579])


There exist several modifications of the YBCO structure obtained bymeans of transformations of the three layers Ba–O4, Cu1–O1, and Ba–O4into a more complicated structure, while keeping the bilayer CuO2–R–CuO2

intact. One such modification has the general formula Pb2ASr2Cu3O8 withTc = 70K, where A = Y, R, Ca [213]. In this structure, the main 123 unitASr2Cu2O6 is preserved upon the replacement of Ba by Sr. The connectionbetween these units is no longer provided by Cu–O chains. It is now dueto a more complicated structure Pb2CuO2, which consists of two PbO lay-ers separated by a layer of copper with two-fold coordination. The structureof this compound is described by an orthorhombic unit cell (space groupCmmm) with the parameters a = 5.40 A, b = 5.43 A, and c = 15.8 A [213].The small distortion of the tetragonal cell is accounted for by an ordering ofoxygen in the PbO plane in noncentrosymmetric positions. In quenching fromthe high-temperature phase at 500◦C, the tetragonal structure P4/mmm isobserved.

A modification of the Y-123 structure, which does not contain any Cu–Ochains, is investigated by Roth et al. [1071]. This is the compound RSr2GaCu2

O7 in which the main element of the Y-123 structure, that is, the CuO2–R–CuO2 bilayer, is preserved. However, instead of the copper chains, thebilayers are now bounded by GaO4 octahedron, which also form chains. Thisconsiderable change in the structure results in quite different electronic prop-erties. Having the stoichiometric composition with respect to oxygen O7, thiscompound possesses semiconducting properties. Substitution of the trivalentions R = Y, Yb, Er by divalent ones, for example, Ca, which can be per-formed only under high oxygen pressure, results in metallic properties andsuperconductivity at Tc = 40–50K in multiphase samples [216].

2.4.3 Rutheno-Cuprates Magneto-Superconductors

In the conventional superconductors, long-range ferromagnetic ordering andsuperconductivity usually exclude each other. Therefore, the observationof ferromagnetic and superconducting properties in the ruthenate–cupratecompound RuSr2GdCu2O8−δ (Ru-1212) by means of zero-field muon-spinrotation and dc magnetization measurements [130] attracted the interest ofmany researches. It has been established that the material exhibits ferromag-netic order of the Ru moments (μRu ∼ 1μB) on a microscopic scale belowTC = 133K and becomes superconducting at a lower temperature Tc = 16K,which can reach 46K depending on the sample preparation. Coexistenceof superconductivity and ferromagnetism was also reported in the morecomplicated structure RuSr2(Gd0.7Ce0.3)2Cu2O10−δ (Ru-1222) [328].

The structures of both Ru-1212 and Ru-1222 compounds can be alsoviewed as modifications of Y-123 structures. In Ru-1212, the CuO2–R–CuO2

bilayer (R = Gd) is preserved but the charge reservoir of CuO1−δ chains inFig. 2.12 are replaced by RuO2−δ sheets. In Ru-1222, the single oxygen freeR-layer between the CuO2 planes is replaced by a three-layer fluorite-type


(R,Ce)–O2–(R,Ce) block. The Ru-1212, Ru-1222 compounds have tetragonalstructures (space group P4/mmm and I4/mmm respectively) with latticeparameters: a = b = 3.822 A, c = 11.476 A for Ru-1212 and a = b = 3.834 A,c = 27.493 A for Ru-1222 crystal.

Though the bulk ferromagnetism in RuO2 layers of Ru-1212 compoundon the microscopic scale was confirmed by the μSR-method and ESR studies,the exact long-range magnetic structure is still controversial. While neutrondiffraction experiments show antiferromagnetic ordering, magnetization stud-ies indicate ferromagnetic component. Specific heat measurements confirmthe bulk nature of superconducting transition, while at the magnetic phasetransition only hump is seen in CP , which indicates inhomogeneous magneticorder. It should be pointed out that the physical properties of these com-pounds strongly depend on the sample preparation conditions and there arestill many controversial results in the literature. A critical review of physicalproperties of these magneto-superconductors is given by Awana et al. [89] (seealso [232]).

In theoretical studies of coexistence of superconductivity and ferromag-netism, usually the Fulde–Farrell–Larkin–Ovchinnikov (FFLO) phase isconsidered [370, 690], where superconducting or (and) ferromagnetic orderparameter develops spatial variation to decrease the total free energy of thesystem. By applying the local density approximation and its generalization forelectron structure calculation, Pickett et al. [984] have concluded that coexis-tence of superconductivity and ferromagnetism is possible within the FFLOphase. This result can be explained by several specific properties of the lay-ered type compounds: small magnetization of RuO2 layer, a certain degree ofisolation of the RuO2 and CuO2 sublattices due to weak chemical coupling ofthe relevant atomic orbitals for Cu, O, and Ru ions in the layers, while theselayers are thin enough to allow three-dimensional coupling for both orderparameters. So, this quite complicated crystal structure could afford coexis-tence of two competing ordering on the microscopic scale, which now is wellestablished on the macroscopic scale for artificial sandwich structures of thinferromagnetic and superconducting films (see, e.g., [545]).

2.5 Bi-, Tl- and Hg-Compounds

Soon after Bednorz and Muller’s discovery, a new class of quinary copper-oxide compounds were discovered: the bismuth Bi2Sr2CaCu2O8+δ (Bi-2212)[767] and the thallium Tl2Ba2CaCu2O8+δ (Tl-2212) [1147] compounds withsuperconducting transition temperatures Tc above 100K. These were majorachievements. Even more exciting was the discovery of mercury cuprate com-pounds HgBa2Can−1CunO2n+2+δ (Hg-12(n − 1)n) [1034, 1035, 1112] with arecord Tc = 134K (n = 3) which can be further enhanced up to Tc = 164Kunder an external pressure P � 30GPa [375]. These compounds can havevarious numbers of copper–oxygen planes and are described by the general

2.5 Bi-, Tl- and Hg-Compounds 41

TI2 Can–1 Ba2 Cun O2n+4

n = 1 n = 2 n = 3

Fig. 2.16. The ideal crystal structure of the Tl2Ba2Can−1CunO2n+4+δ compoundsfor n = 1, 2, 3 copper–oxygen layers (reprinted with permission by APS from Parkinet al. [962], c© 1988)

formula AmB2Can−1CunOx, where A = Bi, Tl, or Hg and B = Sr for Bi andB = Ba for Tl and Hg compounds. For bismuth m = 2, mercury m = 1,while the thallium compounds can be synthesized with one or two Tl layers:m = 1, 2 [962]. Figure 2.16 shows the ideal pseudo-tetragonal unit cell of theTl-22(n − 1)n compounds (space group (I4/mmm − D17

4h). The structure ofthe Bi-22(n− 1)n compounds is the same as that of the Tl-compounds shownin Fig. 2.16. The pseudo-tetragonal unit cell of Bi- and Tl-compounds has thedimensions 3.9×3.9 A in the basal plane. The lattice constant along the c-axisdepends on the number of copper-oxide planes n. It equals to c = 24.4, 30.8,37.1 A and c = 23.2, 29.4, 36 A for n = 1, 2, 3 for Bi- and Tl-compoundsrespectively.

The actual structure of the Bi compounds has orthorhombic distortions.The unit cell (Fmmm − D23

2h) has the dimensions ao � bo � at

√2 � 5.4 A,

co � ct. Moreover, an incommensurate modulation in the Bi–O layer with thewave vector q = 0.21b∗ is observed, which can be roughly approximated in asupercell orthorhombic model with b

′o = 5ao. The average Cu–O distances in

the plane have typical value 1.9 A. The Cu–O distance along the c-axis in theCuO5 pyramid is 2.6 A, which is larger than the corresponding Cu2–O4 dis-tance of about 2.3 A in the YBCO compound. A weak coupling between Bi–Olayers due to the large distance ∼3 A between them is typical for Bi mica-likecompounds. The crystal is easily cleaved between the Bi–O layers produc-ing a clean surface. Although the modulation of the structure within Bi–Olayers does not influence superconductivity, this structure irregularity in Bicompounds greatly complicates the study of the single crystal properties. Tosuppress the distortions of the crystal structure, a mixed Bi–Pb compound


was synthesized, Bi2−xPbxCaSr2Cu2O8+δ. Its orthorhombic structure withlattice parameters a � 5.40, b � 5.38, c � 30.78 A for a large concentrationof lead x ≥ 0.4 shows modulations only for Pb displacements in one direc-tion with much a larger period, which is characterized by the wave vectorq = 0.12b∗ [551]. This configuration allows more regular distances betweenneighboring Bi–O layers and that considerably simplifies the studies of singlecrystal properties.

The structure of the Tl compounds undergoes much less distortion fromthe ideal tetragonal lattice due to the relatively strong coupling betweenTl–O layers. The distance between them is ∼2 A, while the lattice constantis smaller than in the Bi compounds. The length of the Cu–O bonds in theCuO2 planes is 1.92 A, while the copper–apical oxygen distance along thec-axis reaches 2.7 A, which is sensibly larger than the corresponding distancein YBCO. More detailed studies of the structure of Tl compounds have shown,however, a considerable statistical disorder in the Tl–O layers [451]. The Tland O ions in the Tl–O layers have been found to be displaced from there cen-trosymmetric positions in the layer in such a way that they form Tl–O chainswith a shorter bond. However, this structure possesses only short-range orderso that it maintains the tetragonal symmetry on the average [284]. The super-conducting transition temperature in the Bi- and Tl-compounds depends onthe number of the copper–oxygen planes n = 1, 2, 3 and takes the valuesTc � 10, 85, 110K for the Bi compounds, and Tc � 80, 100, 125K for the Tlcompounds, respectively, with the highest Tc for three layer compounds [444].This trend is general for all cuprate superconductors as discussed below formercury compounds.

The crystal structures of mercury compounds with one, two, and threecopper layers are shown in Fig. 2.17. Their tetragonal structure P4/mmm ischaracterized by close values of the lattice parameter a and Cu–O distancesin the plane a/2 � 1.94− 1.93 A with, however, a well resolved decreasing onthe number of planes n (see Fig. 2.18). The c-lattice parameter depends onthe number of CuO2 planes: c � 9.5 + 3.2(n − 1) A. The hole doping of these

Fig. 2.17. Crystal structures of mercury compounds Hg-1201, Hg-1212 and Hg-1223(after [77])

2.5 Bi-, Tl- and Hg-Compounds 43

Fig. 2.18. The dependence of superconducting Tc on the lattice parameter a forthe Hg-12(n − 1)n series for n = 1, 2, 3, 4, and 5 (after [77])

compounds is achieved by annealing them in oxygen atmosphere, which givesexcess oxygen ions with concentration δ in the buffer layers Hg–Oδ, as shownin Fig. 2.17. The structure of the TlBa2Can−1CunO2n+3+δ (Tl-12(n − 1)n)compounds is similar to that of the compounds Hg-12(n− 1)n structure. Themain difference comes from the number of oxygen ions in the Hg and Tl layers:while for Hg2+ only a small content δ ≤ 0.2–0.4 of doping oxygen (dependingon the number n of Cu-planes) is needed to obtain the optimal concentrationof doped holes, for Tl3+ ions the number of oxygen ions in the Tl–O planesequals to 1+ δ, which produces distorted octahedron coordination for Tl ions.

The characteristic feature of the Hg-based compounds is a stable dumb-bell Hg2+ coordination with two strong covalent Hg–O bonds with the nearestoxygen ions in Ba–O layers. This results in a weak coupling with nonstoichio-metric oxygen ions in the Hg–Oδ layers and an ideal tetragonal structure. Themercury compounds show the largest distance between the CuO2 plane andthe apical oxygen of about 2.77–2.8 A. As a result, the buffer layers exert avery weak influence on the copper–oxygen planes, which results in the small-est buckling of the planes among the copper-oxide materials. It is inferredthat just this ideal flat structure of CuO2 plane results in the highest Tc incuprates [77].

The stable structure of the Hg-based cuprates enables a detail investigationof the dependence of the superconducting temperature on the number n ofcopper–oxygen planes [77]. The behavior of Tc,max with the number of layersis similar to that observed in the Bi- and Tl-compounds. It first increases,Tc � 96, 127, 135K for the number of Cu layers n = 1, 2, 3, but then decreases:Tc � 127, 110, 107K for n = 4, 5, 6. The decrease of Tc for the number oflayers n > 3 can be explained by underdoping of inner Cu layers. The effectiveconcentration of holes in these layers appeared to be lower than the optimal


Fig. 2.19. Superconducting transition temperature for Hg-1201 as a function ofoxygen or fluorine concentration (after Abakumov et al. [1])

value of xopt � 0.16 per one CuO2 unit cell in the plane. Figure 2.18 plotsthe Tc dependence on the lattice parameter a for n = 1, 2, 3, 4, 5 along theHgBa2Can−1CunO2n+2+δ homologous series. The functional dependence ofTc as a function of the concentration δ of doped oxygen in the Hg–Oδ layershows parabolic form similar to that of (1.5),

Tc(δ) = Tc,max

[1 − q (δ − δopt)2

], (2.9)

where for Hg-1201, Tc,max � 97K, q � 52, δopt � 0.128. These parameterscan vary from one family of compounds to another but the general parabolicform (2.9) is retained.

Interesting results in this respect were obtained for mercury compoundsby using fluorine instead of oxygen for hole doping. Figure 2.19 plots Tc as afunction of oxygen and fluorine ion concentration in the Hg-1201 compound[1]. The maximum Tc value is reached at fluorine ion (F−) concentrationδF � 0.26 twice as large as the oxygen ion (O2−) concentration δO � 0.13.Taking into account the ratio of the ion valences, this results in the sameoptimal hole doping. If we assume that every doped oxygen ion O2− transferstwo holes into the CuO2 plane, then the optimal concentration of doped holesxopt � 0.26 appears to be larger than for other cuprates: xopt � 0.16 in (1.5).The apparent discrepancy originates in the distribution of the doped chargeover the CuO2 layer and Hg–O2 dumb-bell, which results in transferring only60% of the doped oxygen charge into the CuO2 plane. The comparison ofthe lattice parameters for oxidized and fluorinized samples has revealed thatthe maximal Tc at optimal doping strongly depends on the in-plane Cu–Obond length, while it does not depend on small variations of the copper–apical oxygen distance. Figure 2.20 shows the variation of maximal Tc formercury compounds as a function of the lattice parameter a in the plane [737].

2.6 High-Pressure Effects 45

Fig. 2.20. Superconducting transition temperature for mercury compounds at opti-mal doping as a function of the lattice parameter a in the CuO2 plane for n = 1, 2, 3copper–oxygen materials. Hg-1223F and Hg-1201F are fluorinated compounds (after[737])

While for the Hg-1201 compound the fluorination does not change the latticeparameter a and Tc (but changes the copper–apical oxygen distance), thelattice parameter a decreases and Tc increases up to 138K in the fluorinatedHg-1223 sample. The rate of Tc enhancement with the decrease of a appearsto be very large, dTc/da � 1,300K/A. It is worthwhile to note that the samecoefficient defines the Tc dependence on the lattice parameter a for all thethree mercury compounds, as shown in Fig. 2.20.

From these experiments, we can draw the general conclusion that theCu–Oapical distance (at small variation) is irrelevant for the superconduct-ing Tc value, while the compression of the in-plane Cu–O distance can greatlyenhances Tc. This conclusion holds true provided that other parameters ofthe copper–oxygen bond, in particular the Cu–O–Cu bond angle in the plane,do not change. In mercury compounds, this angle is close to 180◦ and doesnot change under fluorination, while the application of an external pressuresignificantly reduces the angle which results in buckling of the CuO2 plane(e.g., at the pressure of 2 GPa the angle reduces to 175.0◦ from 178.4◦ foroxygenated [177.3◦ for fluorinated] Hg-1223 compound [737]). Therefore, thechemical compression of the structure without changing the buckling angle ismuch more efficient in enhancing Tc than by isotropic external pressure.

2.6 High-Pressure Effects

High-pressure studies of cuprate superconductors play an essential role inour understanding of both the normal state and the superconducting proper-ties of these compounds. It should be mentioned that the idea of enhancingsuperconducting Tc by “chemical pressure” resulted in the discovery of YBCOsuperconductor by Wu et al. [1371] and later on in achieving the record


Tc = 164K under external pressure P � 30GPa [375]. The application ofexternal pressure enables a smooth variation of the lattice parameters withoutuncontrollable side effects, which usually accompany chemical substitution.Therefore, the study of the Tc dependence on the pressure provides an impor-tant approach to the elucidation of the mechanism of high-temperature super-conductivity. There are several reviews on the studies of high-pressure effectsin high-temperature superconductors (see, e.g., [1111,1118,1119,1213,1358]).Here, we discuss only the most important results concerning the pressuredependence of Tc.

In studies of the conventional electron–phonon superconductors, it wasobserved that Tc decreases with the pressure P in most cases. This featurecan be explained if, by using the BCS formula (1.1) for Tc, we write

d ln Tc

dP=

∂ ln ω

∂P+

1λ

∂ ln λ

∂P� ∂ ln ω

∂P+

1λ

(∂ ln η

∂P− 2

∂ ln ω

∂P

). (2.10)

Here, the interaction in (1.2) is written as λ = η/Mω2, where the Hopfieldparameter η = N(0)g2. For conventional s–p superconductors, both the den-sity of states N(0) and the electron–phonon interaction g weakly depend onpressure due to the high electron density. The most important contribution tothe variation of Tc with P originates in the second (negative) term in (2.10)owing to the increase of the lattice stiffness with the pressure: ∂ ln ω/∂P > 0.Therefore, a behavior dTc/dP < 0 is expected to be found in the s–p met-als, a feature confirmed, for example, by studies of the MgB2 superconductor:dTc/dP � −1.1K GPa−1 [1118]. Only for narrow d-band metals in which thedensity of states N(0) may strongly vary with pressure, the second term in(2.10) can be positive, ∂ ln η/∂P > 2 ∂ ln ω/∂P , and Tc would increase withthe pressure.

Concerning the copper-oxide superconductors, we should take into accountthat Tc strongly depends on the density of charge carriers, as given by theuniversal formula (1.5). Therefore, we should distinguish two contributions,the first one due to intrinsic effects, as changing the parameters of the pairinginteraction, and the second one due to charge transfer x under pressure:

dTc

dP=

(∂Tc

∂P

)intrinsic

+(

∂Tc

∂x

) (∂x

∂P

). (2.11)

At optimal doping, x = xopt, we have dTc/dP = dTc,max/dP , which givesthe intrinsic pressure effect since the second term according to the parabolicTc dependence in (1.5) gives no contribution. As many experiments show,contrary to the conventional superconductors, in cuprates dTc,max/dP > 0[1359]. For “normal” cuprate superconductors, dTc,max/dP � 1.5K GPa−1

[1118, 1119]. For underdoped and overdoped hole cuprate superconductors,the Tc dependence on pressure has opposite signs:(

dTc

dP

)underdoped

> 0,

(dTc

dP

)overdoped

< 0. (2.12)

2.6 High-Pressure Effects 47

In these regions, the charge transfer term ∂x/∂P in (2.11) becomes importantand it defines the pressure dependence. Under pressure, the distance betweenthe negatively charged copper–oxygen CuO2 layer and positively charged dop-ing buffer blocks decreases, which results in positive charge transfer to thecopper–oxygen layer with the maximum rate of ∂x/∂P � 0.02GPa−1 [1359].Owing to the parabolic Tc(x) in (1.5), for hole doped cuprates the dependence(2.12) is realized, while for electron doped cuprates it should be opposite. Thisgeneral trend for Tc dependence on pressure in cuprates is confirmed by manyexperiments [1359].

In LaSr-124 compounds, Tc first increases under pressure reaching a certainmaximum but then decreases at higher pressure. In electron doped Nd(Ce)-124compounds, Tc appears to be pressure independent, while in mixed compoundsNd(Sr, Ce)-124 quite a strong pressure dependence is observed [789]. This dif-ference in the Tc(P ) dependence can be related to a special role of the apexoxygen: in the T-phase of La(Sr)-124 compounds, there are two apex oxygenions inside the complete CuO6 octahedron, while in the T

′-phase of Nd(Ce)-

124 compounds the apex oxygen ion is absent (see Fig. 2.10). In the T∗-phaseof Nd(Sr, Ce)-124, there is only one apex oxygen in the CuO5 pyramid. There-fore, the mechanism of charge transfer under pressure from the buffer blocksLa–Sr–O or Nd–Ce–O is quite different.

In some cases, an extremely large pressure dependence is observed forcuprates in the underdoped region, as for YBa2Cu3O7−y for small oxygencontent, y � 0.5, or for YBCO-124, where Tc � 80K at ambient pres-sure increases up to 105K under a pressure of 10GPa [1359]. Strong Tc

dependence on the pressure was also obtained close to structural phase trans-formations, as in La1−xBaxCuO4 system in the vicinity of x = 1/8, where Tc isstrongly suppressed due to transformation to the LTT phase (see Fig. 2.6) anddTc/dP � 12K GPa−1 [1111]. This strong Tc dependence on pressure in theLTT phase was confirmed by Takeshita et al. [1215], who have found a highlyanisotropic pressure dependence in the (a, b) plane for La1.64Eu0.2Sr0.16CuO4

system: (dTc/dP )[110] � 2.5KGPa−1 and much weaker in [100] direction. InEu-doped LSCO-124-compound, the LTT structure occurs in a wide rangeof Sr concentrations (see Fig. 2.6). As discussed in Sect. 2.2, bulk super-conductivity in the LTT phase exists only if the tilting angle Φ of theoctahedron rotation is smaller than the critical one, Φc � 3.6◦ [183, 184].A uniaxial pressure along the [110] direction decreases the amplitude of theoctahedron rotation in the LTT phase (and CDW-stripe amplitude) and thisenhances Tc.

The most accurate studies of Tc dependence on the pressure were per-formed in Hg-based superconductors. Due to a stable crystal structure over awide range of doping, it was possible to investigate pressure effects both in theunderdoped and overdoped regions with high precision (see, e.g., [77,96,231]).For all optimally doped Hg-12(n−1)n compounds, Tc increases with the pres-sure with a rate dTc/dP � 1 − 2KGPa−1 up to 5GPa. Further increaseof pressure leads to saturation and at some specific value, depending on


the number n of copper–oxygen layers, Tc decreases. For compounds withn ≥ 3, the inequivalent inner and outer layers have different doping levelsand the charge transfer under external pressure can be important in enhanc-ing Tc. Therefore, the studies of Tc and structure under pressure in Hg-1201compound with only one copper–oxygen layer by Balagurov et al. [96] wereimportant for the elucidation of the intrinsic pressure effects. By using a high-resolution neutron diffraction under pressure in a wide region of oxygen, Oδ,doping 0.06 < δ < 0.19, the authors were able to find out a correlationbetween Tc and structure changes. In the underdoped δ = 0.06 and optimallydoped δ � 0.13 regions, the compression of the structure is uniform and thecharge transfer plays a minor role. However, in the overdoped region δ = 0.18,a large compression of the apical Cu–O2 and Ba–Oδ distances is observed.These results explain why at low and optimal doping Tc increases with pres-sure (the intrinsic effect in cuprates), while in the overdoped region a chargetransfer occurs under pressure, which leads to Tc decreasing according to theparabolic law (2.9).

The copper-oxide materials are strongly anisotropic systems and, there-fore, uniaxial deformations are required to distinguish between the chargetransfer (which is sensitive to c-axis compression) and the intrinsic pressureeffects in the conducting CuO2 plane caused by (a, b) compression. Thesequite different pressure effects were confirmed in direct measurements of thechanges of Tc induced by uniaxial compression along the a,b, and c axes of anuntwinned crystal of YBa2Cu3O7 by Welp et al. [1350]. It was found that thepressure derivatives are large but have opposite signs for compression alongdifferent axes (in KGPa−1):

dTc

dPa� −2.0,

dTc

dPb� 1.9,

dTc

dPc� −0.3. (2.13)

Under hydrostatic pressure, only a small pressure effect is observed in YBCOat optimal doping due to significant cancellation of these opposite in signderivatives. This experiment also proves that Tc increases by diminishing theorthorhombicity b/a of the crystal: dTc/dPi have opposite signs along thea and b directions. Similar results were obtained for the Bi2Sr2CaCu2O8+x

single crystal [822]. The uniaxial-pressure dependencies of Tc were calculatedfrom the amplitudes of the measured expansivity anomalies. The pressurederivatives occurred also large but of opposite signs for compression alongdifferent axes (in KGPa−1):

dTc

dPa� 1.6,

dTc

dPb� 2.0,

dTc

dPc� −2.8. (2.14)

The crystal structure of the Bi2Sr2CaCu2O8+x compound is close to theorthorhombic one and compressions along a and b axes both increase Tc,while compression along c axes decreases Tc. These results agree quite wellwith the values found in uniaxial stress experiments on single crystals by

2.7 Conclusion 49

Watanabe et al. [1341]: dTc/dPa,b � 1.5, dTc/dPc � −4.5 (K GPa−1). Underhydrostatic pressure, a large cancellation of the uniaxial pressure derivativesoccurs, which results in a small pressure effect on Tc. Interesting results wereobtained for large Tc enhancement in underdoped LSCO films under compres-sive epitaxial strain. As shown by Bozovic et al. [173], the underdoped LSCOfilms are extremely sensitive to oxygen intake and, therefore, a strong increaseof Tc observed in some experiments in these films may be caused by oxygendoping. However, they obtained the record Tc = 51K for LSCO film on theLaSrAlO4 substrates which produce compressive epitaxial strains.

Summarizing the high-pressure investigations, we can draw the generalconclusion that, even for an optimally doped cuprate superconductor, Tc canbe enhanced by compression of the Cu–O bond length in the plane, which isan intrinsic pressure effect. As pointed out by Schilling [1118], for optimallydoped cuprates Tc is approximately proportional to the inverse square of thearea of the CuO2 plane: d lnTc/d lna � −4.5, where a is the lattice parame-ter in the plane. This value should be compared with the results for Hg-basedcompounds considered in the previous section, Figs. 2.18 and 2.20. Applica-tion of chemical pressure by fluorination gives an order of magnitude strongerdependence: d lnTc/d lna � −40. As explained above, under chemical pres-sure in Hg-1223, the Cu–O–Cu bond angle in the plane is not changed, whileapplication of an external pressure causes the buckling of the CuO2 plane. Thepressure effects in cuprates should be also compared with those in electron–phonon superconductors, where, for instance, for MgB2 d ln Tc/d ln a � +16[1118]. Therefore, if we try to explain the large increase of Tc in cuprates withapplied pressure within the BCS electron–phonon mechanism of pairing in(2.10), an extremely strong pressure dependence of the interaction should beinvoked to overcome the negative contribution from the stiffening of the lat-tice. For an electronically driven pairing mechanism with an electronic energyas a prefactor in BCS-like formula (1.1), ω � EF , weakly dependent on pres-sure and the interaction λ increasing with pressure (e.g., the antiferromagneticexchange interaction), one can easily explain large increasing of Tc with com-pression. In Sect. 7.3.3, we consider the antiferromagnetic exchange pairingmechanism that gives a Tc(a) dependence close to experiments.

2.7 Conclusion

We now emphasize the most important results obtained in the study of thecrystal structure of copper-oxide superconductors:

1. The structure of all the copper-oxide superconductors is of a block nature.The main unit determining the metallic and superconducting propertiesof a compound is the CuO2 plane, which is a square lattice formed bycopper ions bound to each other through oxygen ions. Each copper ioncan additionally be coordinated to oxygen ions in the apex positions of


an octahedron; this gives rise to several possible coordinations of cop-per ions equal to four, five, and six. The effective charge of the CuO2

complex is determined by the buffer blocks binding them in a crystalstructure. Only structures with regular CuO2 planes show superconduc-tivity, which appears in the range of the formal copper valence v equalto positive values 2.05 ≤ v ≤ 2.25 for hole-doped and 1.8 ≤ v ≤ 1.9 forelectron-doped compounds. Superconductivity in complex copper-oxideswith three-dimensional network of Cu–O bonds have not yet been detected.

2. The highest Tc was obtained for the mercury compounds due to the idealsquare and flat structure of CuO2 plane with the smallest buckling amongthe copper-oxide materials and the largest copper-apex oxygen distance.The maximal Tc,max in the cuprates increases under external pressure (con-trary to the conventional electron–phonon superconductors) due to thecompression of the Cu–O bond distance in the plane, d ln Tc/d ln a � −4.An order of magnitude larger increase of Tc,max is observed in the mer-cury cuprates under chemical pressure, which preserves the Cu–O–Cu bondangle close to 180◦.

3. However, the crystal structure of copper-oxide compounds does not com-pletely determine their superconducting properties, which crucially dependon structural defects and short-range order. The structural instabilityand tendency to a short-range ordering were observed in the Bi- and Tl-compounds. The arrangement and local ordering of doping atoms in alattice significantly influence the superconducting properties of the sys-tem, as has been clearly proved in YBCO, due to the local nature of chargetransfer from the buffer layer to the CuO2 plane.

4. A peculiar property of the hole-doped cuprates is the formation of incom-mensurate static or dynamic CDWs, which results in an inhomogeneouscharge (and spin) distribution in crystals in the form of stripes. They wereexplicitly detected in the LTT phase of La-Ba-124 and La-Nd(Eu)-124compounds at hole doping close to x = 1/8. The strong suppression ofsuperconductivity at this “magic” hole concentration may be explained bya localization of charge carriers caused by the CDW formation.

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