Superconductivity Theory and Applications Part 2 ppt

A power-law temperaturedependence, observed for both physical quantities in the superconducting state, yields theexistence of nodes zero gap with sign change on the superconducting gap,

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1 Introduction

Two different ground states, superconductivity and magnetism, were believed to beincompatible, and impossible to coexist in a single compound The Ce based heavy Fermionsuperconductor CeCu2Si2, however, was discovered in the vicinity of magnetic phase Steglich

et al (1979) This new class of superconductors, which are referred to as "unconventional"superconductor, demonstrate various novel properties which are not accounted for in theframework of the BCS theory Electrons in unconventional superconductors are stronglycorrelated through the Coulomb interaction, while strong electron-electron correlations arenot preferable for the conventional BCS superconductors Modern theory predicts that therepulsive Coulomb interaction can induce attractive interaction to form superconductingCooper pairs as the result of many-body effect

Unconventional superconductivity is often observed nearby a quantum critical point

(QCP), where magnetic instability is suppressed to T = 0 by some physical parameters

It is invoked that the quantum critical fluctuations, which are enhanced aroundQCP, drive the superconducting pairing interactions, instead of the electron-phononinteraction proposed in the BCS theory In addition to the novel pairing mechanisms,unconventional superconductivity shows various novel superconducting states, such asFulde-Ferrell-Larkin-Ovchinnikov state and spin-triplet pairing state Unveiling novelmechanism and resulting novel properties is the main topic of condensed matter physics.The cobaltate compound is also classified to an unconventional superconductor when we takethe results of nuclear spin-lattice relaxation rate Fujimoto et al (2004); Ishida et al (2003),and specific heat Yang et al (2005) measurements into account A power-law temperaturedependence, observed for both physical quantities in the superconducting state, yields theexistence of nodes (zero gap with sign change) on the superconducting gap, and addressesthe unconventional pairing mechanism Besides, a magnetic instability was found in thesufficiently water intercalated cobaltates Ihara, Ishida, Michioka, Kato, Yoshimura, Takada,Sasaki, Sakurai & Takayama-Muromachi (2005) The close proximity of superconductivity

to magnetism in cobaltates lead us to consider that the same situation as heavy Fermionsuperconductors is realized in cobaltates

Unconventional Superconductivity Realized Near

Magnetism in Hydrous Compound

Yoshihiko Ihara1 and Kenji Ishida2

1Department of Physics, Faculty of Science, Hokkaido university

2Department of Physics, Graduate School of Science, Kyoto university

Japan

Fig 1 Crystal structures of non-hydrate, mono-layer hydrate and bilayer hydrate

compounds

In this chapter, the relationship between superconductivity and magnetism will be exploredfrom the nuclear magnetic resonance (NMR) and nuclear quadrupole resonance (NQR)experiments on superconducting and magnetic cobaltate The principles of experimentaltechnique is briefly reviewed in §3 Then the experimental results are presented inthe following sections, §4 and §5 Finally, we will discuss the superconducting paringmechanisms in bilayer-hydrate cobaltate, showing the similarity between cobaltate and heavyFermion superconductors

2 Water induced superconductivity in Nax(H3O)zCoO2·yH2O

Superconductivity in a cobalt oxide compound was discovered in 2003 Takada et al (2003).The hydrous cobaltate Nax(H3O)zCoO2· yH2O demonstrates superconductivity when watermolecules are sufficiently intercalated into the compound by a soft chemical procedure Incontrast, anhydrous NaxCoO2does not undergo superconducting transition at least above 40

mK Li et al (2004) A peculiarity of superconductivity in Nax(H3O)zCoO2· yH2O compound

is the necessity of sufficient amount of water intercalation between the CoO2 layers, anddepending on the water content, superconducting transition temperatures vary from 2 K to4.8 K This compound is the first superconductor which shows superconductivity only in thehydrous phase

The cobaltate compound has three types of crystal structures with different waterconcentrations as shown in Fig 1 The parent compound NaxCoO2, which is y = 0

and z = 0, contains the randomly occupied Na layer between the CoO2 layers When

Na ions are deintercaleted and water molecules are intercalated between the CoO2 layers,the crystal structure changes to bilayer hydrate (BLH) structure, in which the Na layer

is sandwiched with double water layers to form H2O-Na-H2O block layer Due to theformulation of this thick block layer, the CoO2layers are separated by approximately 10 Å,and is considered to have highly two-dimensional nature Superconductivity is observed inthis composition below 5 K The crystal structure of the superconducting BLH compoundchanges to mono-layer hydrate (MLH) structure, which forms Na-H2O mixed layers betweenthe CoO2layers containing less water molecules than those of BLH compounds The watermolecules inserted between CoO2 layers are easily evaporated into the air at an ambient

condition, seriously affecting the physical properties, namely superconductivity, of BLHcompounds.

A BLH compound left in the vacuum space for three days becomes a MLH compound,and does not demonstrate superconductivity Inversely the crystal structure of the MLHcompound stored in high-humid atmosphere comes back to the BLH structure, andsuperconductivity recovers, although the transition temperature of the BLH compoundafter the dehydration-hydration cycle is lower than that of a fresh BLH compound.Superconducting and normal-state properties in various kinds of samples have beeninvestigated with several experimental methods

In the normal state, spin susceptibility is almost temperature independent above 100 K, which

is a unique behavior irrespective of samples The sample dependence appears below 100

K, for instance, spin susceptibility increases with decreasing temperature in some samples,but some do not From the temperature dependence of spin susceptibility below 100 K,temperature independent susceptibilityχ0, effective momentμeffand Weiss temperatureΘWwere reported to beχ0 =3.02×10−4emu/mol,μeff ∼0.3μBandΘW = −37 K by Sakurai

et al (2003) Different values are reported by Chou, Cho & Lee (2004), where the increase

of susceptibility toward low temperature is hardly observed, and correspondingly, μeff israther small Although the low temperature behavior of susceptibility is strongly sampledependent, we believe that the slight increase below 100 K is an intrinsic behavior because it

is observed in most of the high-quality powder samples and also observed in the Knight shiftmeasured by nuclear magnetic resonance Ihara, Ishida, Yoshimura, Takada, Sasaki, Sakurai &Takayama-Muromachi (2005) and muon spin rotation measurements Higemoto et al (2004)

In the superconducting state, specific heat is intensively measured by several groups Cao et al.(2003); Chou, Cho, Lee, Abel, Matan & Lee (2004); Jin et al (2005); Lorenz et al (2004); Oeschler

et al (2005); Ueland et al (2004); Yang et al (2005) The specific heat jump at superconductingtransition temperature ΔC/γT c is estimated to be approximately 0.7, which is half of theBCS value 1.43 The small jump suggests either the quality of the sample is insufficient, orthe superconductivity is an unconventional type with nodes Below the superconducting

transition temperature, C/T does not follow exponential temperature dependence but follows

power law behavior, which is universally observed in unconventional superconductor The

power law behavior observed from nuclear-spin-lattice relaxation rate 1/T1 measurementalso supports unconventional superconductivity Fujimoto et al (2004); Ishida et al (2003)

The results of 1/T1measurements are presented in § 4 The values of Sommerfeld constant,which are 12∼16 mJ/molK2depending on samples, are comparable to those of anhydrouscompound Na0.3CoO2 and less than those of mother compound Na0.7CoO2 It is curiousthat the BLH compound which has smaller density of state compared to mother compoundsdemonstrates superconductivity, while a single crystal of Na0.7CoO2 with larger density ofstate is not a superconductor The hydrous phases ought to have specific mechanisms toinduce superconducting pairs

The discovery of magnetism in a sufficiently water intercalated BLH compound providesimportant information to understand the origin of superconductivity The superconductingBLH is located in the close vicinity of magnetic phase, as in the case for heavy Fermionsuperconductors This similarity lets us invoke that the magnetic fluctuations near magneticcriticality can induce superconductivity in BLH system The magnetic fluctuations areexamined in detail with nuclear quadrupole resonance and nuclear magnetic resonancetechnique in order to unravel the superconducting mechanisms

3 Nuclear magnetic resonance and nuclear quadrupole resonance

3.1 Nuclear quadrupole resonance measurement

In this section, the fundamental principles of nuclear quadrupole resonance (NQR) arebriefly reviewed Resonance phenomena are observed between split nuclear states and radiofrequency fields with energy comparable to the splitting width To observe the resonance,degenerated nuclear spin states have to be split by a magnetic field and/or an electric-fieldgradient (EFG) For NQR measurements, a magnetic field is not required because the nuclearlevels are split only by the electric-field gradient Under zero magnetic field, nuclear levelsare determined by the electric quadrupole HamiltonianH Q, which describes the interaction

between the electric quadrupole moment of the nuclei Q and the EFG at the nuclear site In

general,H Qis expressed as

H Q=ν zz

6

(3I2− I2) +1

2η(I2++I2−)

,

where eq(=V zz)andη = (V yy − V xx)/V zz are the EFG along the principal axis (z axis) and

the asymmetry parameter, respectively The resonant frequency is calculated by solving theHamiltonian From the measurement of these resonant frequencies,ν zzandη are estimated

separately These two quantities provide information concerning with the Co-3d electronic

state and the subtle crystal distortions around the Co site, because the EFG at the Co site

is determined by on-site 3d electrons and ionic charges surrounding the Co site The ionic

charge contribution is estimated from a calculation, in which the ions are assumed to be point

charges (point-charge calculation) The result of the point-charge calculation indicates that Vzz

is mainly dependent on the thickness of the CoO2layers, because the effect of the neighboring

O2−ions is larger than that of oxonium ions and Na+ ions that are distant from the Co ions.Due to the ionic charge contribution, the resonant frequency was found to increase with thecompression of the CoO2layers

When small magnetic fields, which are comparable to EFG, are applied, the perturbationmethod is no longer valid to estimate the energy level The resonant frequency should

be computed numerically by diagonalizing Hamiltonian, which includes both Zeeman andelectric quadrupole interactions The total Hamiltonian is expressed as

=ν zz

6

(3I2− I2) +1

The results of a numerical calculation is displayed in Fig 2, where the parametersν zz,η are

set to be 4.2 MHz and 0.2, respectively The shift of the resonant frequency depends on thedirection of the small magnetic fields The magnetic fields parallel to the principal axis of EFG

(z axis) affect the transition between the largest m states, while spectral shift of this transition

is small when the magnetic fields are perpendicular to z axis In other word, when internal

magnetic fields appear in the magnetically ordered state, the direction of the internal fields can

be determined from the observation of NQR spectrum with the highest resonant frequencyabove and below the magnetic transition

3.2 Nuclear spin-lattice relaxation rate

The nuclear spin system has weak thermal coupling with the electron system Through thecoupling, heat supplied to the nuclear spin system by radio-frequency pulses flows into the

0.0 0.1 0.20

481216

Magnetic Field ( T )-1/2 1/2

1/2 3/23/2 5/25/2 7/2

++

Fig 2 NQR frequency calculated from equation (3) with small magnetic fields up to 0.2 T.The solid and dashed lines are the resonant frequency in magnetic fields perpendicular and

parallel to the principal axis of EFG (z axis) The parameters ν zzandη were set to the realistic

values of 4.2 MHz and 0.2, respectively

electron system, and the excited nuclear spin system relaxes after a characteristic time scale

T1 The nuclear spin-lattice relaxation rate 1/T1 contains important information concerningwith the dynamics of the electrons at the Fermi surface

When the nuclear spin system is relaxed by the magnetic fields induced by electron spinsδH,

the transition probability is formulated by using the Fermi’s golden rule as

Here, A q andχ  ⊥(q, ω)are the coupling constant inq space and the imaginary part of the

dynamical susceptibility, respectively ω0 in the equation is the NMR frequency, which isusually less than a few hundreds MHz The q dependence of A q is weak, when 1/T1 ismeasured at the site where electronic spins are located

The temperature dependence of 1/T1 has been studied by using the self-consistentrenormalization (SCR) theory Moriya (1991) The dynamical susceptibility assumed in thistheory is formulated as

χ(Q+q, ω) = χ(Q)

1+q2Aχ(Q ) − iωq −θ Cχ(Q). (8)The dynamical susceptibility is expanded inq space around Q, which represents the ordering

wave vector When the magnetic ordering is ferromagnetic(Q=0),θ=1 is used to calculate

χ(q, ω), and when it is anti-ferromagnetic(Q > 0),θ is zero The parameters A and C are

determined self-consistently to minimize the free energy The SCR theory is available when

the electronic system is close to a magnetic instability The temperature dependence of 1/T1 above TC and TN is derived using the renormalized dynamical susceptibility The resultsdepend on the dimensionality and theθ values, which determine that the magnetic ordering

is ferromagnetic (FM) or anti-ferromagnetic (AFM) The temperature dependence anticipatedfrom the SCR theory is listed in Table 1

FM 1/T1∝ T/(T − TC)3/2 1/T1∝ T/(T − TC)

AFM 1/T1∝ T/(T − TN) 1/T1∝ T/(T − TN)1/2

Table 1 Temperature dependence of 1/T1anticipated from the SCR theory

In the superconducting state, thermally exited quasiparticles can contribute to the Knight shift

K and the nuclear spin-lattice relaxation rate 1/T1 Since the quasiparticles do not exist withinthe superconducting gap, the energy spectrum of the quasiparticle density of state is expressedas

N(E; θ, φ) = N0E

E2− Δ2(θ, φ) for E > Δ(θ, φ) (9)

=0 for 0< E < Δ(θ, φ), (10)

where E is the quasiparticle energy, which is determined as E2 = ε2+Δ2, and N0 is the

density of state in the normal state In order to obtain the total density of state N(E), N(E; θ, φ)

should be integrated over all the solid angles N(E)is calculated with considering simpleangle dependence forΔ, and the resulting energy spectra are represented in Fig 3 The angle

dependence of the superconducting gap, which we considered for the calculations, are listedbelow

Δ(θ, φ) =Δ0sinθe iφ for p-wave axial state, (12)

Δ(θ, φ) =Δ0cosθe iφ for p-wave polar state, (13)

Δ(θ, φ) =Δ0cos 2φ for two-dimensional d-wave. (14)The temperature dependence of the Knight shift and the nuclear spin-lattice relaxation rate

in the superconducting state can be computed from the following equations with using N(E)

calculated above The temperature dependence of the gap maximumΔ0was assumed to bethat anticipated from the BCS theory for all the gap symmetry considered above.

Here, f(E)is the Fermi distribution function The term(1+Δ2/E2)in equation (16) is referred

to as the coherence factor, which is derived from the spin flip process of unpaired electronsthrough the interactions between the unpaired electrons and the Cooper pairs Hebel & Slichter(1959)

As an initial state| i , it is assumed that one unpaired electron with up spin and one Cooper

pair have wave numbers k and k , respectively After the interaction between these particles,the wave numbers are exchanged, and the electronic spin of the unpaired electron can flipwith preserving energy The final state| f is chosen to be one electron with down spin andwave number− k  , and one Cooper pair with wave number k This process can contribute to

the relaxation rate by exchanging the electronic spins and the nuclear spins The initial state

and the final state are expressed with using the creation and annihilation operators c ∗ k and c k

Here, the anticommutation relation of c ∗ k and E k = ε2

k+Δ2are used The second term ofequation (21) is canceled out when this term is integrated over the Fermi surface, becauseε k

is the energy from the Fermi energy The counter process that a down spin flips to an upspin should also be considered When this process is included, the transition probabilitybecomes twice of| f |Hp| i |2 The third termΔ2/E k E k  possesses finite value only for an

s-wave superconductor with an isotropic superconducting gap, such as Al metal Hebel &

Slichter (1959) This term vanishes when the superconducting pairing symmetry is p-wave and two-dimensional d-wave type, which are represented in equations (12), (13) and (14).

In unconventional superconductors with an anisotropic order parameter, the temperature

dependence of 1/T1 shows power-law behavior, because the quasiparticle density of statesexist even below the maximum value of the gapΔ0 The existence of the density of states

in the small energy region originates from the nodes on the superconducting gap In the

p-wave axial state, where the superconducting gap possesses point nodes at θ =0,π, N(E)

is proportional to E2near E=0 As a result, 1/T1is proportional to T5far below Tc In the

p-wave polar state and the d-wave state, where the superconducting gap possesses line nodes

atθ =π/2 and φ =0,π/2, respectively, the temperature dependence of 1/T1becomes T3,

which is derived from the linear energy dependence of N(E)in the low energy region The

observation of the power-law behavior in the temperature dependence of 1/T1far below Tcisstrong evidence for the presence of nodes on the superconducting gap, and therefore, for theunconventional superconductivity

4 Ground states of Nax(H3O)zCoO2·yH2O

4.1 Superconductivity

In order to investigate the symmetry of superconducting order parameter, 1/T1was measured

at zero field using NQR signal of the best superconducting sample with T c = 4.7 K The

temperature dependence of 1/T1observed in the superconducting state shows a power-law

behavior as shown in Fig 4 Ishida et al (2003) This power-law decrease starts just below T c

without any increase due to Hebel-Slichter mechanism, and gradually changes the exponent

from ∼ 3 at half of T c to unity below 1 K The overall temperature dependence can be

sufficiently fitted by the theoretical curve assuming the two-dimensional d-wave pairing

state with the gap size 2Δ/kB T c = 3.5 and residual density of state Nres/N0 ∼ 0.32 Theunconventional superconductivity with nodes on the superconducting gap is concluded forthe superconductivity in BLH compounds The residual density of states are generated bytiny amount of impurities inherent in the powder samples, because superconducting gapdiminishes to zero along certain directions in the unconventional superconductors

Next, the normal-state temperature dependence of 1/T1T for BLH compound is compared

with those of non-superconducting MLH and anhydrous compounds in order to identify

the origin of superconductivity Surprisingly, 1/T1T in MLH is nearly identical to that in

anhydrous cobaltate, even though the water molecule content is considerably different In

these compositions, gradual decrease in 1/T1T from room temperature terminates around

100 K, below which Fermi-liquid like Korringa behavior is observed This gradual decrease

in 1/T1T at high temperatures is reminiscent of the spin-gap formation in cuprate Alloul

et al (1989); Takigawa et al (1989) The temperature dependence of the MLH and anhydroussamples are mimicked by

1

0 10

Bilayer hydrate Monolayer hydrate

Fig 4 (a) Temperature dependence of 1/T1measured at zero magnetic field Ishida et al

(2003) Red solid curve is a theoretical fit to the date below T c=4.7 K, for which line nodes

on the superconducting gap are taken into account (b) Temperature dependence of 1/T1T in

superconducting BLH compound, MLH compound, and anhydrous cobaltate Na0.35CoO2.The experimental data of the anhydrous cobaltate was reported by Ning et al (2004) Thedashed line represents the sample-independent pseudogap contribution, which is expressed

in equation (22) Ihara et al (2006)

In contrast to the sample independent high-temperature behavior, the Korringa behavior

is not observed in the superconducting BLH samples and, instead, magnetic fluctuations

increase approaching to T c The sample independent high-temperature behavior andstrongly sample dependent low-temperature behavior lead us to conclude that the pseudogap

contribution robustly exists in all phases, and the increasing part of 1/T1T detected only in

the superconducting BLH sample is responsible for the superconducting pairing interactions.Multi orbital band structure of cobaltate allows to coexist strongly and weakly correlated

bands in a uniform system The sophisticated analyses based on the sample dependent 1/T1T

measurements are made in § 5

Immediately after the water filtration for BLH compounds, some samples do not showsuperconductivity Superconductivity appears, even for these samples, after a few days ofduration NQR experiment on freshly hydrated non-superconducting samples has revealed

that magnetic ordering sets in below T M = 6 K Ihara, Ishida, Michioka, Kato, Yoshimura,Takada, Sasaki, Sakurai & Takayama-Muromachi (2005) This magnetism was evidenced from

divergence of 1/T1at T M(Fig 5(a)) and NQR spectral broadening (Fig 5(b))

In a case of conventional Néel state, in which the same size of ordered moments are arrangedantiferromagnetically, the internal-field strength could be uniquely determined from the splitNQR spectra The frequency separation between the split spectra is converted to internal fieldstrength through the frequency-field relation derived from the Hamiltonian introduced byequation (3) For the cobaltate, however, the NQR spectrum does not split but just broadensdue to the distributing internal fields We have succeeded in extracting the distribution of theinternal fields by taking the process explained below

First, we determined the direction of the internal field to Hint ⊥ c, where the c axis is

the principal axis of the electric field gradient, because the resonance peak arising from

m = ±5/2↔ ± 3/2 transitions become broader than that arising from m = ±7/2↔ ±5/2

transitions If the internal fields were along the c axis, ±7/2 ↔ ±5/2 transition line wouldbecome the broadest within the three NQR lines Obviously, this is not a case The anisotropicbroadening of±7/2↔ ± 5/2 transitions is also consistently explained by assuming Hint⊥ c.

These two results suggest that the internal fields direct to the ab plane.

Next, the intensity of the internal fields is estimated using frequency-field map derived fromequation (3), and shown in Fig 2 The details of the analyses are described in the separatepaper Ihara et al (2008) For the estimation of the fraction, we used the NQR spectra in thefrequency range of 7∼10.5 MHz, because almost linear relationship was observed betweenthe frequency and internal field in this frequency range The internal field profile is exhibited

in the inset of Fig 5

It is found that the maximum fraction is at zero field, and that rather large fraction is in thesmall field region We also point out a weak hump around 0.15 T, which corresponds to 0.1μB

when we adopt 1.47 T/μBas the coupling constant Kato et al (2006) It should be noted thatNQR measurements were performed for Co nuclei, where the magnetic moments are located.The distribution of the hyperfine fields at the Co site suggests that the size of the orderedmoments has spatial distributions The magnetic ordering with the modulating orderedmoments is categorized to the spin-density-wave type with incommensurate ordering vector

In order to investigate the magnetic structure in detail, neutron diffraction measurements arerequired

4.3 Phase diagram

The ground state of Nax(H3O)zCoO2· yH2O strongly depends on the chemical compositions,

Na ion (x), oxonium ion H3O+ (z) and water molecule (y) contents The samples evolve

drastically after water intercalation, as the water molecules evaporate easily into the air,when the samples were preserved in an ambient condition This unstable nature causesthe sample dependence of various physical quantities The sample properties of the fragileBLH compounds have to be clarified in detail both from the microscopic and macroscopicmeasurements before the investigation on ground states

In order to compare the physical properties of our samples to those of others, thesuperconducting transition temperatures reported in the literature Badica et al (2006); Barnes

et al (2005); Cao et al (2003); Chen et al (2004); Chou, Cho, Lee, Abel, Matan & Lee (2004); Foo

et al (2005; 2003); Ihara et al (2006); Jin et al (2003; 2005); Jorgensen et al (2003); Lorenz et al.(2004); Lynn et al (2003); Milne et al (2004); Ohta et al (2005); Poltavets et al (2006); Sakurai

et al (2005); Schaak et al (2003); Zheng et al (2006) are plotted against the c-axis length of each sample in Fig 6(a) A relationship was observed between T c and the c-axis length The scattered data points indicate that the c-axis length is not the only parameter that determines the ground state of the BLH compound The c-axis length, however, behaves as a dominant

parameter, and can be a useful macroscopic reference to compare the sample properties ofvarious reports In the figure, superconductivity seems to be suppressed in some samples with

c ∼19.75 Å, probably due to the appearance of magnetism The magnetism was reported onsamples in the red region Higemoto et al (2006); Ihara, Ishida, Michioka, Kato, Yoshimura,Takada, Sasaki, Sakurai & Takayama-Muromachi (2005); Sakurai et al (2005) We also found,

in some samples, that both the magnetic and superconducting transitions are observed Ihara

et al (2006) It has not been revealed yet how these two transitions coexist in one sample

et al (20 05); Cao et al (20 03); Chen et al (20 04); Chou, Cho, Lee, Abel, Matan & Lee (20 04); Foo

et al (20 05; 20 03); Ihara et al (20 06); Jin et al (20 03; 20 05);... et al (20 03); Lorenz et al. (20 04); Lynn et al (20 03); Milne et al (20 04); Ohta et al (20 05); Poltavets et al (20 06); Sakurai

et al (20 05); Schaak et al (20 03); Zheng et al (20 06) are... ±5 /2< i>↔ ± 3 /2 transitions become broader than that arising from m = ±7 /2< i>↔ ±5 /2

transitions If the internal fields were along the c axis, ±7 /2 ↔ ±5 /2 transition