# Determination of The Spin Contributed Knight Shift in $Na_{0.35} Co O_2 \cdot 1.3 H_2 O$

**ABSTRACT** The spin contributed Knight shift $^{59}K_{s}$ of $^{59}\mathrm{Co}$ in the triangular lattice superconductor, $\mathrm{Na_{0.35}CoO_{2} \cdot 1.3H_{2}O}$, has been estimated by NMR experiments in an oriented 2-dimensional powder sample. The Knight shift in the paramagnetic state was estimated by $\delta\nu/(\gamma\mathrm{_{N}}H_{i})$ vs. $(\gamma\mathrm{_{N}}H_{i})^{-2}$ plot at various temperatures. A Knight shift vs. magnetic susceptibility diagram revealed that at least 0.3 % for $K_{x}$ and 0.1 % for $K_{y}$ are attributed to the spin contribution. Below $T\mathrm{_{c}}$, $\delta\nu/(\gamma\mathrm{_{N}}H_{i})$ vs. $(\gamma\mathrm{_{N}}H_{i})^{-2}$ plot does not give unique Knight shift because of the diamagnetic effect and/or the other ones. We concluded that the invariant behavior at about 7 T (between $H\mathrm{_{c1}}$ and $H\mathrm{_{c2}}$) below $T\mathrm{_{c}}$ possibly indicates the nature of the spin-triplet superconductivity.

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arXiv:cond-mat/0403293v1 [cond-mat.supr-con] 11 Mar 2004

Determination of Spin Contributed Knight Shift of59Co NMR in Na0.35CoO2· 1.3H2O

Chishiro Michioka, Masaki Kato, and Kazuyoshi Yoshimura

Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan

Kazunori Takada, Hiroya Sakurai, Eiji Takayama-Muromachi, and Takayoshi Sasaki

Advanced Materials Laboratory, National Institute for Materials Science, Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan

(Dated: February 2, 2008)

The spin contributed Knight shift

Na0.35CoO2· 1.3H2O, has been estimated by NMR experiments in an oriented 2-dimensional pow-

der sample. The Knight shift in the paramagnetic state was estimated by δν/(γNHi) vs. (γNHi)−2

plot at various temperatures. A Knight shift vs. magnetic susceptibility diagram revealed that at

least 0.3 % for Kx and 0.1 % for Ky are attributed to the spin contribution. Below Tc, δν/(γNHi)

vs. (γNHi)−2plot does not give unique Knight shift because of the diamagnetic effect and/or the

other ones. We concluded that the invariant behavior at about 7 T (between Hc1 and Hc2) below

Tc possibly indicates the nature of the spin-triplet superconductivity.

59Ks of

59Co in the triangular lattice superconductor,

Superconductivity in Na0.35CoO2· 1.3H2O with Tc∼ 5

K was discovered recently. [1] In this compound, the tri-

angular configuration in CoO2plane has a possibility of

an unconventional superconductivity in the novel quan-

tum state. Indeed, many theoretical models for the CoO2

plane have been investigated, and pointed out a possibil-

ity that this superconductivity originates in a marvelous

superconducting mechanism. For example, Tanaka et al.

indicated that the spin-triplet superconductivity may be

realized with similar pairing process to Sr2RuO4 since

the hexagonal structure in Na0.35CoO2· 1.3H2O gives a

stronger spin-triplet pairing tendency. [2] On the other

hand, some groups studied a single band t-J model as

an appropriate model to understand the low energy elec-

tronic phenomena resulting that the superconductivity

in Na0.35CoO2· 1.3H2O occurs related to the resonating-

valence-bond (RVB) state.

Ikeda et al.investigated superconducting instabilities

on a 2D triangular lattice with the repulsive Hubbard

model. Their result shows that the f-wave pairing with

the triplet superconductivity is stable in the wide range

of the phase diagram. [7] Quite recently, Kuroki et al.

introduced a single band effective model with taking into

account the pocket-like Fermi surfaces along with the

van Hove singularity near the K point, and showed that

the large density of states near the Fermi level gives rise

to ferromagnetic spin fluctuations, leading to a possible

realization of the f-wave superconductivity due to the

disconnectivity of the Fermi surfaces near the Γ point.

[8, 9, 10, 11]

[3, 4, 5, 6] Furthermore,

In order to clarify the superconducting mechanism in

Na0.35CoO2· 1.3H2O, it is very important to investigate

the Cooper-pair symmetry. Nuclear magnetic resonance

(NMR) is known as the powerful probe to decide the

pairing symmetry in the superconducting state. The nu-

clear quadrupole resonance (NQR) studies of59Co re-

vealed that there is no coherence peak in the temper-

ature dependence of the nuclear spin-lattice relaxation

rate 1/T1suggesting that the unconventional supercon-

ductiving state with line-node gap. [12, 13] The present

authors studied 1/T1 up to 7.12 T by59Co NMR, and

found that the transition temperature at 7.12 T is about

4.4 K. [14] This anomalously small decrease of Tcsuggests

large Hc2, which is consistent with macroscopic proper-

ties. [15] Moreover, field-swept NMR spectra suggested

that the resonance magnetic fields in constant frequencies

do not change below Tcat about H = 7 T which is far

below Hc2. This means that the Knight shift is invariant

below Tc, which results in the spin-triplet superconduc-

tivity in this system. However, this result is inconsistent

with those reported by Kobayashi et al., which suggested

spin-singlet s-wave superconductivity. [16] It is generally

difficult to determine NMR parameters including Knight

shift uniquely from a field-swept spectrum taken at only

one constant frequency. In this paper, we first present

the temperature dependence of the Knight shift above

Tcand show how to estimate its spin part, which is very

important for the superconducting state as well. Next

we discuss the spin contributed Knight shift below Tc, in

which the diamagnetic effect would be considered.

The powder sample of Na0.35CoO2· 1.3H2O was pre-

pared by the oxidation process from the mother com-

pound, Na0.7CoO2. [1] The superconducting transition

temperature (Tc) is estimated by the macroscopic mag-

netization using SQUID magnetometer.59Co NMR spec-

tra were measured by the spin-echo method with a phase

coherent-type pulsed spectrometer. The powder sample

was oriented and fixed by using an organic solvent Hex-

ane under the condition of H = 8 T. [17] The X-ray

diffraction measurement indicated that c-axes of pow-

ders were oriented perpendicular to the external mag-

netic field H, and random orientation in the basal plane

was obtained along H. Therefore our NMR measure-

ments were done on the two-dimensional (2D) powder

sample in the condition with the magnetic field being

perpendicular to the c-axis.

From the following two reasons, we took care of the

following experimental condition that once the sample

Page 2

2

Intensity(arb.unit)

7.187.167.14 7.127.107.08

H(T)

T=50 K, ν=74.2 MHz

FIG. 1: Field-swept NMR spectrum of Na0.35CoO2· 1.3H2O

at T = 50 K and ν = 74.2 MHz.

decomposed by the Gaussian fitting indicated by the dashed

lines, correspond to the central resonance of Iz = −1/2 ↔ 1/2

transition. The analysis of the anisotropic Knight shifts Kx

and Ky were done using the low-field smaller peak and the

high-field larger peak, respectively (see the text).

Two peaks, which are

has cooled, and then the temperature of the chamber

was kept below 50 K, until all the experiments had been

finished. The first reason is that the specific resonance

magnetic field corresponding to a peak on the field-swept

NMR spectrum should be changed when the condition of

the sample orientation was changed. In other words, a

degree of the orientation should not be changed during

whole measurements. The second is that the change of

the site and its occupancy of the water molecules within

the sample may change the electronic state as well as the

quadrupole frequency νQbecause the increase of the tem-

perature enables water molecules to move. In this case,

the intrinsic Knight shift cannot be obtained correctly.

Figure 1 shows the typical central peak of the field-

swept spectrum, which corresponds to the central transi-

tion (Iz= −1/2 ↔ 1/2) at T = 50 K and ν = 74.2 MHz.

The whole spectrum and its assignment are explained in

detail in a different paper. [14] In this paper, we carefully

discuss the resonance field depending upon the resonance

frequency in order to determine the value of Knight shift

uniquely.

In this process, we assume that principal axes of the

Knight shift tensor coincide with those of the electric-

field gradient (EFG) tensor.

since we measured on a 2D-powder sample, we could not

estimate the Z component of the Knight shift. In gener-

ally, five specific resonance frequencies of the central tran-

sition appear in a constant field on the powder sample for

the case of η <1

3, where η is the asymmetric parameter

of the EFG tensor defined as η = |VXX− VY Y|/|VZZ|,

in which Vii(ii = XX, Y Y , ZZ) are the principal com-

ponents of the EFG tensor with the relation, |VXX| ≤

|VY Y| ≤ |VZZ|.[18] That is, five specific fields (H1-

H5) appear as two peaks, one step and two edges in a

constant frequency. In the case of the 2D-powder, two

specific resonance fields (H1and H2) appear as singular

points, and the rest specific fields disappear because of

the 2D orientation. Hence the fields, at which the field-

swept spectrum shows peaks as shown in Fig. 1, should

correspond to H1 and H2. If the quadrupole effect is

Under this assumption,

4.8

4.6

4.4

4.2

4.0

3.8

3.6

3.4

10-2δν/(γHi)

6543

1/(γHi)2 (10-4MHz-2)

2

Hi (4.2 K)

Hi (10 K)

Hi (20 K)

Hi (30 K)

Hi (40 K)

Hi (50 K)

H1

H2

4.4

4.0

3.6

3.2

10-2δν/(γHi)

6420

1/(γHi)2 (10-4MHz-2)

H1

H2

T = 4.2 K

FIG. 2:

temperatures above Tc and 4.2 K (just below Tc) in the fre-

quency range 44.2 - 74.4 MHz. H1 and H2 are the resonance

fields at which the59Co NMR spectrum with 2D-powder pat-

tern of the central transition show two singular points. Inset

shows data at 4.2 K with the fitted lines using the NMR pa-

rameters, νQ = 4.15 MHz, η = 0.21, Kx = 3.56 % and Ky =

3.21 %.

δν/(γNHi) vs. (γNHi)−2(i = 1, 2) plot at various

treated by the second-order perturbation, the linear re-

lation between δνi/γNHiand (γNHi)−2can be obtained

as follows,

δνi

γNHi

= Kj+

Ci

(1 + Kj)(γNHi)2,(1)

δν = ν0− γNHi

(2)

C1=R(3 + η)2

144

,C2=R(3 − η)2

144

, (3)

R = ν2

Q[I(I + 1) −3

4],(4)

where the values with j = X,Y correspond to those with

i = 1, 2, respectively, γNis the nuclear gyromagnetic ra-

tio (10.054 MHz/T for59Co with the nuclear spin of I =

7/2), ν0the resonance frequency and νQthe quadrupole

frequency. Figure 2 shows δν/(γNHi) vs. (γNHi)−2(i

= 1, 2) plot of Na0.35CoO2· 1.3H2O above Tc and 4.2

K in the frequency range 44.2 - 74.4 MHz.

can estimate anisotropic Knight shift uniquely as an in-

terception of the δνi/γNHi axis in this plot under the

assumption that the Knight shift is independent of H.

The resonances of H1and H2show clear linearity in this

plot, and Kxand Ky(interceptions of the δνi/γNHiaxis)

Here, we

Page 3

3

3

2

1

0

Kx, Ky (% )

600x10-6

400200

χ(emu/mol f.u.)

0

Kx

Ky

Kvv,y

Kvv,x

χvv,x

χvv,y

χdia

3.5

3.4

3.3

3.2

Kx, Ky (%)

600x10-6

500

χ(emu/mol f.u.)

T = 4.2 K

χspin

Kspin

FIG. 3:

Na0.35CoO2· 1.3H2O. The data at 4.2 K (below Tc) is shown

for comparison but not used for the line fitting. Inset shows

the expansion.The numerical details for constructing the

diagram are given in the text.

Knight shift vs. susceptibility diagram in

change with T while the slopes are almost independent

of T, which is consistent with the fact that the NQR

frequency is almost constant in these temperatures, be-

cause the slope Ci is determined mainly by quadrupole

parameters. [19] The inset in Fig. 2 shows the fitting

using Eqs. (1)-(4) at 4.2 K. Slopes of the two lines de-

pending on Eq. (2) give parameters of the quadrupole

interaction, νQ= 4.15MHz and η = 0.21, and these val-

ues are consistent with the resonance fields of the steps of

the first, second and third satellites appeared in a whole

spectrum. [14] As a result, Kx and Ky decrease with

increasing temperature, which have scaling relationship

with the magnetic susceptibility. Therefore we can esti-

mate the contribution of each interaction for the Knight

shift from the Knight shift vs. susceptibility (K−χ) plot

shown in Fig. 3.

In terms of the corresponding susceptibilities three

contributions to the Knight shift can be expressed as

K(T) = Kspin(T) + Kce+ KV V, (5)

where the temperature-depending term Kspin(T) and the

temperature independent term Kceare attributed to the

spin susceptibilities of d spins and conduction electrons

or holes, respectively, and the last term is due to the

orbital susceptibility of d electrons.

susceptibilities of core electrons can be estimated from

relativistic Hartree-Fock calculations, in which χH

−0.4 × 10−9µB/atom, χO

χNa

µB/atom. [20] The total diamagnetic susceptibility of

Na0.35CoO2· 1.3H2O was described in Fig. 3 as χdia.

The diamagnetic

dia=

dia= −1.6 × 10−9µB/atom,

dia= −5.5 × 10−9

dia= −3.8 × 10−9µB/atom and χCo

Intensity (arb.unit)

4.284.24 4.204.16

H (T)

= 44.2 MHz

ν

2K

4.2K

Intensity (arb.unit)

4.664.64 4.624.60

H (T)

4.584.56

= 48.2 MHz

ν

2K

4.2K

Intensity (arb.unit)

5.125.08

H (T)

5.04

= 53.2 MHz

ν

2K

4.2K

Intensity (arb.unit)

5.725.685.64 5.60

H (T)

= 59.2 MHz

ν

2K

4.2K

Intensity (arb.unit)

6.386.366.34

H (T)

6.326.30

= 66.2 MHz

ν

2K

4.2K

Intensity (arb.unit)

7.167.147.12

H (T)

7.10 7.087.06

= 74.2 MHz

ν

2K

4.2K

FIG. 4: Field-swept 2D-powder spectra for the central tran-

sition at 2.0 and 4.2 K in several constant frequencies ν.

The hyperfine coupling constant for the d orbital mo-

ment, AV V

hf=

as a product of 2/?r−3? = 9.05 × 1025cm−3for Co3+

and the reduction factor of 0.8 for the metallic state

compared with the free atom. Because the strict band

structure of Na0.35CoO2· 1.3H2O is unknown, Kcecan-

not be estimated correctly. We ignore the small con-

tribution by the conduction electrons because the con-

tribution is much smaller than the uncertainty of the

reduction factor. The coupling constants for the tem-

perature dependent Knight shift components, Ax

Ay

hfare estimated as 34 kOe/µB and 14.7 kOe/µB by

the least-square fitting shown as solid lines in Fig. 3.

Then χV V,xand χV V,yare estimated as 2.0 × 10−6and

1.8 × 10−6emu/mol f.u., respectively, and KV V,x and

KV V,y, 3.3 and 3.1 %, respectively. Therefore spin con-

tributed Knight shifts are at least 0.3 % for Kxand 0.1

% for Ky.

Next we discuss about the temperature dependence of

the Knight shift below Tc. Figure 4 shows the central

peaks in the field-swept spectra at 2.0 and 4.2 K at sev-

eral constant frequencies (44.2 - 74.2 MHz). While the

resonance fields are almost the same between 2.0 and 4.2

K at ν = 74.2 MHz, it changes drastically at ν = 44.2

MHz. The differences of the resonance fields between 2.0

and 4.2 K increase gradually with decreasing the reso-

nance frequency. In the previous paper, we showed that

the field H = 7 T is obviously smaller than Hc2from the

temperature dependence of 1/T1T, in which the marked

decrease of 1/T1T due to the decrease of the density of

states around Fermi surface in the superconducting state

appears below 4.4 K. As well as in the paramagnetic

state, we discuss about the δν/(γNH2) vs. (γNH2)−2

plot in the superconducting state shown in Fig. 5. We

KV V

χV V= 6.7 × 102kOe/µB was estimated

hfand

Page 4

4

4.0

3.8

3.6

3.4

3.2

10-2δν/(γHi)

6543210

1/(γHi)2 (10-4MHz-2)

4.4K

4.0K

3.6K

3.2K

2.8K

2.4K

2.0K

FIG. 5:

tures below Tc. Lines in the figure are guide for eyes. The

linear correlation which is characteristic above Tc disappears

below Tc. This is not due to the change of νQ or η but due

to the effect of the diamagnetization.

δν/(γNH2) vs. (γNH2)−2plot at various tempera-

cannot detect H1because of the difficulty in decomposing

these spectra especially at lower frequencies. The plot is

almost linear at 4.4 K as similar to those above Tc, in

which the slope is consistent with the fact that νQ and

η do not change in these temperatures. Since νQand η

do not change also below Tc, it makes no physical sense

to fit the data by the line having different slope below

Tc. In this case, if one estimates the Knight shift from

the resonance field taken at only one constant frequency,

the intrinsic value of the Knight shift should not be es-

timated accurately. Two explanations can be adopted

about the anomalous behavior below Tc. One is that the

Knight shift depends on H and the other is the diamag-

netic or the other unexpected effects. The possibility of

the former case is, however, excluded because the intrin-

sic Knight shift in the superconducting state should not

change even if the resonance field on appearance changes

due to the distribution between Knight shift of the in-

trinsic superconducting and normal state, in which H

penetrates. The fact that the difference of the resonance

fields between 2.0 and 4.4 K is decreasing with increas-

ing the resonance frequency coincides qualitatively to the

diamagnetic effects. In addition to this fact, since the res-

onance field is almost the same between at 2.0 and 4.2

K at ν = 74.2 MHz, the intrinsic behavior of the Knight

shift in the superconducting state is invariant, suggesting

that the symmetry of the Cooper pair of the supercon-

ducting state in Na0.35CoO2· 1.3H2O is of the p or f

wave with the triplet one. The result is consistent with

the µSR study on Na0.35CoO2· 1.3H2O. [21]

We estimated the spin contributed Knight shift of

Na0.35CoO2· 1.3H2O. In the paramagnetic region, the

spin contributed Knight shifts are at least 0.3 and 0.1

% for Kx and Ky. In the superconducting state, the

Knight shift is not estimated correctly, however the in-

trinsic Knight shift should not change at the external field

up to at least 7 T. From these results, we concluded that

the p or f wave pairing state with triplet spin symmetry

may be most preferable in the superconducting state of

Na0.35CoO2· 1.3H2O.

cussions. This study was supported by a Grant-in-Aid

on priority area ’Novel Quantum Phenomena in Transi-

tion Metal Oxides’, from Ministry of Education, Science,

Sports and Culture (12046241), and also supported by

a Grant-in-Aid Scientific Research of Japan Society for

Promotion of Science (12440195, 12874038)

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