chniquem is stilld

in mixedtrum,cation ofextent of

e studyency-heing gap..lves inion andampedenergy

Physics Letters A 324 (2004) 203–210

www.elsevier.com/locate/pla

Equations for collective modes spectrum in a mixedd-wave stateof unconventional superconductors

C.Y. Lee

Department of Physics, Hannam University, Taejon 300-791, South Korea

Received 1 February 2004; received in revised form 15 February 2004; accepted 16 February 2004

Communicated by V.M. Agranovich

Abstract

Direct observation of the collective modes in unconventional superconductors (USC) by microwave impedance teexperiments has made the very important study of the collective excitations in these systems. One of the problethe exact form of the order parameter of unconventional superconductors. Among the possibilities there are extendes-wavepairing, mixture ofs- andd-states, as well as of differentd-wave states. I consider the mixed(1−γ )dx2−y2 +iγ dxy state in hightemperature superconductors (HTSC) and derive for the first time a full set of equations for collective modes spectrumd-wave state with arbitrary admixture ofdxy state. Obtained results allow to calculate the whole collective mode specwhich could be used for interpretation of the sound attenuation and microwave absorption data as well as for identifithe type of pairing and order parameter in unconventional superconductors. In particular, they allow to estimate theadmixture ofdxy state in a possible mixed state. 2004 Elsevier B.V. All rights reserved.

PACS: 74.20.-z; 74.25.-q; 74.76.Bz

1. Introduction

Recently Northwestern University group [1] has presented results of a microwave surface impedancof the heavy fermion superconductor UBe13. They clearly have observed an absorption peak whose frequand temperature-dependence scales with the BCS gap function∆(T ). This was the first direct observation of tresonant absorption into a collective mode, with energy approximately proportional to the superconductThis discovery opens a new page in study of the collective excitations in unconventional superconductors

The significance of studying of collective modes connects to the fact that they exhibit themseultrasound attenuation [2] and microwave absorption [1] experiments, neutron scattering, photoemissRaman scattering [3]. The large peak in the dynamical spin susceptibility in HTSC arises from a weakly dspin density-wave CM. This gives rise to a dip between the sharp low-energy peak and the higher binding

E-mail address: cylee@mail.hannam.ac.kr (C.Y. Lee).

0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2004.02.052

204 C.Y. Lee / Physics Letters A 324 (2004) 203–210

ve

omet

time

n: do we

e certain] which

thatossible

with theobe ofcrowave

r thef

lex

hump in the ARPES spectrum. Also, the CM of amplitude fluctuation of thed-wave gap yields a broad peak abothe pair-breaking threshold in theB1g Raman spectrum [4].

While now the type of pairing is established for a lot of unconventional superconductors (s-pairing is realized inelectron-type HTSC (Nd2−xCexCuO4) and MgB2; d-pairing in a hole-type HTSC, organic superconductors, sHFSC (UPd2Al3, CePd2Si2, CeIn3, CeNi2Ge2, etc.) andp-pairing in Sr2RuO4 (HTSC), UPt3 (HFSC)) the exacform of the order parameter is still unknown for the most of unconventional superconductors.

The most scientists believe that there is ad-wave pairing in a hole-type oxide superconductors. At the samethe different ideas concerning extendeds-wave pairing, mixture ofs- andd-states, as well as of differentd-wavestates still discuss actively. One of the cause of such a situation is the uncertainty in answer the questiohave exact zero gap along some chosen lines in momentum space (like the case ofdx2−y2) or gap is anisotropicbut nonzero everywhere (except maybe some points). Existing experiments (tunneling etc.) do not give thanswer this question while the answer is quite principle. From other side there are some experiments [5could be explained [6,7] under suggestion about realization in HTSC of a mixed states, likedx2−y2 + idxy . Annettet al. [7] considered the possibility of mixture of differentd-wave states in HTSC and came to conclusionmixture ofdx2−y2 anddxy states is the most likely state. Brusov and Brusov [8] have suggested one of the pways to distinguish the mixture of twod-states from pured-states. For this they considered the mixeddx2−y2 + idxystate and calculated the spectrum of collective modes in this state. The comparison of this spectrumspectrum of a pured-wave states of HTSC shows that they are significantly different and could be the prthe symmetry of the order parameter in HTSC. Thus the probe of the spectrum in ultrasound and/or miabsorption experiments could be used to distinguish the mixture of twod-wave states from pured-wave states.

2. Model for mixed state

I have generalized Brusov and Brusov [8] consideration for the case of arbitrary admixture ofdxy state. Iconsider the mixed(1 − γ )dx2−y2 + iγ dxy state in high temperature superconductors (HTSC) and derive fofirst time a full set of equations for collective modes spectrum in mixedd-wave state with arbitrary admixture odxy state.

I have used the model ofd-pairing in HTSC and HFSC, created by Brusov [3].It is described by the effective functional of actionSeff

(1)Seff = g−1∑p,i,a

c+ia(p)cia(p)+ 1

2lndet

M(cia, c+ia)

M(c(0)ia , c

(0)+ia )

,

wherec(0)ia is the condensate value of Bose-fieldscia (symmetric traceless matrix) andM(cia, c+ia) is the 4× 4

matrix depending on Bose-fields and parameters of quasifermions.The number of degrees of freedom in the case ofd-pairing is equal to 10, i.e., one must have five comp

canonical variables, which can be naturally chosen in the form

c1 = c11 + c22, c2 = c11 − c22, c3 = c12 + c21, c4 = c13 + c31, c5 = c23 + c32.

In the canonical variables, the effective action has the form

(2)Seff = (2g)−1∑p,j

c+j (p)cj (p)(1+ 2δj1)+ 1

2lndet

M(c+j , cj )

M(c+(0)j , c

(0)j )

,

where

M11 =Z−1[iω+ ξ −µ(Hσ )]δp1p2, M22 = Z−1[−iω+ ξ +µ(Hσ )

]δp1p2,

C.Y. Lee / Physics Letters A 324 (2004) 203–210 205

to

the

the

(3)

M12 =M∗21 = (βV )−1/2

(15

32π

)1/2[c1

(1− 3 cos2 θ

) + c2 sin2 θ cos2ϕ + c3 sin2 θ sin2ϕ

+ c4 sin 2θ cosϕ + c5 sin2θ sinϕ],

wherep = (k,ω), ω = (2n+ 1)πT are Fermi-frequencies andx = (x, τ ), ξ is the kinetic energy with respectFermi-level,µ is magnetic moment of quasifermion,H is magnetic field,σ = (σ1, σ2, σ3)—Pauli-matrices.

This functional determines all the properties of the model superconducting Fermi-system withd-pairing.Let us consider the mixed(1 − γ )dx2−y2 + iγ dxy state of HTSC. The order parameter in this state takes

following form

(4)∆0(T )

(1− γ )

1 0 0

0 −1 00 0 0

+ iγ

0 1 0

1 0 00 0 0

,

or in canonical variables

(5)∆0(T )(0; (1− γ )sin2 θ cos2ϕ; iγ sin2 θ sin 2ϕ;0;0

).

The gap equation has the following form

(6)g−1 + α2Z2

2βV

∑p

sin4Θ[γ 2 + cos2 2φ(1− 2γ )]ω2 + ξ2 +∆2

0 sin4Θ[γ 2 + cos2 2φ(1− 2γ )] = 0,

where

(7)∆0 = 2cZα, α = (15/32π)1/2, and gap ∆2(T )=∆20 sin4Θ

[γ 2 + cos2 2φ(1− 2γ )

].

For limited caseγ = 0 one getsdx2−y2 state with order parameter(0;sin2 θ cos2ϕ;0;0;0).The gap equation in this case has the following form

(8)g−1 + α2Z2

2βV

∑p

sin4Θ cos2 2φ

ω2 + ξ2 +∆20 sin4Θ cos2 2φ

= 0, and gap

(9)∆2(T )=∆20 sin4Θ cos2 2φ.

For limited caseγ = 1 one getsdxy state with order parameter∆0(T )(0;0; i sin2 θ sin 2ϕ;0;0).The gap equation has the following form

(10)g−1 + α2Z2

2βV

∑p

sin4Θ sin2 2φ

ω2 + ξ2 +∆20 sin4Θ sin2 2φ

= 0, and gap

(11)∆2(T )=∆20 sin4Θ sin2 2φ.

Brusov and Brusov [8] case of equal admixtures ofdx2−y2 anddxy states in our consideration corresponds tocaseγ = 1/2. The order parameter takes the following form

(12)∆0(T )(0;sin2 θ cos2ϕ; i sin2 θ sin2ϕ;0;0

).

The gap equation has the following form

(13)g−1 + α2Z2

2βV

∑p

sin4Θ

ω2 + ξ2 +∆20 sin4Θ

= 0, and gap ∆(T )=∆0(T )sin2 θ.

206 C.Y. Lee / Physics Letters A 324 (2004) 203–210

of

ions of

esultingof the

3. Equations for spectrum of collective excitations

The spectrum of collective excitations in the first approximations is determined by the quadratic partSeff,obtained after shiftcj → cj + c0

j wherec0j are the condensate values ofcj , which take the following form [3],

c0j (p)= (βV )1/2cδp0b

0j and b0

2 = 2(1− γ ), b03 = 2iγ

with all remaining components ofb0j equal to zero.

Excluding terms involvingg−1 by gap equation, one obtains the following form for the quadratic part ofSeff

Seff = α2Z2

8βV

∑p

[c0Y ∗][c+0Y ]ω2 + ξ2 + [c0Y ∗][c+0Y ]

∑j

(1+ 2δj1)c+j (p)cj (p)

(14)

+Z2/4βV∑

p1+p2=p

1

M1M2

(iω1 + ξ1)(iω2 + ξ2)

([c+(p)Y (p2)

][c(p)Y ∗(p1)

]+ [

c+(p)Y (p1)][c(p)Y ∗(p2)

])−∆2[c+(p)Y (−p1)][c+(−p)Y (−p2)

]−∆+2[c(p)Y ∗(−p1)

][c(−p)Y ∗(−p2)

].

Here[cY ∗] = c1(1− 3 cos2 θ)+ c2 sin2 θ cos2ϕ + c3 sin2 θ sin2ϕ + c4 sin2θ cosϕ + c5 sin 2θ sinϕ.The coefficients of the quadratic form are proportional to the sums of the products of Green’s funct

quasifermions. At low temperatures (Tc − T ∼ Tc) we can go from a summation to an integration by the rule

(15)1

βV

∑p

→ 1

(2π)4k2F

cF

∫dωdξ dΩ.

To evaluate these integrals it is useful to use Feynman equality

(16)[(ω2

1 + ξ21 +∆2)(ω2

2 + ξ22 +∆2)]−1 =

∫dα

[α(ω2

1 + ξ21 +∆2) + (1− α)

(ω2

2 + ξ22 +∆2)]−2

.

It is easy to evaluate the integrals with respect to variablesω andξ and then with respect to parameterα and theangular variables.

After calculating all integrals except over the angular variables and equating the determinant of the rquadratic form to zero one gets the following set of ten equations, which determine the whole spectrumcollective modes for(1− γ )dx2−y2 + iγ dxy state at arbitraryγ :

1∫0

dx

∫dϕ

√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )]

ωln

√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] +ω√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] −ω

× (1− 3x2)2 +

[(1− 3x2)2 − 3

8

((1− 3x2)2 + 3

(1− x2)2 cos2 2φ

)]

× ln(1− x2)2[

γ 2 + cos2 2φ(1− 2γ )] = 0,

1∫0

dx

∫dϕ

ω√

ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] ln

√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] +ω√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] −ω

× (1− 3x2)2 +

[(1− 3x2)2 − 3

8

((1− 3x2)2 + 3

(1− x2)2 cos2 2φ

)]

× ln(1− x2)2[

γ 2 + cos2 2φ(1− 2γ )] = 0,

C.Y. Lee / Physics Letters A 324 (2004) 203–210 207

1∫0

dx

∫dϕ

√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )]

ωln

√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] +ω√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] −ω

× (1− x2)2 cos2 2φ + [(

1− x2)2 cos2 2φ − 2(1− x2)2

x2 cos2φ]

× ln(1− x2)2[

γ 2 + cos2 2φ(1− 2γ )] = 0,

1∫0

dx

∫dϕ

ω√

ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] ln

√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] +ω√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] −ω

× (1− x2)2 cos2 2φ + [(

1− x2)2 cos2 2φ − 2(1− x2)2

x2 cos2φ]

× ln(1− x2)2[

γ 2 + cos2 2φ(1− 2γ )] = 0,

1∫0

dx

∫dϕ

√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )]

ωln

√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] +ω√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] −ω

× 4(1− x2)2

x2 cos2φ + [4(1− x2)2

x2 cos2φ − 2(1− x2)2

x2 cos2φ]

× ln(1− x2)2[

γ 2 + cos2 2φ(1− 2γ )] = 0,

1∫0

dx

∫dϕ

ω√

ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] ln

√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] +ω√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] −ω

× 4(1− x2)2

x2 cos2φ + [4(1− x2)2

x2 cos2φ − 2(1− x2)2

x2 cos2φ]

× ln(1− x2)2[

γ 2 + cos2 2φ(1− 2γ )] = 0,

1∫0

dx

∫dϕ

√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )]

ωln

√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] +ω√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] −ω

× 4(1− x2)2

x2 sin2φ + [4(1− x2)2

x2 sin2φ − 2(1− x2)2

x2 cos2φ]

× ln(1− x2)2[

γ 2 + cos2 2φ(1− 2γ )] = 0,

1∫0

dx

∫dϕ

ω√

ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] ln

√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] +ω√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] −ω

× 4(1− x2)2

x2 sin2φ + [4(1− x2)2

x2 sin2φ − 2(1− x2)2

x2 cos2φ]

× ln(1− x2)2[

γ 2 + cos2 2φ(1− 2γ )] = 0,

208 C.Y. Lee / Physics Letters A 324 (2004) 203–210

eell as forar, they

e

is

1∫0

dx

∫dϕ

√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )]

ωln

√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] +ω√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] −ω

× (1− x2)2 sin2φ + [(

1− x2)2 sin2φ − 2(1− x2)2

x2 cos2φ]

× ln(1− x2)2[

γ 2 + cos2 2φ(1− 2γ )] = 0,

1∫0

dx

∫dϕ

ω√

ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] ln

√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] +ω√ω2 + 4(1− x2)2[γ 2 + cos2 2φ(1− 2γ )] −ω

× (1− x2)2 sin2φ + [(

1− x2)2 sin2φ − 2(1− x2)2

x2 cos2φ]

(17)× ln(1− x2)2[

γ 2 + cos2 2φ(1− 2γ )] = 0.

The following substitutions have been used: cosθ = x, ω = ω/∆0.These equations determine the whole spectrum of collective modes in mixed(1 − γ )dx2−y2 + iγ dxy state of

high temperature superconductors (HTSC) with arbitrary admixture ofdxy state. Knowledge of the collective modspectrum could be used for interpretation of the sound attenuation and microwave absorption data as widentification of the type of pairing and order parameter in unconventional superconductors. In particulallow to estimate the extent of admixture of adxy state in a possible mixed state.

The most interesting case is the case of smallγ : we suppose that dominant state isdx2−y2 state and admixturof dxy state is small, say 3–10%. In these case we could expand all expressions in powers of smallγ and obtainthe corrections to the spectrum of puredx2−y2 state, which has been found before [3,8].

4. Equal admixtures of dx2−y2 and dxy states

Brusov and Brusov [8] have supposed the equal admixtures ofdx2−y2 anddxy states—in our consideration thcorresponds to the caseγ = 1/2—and derived the following equations:

i = 1,1∫

0

dx

∫dϕ

√ω2 + 4f

ωln

√ω2 + 4f +ω√ω2 + 4f −ω

g1 +(g1 − 3

2f1

)lnf

= 0,

(18)

1∫0

dx

∫dϕ

ω√

ω2 + 4fln

√ω2 + 4f +ω√ω2 + 4f −ω

g1 +(g1 − 3

2f1

)lnf

= 0,

i = 2,3,4,5,1∫

0

dx

∫dϕ

√ω2 + 4f

ωln

√ω2 + 4f +ω√ω2 + 4f −ω

gi +(gi − 1

2g

)lnf

= 0,

(19)

1∫0

dx

∫dϕ

ω√

ω2 + 4fln

√ω2 + 4f +ω√ω2 + 4f −ω

gi +(gi − 1

2g

)lnf

= 0.

C.Y. Lee / Physics Letters A 324 (2004) 203–210 209

ch statedes with

nd

,s arequencies)

ureate).d state it

ilityle these]) theredes are

of the

orderand

te

Here

g1 = (1− 3x2)2

, g2 = (1− x2)2

cos2 2ϕ, g3 = g = 4(1− x2)x2 cos2ϕ,

g4 = 4(1− x2)x2 sin2ϕ, g5 = (

1− x2)2sin2ϕ, f1 = 1

4

[(1− 3x2)2 + 3

(1− x2)2

cos2 2ϕ],

(20)f = (1− x2)2

.

They have solved [3,8] these equations numerically and have found five high frequency modes in eaobtained from the second equations while the first ones appear to give either Goldstone modes or movanishing energies (of order 0.03∆0(T )–0.08∆0(T )).

Below we give their results for high frequency modes (Ei is the energy (frequency) ofith branch).

(21)E1,2 =∆0(T )(1.93− i0.41), E3 =∆0(T )(1.62− i0.75), E4,5 =∆0(T )(1.59− i0.83).

Comparison of these results with spectrum of puredx2−y2 anddxy states, obtained by Brusov [3], and Brusov aBrusov [8],

E1 =∆0(T )(1.88− i0.79), E2 =∆0(T )(1.66− i0.50), E3 =∆0(T )(1.40− i0.68),

(22)E4 =∆0(T )(1.13− i0.71), E5 =∆0(T )(1.10− i0.65)

led them to conclusion, that in spite of the fact that spectra in both pure statesdx2−y2 anddxy turn out to be identicalspectrum in mixeddx2−y2 + idxy state is quite different from that in pure states. In pure states all modenondegenerated while in mixed state two high frequency modes are twice degenerated. The energies (freof high frequency modes are ranged between 1.1∆0(T ) and 1.88∆0(T ) while in mixed state between 1.59∆0(T )

and 1.93∆0(T ), i.e., the collective modes have higher frequencies. Also damping of collective modes in pd-states is more than in mixed state (ImEi is from 30% to 65% in pure states and from 20% to 50% in mixed stIt could be easy understood, because in pure states the gap vanishes along chosen lines while in mixevanishes just at two points (poles) [8].

The difference of spectrum of collective excitations in pured-wave states and in mixed state give us a possibto probe the state symmetry by ultrasound and/or microwave absorption experiments. Note that whiexperiments could require high frequencies (of order of tens GHz as in case of Feller et al. experiment [1is no principle restrictions for ultrasound (microwave) frequencies: because frequencies of collective moproportional to∆0(T ), which vanishes atTc, one could use in principle any frequency approaching toTc.

So, obtained results could allow to answer three very important questions:

(1) Does the gap disappear along some chosen lines?(2) Do we have a pure or mixedd-wave state in HTSC?(3) How large is the admixture ofdxy state in a possible mixed state?

Note, that case ofdx2−y2 + idxy state has been considered by Balatsky et al. [9] as well, who studied onepossible collective mode in this state.

5. Conclusions

In order to solve one of the problem of unconventional superconductivity—the exact form of theparameter—I consider the mixed(1 − γ )dx2−y2 + iγ dxy state in high temperature superconductors (HTSC)derive for the first time a full set of equations for collective modes spectrum in a mixedd-wave state with anarbitrary admixture ofdxy state. The most interesting case is the case of smallγ : we suppose that dominant sta

210 C.Y. Lee / Physics Letters A 324 (2004) 203–210

ons inre.

orm ofdmixture

retation

rsity.

1999.

re, 1996.

is dx2−y2 state and admixture ofdxy state is small, say 3–10%. In this case one could expand all expressipowers of smallγ and obtain the corrections to the spectrum of puredx2−y2 state, which has been found befoI am planning to realize this program in future publications.

Obtained results could be useful for identification of the type of pairing and determination of the exact fthe order parameter in unconventional superconductors. In particular, they allow to estimate the extent of aof a dxy state in a possible mixed state.

Derived equations allow to calculate the whole collective mode spectrum, which could be used for interpof the sound attenuation and microwave absorption data.

Acknowledgements

I thank Peter Brusov for useful discussions. I acknowledge the support by 2004 Kyobi of Hannam Unive

References

[1] J.R. Feller, C.-C. Tsai, J.B. Ketterson, et al., Phys. Rev. Lett. 88 (2002) 247005.[2] H. Matsui, et al., Phys. Rev. B 63 (2001) 060505.[3] P.N. Brusov, Mechanisms of High Temperature Superconductivity, vols. 1, 2, Rostov State University Publishing, Rostov on Don,[4] T. Dahm, D. Manske, L. Tewordt, Phys. Rev. B 58 (1998) 12454.[5] K. Krishana, et al., Science 277 (1997) 83.[6] R.B. Laughlin, Phys. Rev. Lett. 80 (1998) 5188.[7] J.F. Annett, et al., in: D.M. Ginsberg (Ed.), Physical Properties of High Temperature Superconductors V, World Scientific, Singapo[8] P.P. Brusov, P.N. Brusov, Physica B 281–282 (2000) 949.[9] A. Balatsky, P. Kumar, J.R. Schrieffer, Phys. Rev. Lett. 84 (2000) 4445.