Pairing Symmetry and Upward Curvature of Upper Critical Field in Superconducting $Na_{0.35} Co O_2 \cdot y H_2 O$

0.13 MB·English

by  Jun-Ting Kao

Pairing Symmetry and Upward Curvature of Upper Critical Field in Superconducting
Na CoO yH O
0.35 2 2
·
Jun-Ting Kao,1 Jiunn-Yuan Lin,2 and Chung-Yu Mou1,3
1 Department of Physics, National Tsing Hua University, Hsinchu 30043, Taiwan
2 Institue of Physics, National Chao Tung University, Hsinchu 30043, Taiwan
7 3Physics Division, National Center for Theoretical Sciences, P.O.Box 2-131, Hsinchu, Taiwan
0
(Dated: February 6, 2008)
0
2 Theoriginfortheupwardcurvatureoftheuppercriticalfield(Hc2)observedinhydratecobaltate
n Na0.35CoO2 ·yH2O is investigated based on the microscopic gap equation. It is shown that the
a observed upward curvature results from the transition between two different pairing symmetries
J that occur on different energy bands. Furthermore, different pairing symmetries involved in the
8 transitionresultsindifferentupwardcurvatures. Byconsideringtransitionsamongalllowestpossible
pairingsymmetries,itisfoundthatthetransitionofthepairingsymmetryfroms-waveatlowfields
] to dx2−y2 +idxy at high fields is the best fit to the experimental data. Our results provide an
n important clue to theunderstanding of thesuperconductivity in hydratecobaltate.
o
c PACSnumbers: 73.20.-r,73.20.At,73.21.Hb
-
r
p
u Since the discovery of superconductivity in hydrate due to mixing of two pairing order parameters. The rea-
s cobaltateNa CoO yH O[1],extensivetheoreticaland son why two approaches give different results lies in the
. 0.35 2· 2
t experimental studies have been devoted to elucidate the fact that in the Ginzburg-Landau theory, H T T
a c2 c
∝ −
m mechanism of superconductivity. To unravel the mecha- andphenomenologically,bothTcandslopesofHc2areof-
nism,identifyingtheunderlyingpairingsymmetrywould tenchosenarbitrarily. IflargerslopeofH is chosenfor
- c2
d be the first step. The two-dimensional triangle lattice smallerT ,H oftwopairingsymmetriesneartheirT ’s
c c2 c
n formedbyCoO providesanalternativelatticesymmetry essentiallyshowstheintersectionoftwostraightlines. A
2
o
to the square CuO lattice in high Tc materials and has kink in the slope thus arises and the upward curvature
c 2
[ ledtomanyproposalsforunconventionalpairingsymme- can be easily simulated. In real materials, however, the

tries. For singlet pairing, the lowest possible unconven- slope of Hc2 and Tc both depend on microscopic details
1
tional symmetry without breaking rotational symmetry and are not independent from each other. In fact, in the
v
5 is dx2 y2 idxy, and mechanism based on the correla- Gorkov’smicroscopicderivationofthe Ginzburg-Landau
3 tioneff−ect±sforsuchsymmetryhasbeenproposed[2,3,4]. equation[16], the slope is proportional to m∗Tc/ǫF with
1 Possible spin-triplet f-wave pairing was also proposed m being the effective mass of the electron and ǫ be-
∗ F
1 based on ferromagnetic fluctuations[5, 6, 7]. Exper- ing the Fermi energy. For a single band, m /ǫ are the
∗ F
0
imentally, however, data reported show contradictory same for different pairing symmetries, hence smaller T
7 c
conclusions[8], indicating the fragility of the supercon- goes with smaller slope, in the opposite trend adopted
0
/ ductivity in this system. Furthermore, there are evi- in the Ginzburg-Landau equation. Hence in this case,
t
a dences indicating that two pairing symmetries may be joining two pairing symmetries with different Tc would
m involved in this system [9, 10, 11]. The issue of whether not yield the upward curvature. This picture essentially
- the pairing symmetry is conventional or not is still un- explains why the upward curvature is not reproduced in
d
settled and needs to be clarified. the calculation of Kim et al. [15] for high Tc materi-
n
als. Then,whatisthe originfortheupwardcurvaturein
o In this paper, we shall focus on the data of the upper
c hydrate colaltate ? It is known that multi-orbitals near
criticalfields,measuredbythespecificheat. Thespecific-
: the Fermi surface might be involved for the occurrence
v heattechnique probesthe bulk propertiesofthe samples
Xi and has been proved to be a powerful tool for inves- of superconductivity[17]. Hence m∗/ǫF can no longer be
treated as a fixed parameter for different pairing sym-
tigating the pairing state of many superconductors[12].
r metries if different pairings occur on different bands. In-
a Specifically,forthehydratecobaltateNa CoO yH O,
0.35 2 2
· deed, our calculationin below shows that upwardcurva-
upward curvature (a kink structure in the slope) of H
c2
ture feature can result from the two-band calculation in
wasobserved[9,10]. Similarstructurewasalsoobserved
whichdifferentvaluesofm /ǫ areassumedfordifferent
in early studies of high Tc materials[13]. Based on the ∗ F
energy bands where different pairing symmetry occurs.
Ginzburg-Landau theory, Joynt[14] attributed the up-
Furthermore, mixing of different pairing symmetries re-
ward curvature to the transition between two different
sults in coupling of the ground state to different Landau
pairing symmetries with different critical temperatures.
levelsinthepresenceofmagneticfieldsandcausesdiffer-
However, in a later investigation based on microscopic
ent upward curvatures. By direct comparison of experi-
formulation of the gap equation [15], negative results
mental data with H obtained by mixing of the lowest
were found, indicating that the upward curvature is not c2
2
possible pairing symmetries, it is possible to pin down amplitudes in real space and φˆ (φ) are the projection
α
the pairing symmetries of the system. We find that the operators in real space: φ (φ) = 1, φ (φ) = e iφ,
s p ip ±
upward curvature observed in the experimental data is φ (φ) = e 2iφ, and φ (φ) = cos3φ ±3cosφsin2φ =
d id ± f
mostly consistent with a transition of the pairing sym- cos±3φ. Kˆ (r,ω) is the kernel operator,−given by
i
metry from s to dx2 y2 +idxy.
baWndes.staTrhtebytwfior-sdticm−oennssidioenrianlgBpCaSir-ilnikgethHaatmoiclctounrsiainntcwano Kˆi(r,ω)=Ki0(r,ω)exp r·(▽R+ 2¯hiceA(R)) (4)
(cid:20) (cid:21)
be written as
with
2
Hˆ =Xi=1kXi,σξkiCˆk+iσCˆkiσ Ki0(r,ω)=kBTNi(0)2k2π exph−rv2Fr|iω|i, (5)
Fi
+ Xi,j kXi,kj′ Vkikj′Cˆk+i↑Cˆ−+ki↓Cˆ−kj′↓Cˆkj′↑, (1) wFehremrei vkeFloi,civtyFianadndthNeit(w0)oadrime etnhseioFnearlmdienwsaitvyenoufmstbaetre,
which was first investigated by Suhl, Matthias, and for the ith band. In the absence of V , Eqs.(2) and (3)
I
Walker[18]. Here k and k are wave vectors on two decouples, and their solutions for constant ∆ and A=0
1 2
Fermi surface sheets indexed by 1 and 2. The electron- yield relations between T and V [19]
c α
electron interaction Vk1k′1 and Vk2k′2 are the intrasheet 1 1 Ti
contributions and Vk1k′2 is an intersheet contribution. In = ln c, (6)
general, the superconducting pairing symmetry is differ- ΓiViNi 2ν+1 − T
ν | |
entfordifferentband. Hencewecanwritetheprojection X
of the interaction in the form Vk1k′1 = Vαφˆα(k1)φˆ∗α(k′1), where i = α or β with Tci being the correspond-
Vk2k′2 = Vβφˆβ(k2)φˆ∗β(k′2), and Vk1k′2 = VIφˆα(k1)φˆ∗β(k′2). in2gπ dtφraφˆns(iφti)on2. teDmipviedriantgureEqisn.(2z)eraondfie(l3d) abnydVΓian=d
Hereαandβ areindicesforthepairingsymmetry. φˆα(k) V0 r2eπsp|eictiv|ely and using Eq.(6), V and V cαan be
is an operator that projects out the corresponding pair- Rβ α β
eliminated. Following Ref.[19], if we further adopt di-
ing symmetry α. For the lowest possible pairings, we
mensionless variables, t =T/Ti, r = ρ/ 2eH/¯hc and
shallconsiderpossiblemixingofpandf wavesfortriplet i c
h =2eH/¯hc(h¯v /2πk Ti)2, the gap equations become
pairingandmixingofsanddforsingletpairing. Forthe i Fi B c p
triangle lattice, assuming that the rotational symmetry
1 1
isnotbroken,theappropriatepanddpairingamplitudes Γ ln ∆ (R)=
α α
jaφˆercetipoxn(k±)opi=peyrkaˆat2nodrsdkˆax2r2e−yφˆ22si±(kˆk)ikˆdx=,ya.1nT,dhφˆφeˆp±c(iokpr()rke=)sp=kˆo3nkˆdxin3±gkˆpikkˆˆroy2-,. Kˆα α∆Xνα(R|2)ν−+ξ1α|γ−Kˆαβt∆αβ!(R) (7)
Ndo±tiedthat thexre−areyt±wo poxssyible f-wafves;howxev−er,sxincye Γ 1 ln 1 ∆ (R)=
β β
they are related by a rotation of π/6 and it suffices to ν |2ν+1| − tβ!
X
consider one of them (see below for details). Following Kˆ ∆ (R) ξβKˆ ∆ (R), (8)
Kim et al.[15], the real-space linearized gap equation in ββ β − γ βα α
the presence of the magnetic field A can be written
as[15] ∇× where ξi =VI/Vi and γ =Nβ/Nα. The operators Kˆ are
given by
∆ (R)=V drφˆ (φ)φˆ (φ)Kˆ (r,ω)∆ (R)
α αXω Z ∗α α 1 α Kˆnm = 21π√thm ∞dρ exp α−mρ exp −4ρ2
+V drφˆ (φ)φˆ (φ)Kˆ (r,ω)∆ (R), (2) m Xω Z0 (cid:20) ω (cid:21) (cid:20) (cid:21)
I ∗α β 2 β 2π ρ ρ
Xω Z × dφφˆ∗n(φ)φˆm(φ)exp √−2aˆ+eiφ exp √2aˆ−e−iφ ,
and Z0 (cid:20) (cid:21) (cid:20) (cid:21)
(9)
∆β(R)=Vβ drφˆ∗β(φ)φˆβ(φ)Kˆ2(r,ω)∆β(R) where αm = √2h¯ecHv and aˆ = h¯c [( +2ieA)
Xω Z ω 2|ω| Fm ± 4eH ▽ x±
+VI drφˆ∗β(φ)φˆα(φ)Kˆ1(r,ω)∆α(R). (3) i(▽Eq+s.2(7ie)Aa)nyd]. (8) are the governqing equations for the
ω Z
X situationwhentwopairingsymmetriesoccurondifferent
Here R represents the position for the center of mass of energy bands. However,it also covers the case when the
theCooperpair,risthedisplacementoftheCooperpair two pairing symmetries occur in the same energy band.
and φ is the corresponding angle of r. ∆’s are pairing In that case, one simply sets γ =1, ξ =1, and drop the
i
3
index i. Eqs.(7) and (8) can be solved by expanding ∆ and
α
and ∆ in terms of the set of eigenfunctions ψ of the
β n
| i 1 ξ (n 4)!
tAwo=-d(i0m,Henxsi,o0n)alSchrodinger equationin the presence of (cid:18)lntp +βnp(cid:19)Apn±ip− p2±γipr ∓n! αf(±4,n∓4)A1nf∓4
ξ (n 2)!
∆ = ∞ Ai ψ , (10) + p2±γipr ±n! αf(∓2,n±2)Afn±2 =0. (14)
i n| ni
n=0 Ontheotherhand,forsingletpairing,recursionrelations
X
where i = α or i = β. Note that the eigenfunctions for the mixing of s-wave with dx2−y2±idxy are givenby
ψ are essentially the Landau levels. The operators
n 1 ξ (n 2)!
Kp|ˆlincigouispldeetdeirffmerinenedt bLyantdhaeubil-epvreolds.uctTphreojdecettiaoinledopceorua--(cid:18)lnts +βns(cid:19)Asn+ γsr ±n! αd±id(∓2,n±2)Adn±±i2d =0,
tor,φˆ (φ)φˆ (φ),inEq.(9),which,afterbeingintegrated, (15)
∗n m
selects the correct combinations of aˆ+ and aˆ that sur-
− and
vive. The selected combination of aˆ+ and aˆ then de-
−
dte+rmiidn,es{Ahosnw} cAoinupcloeuspwlei.thFo{rAtdnh++ei2dm};ixwinhgileoffosr-wthaveemainxd-(cid:18)lnt1d +βnd(cid:19)Adn±id+ξd±idγr(Nn∓!2)!αs(±2,n∓2)Asn∓2 =0.
ing of s-wave and d id, As couples with Ad+id . (16)
On the other hand, fo−r the m{ ixni}ng of p ip and{f-nw−a2v}e,
because φˆ (φ) contains both e3iφ and e±3iφ, in addition When applying the above analysis to the calculation
f −
to the coupling between Ap ip and Af, there are also of Hc2 with mixing of two pairing symmetries, α and
n± n
couplings among Af . Since the bi-product projection β, one assumes that the pairing symmetries are associ-
{ n} taed with different T ’s with Tα > Tβ. In other word,
operator in Kˆ , φˆ (φ)φˆ (φ), contains e i6φ, Af cou- c c c
ff ∗f f ± n the pairing symmetry α is the stable bulk pairing state
ples with Af for each n; while because the bi-product
n+6 at low fields while β is the stable pairing state at high
cpt{aoArioennpffij+s+ecci4epite±}ioni4natφsnodaaprne{edrAamefnto±o}rsis2tcφoic,nuowpnolveeffe-fnwdoiuieiatnnhgtdoly{ntAahelpnax−+tKpˆi2r{pαe}Aβs.sfne(}dTαchi6=oneuctpβeol)reu,mpwclsoiintnohg-f fiiwanneigltadhltsyo.snistTh≥ihemerpo1etl.fiheoesrSreti,nshyocaemntem{osAnetetasnrr}ycta,scnβofr,uoopntmlhleyes∆nhwαacviot=euhptAh{leeAα0s|dntψA±±r0aiα02idn.}st,iotMitoAhinxiβnss-
the following functions[15] fromhigh-temperatures-waveto low-temperatured+id
or from high-temperature d id to low-temperature s-
βi = ti ∞dρ e−ρ42Ln ρ22 −1 (11) awnadvef.-wOavne,tbheecaoutsheeφrˆ h(φan)dc,o−nftoarintshbeomthixei3niφgaonfdpe±3iiφp,
n √hi Z0 sinh (cid:16)√tihρi(cid:17) both transitions fromfhigh-temperature p ± ip to−low-
(cid:16) (cid:17) temperature f-wave and from high-temperature f-wave
and to low-temperature p ip are possible. Note that there
are two possible f-wa±ves, kˆ3 3kˆ kˆ2 and kˆ3 3kˆ kˆ2.
αi(2k,n)= √thii Z0∞dρ e−4ρ2 (cid:16)si−n2ρh2(cid:17)k√tiLhρi2nk(cid:16)ρ22(cid:17), (12) tkSˆoyincwcoehnitschihdeestrimwoopnlefy-oewfxactvhheeasmnag.reesprexxl+a−tiepdybaxyndyexpcxh−ainpgyyi,ni−gtskˆuxffiyacnexds
(cid:16) (cid:17) When solving H , it is important to note that there
c2
where i is the index for the pairing symmetry, Ln(x) are many eigenvalues H(T) satisfying Eqs.(7) and (8)
is the Laguerre polynomial and L2nk is the associated and only the largest one defines Hc2. To compare the
Laguerre polynomial in Rodrigues representation. For calculated H with experimental data, one needs to fix
c2
triplet pairing, we find that recursion relations for the scales of temperatures and magnetic fields. The tran-
mixing of f-wave with px±ipy are given by sition temperature Tc of the most stable bulk pairing
state determines the temperature scale. On the other
−1 (n−6)!αf(6,n 6)Af + ln 1 +βf Af hand, the scale of magnetic fields can be fixed by the
2 r n! − n−6 (cid:18) tf n(cid:19) n data points with lower magnetic fields. The remaining
parameters are the ratio of Fermi velocities, the ratio of
1 (n+6)!
αf( 6,n+6)Af T , ξ and γ. At this point, it is important to note that
−2 n! − n+6 c i
r accordingto Eqs. (13)-(16), for different mixing scheme,
+ξfγr(n∓n!2)!αp±ip(±2,n∓2)Apn±∓i2p denifftemreinxtinLgasncdhaemuelerveeslusltasrienmdiiffxeedre.ntAuspwaarredsuclutr,vdatiffuerre-.
(n 4)! To find the best fit to the experimental data, after fix-
−ξfγr ±n! αp±ip(∓4,n±4)Apn±±i4p =0, (13) ing scales of temperatures and magnetic fields, we vary
4
the remaining parameters to find the best. We find that
the transition from s-wave at low fields to d+id at high
fields is the best fit to the data. In Fig. 1, we show nu-
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F F c c s d+id
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2d
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c2
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0.35 2 2
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This work is supported by the NSC of Taiwan. We T.Moyoshi,Y.MoriandM.Sato,J.Phys.Soc.Jpn.74,
1800 (2005).
thank Prof. Hsiu-Hau Lin for useful discussions.