Upper critical field H_c2 in Bechgaard salts (TMTSF)_2PF_6

0.18 MB·English

by  A. Dominguez Folgueras

Europhysics Letters PREPRINT
Upper critical field H in Bechgaard salts (TMTSF) PF
6 c2 2 6
0
0
Ana Dom´ınguez Folgueras1,2,3 and Kazumi Maki3
2
1 Departamento de F´ısica, Universidad de Oviedo, 33007 Oviedo, Spain
n
a 2 Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, 28049 Madrid,
J Spain

4 3 Departmentof Physics &Astronomy, University of SouthernCalifornia, Los Angeles,
CA 90089-0484, USA

]
n
o
c PACS.74.70.Kn – Organic Superconductors.
-
r PACS.74.20.Rp – Pairing symmetries.
p PACS.74.25.Op – Mixed states, critical fields, and surface sheaths.
u
s
.
t
a
m Abstract. – ThesymmetryofthesuperconductingorderparameterinBechgaardsaltsisstill
- unknown, though the triplet pairing is well established by NMR data and large upper critical
d field Hc2(0) ∼ 5 Tesla for H~ k a and H~ k b′. Here we examine the upper critical field of a
n
few candidate superconductors within the standard formalism. The present analysis suggests
o
strongly chiral f-wave superconductor somewhat similar to the one in Sr RuO is the most
c 2 4
[ likely candidate.

1
v
5
6
Introduction. – The Bechgaard salts (TMTSF) PF is the first organic superconductor
0 2 6
1 discovered in 1980 [1]. For a long time the superconductivity is believed to be conventional
0 s-wave [2]. Recently the symmetry of the superconducting energy gap becomes the central
6 issue [3,4]. The upper critical field at T = 0K, H 5 Tesla for both H~ a and H~ b
c2 ′
0 ∼ k k
for both (TMTSF) PF [5] and (TMTSF) ClO [6] are clearly beyond the Pauli limit [7,8]
/ 2 6 2 4
t indicating the triplet pairing. More recently the NMR data from (TMTSF) PF [9]indicates
a 2 6
m clearly the triplet pairing. Therefore the candidates for the superconductivity in Bechgaard
salts are more likely within p-wave and f-wave superconductors. In the following we shall ex-
-
d aminetheuppercriticalfieldofthesesuperconductorsfollowingthestandardmethodinitiated
n by Gor’kov [10] and extended by Luk’yanchuk and Mineev [11] for unconventional supercon-
o
ductors. Also we take the quasiparticle energy in the normal state as in the standard model
c
: for Bechgaardsalts [2]
v
i
X ξ(k)=v(ka kF) 2tbcosbk 2tccosck (1)
| |− − −
r
a with v :v :v 1 :1/10:1/300 and v = √2t b and v = √2t c. There are earlier analysis
b c b b c c
∼
of H of Bechgaard salts starting from the one dimensional models [12,13]. However, those
c2
models predict diverging H (T) for T 0K or the reentrance behaviour, which have not
c2
→
beenobservedinthe experiments[5,6]. Also,thequasilinearTdependence ofH (T)inboth
c2
(TMTSF) PF and (TMTSF) ClO is very unusual. Among the models we have considered,
2 6 2 4
(cid:13)c EDPSciences
2
EUROPHYSICSLETTERS
Fig. 1 – |∆(~k)| of chiral f-wave and chiral f′-waveSC are sketched in a) and b) respectively.
the chiral f-wave superconductor with ∆ ~k 1 sgn(k )+ı sinχ cosχ , looks most
′ (cid:16) (cid:17) ∼ (cid:16)√2 a 2(cid:17) 2
promising, where χ =~b~k and χ =~c~k where~b and~c are crystal vectors.
1 2
Also if the superconductor belongs to one of the nodal superconductors [14] and if nodes
layparalleltok~ within the twosheets ofthe Fermisurface,the angledependent nuclearspin
c
relaxation rate T1−1 in a magnetic field rotated within the b′ −c∗ plane will tell the nodal
directions.
Beforeproceeding,weshow ∆(~k) ostwochiralf-wavesuperconductorsinFig.1a)andb).
| |
1 1
where ∆(k) (1+cos2χ )(1 1cos2χ ) 2 and ∆(k) (1+cos2χ )(1 1cos2χ ) 2
| | ∼ 1 − 2 2 | | ∼ 2 − 2 2
for chiral f1 and(cid:2)chiral f2 respectively. (cid:3) (cid:2) (cid:3)
Upper critical field for H~ b~. – Inthefollowingweneglectthe spincomponentof∆~(~k).
k ′
Most likely the equal spin pairing is realised in Bechgaard salts as in Sr RuO [4]. In this
2 4
case the spin component is characterised by a unit vector dˆ. Also dˆis most likely oriented
parallel to c~. Let’s assume dˆ c~, though H (T) is independent of dˆas long as the spin
∗ k ∗ c2
orbitinteractionis negligible. Experimentaldata frombothUPt andSr RuO indicate that
3 2 4
the spin-orbit interactions in these systems are not negligible but extremely small [15]. We
consider a variety of triplet superconductors (see Fig. 1):
A. Simple p-wave SC: ∆~(k) sgn(k ). Following [16] the upper critical field is deter-
a
∼
mined by
∞ du
lnt= (1 K ) (2)
1
− Z sinhu −
0
∞ du
C lnt= (C K ) (3)
2
− Z sinhu −
0
where
K1 = e−ρu2|s|2 1+2Cρ2u4s4 (4)
h i
(cid:0) (cid:1)
AnaDom´ınguezFolguerasandKazumiMaki:UppercriticalfieldHc2 inBechgaardsalts(TMTSF)2PF63
0.4 0
0.35 cchhiirraall fp’--wwaavvee -0.05 cph-wiraalv ep-wave
pwave
0.3 -0.1
0.25 -0.15
ρ0 0.2 C-0.2
0.15 -0.25
0.1 -0.3
0.05 -0.35
00 0.2 0.4 0.6 0.8 1 -0.40 0.2 0.4 0.6 0.8 1
a) t b) t
Fig.2–NormalisedHc2(t)andC(t)forH~ k~b′ areshownina)andb)respectively. Heresolid,dashed
and dotted lines are chiral f′-wave,chiral p-waveand simple p-waverespectively.
1 16 2
K = e ρu2s2 ρ2u4s 4+C 1 8ρu2 s2+12ρ2u4 s4 ρ3u6 s6+ ρ4u8 s8 (5)
2 − | | ∗
h (cid:18)6 (cid:18) − | | | | − 3 | | 3 | | (cid:19)(cid:19)i
and t = T , ρ = vavbeHc2(T), s = 1 sgn(k )+ısinχ , χ =~c~k and ... means average over
Tc 2(2πT)2 √2 a 2 2 h i
χ . Here v , v are the Fermi velocities parallel to the a axis and the c axis respectively.
2 a c
Here we assumed that ∆(~r) is given by [16]
∆(~r) 1+C(a+)4 (6)
∼ i
(cid:0) (cid:1)
wherei= Cne−eBx2−nk(x+ız)−(n4ekB)2 istheAbrikosovstate[17]anda+ = √21eB (−ı∂z−∂x+2ıeHz)
is the raising oPperator.
Then in the vicinity of t 1 we find ρ = 2 ( lnt) = 0.237697( lnt) and C =
→ 7ζ(3) − −
93ζ(5) ρ.
−647ζ(3)
For t 0 on the other hand we obtain
→
v v eH (0)
ρ = limρt2 = a c c2 =0.1583 (7)
0 t 0 2(2πT )2
→ c
and C = 0.031. From these we obtain
−
H (0)
c2
h(0)= =0.6659 (8)
∂Hc2(t)
∂t |t=1
Both ρ (t) and C(t) are evaluatednumerically and shownin Fig.2 a)and b) respectively.
0
Here ρ (t)=t2ρ(t)=vv eH (t)/2(2πT )2.
0 c c2 c
B.Chiralp-waveSC:∆~(k)=1/√2sgn(k )+isin(χ ). Here 1 sgn(k )+isin(χ )isthe
a 2 √2 a 2
analogue of eıφ in the 3D systems in the quasi 1D system. For a chiral state the Abrikosov
function is written as [18]
∆(~r,~k) (s+Cs∗(a†)2) (9)
∼ i
4
EUROPHYSICSLETTERS
where s= 1 sgn(k )+ısin(χ ). Then we obtained eq. 3 with
√2 a 2
K1 = e−ρu2|s|2 s2 2C s4 (10)
h | | − | | i
K = e ρu2s2 s4+C s2 1 4ρu2 s2(cid:0)+2ρ2u4 s4 (cid:1) (11)
2 − | |
h −| | | | − | | | | i
(cid:0) (cid:0) (cid:1)(cid:1)
and the same expressions for t, ρ,...
For t 1 we find C =1 √1.5= 0.2247 and ρ=0.3838( lnt).
→ − − −
On the other hand, for t 0 we obtain C = 0.3660 and ρ =0.27343.
0
→ −
From these we obtain h(0) = 0.71324. We obtain ρ(t) and C(t) numerically. They are
shown in Fig. 2 a) and b) respectively.
C. Chiral f-wave SC: ∆ˆ(k) dˆscosχ . H (t) is determined from eq. 3 where now:
1 c2
∼
K = (1+cos2χ )e ρu2s2 s2 2ρu2 s4 (12)
1 1 − | |
h | | − | | i
K = (1+cos2χ )e ρu2s2 ρu2 s4+C s2 1 4ρu2 s(cid:0)2+2ρ2u4 s4 (cid:1) (13)
2 1 − | |
h − | | | | − | | | | i
(cid:0) (cid:0) (cid:1)(cid:1)
Here now ... means the average over both χ and χ . Then it is easy to see that the
1 2
h i
chiral f-wave SC has the same H (t) and C(t) as the chiral p-wave SC, since the variable χ
c2 1
is readily integrated out.
D. Chiral f-wave SC: ∆ˆ(k) dˆscosχ . Now we have a set of equations similar to the
′ 2
∼
chiral f-wave except 1+cos2χ in both eqs. 13 has to be replaced by 4(1+cos2χ ). Then
1 3 1
we obtain for t 1 C = 0.2247 and ρ = 0.5181( lnt). On the other hand, for t 0 we
→ − − →
find C = 0.3660 and ρ =0.3734.
0
−
We show ρ and C(t) of the chiral f-wave in Fig.2 a) and b) respectively.
0 ′
Note that C(t) is the same for three chiral states (chiral p-wave, chiral f-wave and chiral
f-wave) as well as chiral p-wave studied in [18]
′
Therefore for the magnetic field H~ b, the chiral f-wave have the largest H (t) if we
′ ′ c2
k
assume T and v, v are the same. Also H (t) of these states are closest to the observation.
c c c2
Upper critical field for H~ ~a. –
k
A. Simple p-wave SC: ∆ ~k = sgn(k ). The equation for H (t) is given by [16] and
a c2
(cid:16) (cid:17)
can be written as in eq.3 with
K1 = e−ρu2|s|2 1+2Cρ2u4 s4 (14)
h | | i
16 (cid:0) 2 (cid:1)
K = e ρu2s2 ρ2u4 s4+C 1 8ρu2 s2+12ρ2u4 s4 ρ3u6 s6+ ρ4u8 s8 (15)
2 − | |
h (cid:18) | | (cid:18) − | | | | − 3 | | 3 | | (cid:19)(cid:19)i
where t= T , ρ= vavbeHc2(t) and s=sinχ +ısinχ with χ =~b~k and χ =~c~k.
Tc 2(2πT)2 1 2 1 2
Then for t 1, we find C = 93ζ(5) ρ and ρ = 2 ( lnt) = 0.2377( lnt). While for
→ −508ζ(3) 7ζ(3) − −
2
t 0 C = 3 3 + 1 = 0.0170129 and ρ = vavbeHc2(0) = 1 exp[α +2Cβ ] =
→ 2β0 −r(cid:16)2β0(cid:17) 12 − 0 2(2πTc)2 4γ 0 0
0.1751209, where α = ln s2 = 0.220051 and β = s4 = 4 1 = 0.0170. From these
0 −h | | i 0 −h s4 π −
| |
we obtain h(0)=0.73673.
Both h(t) and C(t) are evaluated numerically and we show them in Fig. 3 a) and b)
respectively.
AnaDom´ınguezFolguerasandKazumiMaki:UppercriticalfieldHc2 inBechgaardsalts(TMTSF)2PF65
0.25 0
p-wave
chiral p-wave
0.2 chiral f-wave -0.025
chiral f’-wave
0.15 -0.05
ρ0 C
0.1 -0.075
0.05 -0.1 p-wave
chiral p-wave
chiral f-wave
chiral f’-wave
0 -0.125
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
a) t b) t
Fig. 3 – Normalised Hc2(t) and C(t) for H~ k ~a are shown in a) and b) respectively. Here solid,
dashed, dashed-dotted and dotted lines are chiral f′-wave, chiral f-wave, chiral p-wave and simple
p-waverespectively.
B. Chiral p-wave SC: ∆(k) 1 sgn(k )+ısinχ . Now H (t) is determined by
∼ (cid:16)√2 a 2(cid:17) c2
a similar set of equations as in sec. 1.B. In particular we find for t 1 C = 0.027735
→ −
and ρ = 0.212598(lnt) while for t 0 C = 0.067684 and ρ = 0.139672. We obtain
0
→ −
h(0)=0.6566. We show h(t) and C(t) in Fig. 3 a) and b) respectively.
C. Chiral f-wave SC: ∆ˆ(k) dˆscosχ . Again we use a similar set of equations as those
1
∼
discussed in sec. 1.C, we find for t 1 C = 0.0356236 and ρ = 0.2744495(lnt) while for
→ −
t 0 C = 0.066 and ρ = 0.1920 and h(0) = 0.6997. Both h(t) and C(t) are evaluated
0
→
numerically and shown in Fig. 3 a) and b).
D. Chiral f-wave SC: ∆ˆ(k) dˆscosχ . Now we find for t 1 C = 0.05 and ρ =
′ 2
∼ → −
0.2910(lnt), while for t 0 C = 0.1019 and ρ =0.2090.
0
− → −
We have shown again h(t) and C(t) in Fig. 3 a) and b) respectively.
Comparing these results with H (T) from (TMTSF) PF and (TMTSF) ClO [4,5], we
c2 2 6 2 4
canconcludebothH~ b~ andH~ ~athechiralf-waveSCismostconsistentwithexperimental
k ′ k ′
data. In particular these states have relatively large h(0) (see Table I).On the other hand
almost the same H (0) for H~ b~ and H~ ~a has to be still accounted.
c2 k ′ k
Nodal lines in ∆(~k). – We have seen that from the temperature dependence of H (T),
c2
we deduce the chiral f-wave and chiral f are the most favourable. They have nodal lines on
′
the Fermi surface (i.e. the χ χ plane), the chiral f-wave SC at χ = π, while chiral
f-wave SC at χ = π. 1 − 2 1 ±2
′ 2 ±2
These nodal lines may be detected if the nuclear spin relaxation rate is measured in a
magnetic field rotated within the b c plane.
′ ∗
−
Following the standard procedure given in [14] the quasiparticle density of states in the
vortex state for T <<T and E =0 is given by
c
N 0,H~ = 2 v2√eH 1+cosθ2sinχ 2 21 (16)
π2 10
(cid:16) (cid:17) (cid:0) (cid:1)
6
EUROPHYSICSLETTERS
Table I – Summary of results. Here ρ (0)= vˆ2eHc2(0) and h(0)= Hc2(0)
0 2(2πTc)2 ∂H∂c2t(t)|t=1
symmetry C(0) C(1) −∂∂ρt|t =1 ρ0(0) h(0)
p-wave -0.031 0 0.2377 0.1583 0.6659
H~ kb′ chiral p-wave -0.2247 -0.3660 0.3838 0.2734 0.71324
chiral f′-wave -0.2247 -0.3660 0.5181 0.3734 0.72073
p-wave -0.017 0 0.2377 0.1751 0,7366
H~ ka chiral p-wave -0.066 -0.028 0.2126 0.1396 0,6566
chiral f-wave -0.066 -0.035 0.2744 0.1920 0.6997
chiral f′-wave -0.1019 -0.05 0.2910 0.2090 0,7182
where χ is the position of the nodal line on the χ axis. So for the chiral f-wave SC we
10 10
findχ = π andN 0,H~ exhibits the simple angulardependence. Onthe other handwhen
10 2
(cid:16) (cid:17)
nodal lines are on the χ axis, the θ dependence will be too small to see. Finally this gives
2
2
2
T1−1(cid:16)H~(cid:17)/T1−N1 =(cid:18)π2(cid:19) ~v2(eH)(cid:0)1+cosθ2(cid:1) (17)
for the chiral f-wave SC.
We show the θ dependence of T1−1 in Fig. for a few candidates. The chiralf-wave SC has
thestrongestθ dependence(solidline)whilethechiralh-waveSC(dashedline)andthechiral
p-wave SC (dotted line) have a similar θ dependence.
Concluding remarks. – We have computes the upper critical field of Bechgaard salts
for a variety of model superconductors with the standard microscopic theory. We find:
a) Assuming all these superconductors have the same T , the chiral f-wave SC (∆~(k)
c ′
∼
1 sgn(K )+ısinχ cosχ ) appears to be the most favourablewith largestH ’s for both
√2 a 2 2 c2
(cid:16) (cid:17)
H~ b and H~ a; b) however, non of these states exhibit the quasi T linear dependence of
′
k k
Hc2(T)asobservedin [4]; c)Alsothe presenttheorypredictsHc2(0) (vvc)−1 and(vbvc)−1
∼
for H~ b and H~ a respectively. This means H (0) for H~ a is about 5 time larger than
′ c2
k k k
theoneforH~ b contrarytoobservation;d)fromH (0) 5TandT =1.5Kwecanextract
′ c2 c
k ∼
V2 =√vvc 1.5 104cms−1, consistent with the known values of v, vc.
∼ ∗
∗∗∗
WethankStuartBrownandPaulChaikinforusefuldiscussiononpossibledetectionofthe
nodalstructureof∆(~k)sinBechgaardsaltsthroughNMR.Wehavebenefitedfromdiscussion
with Stephan Haas and David Parker. ADF acknowledges gratefully the financial support of
Ministeriode Educaci´onyCiencia(Spain)(AP2003-1383)andJaimeFerrerandPacoGuinea
for useful discussion.
REFERENCES
[1] Jerome D., Mazard A., Ribault M. and Bechgaard K., J. Phys (France) Lett, 41 (1980)
L95
[2] IshiguroT.,YamajiK.andSaitoG.,OrganicSuperconductors, Springer-Verlag(Berlin1999)
[3] Sigrist M. and Ueda K., Rev. Mod. Phys., 63 (1991) 239
[4] Maki K., Haas S., Parker and Won H., Chinese J. Phys, 43 (2005) 532
AnaDom´ınguezFolguerasandKazumiMaki:UppercriticalfieldHc2 inBechgaardsalts(TMTSF)2PF67
2
1.8
1.6
1.4
1.2
1
0 0.5 1 1.5
θ
Fig. 4 – The angle dependentnuclear spin relaxation rate for a few nodal superconductors is shown.
(Chiral f-wave, chiral h-waveand chiral p-waveare represented in solid, dashed and dotted lines.)
[5] Lee I.J., Chaikin P.M. and Naughton M.J., Phys. Rev. B, 63 (2002) R180502
[6] Oh J.I. and Naughton M.J., Phys. Rev. Lett., 92 (2004) 067001
[7] Clogston A.M., Phys. Rev. Lett., 9 (1967) 266
[8] Chandrasekhar B.S.,Appl. Phys. Lett., 1(1962) 7
[9] LeeI.J.,BrownS.E.,ClarkW.G.,StrouseM.J.,NaughtonM.J.,KangW.andChaikin
P.M., Phys. Rev. Lett., 88 (2002) 017004
[10] Gor’kov L.P., Soviet Phys. JETP, 10 (1960) 59
[11] Luk’yanchuk I. and Mineev V.P., Soviet Phys. JETP, 66 (1987) 1168
[12] Lebed A.G.,JETP Lett, 44 (1986) 114
[13] Depuis N., Mantambaux G. and Sa de Melo C.A.R.,Phys. Rev. Lett., 70 (1993) 2613
[14] Won H., Haas S. Parker D., Telang S., Vanyolos A. and Maki K., Lectures on the
Physics of Highly Correlated Electron Systems IX, AIP Conference Proceedings 789 (Melville
2005)
[15] Maki K., Haas S., Parker D. and Won H.,cond-mat/0504635, (2005)
[16] Won H. and Maki K., Europhys. Lett., 30 (1995) 421 Phys. Rev. B,53 (1996) 5927
[17] Abrikosov A.A., Soviet Phys. JETP, 5 (1957) 1174
[18] Wang G.F. and Maki K.,Europhys. Lett., 45 (1999) 71