The phase diagram of an orthorhombic d-wave superconductor

Physica C 312 Ž1999. 304–312

The phase diagram of an orthorhombic d-wave superconductor Ch. Jurecka, E. Schachinger

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Institut fur ¨ Theoretische Physik, Technische UniÕersitt Graz, 8010 Graz, Austria Received 24 August 1998; revised 1 December 1998; accepted 8 December 1998

Abstract Possible pure and mixed symmetry phases of the superconducting order parameter are studied for a two dimensional system of orthorhombic symmetry which is assumed to be a representative of the CuO 2 planes typical for high Tc superconductors. The anisotropy of the Fermi surface due to the orthorhombicity is described by an effective mass model. The study is based on the BCS-theory of anisotropic superconductors and reveals that an Ž s q d .-symmetric order parameter is the stable solution for the biggest part of the available parameter space. A small pocket in which an w s q iŽ s q d .x-symmetric order parameter is the stable solution has also been found. q 1999 Elsevier Science B.V. All rights reserved. PACS: 74.20.Fg; 74.25.Nf; 74.72.y h Keywords: d-Wave superconductor; Orthorhombic symmetry; Phase diagram

Optimally doped YBa 2 Cu 3 O6.95 ŽYBCO. with its orthorhombic structure is probably the most intensively studied material of the high-Tc oxides. There is quite some experimental evidence based on tunneling experiments w1–5x that the superconducting order parameter of this material is of Ž s q d .-symmetry. Covington et al. w6x observed in another, specific tunneling experiment a surface-induced broken time-reverse symmetry order parameter in YBCO. This manifested itself by a split zero bias conductance peak; such a result was predicted theoretically by Fogelstrom ¨ et al. w7x as an Andreev bound state which is shifted to finite energy and thus results in such a split zero conductance peak. This broken time-reverse symmetry state has been identified as a Ž d q is .-symmetric state of the superconducting order parameter which develops in YBCO close to the boundary with an insulator. In YBCO there are two CuO 2 planes and one CuO chain per unit cell and a theoretical description of the quasiparticle states involves three bands w8x. Hybridization resulting from the transverse hopping between planes leads to two orthorhombic bands and one tetragonal. The superconducting gap will then acquire in addition to a dominant d x 2yy 2 part an s and an extended s-wave component because all belong to the same irreducible representation of the C2 Õ space group and thus will mix freely w9,10x. The simplest approach to band anisotropy though is a free electron band model but with different effective masses along the a- and b-direction of a two dimensional lattice representing the CuO 2 planes which are commonly believed to be most important for

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Corresponding author. Tel.: q43-316-873-8176; Fax: q43-316-873-8678; E-mail: [email protected]

0921-4534r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 8 . 0 0 6 9 4 - 7

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superconductivity in the cuprates. Such a model has already been proved to be useful in a first understanding of some aspects of the in-plane Ž a,b .-anisotropy in YBCO w11,12x. Other model approaches have been studied by Musaelian et al. w13x and by Ferrer et al. w14x. The phase diagram of the order parameter in pure tetragonal systems has been studied by Matsumoto and Shiba w15x and this work was extended by Preosti and Palumbo w16x to encompass the effect of nonmagnetic impurity-induced time-reversal breaking in unconventional superconductors. It is the aim of this paper to discuss the phase diagram of the superconducting order parameter DŽ k,T . of an orthorhombic d-wave superconductor in the clean limit and we start with the well-known BCS equation for an anisotropic superconductor in an imaginary axis notation V Ž k , kX . D Ž kX ,T .

D Ž k ,T . s yT

Ý < v n
¦(

v n2 q D 2 Ž kX ,T .

;

. k

Ž 1.

X

Here, the v n s p T Ž2 n q 1., n s 0, "1, "2, . . . , are the charge carrier Matsubara frequencies, v c is a high energy cutoff usually an integer multiple of some characteristic exchange boson energy, T is the temperature, and V Ž k, kX . is the anisotropic pairing potential responsible for the formation of Cooper pairs. Finally, ² PPP : k stands for the Fermi surface average. A mixed s and d x 2yy 2 symmetry of the order parameter can be introduced using the following ansatz w13,17x

D Ž k ,T . s D0 Ž T . q e iqD d Ž w ,T . , D d Ž w ,T . s D1 Ž T . '2 cos2 w ,

Ž 2.

with D0 ŽT . the isotropic, s-wave contribution and D d Ž w . the d x 2yy 2-symmetric contribution to the order parameter. w is the polar angle in the two dimensional CuO 2 plane, q describes the phase between the two components, and D1ŽT . is the amplitude of the d-wave symmetric contribution. We will concentrate particularly on q s 0, the Ž s q d .-symmetric case, and on q s pr2, the Ž s q id .-symmetric case. Following the work by Millis et al. w18x and by Schachinger and Carbotte w19x the pairing potential is assumed to be separable and of the form yV Ž k , kX . s yV Ž w , w X . s A 0 q 2 A1cos2 w cos2 w X q . . . ,

Ž 3.

where A 0 and A1 are two free parameters of the theory. They define the anisotropy in the pairing potential and they can, of course, be of either sign with the negative sign indicating a repulsive interaction between the charge carriers. The function 62cos2 w plays the role of a Fermi Surface Harmonic ŽFSH. function w20x and Eq. Ž3. can also formally be understood as an FSH expansion of the separable pairing potential. The orthorhombicity of the lattice is described by an ellipsoidal Fermi surface

´Fs

k F2 , a

q

2 ma

k F2 , b

Ž 4.

2 mb

with ´ F the Fermi energy, k FŽ a, b. the Fermi momentum in a and b-direction, respectively, and mŽ a, b. the corresponding effective mass. This Fermi surface can be transformed into a circular one using a coordinate transformation from Ž k i , w . to Ž pi , f .-space w11x: pi s k i

ma q mb

( (

tan f s

2 mi ma mb

,

tan w ,

i s a,b,

Ž 5a . Ž 5b .

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306

which results in the Fermi surface p F2 , a q p 2F , b

´Fs

ma q mb

.

Ž 6.

This transformation leaves the isotropic part of the order parameter unaltered, the d x 2yy 2-symmetric part is modified because of cos2 w ™

a q cos2 f 1 q a cos2 f

,

Ž 7.

with the effective mass anisotropy parameter ma y mb as . ma q mb

Ž 8.

We introduce the ansatz Ž2. into the gap equation Ž1. and separate for D0 and D1. This results, using a vector notation, in

GsDy

 

A0T

Ý

v n2 q D 2

< v n
A 1T

D Ž f ,T .

¦( ¦(

D Ž f ,T . f Ž f .

Ý < v n
; ;

Ž f ,T .

v n2 q D 2 Ž f ,T .

f

f

0

Ds

s 0,

D0 Ž T .

ž / D1 Ž T .

,

Ž 9.

for the Ž s q d .-symmetry with DŽ f ,T . s D0 ŽT . q D1ŽT . f Ž f .. The two components of the gap function in Ž s q id .-symmetry are determined by

GsDy

A0T

Ý

¦( ¦(

Ý < v n
; ;

v n2 q D02 Ž T . q D12 Ž T . f 2 Ž f .

< v n
A 1T

D0 Ž T .

D1 Ž T . f Ž f .

v n2 q D02 Ž T . q D12 Ž T . f 2 Ž f .

f

f

0

s 0.

Ž 10 .

Here ² PPP :f denotes the Fermi surface average over the circular Fermi surface, f Ž f . is defined by fŽf. s

1 cos2 f q a N 1 q a cos2 f

,

Ž 11 .

and N is the normalization factor

¦ž

N2s

cos2 f q a 1 q a cos2 f

2

/;

.

Ž 12 .

f

If more than one symmetry of the order parameter exists at a given temperature it is necessary to determine the free energy difference D F ŽT . between the normal and the superconducting state w21,22x D F Ž T . s 2p TN Ž 0 .

Ý n)0

¦(

D 2 Ž f ,T .

v n2 q D 2 Ž f ,T .

(

;

y 2 v n2 q D 2 Ž f ,T . q 2 v n

, f

Ž 13 .

Ch. Jurecka, E. Schachingerr Physica C 312 (1999) 304–312

307

with DŽ f ,T . either determined from Eq. Ž9. or Eq. Ž10. depending on the symmetry and N Ž0. the quasiparticle density of states at the Fermi surface. The gap symmetry which results in the greater value of < D F ŽT .< will then be the stable solution at the given temperature. It is obvious that ² f Ž f .:f / 0 as long as a / 0 and thus DŽ f ,T . contains an isotropic admixture. In order to separate the symmetries fully we introduce w12x FŽ f . sf Ž f . yf ,

f s ² f Ž f . :f ,

Ž 14.

which ensures ² F Ž f .:f s 0, and we write either

D Ž f ,T . s DX0 Ž T . q DX1 Ž T . F Ž f . ,

DX0 Ž T . s D0 Ž T . q f D1 Ž T . ,

DX1 Ž T . s D1 Ž T . ,

Ž 15 .

in the case of Ž s q d .-symmetry, or

D Ž f ,T . s DX0 Ž T . q i DX1 Ž T . F Ž f . ,

DX0 Ž T . s D0 Ž T . q if D1 Ž T . ,

DX1 Ž T . s D1 Ž T . ,

Ž 16 .

for the Ž s q id .-symmetry which now turns out to be rather of w s q iŽ s q d .x-symmetry. We would like to emphasize at this point that it is a typical feature of this model that there is a Ž s q d .-symmetric part of the order parameter at all temperatures T - Tc reflecting the orthorhombicity of the system. This also makes the pairing potential Ž3. which is used in the gap Eqs. Ž9. and Ž10. effectively orthorhombic. We start our analysis in concentrating on the Ž s q d .-symmetry. It becomes immediately apparent that the case D0 ŽT . / 0 and D1ŽT . s 0 contradicts Eq. Ž9. for any temperature T as long as f / 0. The same holds for the ‘orthorhombic d-wave’ case with D0 ŽT . s 0 and D1ŽT . / 0. ŽWe keep the term ‘orthorhombic d-wave’ to characterize the case with D0 ŽT . s 0 and D1ŽT . / 0. This state is according to Eq. Ž15. or Eq. Ž16. not a pure d-wave state as there will always be some isotropic contribution to the order parameter.. Therefore, it is to conclude that D0 ŽT . and D1ŽT . and thus DX0 ŽT . and DX1ŽT . can only become zero at the same temperature, namely the critical temperature Tc . This temperature can be calculated by applying the implicit function theorem to Eq. Ž9. and by calculating the functional matrix of Eq. Ž9. at the point Ž D0 ŽT . s D1ŽT . s 0,T .:

EG ED

s Ž0,0,T .



1

1 y A0T

Ý < v n
yA1T

< vn < f

Ý < v n
< vn <

f

yA 0 T

Ý < v n
1 y A 1T

< vn < 1

Ý < v n
< vn <

0

.

Ž 17 .

At the critical temperature the determinant of the matrix Ž17. is equal to zero and this results in the Tc equation of the Ž s q d .-symmetric case: Tc 1 ,2 s 1.13 vc eyx 1 ,2 ,

Ž 18 .

with x 1,2 s p

(

Ž A 0 q A1 . . A20 q A21 q 2 A 0 A1 Ž 2 f y 1 . 2 A 0 A1 Ž 1 y f 2 .

,

Ž 19 .

which is symmetric in A 0 and A1 with Tc depending on both parameters. In the case f s 0 Ž a s 0, tetragonal lattice symmetry. Tc is either determined by A 0 or A1 Žsee also Ref. w15x.. We emphasize again that as a result of orthorhombicity the pure s-wave or the pure d-wave solution are forbidden for all temperatures T - Tc . Thus a mixture between s and d-wave always occurs resulting in either s-wave Ž DX0 ŽT . ) DX1ŽT .. or d-wave Ž DX0 ŽT . - DX1ŽT .. dominated Ž s q d .-symmetric solutions.

Ch. Jurecka, E. Schachingerr Physica C 312 (1999) 304–312

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Successive approximation is a very convenient numerical method in solving Eq. Ž9. and the gap D lq1 of step l q 1 is found from the solution D l of the l-th step by solving

D lq1 s g Ž D l .

Ž 20 .

for any temperature T - Tc . Thus, Eq. Ž20. establishes a system of difference equations and the fixed point of Eq. Ž9. is determined by

D lq1 s D l .

Ž 21 .

Furthermore, the theory of difference equations offers a theorem to the content that a fixed point is stable if and only if the absolute value of the diagonal elements of the diagonalized matrix E g Ž D .rE D are less than one. Thus, for instance, the fixed point Ž D0 s D1 s 0. is stable for all temperatures T G Tc and becomes unstable for T - Tc . But unstable fixed points can also occur at temperatures below Tc as Fig. 1 demonstrates for the parameter set A 0 s 0.87, A1 s 0.9, vc s 200 meV, and a s 0.05. ŽThis set of parameters ensures a critical temperature of the order of 90 K.. The coupling potential is dominated by its d-wave symmetric part Ž A1 ) A 0 .

X X Fig. 1. The temperature dependence of the s-wave Ž D0 , solid line. and the d-wave Ž D1 , dashed line. contribution to an order parameter of Ž sq d . symmetry according to Eqs. Ž9. and Ž12.. The parameters are A 0 s 0.87, A1 s 0.9, vc s 200 meV, and a s 0.05. Top frame: the d-wave dominated Ž sq d .-symmetric solution. Bottom frame: below the temperature T ) which marks the unstable fixed point an s-wave dominated Ž sq d .-symmetric solution can be found. For T )T ) only the d-wave dominated Ž sq d .-symmetric solution can by found by applying the method of successive approximation in solving Eq. Ž9..

Ch. Jurecka, E. Schachingerr Physica C 312 (1999) 304–312

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and the top frame of Fig. 1 shows the temperature dependence of the s-wave part Ž DX0 , solid line. and of the d-wave part Ž DX1 , dashed line. of the order parameter. As to be expected, we find a d-wave dominated Ž s q d .-symmetric solution of the gap Eq. Ž9.. The bottom frame of this figure demonstrates that an s-wave dominated Ž s q d .-symmetric order parameter is also a solution of Eq. Ž9. which can be found by using a different set of starting values. This solution becomes an unstable fixed point at the temperature T ) ; 50 K above which only the d-wave dominated Ž s q d .-symmetric solution can be found using this numerical method. Nevertheless, it is important to note that calculating the free energy difference reveals the d-wave dominated solution to be the stable one for all temperatures. Fig. 2 presents the phase diagram of the order parameter in Ž A 0 ,T .-parameter space for A1 s 0.9, vc s 200 meV, and a s 0.1. We see that below a certain value of A 0 Žwhich, of course, depends on a . only a d-wave dominated Ž s q d .-symmetry can be found. In a certain, rather small, region of values for A 0 a first order phase transition from a d-wave dominated gap symmetry to an s-wave dominated one can be observed. With A 0 approaching the value of A1 and A 0 ) A1 only s-wave dominated solutions of Eq. Ž9. are stable. We are now in a position to apply the methods developed above to the analysis of possible solutions of the gap equations Ž10. which are valid for the w s q iŽ s q d .x-symmetry. Here, the sets Ž D0 / 0, D1 s 0,T . and Ž D0 s 0, D1 / 0,T . are independent solutions of Eq. Ž10. and thus both ‘pure’ phases can exist. Symmetry arguments imply that there will be a certain temperature T - Tc below which a mixed, w s q iŽ s q d .x-symmetric phase can exist. ŽIn the tetragonal case, a s 0, this will then reduce to a Ž s q id .-symmetric phase w15,16x.. Tc is calculated from the functional matrix of Eq. Ž10. at Ž D0 s D1 s 0, T .:

EG ED

s Ž0,0,T .



1

1 y A0T

Ý < v n
0

0

< vn < 1 y A 1T

1

Ý < v n
< vn <

0

,

Ž 22 .

and Tc is found using the same arguments as before by: Tc 0 ,1 s 1.13 vc eyp r A 0 ,1 .

Ž 23 .

This result implies that for A 0 ) A1 the system starts off with a gap function of pure s-wave symmetry and for A1 ) A 0 we can expect to observe orthorhombic d-wave Ž D1 / 0. symmetry. For A 0 s A1 it depends on the free energy difference which of the two phases will be stable. Furthermore, it can be shown using, again, the implicit function theorem that if the system starts off in pure s-wave symmetry there exists no temperature T - Tc which

Fig. 2. Phase diagram for an Ž sq d .-symmetric order parameter in Ž A 0 ,T . parameter space. The other parameters are: A1 s 0.9, Dc s 200 meV, and a s 0.1. Note that Tc changes as a function of A 0 according to Eqs. Ž18. and Ž19..

Ch. Jurecka, E. Schachingerr Physica C 312 (1999) 304–312

310

would allow a transition to a symmetry with D1 / 0. On the other hand, if the system starts off in orthorhombic d-wave symmetry there does exist a transition to solutions with D0 / 0 and D1 / 0 at some transition temperature T ) which can be determined by solving 1 y A0T )

Ý < v n
1 y A 1T )

Ý < v n
¦(

1

v n2 q D12

¦(

;

s 0,

Ž 24a.

;

s 0,

Ž 24b.

ŽT . f 2Ž f.

f 2Žf.

v n2 q D12 Ž T . f 2 Ž f .

f

f

simultaneously. Eqs. Ž24a. and Ž24b. establish—for a given value of a —the Ž A 0 , A1 .-parameter space with w s q iŽ s q d .x-symmetric solutions of Eq. Ž10. at temperatures T - T ). Of course, at the same temperature and within the same parameter space Ž s q d .-symmetric solutions can also be possible and thus calculating the free energy difference is again required to determine the stable phase. Fig. 3 shows the temperature dependence of the isotropic part Ž D0 , solid line. and of the anisotropic part Ž D1 , dashed line. of the order parameter according to Eq. Ž10. for the w s q iŽ s q d .x-symmetric case. In the temperature region T ) - T - Tc the gap is orthorhombic d-wave symmetric. The isotropic part Ž D0 / 0. appears for all temperatures below T ) only after the orthorhombic d-wave part became sufficiently suppressed. This is required because the w s q iŽ s q d .x-symmetric phase is a broken time-reverse symmetry phase of the order parameter. The parameters for this figure are A 0 s 0.875, A1 s 0.9, vc s 200 meV, and a s 0.05. Inspection of the free energy difference reveals that at T ) a second order phase transition occurs when the system changes from the orthorhombic d-wave symmetry to the w s q iŽ s q d .x-symmetric state. ŽThe existence of an s q id-symmetric order parameter in an orthorhombic system was also reported by O’Donovan and Carbotte w23x using a two-dimensional tight binding model.. Finally, Fig. 4 presents the phase diagram of the superconducting order parameter for A1 s 0.9, vc s 200 meV, and a s 0.05 in Ž A 0 ,T .-parameter space. It shows that there is at low temperatures a ‘pocket’ of A 0 values at which an w s q iŽ s q d .x-symmetric order parameter is allowed to exist. This pocket becomes smaller in A 0 variation and moves to lower temperatures with increasing values of a until it vanishes completely as has

Fig. 3. The temperature dependence of the s-wave Ž D0 , solid line. and of the ‘d-wave’ Ž D1 , dashed line. contributions to an order parameter of w sq iŽ sq d .x-symmetry according to Eq. Ž10.. The parameters are: A 0 s 0.875, A1 s 0.9, vc s 200 meV, and a s 0.05. The onset of the s-wave part appears at the temperature T ) as determined by Eqs. Ž24a. and Ž24b..

Ch. Jurecka, E. Schachingerr Physica C 312 (1999) 304–312

311

Fig. 4. Complete phase diagram of the superconducting order parameter in the Ž A 0 ,T . parameter space. The other parameters are: A1 s 0.9, vc s 200 meV, and a s 0.05.

already been demonstrated in Fig. 2 for a s 0.1. We find in particular that for A 0 - 0.846 the d-wave dominated Ž s q d .-symmetric solution is the stable one for all temperatures. Within 0.876 F A 0 F 0.884 we find first order transitions from a d-wave dominated Ž s q d .-symmetric to an s-wave dominated one, while for A 0 ) 0.884 this solution is the only stable one for all temperatures. A first order phase transition w24x from the d-wave dominated Ž s q d . symmetry to an w s q iŽ s q d .x symmetry occurs in the range 0.846 F A 0 F 0.876. In the limit a ™ 0 our results agree completely with the phase diagram of a system with tetragonal symmetry as it was discussed by Matsumoto and Shiba w15x. They also showed a pocket in which an Ž s q id .-symmetric order parameter can exist. The triple point ŽFig. 4. moves to the critical temperature of the d-wave phase and the pocket is significantly larger in Ž A 0 ,T .-space compared to the one shown in Fig. 4. Furthermore, no phase transition from pure d-wave to pure s-wave is allowed in the tetragonal case while transitions from d-wave dominated Ž s q d .-wave symmetry to the s-wave dominated one are certainly possible in systems of orthorhombic symmetry ŽFigs. 2 and 4.. If one wants to compare our results with experiment it has to be noted that measuring the anisotropy of the London penetration depth l at low temperatures immediately gives access to the effective mass anisotropy parameter w25x:

la lb

1qa s T™0

1ya

.

Ž 25 .

Using the experimental data by Bonn et al. w26x for optimally doped, nominally clean YBCO single crystals we find a s 0.41. Tunneling experiments w4x suggest the s-wave contribution to the superconducting order parameter to be of the order of 10 to 15%. This can only be achieved if A 0 is negative with < A 0 < - < A1 < which is not surprising as A 0 plays the role of an on-site potential. Such a parameter set will result in a d-wave dominated Ž s q d .-symmetric order parameter for all temperatures and no phase transition will be observed which agrees well with specific heat data w27x. We studied possible symmetries of the superconducting order parameter for a two dimensional superconductor of orthorhombic lattice symmetry. We found that in compliance with the irreducible representations of the C2 Õ space group the order parameter can indeed be of pure s-wave, orthorhombic d-wave, s-wave dominated Ž s q d ., d-wave dominated Ž s q d ., and of w s q iŽ s q d .x-symmetry. Checking the free energy difference between the normal and the superconducting state revealed furthermore that there is only a very small pocket in the Ž A 0 , A1 , a ,T .-parameter space in which the w s q iŽ s q d .x-symmetry is the stable solution. In all cases of rather large effective mass anisotropy the d-wave dominated Ž s q d .-symmetry is the only stable solution for all temperatures. In this case there will be at all temperatures T - Tc an isotropic admixture to the predominantly d-wave symmetric order parameter and no phase transitions will be observed.

312

Ch. Jurecka, E. Schachingerr Physica C 312 (1999) 304–312

Acknowledgements The authors acknowledge fruitful discussions with Dr. I. Schurrer and thank Dr. J.P. Carbotte for his great ¨ interest in this work. This research was supported in part by Fonds zur Forderung der wissenschaftlichen ¨ Forschung ŽFWF., Vienna, Austria under contract No. P11890-NAW.

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