Band Jahn–Teller effect on the density of states of the magnetic high-Tc superconductors: A model study

Physica C 475 (2012) 14–19

Contents lists available at SciVerse ScienceDirect

Physica C journal homepage: www.elsevier.com/locate/physc

Band Jahn–Teller effect on the density of states of the magnetic high-Tc superconductors: A model study B. Pradhan a, K.L. Mohanta b, G.C. Rout c,⇑ a

Department of Physics, Govt. Science College, Malkangiri 764 048, India Department of Physics, ITER, Siksha ‘O’ Anusandhan University, Bhubaneswar 751 030, India c Condensed Matter Physics Group, Dept. of Applied Physics and Ballistics, F.M. University, Balasore 756 019, India b

a r t i c l e

i n f o

Article history: Accepted 27 December 2011 Available online 31 January 2012 Keywords: High-Tc cuprate superconductors Antiferromagnetism Jahn–Teller effect Density of states

a b s t r a c t We report here a mean-field study of competing antiferromagnetism, superconductivity and lattice strain phases and their effect on the local density of states of the cuprate system. Our model Hamiltonian incorporating these interactions is reported earlier [G.C. Rout et al., Physica C, 2007]. The analytic expression for superconducting, antiferromagnetism and lattice strain order parameters are calculated and solved self-consistently. The interplay of these order parameters is investigated considering the calculated density of states (DOSs) of the conduction electrons. The DOS displays multiple gap structures with multiple peaks. It is suggested that the tunneling conductance data obtained from the scanning tunneling microscopy (STM) measurements could be interpreted by using the quasi-particle bands calculated from our model Hamiltonian. We have discussed the mechanism to calculate the order parameters from the conductance data. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction The pairing mechanism of high temperature superconductivity is still not known. The intermediate region between the antiferromagnetism and superconductivity is crucial for the understanding of the pairing in copper oxide systems. The interplay between different phases, i.e., antiferromagnetic (AFM) insulator, lattice distortion, superconductivity, spin density wave (SDW), charge density wave (CDW) and so on, seems to be essential to understand many of the physical properties [1,2] of high temperature superconducting (SC) materials. The experimental techniques involving high-Tc cuprate superconductors have shown an ubiquitous phase separation phenomenon at nanometer scale [3–5]. A number of high temperature superconductors (HTSs) have been discovered, all of which, have shown evidence of the opening of pseudogap (PG) in the under-doped region of the electronic excitation spectrum of HTSs above the critical temperature Tc. These techniques measure the density of the single electronic states as a function of energy and momentum of the occupied states in the angle resolved photoemission spectroscopy (ARPES) [6–8] and both occupied and unoccupied states in the tunneling measurements [8–10]. These techniques are particularly sensitive to the density of states (DOSs) near Fermi surface (FS), with the unique capability to probe the states above and below FS. The ⇑ Corresponding author. Tel.: +91 9937981694. E-mail address: [email protected] (G.C. Rout). 0921-4534/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2011.12.041

tunneling spectroscopy is thus especially sensitive to any gap in the quasi-particle excitation spectrum at FS, which shows up directly as a characteristic feature near zero bias. So far various theoretical interpretations of DOS and PG phenomena have emerged from various literatures. The phenomenological interpretations are that, the PG is the manifestation of some order, static or fluctuating, generally of magnetic origin, but unrelated to and/or in competition with the SC order [11–15] and the PG is the precursor of the SC gap, and reflects the pair fluctuation above Tc [16–19]. The band Jahn–Teller (BJT) effect is assumed to manifest as PG in high-Tc systems. The AFM and the SC phases are individually well understood phenomenologically. The intermediate regime interpolating between AFM and SC remains mysterious even after several years of research. The much debated PG appearing in the DOS may be regarded as a manifestation of hidden order. It is also suggested that this strange phase may be characterized by strong attraction between the charge carriers leading to the condensed pairs [20,21]. The pairing symmetry of the superconducting energy gap in highTc superconductors still remain an open problem. The experimental observation which are sensitive to the phase and nodes [22–26] provide strong evidences for d-wave like pairing symmetry. On the other hand the experiments on Y1Ba2Cu3O7 (YBCO) material indicate the existence of a significant s-component [27–29]. There are strong evidences that the electron doped Nd2xCexCuO4 [30–32] superconductors are s-wave type. Some theories and experiments indicate that the high-Tc material may have a mixed

B. Pradhan et al. / Physica C 475 (2012) 14–19

pairing symmetry, i.e d ± s or d + is/dxy in presence of external magnetic field and interface effects [33–42]. The angle resolved photoemission (ARPES) study provides strong indication Bi2Sr2CaCu2O8+d compound is d-wave like in the under and optically doped regime, where as not d-wave like in the slightly over doped high-Tc samples [43]. On the other hand, pairing mechanism based on electron– phonon interactions and polarons may be compatible with pure s-wave and pure d-wave or an admixture of the two [44]. Thus it is evident that the symmetry of the order parameter and the pairing mechanism are not clear in high-Tc cuprates. The high-Tc cuprates are regarded as the ideal systems to exhibit the interplay between BJT effects, superconductivity and the effects of strong correlation. One of the manifestations of the later being the presence of a long range AFM order in the undoped systems and strong AFM fluctuations in the doped ones, where these may coexist with and even can be responsible for SC [45,46]. The BJT effect arises due to the doping of impurities as well as the partially filled electronic states of the copper ions in high-Tc superconductors. Recently Rout et al. [47,48] have studied the SC and AFM orders in cuprate systems in presence of an impurity f-level of the rare-earth ions and the hybridization of it with the 3d-electrons of the copper atoms. Earlier Ghosh et al. [49] have considered the BJT distorted degenerate conduction band and have studied its effect on the SC transition temperature. Recently Rout et al. have reported the BJT effect on the SC pairing in high-Tc systems through the Raman spectra [50,51]. More recently, Rout et al. have studied the effect of lattice strain and antiferromagnetism on superconducting gap in copper oxide systems [52]. It is observed that antiferromagnetism and superconducting gaps are suppressed considerably in the co-existence phase with the lattice strain. Further the antiferromagnetic coupling enhances considerably the superconducting transition temperature. To carry out this model study further we, in this communication, calculate the single particle conduction electron DOS and attempt here to interpret the tunneling conductance data by using our model calculation. The Section 2 discusses the basic back ground of the model and presents the expressions for the SC order parameter, the lattice strain parameter and the sub lattice magnetization. The results and discussion are given in Section 3 and finally conclusion is given in Section 4. 2. Model and calculations It is well known that the high-Tc cuprate systems exhibit superconductivity in presence of antiferromagnetism. On doping the system, the long range strong AFM order is destroyed giving rise to short range AFM order. Further, the system transfers from high temperature orthorhombic to low temperature tetragonal structure. In this process the lattice gets deformed giving rise to the static Jahn– Teller distortion. We attempted here to frame a model to study the interplay of superconductivity, antiferromagnetism and lattice strain. In order to simulate the AFM sub-lattice magnetization, we introduce two sub-lattices in the crystal system with electron operators air and bjr respectively corresponding to spin up and spin down states and vice versa. Further the degenerate energy levels of the electrons at each site splits into two, designating two orbits (1 and 2) at the same site. In the present case we consider our earlier model Hamiltonian [52] which describes the superconductivity, antiferromagnetic and lattice strain interactions. The Hamiltonian Hc describing the hopping of the conduction electrons between the neighboring sites of the two degenerate orbits of Cu2+ is given by

X  y  Hc ¼  tij a1ir b1jr þ ay2ir b2jr þ h:c: :

ð1Þ

ijr y

y

Here aair ; bajr ðaair ; bajr Þ, for a = 1 and 2, are the creation (annihilation) operators of the conduction electrons of spin r for copper ions

15

at two orbitals 1 and 2 respectively. The lattice sites are denoted by i and j. The hopping of the electrons takes place between nearest neighboring sites of the copper ions with dispersion k = 2t0 (cos kx + cos ky), where 2t0 is the nearest neighbor hopping integral through the tight binding dispersion in CuO2 planes. In the insulating phase of La2CuO4 system, the usual Jahn–Teller effect for the eg manifold is responsible for the tetragonal to the orthorhombic transition at high temperatures. The splitting of the 3dz2 and 3dx2 y2 orbitals of Cu ions is quite large. On doping the splitting between orbitals reduces as discussed by Khomskii et al. [54] due to the formation of band Jahn–Teller effect. The band Jahn–Teller effect is the inverse Jahn–Teller effect arising out of the hole doping. Under a suitable doping concentration, the magnitudes of superconducting gaps coexist and interact strongly. It is to mentioned further that there is staggered filling of the oxygen octahedral in CuO2 planes arising due to doping. In our model study, we have included the possible lattice strain. For that purpose, two degenerate orbitals which couple differently to the distortions are used below. In presence of lattice distortion, the electron density in the degenerate conduction band interacts with the lattice and creates a population difference between the two bands. Thus a static Jahn–Teller distortion is created in the crystal lattice. So the interaction between the electron band and the static lattice strain is described by

HeL ¼ Ge

Xh

i y y ðay1ir b1jr þ b1jr a1ir Þ  ðay2ir b2jr þ b2jr a2ir Þ :

ð2Þ

ijr

As the population difference increases, the lattice strain splits the single degenerate band into two with band energies 1,2k = k ± Ge. Here, G is the strength of the electron–lattice interaction and e the strength of the static strain. The two Jahn–Teller distorted bands are separated by the energy 2Ge. The antiferromagnetism present in the lattice can also be described well by Heisenberg exchange interaction between the magnetic moments of electrons of the nearest neighboring lattice sites. In the present model study, we assume that the spins of the electrons are alternately up and down in the CuO2 planes. This introduces a sub-lattice magnetization in the CuO2 plane with strength h. This magnetization acts as magnetic order parameter. The sub-lattice magnetization simulates a strong magnetic correlation, which consequently suppresses the electron fluctuation in the CuO2 planes. The interaction Hamiltonian representing the sub-lattice magnetization can be written as

Hs ¼

i hg L lB X h y r ðaair aair  byajr bajr Þ ; 2 aijr

ð3Þ

where r = ±1, depending on the spin up and spin down states of the electrons respectively. Further gL and lB are the Lande-g factor and Bohr magnetron respectively and they are taken as 1 each. The total Hamiltonian describing the physical system is written as H = HC + HeL + Hs + HI. Under suitable doping concentrations and temperatures, the superconductivity develops in the system. In presence of sub-lattice magnetization in the strained lattice, it is assumed here that the BCS type pairing interaction exists only within the same orbitals with the same strength of superconducting interaction in both the orbitals. The inter orbital pairing is not taken into consideration in the present model. The mean-field superconducting Hamiltonian is written as

HI ¼ D

Xh

i y y ayak" aya;k# þ bak" ba;k# þ h:c ;

ð4Þ

a;k

where a = 1, 2. The superconducting order parameter defined in our earlier paper [52]. From the discussion in the introduction it is found there are conflicting conclusion regarding the theory of

16

B. Pradhan et al. / Physica C 475 (2012) 14–19

pairing mechanism for high-Tc cuprates. Since we are interested in explaining the multiple peak structures in the tunneling conductance spectra of high-Tc cuprates, we consider here the isotropic BCS type pairing mechanism for the present calculation for its simplicity. In this case the effective attractive interaction V0 is assumed to be anisotropic constant within the interval of ⁄xc around the Fermi level, where xc is the cut-off energy due to some boson fluctuation. One electron Green’s functions are calculated using the total Hamiltonian H for the superconducting cuprate systems. The Green’s functions for orbital 1 of the copper ions are defined as

A1 ðk; xÞ ¼ hha1k" ; ay1k" iix ; A3 ðk; xÞ ¼ hhb1k" ; ðay1k" iix ; y

A2 ðk; xÞ ¼ hhay1;k# ; ay1k" iix ; A4 ðk; xÞ ¼ hhb1;k# ; ay1k" iix : Similarly other four Green’s functions are defined for the orbital 2 of the Cu ions. These Green’s functions are calculated by using Zubarev’s Green’s functions technique of equations of motion method [53]. Detail calculations are given in our earlier paper [52]. The poles of these Green’s functions give eight quasi-particle energy bands, i.e., ±xik (i = 1–4). They are written as

x1k;2k

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2 h h 2 ¼ 1k þ D  and x3k;4k ¼ 22k þ D  ; 2 2

ð5Þ

with 1k,2k = (k ± Ge). Each of these bands contains the superconducting gap parameter D, J–T distortion energy parameter Ge, and the AFM parameter h. From our earlier calculations [52], the superconducting order parameter D, the lattice strain parameter, e, and the sub-lattice magnetization h can be written as

" ) ( Z tanh bx2 1 tanh bx2 3 g xD h dk D  þ 2 xD 2 x1 x3 )#  ( bx2 bx4 tanh 2 tanh 2 h ; þ þ Dþ 2 x2 x4

D¼

G e¼ C0 2k

Z

"

þW=2

dk

1k

( tanh bx2 1

x1

W=2

( tanh bx2 3

x3

þ

tanh bx2 4

)#

þ

tanh bx2 2

ð6Þ

)

x2

x4

ð7Þ

3. Results and discussion It is known that the BCS theory is valid for the metallic superconductors with the reduced gap size of 2D(0)/kBTc ’ 3.52. This reduced gap size of BCS theory is not true for high-Tc superconductors. Ekino et al. [55] have used the BCS reduced equation to find out the SC gap magnitude for the high-Tc systems, exhibiting the competing features between different kinds of orders and predicted possible origins of the observed anomalous high-Tc gaps. The possible gaping condensations are antiferromagnetic correlations, the CDW states and short range interactions, and charge stripes in the CuO2 layers. There is no practical theory to determine the SC gap magnitude of the copper oxide superconductors, which display a multiple competing interactions. In the present report, we consider the competition between the superconducting, antiferromagnetic and lattice strain orders and attempt to interpret the tunneling conductance data. For this purpose we have made dimensionless the different physical quantities dividing them by the hopping integral 2t0 and considered the width of the conduction band W = 8t0. The dimensionless parameters are the superconducting order parameter, z = D/2t0, the lattice strain, ~ e ¼ e=e0 , the AFM order parameter, h = h/2t0, the reduced temperature, t = kBT/2t0 and the distortion energy, e0 ð¼ ~e  g 1 Þ, is the product of the lattice strain, ~e and the Jahn–Teller coupling constant, g1. For the self-consistent calculations, we have considered the SC coupling parameter, g = 0.016, the Jahn–Teller coupling parameter, g1 = 0.151 and the AFM coupling parameter, g2 = 0.037. The selfconsistent plots of the temperature dependence of the SC gap, z, the J–T energy, e0 and the AFM order, h are shown in Fig. 1. We observe the SC transition temperature, tc ’ 0.0122 (Tc ’ 31 K), the lattice distortion temperature, td ’ 0.0058 (Td ’ 15 K), the AFM Néel temperature, tN ’ 0.0036 (TN ’ 9 K) and the hopping integral, 2t0 ’ 0.25 eV is taken for this calculation such that tc > td > tN. All the three long range orders coexist for the temperatures t < tN. The SC and J–T states coexist for temperatures tN < t < td and the SC order exists for temperatures td < t < tc. The temperature dependence of the SC gap shows mean-field behavior at higher temperature t > td and begins to suppress at low temperatures in the presence of the insulating J–T distortion phase. At still low temperature t < tN, the SC gap is more suppressed, but both the J–T and AFM long range orders exhibit the mean-field behavior through out the temperature range. The reduced SC gap size gives 2D(0)/ kBTc ’ 1.38, which is smaller as compared to the universal BCS

and

" ) ( Z tanh bx2 1 tanh bx2 3 g 2 þW=2 h dk D  þ 2 2 W=2 x1 x3 )#  ( bx 2 bx4 tanh 2 tanh 2 h þ :  Dþ 2 x2 x4

0.014

h¼

e’

0.012

z

h 0.01

The superconducting coupling constant is g = N(0)V0, with N(0) as the DOS of the conduction electrons at the Fermi level with conduction band with x. Here V0 and xD are the effective Coulomb interaction and the Debye energy respectively. The free energy of the electrons includes the lattice energy 12 C 0 e2 , where C0 is the elastic constant. The minimization of the free energy gives the value of lattice strain at any temperature (see Eq. (7)). The other coupling parameters are the Jahn–Teller coupling, g1 = G/2t0 and the AFM coupling, g2 = gLlBN(0). where 2t0 is the nearest neighbor hoping integral. The expressions for the parameters D, e and h written above are the involved functions in the integral form. In order to study the interplay between them, they are to be solved numerically and self consistently.

z, e’, h

ð8Þ

0.008 0.006 0.004 0.002 0

0

0.002

0.004

tN

0.006

td

0.008

0.01

0.012

tc

Fig. 1. The self-consistent plots of superconducting gap parameter, z, Jahn–Teller distortion energy, e0 , and antiferromagnetic order parameter, h, vs. reduced temperature, t, for the fixed values of SC coupling constant, g = 0.016, J–T coupling constant, g1 = 0.151 and AFM coupling constant, g2 = 0.037.

17

B. Pradhan et al. / Physica C 475 (2012) 14–19

400 t=0.006 t=0.004

DOS

300

I

I

200

100 O

O

0 -0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

~ ω Fig. 2. DOS plots for conduction electrons vs. quasi-particle band energy, x, for different reduced temperatures, t = 0.006 and 0.004.

250 I

I

t=0.004 t=0.003

200

I2

150

I2

DOS

value 2D(0)/kBTc ’ 3.52. The present model calculation of the interplay of the three order parameters indicates that the insulating J–T phase and the insulating magnetic phase suppress the SC gap magnitude at lower temperatures but enhances the SC transition temperature. The tunneling conductance measured by the electron tunneling measurements provides the direct measurement of the quasi-particle density of states (DOSs) of a superconductors. The detailed temperature dependence of gap features can be obtained by this technique. The electronic DOS of the conduction electrons is defined P as q ¼ 2p k;r ImGk ðx þ igÞ, where Gk(x) is the Green’s function for the conduction electrons of the present system under consideration. The DOS is plotted against the conduction electron band energy for different temperatures. The SC gap, z, the AFM gap, h and the J–T gap energy, e0 , parameters are obtained from the self-consistent plots of Fig. 1 and the graphs of the DOS vs. the reduced conduc~ , are plotted for different temperatures tion electron band energy, x as shown in Figs. 2 and 3. It is to note that the tunneling spectra for some of the high-Tc cuprates exhibit V-shaped energy dependence near the Fermi level and the very large peaks are at the gap edges. The V-shaped spectra at the low biased conductance indicates that there is a node in the gap. This is consistent with the d-wave pairing symmetry. On the other hand the tunneling conductance spectra for low-Tc superconductors exhibit U-shaped conductance with zero conductivity at the Fermi level and two symmetry square root singularities at the gap edges. Such U-shaped conductance spectra is the characteristics of the s-wave pairing symmetry. Since we have considered the s-wave pairing symmetry for our calculation we expect U-shaped multiple spectra arising due to interaction of the three interacting order parameters. The multiple conductance spectra can be interpreted below based upon our model calculation. In the pure SC phase at temperature t = 0.006, the DOS graph shows one gap with gap edges at ±z ’ ±0.0106. Hence the total SC gap, 2z, can be determined. In the co-existence of the SC and the J–T phases at temperature t = 0.004, the DOS displays two gaps: the inner gap (I–I) appears with gap edge at energy, ±z ’ ±0.0098 and the outer gap (O–O) appears with gap edge at energy, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð z2 þ e02 Þ ’ 0:0146. It is seen here that the SC gap edge is shifted from I to O due to J–T distortion by an amount of ±Ge. Thus the lattice energy can be measured from the gap separation between (I–I) and (O–O). In the co-existence phases of all the three order parameters at temperature t = 0.003 (see Fig. 3), the above two gaps (I–I) and (O–O) split into two gaps each, i.e., (I–I) splits to (I1–I1) and (I2–I2) and (O–O) splits to (O1–O1) and (O2–O2) gaps. Here the gap (I1–I1) exists with gap edge at energy ±(z  h/2) ’ ±0.0064 and the gap (I2–I2) exists with gap edge at energy ±(z + h/2) ’ ±0.0122.

100

I1 O

O1

I1

O2

O2

O O1

50

0 -0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

~

ω Fig. 3. DOS plot for conduction electrons vs. quasi-particle band energy, x, for different reduced temperatures, t = 0.004 and 0.003.

From these gap edge values, the SC and the AFM gap magnitudes can be determined. The calculated values are z = 0.0093 and h = 0.0058 at temperature t = 0.003. Out of the other two outer gaps, the extreme outer gap (O1–O1) exists with gap edge at energy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  e02 þ ðz þ h=2Þ2 ’ 0:0173 and the other gap (O2–O2) exists qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with gap edge at energy  e02 þ ðz  h=2Þ2 ’ 0:01393. The magnitudes of z and h are known at temperature t = 0.003 and hence e, can be calculated from the above expresthe J–T energy, e0 ¼ g 1 ~ sions derived in our model calculation. Since J–T coupling g1 is known, the magnitude of the lattice strain can be calculated. Thus, it is concluded that the magnitudes of all the three order parameters can be calculated from the measured tunneling spectra of the system at any given temperature. We have discussed above how to calculate the gap parameters from the tunneling conductance data in Figs. 2 and 3. These gap parameters can also be calculated easily from the quasi-particle band energies given in Eq. (5). We have plotted the quasi-particle band energies, xik (i = 1–4), against the conduction band energy   x ¼ 2tk0 as shown in Fig. 4 for different interplay regions corresponding to different temperatures. The pure SC phase exists at temperature, t = 0.006, where the quasi-particle dispersion band qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi splits into two, i.e., xk ¼  2k þ z2 , and are shown in Fig. 4a. The magnitude of the SC gap at Fermi-level is given by AA = 2z. When static J–T effect is introduced to the SC system, at temperature, t = 0.004 the above two quasi-particle bands split into two each as shown in Fig. 4b, and each band is shifted by the magnitude of Ge, on either side of the Fermi-level. The magnitude of the J–T gap is BB = 2Ge and the magnitude of the SC gap is BC = 2z. In the interplay region of the SC and lattice interaction, the magnitude pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the gap is given by e02 þ z2 . At still lower temperatures, at t = 0.003, all the three phases, i.e., superconductivity, antiferromagnetism and static lattice strain coexist. When the AFM interaction is introduced at this temperature, the four quasi-particle bands shown in Fig. 4b split into two each shifted by an amount of 2h. From the dispersion of the bands shown in Fig. 4c, we find DE ¼ z  2h ¼ 0:0064; FF ¼ z þ 2h ¼ 0:0122 and J–T energy, DD = 2e0 = 0.025. From these data, the magnitudes of the SC gap, AFM gap and J–T gap energies can be calculated.

4. Conclusion We consider here a model Hamiltonian in order to describe the interplay of three phenomena, i.e., the superconducting, the sub-

18

B. Pradhan et al. / Physica C 475 (2012) 14–19

0.02

~ ω

0.01

(a)

A

0 0.02

A

-0.01

(c)

F

-0.02 -0.02

0.01 -0.01

0

0.01

0.02

~ ω

(b) B

-0.01 F

B

-0.02 -0.02

0

-0.01

0

0.01

0.02

x C

-0.01 -0.02 -0.02

0 E

0.02 0.01

D

D

~ ω

x

-0.01

0

0.01

0.02

x Fig. 4. The curves of the dispersion for the quasi-particle bands vs. conduction electron band energy for different reduced temperatures, t = 0.006, 0.004 and 0.003.

lattice magnetization and the lattice strain in high-Tc cuprate superconductors. The interplay shows that the presence of the sub-lattice magnetization suppresses the magnitude of the SC gap, but enhances the SC critical temperature. However, the sublattice magnetization and the lattice strain show mean-field behavior. For the superconductor–insulator-superconductor (SIS) tunneling junction of a conventional superconductor, the peakto-peak separation of dI/dV corresponds to 4D(T) [10], where D(T) is the gap parameter measured from the Fermi-energy at a given temperature T. Ekino et al. [55] have adopted this definition to calculate the gap magnitudes, because no practical theory exists to determine the gap magnitudes of the copper oxide superconductors. The present model calculation involving the interplay of the three competing order parameters can be used to interpret the multiple peak structure observed in the scanning tunneling spectra (STS). The conductance spectra calculated based upon our model calculation using s-wave pairing symmetry exhibit multiple Ushaped conductance spectra. It is possible to calculate the individual order parameter values from the quasi-particle energies at the gap edges. In more realistic calculations the conductance spectra will be exhibit multiple V-shaped spectra taking d-wave pairing symmetry for high-Tc cuprates. This type of work are in progress which will be reported else where. Acknowledgment The authors gracefully acknowledge the research facilities offered by the Institute of Physics, Bhubaneswar, India during their short stay. References [1] H. Mohottala, B.O. Wells, J. Budnick, W. Hines, C. Niedermayer, L. Udby, C. Bernhard, A. Moodenbaugh, F.C. Chou, Nat. Mater. 5 (2006) 377. [2] E.V.L. de Mello, E.S. Caixeiroe, J.L. González, Phys. Rev. B 67 (2003) 024502. [3] K.M. Lang, V. Madhavan, J.E. Hoffman, E.W. Hudson, H. Eisaki, S. Uchida, J.C. Davys, Nature 415 (2002) 412. [4] S.H. Pan, J.P. Oneal, R. Badzey, C. Chamon, H. Ding, J. Engelbrecht, Z. wang, H. Eisaki, S. Uchida, A. Guptak, K.W. Ng, E. Hudson, K. Lang, J.C. Davys, Nature 413 (2001) 282. [5] I. Iguchi, T. Yamaguchi, A. Sugimoto, Nature 4112 (2001) 420. [6] M. Randeria, J-C. Compuzano, cond-mat/9709107 (1997). [7] A. Damascelli, Z. Hussain, Z.-X. Shen, Rev. Mod. Phys. 75 (2003) 473.

[8] T. Timsuk, B. Statt, Rep. Prog. Phys. 62 (1999) 61. [9] Ø. Fischer, M. Kugler, I. Maggio-Aprile, C. Berthod, Rev. Mod. Phys. 79 (2007) 353. [10] E.L. Wolf, Principles of Electtron Tunneling spectroscopy, Oxford University Press, Oxford, 1989. [11] P.A. Lee, N. Nagaosa, X.-G. Wen, Rev. Mod. Phys. 78 (2006) 17. [12] A. Paramakanti, M. Randeria, N. Trivedi, Phys. Rev. Lett. 87 (2001) 217002. [13] P.W. Anderson, P.A. Lee, M. Randeria, T.M. Rice, N. Trivedi, F.C. Zhang, J. Phys.:Condens. Matter 16 (2004) R1755. [14] S.A. Kivelson, I.P. Bindloss, E. Fradkin, V. Oganesyan, J.M. tranquada, A. Kapitulnik, C. Howald, Rev. Mod. Phys. 75 (2003) 1201. [15] E.W. Carlson, V.J. Emery, S.A. Kivelson, D. Orgad, in: K.H. Bennemann, J.B. Ketterson (Eds.), Physics of Superconductors, vol. II, Springer-Verlag, Berlin, 2004. [16] Q.J. Chen, J. Stajic, S. Tan, K. Levin, Phys. Rep. 412 (2005) 1. [17] B. Giovannini, C. Berthod, Phys. Rev. B 63 (2001) 144516. [18] P.E. Lammart, D.S. Rokhsar, e-print cond-matt/0108146 (2001). [19] T. Eckle, D.J. Scalapino, E. Arrigoni, W. Hanke, Phys. Rev. B 66 (2001) 140510. [20] V. Emery, S. Kivelson, Nature (London) 374 (1995) 434. [21] A.W. Hunt, P.M. Singer, A.F. Cederstrom, T. Imami, Phys. Rev. B 64 (2001) 134525. [22] D.A. Wollman, D.J. Van Harlingen, W.C. Lee, D.M. Ginsberg, A.J. Leggett, Phys. Rev. Lett. 71 (1993) 2134. [23] D.A. Brawner, H.R. Ott, Phys. Rev. B 50 (1994) 6530. [24] C.C. Tsuei, J.R. Kirtley, C.C. Chi, L.S. Yu-Jahnes, A. Gupta, T. Shaw, J.Z. Sun, M.B. Ketchen, Phys. Rev. Lett. 73 (1994) 593. [25] W.N. Hardy, D.A. Bonn, D.C. Morgan, R. Liang, K. Zhang, Phys. Rev. Lett. 70 (1990) 3999. [26] L.A. de Vaulchier, J.P. Vieren, Y. Guldner, N. Bontemps, R. Combescot, Y. Lemaitre, J.C. Mage, Europhys. Lett. 33 (1996) 153. [27] P. Chaudhari, S-Y. Lin, Phys. Rev. Lett. 72 (1994) 1084. [28] A.G. Sun, D.A. Gajewski, M.B. Maple, R.C. Dynes, Phys. Rev. Lett. 72 (1994) 2267. [29] R. Kleiner, A.S. Katz, A.G. Sun, R. Summer, D.A. Gajewski, S.H. Han, S.I. Woods, E. Dantsker, B. Chen, K. Char, M.B. Maple, R.C. Dynes, John. Clarke, Phys. Rev. Lett. 76 (1996) 2161. [30] D-H. Wu, J. Mao, S.N. Mao, J.L. Peng, X.X. Xi, T. Venkatesan, R.L. Greene, S.M. Anlage, Phys. Rev. Lett. 70 (1993) 85. [31] S.M. Anlage, D-H. Wu, J. Mao, S.N. Mao, X.X. Xi, T. Venkatesan, J.L. Peng, R.L. Greene, Phys. Rev. B 50 (1994) 523. [32] A.F. Annett, N. Goldenfield, A.J. Leggett Jr., Low Temp. Phys. 105 (1996) 473. [33] K. Krishna, N.P. Ong, Q. Li, G.D. Gu, N. Koshizuka, Science 277 (1997) 83. [34] K.A. Kouznetsov, A.G. Sun, B. Chen, A.S. Katz, S.R. Bahcall, John Clarke, R.C. Dynes, D.A. Gajewski, S.H. Han, M.B. Maple, J. Giapintzakis, J.-T. Kim, D.M. Ginsberg, Phys. Rev. Lett. 79 (1997) 3050. [35] A.V. Balatsky, Phys. Rev. Lett. 80 (1998) 1972. [36] R. Movshvich, M.A. Hubbard, M.B. Salamon, A.V. Balatsky, R. Yoshizaki, J.L. Sarrao, M. Jaime, Phys. Rev. Lett. 80 (1998) 1968. [37] R.B. Laughlin, Phys. Rev. Lett. 80 (1998) 5158. [38] T.V. Ramakrishnan, Preprint cond-mat/9803069 (1998). [39] Haranath. Ghosh, Europhys. Lett. 43 (1998) 707. [40] M.M. Covington, G. Aprili, E. Paraoanu, L.H. Greene, F. Xu, J. Zhu, C.A. Mirkin, Phys. Rev. Lett. 79 (1997) 277.

B. Pradhan et al. / Physica C 475 (2012) 14–19 [41] [42] [43] [44] [45] [46]

M. Forgelstrom, D. Rainer, J.A. Sauls, Phys. Rev. Lett. 79 (1997) 281. M.E. Zhitomirsky, M.B. Walker, Phys. Rev. Lett. 79 (1997) 1734. C. Kendziora, R.J. Kelley, M. Onellion, Phys. Rev. Lett. 77 (1996) 727. G. Varelogiannis, Phys. Rev. B 57 (1998) R723. P. Monthoux, D. Pines, Phys. Rev. B 47 (1993) 6069. Debananda Sa, S.N. Behera, Physica C 203 (1992) 347; Debananda Sa, S.N. Behera, Physica C 226 (1994) 85. [47] G.C. Rout, B.N. Panda, S.N. Behera, Physica C 333 (2000) 104.

[48] [49] [50] [51] [52] [53] [54] [55]

19

G.C. Rout, B. Pradhan, S.N. Behera, Physica C 468 (2008) 1085. Haranath Ghosh, S.N. Behera, S.K. Ghatak, D.K. Ray, Physica C 274 (1997) 107. G.C. Rout, B. Pradhan, S.N. Behera, Phys. Status Solidi (c) 3 (2006) 3617. G.C. Rout, B. Pradhan, S.N. Behera, Physica C 444 (2006) 23. G.C. Rout, B. Pradhan, Physica C 451 (2007) 59. D.N. Zubarev, Sov. Phys. Usp. 3 (71) (1960) 320. D.I. Khomskii, E.I. Neimark, Physica C 173 (1990) 342. T. Ekino, Y. Sezaki, H. Fujii, Phys. Rev. B 60 (1999) 6916.