pressure dependence of tc in high temperature superconductors: role of interlayer interactions

Journal of Superconductivity: Incorporating Novel Magnetism, Vol. 13, No. 1, 2000

Pressure Dependence of Tc in High TemperatureSuperconductors: Role of Interlayer Interactions

Govind,1 A. Pratap,1 and R. S. Tripathi1

Received 12 December 1998

We have studied the role of interlayer interactions (W) in the pressure dependence of Tcof layered superconductors. The expressions for dTc/dP are obtained by including the effectsof layered structure within the framework of two different proposed models, namely thenegative-U Hubbard model and the Hirsch model. We observe that the inclusion of interlayerinteraction provides better explanation of pressure dependence of Tc. Our numerical resultsshow that the systems having one CuO2 layer per unit cell may be well described by smallvalues of W while the larger values of W accounts for the systems having two or moresuperconducting layers in a unit cell. The calculated values of dTc/dP vs. W are found to bein good agreement with those of experimental results obtained for various high Tc supercon-ductors of cuprate family.

KEY WORDS: Superconductivity; pressure dependence; interlayer interactions.

1. INTRODUCTION

In the study of high Tc cuprate superconductors,it has been observed that the most common featureamong them is their layered structure. The layeredstructure possesses one or more copper-oxide planesin a unit cell. The superconducting transition temper-ature (Tc) increases with the number of layers in theunit cell [17]. The other common feature of thesesystems is that Tc increases with the applied pressure[811]. This may be due to the changes in the volume(i.e., the lattice parameters), the carrier concentra-tion, and the phonon frequency. It has been arugedthat any theory capable of explaining the origin ofhigh transition temperature should in principle pre-dict the pressure dependence of Tc up to the correctorder of magnitude but the theories like 2D and 3DBCS model [12], RVB theory [13], the model of Fu-kuyama and Yosida [14], and the model of Cyrot[1517] do not predict the correct order of dlnTc/dlnV and hence dTc/dP. However, the other modelssuch as Hubbard-like models, many polaranic model[18], and the BCS model based on non-electron

1Department of Physics, G.B. Pant University of Ag. and Technol-ogy, Pantnagar 263 145 India.

61

0896-1107/00/0200-0061$18.00/0 2000 Plenum Publishing Corporation

phonon coupling [19] are found to be capable ofexplaining the pressure dependence of Tc to someextent.

These facts motivated us to examine the pressuredependence of Tc in high Tc layered systems withinthe framework of different models. Recently Marsig-lio and Hirsch [20], considering the isotropic intersiteinteraction and different hopping interactions withinand between the planes, studied the pressure depen-dence of Tc and concluded that Tc increases onapplying pressure along ab-plane while it decreaseson applying pressure along c-axis. However, severalstudies have confirmed that Tc always increaseswhether pressure is applied along c-axis or ab-plane[2129]. To explain the increase in Tc with pressurealong c-axis, the interlayer interaction may play im-portant role and therefore it should be included intothe model Hamiltonian. In fact it has been shown bymany authors [3035] that the transition temperaturedoes increase with interlayer interactions. More re-cently, Ajay and Tripathi [36] and Pratap et al. [37,38] also investigated the effect of interlayer interac-tions on Tc and specific heat. These authors reportthat Tc increases but the jump height in specific heatdecreases with interlayer interactions.

In view of the above facts, in the present

62 Govind, Pratap, and Tripathi

paper we examine the pressure dependence of Tcin layered systems. For this, we confine our studywithin two models of superconductivity [3638].The rest of the paper is organized in the followingway: In Sections 2.1 and 2.2 we briefly reproducethe work of these authors and perform furthercalculations and obtain the expression for dTc/dP.In Section 3, we compare our numerical calculationswith the existing theoretical and experimental re-sults. Section 4 deals with our summary and conclu-sions.

2. THEORETICAL FORMULATION

2.1. Model Hamiltonian

In a recent study Ajay and Tripathi [36] investi-gated the effects of interlayer interactions usingone-band two-layer model and explained its effectson Tc. The model Hamiltonin may be described as

(1)H 5 Hintra 1 Hinter

where,

Hintra 5 Orks

k C1rksCrks 1 U Orkk9qss9

C1rk1qs9C1rk92qs9Crk9s9Crks

(1a)

and

Hinter 5 2hOrsks

C1rksCsks 1 W Orskk9ss9q

C1rk1qsC1sk92qs9Csk9s9Crks

(1b)

where r, s are the layer indices, C1(C) are thecreation (annihilation) operators for the charge car-riers with the wave vectors k, k9, and spin s,s9.In Eq. (1a) the first term represents the energy ofcharge carriers and U is the strength of attractiveinteraction which originates from any proposedmechanism.

In Eq. (1b) h is the direct hopping between thelayers while the last term represents the attractiveinterlayer interaction which arises from any knowntransitions between the layers.

Applying the double time Greens function tech-nique within the mean field approximation the linear-ized equations of motion have been obtained. Theseequations have been solved to obtain the variouscorrelation functions. The equation for Tc has beenobtained as

(2)4UWI1I2 1 (U 1 W)(I1 1 I2) 1 1 5 0

where,

I1,2 51

4N Oktanh(bE1,2k/2)

E1,2k(2a)

with

E1,2k 5 h(k 7 h)2 1 D21,2j1/2

where

k 5 k 2 e 1 n(U 1 W)

and

D1,2 5 Da 6 Db; h 5 h 2 Wcc

with cc as excitonic type correlations and its effecton Tc has already been studied by Ajay and Tripathi[33]. In the present calculations we have not includedthe effect of cc explicitly as it may be taken into bythe hopping h.

It has been reported [30, 37, 38] that in the pres-ence of interlayer interactions single particle hopping(h) gets suppressed. Therefore we neglect h in E1,2kand solve Eq. (2a) at Tc by converting the summationover k-values into an integral with cutoff energy 6D. Before solving Eq. (2a), on the numerical grounds,we make the assumption that for small h, I1 P I2.We obtain

I 51

4N(0)ln HEcbcD2 q(2 2 q)J

where Ec 5 Eulers constant with q 5 n h1 2 (U 1W)/Dj.

Under this approximation the Tc Eq. (2) hasgives two roots

X1,2 522(U 1 W) 1 4(U 1 W)2 2 16UW

8UW

Corresponding to these two roots we obtain twoequations for Tc as

Tc 5EcD

2q(2 2 q) exp S22N(O)U D (3a)

and

Tc 5EcD

2q(2 2 q) exp S22N(O)W D (3b)

Now, we differentiate Eq. (3) with respect to volume(V) and after performing the simple algebra thesimplified expression for average dlnTc/dlnV is ob-tained as

Pressure Dependence of Tc in High Temperature Superconductors 63

dlnTcdlnV

5dlnDdlnV H1 1 nx(U 1 W)D J1 xq dlnndlnV2

dlnWdlnV Fnx WD 2 N(O)W G

2dlnUdlnV Fnx UD 2 N(O)U G

1 N(O)dlnN(O)

dlnV F1U 1 1WG (4)where x 5 (1 2 q)/q(2 2 q).

2.2. Hirsch Model

In this section we briefly describe the work pre-sented in [34]. The authors in their paper studied theeffect of interlayer interaction within Hirsch model[39, 40]. The model Hamiltonian which consists ofintra and inter layer interactions may be described as

(5)H 5 Hintra 1 Hinter

where

Hintra 5 2 Orljs

Tij C1risCrjs 1 U Orls

nrisnri2s 2 e Orls

nris

2 Orljs

Vij C1risCrjs(nri2s 1 nrj2s) (5a)

where Tij is the hopping matrix, U is the on-site repul-sion, e is the chemical potential, and Vij is the hybridCoulomb interaction and Hinter reads

Hinter 5 21/2 Orsljs

hij (C1ris Csjs 1 h.c.)

1 W/2 Orslss9

nrisnsis9 (5b)

where hij is the single particle hopping and W is theattractive interlayer interaction between the two cop-per oxide planes.

Applying the double time Greens function tech-nique within mean field decoupling scheme, thequasi-particle energy is obtained as

E1,2k 5 h(k 7 h)2 1 D21,2j1/2

with

D1,2 5 Da 6 Db

h 5 h 2 Wcc

andk 5 k 2 e 1 nW

The equation that determines the transition tem-perature is given by

1 5 2KP1 1 K 2 (P0P2 2 P21) 2 UP0 1W Q20

1 1 WP0(6)

where

K 5 34.4tp

PL 51N Ok FCoskx 1 Cosky2 G

L

( f(E1k) 1 f(E2k))

QL 51N Ok FCoskx 1 Cosky2 G

L

( f(E1k) 2 f(E2k))

f(E1,2k) 5tanh(bE1,2k/2)

4E1,2k

The summation over k values is converted intoan integration with the same assumption as consid-ered in Section 2.1. Under this approximation Q0 R0 and the approximate integrations of PL are ob-tained as

P1 5 (1 2 q) SP0 2 18tpD (7a)and

P2 5 (1 2 q)2 P0 21 2 3q 1 3q2/2

8tp(7b)

where P0 is given by

P0 51

8tpln (4.53bctp q(2 2 q)) (7c)

with q 5 n (1 2 W/4tp)Substituting Eq. (7) in Eq. (6), after performing

simple algebra, the expression for Tc comes out to be

Tc 5 4.53tp q(2 2 q)

exp S2 [1 1 k(1 2 q)]2hk(k 1 2)(1 2 q) 1 (kq)2/2 2 ujD (8)where k 5 K/8tp and u 5 U/8tp

Now, we differentiate Eq. (8) with respect tovolume and obtain

dlnTcdlnV

5dlntpdlnV

h1 1 X 2Z 22uj 1 dqdlnV

F (1 2 q)q(2 2 q) 2 X 2Z 22k(k 1 2) 1 X 2Z 22k2q 1 2XZ 21kG(9)


Fig. 1. Plot of dTc/dP vs. n for U/N(O) 5 20.48, W/U 5 0.0 (curve a), U/N(O) 520.59, W/U 5 0.0 (curve b), and U/N(O) 5 20.65, W/U 5 0.0 (curve c) with D/N(O)5 0.15, dlnn/dlnV 5 1.0 and dlnN(O)/dlnV 5 21.5.

with

X 5 [1 1 k(1 2 q)]Z 5 k(k 1 2)(1 2 q) 1 (kq)2/2 2 u

and

dqdlnV

5 qdln(n)dlnV

1n

4tpHW dlntpdlnV 2 W dlnWdlnVJ (10)

Fig. 2. Plot of dTc/dP vs. n for U/N(O) 5 20.59, W/U 5 0.0 (curve a), U/N(O) 520.59, W/U 5 0.5 (curve b), and U/N(O) 5 20.59, W/U 5 1.0 (curve c).

using Eqs. (4) and (9), the dTc/dP within two modelsmay be calculated from

dTcdP

5 2TcB

dlnTcdlnV

(11)

where B 5 bulk modulus of the material and in the


Fig. 3. Plot of dTc/dP vs. n for U/N(O) 5 20.65, W/U 5 0.0 (curve a), and U/N(O)5 20.65, W/U 5 1.0 (curve b).

present calculations, we take B 5 180 GPa as sug-gested in [41].

3. RESULTS AND DISCUSSION

We have obtained the expressions for dTc/dPwithin the framework of two proposed models. Theseexpressions are given by Eqs. (4), (9), and (11). First,

Fig. 4. Plot of dTc/dP vs. n for tP 5 0.037 ev, W 5 0.0 ev (curve a), tP 5 0.042 ev, W5 0.0 ev (curve b), and tP 5 0.045 ev, W 5 0.0 ev (curve c), with D/N(O) 5 0.15 dlnn/dlnV 5 1.0 and dlnN(O)/dlnV 5 21.5.

we discuss the result obtained from Eq. (4). To solvethis equation numerically we set D/N(O) 5 0.15,dlnn/dlnV 5 1.0 and dlnN(O)/dlnV 5 21.5. Further,we choose U/N(O) 5 20.48, 20.59, and 20.65. TheTc value corresponding to these parameters are 38K, 92 K, and 125 K, respectively. Thus the resultsobtained with these parameters may represent theLa22xMxCuO4; YBa2Cu3O7d and Bi-, T1-, and Hg-based cuprates, respectively. Next to solve the equa-


Fig. 5. Plot of dTc/dP vs. n for tP 5 0.042 ev, W 5 0.0 ev (curve a), tP 5 0.042 ev, W5 0.05 ev (curve b), and tP 5 0.042 ev W 5 0.1 ev (curve c).

tion for dTc/dP, we require the value of dlnU/dlnVand dlnW/dlnV. However, to our knowledge the vari-ation of attractive U and W with either pressure,volume, or temperature is not available. Therefore,during the present calculations, we assume that dlnU/dlnV and dlnW/dlnV contribute only 10% of U andW. In Fig. 1 we have plotted dTc/dP vs. n for theabove mentioned parameter. It is clear from Fig. 1that curve (a) predicts the correct order [41] whilecurves (b) and (c) do not agree with the experimental

Fig. 6. Plot of dTc/dP vs. n for tP 5 0.045 ev, W 5 0.0 ev (curve a), and tP 5 0.045 evW 5 0.1 ev (curve b).

values of dTc/dP for respective cuprates. Here,we would like to emphasize that the systems likeYBa2Cu3O72d and Bi-, T1-, and Hg cuprates are lay-ered in structure and therefore the interlayer interac-tions in these systems are of immense importance.In Figs. 2 and 3, we include the effects of interlayerinteractions and observe that dTc/dP decreases rap-idly with W/U. Its observed magnitude is inagreement with the experiments for layered systems[22, 23, 41]. Moreover, in Fig. 3, dTc/dP changes its


sign in the overdoped region (n . nop). This is inaccord with the recent observations for Hg cuprates,where it has been shown that on applying pressurein the overdoped region Tc decreases [22, 23].

Now we turn toward the numerical calculationsof dTc/dP from Eq. (9). For this we choose dlnn/dlnV 5 1.0, dlnN(O)/dlnV 5 21.5 and U 5 5.0 evand tp 5 0.037, 0.042, and 0.045, the correspondingTc values are 38 K, 90 K, and 130 K, respectively. InFig. 4, we plot dTc/dP vs. n for different values ofparameter and observed that for 2D case (curve a)the results are in good agreement with the experi-mentally observed values [41]. On the other hand,curves (b) and (c) show that the estimated valueof dTc/dP for layered cuprates is higher than theobserved values [22, 23, 41]. Further the inclusion ofinterlayer interactions (Figs. 5 and 6) provides bettergrounds for predicting the correct order of dTc/dP inlayered systems. However, in the present parameterregimes we do not obtain any negative value of dTc/dP as obtained in Fig. 3.

Here, we would like to point out that our resultsdiffer from the previous calculations of Marsiglio andHirsch [20] in the following way. As mentioned, theseauthors have shown that the single particle hoppingalong z-direction acts as a pair breaking agency andreduces Tc. It is important to note that these authorshave not considered the role of interlayer interactionsin their formalism. On the other hand our analysisshows that the inclusion of such interactions enhancesTc and dTc/dP remains positive till optimum doping.Our results are in agreement with the experimentalobservations [22, 23].

4. CONCLUSION

The effect of interlayer interactions on pressuredependence of Tc of high Tc superconductor has beeninvestigated in this paper. We have used two models:(i) negative U Hubbard model and (ii) Hirsch model.We find that dTc/dP as calculated from these modelsfor 2D systems like La2-xSrxCuO4 agrees well withexperimental observations. But for systems havingtwo or more layers per unit cell, these models givehigher values of dTc/dP and do not agree with experi-ments. We have calculated dTc/dP for layered cu-prate families by including interlayer interactions inthe above models. This gives a better agreement withexperimental observations. Moreover, in Fig. 3 weobserve negative value of dTc/dP in the overdoped

region. This is in accordance with the experimentalobservations for Hg-based systems.

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pressure dependence of tc in high temperature superconductors: role of interlayer interactions

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