# Finite-wave vector effect of plasmon dispersion on critical

## Transcript Of Finite-wave vector effect of plasmon dispersion on critical

FINITE-WAVE VECTOR EFFECT OF PLASMON DISPERSION ON CRITICAL TEMPERATURE IN La1.85 Sr0.15 CuO4

Samnao Phatisena1* and Nouphy Hompanya2

Received: Jul 6, 2005 Revised: Sept 9, 2005 ; Accepted: Sept 21, 2005

Abstract

Using the plasmon exchange model in the framework of the Eliashberg theory for strong-coupling superconductors, the critical temperature of La1.85 Sr0.15 CuO4 has been calculated. The finite-wave vector effect on acoustic plasmon dispersion has been included in the expression for the effective interaction between charge carriers. This effect is shown to enhance the critical temperature significantly as compared with the result without this term. The critical temperature is sensitively dependent on the

spacer dielectric constant εM which is not known precisely. The Coulomb repulsion strength µ∗ has been tested around the commonly used value of 0.1. It is found that their proper values of them are εM ≈ 8.0

and µ∗ ≈ 0.1.

Keywords: Plasmon exchange model, finite-wave vector effect, acoustic plasmons, critical temperature

Introduction

After the discovery of high-temperature superconductivity in cuprates (Bednorz and Mu¨ller, 1986), various kinds of theoretical models for the mechanism of high-temperature superconductivity were proposed. Even today there is no consensus among theoretical physicists as to how to develop a more detailed theoretical description of the cuprates. It is known that all cuprate high-temperature superconductors (HTS) have a layered structure. The layers are composed of Cu-O planes (or sheets) separated from each other by planes of various other oxides and rare earths. The Cu-O layer is assumed to form a two-dimensional electron gas (2DEG) and the electrons in a given layer can interact with each other within the same layer as well as from layer to layer via an effective interaction. An isolated layer has only one plasmon mode with a dispersion relation ωp ∝ q1/2. Interlayer interaction

leads to a noticeable modification of the pure two-dimensional (2D) dispersion relation, namely, to the formation of plasmon bands.

Indeed, it is a well-known fact that the spectrum of a layered electron gas contains low-energy electronic collective modes, often called acoustic plasmons, with a dispersion relation ωρ ∝ q. That such modes could not be observed experimentally at finite q so far is related to the fact that the only technique known to date to determine the plasmon energy as a function of its wave-vector (i.e., electron energy loss spectroscopy), has a resolution of 0.2 - 0.5 eV at best (Nu¨cker et al., 1989; Sto¨ckli et al., 2000). It remains thus an experimental challenge to measure collective charge excitations down to very low energies at finite q. It is also worth noting that the largest contribution of acoustic plasmons to physical quantities such as

1 School of Physics, Institute of Science, Suranaree University of Technology, Nakhonratchasima, 30000 Thailand 2 Department of Physics, Faculty of Science, National University of Lao, Vientiane, Laos * Corresponding author

Suranaree J. Sci. Technol. 12(3):231-237

232 Finite-wave Vector Effect of Plasmon Dispersion on Critical Temper Ture in La1.85 Sr0.15 CuO4

condensation energy is expected to come from finite but rather small values of q with respect to the Fermi wave-vector. To study the effect of acoustic plasmons on superconductivity requires thus to probe finite q,s. Recently, the effect of temperature dependence and the inclusion of the leading higher order in wave-vector q on the plasmon dispersion relation in layered superconductors has been reported (Hompanya and Phatisena, 2005). At a low temperature limit the slope of the layered plasmon dispersion was shown to increase significantly due to the inclusion of the higher order in q while the thermal enhancement is nearly negligible.

The influence of acoustic modes on superconducting properties has been studied within the strong-coupling phonon-plasmon scheme (Bill et al., 2000). It was shown that the density of states is peaked at qz = and qz = 0, where qz is the wave-vector perpendicular to the planes. The optical branch (qz = 0) has a smaller attractive interaction than the acoustic branch. Therefore, the largest collective-mode contribution to Tc is provided by the lowest acoustic branch. Screening of the Coulomb interaction in a layered conductor is incomplete due to the nature of layering (Visscher and Falicov, 1971; Fetter, 1974). The response to a charge fluctuation is time dependent and the frequency dependence of the screened Coulomb interaction becomes important. The additional impact of dynamic screening on pairing in layered superconductors has been evaluated (Bill et al., 2003). The plasmonsí contribution in conjunction with the phonon mechanism was used. The presence of only phonons is assumed to be sufficient to overcome the static Coulomb repulsive interaction and the dynamic screening acts as an additional factor. The full temperature, frequency and wave-vector dependence of the dielectric function was used to calculate Tc of three classes of layered superconductors. In metal-intercalated halide nitrides the contribution arising from acoustic plasmons is dominant while the contribution of phonons and acoustic plasmons is of the same order in layered organic superconductors and the contribution of acoustic plasmons is significant but not dominant in high-temperature oxides.

In this paper the Eliashberg theory for strong-coupling superconductors as modified by McMillan (McMillan, 1968) and Kresin (Kresin, 1987) will be used to calculate the superconducting transition temperature, Tc. The plasmon exchange model will be reconsidered and the effective interaction between electrons is described within the random phase approximation (RPA). This model was previously used to calculate the Tc in HTS (Longe and Bose, 1992). Here, the finite-wave vector effect on the plasmon dispersion relation in a layered superconductor will be included in our calculation. The appropriated values of the dielectric constant εM and the effective Coulomb repulsion µ* for the cuprate superconductor La1.85Sr0.15CuO4 will be determined. Numerical results and discussions will be presented.

Acoustic Plasmon Exchange Model

The simple layered electron gas system consists of two conducting sheets along the z-axis separated by a dielectric spacer with the dielectric constant εM and with the interlayer distance L. The description of layered conductors can be made by neglecting the small interlayer hopping in a first approximation. The electrons in a Cu-O plane interact via the Coulomb interaction with charge carriers both within and between the planes. The effective interaction between the electrons are described within the RPA and can be written in the standard form (Bose and Longe, 1992)

(1)

where V0 (q) =2 e2 /εMq is the bare 2D Coulomb interaction, ωp (q, qz) is the plasmon frequency and⎥M (q, qz)⎥2 is the square of the electron-plasmon matrix element. The plasmon frequency in the present model is shown to be (Hompanya and Phatisena, 2005)

(2)

(3)

Suranaree J. Sci. Technol. Vol. 13 No. 3; July-September 2005

233

(4)

(5)

n is 2D electron density, m* is the electron effective

mass, and R (q, qz) is the layer form factor which reflects the layered nature of the system. Note

that for L → ∞, the effective interaction V (q, ω)

becomes

which is the

effective electron-electron interaction in a single

2DEG.

Also to be noted is the appearance of the

second term in the parenthesis of Eqn. (2). This

term reflects the finite-wave vector effect on

plasmon dispersion relation, which is always

missing in the otherís calculation. This paper

shows the significance of this term to Tc of the

layered superconductors.

Indeed, it has been shown (Allen and

Dynes, 1975) that if the effective interaction

between electrons in a superconductor can be

written as given by Eqn. (1), then the coupling

strength λ due to the attractive part of the effective

interaction and the average value 〈ω〉 of the

plasmon frequency can be obtained from

(6)

(7)

where N(0) is the density of states of the electrons at the Fermi surface and 〈...〉FS indicates that an average of the expression is taken over the Fermi surface. Eqns. (6) and (7) can be shown (Longe and Bose, 1992) to be

(8)

and (9)

It is interesting to note that λ, as given by Eqn. (8), does not depend on the interlayer distance L. This is due to the analytic properties of the RPA potential given by Eqn. (1). Another important point is that the integrals (8) and (9) diverge for small momentum transfer q. The technique to avoid this difficulty is to introduce a cutoff qm to obtain finite results. Physically one

would expect that the effective range of should

qm be of the order of inverse coherence length ξ since charge carriers at distances larger than ξ do

not contribute significantly to Cooper pairing.

Therefore, one can write qm =1/ξ and thus Tc must obviously depend on the value of ξ.

It is also interesting to note that even though

integrals (8) and (9) diverge for small qm, but their ratio, i.e. 〈ω2〉, however does not. For small qm, 〈ω2〉 tends rapidly to the lower limit of σ/L which

is the 3D electron density. On the other hand, for

qm large orq 2kF, the average 〈ω2〉 tends rather slowly to the upper limit 2kF σ coth (2kF L)≈ 2kFσ. Hence the range of variation of 〈ω2〉 as a function

of qm is not very extended. This is not the case for λ which diverges linearly for small qm.

It is simpler to scale the parameter,

y = q / kF. Eqn. (8) and (9) then become

(10)

(11)

where N(0) = m* and k2 = 2πn . The average

plasmon

2π frequency,

〈ω2〉,

F

is

given

by

the

square

root of the ratio of (10) and (11).

It is seen from Eqns. (10) and (11) that the

two parameters obviously depend on the

dielectric constant εM, the effective mass m*, the surface density n of the electron gas (or

equivalently the Fermi wave-vector kF and hence

the Fermi energy εF), and the coherence length ξ.

Critical Temperature of La1.85 Sr0.15 CuO4

In this section we will focus on the La1.85 Sr0.15 CuO4 for which most parameters have been determined and it deserves special attention because of the simplicity of its structure. This system plays a role similar to the hydrogen atom in atomic physics. It is the best test system for understanding the basic principles of high-temperature superconductivity.

Following are the normal state parameters (Bill et al., 2003):

the interlayer distance L = 6.5Å the Fermi wave-vector kF = 3.5 × 107cm-1

234 Finite-wave Vector Effect of Plasmon Dispersion on Critical Temper Ture in La1.85 Sr0.15 CuO4

the dielectric constant εM ≈ 5 - 10 the effective mass m* = 1.7me the coherence length ξ = 35Å and the Coulomb pseudopotential is taken to be µ* = 0.1 (here, me being the mass of the bare electron). Two equations, McMillan,s equation and Kresin,s equation, both of which were modified from the Eliashberg theory for strong-coupling superconductors will be used for the calculation of Tc of this material. The McMillan,s equation for the plasmon exchange model has the form

(12)

where the Debye frequency θD is replaced by the average frequency of plasmon, 〈ω〉, the exchange of which is responsible for superconductivity.

Kresinûs equation for the plasmon exchange model to calculate the value of Tc is given by

(13)

where the effective interaction strength λeff is given by

(14)

and the analytical expression for the function t (λ) is given (Longe and Bose, 1992) by

(15)

The results of Tc obtained by these two equations will be compared with the recent

work by Bill et al. (2003). We will start with the

calculation of λ and 〈ω〉 given by Eqns. (10) and

(11) respectively. It can be seen from the given

parameters that the value of dielectric constant

εM is in the range 5 - 10, and λ and 〈ω〉 are obviously sensitive to this choice of εM. We, therefore, calculate the value of λ, 〈ω〉 and then Tcpl by using different values of εM. The result is shown in Table 1. The finite-wave vector (higher

order in q) effect of the plasmon dispersion

relation given by Eqn. (2), which is the term (3/

4) h2q / me2,

on

pl

T is

also

shown

in

the

Table.

It

c

is seen that the values of Tcpl obtained by using Kresin,s equation are higher than those by McMillan,s equation and the finite-wave vector

effect enhances the values of Tcpl significantly. As reported by Bill et al. (2003), the

experimental value of Tc of La1.85 Sr0.15 CuO4 is

T exp c

≈

38K

.

Their

numerical

result

is

Tc =

36.5

K

whereas in the absence of acoustic plasmons it is Tcph = 30 K. Therefore, the value of Tc due to

acoustic plasmons is Tcpl ≈ 8K. It is seen from

Table 1 that the expected results correspond to

the dielectric constant εM = 8 (8.38K and 8.97K by Kresin,s equation, 6.96K and 7.45K by McMillan,s equation). These results are quite

different from the values that correspond to εM = 7 and εM = 9. It is, therefore, necessary to obtain Tcpl that corresponds to the dielectric constant around εM = 8. The result is shown in Table 2. It

is seen from Table 2 that the appropriate value of

the dielectric constant is εM ≈ 8.0 forµ* = 0.1.

Table 1.

The calculated values for Tcpl (in K) as obtained from Kresin,s equation (Tc1 and Tc2) and from McMillan,s equation (Tc3 and Tc4). Tc1 and Tc3 are the values without the finite-wave vector effect whereas Tc2 and Tc4 are the values including that effect

By Kresin,s equation

By McMillan,s equation

εM

Tc1

Tc2

Tc3

Tc4

5

136.821

146.540

130.213

139.463

6

57.943

62.059

53.459

57.256

7

22.978

24.610

20.279

21.720

8

8.377

8.973

6.960

7.454

9

2.753

2.948

2.109

2.259

10

0.797

0.854

0.548

0.587

Suranaree J. Sci. Technol. Vol. 13 No. 3; July-September 2005

235

Finally, to find the proper value of the effective repulsive strength µ* (rather than 0.1) that fit the expected result of Tcpl ≈ 8K, we use it here as a second parameter varying from 0.07 to 0.13 in steps of 0.01. The critical temperature Tcpl as a function of the dielectric constant εM for 7 values of µ* is shown in Table 3 and Figure 1. It is seen from the Figure that the proper value of εM and µ* that fit the expected result of Tcpl ≈ 8K are εM ≈ 8.0 and µ* = 0.1.

Discussion and Conclusions

Using the plasmon exchange model in the framework of the Eliashberg theory for strong coupling superconductors, the plasmon contribution to the critical temperature Tcpl could be obtained. In this model the plasmons are assumed to be attractive bosons in the pairing effect. The effective interactions between the electrons are described within the RPA. The electrons interact with each other within the same layer as well as from layer to layer via an effective interaction involving

plasmon exchanges among all layers. Eliashberg,s equation for the calculation of Tc has been modified into McMillan,s equation and Kresin,s

equation. These two equations contain two basic parameters to be evaluated, λ and 〈ω〉. The quantity λ represents the attractive strength between electrons, which in this model is

essentially mediated by plasmons. The quantity 〈ω〉 is the average value of the frequency of the plasmons, the exchange of which is responsible for superconductivity. Both λ and 〈ω〉 obviously depend on the dielectric constant εM, the effective massm*, the Fermi wave-vector kF, the interlayer dist ance L, and the coherence length ξ (to specify the lower limit of integration for λ and 〈ω〉). The third parameter entered in the two equations for Tcpl is the Coulomb repulsion strength µ*. This parameter is generally not well known, but one knows that it is limited by the condition 0 < µ* < 0.5. Many other investigators take its numerical value to be 0.1. In this work, µ* is kept as an undefined parameter around 0.1.

Table 2. The calculated values for Tcpl (in K) as obtained from the same process as Table 1

with dielectric constant around εM = 8

By Kresin,s equation

By McMillan,s equation

εM

Tc1

Tc2

Tc3

Tc4

7.5

14.0379

15.0351

12.0473

12.9031

7.75

10.8781

11.6508

9.19075

9.8436

8

8.3779

8.9730

6.96018

7.4545

8.25

6.4107

6.8661

5.23028

5.6018

8.5

4.8721

5.2182

3.89835

4.1752

Table 3.

The calculated values for Tcpl (in K) as a function of εM by using Kresin,s equation for various values of effective repulsive strength around µ* = 0.1. Kresin,s equation without finite-wave vector effect has been used

εM µ*

7.5

7.75

8.0

8.25

8.5

0.07

32.286

26.591

21.845

17.899

14.625

0.08

24.980

20.211

16.296

13.091

10.477

0.09

18.940

15.018

11.853

9.310

7.274

0.10

14.038

10.878

8.378

6.411

4.872

0.11

10.141

7.654

5.731

4.255

3.131

0.12

7.116

5.210

3.776

2.706

1.917

0.13

4.830

3.414

2.381

1.637

1.109

236 Finite-wave Vector Effect of Plasmon Dispersion on Critical Temper Ture in La1.85 Sr0.15 CuO4 µ* = 0.07

40 µ* = 0.08

µ* = 0.09

30

µ* = 0.1 20 µ* =0. 11

µ* = 0.12 3078 µ* = 0.13

Figure 1. The critical temperrature as a function of constant εM for seven values of electron-electron repulsive strength µ*, varying from 0.07 to 0.13 in steps of 0.01

A specific cuprate superconductor, La1.85 Sr0.15 CuO4, for which most parameters have been determined, is selected for the calculation of Tcpl.

Since the experimental value of Tc of this

material

is

T exp c

≈

38K

and

the

phonon

contribution

to the Tc is shown to be Tcph = 30 k, hence the

plasmon contribution should be Tcpl ≈ 8K. Indeed the critical temperature is sensitively dependent

on parameters mentioned above. However, only

εM is not known precisely and the value of µ* should be tested around the value of 0.1. Variation of εM and µ* shows that their proper

values for Tcpl ≈ 8K are εM ≈ 8 and µ* ≈ 0.1. The plasmon exchange model is very

simple since the microstructure of the

superconductors is completely neglected. The

model is characterized by four parameters only.

For reasonable values of these parameters the

calculated value of Tcpl is found to be in reasonable agreement with the experimental values of the

materials. In the case of high-temperature oxides,

the contribution of low-energy plasmons to

the critical temperature is significant but not

dominant. The phonon contribution is still

largest in this model. In some classes of layered

superconductors, the acoustic plasmon contribution is shown to be dominant or of the same order as the phonon contribution.

References

Allen, B.P., and Dynes, C.R. (1975). Transition temperature of strong-coupled superconductors reanalyzed. Physical Review B., 12(3):905.

Bednorz, G.J., and Muller, K. A. (1986). Electronic band pro¨perties and supercon ductivity in. Journal of Physics B., 64:189.

Bill, A., Morawitz, H., and Kresin, Z. V. (2000). Plasmons in layered superconductors. Elsevier Science Physica B., 284-288 (2,000):433-434.

Bill, A., Morawitz, H., and Kresin, Z.V. (2003). Electronic collective modes and superconductivity in layered conductors. Physical Review B., 68:144,519.

Bose, M. S., and Longe, P. (1992). Acoustic plasmon exchange in multilayered systems: I. The effective interaction potential. Journal of Physic C, 4:1,799.

Fetter, L. A. (1974). Electrodynamics of a layered electron gas. II. Periodic array,

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Annual Physics (New York), 88:1-25. Hompanya, N., and Phatisena, S. (2005).

Plasmons dispersion relation in layered superconductors at finite temperatures. Suranaree J. Sci. Technol., 12(2):117-124. Kresin, Z.V. (1987). Critical temperatures of superconductors with low dimen-sionality. Physical Review B., 35(16):8,716. Longe, P., and Bose, M.S. (1992). Acoustic plasmon exchange in multilayered systems: II. Application to high-Tc superconductors. Journal of Physics C, 4:1,811. McMillan, L.W. (1968). Transition temperature

of strong-coupled superconductors. Physical Review, 167(2):331. Nu¨cker, N., Romberg, H., Nakai, S., Sheerer, B., and Fink, J. (1989). Plasmons and interband transition in. Bi2Sr2CaCu2O8. Physical Review B., 39(16):12,379. Sto¨ckli, T., Bonard, M.J., and, Chatelaing A. (2000). Plasmon excitations in graphitic carbon spheres measured by EELS. Physical Review B., 61(8):5,751. Visscher, B.P., and Falicov, M.L. (1971). Dielectric screening in layered electron gas. Physical Review B., 3(8):2,541.

Samnao Phatisena1* and Nouphy Hompanya2

Received: Jul 6, 2005 Revised: Sept 9, 2005 ; Accepted: Sept 21, 2005

Abstract

Using the plasmon exchange model in the framework of the Eliashberg theory for strong-coupling superconductors, the critical temperature of La1.85 Sr0.15 CuO4 has been calculated. The finite-wave vector effect on acoustic plasmon dispersion has been included in the expression for the effective interaction between charge carriers. This effect is shown to enhance the critical temperature significantly as compared with the result without this term. The critical temperature is sensitively dependent on the

spacer dielectric constant εM which is not known precisely. The Coulomb repulsion strength µ∗ has been tested around the commonly used value of 0.1. It is found that their proper values of them are εM ≈ 8.0

and µ∗ ≈ 0.1.

Keywords: Plasmon exchange model, finite-wave vector effect, acoustic plasmons, critical temperature

Introduction

After the discovery of high-temperature superconductivity in cuprates (Bednorz and Mu¨ller, 1986), various kinds of theoretical models for the mechanism of high-temperature superconductivity were proposed. Even today there is no consensus among theoretical physicists as to how to develop a more detailed theoretical description of the cuprates. It is known that all cuprate high-temperature superconductors (HTS) have a layered structure. The layers are composed of Cu-O planes (or sheets) separated from each other by planes of various other oxides and rare earths. The Cu-O layer is assumed to form a two-dimensional electron gas (2DEG) and the electrons in a given layer can interact with each other within the same layer as well as from layer to layer via an effective interaction. An isolated layer has only one plasmon mode with a dispersion relation ωp ∝ q1/2. Interlayer interaction

leads to a noticeable modification of the pure two-dimensional (2D) dispersion relation, namely, to the formation of plasmon bands.

Indeed, it is a well-known fact that the spectrum of a layered electron gas contains low-energy electronic collective modes, often called acoustic plasmons, with a dispersion relation ωρ ∝ q. That such modes could not be observed experimentally at finite q so far is related to the fact that the only technique known to date to determine the plasmon energy as a function of its wave-vector (i.e., electron energy loss spectroscopy), has a resolution of 0.2 - 0.5 eV at best (Nu¨cker et al., 1989; Sto¨ckli et al., 2000). It remains thus an experimental challenge to measure collective charge excitations down to very low energies at finite q. It is also worth noting that the largest contribution of acoustic plasmons to physical quantities such as

1 School of Physics, Institute of Science, Suranaree University of Technology, Nakhonratchasima, 30000 Thailand 2 Department of Physics, Faculty of Science, National University of Lao, Vientiane, Laos * Corresponding author

Suranaree J. Sci. Technol. 12(3):231-237

232 Finite-wave Vector Effect of Plasmon Dispersion on Critical Temper Ture in La1.85 Sr0.15 CuO4

condensation energy is expected to come from finite but rather small values of q with respect to the Fermi wave-vector. To study the effect of acoustic plasmons on superconductivity requires thus to probe finite q,s. Recently, the effect of temperature dependence and the inclusion of the leading higher order in wave-vector q on the plasmon dispersion relation in layered superconductors has been reported (Hompanya and Phatisena, 2005). At a low temperature limit the slope of the layered plasmon dispersion was shown to increase significantly due to the inclusion of the higher order in q while the thermal enhancement is nearly negligible.

The influence of acoustic modes on superconducting properties has been studied within the strong-coupling phonon-plasmon scheme (Bill et al., 2000). It was shown that the density of states is peaked at qz = and qz = 0, where qz is the wave-vector perpendicular to the planes. The optical branch (qz = 0) has a smaller attractive interaction than the acoustic branch. Therefore, the largest collective-mode contribution to Tc is provided by the lowest acoustic branch. Screening of the Coulomb interaction in a layered conductor is incomplete due to the nature of layering (Visscher and Falicov, 1971; Fetter, 1974). The response to a charge fluctuation is time dependent and the frequency dependence of the screened Coulomb interaction becomes important. The additional impact of dynamic screening on pairing in layered superconductors has been evaluated (Bill et al., 2003). The plasmonsí contribution in conjunction with the phonon mechanism was used. The presence of only phonons is assumed to be sufficient to overcome the static Coulomb repulsive interaction and the dynamic screening acts as an additional factor. The full temperature, frequency and wave-vector dependence of the dielectric function was used to calculate Tc of three classes of layered superconductors. In metal-intercalated halide nitrides the contribution arising from acoustic plasmons is dominant while the contribution of phonons and acoustic plasmons is of the same order in layered organic superconductors and the contribution of acoustic plasmons is significant but not dominant in high-temperature oxides.

In this paper the Eliashberg theory for strong-coupling superconductors as modified by McMillan (McMillan, 1968) and Kresin (Kresin, 1987) will be used to calculate the superconducting transition temperature, Tc. The plasmon exchange model will be reconsidered and the effective interaction between electrons is described within the random phase approximation (RPA). This model was previously used to calculate the Tc in HTS (Longe and Bose, 1992). Here, the finite-wave vector effect on the plasmon dispersion relation in a layered superconductor will be included in our calculation. The appropriated values of the dielectric constant εM and the effective Coulomb repulsion µ* for the cuprate superconductor La1.85Sr0.15CuO4 will be determined. Numerical results and discussions will be presented.

Acoustic Plasmon Exchange Model

The simple layered electron gas system consists of two conducting sheets along the z-axis separated by a dielectric spacer with the dielectric constant εM and with the interlayer distance L. The description of layered conductors can be made by neglecting the small interlayer hopping in a first approximation. The electrons in a Cu-O plane interact via the Coulomb interaction with charge carriers both within and between the planes. The effective interaction between the electrons are described within the RPA and can be written in the standard form (Bose and Longe, 1992)

(1)

where V0 (q) =2 e2 /εMq is the bare 2D Coulomb interaction, ωp (q, qz) is the plasmon frequency and⎥M (q, qz)⎥2 is the square of the electron-plasmon matrix element. The plasmon frequency in the present model is shown to be (Hompanya and Phatisena, 2005)

(2)

(3)

Suranaree J. Sci. Technol. Vol. 13 No. 3; July-September 2005

233

(4)

(5)

n is 2D electron density, m* is the electron effective

mass, and R (q, qz) is the layer form factor which reflects the layered nature of the system. Note

that for L → ∞, the effective interaction V (q, ω)

becomes

which is the

effective electron-electron interaction in a single

2DEG.

Also to be noted is the appearance of the

second term in the parenthesis of Eqn. (2). This

term reflects the finite-wave vector effect on

plasmon dispersion relation, which is always

missing in the otherís calculation. This paper

shows the significance of this term to Tc of the

layered superconductors.

Indeed, it has been shown (Allen and

Dynes, 1975) that if the effective interaction

between electrons in a superconductor can be

written as given by Eqn. (1), then the coupling

strength λ due to the attractive part of the effective

interaction and the average value 〈ω〉 of the

plasmon frequency can be obtained from

(6)

(7)

where N(0) is the density of states of the electrons at the Fermi surface and 〈...〉FS indicates that an average of the expression is taken over the Fermi surface. Eqns. (6) and (7) can be shown (Longe and Bose, 1992) to be

(8)

and (9)

It is interesting to note that λ, as given by Eqn. (8), does not depend on the interlayer distance L. This is due to the analytic properties of the RPA potential given by Eqn. (1). Another important point is that the integrals (8) and (9) diverge for small momentum transfer q. The technique to avoid this difficulty is to introduce a cutoff qm to obtain finite results. Physically one

would expect that the effective range of should

qm be of the order of inverse coherence length ξ since charge carriers at distances larger than ξ do

not contribute significantly to Cooper pairing.

Therefore, one can write qm =1/ξ and thus Tc must obviously depend on the value of ξ.

It is also interesting to note that even though

integrals (8) and (9) diverge for small qm, but their ratio, i.e. 〈ω2〉, however does not. For small qm, 〈ω2〉 tends rapidly to the lower limit of σ/L which

is the 3D electron density. On the other hand, for

qm large orq 2kF, the average 〈ω2〉 tends rather slowly to the upper limit 2kF σ coth (2kF L)≈ 2kFσ. Hence the range of variation of 〈ω2〉 as a function

of qm is not very extended. This is not the case for λ which diverges linearly for small qm.

It is simpler to scale the parameter,

y = q / kF. Eqn. (8) and (9) then become

(10)

(11)

where N(0) = m* and k2 = 2πn . The average

plasmon

2π frequency,

〈ω2〉,

F

is

given

by

the

square

root of the ratio of (10) and (11).

It is seen from Eqns. (10) and (11) that the

two parameters obviously depend on the

dielectric constant εM, the effective mass m*, the surface density n of the electron gas (or

equivalently the Fermi wave-vector kF and hence

the Fermi energy εF), and the coherence length ξ.

Critical Temperature of La1.85 Sr0.15 CuO4

In this section we will focus on the La1.85 Sr0.15 CuO4 for which most parameters have been determined and it deserves special attention because of the simplicity of its structure. This system plays a role similar to the hydrogen atom in atomic physics. It is the best test system for understanding the basic principles of high-temperature superconductivity.

Following are the normal state parameters (Bill et al., 2003):

the interlayer distance L = 6.5Å the Fermi wave-vector kF = 3.5 × 107cm-1

234 Finite-wave Vector Effect of Plasmon Dispersion on Critical Temper Ture in La1.85 Sr0.15 CuO4

the dielectric constant εM ≈ 5 - 10 the effective mass m* = 1.7me the coherence length ξ = 35Å and the Coulomb pseudopotential is taken to be µ* = 0.1 (here, me being the mass of the bare electron). Two equations, McMillan,s equation and Kresin,s equation, both of which were modified from the Eliashberg theory for strong-coupling superconductors will be used for the calculation of Tc of this material. The McMillan,s equation for the plasmon exchange model has the form

(12)

where the Debye frequency θD is replaced by the average frequency of plasmon, 〈ω〉, the exchange of which is responsible for superconductivity.

Kresinûs equation for the plasmon exchange model to calculate the value of Tc is given by

(13)

where the effective interaction strength λeff is given by

(14)

and the analytical expression for the function t (λ) is given (Longe and Bose, 1992) by

(15)

The results of Tc obtained by these two equations will be compared with the recent

work by Bill et al. (2003). We will start with the

calculation of λ and 〈ω〉 given by Eqns. (10) and

(11) respectively. It can be seen from the given

parameters that the value of dielectric constant

εM is in the range 5 - 10, and λ and 〈ω〉 are obviously sensitive to this choice of εM. We, therefore, calculate the value of λ, 〈ω〉 and then Tcpl by using different values of εM. The result is shown in Table 1. The finite-wave vector (higher

order in q) effect of the plasmon dispersion

relation given by Eqn. (2), which is the term (3/

4) h2q / me2,

on

pl

T is

also

shown

in

the

Table.

It

c

is seen that the values of Tcpl obtained by using Kresin,s equation are higher than those by McMillan,s equation and the finite-wave vector

effect enhances the values of Tcpl significantly. As reported by Bill et al. (2003), the

experimental value of Tc of La1.85 Sr0.15 CuO4 is

T exp c

≈

38K

.

Their

numerical

result

is

Tc =

36.5

K

whereas in the absence of acoustic plasmons it is Tcph = 30 K. Therefore, the value of Tc due to

acoustic plasmons is Tcpl ≈ 8K. It is seen from

Table 1 that the expected results correspond to

the dielectric constant εM = 8 (8.38K and 8.97K by Kresin,s equation, 6.96K and 7.45K by McMillan,s equation). These results are quite

different from the values that correspond to εM = 7 and εM = 9. It is, therefore, necessary to obtain Tcpl that corresponds to the dielectric constant around εM = 8. The result is shown in Table 2. It

is seen from Table 2 that the appropriate value of

the dielectric constant is εM ≈ 8.0 forµ* = 0.1.

Table 1.

The calculated values for Tcpl (in K) as obtained from Kresin,s equation (Tc1 and Tc2) and from McMillan,s equation (Tc3 and Tc4). Tc1 and Tc3 are the values without the finite-wave vector effect whereas Tc2 and Tc4 are the values including that effect

By Kresin,s equation

By McMillan,s equation

εM

Tc1

Tc2

Tc3

Tc4

5

136.821

146.540

130.213

139.463

6

57.943

62.059

53.459

57.256

7

22.978

24.610

20.279

21.720

8

8.377

8.973

6.960

7.454

9

2.753

2.948

2.109

2.259

10

0.797

0.854

0.548

0.587

Suranaree J. Sci. Technol. Vol. 13 No. 3; July-September 2005

235

Finally, to find the proper value of the effective repulsive strength µ* (rather than 0.1) that fit the expected result of Tcpl ≈ 8K, we use it here as a second parameter varying from 0.07 to 0.13 in steps of 0.01. The critical temperature Tcpl as a function of the dielectric constant εM for 7 values of µ* is shown in Table 3 and Figure 1. It is seen from the Figure that the proper value of εM and µ* that fit the expected result of Tcpl ≈ 8K are εM ≈ 8.0 and µ* = 0.1.

Discussion and Conclusions

Using the plasmon exchange model in the framework of the Eliashberg theory for strong coupling superconductors, the plasmon contribution to the critical temperature Tcpl could be obtained. In this model the plasmons are assumed to be attractive bosons in the pairing effect. The effective interactions between the electrons are described within the RPA. The electrons interact with each other within the same layer as well as from layer to layer via an effective interaction involving

plasmon exchanges among all layers. Eliashberg,s equation for the calculation of Tc has been modified into McMillan,s equation and Kresin,s

equation. These two equations contain two basic parameters to be evaluated, λ and 〈ω〉. The quantity λ represents the attractive strength between electrons, which in this model is

essentially mediated by plasmons. The quantity 〈ω〉 is the average value of the frequency of the plasmons, the exchange of which is responsible for superconductivity. Both λ and 〈ω〉 obviously depend on the dielectric constant εM, the effective massm*, the Fermi wave-vector kF, the interlayer dist ance L, and the coherence length ξ (to specify the lower limit of integration for λ and 〈ω〉). The third parameter entered in the two equations for Tcpl is the Coulomb repulsion strength µ*. This parameter is generally not well known, but one knows that it is limited by the condition 0 < µ* < 0.5. Many other investigators take its numerical value to be 0.1. In this work, µ* is kept as an undefined parameter around 0.1.

Table 2. The calculated values for Tcpl (in K) as obtained from the same process as Table 1

with dielectric constant around εM = 8

By Kresin,s equation

By McMillan,s equation

εM

Tc1

Tc2

Tc3

Tc4

7.5

14.0379

15.0351

12.0473

12.9031

7.75

10.8781

11.6508

9.19075

9.8436

8

8.3779

8.9730

6.96018

7.4545

8.25

6.4107

6.8661

5.23028

5.6018

8.5

4.8721

5.2182

3.89835

4.1752

Table 3.

The calculated values for Tcpl (in K) as a function of εM by using Kresin,s equation for various values of effective repulsive strength around µ* = 0.1. Kresin,s equation without finite-wave vector effect has been used

εM µ*

7.5

7.75

8.0

8.25

8.5

0.07

32.286

26.591

21.845

17.899

14.625

0.08

24.980

20.211

16.296

13.091

10.477

0.09

18.940

15.018

11.853

9.310

7.274

0.10

14.038

10.878

8.378

6.411

4.872

0.11

10.141

7.654

5.731

4.255

3.131

0.12

7.116

5.210

3.776

2.706

1.917

0.13

4.830

3.414

2.381

1.637

1.109

236 Finite-wave Vector Effect of Plasmon Dispersion on Critical Temper Ture in La1.85 Sr0.15 CuO4 µ* = 0.07

40 µ* = 0.08

µ* = 0.09

30

µ* = 0.1 20 µ* =0. 11

µ* = 0.12 3078 µ* = 0.13

Figure 1. The critical temperrature as a function of constant εM for seven values of electron-electron repulsive strength µ*, varying from 0.07 to 0.13 in steps of 0.01

A specific cuprate superconductor, La1.85 Sr0.15 CuO4, for which most parameters have been determined, is selected for the calculation of Tcpl.

Since the experimental value of Tc of this

material

is

T exp c

≈

38K

and

the

phonon

contribution

to the Tc is shown to be Tcph = 30 k, hence the

plasmon contribution should be Tcpl ≈ 8K. Indeed the critical temperature is sensitively dependent

on parameters mentioned above. However, only

εM is not known precisely and the value of µ* should be tested around the value of 0.1. Variation of εM and µ* shows that their proper

values for Tcpl ≈ 8K are εM ≈ 8 and µ* ≈ 0.1. The plasmon exchange model is very

simple since the microstructure of the

superconductors is completely neglected. The

model is characterized by four parameters only.

For reasonable values of these parameters the

calculated value of Tcpl is found to be in reasonable agreement with the experimental values of the

materials. In the case of high-temperature oxides,

the contribution of low-energy plasmons to

the critical temperature is significant but not

dominant. The phonon contribution is still

largest in this model. In some classes of layered

superconductors, the acoustic plasmon contribution is shown to be dominant or of the same order as the phonon contribution.

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