We address the T c ( s) and multiple gaps of La 2CuO 4 (LCO) via generalized BCS equations incorporating chemical potential. Appealing to the structure of the unit cell of LCO, which comprises sub- lattices with LaO and OLa layers and brings into play two Debye temperatures, the concept of itinerancy of electrons, and an insight provided by Tacon et al.’s recent experimental work concerned with YBa 2Cu 3O 6.6 which reveals that very large electron-phonon coupling can occur in a very narrow region of phonon wavelengths, we are enabled to account for all values of its gap-to- T c ratio (2Δ 0/k BT c), i.e., 4.3, 7.1, ≈8 and 9.3, which were reported by Bednorz and Müller in their Nobel lecture. Our study predicts carrier concentrations corresponding to these gap values to lie in the range 1.3 × 10 21 - 5.6 × 10 21 cm -3, and values of 0.27 - 0.29 and 1.12 for the gap-to- T c ratios of the smaller gaps.
It is well known that La2CuO4 (LCO) is an insulator. It becomes superconducting when suitably doped, (La0.925Sr0.075)2CuO4 being an example which has a
1) Being the first SC to transcend the BCS barrier of
2) Unlike all the other HTSCs mentioned above, LCO contains predominantly only one species of ions that can give rise to pairing: La (strictly speaking, La0.925Sr0.075), which implies that pairing in it is governed by only one interaction parameter. This prima facie poses a problem because application of the generalized BCS equations [
3) When the problem mentioned above is addressed―by appealing to the structure of LCO as will be seen below―we find that the input of its Tc alone yields an interaction parameter that enables one to calculate both its gaps. This is in contrast with all the other HTSCs for which the input of two parameters from the set
The work reported here is motivated by the need to:
a) Bring the understanding of LCO in the framework of GBCSEs at par with that of all the other HTSCs noted above. This is done in the next section where for any Tc in the range 37 - 40 K, it is shown, in accord with experiment, that 2∆20/kBTc = 4.3, where kB is the Boltzmann constant, and
b) Explain experimental values for the gap-to-Tc ratio other than 4.3, i.e., 7.1, ≈8 and 9.3, attention to which was drawn by Bednorz and Müller [
The final section is devoted to a discussion of our findings.
As noted [
where
The observation that
Applying Equations (1) and (2) to the sub-lattice of LCO comprising OLa layers in which La is the lower of the two bobs, we find that [
where
For the reason given above, GBCSE for Tc of LCO in the two sub-lattice (two-phonon) scenario, obtained by putting
Note that the equivalent of Equation (5) for YBCO, for example, had two different interaction parameters:
In writing Equation (5) we have dropped the superscripts of the two θs in the original equation. Solution of this equation with the input of
This leads to two values for the smaller gap and one for the larger gap via the following equations (Equations (3) and (5) of [
In these equations
Because Tc values of LCO reported in the literature vary from 36 - 40 K, we have given in
It was mentioned above that in giving an account of the properties of LCO, Bednorz and Müller in their Nobel lecture [
Tc (K) | λ | ||||||
---|---|---|---|---|---|---|---|
36 | 0.26103 | 6.573 | 4.24 | 0.400 | 0.26 | 1.647 | 1.06 |
37 | 0.26461 | 6.784 | 4.26 | 0.422 | 0.26 | 1.737 | 1.09 |
38 | 0.26818 | 6.995 | 4.27 | 0.445 | 0.27 | 1.829 | 1.12 |
39 | 0.27173 | 7.207 | 4.29 | 0.467 | 0.28 | 1.922 | 1.14 |
40 | 0.27527 | 7.420 | 4.31 | 0.491 | 0.28 | 2.018 | 1.17 |
employed in our study is 370 K, whereas values both smaller (360 K [
From these results we conclude that all the different observed values of
This study of LCO in the framework of Equations (5) and (8) parallels earlier studies [
It was remarked in [
These considerations strongly suggest the need to study HTSCs in a more general framework of μ-incorpo- rated equations that are not constrained by the above inequality. Some of these equations are already available for studying pairing in the OPEM scenario; as was noted above, they were obtained in studies concerned with crossover physics and the superconductivity of SrTiO3. In this section we give a complete account of such equations in the OPEM scenario.
In the OPEM scenario, the μ-incorporated GBCSE at
where V0 is the BCS interaction parameter as in
Substituting these into Equation (12) and multiplying with
where
and
From the following relations involving the number density of conduction electrons and other relevant parameters
we obtain
where
The last step is a consequence of splitting the region of integration into two parts (when
From Equations (13) and (16) we obtain
Equation (18), valid at
where
and
In the above equations μ1 is the chemical potential and V1 the value of V in
If μ1 and Tc are known, then we can determine
Equations (18) and (19) can be generalized to the TPEM scenario by replacing the “propagator” V used in the former scenario by a “superpropagator”
where the superscript (2) of a symbol denotes that it pertains to the TPEM scenario, and
In these equations θ1 and θ2 are the values of
Similarly, we obtain the desired generalization of Equation (19) as
where
and
While our treatment of the pairing problem in both OPEM and TPEM scenarios has been perfectly general, it has led to an undetermined set of equations. In the latter case we have only two equations, Equations (24) and (27), containing six variables: λ0, λ1, μ0, μ1, W20, and Tc. Therefore we now assume
We could have made this assumption at the outset because it is in accord with a tenet of BCS theory. We chose not to do so in order to have readily available a set of equations that might be useful, should it be considered worthwhile to follow up this work with one of greater generality. All calculations in the following are carried out by assuming Equation (30).
Since Equations (24) and (27) are the μ-incorporated versions of Equations (8) and (5), respectively, we need to show that when the constraint embodied in Equation (11) is imposed they yield solutions in agreement with those obtained by solving the latter equations. To this end, we solve Equation (27) for λ with the input of θ1 = 104.8 K, θ2 = 431.1 K (see Equations (3) and (4)), Tc = 38 K and different values of μ. We begin with μ = 300 kBθ2, which manifestly satisfies Equation (11), and find that λ = 0.26818, which is the result that we had obtained earlier via Equation (5) and noted in Equation (6). Solution of Equation (24) with μ = 300 kBθ2, and λ = 0.26818 then yields
Even after incorporating μ in the equations for Tc and ∆, our considerations so far have succeeded in explaining only one of the observed values of the gap-to-Tc ratio for LCO. Therefore there would seem to be a need for a new idea to explain the other values. It seems to us that the recent findings of Tacon et al. [
The work reported in this section is based on an idea inspired by Tacon et al.’s experimental findings about the role of low-energy phonons in YBa2Cu3O6.6 (superconducting transition temperature Tc = 61 K) determined by employing high-resolution inelastic X-ray scattering. These experiments revealed features that were interpreted by the authors as signifying that (a) extremely large superconductivity-induced line-shape renormalizations are caused by phonons in a narrow range of momentum space and (b) the electron-phonon interaction has sufficient strength to generate various anomalies in electronic spectra, but does not contribute significantly to Cooper pairing. Having probed the electron-phonon coupling via their ingenious two-level approach of X-ray scattering, they further noted that “in terms of its amplitude, the coupling is actually by far the biggest ever observed in a superconductor, but it occurs in a very narrow region of phonon wavelengths and at a very low energy of vibration of the atoms”. This statement induces us to review below our earlier treatment of the number equation.
We recall that the limits of the number equation in the previous section―both at T = Tc and T = 0―were taken as −μ and kBθ2. We now call attention to the facts that a) in dealing with an SC that has Tc ≈ 38 K, we need to invoke TPEM, which b) brings into play two Debye temperatures θ1 and θ2 > θ1, and c) if the SC were a 1-component SC characterized by θ1, the number equation at T = 0 for it would comprise two terms: one corresponding to where
Guided by the insight provided by the findings of Tacon et al. [
Our procedure above comprised solving Equation (27) for λ with the input of Tc and different values of μ and then using these (λ, μ) values to solve Equation (24) for
Eliminating λ from Equations (24) and (27) by appealing to Equation (30), we have
Attempting to solve Equation (31), we find that it has no solution for
A review of our procedure so far is as follows. By including μ in GBCSEs as applicable to LCO, appealing to an idea inspired by the work of Tacon et al. [
TPEM scenario | OPEM scenario | |||||
---|---|---|---|---|---|---|
λ | ||||||
4.3 | 39.72 | 0.27388 | 0.480 (3.1) | 0.293 | 1.85 (11.9) | 1.13 |
7.1 | 104.4 | 0.26886 | 0.449 (2.9) | 0.274 | 1.83 (11.8) | 1.12 |
8.0 | 127.8 | 0.26862 | 0.447 (2.9) | 0.273 | 1.83 (11.8) | 1.12 |
9.3 | 165.2 | 0.26846 | 0.446 (2.9) | 0.273 | 1.83 (11.8) | 1.12 |
We now note that the (μ, λ)-values that we have been led to, enable us to calculate
We can calculate the carrier concentration n via the following equation
which is the second equation after Equation (15), with EF replaced by μ and the factor of ħ2 inserted. However, before we can use this equation we need to find the band effective mass of electrons in LCO. This can be done by using the following expressions for
where we have put the band effective mass as s times the free electron mass me, and
where γ is the experimentally obtained electronic specific heat constant (also known as the Sommerfeld constant) and v the gm-at volume.
From Equations (33) and (34) we obtain
We now use: γ = 4.5 mJ/mol K2 (given in [
Upon putting m = s times the free electron mass in Equation (32) we obtain
Since the value of the carrier concentration noted in [
1) Strictly speaking, Fermi energy is a term applicable to non-interacting systems. In this note however it has been used interchangeably with the chemical potential, as is usually done in the literature on superconductivity.
2) It is well known that HTSCs are characterized by multiphase and multi-scale complexity and that their Fermi surfaces have highly complex structures comprising many sheets that span different bands. In the light of this observation it would seem that our treatment of LCO based on multiple Debye temperatures and TPEM is rather naive. We are therefore impelled to draw attention to the following:
a) As can be seen in [
b) It is evident that the values of μ corresponding to different values of ∆2 and Tc that we have determined reflect the structure of the Fermi surface of the SC. Implicit in our approach is the concept of locally spherical Fermi surfaces (LSFSs), which takes the complexity of the HTSC into account in a rather simple (if not the simplest possible) manner. Such simplicity of approach for the kind of system we are dealing with is a goal every model strives to achieve. We note that LSFSs come into play because of itinerancy of electrons and the adoption of MFA.
c) Itinerancy of electrons in a multi-band SC is a much-invoked concept since at least the time of Suhl et al.’s paper [
d) Consider a convoy on a road passing through a range of mountains. As the road twists and turns through a series of valleys and mountains, the amount of sunlight it receives will vary from a maximum at the highest point of the range to a minimum at a place determined by the topography of the entire range. Itinerant conduction electrons in a solid on a 3-D Fermi surface are akin to such a convoy: there will be places where they can exchange phonons with the A- or the B-ions separately, and a place or places where they can exchange phonons with both of them simultaneously. For the last of these cases, one can then envisage a situation where phonon energies span not the usual range from −μ to kBθ2, but a depleted range from −kBθ2 to kBθ2. The mean value of μ for the electrons under consideration is equivalent to the word “place” used for the convoy.
e) It seems interesting to point out that the concept of LSFSs is similar to the concept of locally inertial coordinate frames employed in the general theory of relativity [
3) In connection with the remark by Tacon et al. [
Our findings provide a confirmation of the idea that low values of μ play an important role in HTSCs. Besides, we have for the first time given a plausible quantitative explanation of the different values of gap-to-Tc ratio that have been reported for LCO in the literature.
In effect, the approach followed in this note can also be viewed as a new direct method for relating the ∆0 and Tc of an HTSC with its EF.
G. P. M. acknowledges that his correspondence with Dr. D. M. Eagles and Professor M. de Llano has been invaluable in this study. He thanks Professor G. Szirmai for a perceptive remark during EUROQUAM 2010 on the approach followed herein and Professor D. C. Mattis for encouragement.
G. P.Malik,V. S.Varma, (2015) A Study of Superconducting La2CuO4 via Generalized BCS Equations Incorporating Chemical Potential. World Journal of Condensed Matter Physics,05,148-159. doi: 10.4236/wjcmp.2015.53017