Theory of thermal conductivity of cuprate superconductors

Materials Today: Proceedings xxx (xxxx) xxx

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Theory of thermal conductivity of cuprate superconductors Nitin P. Singh Physics Department, Jaipur National University, Jaipur 302017, India

a r t i c l e

i n f o

Article history: Received 10 January 2020 Received in revised form 22 January 2020 Accepted 24 January 2020 Available online xxxx Keywords: High temperature superconductors Lattice thermal conductivity Relaxation time Interference scattering Boundary scattering

a b s t r a c t The thermal conductivity of high temperature superconductor (HTS) cuprates is successfully analyzed in three temperature regions, namely; (i) low temperature regime (T < Tmax), (ii) broad peak regime (T
T < Tdip), (iii) dip region (T > Tdip), where Tmax and Tdip are the temperatures corresponding to the conductivity maximum and to the dip in the thermal conductivity curve, respectively. Various theories (which suffer from a large number of inadequacies) beginning from the pioneering work of Bardeen, Rickayzen and Tewordt till now have been proposed to analyze the thermal conductivity curves but a satisfactory approach is yet to come. In general, several scattering events such as boundary scattering, point defect scattering, dislocation scattering, tunneling states scattering, phonon–phonon scattering, electron–phonon scattering, etc. have been reported to explain the phenomenon but the curves above conductivity maximum could not be explained successfully. The explanations to the various anomalies have been reported in this work via the new quantum dynamical approach. The crucial dip region has been reportedly found related to transition temperature. This approach successfully estimates the temperature at and above which the contribution from one scattering event emerges to be dominant over the others, which helps to assign the specific temperature ranges for different collision events where they register their significant presence. Ó 2020 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of the scientific committee of the Innovative Advancement in Engineering & Technology.

1. Introduction The theoretical as well as experimental studies of thermal conductivity of conventional superconductors has gained heightened interest for a long time and served an important problem to investigate transport properties [1–4]. A bulk of literature has been created on the thermal properties of high Tc superconductors based on the oxygen-deficient perovskite structures. Based on BCS theory the classic paper of Bardeen–Rickayazen–Tewordt [1] (BRT) reported the method to calculate the electronic contribution to thermal conductivity when dominant scatterers are impurities and electrons. This provided away to study the thermal conductivity in the form of phonon thermal conductivity jph and electronic thermal conductivity je :

j ¼ jph þ je

ð1Þ

Later Callaway’s phenomenological model [2,3] greatly simplified the problem and the BRT function in this model appears in some part of relaxation time function. Although the Callaway the-

ory is amenable to calculate thermal conductivity but it suffers from certain serious drawbacks namely; (i) additivity of inverse relaxation times for different scattering events (since the Matthiessen’s rule applies only to the independent scattering processes) (ii) the absence of phonon dispersion effects and (iii) non inclusion of phonon polarization indices due to isotropicity considerations. As one includes the effects of anharmonicities and defects the interactions among different modes (anharmonicity modes, impurity modes, etc.) cannot be overlooked even at very low temperatures and rules out the suitability of Matthiessen’s rule. In the present work, we have examined the role of various scattering phenomena and the temperature dependence of the phonon conductivity. We have not considered the separation of electronic and lattice contributions in the heat conductivity of a superconducting material YBa2Cu3O7 for the purpose of analysis, because such separation goes against the foundations of WiedemannFranz law. In the present formulation the idea of relaxation times and different scattering mechanisms has been developed from the frequency (energy) line widths, which removes most of the above discrepancies. Based on this model the numerical analysis of thermal conductivity of YBa2Cu3O7 below and above the transi-

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N.P. Singh / Materials Today: Proceedings xxx (xxxx) xxx Table 1 Various scattering relaxation time. Scattering process

Inverse relaxation time

Boundary Scattering [9,10]

s1 cB ¼ v =LðBÞ; LðBÞ ¼ B1 L D 4 s1 D ðxÞ ¼ Ck ðxÞ ¼ Ax 3 4 2 2 2 s1 ph ¼ Ck ðxÞ þ Ck ðxÞ ¼ Bx T þ BH x T 2 2 2 1 sdis ¼ 6  10 Nd b c x 4D 0 4 2 4 CDk ðxÞ ¼ C3D k ðxÞ þ Ck  Dx T þ D x T h i 1 2 2 4 2 1 2 s1 ¼ C ¼ 2g p h x k þ 4g cothðbhx=2Þ ebhx=2 þ 1 ep B eph

Point Defect Scattering [6,7,11] Phonon–Phonon Scattering [6,7] Dislocation Scattering [12] Interference Scattering [6,12] Electron–Phonon Scattering [13–15]

Table 2 Parameters used in the analysis of thermal conductivity of YBaCuO. Tc (K)

hD (K)

c

v ðcm=sec  105 Þ

L (B) (cm)

A ðsec3  1043 Þ

B ðsK1  1022 Þ

D ðs3 K1  1045 Þ

N d ðcm2  1015 Þ

92

383

2

3.56

0.07

4.1

4.67

6.32

3.52

tion temperature shows fairly good agreement with the experimental observations. The experimental data of Uher and Kaiser [4] have been utilized for the analysis in the temperature range 5 K–120 K. The organization of the paper includes the formulation of the problem in Section 2, the involvement of different scattering mechanisms and concerned energy line widths is described in Section 3. The analysis of thermal conductivity of YBaCuO has been reported in Section 4 with discussion. 2. Formulation of the problem Let us start with the Callaway’s expression for heat conductivity of the form [2]

j ¼ ðkB =2p2 v Þðb hÞ2

xD Z

2

sðxÞx4 eb —hx ðeb —hx  1Þ dx

ð2Þ

0

with reduced frequency x ð¼  hx=kB TÞ. Where sðxÞ is the total relaxation time for all scattering processes, xD is Debye frequency, v is 1

the phonon velocity and b ¼ ðkB TÞ . In order to avoid the crux related to the inverse additivity of relaxation times, without the loss of generality we can use the following relation:

s1 ðxÞ ¼ Ck ðxÞ

4. Analysis of thermal conductivity of YBaCuO The thermal conductivity of YBa2Cu3O7 cuprate superconductors has been analyzed from 5 K to 120 K. The heat is mainly carried out by acoustic phonons. The scattering of phonons is considered due to the combined boundaries, defects, anharmonicities, interference and dislocation modes. The experimental data has been taken from the work of Uher and Kaiser [4]. For the purpose of analysis we use the data given in Table 2 to characterize the strengths of the above mentioned scattering processes. It is observed that at low temperatures (T < Tmax) boundary scattering and dislocation scattering are found to dominate over the other scattering mechanisms. As the temperature starts rising towards Tmax (i.e., towards conductivity maximum) the scattering due to isotopes, point defects, etc., then become significant. At such temperature the high frequency phonons are not exited to a large extent. In the temperature range (Tmax
T < Tdip) below Tc electron–phonon interaction results a broad peak regime. At and above conductivity maximum the role of boundaries and faults becomes insignificant and phonons start interacting with the localized fields giving rise to impurity-anharmonicity interaction modes [12] to effectively bring down the conductivity curve. Fig. 1 shows a fairly

ð3Þ

where Ck ðxÞ is the phonon frequency line width and is obtainable from the response function

Pðk; x; ieÞ ¼ lim Dk ðxÞ  iCk ðxÞ

ð4Þ

s!0

where Dk (x) is frequency (energy) shift. The detailed expression for Dk (x) and Ck (x) are given in the references elsewhere [5–8]. 3. The scattering mechanism The relaxation time for various scattering processes can be given by ð3Þ ð4Þ ðintÞ D 1 s1 ðxÞ ¼ s1 ð xÞ B ðxÞ þ sdis ðxÞ þ Ck ðxÞ þ Ck ðxÞ þ Ck ðxÞ þ Ck

ð5Þ In above equation various symbols s C xÞ, ð4Þ ðintÞ 1 CAk ðxÞ ½¼ Cð3Þ ð x Þ þ C ð x Þ, C ð x Þ and s ð x Þ stand for bounddis k k k 1 B ,

D kð

ary scattering, defect scattering, anharmonic (cubic and quartic) phonon scattering, interference scattering and dislocation scattering processes [9]. These scattering relaxation times are separately given in Table 1.

Fig. 1. Analysis of thermal conductivity of YBa2Cu3O7.

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N.P. Singh / Materials Today: Proceedings xxx (xxxx) xxx

Fig. 4. Role of electron–phonon scattering. Fig. 2. Role of boundary & dislocation scattering.

Fig. 5. Variation of

Fig. 3. Role of Impurity, phonon & interference scattering.

good agreement between experimental observations and theoretical calculations based on the present model. Our fit suggests that electron–phonon scattering is large for T > Tc and then falls off rapidly for T < Tc as the carriers condense in to Cooper pairs untill the contribution is negligible near Tc/2. Fig. 2 describes the contributions to thermal conductivity due to boundary scattering and dislocation scattering, obviously their role is significant well below Tmax. Also the Fig. 3 exhibits the contribution of anharmonic phonon (cubic) processes, impurity scattering and interference scattering, which reveals that we can not take these scattering scattering processes independently because the sum of their individual contributions do not contribute to the correct value of thermal conductivity. This suggest that additivity of inverse relaxation times is not viable because in the real systems the scattering processes can not be taken as independent of each other. Also it is clear from Figs. 2 and 3 that at low temperature

j with xð— hx=kB T Þ and T.

the boundary scattering and dislocation play an important role whereas the contribution of phonon–phonon scattering is constant along with that interference scattering and impurity scattering brings down the curve and becomes more effective at temperature near conductivity maximum. The role of electron–phonon processes is depicted in Fig. 4 which shows that the presence of a cusp in the thermal conductivity curve can be explained by these processes. The three dimensional plot for the variation of thermal conductivity j with reduced frequency x and temperature T in the range 0.1 to 10 and 0 to 300 K respectively has been depicted in Fig. 5. It is clear from figure that the thermal conductivity depends on both x and T. Initially thermal conductivity increases with increase in x and T, until x and T reaches its maximum value x  0.1 and T  125 K at x ¼ 1:64  1012 sec1 . After that thermal conductivity starts decreasing either decreasing in x or T. It can be concluded here that at high temperature or at high reduced frequency, j tends to be minimum. This phenomenon can be seen more clearly by the Figs. 6 and 7. The significant role of boundary scattering and dislocation scattering at low temperatures is shown in Fig. 8. Their contribution to thermal conductivity first increases up to the maximum value at x  4.0, then the contribution to these processes appears to be unaffected by temperature.

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N.P. Singh / Materials Today: Proceedings xxx (xxxx) xxx

Fig. 6. Variation of j with T for different values of xð— hx=kB T Þ.

Fig. 7. (a & b): Variation of

j with x for different values of T

Fig. 8. Variation of

j with xð— hx=kB T Þ and T in the presence of (a) boundary scattering (b) dislocation scattering.

Fig. 9. Variation of

j with xð— hx=kB T Þ and T in the presence of (a) impurity scattering (b) interference scattering.

Fig. 9 Shows the effect of impurity and interference scattering on the temperature and reduced frequency dependence of thermal conductivity. From these figures, it is clear that the impurity and interference scattering plays an important role near and above thermal conductivity maximum and brings down the conductivity

curve more effectively. The higher values of x does not contribute significantly in this region of thermal conductivity. The variation of thermal conductivity with frequency and temperature for phonon scattering processes is depicted in Fig. 10 and is self explanatory.

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Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements Author is very thankful to Prof. B.D Indu Ex- Professor I.I.T. Rookee for providing the support in completion of this work. References

Fig. 10. Variation of j with xð— hx=kB T Þ and T in the presence of phonon scattering.

5. Conclusion From the above discussion it has been clear that various scattering process play an important role in the analysis of lattice thermal conductivity of HTS and we cannot take them as independent. It has been observed that electron–phonon scattering found highly oscillatory near the critical temperature and offers negative resistance. It emerges from the analysis that the present model successfully explains the experimental observations using various scattering process.

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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CRediT authorship contribution statement Nitin P. Singh: Conceptualization, formal analysis, investiga tion,methodology, software, validation, original drafting, writingreview & editing.

Please cite this article as: N. P. Singh, Theory of thermal conductivity of cuprate superconductors, Materials Today: Proceedings, https://doi.org/10.1016/j. matpr.2020.01.438