Electrons and Phonons in High Temperature Superconductors

Abstract

The defect-induced anharmonic phonon-electron problem in high-temperature superconductors has been investigated with the help of double time thermodynamic electron and phonon Green’s function theory using a comprehensive Hamiltonian which includes the contribution due to unperturbed electrons and phonons, anharmonic phonons, impurities, and interactions of electrons and phonons. This formulation enables one to resolve the problem of electronic heat transport and equilibrium phenomenon in high-temperature superconductors in an amicable way. The problem of electronic heat capacity and electron-phonon problem has been taken up with special reference to the anharmonicity, defect concentration electron-phonon coupling, and temperature dependence.

1. Introduction

With the remarkable discovery of high-temperature superconductivity (HTSC) in the Ba-La-Cu-O system with 𝑇_𝑐∼30 K by Bednorz and Mullers there begins a new exciting era in condensed matter physics because of their variety of applications in science and technology. The pairing mechanism in high temperature superconductors (HTS), however, being an unresolved problem, there are large number of experimental evidences that the electron-phonon (e-p) interaction together with strong electronic correlations plays a decisive role in understanding the phenomenon of superconductivity [1]. In the literature, it is reported that e-p coupling plays a crucial role in determining the electron density of states (EDOS) and electronic heat capacity (EHC). The specific heat which can be determined from temperature dependence and the spectrum of electrons and phonons has always been a central one in view of its importance in understanding the low-temperature phenomenon in solids. The total heat capacity of HTS is contributed by lattice heat capacity (LHC) and EHC [2]. The EHC (~𝛾𝑇) is only appreciable at low-temperatures and changes dramatically at the superconducting transition, whereas the phonon contribution dominates at room temperature and is generally undisturbed by the transition at 𝑇_𝑐. The Sommerfeld constant 𝛾=(2𝜋𝑘²_𝐵/3)𝐷(𝜖_𝐹) provides an important test for proposed theories [3–5], where 𝐷(𝜖_𝐹) is the EDOS evaluated at Fermi energy 𝜖_𝐹.

In the present work, the expressions for EDOS and EHC have been obtained with the help of many body Green's function theory which uses an almost complete Hamiltonian via quantum dynamics of electrons and phonons.

2. The Hamiltonian and Green’s Functions

In order to formulate the problem with special reference to HTS, we consider an almost complete (without BCS type) Hamiltonian [6, 7] in the form: 𝐻=𝐻_𝑒+𝐻_𝑒𝑝+𝐻_𝐴+𝐻_𝑝+𝐻_𝐷=_𝑞𝜖_𝑞↑𝑏^∗_𝑞↑𝑏_𝑞↑+𝜖_𝑞↓𝑏^∗_𝑞↓𝑏_𝑞↓+𝜖_−𝑞↑𝑏^∗_−𝑞↑𝑏_−𝑞↑+𝜖_−𝑞↓𝑏^∗_−𝑞↓𝑏_−𝑞↓+_𝑞𝑘𝑔_𝑘𝑏^∗_𝑄↑𝑏_𝑞↑+𝑔^∗_𝑘𝑏^∗_𝑞↑𝑏_𝑄↑+𝑔_𝑘𝑏^∗_𝑄↓𝑏_𝑞↓+𝑔^∗_𝑘𝑏^∗_𝑞↓𝑏_𝑄↓𝐵_𝑘+_{𝑠≥3𝑘1⋯𝑘𝑠}𝑉_𝑠𝑘₁,𝑘₂⋯𝑘_𝑠𝐴_𝑘1𝐴_𝑘2⋯𝐴_𝑘𝑠+_𝑘𝜖_𝑘4𝐴^∗_𝑘𝐴_𝑘+𝐵^∗_𝑘𝐵_𝑘+_𝑘1,𝑘2−𝐶𝑘₁,𝑘₂𝐵_𝑘1𝐵_𝑘2+𝐷𝑘₁,𝑘₂𝐴_𝑘1𝐴_𝑘2, (1) where 𝐻_𝑒, 𝐻_𝑒𝑝, 𝐻_𝐴, 𝐻_𝑝, and 𝐻_𝐷 describe the contributions to the Hamiltonian coming from unperturbed electrons, electron-phonon coupling, anharmonicities, harmonic phonons and defects, respectively. 𝐴_𝑘, 𝐵_𝑘, and 𝑏_𝑞(𝑏^∗_𝑞) are the phonon field, phonon momentum, and electron annihilation (creation) operators with spins up (↑) or down (↓), respectively. 𝑄^→=^→𝑘+^→𝑞. 𝑉_𝑠(𝑘₁,𝑘₂⋯𝑘_𝑠) represents anharmonic coupling cofficients [8] and 𝐶(𝑘₁,𝑘₂) and 𝐷(𝑘₁,𝑘₂) are the mass and force constant change parameters [9], respectively.

Now we consider the evaluation of one electron Green’s function: 𝐺_𝑞,𝑞′𝑡−𝑡^′=𝑏_𝑞𝜎(𝑡);𝑏^∗_{𝑞′𝜎′}𝑡^′=−𝑖𝜃𝑡−𝑡^′𝑏_𝑞𝜎(𝑡),𝑏^∗_{𝑞′𝜎′}𝑡^′ (2) via Hamiltonian (1) and adopting the quantum dynamical approach of electrons and phonons [10]. After some simplifications and Dyson equation approach, this can be obtained in the form: 𝐺_𝑞,𝑞′(𝜖)=3𝜖_𝑞+𝜖^𝑐_𝑞(2𝜋)⁻¹×𝜖²−𝜖²_𝑞+𝑖3𝜖_𝑞+𝜖^𝑐_𝑞Γ_𝑞(𝜖)⁻¹𝛿_𝑞𝑞′𝛿_𝜎𝜎′. (3) with 𝜖²_𝑞=𝜖̃²_𝑞+(3𝜖_𝑞+𝜖^𝑐_𝑞)Δ_𝑞(𝜖). In the above expressions  𝜖̃_𝑞, 𝜖_𝑞 and 𝜖^𝑐_𝑞 are renormalized mode, perturbed mode, and pairon energies, respectively. Δ_𝑞(𝜖) and Γ_𝑞(𝜖) are the electron energy line shifts and widths, respectively [11]. The electron energy line width Γ_𝑞(𝜖) has the form: Γ_𝑞(𝜖)=Γ^𝐷_𝑞(𝜖)+Γ_𝑞^3𝐴(𝜖)+Γ_𝑞^𝑒𝑝(𝜖),Γ^𝐷_𝑞(𝜖)=512𝜋_𝑘,𝑘1𝐷𝑘,𝑘₁²×𝜉(𝜖)𝜖_𝑘1𝑁𝜖_𝑘𝑐𝛿𝜖²−𝜖̃²_𝑘1+𝑛_𝑘1𝛿𝜖−𝜖_𝑞𝑐×𝜖_𝑞𝑐⁻²,Γ_𝑞^3𝐴(𝜖)=512𝜋_{𝑘,𝑘1,𝑘2}𝑉₃𝑘,𝑘₁,𝑘₂²×𝜉(𝜖)𝜖_𝑘1𝜂₁𝑁𝜖_𝑘𝑐𝐴_𝛼+𝑛_𝑘1𝑛_𝑘2𝛿𝜖−𝜖_𝑞𝑐×𝜖_𝑞𝑐⁻²,Γ_𝑞^𝑒𝑝(𝜖)=16𝜋_𝑘𝑔_𝑘²𝜉(𝜖)𝑁𝜖_𝑘𝑐𝛿𝜖²−𝜖̃²_𝑘Ω+Ω𝛿𝜖−𝜖_𝑞𝑐. (4) In the present work, we have taken the contribution of anharmonicities up to cubic order because the higher-order anharmonicities are appreciable at high temperatures.

3. Electron Density of States

Using Lehman’s representation and Green’s function formalism with some algebra, the EDOS can be obtained in the form: 𝐷(𝜖)=(2𝜋)⁻¹_𝑞3𝜖_𝑞+𝜖^𝑐_𝑞²Γ_𝑞(𝜖)×𝜖²−𝜖²_𝑞²+3𝜖_𝑞+𝜖^𝑐_𝑞²Γ²_𝑞(𝜖)⁻¹𝛿_𝑞𝑞′𝛿_𝜎𝜎′. (5) For low values of Γ_𝑞(𝜖), 𝐷(𝜖) can be approximated via Breit-Wigner approximation. In the limiting case, when electron line width is very small but finite, 𝐷(𝜖) shows a steep maximum at 𝜖=𝜖_𝑞 and density of state behaves as Lorentzian line shape function peaked at 𝜖=𝜖_𝑞.

4. The Electron Energy

Using (5), one can get the energy of an electron in the form: 𝐸=𝐸^𝐷+𝐸^3𝐴+𝐸^𝑒𝑝. (6) The different terms appeared in the above energy equation are 𝐸^𝐷=128_𝑘,𝑘1𝑔_𝑘²𝐷𝑘₁,𝑘²×𝑁𝜖_𝑘𝑐𝜖_𝑘1𝑛𝜖̃_𝑘1𝐷𝜖̃_𝑘1,𝜖̃_𝑞+2𝑛_𝑘1𝜖̃_𝑞𝑐𝑛𝜖̃_𝑞𝑐𝐷𝜖̃_𝑞𝑐,𝜖_𝑞,𝐸^3𝐴=288_{𝑘1,𝑘2,𝑘}𝑉₃𝑘₁,𝑘₂,𝑘²×𝜂₁𝑁𝜖_𝑘𝑐𝐷_𝑠𝜖̃_±𝛼,𝜖̃_𝑞+2𝑛_𝑘1𝑛_𝑘2𝜖̃_𝑞𝑐𝑛𝜖̃_𝑞𝑐𝐷𝜖̃_𝑞𝑐,𝜖_𝑞,𝐸^𝑒𝑝=4_𝑘𝑔_𝑘²𝜖²_𝑞𝑐𝑁𝜖_𝑘𝑐𝑛𝜖̃_𝑘𝐷𝜖̃_𝑘,𝜖̃_𝑞Ω+2𝜖̃_𝑞𝑐𝑛𝜖̃_𝑞𝑐Ω𝐷𝜖̃_𝑞𝑐,𝜖_𝑞. (7)

5. The Electronic Heat Capacity

We now readily obtain the expression for EHC in the following form: 𝐶_𝑒𝑙(𝑇)=𝐶^𝐷_𝑒𝑙(𝑇)+𝐶^3𝐴_𝑒𝑙(𝑇)+𝐶^𝑒𝑝_𝑒𝑙(𝑇),𝐶^𝐷_𝑒𝑙(𝑇)=128𝐾_𝐵𝑇²_𝑘,𝑘1𝑔_𝑘²𝐷𝑘₁,𝑘₂²×𝐷𝜖̃_𝑘1,𝜖_𝑞𝜖_𝑘1×𝑁𝜖_𝑘𝑐𝑛𝜖̃_𝑘1+𝑁𝜖_𝑘𝑐𝜖̃_𝑘1𝑛𝜖̃_𝑘1+𝜖̃_𝑞𝑐𝐷𝜖̃_𝑞𝑐,𝜖_𝑞×𝜖_𝑘1𝑛²_𝑘1−1𝑛𝜖̃_𝑞𝑐+2𝑛_𝑘1𝑛𝜖̃_𝑞𝑐,𝐶^3𝐴_𝑒𝑙(𝑇)=288𝐾_𝐵𝑇²_{𝑘,𝑘1,𝑘2}𝑔_𝑘²𝑉₃𝑘₁,𝑘₂,−𝑘²×𝜂₁𝑁𝜖_𝑘𝑐𝐴^(±𝛼)+𝐷_𝑠⁽¹⁾𝜖̃_±𝛼,𝜖̃_𝑞+2𝜖̃_𝑞𝑐𝐷𝜖̃_𝑞𝑐,𝜖_𝑞𝑛_𝑘1𝑛_𝑘2𝑛𝜖̃_𝑞𝑐+𝑛𝜖̃_𝑞𝑐𝑛₁₂,𝐶^𝑒𝑝_𝑒𝑙(𝑇)=4𝑘_𝐵𝑇²_𝑘𝑔_𝑘²𝜖²_𝑞𝑐×𝐷𝜖̃_𝑘,𝜖̃_𝑞Ω𝑁𝜖_𝑘𝑐𝑛𝜖̃_𝑘+𝜖̃_𝑘𝑛𝜖̃_𝑘𝑁𝜖_𝑘𝑐+𝜖³_𝑘𝑛²_𝑘−1𝜖²_𝑞𝑐+𝜖̃_𝑘𝑛²_𝑘−1𝜖_𝑘𝜖⁻¹_𝑞𝑐𝜖̃+1×𝑛_𝑞𝑐+Ω𝜖̃_𝑞𝑐𝑛𝜖̃_𝑞𝑐𝜖̃_𝑞𝑐𝐷𝜖̃_𝑞𝑐,𝜖_𝑞. (8) The various symbols used in above expressions are defined as follows: 𝑁𝜖_𝑘𝑐=12𝑁3𝜖_𝑘𝐹+𝑛𝜖_𝑐,𝑁𝑥_𝑖=𝑒^𝛽𝑥𝑖+1⁻¹,𝑁𝜖_𝑘𝑖=𝑁𝜖_𝑘𝑖1−𝑁𝜖_𝑘𝑖𝑛𝑥_𝑖=𝑒^𝛽𝑥𝑖−1⁻¹,𝑛𝜖_𝑘𝑖=𝑛𝜖_𝑘𝑖1+𝑛𝜖_𝑘𝑖,𝜂_𝑖−1=𝜖_𝑘1𝜖_𝑘2⋯𝜖_𝑘𝑖𝜖̃_𝑘1𝜖̃_𝑘2⋯𝜖̃_𝑘𝑖,𝑆_±𝛼=𝑛_𝑘2±𝑛_𝑘1,𝑆⁽¹⁾_±𝛼=𝑛²_𝑘2−1𝜖_𝑘2±𝑛²_𝑘1−1𝜖_𝑘12,𝐴^(±𝛼)=𝑆_+𝛼𝜖̃_+𝛼𝑛(+𝛼)𝐷𝜖̃_+𝛼,𝜖_𝑞×𝑁𝜖_𝑘𝑐+𝑁𝜖_𝑘𝑐𝜖_+𝛼𝑛𝜖_+𝛼+1+𝑆_−𝛼𝜖̃_+𝛼𝑛(−𝛼)×𝐷𝜖̃_+𝛼,𝜖_𝑞×𝑁𝜖_𝑘𝑐+𝑁𝜖_𝑘𝑐𝜖_−𝛼𝑛𝜖_−𝛼,𝑛+1₁₂=𝑛_𝑘2𝜖_𝑘1𝑛²_𝑘1−1+𝑛_𝑘1𝜖_𝑘2𝑛²_𝑘2,𝐴−1_𝛼=𝑆_+𝛼𝜖_+𝛼𝛿𝜖²−𝜖̃²_+𝛼+𝑆_−𝛼𝜖_−𝛼𝛿𝜖²−𝜖̃²_−𝛼,𝐷_𝑠⁽¹⁾𝜖̃_±𝛼,𝜖_𝑞=𝑆⁽¹⁾_+𝛼𝜖̃_+𝛼𝑛𝜖_+𝛼𝐷𝜖̃_+𝛼,𝜖_𝑞+𝑆⁽¹⁾_−𝛼𝜖̃_−𝛼𝑛𝜖_−𝛼𝐷𝜖̃_−𝛼,𝜖_𝑞,𝐷_𝑠𝜖̃_±𝛼,𝜖_𝑞=𝑆_+𝛼𝜖̃_+𝛼𝑛𝜖_+𝛼𝐷𝜖̃_+𝛼,𝜖_𝑞+𝑆_−𝛼𝜖̃_−𝛼𝑛𝜖_−𝛼𝐷𝜖̃_−𝛼,𝜖_𝑞,𝑁𝜖_𝑘𝑐=123𝜖_𝑘𝐹𝑁3𝜖_𝑘𝐹+𝜖̃_𝑐𝑛𝜖_𝑐,𝑛𝜖̃_𝑞𝑐=𝑁3𝜖̃_𝑞𝑛²𝜖̃^𝑐_𝑞×𝑛𝜖̃^𝑐_𝑞−𝑛𝜖̃^𝑐_𝑞𝑁3𝜖̃_𝑞⁻¹,𝑛𝜖̃_𝑞𝑐=𝜖̃_𝑞𝑐𝑁3𝜖̃_𝑞𝑛𝜖̃^𝑐_𝑞𝑛²𝜖̃^𝑐_𝑞×𝑛𝜖̃^𝑐_𝑞−𝑛𝜖̃^𝑐_𝑞𝑁3𝜖̃_𝑞⁻²,𝐷𝜖̃_𝑖,𝜖_𝑞=𝜖̃²_𝑖−𝜖²_𝑞⁻²,𝜖̃_±𝛼=𝜖̃_𝑘2±𝜖̃_𝑘1,𝜖̃_±𝛽=𝜖̃_𝑘1±𝜖̃_𝑘2±𝜖̃_𝑘3,𝑛_𝑘𝑖=coth𝛽𝜖_𝑘𝑖2,𝑛_𝑘𝑖=𝜖̃_𝑘𝑖𝜖_𝑘𝑖coth𝛽𝜖_𝑘𝑖2,𝜖̃Ω=−8²_𝑘𝜖_𝑘+2𝜖³_𝑘𝜖²_𝑞𝑐,𝜖Ω=^2𝑘𝜖²_𝑞𝑐+4𝜖_𝑘𝑛_𝑘𝜖_𝑞𝑐+𝑛_𝑘,𝜖_𝑞𝑐=3𝜖_𝑞+𝜖^𝑐_𝑞. (9)

6. Discussion and Conclusions

Above investigations obviously exhibit that the EDOS not only depends on electron energy but also becomes a function of various renormalized/perturbed mode energies, pairon energies, temperature, anharmonicity, and defect concentration. Based on this model, the electron phonon contribution to EDOS has been depicted in Figure 1. This work investigates the general theory of EHC for HTS and reveals that the EHC is not a simple quantity as (~𝛾𝑇) but comprises of defect contribution 𝐶^𝐷_𝑒𝑙(𝑇), anharmonic contribution 𝐶^3𝐴_𝑒𝑙(𝑇), and electron-phonon contribution 𝐶^𝑒𝑝_𝑒𝑙(𝑇) through electron-phonon coupling constant g(k). A careful examination of these terms infers that 𝐶^𝐷_𝑒𝑙(𝑇) varies with temperature as ~𝑇⁻² along with defect concentration, pairon distribution functions 𝑁(𝜖_𝑘𝑐), 𝑛(𝜖̃_𝑞𝑐), phonon distribution, and electron, phonon, and pairon frequencies, which ensures high sensitivity at low-temperatures. The anharmonic contribution depends on the nature of anharmonic forces as well as on various temperature dependent terms. The contribution to EHC by electron-phonon interaction is heavily influenced by pairon distribution functions, renormalized phonon distribution functions with sophisticated variation of electron, phonon, and pairon frequencies in fundamental, renormalized, and perturbed modes.

Figure 1 

Electronic excitation due to electron-phonon interaction.

It emerges from the present study that the present formulation is capable to explain the EHC with the signatures of electron-phonon interaction in HTS along with their normal phase. This theory can be applied to model calculations of HTS.

Acknowledgments

The authors (A. Singh and H. Singh) are thankful to CSIR and MHRD, New Delhi, India, for the financial support to carry out this research work.