1. Introduction
Due to the direct relation between the structure of the phonon spectrum in an impure crystal and the absorption line shape the problem of the optical absorption of impurity induced crystal in the infrared and far infrared, appears as a very interesting problem [1]. We have not come across the sufficient experimental results which has offered as much information about the vibrational spectra of crystals as have been endorsed by the infrared absorption experiments. As a result of interaction of radiation (photon) the light is absorbed by the crystal without any change in the electronic state but excites or de-excites one or two phonons. The appropriate selection rule for such interactions is that only modes whose wave vector 
adds up to zero interact with the light incident on the crystal. The study of infrared properties of high temperature superconductors has equally attracted the theorists and experimentalists to establish the phenomenon and dynamical behavior. The far-infrared spectroscopy is used to explain the energy gap in BCS superconductors [2] as well. This interesting problem of infrared absorption with low impurity concentration using the method of Green's function of many body theory has been theoretically investigated by many authors [3-8][3]. The absorption coefficient of a harmonic crystal is temperature independent, while experimental evidences show that temperature plays an important role to study the absorption spectra. This arises only when nonlinear terms in the dipole moment or anharmonic terms in the potential energy are used [9]. The crystal structure is linked with absorption bands and these bands are often observed in the infrared spectra of crystalline solids [10]. If an ionic cubic diatomic crystal absorbs the infrared light, a single band coupled with optical mode of zero propagation constant will occur. As anharmonicity alone does not give any mechanism for absorption so the intrinsic lattice absorption can be explained on the basis of second order electric moment [11]. Czerny [12] and Barnes [13] make the observations on alkali halide crystals significantly provided the existence of side band (the short wavelength side) of the chief reflection or absorption band. Some authors tried to explain these side bands using the anharmonicity approach in which the potential energy (connected with lattice vibrations) contains the higher order terms like cubic and quartic [14, 15, 16]. An anharmonic quantum dynamical approached is used via double time temperature dependent Green’s functions to develop the theory of infrared absorption for high temperature superconductors in the following sections of the present paper:
2. The Hamiltonian
To investigate the quantum dynamics of phonons let us consider almost complete crystal Hamiltonian in the form [17, 18, 19, 20]
 | (1) |
 | (1a) |
 | (1b) |
 | (1c) |
 | (1d) |
 | (1e) |
where 
, 
, 
, 
and 
, respectively are unperturbed electron-, unperturbed phonon-, electron-phonon-, anharmonic (upto quartic terms)-, and defect contributions to the Hamiltonian 
. In the above equations 
and 
are the electron creation (annihilation) and phonon field and momentum operators, respectively. Also,
stand for spin up(down) and 
(
and 
being phonon and electron wave vectors, respectively) and 
stand for electron-phonon coupling coefficient. The symbols
, 
and 
stand for the anharmonic coupling coefficients, mass and force constant change parameters [17, 18, 19, 20][17, 18, 19, 20].
3. The Phonon Green’s functions
To obtain the phonon line spectrum, we evaluate the double time temperature dependent retarded phonon Green’s function
 | (2) |
where 
is the Heaviside unit step function. Double differentiation of equation (2) with respect to 
followed by the Fourier transformation yields
 | (3) |
with
 | (4) |
Which, in turns, resolves the Green’s function to be obtained in terms of phonon frequency shift 
and line width
as
 | (5) |
After appropriate algebraic simplification the Green’s function takes the form
 | (6) |
 | (7) |
and 
are renormalized and perturbed mode frequencies.
4. The Phonon Frequency Width
The phonon line width enables one to investigate many dynamical properties of crystalline solids, e.g., life times, the phonon frequency (energy) spectrum and density of states which can be obtained in the following form [17, 21]:
 | (8) |
 | (8a) |
 | (8b) |
 | (8c) |
 | (8d) |
various symbols are well defined in the references elsewhere [17, 21].
5. The Infrared Absorption Coefficients
The optical absorption coefficient which is related to Green's function can be given by [5],
 | (9) |
In the above equation 
is the Fourier transform of phonon Green’s function, 
is the polarization vector of incident radiation, c is the speed of electromagnetic radiation, 
is the refractive index and 
is the charge on the i-th atom in the l-th unit cell. For diatomic crystal and 
is the reduced mass. Eq. (9) can be written in the form [22]
 | (10) |
Using Eq. (6) in (10) the absorption coefficient can be obtained in the following form
 | (11) |
The Absorption coefficient given in Eq. (11) can be separated into diagonal and non-diagonal contributions, namely;
 | (12) |
 | (13) |
where 
. For 
the non-diagonal contribution arises and it chiefly depends on the mass change parameter 
. This contribution is momentous only for impurity induced crystal and vanishes for a pure crystal. Now making use of Eq. (8) in Eq. (12) and after some algebra we can derived the expression for diagonal contribution in the following form
 | (14) |
 | (15) |
 | (16) |
 | (17) |
Where
The terms in the square brackets of Eqs. (14) to (17) can be further simplified to give
 | (18) |
 | (19) |
 | (20) |
 | (21) |
Which after further simplification yield,
 | (22) |
 | (23) |
 | (24) |
 | (25) |
The higher order terms with cubic and higher powers add negligibly small contribution [7, 19, 20] and can be dropped from above Eqs. (22) to (25). Thus making use of substantial contributions only the Eqs. (22) to (25) can be further simplified to give
 | (26) |
 | (27) |
 | (28) |
 | (29) |
Thus the final form of absorption coefficients can be obtained as
 | (30) |
 | (31) |
 | (32) |
 | (33) |
And similarly the non-diagonal contribution can be obtained as
 | (34) |
 | (35) |
 | (36) |
 | (37) |
Where
6. Results and Discussions
The results thus obtained can be analyzed for a model cuprate crystal 
. For the purpose of numerical estimation following physical constants have been used: 
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Figure 1. Nature of
for
in superconducting region
A careful investigation reveals that the diagonal contribution to infrared absorption coefficient exhibits 11 peaks corresponding to the energies 19.85 meV, 20.51 meV, 27.37 meV, 34.80 meV, 40.79 meV, 52.01 meV, 53.78 meV, 55.17 meV, 55.56 meV, 56.36 meV and 60.75 meV in the superconducting region which is depicted in Figure 1.
Correspondingly, the contribution due to various individual processes is shown in Figure 2, which confirms that the contribution due to anharmonic processes and impurities is extremely sensitive as compared to others. The emergence of several peaks at various k-values with different magnitudes due to individual processes is self-explanatory. The three dimensional behaviour of absorption coefficient with varying temperature and phonon wave vector is depicted in Figure 3.
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Figure 2. Various contributions to
for
in superconducting region
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Figure 3. 3D depiction of
for
with k and T in superconducting region
Below transition temperature this graphics clearly exhibits sharp five absorption peaks of variable heights at different temperatures appearing at higher side of the phonon wave vector. Similarly, Figure 4 portrays the typical 3-D behaviour of 
with 
and phonon wave vector at the higher end of 
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Figure 4. 3D depiction of
for
with
and
in superconducting region
Further, the three dimensional variation of 
has been depicted with 
and temperature in Figure 5 where one can see large number of peaks with different heights and nature. The appearance of large number of peaks, however have not been observed through any experiment but one may observe and confirm them with the help of high resolution experiments in future.
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Figure 5. 3D Nature of
for
with
and T in superconducting region
After a careful inspection of diagonal contribution it can be plotted to observe its significance which is portrayed as Figure 6. The infrared absorption coefficient shows 8 various peaks at 18.69 meV, 21.28 meV, 27.88 meV, 40.7 meV, 52.01 meV, 55.73 meV, 56.55 meV and 62.09 meV normal region.
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Figure 6. Behaviour of
for
in normal region
Comparatively this contribution is of lower magnitude as compared to the diagonal part, but the sensitivity is of similar intensity.
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Figure 7. Various contributions to
for
in normal region
Figure 7 also shows the various contributions for diagonal infrared absorption coefficient in normal region with various peaks.
7. Conclusions
Taking into account the anharmonicities, impurity and electron-phonon effects, the infrared absorption coefficient has been derived in terms of diagonal and non-diagonal contributions. The various components of infrared absorption coefficient depend upon electron-phonon coupling, temperature and concentration. As we go from superconducting to normal region the nature of curves change sharply with substantial modifications in the phonon spectra. It has been confirmed that infrared absorption coefficient not only depends upon defect concentration but also on temperature, renormalized mode and pairons frequencies which emerges as a new feature of present theory. This theory which is very unique and complex, gives a very good treatment of high temperature superconductors and is applicable for other systems also. The peculiar feature of the theory is the auto-emergence of pairons during the quantum dynamical developments where no use of BCS Hamiltonian has been made. It emerges from the present study that the optical absorption (infrared absorption) for high temperature superconductors can be made successfully with the help of present theory in both superconducting and normal regions.
Acknowledgement
One of the author Hempal Singh is thankful to MHRD for financial support to carry out this research work.
References
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. The temperature dependence of infrared absorption coefficients and automatic emergence of pairons appears as a special feature of the theory.
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