Band electron spectrum and optical properties of KDP-crystals under the external hydrostatic pressure

The electron spectrum of the KDP-type crystals has been investigated as a function of the external hydrostatic pressure using the tight-binding approximation. The joint density of electron states, real and imaginary parts of the dielectric permeability, refraction indices, the gyration coefficient, absorption and reflection coefficients for different polarizations of light are determined. Their pressure and frequency dependencies are investigated. The results are discussed by comparing them with the experimental data on piezooptic coefficients. The anomalous behaviour of the optical constants at the pressure p ≃ 17 kbar is due to the transformation of the hydrogen bond potential from a double minimum one to a single minimum one, where the proton is localised at the midpoint of the hydrogen bond.


Introduction
The physical properties of ferroelectric systems with hydrogen bonds are mainly functions of the bond length and the character of proton motions.Above T c , the protons move in a symmetric double well potential along the O − H − O bonds.Below the ferroelectric transition temperature T c , an ordering of protons in one of these two minima takes place.The most important effect, connected with the decreasing of the bond length, is the decreasing and ultimate vanishing of the transition temperature T c [1].So, it is interesting to investigate the physical properties of such systems with a change of the structural parameters of hydrogen bonds, which gives us a possibility to examine the mechanism of the ferroelectric phase transition.One can change the structure of hydrogen bonds by external pressure, deuteration or other isomorphic changes, for example, substitution of KH 2 PO 4 by RbH 2 PO 4 , KH 2 AsO 4 , etc. Application of pressure is the simplest of them, because the system itself does not change.It is known (see [2,3]) that both KDP and DKDP show a large negative pressure dependence of T c , and T c of KDP falls to 0K at 17 kbar.According to Nelmes and co-authors [4][5][6][7][8][9], above 17 kbar, at 0K, the along-bond (zero-point) thermal motion of the H atom about each site (i.e. the two minima of the potential well with the distance δ between them), U 22 (H) is larger than (δ/2) 2 .Strictly, U 22 (H) is a mean-square amplitude along the y-axis; but this axis is very close to the line joining the two H sites.There then seems to be a simple structural explanation for the loss of the ordered phase that does not necessarily entail tunnelling effects: above 17 kbar, the thermal amplitude of the H atom is always greater than δ/2, the distance from each of the sites to their midpoint, and it seems reasonable to expect that the H atom cannot be localised on one site only.The experimental data of [4][5][6][7][8][9] describe the changes of the KDP structure with pressure in the pressure region 1 bar p 17 kbar.In paper [12], similar investigations for the pressures 1 bar p 54 kbar were carried out, but the proton positions were not determined.Thus, the behaviour of the protons in the KDP structure at pressure p > 17 kbar is not experimentally studied.According to [10,11], at further decrease of the bond length (for example, with the increase of pressure, for p > 17 kbar), the δ value falls to zero.This change may be either abrupt [11] or smooth but in the latter case δ rapidly nonlinearly falls to zero [10].According to [10,12], this change is expected in the pressure region 17 kbar p 27 kbar.The value of pressure at which δ → 0 can be determined only experimentally.It is interesting to calculate the changes of optical functions which are induced by an abrupt change of δ → 0. We have taken δ → 0 at p = 17 kbar in our calculations of optical constants.The results will not change qualitatively if we put δ → 0 at other pressure values of the described region up to 27 kbar, only the corresponding anomaly of optical functions will be shifted to higher values of pressure.
There are anomalous changes of some structural parameters at p = 27 kbar.The bond length decreases almost linearly with the increase of pressure, but at pressure values larger than 27 kbar it becomes much longer.The rotation angle θ increases abruptly at 27 kbar, too.These changes are probably connected with the softening of PO 4 groups, which was observed experimentally at pressure values p > 27 kbar [12].
In this work we have investigated an electron band structure of the KDPtype crystals as a function of the external hydrostatic pressure.We have used the experimental data on the structural changes induced by pressure.The joint density of electron states, real and imaginary parts of the dielectric permeability, refraction indices, absorption and reflection coefficients for different polarizations of light are determined.Their pressure and frequency dependencies are investigated.
There are no other band structure calculations in the literature to compare with, and the dielectric response has not been studied by other authors at frequencies which correspond to the energies of electron transitions.Up till now, calculations of separated groups H n PO 4 (n=0,1,...,4) electronic spectra have been presented [13,14,15].The obtained [14,15] spectra were used to calculate the contribution of H n PO 4 groups to the refractive index and the gyration effect.

Band electron spectrum of the KH 2 PO 4 crystal
The electron energy spectrum of the KDP-type crystals is calculated using the tight-binding approximation.The electron wave functions are presented as linear combinations of the Bloch functions which were constructed from the 3s and 3p orbitals of phosphorus, 2p oxygen and 1s hydrogen orbitals: Here n is a unit cell number, k denotes an ion type and s is an index of electron states.The atomic orbitals ϕ nks were chosen in the form of Slater functions.A set of linear equations for the expansion coefficients C ks ( q) is obtained from the solution of the Schrödinger equation with wave functions (2.1) using an extended Huckel method (see [16]): Secular equations (2.3) provide a band energy spectrum where The summation in equations (2.4) and (2.5) was done over the nearest neighbours only.Secular equation (2.3) of the 36th order with complex matrix elements was solved for the arbitrary wave vector q.Here we take into account the structure changes induced by the external hydrostatic pressure.The dependence of the R O−H distance as well as the rotation angle of PO 4 -groups around the c-axis on pressure results in a pressure dependence of overlap integrals (2.6) and matrix elements of the Hamiltonian (2.7) and, thus, the band electron spectrum becomes a function of the external hydrostatic pressure.The calculations of the overlap integrals (2.6) and the matrix elements (2.7) are described in detail in our previous paper (see [16]).The results of the band structure calculation at the atmospheric pressure (p ≈ 1 bar) for q directed along the principal directions of the first Brillouin zone (BZ) are shown in figure 1.The dependence of the electron spectrum on pressure at Γ, M and X-points in BZ is presented in table 1.The first BZ for a KH 2 PO 4 crystal is shown in figure 2. The state numbers are classified according to energy values.The bands from (1) up to (24) are filled in the ground state.The obtained energy gap is 6.17 eV, which is somewhat less than the experimental results: 7.0 eV [17], 7.62 eV [18].
The most essential changes under the influence of hydrostatic pressure are observed for the electron energy bands which are formed mostly by 1s-orbitals of hydrogen atoms (their numbers are 25-28).There are groups of bands which do not change with pressure (numbers 1,2; 7,8; 13,14; 31,32).We can include in these groups the upper occupied bands (numbers 17-24) the changes of which are smaller than 2 • 10 −3 eV, with the pressure change from 1 bar to 18 kbar.The dependence on the wave vector q for these bands is very weak.The widths of these bands are less than 2 • 10 −2 eV, except for numbers 13, 14, 31, 32 which are of the order 4 • 10 −2 ÷ 1 • 10 −1 eV.It should be noted that at q=0 the energy values for the majority of these bands coincide with the energies of the corresponding levels (2B, 4A, 4B, 8B) of a separate group H 2 PO − 4 with the point symmetry C 2 , up to 1 • 10 −3 eV [14].

Optical properties
The imaginary part of dielectric susceptibility was calculated from the following relation [19]: The indices j and i denote the occupied and empty bands, respectively, and e is a unit vector of the polarization of light.Integration in equation (3.1) is performed over the first BZ.The real part of the dielectric susceptibility ε 1 for all frequencies is determined by the Kramers-Kronig dispersion relation: Besides that, ε 1 within the transparency region was also calculated from the following relation [19]: . (3.4)The optical activity of crystals (i.e.gyrotropy) is a particular manifestation of spatial dispersion and is described by the linear term in the expansion of the dielectric permeability tensor ε αβ in powers of wave vector q [20]: We have calculated yz,x -a component of the gyration tensor, using the relation where The joint density of electron states is given by [19]: It provides the density of the pairs of states; one of them is occupied and the other is vacant, their energies differ by ω.
The integrals in equations (3.1), (3.4), (3.6), (3.8) were replaced by the sums over points q in the first BZ.We included 1836 points in the BZ, then the accuracy of the results is within 10%.At each q point, the band spectrum E( q) and matrix elements of the electric e d and magnetic e 2mc m dipole electron momenta, defined by equations (3.2) and (3.7), respectively, were calculated.The values of ε 1 (ω) obtained from equation (3.4) and from the Kramers-Kronig relation (3.3) differ by less than 1%.For the calculation of ε 2 from equation (3.1) and in equation (3.8) we approximated the δ-function in the following way: where parameter ∆ is taken equal to 0.05 eV.We calculated the values for refractive indices n, extinction coefficients K, absorption coefficients α and reflection R. The formulas presented below were used: (3.10)We have found that the refractive indices n 0 and n e change almost linearly with pressure in the pressure region 1 bar p < 17 kbar.The calculated pressure and frequency dependencies of the optical constants are shown in figures 3-8.Their anomalous behaviour at the pressure values p = 17 kbar and p = 27 kbar are due to the changes of an electron spectrum and wave functions (see table 1) which are induced by the structure changes described above.The obtained changes of the optical constants at p = 27 kbar are less essential than at p = 17 kbar.It was taken into account that for pressure values above p ≃ 17 kbar the hydrogen bond is transformed from two-minimum to one-minimum bond, when the proton is localised at the midpoint of the hydrogen bond.The correlation between the character of a proton distribution and the changes of n 0 , n e and g 11 essential, so it is interesting to study experimentally the behaviour of optical constants with the pressure change in order to investigate their possible anomalies at high pressures (p 17 kbar).Such investigations will give us a possibility to learn the behaviour of protons at high pressure and, thus, to examine the nature of the ferroelectric phase transition from a new point of view.The obtained dispersion of the refractive indices is stronger than the one observed experimentally (see figure 5).One can predict such results because we have obtained from the calculations the energy gap somewhat less than the experimental data (see above).It may be caused by the single-electron approximation used for band structure calculations.Besides, we did not use any fitting parameters in the calculation of the electron spectrum and optical constants.[24], [25].
The main features of the obtained absorption and reflection spectra at the atmospheric pressure (p ≈ 1 bar) are in agreement with the experimental data [13,17] (see figure 8).A wide absorption band at energies 8.5−10.0eV corresponds to the A band in the experiment, while the absorption bands at 10.5−11.2eV and 11.8− 12.7 eV correspond to the B and C bands, respectively.A huge absorption band at 14.0 − 15.8 eV coincides with the D band obtained experimentally.It should be noted that we have obtained the absorption bands in the energy region 6 − 8 eV, which are not observed experimentally.
Let us consider in more detail the dependence of the absorption spectra on hydrostatic pressure.An increase of the main maxima of the absorption and reflection spectra with pressure takes place.It is characteristic especially of the A maximum which lies in the frequency region ω ∼ 9 − 10 eV.That effect is more essential for the light polarized perpendicularly to the c-axis.The dependence of the optical functions which describe the absorption and reflection of the light polarized parallelly to the c-axis (functions K 3 , ε zz 2 etc) on pressure is less essential.The anomalous increase of functions K 1 , ε xx 2 at pressure passing the value p ≃ 17 kbar is larger than the increase of these functions when the pressure is changed from 1 bar to 17 kbar (see figure 7).
It is possible to determine piezooptic coefficients using the obtained pressure dependencies of the coefficients n 0 and n e .In order to describe the change of the dielectric susceptibility ε α,β with the pressure change we shall utilize the following relation: where π ijke are piezooptic coefficients, σ ke is a tensor of mechanical strain.In the case of a hydrostatic pressure we have: We have found that the piezooptic coefficients d(ε −1 xx )/dσ and d(ε −1 zz )/dσ do not depend on pressure.Their changes with the change of pressure from 1 bar to 17 kbar at fixed frequency are less than 2.5%.The calculated values of the piezooptic coefficients are presented in table 2. On the other hand, using the experimental data for π ij (for λ = 632.8)nm [21,22] we have: d(ε −1 xx )/dσ = 9.35•10 −13 cm 2 /dyn; d(ε −1 zz )/dσ = 7.64 • 10 −13 cm 2 /dyn.So, the obtained piezooptic coefficients are in agreement with the experimental data.The g 11 component of the gyration tensor practically does not change with pressure, except for the anomalies at p=17 kbar, when g 11 abruptly increases approximately by 10%.In the case of ionic contributions to the optical properties of a KH 2 PO 4 crystal we also obtained earlier a weak increase of the g 11 component of the gyration tensor with the pressure change from 1 bar to 17 kbar [23].

Figure 1 .
Figure 1.Electron band spectrum for a KH 2 PO 4 crystal.

Figure 2 .
Figure 2. The first Brillouin zone for a KH 2 PO 4 crystal.The line ZM ′ is equivalent to the Z ′ M line.

Figure 8 .
Figure 8. Spectral dependence of the reflection coefficient R 1 for KH 2 PO 4 at the atmospheric pressure, solid line -experimental data [13], dashed line -calculated.

Table 2 .
The calculated values of the piezooptic coefficients for a KH 2 PO 4 crystal for various lengths of a light wave (in 10 −13 cm 2 /dyn).