Magnetooptical Effect In Jahn-Teller Crystals With Kramers Rare-Earth Ions

The investigation of the Faraday eeect in DyVO 4-type Jahn-Teller crystals, containing two Kramers doublets E 0 and E 00 in the low-energy part of the electron spectrum, is performed. The components of the tensor of linear magnetooptical eeect are obtained in the mean eld approximation. The ionic contributions into the eeect are separated, the features of their temperature dependencies are studied. The estimate of the so-called crystalline contributions into the Faraday eeect is performed. It is shown that the crystalline contributions increase and can be of the same order as the ionic one when we approach to the absorption edge. One of the most topical problems of physics is the investigation of the structural phase transitions (SPT) which are interesting both theoretically and from the point of view of searching for the new physical eeects promising in their practical use. The investigators attention is increasingly drawn by the SPT in which Jahn-Teller eeect is evident. Such transitions are observed, particularly in the rare-earth vanadate crystals. The change of the lattice symmetry in the crystals of this type is accompanied by the homogeneous deformation and considerable restructuring of the electron subsystem which results, as a rule, in the anomalies of their elastic, magnetic and optical properties 1{4]. The microscopic theory of the magnetooptical eeect in dielectric ionic crystals was developed in 5,6]. The methods suggested in them is used in this paper to investigate the Faraday eeect in Jahn-Teller crystals of the DyVO 4 type with Kramers rare-earth ions. The linear magnetooptical eeect is described by the component of the tensor of dielectric permeability proportional to the external magnetic eld ~ H. For tensor " (!) as a starting we assume the expression in terms of the two time retarding Green functions hh ^

The investigation of the Faraday e ect in DyVO 4 -type Jahn-Teller crystals, containing two Kramers doublets E 0 and E 00 in the low-energy part of the electron spectrum, is performed.The components of the tensor of linear magnetooptical e ect are obtained in the mean eld approximation.The ionic contributions into the e ect are separated, the features of their temperature dependencies are studied.The estimate of the so-called crystalline contributions into the Faraday e ect is performed.It is shown that the crystalline contributions increase and can be of the same order as the ionic one when we approach to the absorption edge.
One of the most topical problems of physics is the investigation of the structural phase transitions (SPT) which are interesting both theoretically and from the point of view of searching for the new physical e ects promising in their practical use.
The investigators attention is increasingly drawn by the SPT in which Jahn-Teller e ect is evident.Such transitions are observed, particularly in the rare-earth vanadate crystals.The change of the lattice symmetry in the crystals of this type is accompanied by the homogeneous deformation and considerable restructuring of the electron subsystem which results, as a rule, in the anomalies of their elastic, magnetic and optical properties 1{4].The microscopic theory of the magnetooptical e ect in dielectric ionic crystals was developed in 5,6].The methods suggested in them is used in this paper to investigate the Faraday e ect in Jahn-Teller crystals of the DyVO 4 type with Kramers rare-earth ions.
The linear magnetooptical e ect is described by the component of the tensor of dielectric permeability proportional to the external magnetic eld H.For tensor " (!) as a starting we assume the expression in terms of the two time retarding Green functions hh P j P ii 0;! , were is the operator of the total electric dipole moment of the crystal ( = x; y:z) which includes electric and ionic components.
V.P.Tupychak is the operator of the electron dipole moment of the ion k in the unit cell n; Xss 0 nk are the Hubbard operators acting in the space of s electron states of the k-th ion; ss 0 k are the corresponding matrix elements; u nk are the vector components of ionic displacements; e, z k are the electron and ion charges, respectively.
In the Hamiltonian of the problem, the electronic excitations of separate ions, their multipole interactions both with each other and with the lattice deformation, phonon vibrations as well as the interaction of the crystal with the external magnetic eld were taken into account.The investigations were carried out at the frequencies corresponding to the crystal transparence region near the short-wave absorption edge.In this case one can only deal with the electron components of the Green function.The function hh P j P ii 0;! = e 2 X kk 0 hhD k jD k 0 ii 0;! (3)   is determined using the equations of motion, decoupled in the random phase approximation.As a result, for the component of the tensor of the linear magnetooptical e ect the following expression is achieved: where Ĝ0 = 1 i ( @ @H hh Dj Dii) H =0 .It is shown that apart from the main ion contributions B ion ; (!) in Faraday e ect caused by the change of the electron dipole susceptibility of ions under the in uence of the magnetic eld there exist additional, so-called crystal, contributions, connected with the interactions of the electronics dipole and quadrupole moments of neighbouring ions ( ~ is the energy of the electron state of j i in the crystal eld).
It should be noted that ionic contributions B ion ; , in which the crystal eld e ect is taken into account, are, as in the case of isolated molecules 7], the sum total of three components caused by 1) the change of the wave functions of electronic states B (1)ion 2) the change of the energy spectrum B (2)ion At the decreasing of temperature (T c 14 K) the crystals turn into the orthorhombic phase with the appearance of the spontaneous deformation u xy of the B 1g symmetry (here the system of co-ordinates turned at =4 around the z axe in relation to the crystallographic axes of lattice is used).
The low-energy part of the electron spectrum of Dy 3+ ions consists of two Kramers doublets E 0 ; E 00 (? 3 ; ?6 ) the distance between which is 9 cm ?1 .Let us consider the magnetooptical e ect in the crystals with the above pattern of the electronic states of the active ions when the magnetic eld is directed along the z axe.The mean eld Hamiltonian has the following expression on the basis of wave-functions of the doublets E 0 and E 00 : where = F z + qP xy is the parameter characterising the internal elds F z and P xy in uenced on the electronic dipoles and quadrupoles, respectively; 2mc m E 00 E 00 z H z ; ; q; m 0 z are the matrix elements of the dipole, quadrupole and magnetodipole moments, respectively.The upper unpopulated electron levels as well as the in uence of the magnetic eld on them is not taken into account.
The eigenvalues of the mean-eld Hamiltonian (9) show that the H z eld splits the doublets E 0 , E 00 eliminating spin degeneration.They cause the contributions (7), (8) into the magnetooptical e ect (population of the sublevels of the doublets E 0 and E 00 are determined as h X i = e ?~ = P 0 e ?~ 0 ).
To calculate the contribution (6) let us introduce an unitary matrix Û.By solving the system of equations The temperature and eld dependencies of parameter , which is included into the expressions for eigenvalues and components of the unitary matrix U, are determined from the minimum of the crystal free energy.
As the result, the following expressions for components of the tensor of linear magnetooptical e ect at T > T c have been obtained: B (1)ion xy;z = 0 where C (!) = 4 e 3 h!m z =(mcv); Z = P e ?; = E 0 ; E 00 .Factors I (!) describe electron dipole transitions from the occupied states into the symmetry allowed upper free states and have the following from x j 2 h 2 ! 2 ?( n ? ) 2 (16) In the transparence region near the short-wave absorption edge the following estimates I (2) (!) I (3) (!) and I ( E 00 I 3 are obtained for the wide temperature range (when kT n ?E ).Therefore, the magnetooptical e ect is, in general, determined by the contribution (15).
In the low-temperature phase (T < T c ) all contributions ( 6)-( 8) are nonzero but the main are only two of them B (1)ion xy;z (!) = 2C(!) W 2 I 1 (!) 0 sin(2' 0 ) th 1 2 W (17) and B (3)ion xy;z (!) = C(!) where W = p ( E 00 ?E 0 ) 2 + 4 2 0 , ' = 1 2 arctg 2 0 E 00 ?E 0 , 0 = t(B 1g )hq 0 i, t(B 1g ) is the quadrupole-quadrupole interaction constant renormalized due to the lattice deformation as well as dipole interaction; hq 0 i is the averaged quadrupole moment (the order parameter) determined from the minimum of the lattice free energy.In g. 1 the temperature dependencies of the reduced component The crystalline contributions in the Faraday e ect for the rare-earth vanadate crystals are also analysed.It is shown that in transparence region of the DyVO 4 crystals at the frequencies distant by 2 eV and more from the short-wave absorption edge crystalline contributions is up to 50% of the ionic contribution and are caused mainly by the electronic dipole moment interactions.In the crystals with the non-Kramers rare-earth ions (TbVO 4 , TmAsO 4 , etc.) crystalline contributions are small and can be observed only in the close vicinity of the absorption edge.MAGN 8)The DyVO 4 -type crystals in the high-temperature phase have a tetragonal structure with the space symmetry D 19 4h .The unit cell contains two ions Dy 3+ connected by the inversion operation and posses D 2d site symmetry.
a new wave function basis corresponding the diagonal Hamiltonian H MF which determines the matrix elements of the electron dipole moments changed by the magnetic eld ^ = Û ^ Û?1 (13)

Figure 1 .
Figure 1.Temperature dependence of the Bion xy;z components for DyVO 4 crystals at di erent N = I 1 =I 3 ratio values: (1) N = ?2;(2) N = ?1;(3) N = 0; (4) N = 0:2; (5) N = 0:5; (6) N = 1; (7) N = 2. Dashed line corresponds to the B (1)ion xy;z contribution. of the experimental investigations of KMnF 3 crystals which also possess SPT 9].At T = T c the dependence Bion xy;z (T ) for KMnF 3 crystal has peak-like anomaly similar to the one presented by curves 1{3 in g. 1.The crystalline contributions in the Faraday e ect for the rare-earth vanadate crystals are also analysed.It is shown that in transparence region of the DyVO 4 crystals at the frequencies distant by 2 eV and more from the short-wave absorption edge crystalline contributions is up to 50% of the ionic contribution and are caused mainly by the electronic dipole moment interactions.In the crystals with the non-Kramers rare-earth ions (TbVO 4 , TmAsO 4 , etc.) crystalline contributions are small and can be observed only in the close vicinity of the absorption edge.