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Volume 11, No. 1, pages 14-19, 2009
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Editors-in-Chief Jean Jeener (Universite Libre de Bruxelles, Brussels) Boris Kochelaev (KSU, Kazan) Raymond Orbach (University of California, Riverside)
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Editors
Vadim Atsarkin (Institute of Radio Engineering and Electronics, Moscow) Detlef Brinkmann (University of Zürich, Zürich) Yurij Bunkov (CNRS, Grenoble) John Drumheller (Montana State University, Bozeman) Mikhail Eremin (KSU, Kazan) Yoshio Kitaoka (Osaka University,
Osaka)
Boris Malkin (KSU, Kazan) Haruhiko Suzuki (Kanazawa University, Kanazava) Murat Tagirov (KSU, Kazan)
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In Kazan State University the Electron Paramagnetic Resonance (EPR) was discovered by Zavoisky E.K. in 1944.
Electron-phonon interaction in the 4f125d electronic configuration of the
Tm2+ ion in CaF2
O.V. Solovyev
Kazan State University, Kremlevskaya, 18, Kazan 420008, Russia E-mail: [email protected]
(Received October, 28, 2009; accepted November, 8, 2009)
The low temperature 4f13-4f125d absorption and emission band shapes of the impurity Tm2+ ions have been simulated within adiabatic approximation on the basis of a microscopic model operating with the real phonon spectrum of the host CaF2 lattice. Crystal field parameters and electron-phonon coupling constants were treated in the framework of the exchange charge model. Results of simulations of the spectral envelopes agree satisfactorily with the experimental data available from literature.
PACS: 78.20.Bh; 78.40.Ha
Keywords: 4f-5d spectra, electron-phonon interaction, CaF2
1. Introduction
A method for calculation of energy level patterns and transition intensities in the 4fn-4fn-15d spectra of divalent impurity rare earth ions in cubic crystal field was developed in [1,2]. However, to simulate the 4fn-4fn-15d optical spectra, it is necessary to take into account interaction between the electronic and lattice excitations. Most of intensity of the 4fn-4fn-15d transitions is in broad vibrational bands with widths up to a thousand of wave numbers even at liquid helium temperature due to interaction of a 5d electron with the host lattice. Simulations of such vibrational bands have not been presented in literature, bands are usually approximated as Gaussians with three adjustable parameters: the offset, the bandwidth and the intensity ratio of the zero-phonon line to the vibrational band [3,4].
The goal of the present work is to simulate uniformly the energy level structure and the electron-vibrational band shapes for absorption and emission 4f13-4f125d spectra of the impurity Tm2+ ions in CaF2 crystal and to compare the results with the recent experimental data [5].
2. Simulation of 4f13-4f125d optical spectra of the Tm2+ ion in CaF2 crystal
To simulate inter-configuration 4f13-4f125d spectral envelopes for CaF2:Tm2+ crystal, we used the approach derived in [6] for trivalent impurity rare earth ions in LiYF4 crystal. This approach involves calculations of crystal field parameters for a 5d electron as explicit functions of lattice ion’s coordinates in the framework of the exchange charge model [7], numerical diagonalization of the effective impurity ion Hamiltonian containing energies of electrostatic Coulomb and exchange interactions between electrons, spin-orbit interactions and the crystal field interactions for the ground (4f13) and excited (4f125d) electronic configurations; calculations of the 5d electron-phonon coupling constants, and simulations of the band shapes within adiabatic Condon approximation for low temperatures by making use of the realistic phonon spectrum of the host crystal lattice. Since a 5d electron, having a more extended orbital, interacts with the lattice much stronger than 4f electrons, it is possible to neglect interaction of 4f electrons with phonons.
Impurity Tm2+ ions substitute for Ca2+ ions in CaF2 crystal in sites with the Oh point symmetry. In the nearest surrounding of the Ca-site, there are eight fluorine ions which form a cube with distance R = 2.3642 A from the Ca2+ ion. Since the effective ionic radii of Tm2+ and Ca2+ do not differ much, no local deformations of CaF2 lattice were introduced.
The crystal field energy of the localized electron can be defined as
=1 Bpc(p). (1)
p ,k
Here Bp are the crystal field parameters, C(kp) are the components of one-electron spherical tensor operators C(p). Crystal field parameters for the 5d electron, as functions of ligand coordinates, were calculated as a sum of contributions from lattice point charges and exchange charges, which account for overlap with the ligands’ outer electronic shells ns2 and np6:
Bp(5d) = e2X{ - qLs < 5d|rp|5d >fip (R,) + 2(2P + 1) RlS^^Ls)} • (- 1)^^,PP)/R£+1, (2)
L ,s 5
where the sum is taken over all ions in the host lattice with charges eqLs and the spherical coordinates RLs, &Ls, (pLs (unit cells and ions in the cell are labeled by L and s, respectively); < 5d|rp |5d > are the moments of the 5d electron density. The field of exchange charges is defined by quadratic forms of the overlap integrals <5dlz|nllz> (Ss = <5d0|ns0>, Sa = <5d0|np0>, Sn = <5d1|np1>)
Sp5d)(Ls) = GsS2s (Ls) + OX(Ls) + YfiX(LS), y2 = 1, = - 4/3; (3)
Gs, Ga, Gn are the phenomenological parameters of the model. The coefficients Pv(RLs) [8] account
for extended charge distributions of an impurity ion 5d electron and its ligands’ outer electrons and were calculated exactly utilizing bipolar expansion.
Ion charges qLs were fixed as +2 (Ca2+), -1 (F-). The moments of the 5d electron density < 5d|rp |5d > , overlap integrals <5dlz|2llz> and the f3p (RLs) coefficients were calculated using the analytical radial 5d function of the Tm2+ ion presented in [9] and the 2s, 2p functions of the F" ion from [10]. The calculated moments equal < 5d|r2|5d >= 1.6546 A2, < 5d|r4|5d > = 4.7826 A4. The dependences of overlap integrals on the distance R (for 2.2A < R < 2.5A) between the Tm2+ and the
F- ions were approximated as
Ss =< 5d0|2s0 >= 1.23302 - 0.68714 • R + 0.1-R2, (4a)
Sa =< 5d0| 2p0 >= 0.00443 + 0.18757 • R - 0.05429 • R2, (4b)
Sn =< 5d1| 2p1 >= 1.00553 - 0.59507 • R + 0.09286 • R2. (4c)
The following approximations were obtained for the f3p(R) coefficients (for 2.2A < R < 2.5A):
/32(R) = -5.73565 + 4.09841 • R - 0.64083 • R2, (5a)
pA(R) = -1.5127 - 0.72504 • R + 0.46358 • R2. (5b)
The calculated f3A( RLs) coefficients are negative, thus indicating the strong overlap between charge distributions of a 5d electron and ligands’ outer electrons is to be considered strictly.
Integral intensities of 4f13-4f125d transitions, proportional to the squared matrix elements of the electronic dipole moment, were calculated by making use of the impurity ion Hamiltonian eigenfunctions.
The 5d electron-phonon interaction, linear in dynamic displacements of the lattice ions, was considered within the cluster approximation: modulation of the crystal field by ligand vibrations was considered only. The 5d electron-phonon coupling constants were obtained by direct differentiation of corresponding crystal field parameters with respect to the lattice ion coordinates.
Lattice vibrations were considered in the harmonic approximation. The vibration spectrum of the CaF2 crystal lattice was studied in [11]. The maximum phonon frequency of this crystal equals 477 cm-1. Results of optical absorption and neutron inelastic scattering experiments were described successfully in the framework of the shell model [12]. Using parameters of this model, we computed frequencies and polarization vectors of vibrations for 216000 wave vectors distributed over the Brillouin zone. Imaginary parts of the lattice advanced Green’s functions for the displacements of ions in the cluster Ca2+F-8 were calculated at the equally spaced 954 points on the frequency axis by numerical integration over the Brillouin zone. Spectral density for shell motion relative to the ion core is much smaller than spectral density for ion vibrations, thus we neglected the difference in Green’s functions for the displacements of ions and shells.
Symmetrized displacements of a cubic cluster are given in [13]. We needed the explicit expression of the only full symmetric displacement of the cluster: it corresponds to radial displacements towards the central ion of all eight fluorine ions.
3. Results and discussion
The ground configuration 4f13 of the Tm2+ ion in CaF2 crystal consists of 5 levels, which correspond to Oh group irreducible representations T7u, T8u, T6u (term 2F7/2), T7u', T8u' (term 2F5/2) in the order of energy increase. Level energies are determined by three parameters: crystal field parameters B04(4f) and B06(4f) (in the crystallographic system of coordinates) and spin-orbit interaction constant Z(4f). These parameters were varied to fit the level energies, obtained in [14] by measurements of
CaF2:Tm2+ (0.05%) crystal 4f - 4f optical spectra. The calculated 4f13 level energies, relative to the ground r7u level energy, equal (in cm-1, the measured values [14] are in parenthesis): T8u 554 (557), r6u 672 (symmetry forbidden, was not observed), T7u' 8933 (8961), T8u' 9386 (9377). The obtained values of the varied parameters equal Z(4f) = 2508 cm-1, B04(4f) = -1647 cm-1, B^(4f) = 333 cm-1.
The Tm2+ excited configuration 4f125d consists of 303 levels (910 states), which correspond to Oh group irreducible representations T6g (75 levels), r7 (76 levels), T8g (152 levels). Level energies are
determined by three parameters, discussed above, and eleven more parameters: crystal field parameter B04(5d) , spin-orbit interaction constant Z(5d) , parameters of electrostatic interaction F(2)(ff) , F(4)(ff), F(6)(ff), F(2)(fd) , F(4)(fd) , G(1)(fd) , G(3)(fd), G(5)(fd) , energy shift of the excited configuration A.
We used the values of electrostatic interaction parameters F(2)(fd), F(4)(fd), G(1)(fd), G(3)(fd), G(5)(fd), obtained for Lu3+ ion in LiYF4 crystal [6]. The value of spin-orbit interaction constant Z(5d) was fixed at 1719 cm-1 (that is 90% of the value 1910 cm-1, calculated in [15] with the use of standard programs of atomic physics for the isoelectronic ion Yb3+).
Values of the other parameters - A, F(2)(ff), F(4)(ff), F(6)(ff) and B04(5d) - were varied to fit the experimental data on inter-configuration 4f13-4f125d optical spectra of CaF2:Tm2+ crystal [5]. The obtained values equal F(2)(ff) = 122544 cm-1, F(4)(ff) = 64210 cm-1, F(6)(ff) = 46243 cm-1,
Bg(5d) = -41000 cm-1. The obtained values of the phenomenological parameters of the exchange charge model equal Gs = 2 , Ga = 2.563, Gn = 0.2 . These values do not differ much from the values established for Ce3+ and Lu3+ ions in LiYF4 crystal [6]. The following contributions to the crystal field parameter B04(5d) can be distinguished: exchange charge field (-58219 cm-1), ligands’ point charge field (14869 cm-1), lattice point charge field excluding ligands’ contribution (2350 cm-1).
It is possible to interpret levels of the Tm2+ ion excited configuration by considering a superimposition of a 5d electron spectrum and a 4f12 electronic configuration spectrum and afterwards taking into account electrostatic interaction between a 5d electron and 4f electrons. Cubic crystal field splits the 5d states into a term eg (ground level in eightfold surrounding) and a term t2g . Terms of the
4f12 configuration are well known [16]: 3H6, 3F4, 3H5, 3H4, 3F3, 3F2, 1G4, 1D2, 1I6, 3P0, 3P1, 3P2
and 1S0, in the order of energy increase.
Calculated crystal field energies (relative to the ground 4f125d state energy) and the Huang-Rhys parameters for the Tm2+ 4f125d states, dependent on a state ordinal number, are given in Fig. 1. As
seen from Fig.1, the lowest 200 4f 5d states originate mainly from the 5d eg term: the eg states have the Huang-Rhys parameter nearly twice larger than the t2g states (this was confirmed by simulation of
a 4f-5d absorption spectrum for a cubic Ce3+ impurity center in CaF2 crystal using the same approach). Transitions to these 200 states determine the 4f13-4f125d absorption spectrum of CaF2:Tm2+ crystal measured in [5]. These 200 states can be divided into 5 groups, corresponding to the 4f12 terms 3H6, 3F4, 3H5, 3H4 and 3F3, superimposed on the 5d eg term, as shown in Fig. 1. Thus, we can interpret the bands, observed in CaF2:Tm2+ absorption spectrum [5], as shown in Fig. 2.
The 4f12 configuration 3 F2 term energy (~20000 cm-1 [16]) is close to the energy gap between the eg and t2 5d terms, therefore a continuum of electronic states with different Huang-Rhys parameters
is observed for the 4f125d states with ordinal numbers from 200 to 300 (see Fig. 1). The 4f125d electronic states with ordinal numbers from 300 to 450 and from 740 to 900 originate mainly from the 5d t2 term.
We calculated the 4f13-4f125d absorption envelopes for the lowest 200 Tm2+ 4f125d states in adiabatic Condon approximation at zero temperature. Only interaction with the full symmetric
K
u
fa
3
CIQ
c/o
p
pa
3
tt>
^t
a>
State number
Figure 1. Calculated energies (black line, left scale) and Huang-Rhys parameters (blue line, right scale) of 4f125d states of the Tm2+ ion in CaF2 crystal
displacement of the F" cubic cluster was taken into account. The calculated band shapes were convoluted with the Lorentzian to take into account relaxation broadening. To estimate relaxation broadening width we calculated probabilities of spontaneous one" phonon transitions from each 4f125d state, induced by non-full symmetric phonons at zero temperature. In Fig.
2 calculated and measured [5] 4f13-4f125d absorption spectra of CaF2:Tm2+ crystal are compared.
Agreement with experimental data is satisfactory; for example, calculated band widths are close to the measured ones. Pronounced fine structure is observed in the calculated spectrum, as it corresponds to transitions with the Huang-Rhys parameters close to 1.
The calculated spectrum should be convoluted with the Lorentz distribution with a width ~250 cm"1 to conform experimental spectrum, which exhibits no fine structure (see Fig. 2). Calculated relaxation broadening widths are, in average, much smaller than 250 cm"1.
As follows from our calculations, the lowest Tm2+ 4f125d states are a Kramers doublet r6g and a quadruple r8g. Electric dipole transition to the ground Tm2+ 4f13 state Tlu is symmetry forbidden for the former and spin forbidden for the latter. calculated and measured [5] 4f125d-4f^ emission spectra of CaF2:Tm2+ crystal are compared. Agreement with experimental data is very good (the calculated spectrum was convoluted with the Gauss distribution with the width 19 cm-1 to take into account inhomogeneous broadening). The vibrational band observed in emission spectrum [5] corresponds to transitions with the birth of one pho-non. Shape of this band, with the maxima on 225 cm-1 and 315 cm-1 counting from the zero-phonon transition energy, is determined by spectral density of the full symmetric displacement of the F-cubic cluster.
Figure 2. 4f13-4f125d absorption spectra of CaF2:Tm2+ crystal: calculated at zero temperature (1 - relaxation broadening widths are proportional to calculated probabilities of spontaneous one-phonon transitions, 2 - relaxation broadening width equals 250 cm-1) and measured (3) at 10 K [5]. Bands interpretation obtained in the present paper is indicated
In Fig.
13
3
12
14
16
18
20
£
C
o
ff S 3
c *
O -Q
Xfi 03
à
1 Ï » 1 1 | 1 I 1 I 1
1
Energy {103cm'*)
Figure 3. 4f125d-4f13 emission spectra of CaF2:Tm2+ crystal, measured (1) at 10 K [5] and calculated (2) at zero temperature. Arrow marks the top of the calculated zero-phonon line
4. Summary
The following results have been obtained in this work. The 4f13-4f125d absorption spectrum of CaF2:Tm2+ crystal in the range 14000 - 34000 cm-1 is determined by transitions involving 4f125d electronic states that originate mainly from the eg states of a 5d electron. The Huang-Rhys parameters
of these transitions are close to 1 (intermediate electron-lattice coupling). Terms (2 S+1 LJ, eg), where
2s+1 LJ are terms of the 4f12 configuration, were put in correspondence to the bands, observed in CaF2:Tm2+ absorption spectrum [5]. CaF2:Tm2+ crystal emission at low temperatures has a Condon shape. Agreement between calculated and measured [5] 4f13-4f125d absorption and emission spectra of CaF2:Tm2+ crystal is satisfactory. It is necessary to consider multiphonon relaxation and interaction with the non-full symmetric phonons in simulation of 4f13-4f125d CaF2:Tm2+ absorption spectrum to achieve better agreement with the experimental data.
Acknowledgements
The author is grateful to B.Z. Malkin for valuable discussions. This work was supported by the Russian Foundation of Basic Research (Grant 09-02-00930) and by the Ministry of education and science of Russian Federation (Projects RNP 2.1.1.7348 and 2.1.1/2985).
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