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Jahn–Teller effect and electron–phonon interaction in the 4T2g excited state of Cr3+ ion in K2LiAlF6 crystal
Solid State Communications 149 (2009) 2070–2073
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Jahn–Teller effect and electron–phonon interaction in the 4 T2g excited state of Cr3+ ion in K2 LiAlF6 crystal N.M. Avram a,b,∗ , M.G. Brik c , C.N. Avram a , I. Sildos c , A.M. Reisz a a
Department of Physics, West University of Timisoara, Bd. V. Parvan 4, Timisoara 300223, Romania
b
Academy of Romanian Scientists, Splaiul Independentei Nr. 54, 050094 Bucharest, Romania
c
Institute of Physics, University of Tartu, Riia 142, 51014 Tartu, Estonia
article
info
abstract
Article history: Received 22 July 2009 Accepted 13 August 2009 by E.L. Ivchenko Available online 19 August 2009
The electron–phonon coupling and Jahn–Teller effect in the first excited state 4 T2g of Cr3+ in K2 LiAlF6 crystal (which has the potential for tunable solid-state lasers applications) are considered. The force constants for the a1g and eg normal modes are calculated with the FG matrix method for the octahedral [CrF6 ]3− cluster and the Huang–Rhys factors Sa1g and Seg from the experimental value of the total Huang–Rhys factor
PACS: 61.72.S71.70.Ch 71.70.Ej 63.20.Kd
and Stokes shift. Using these data, the changes of the metal–ligand bond lengths 1xeq , 1yeq = 0.072 Å (equatorial expansion) and 1zeq = −0.015 Å (axial compression) were estimated. The Jahn–Teller stabilization energy has been evaluated from the contour plot of the 4 T2g electronic state potential energy surface. The obtained results are compared with those for other fluorides; the common trends are found and discussed. © 2009 Elsevier Ltd. All rights reserved.
Keywords: C. Point defects D. Crystal and ligand fields D. Electron–phonon interaction D. Optical properties
1. Introduction The K2 LiAlF6 fluoride belongs to a wide class of A2 BMX6 materials, where A (Cs, K) and B (Na) are monovalent cations, M is a trivalent cation (Al3+ , In3+ , Y3+ , Sc3+ , Ga3+ ) and X is a monovalent anion (F− , Cl− , Br− ). After doping, the Cr3+ ions substitute for octahedral coordinated trivalent metal ions without any need for charge compensation. Apart from their potential as laser materials, chromium-doped elpasolites are model systems for investigating crystal field and vibronic coupling effects, since the deformations of the crystal lattice after doping can be neglected and the symmetry of the formed [CrF6 ]3− complex is close to Oh . The optical properties of the Cr3+ -doped elpasolites are discussed in [1]. The electronic states of main interest are the 4 A2g and 4 T2g states (since the position of the 4 T2g state basically determines the crystal field effects and the overall appearance of the emission and absorption spectra). All these states are derived from the free-ion 3d3 ground configuration of the Cr3+ ion. In this paper we consider
∗ Corresponding author at: Department of Physics, West University of Timisoara, Bd. V. Parvan 4, Timisoara 300223, Romania. Tel.: +40 722331696; fax: +40 256592108. E-mail address: [email protected] (N.M. Avram). 0038-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2009.08.020
the electron-vibrational interaction in the K2 LiAlF6 :Cr3+ crystal, since this system has a real potential for the tunable solid-state laser applications. Based on the data available in the literature on the frequencies of the normal vibration of the CrF36− complex and Huang–Rhys factors, we estimate the force constants for these normal modes, the magnitudes of the normal modes’ displacements, the changes in the ‘‘Cr–F’’ chemical lengths and the Jahn–Teller stabilization energy for the above mentioned system. Main attention is focused on the geometry of the first excited electronic state 4 T2g , influenced by its coupling with the total symmetric a1g and doubly degenerated eg normal vibrations of the CrF36− complex due to the dynamic Jahn–Teller effect. Such an analysis permits us to estimate the changes of the chemical bond lengths originating from the combined effect of the a1g and eg normal modes, plot the potential energy surface as a function of the ionic displacements, and deduce the value of the Jahn–Teller stabilization energy. 2. Experimental support Spectroscopic studies of the K2 LiAlF6 :Cr3+ system were performed in [2]. From the experimental spectra of absorption and luminescence the following values of the crystal field strength Dq and Racah parameters B and C (all in cm−1 ) were deduced for Cr3+ : 1608, 710, 3341, respectively. The lowest Cr3+ energy levels with
N.M. Avram et al. / Solid State Communications 149 (2009) 2070–2073
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restoring force Ki 1Qi at the distorted geometry 1Qi = −Fi /Ki with 1Qi denoting the difference between the equilibrium and distorted coordinates. This distortion lowers the energy of the electronic state by an amount Ei = (1/2)Ki (1Qi )2 comparing to the equilibrium position. Only distortions along the a1g , eg , t2g octahedral modes are important for the 4 T2g electron state. The Hamiltonian (1) can be rewritten in basis consisting of the real 3d orbitals [4]: HVIB
Fig. 1. Crystal structure of K2 LiAlF6 [3]. K ions are shown by black spheres, Li and Al ions are at the centers of octahedrals composed of fluorine ions.
these parameters are as follows (all in cm−1 ): 4 A2g − 0, 2 Eg − 15 219, 2 T1g − 15 851, 4 T2g − 16 080, 2 T2g − 22 622, 4 T1g − 22 994, Such a sequence of the energy levels (together with the Dq/B ratio) shows that the Cr3+ experiences strong crystal field in this host. The frequencies of the a1g and eg normal modes are (in cm−1 ) 558 and 469, respectively [2]; the Stokes shift (the difference between the maxima of the lowest emission and absorption bands) is 2420 cm−1 , and the Huang–Rhys factors for these modes are Sa1g = 1.3 and Seg = 1.0 [2]. The considered crystal, at low temperature, has a cubic structure, belonging to the Fm3m space group, having the lattice parameters a = 7.842 Å, with four unit formula in unit cell [3].The K2 LiAlF6 structure, as shown in Fig. 1, represents a sequence of alternating octahedral made of six fluorine ions centered at Li and Al ions [3]. 3. Method of calculations The ground electronic state of trivalent chromium ions at the octahedral site is 4 A2g [4]. Due to the coupling of the electronic state 4 T2g (the first excited spin-quartet) and the Jahn–Teller active modes a1g , eg and t2g of the host matrix and the different bonding properties in the ground and excited states, they may have different geometries (in other words, different coordinates of the atoms in an octahedral complex with Cr3+ at the center), what is revealed in the shift of the potential surfaces of the considered electronic states. The effective Hamiltonian for Cr3+ ion, which describes the molecular vibrations of the CrF36− octahedron is given by the following equation [5]: HVIB =
X P2 1 ∂V i + Ki Qi2 + Qi . 2µi 2 ∂ Qi 0 i
(1)
The first two terms (expressed in terms of the ligands’ masses µi , their momenta Pi , force constants Ki for the corresponding normal modes Qi ) transform the electronic levels into potential energy manifolds in the coordinates of the √ octahedral normal modes with vibrational frequencies ωi = Ki /µi , and the complete wave functions in the Born–Oppenheimer approximation can be written as a product of an electronic and a vibrational parts. The third term can be understood as a force, which acts along the vibrational mode associated with the electronic state denoted by Γ :
∂V Γ Fi = − Γ ∂ Qi 0
(2)
where the subscript means that the derivative is to be found at the equilibrium configuration. This force is balanced by the harmonic
X P2 1 i 2 = + Ki Qi 2µi 2 i
* + Q
∂V a1g
+ 4 T2g
4 T2g 0
∂ Qa1g 0
∂V 4
+ 4 T2g
∂ Q T2g eg √ 3 2 Qegθ − 2 Qegε 1
×
0
2
0
Qegθ + 0
* + 0
∂V
+ 4 T2g
4 T Qt2gζ
∂ Qt2g 2g Q
t2gη
0 0 Qa1g
0
1
0 Qa1g 0
0
√ 3 2
Qegε
Qt2gζ 0 Qt2gξ
0 −Qegθ
Qt2gη Qt2gξ . 0
(3)
The reduced matrix elements in front of all matrices are the constants of the electron-vibrational interaction and can be expressed in terms of the crystal field strength Dq and equilibrium distance R0 as follows [6,7]:
∂V 4 25
T2g T2g = − √ Dq,
∂ Qeg R0 3
* +
∂V 50
4 4 T2g Dq, = −√
T
∂ Qa1g 2g 6Ro
+ * √
∂V 18 2 1 5
4 4 T2g =− Dq − ,
T
∂ Qt2g 2g 7R0 η 9
2 2 ! −1 r R0 3 3 − 4eπ /eσ
η= = . 5 1 + eπ /eσ r4
4
The eπ and eσ entries in the last equation are the parameters of the angular overlap model of crystal field and are described in [7]. The magnitudes 1Q of the ligands displacements from the equilibrium position along the a1g and eg active modes can be estimated from the following expression [8]:
|1Qi | =
2Si h¯ ωi Ki
1/2
,
(4)
where S is the Huang–Rhys factor, K is the force constant, and h¯ ω is the energy of the corresponding normal mode. The force constants K for an octahedral complex can be easily evaluated using the FG method [9]. To facilitate the further consideration, the coordinate system in the (Qε , Qθ ) space can always be chosen in such a way, that the potential minimum of the considered 4 T2g component (anyone from the ξ , η, ζ set) lies on the Qθ axis (this means, no distortion takes to place along the Qε axis). Then it is possible consider the 1Qeg eq values as corresponding to 1Q eg θ only, eq
whereas 1Q eg ε is zero. Using the transformation matrix given eq below [10], the changes of 1Q can be easily expressed in terms
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N.M. Avram et al. / Solid State Communications 149 (2009) 2070–2073
Table 1 The force constants Ki , magnitudes of the normal mode displacements, changes in the chemical bond lengths and Jahn–Teller energy EJT for several crystals doped with Cr3+ ions.
K2 NaScF6 [11] Cs2 NaYF6 [11] K2 LiAlF6 (this work)
1Qa1g (Å) 1Qeg (Å)
a1g mode
eg mode
h¯ ω (cm−1 ) K (N/m) S
h¯ ω (cm−1 ) K (N/m) S
542 501 558
449 402 469
241 206 255
0.9 3.2 1.3
165 133 180
Combined result of both a1g and eg modes
1x, 1y (Å) 1z (Å)
1.3 0.64 1.0
0.09 0.18 0.11
−0.11 −0.09 −0.10
0.07 0.09 0.072
−0.03 0.02 −0.015
Result of a1g mode only
EJT (cm−1 )
1x, 1y, 1z (Å) 0.037 0.072 0.043
584 257 469
0.15
(Cr3+ – F–(z)), Å
0.1
0.05
z0
–0.05
–0.1
Fig. 2. Distortion of the [CrF6 ]3− complex in the 4 T2g excited state with respect to the ground state (direction of displacements are indicated by arrows; their magnitudes are given in Å).
of the chemical bond length changes 1x, 1y, 1z (which can be measured, in principle):
r 1x 1y 1z
!
2
3 r 1 2 = 2 r 3 2 3
r −
1
r3 1
− r 3 4 3
−1 1Qa1g 1 1Qeg θ . 1Q eg ε
(5)
0
After the force constants and amplitudes of the ionic displacements are found, it is possible to draw the potential energy surface of the 4 T2g excited electronic state. In the harmonic approximation, this energy is described by the following expression: V =
1 2
Ka1g 1Qa1g − 1Qa1g ,eq
2
1
+ Keg 1Qeg − 1Qeg ,eq 2
2
.
(6)
Using the transformation inverse to that one in Eq. (5), it is possible to visualize the potential energy dependence on the changes of the chemical bond lengths 1x, 1y, 1z, and, estimate the Jahn–Teller stabilization energy. 4. Results of calculations and discussion Consistent application of all given above equations to the K2 LiAlF6 :Cr3+ crystal allowed to obtain the values of the electronvibrational interaction and interionic distances changes collected in Table 1. We note that vibronic coupling constants of the 4 T2g electron state with triple degenerate vibrations of host lattice is very small compare with that of total symmetric and respective double degenerate vibration, so we neglect former one. In Table 1 we give only the parameters regarding vibronic coupling of the 4 T2g electron state of Cr3+ with total symmetric and respective, double
–0.15 –0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 (Cr3+ – F–(x,y)), Å Fig. 3. A contour plot of the harmonic 4 T2g potential energy surface (only one of the three parts is shown) for the K2 LiAlF6 :Cr3+ system as a function of changes of the Cr3+ – F− (x, y) and Cr3+ –F− (z) chemical bonds lengths. The energies of individual contours are given in hundreds of wave numbers. The open circle at the origin corresponds to the equilibrium position of the ground 4 A2g potential energy surface; the black square indicates the equilibrium position of the 4 T2g potential energy surface shifted with respect to the ground state as a combined result of the a1g and eg normal vibrations. The black circle shows the hypothetical position of the 4 T2g potential energy surface minimum if there were no eg normal vibration (i.e. in the absence of the Jahn–Teller distortion). The value on the potential energy surface of the 4 T2g state at this point (between 400 and 500 cm−1 from the figure) corresponds to the Jahn–Teller energy for the considered complex.
degenerate vibration of the host matrix. For comparison, the analogous values for Cr3+ ions in two other fluoroelpasolites [11] are also given. As seen from this Table 1, the force constants for the K2 LiAlF6 :Cr3+ crystal are the greatest among all presented in the Table 1, which is in line with the greatest frequencies of the a1g and eg normal modes for this host. Generally, the increasing order of the force constants in the Cs2 NaYF6 → K2 NaScF6 → K2 LiAlF6 series (Table 1) is in line with decreasing order of the lattice constants in the same direction (9.075 Å for Cs2 NaYF6 [13], 8.482 Å for K2 NaScF6 [14], and 7.842 Å for K2 LiAlF6 [3]), which means that moving ions closer to each other makes chemical bonds harder. Regarding the magnitudes of the normal displacements and Jahn–Teller stabilization energies, the trend is not that straightforward, which can be explained by interplay of crystal field effects (like splitting of the orbital triplets etc), which reveal themselves in different Stokes shifts for each of the considered materials. However, a quick glance at the deformation magnitudes shows that in all these crystals the CrF36− complex is experiencing an equatorial expansion in the xy plane and an axial compression along the z-axis in K2 NaScF6 and K2 LiAlF6 crystals, whereas in Cs2 NaYF6 the complex undergoes a slight elongation along the z axis. Similar results are obtained by first principle calculations of parameters of electron-vibration interactions in elpasolite crystals
N.M. Avram et al. / Solid State Communications 149 (2009) 2070–2073
doped with Cr3+ ions [12]. Fig. 2 visualizes the character of the Jahn–Teller deformation of the CrF36− cluster in K2 LiAlF6 . Finally, Fig. 3 shows a cross-section of one of three parabolas of potential energy arising from the 4 T2g state after it is split due to the Jahn–Teller effect. Such a contour plot allows for a direct estimation of the Jahn–Teller stabilization energy, if the following circumstance is taken into account: if only interaction with the fully symmetric a1g mode is considered, then the 4 T2g potential surface will be only shifted (with equal changes of 1X , 1Y , 1Z shown in Table 1), without any splitting (the splitting is actually produced by the eg vibrational mode, which effectively lowers the symmetry). The point of this hypothetical minimum is shown in Fig. 3 by an open circle, and the value of the potential energy at this point (which lies between 400 and 500 cm−1 in the Figure) is exactly the Jahn–Teller stabilization energy. Numerical estimations with Eq. (6) and all calculated force constants and displacements yield the value of 469 cm−1 for K2 LiAlF6 . 5. Conclusions The vibronic interaction between the 4 T2g excited state of Cr3+ and the a1g , eg normal vibration modes of the host matrix modifies the geometry of the 4 T2g state relative to the ground state. Using the harmonic approximation for vibrations and linear approximation for the electron-vibrational interaction, we have calculated the equilibrium displacements of the 4 T2g potential surface minimum from the ground state along the a1g and eg Jahn–Teller active modes The Ki force constants for the a1g and eg modes are calculated with the FG matrix method and are equal to 255.3 and 180.3 N/m. The Huang–Rhys factors Sa1g and Seg are obtained from the experimental value of the total Huang–Rhys factor and Stokes shift. Using these data and frequencies of the normal modes, we
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estimated 1Qa1g = 0.10 Å and 1Q eg θ = 0.10 Å. These eq eq values have been converted into the changes of the metal–ligand bond lengths 1xeq , 1yeq = 0.072 Å and 1zeq = −0.015 Å. The Jahn–Teller stabilization energy EJT = 469 cm−1 has been estimated from the contour plot of potential surface of the 4 T2g electronic state. The obtained results are compared with those for other fluorides [9]; a few common trends are discussed and related to the corresponding changes of the lattice constants of the considered hosts. Acknowledgement N.M. Avram appreciates the financial support from Romanian National Authority for Research in the frame of CEEX Program (Project Nr. 51/2006/MATNANTECH). References [1] P.A. Tanner, Chem. Phys. Lett. 388 (2004) 488. [2] M.A.F.M. da Silva, R.B. Barthem, L.P. Sosman, J. Solid State Chem. 179 (2006) 3718. [3] J. Graulich, S. Drueeke, D. Babel, Z. Anorg. Allg. Chem. 624 (9) (1998) 1460. [4] S. Sugano, Y. Tanabe, H. Kamimura, Multiplets of Transition-Metal Ions in Crystals, Academic Press, New York, 1970. [5] M.D. Sturge, Advanced in Solid State Physics, vol. 20, Academic Press, New York, 1967, 91–211. [6] K. Wissing, J. Degen, Mol. Phys. 95 (1998) 51. [7] T. Schönherr, Topics Current Chem. 191 (1997) 87. [8] T.C. Brunold, H.U. Güdel, Inorganic Electronic Structure and Spectroscopy, in: Methodology, vol. I, Wiley, New York, 1997. [9] K. Venkateswarlu, S. Sundaram, Z. Phys. Chem. Neue Folge 9 (1956) 174. [10] R.B. Wilson, E.I. Solomon, Inorg. Chem. 17 (1978) 1729–1736. [11] N.M. Avram, M.G. Brik, J. Mol. Struct. 838 (2007) 198. [12] M.G. Brik, N.M. Avram, J. Mol. Struct. 838 (2007) 193–197. [13] A. Vedrine, J.P. Besse, G. Baud, M. Capstan, Rev. Chim. Miner. 7 (1990) 593. [14] B.F. Aull, H.P. Jenssen, Phys. Rev. B 34 (1986) 6647.
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Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Jahn–Teller effect and electron–phonon interaction in the 4 T2g excited state of Cr3+ ion in K2 LiAlF6 crystal N.M. Avram a,b,∗ , M.G. Brik c , C.N. Avram a , I. Sildos c , A.M. Reisz a a
Department of Physics, West University of Timisoara, Bd. V. Parvan 4, Timisoara 300223, Romania
b
Academy of Romanian Scientists, Splaiul Independentei Nr. 54, 050094 Bucharest, Romania
c
Institute of Physics, University of Tartu, Riia 142, 51014 Tartu, Estonia
article
info
abstract
Article history: Received 22 July 2009 Accepted 13 August 2009 by E.L. Ivchenko Available online 19 August 2009
The electron–phonon coupling and Jahn–Teller effect in the first excited state 4 T2g of Cr3+ in K2 LiAlF6 crystal (which has the potential for tunable solid-state lasers applications) are considered. The force constants for the a1g and eg normal modes are calculated with the FG matrix method for the octahedral [CrF6 ]3− cluster and the Huang–Rhys factors Sa1g and Seg from the experimental value of the total Huang–Rhys factor
PACS: 61.72.S71.70.Ch 71.70.Ej 63.20.Kd
and Stokes shift. Using these data, the changes of the metal–ligand bond lengths 1xeq , 1yeq = 0.072 Å (equatorial expansion) and 1zeq = −0.015 Å (axial compression) were estimated. The Jahn–Teller stabilization energy has been evaluated from the contour plot of the 4 T2g electronic state potential energy surface. The obtained results are compared with those for other fluorides; the common trends are found and discussed. © 2009 Elsevier Ltd. All rights reserved.
Keywords: C. Point defects D. Crystal and ligand fields D. Electron–phonon interaction D. Optical properties
1. Introduction The K2 LiAlF6 fluoride belongs to a wide class of A2 BMX6 materials, where A (Cs, K) and B (Na) are monovalent cations, M is a trivalent cation (Al3+ , In3+ , Y3+ , Sc3+ , Ga3+ ) and X is a monovalent anion (F− , Cl− , Br− ). After doping, the Cr3+ ions substitute for octahedral coordinated trivalent metal ions without any need for charge compensation. Apart from their potential as laser materials, chromium-doped elpasolites are model systems for investigating crystal field and vibronic coupling effects, since the deformations of the crystal lattice after doping can be neglected and the symmetry of the formed [CrF6 ]3− complex is close to Oh . The optical properties of the Cr3+ -doped elpasolites are discussed in [1]. The electronic states of main interest are the 4 A2g and 4 T2g states (since the position of the 4 T2g state basically determines the crystal field effects and the overall appearance of the emission and absorption spectra). All these states are derived from the free-ion 3d3 ground configuration of the Cr3+ ion. In this paper we consider
∗ Corresponding author at: Department of Physics, West University of Timisoara, Bd. V. Parvan 4, Timisoara 300223, Romania. Tel.: +40 722331696; fax: +40 256592108. E-mail address: [email protected] (N.M. Avram). 0038-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2009.08.020
the electron-vibrational interaction in the K2 LiAlF6 :Cr3+ crystal, since this system has a real potential for the tunable solid-state laser applications. Based on the data available in the literature on the frequencies of the normal vibration of the CrF36− complex and Huang–Rhys factors, we estimate the force constants for these normal modes, the magnitudes of the normal modes’ displacements, the changes in the ‘‘Cr–F’’ chemical lengths and the Jahn–Teller stabilization energy for the above mentioned system. Main attention is focused on the geometry of the first excited electronic state 4 T2g , influenced by its coupling with the total symmetric a1g and doubly degenerated eg normal vibrations of the CrF36− complex due to the dynamic Jahn–Teller effect. Such an analysis permits us to estimate the changes of the chemical bond lengths originating from the combined effect of the a1g and eg normal modes, plot the potential energy surface as a function of the ionic displacements, and deduce the value of the Jahn–Teller stabilization energy. 2. Experimental support Spectroscopic studies of the K2 LiAlF6 :Cr3+ system were performed in [2]. From the experimental spectra of absorption and luminescence the following values of the crystal field strength Dq and Racah parameters B and C (all in cm−1 ) were deduced for Cr3+ : 1608, 710, 3341, respectively. The lowest Cr3+ energy levels with
N.M. Avram et al. / Solid State Communications 149 (2009) 2070–2073
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restoring force Ki 1Qi at the distorted geometry 1Qi = −Fi /Ki with 1Qi denoting the difference between the equilibrium and distorted coordinates. This distortion lowers the energy of the electronic state by an amount Ei = (1/2)Ki (1Qi )2 comparing to the equilibrium position. Only distortions along the a1g , eg , t2g octahedral modes are important for the 4 T2g electron state. The Hamiltonian (1) can be rewritten in basis consisting of the real 3d orbitals [4]: HVIB
Fig. 1. Crystal structure of K2 LiAlF6 [3]. K ions are shown by black spheres, Li and Al ions are at the centers of octahedrals composed of fluorine ions.
these parameters are as follows (all in cm−1 ): 4 A2g − 0, 2 Eg − 15 219, 2 T1g − 15 851, 4 T2g − 16 080, 2 T2g − 22 622, 4 T1g − 22 994, Such a sequence of the energy levels (together with the Dq/B ratio) shows that the Cr3+ experiences strong crystal field in this host. The frequencies of the a1g and eg normal modes are (in cm−1 ) 558 and 469, respectively [2]; the Stokes shift (the difference between the maxima of the lowest emission and absorption bands) is 2420 cm−1 , and the Huang–Rhys factors for these modes are Sa1g = 1.3 and Seg = 1.0 [2]. The considered crystal, at low temperature, has a cubic structure, belonging to the Fm3m space group, having the lattice parameters a = 7.842 Å, with four unit formula in unit cell [3].The K2 LiAlF6 structure, as shown in Fig. 1, represents a sequence of alternating octahedral made of six fluorine ions centered at Li and Al ions [3]. 3. Method of calculations The ground electronic state of trivalent chromium ions at the octahedral site is 4 A2g [4]. Due to the coupling of the electronic state 4 T2g (the first excited spin-quartet) and the Jahn–Teller active modes a1g , eg and t2g of the host matrix and the different bonding properties in the ground and excited states, they may have different geometries (in other words, different coordinates of the atoms in an octahedral complex with Cr3+ at the center), what is revealed in the shift of the potential surfaces of the considered electronic states. The effective Hamiltonian for Cr3+ ion, which describes the molecular vibrations of the CrF36− octahedron is given by the following equation [5]: HVIB =
X P2 1 ∂V i + Ki Qi2 + Qi . 2µi 2 ∂ Qi 0 i
(1)
The first two terms (expressed in terms of the ligands’ masses µi , their momenta Pi , force constants Ki for the corresponding normal modes Qi ) transform the electronic levels into potential energy manifolds in the coordinates of the √ octahedral normal modes with vibrational frequencies ωi = Ki /µi , and the complete wave functions in the Born–Oppenheimer approximation can be written as a product of an electronic and a vibrational parts. The third term can be understood as a force, which acts along the vibrational mode associated with the electronic state denoted by Γ :
∂V Γ Fi = − Γ ∂ Qi 0
(2)
where the subscript means that the derivative is to be found at the equilibrium configuration. This force is balanced by the harmonic
X P2 1 i 2 = + Ki Qi 2µi 2 i
* + Q
∂V a1g
+ 4 T2g
4 T2g 0
∂ Qa1g 0
∂V 4
+ 4 T2g
∂ Q T2g eg √ 3 2 Qegθ − 2 Qegε 1
×
0
2
0
Qegθ + 0
* + 0
∂V
+ 4 T2g
4 T Qt2gζ
∂ Qt2g 2g Q
t2gη
0 0 Qa1g
0
1
0 Qa1g 0
0
√ 3 2
Qegε
Qt2gζ 0 Qt2gξ
0 −Qegθ
Qt2gη Qt2gξ . 0
(3)
The reduced matrix elements in front of all matrices are the constants of the electron-vibrational interaction and can be expressed in terms of the crystal field strength Dq and equilibrium distance R0 as follows [6,7]:
∂V 4 25
T2g T2g = − √ Dq,
∂ Qeg R0 3
* +
∂V 50
4 4 T2g Dq, = −√
T
∂ Qa1g 2g 6Ro
+ * √
∂V 18 2 1 5
4 4 T2g =− Dq − ,
T
∂ Qt2g 2g 7R0 η 9
2 2 ! −1 r R0 3 3 − 4eπ /eσ
η= = . 5 1 + eπ /eσ r4
4
The eπ and eσ entries in the last equation are the parameters of the angular overlap model of crystal field and are described in [7]. The magnitudes 1Q of the ligands displacements from the equilibrium position along the a1g and eg active modes can be estimated from the following expression [8]:
|1Qi | =
2Si h¯ ωi Ki
1/2
,
(4)
where S is the Huang–Rhys factor, K is the force constant, and h¯ ω is the energy of the corresponding normal mode. The force constants K for an octahedral complex can be easily evaluated using the FG method [9]. To facilitate the further consideration, the coordinate system in the (Qε , Qθ ) space can always be chosen in such a way, that the potential minimum of the considered 4 T2g component (anyone from the ξ , η, ζ set) lies on the Qθ axis (this means, no distortion takes to place along the Qε axis). Then it is possible consider the 1Qeg eq values as corresponding to 1Q eg θ only, eq
whereas 1Q eg ε is zero. Using the transformation matrix given eq below [10], the changes of 1Q can be easily expressed in terms
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N.M. Avram et al. / Solid State Communications 149 (2009) 2070–2073
Table 1 The force constants Ki , magnitudes of the normal mode displacements, changes in the chemical bond lengths and Jahn–Teller energy EJT for several crystals doped with Cr3+ ions.
K2 NaScF6 [11] Cs2 NaYF6 [11] K2 LiAlF6 (this work)
1Qa1g (Å) 1Qeg (Å)
a1g mode
eg mode
h¯ ω (cm−1 ) K (N/m) S
h¯ ω (cm−1 ) K (N/m) S
542 501 558
449 402 469
241 206 255
0.9 3.2 1.3
165 133 180
Combined result of both a1g and eg modes
1x, 1y (Å) 1z (Å)
1.3 0.64 1.0
0.09 0.18 0.11
−0.11 −0.09 −0.10
0.07 0.09 0.072
−0.03 0.02 −0.015
Result of a1g mode only
EJT (cm−1 )
1x, 1y, 1z (Å) 0.037 0.072 0.043
584 257 469
0.15
(Cr3+ – F–(z)), Å
0.1
0.05
z0
–0.05
–0.1
Fig. 2. Distortion of the [CrF6 ]3− complex in the 4 T2g excited state with respect to the ground state (direction of displacements are indicated by arrows; their magnitudes are given in Å).
of the chemical bond length changes 1x, 1y, 1z (which can be measured, in principle):
r 1x 1y 1z
!
2
3 r 1 2 = 2 r 3 2 3
r −
1
r3 1
− r 3 4 3
−1 1Qa1g 1 1Qeg θ . 1Q eg ε
(5)
0
After the force constants and amplitudes of the ionic displacements are found, it is possible to draw the potential energy surface of the 4 T2g excited electronic state. In the harmonic approximation, this energy is described by the following expression: V =
1 2
Ka1g 1Qa1g − 1Qa1g ,eq
2
1
+ Keg 1Qeg − 1Qeg ,eq 2
2
.
(6)
Using the transformation inverse to that one in Eq. (5), it is possible to visualize the potential energy dependence on the changes of the chemical bond lengths 1x, 1y, 1z, and, estimate the Jahn–Teller stabilization energy. 4. Results of calculations and discussion Consistent application of all given above equations to the K2 LiAlF6 :Cr3+ crystal allowed to obtain the values of the electronvibrational interaction and interionic distances changes collected in Table 1. We note that vibronic coupling constants of the 4 T2g electron state with triple degenerate vibrations of host lattice is very small compare with that of total symmetric and respective double degenerate vibration, so we neglect former one. In Table 1 we give only the parameters regarding vibronic coupling of the 4 T2g electron state of Cr3+ with total symmetric and respective, double
–0.15 –0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 (Cr3+ – F–(x,y)), Å Fig. 3. A contour plot of the harmonic 4 T2g potential energy surface (only one of the three parts is shown) for the K2 LiAlF6 :Cr3+ system as a function of changes of the Cr3+ – F− (x, y) and Cr3+ –F− (z) chemical bonds lengths. The energies of individual contours are given in hundreds of wave numbers. The open circle at the origin corresponds to the equilibrium position of the ground 4 A2g potential energy surface; the black square indicates the equilibrium position of the 4 T2g potential energy surface shifted with respect to the ground state as a combined result of the a1g and eg normal vibrations. The black circle shows the hypothetical position of the 4 T2g potential energy surface minimum if there were no eg normal vibration (i.e. in the absence of the Jahn–Teller distortion). The value on the potential energy surface of the 4 T2g state at this point (between 400 and 500 cm−1 from the figure) corresponds to the Jahn–Teller energy for the considered complex.
degenerate vibration of the host matrix. For comparison, the analogous values for Cr3+ ions in two other fluoroelpasolites [11] are also given. As seen from this Table 1, the force constants for the K2 LiAlF6 :Cr3+ crystal are the greatest among all presented in the Table 1, which is in line with the greatest frequencies of the a1g and eg normal modes for this host. Generally, the increasing order of the force constants in the Cs2 NaYF6 → K2 NaScF6 → K2 LiAlF6 series (Table 1) is in line with decreasing order of the lattice constants in the same direction (9.075 Å for Cs2 NaYF6 [13], 8.482 Å for K2 NaScF6 [14], and 7.842 Å for K2 LiAlF6 [3]), which means that moving ions closer to each other makes chemical bonds harder. Regarding the magnitudes of the normal displacements and Jahn–Teller stabilization energies, the trend is not that straightforward, which can be explained by interplay of crystal field effects (like splitting of the orbital triplets etc), which reveal themselves in different Stokes shifts for each of the considered materials. However, a quick glance at the deformation magnitudes shows that in all these crystals the CrF36− complex is experiencing an equatorial expansion in the xy plane and an axial compression along the z-axis in K2 NaScF6 and K2 LiAlF6 crystals, whereas in Cs2 NaYF6 the complex undergoes a slight elongation along the z axis. Similar results are obtained by first principle calculations of parameters of electron-vibration interactions in elpasolite crystals
N.M. Avram et al. / Solid State Communications 149 (2009) 2070–2073
doped with Cr3+ ions [12]. Fig. 2 visualizes the character of the Jahn–Teller deformation of the CrF36− cluster in K2 LiAlF6 . Finally, Fig. 3 shows a cross-section of one of three parabolas of potential energy arising from the 4 T2g state after it is split due to the Jahn–Teller effect. Such a contour plot allows for a direct estimation of the Jahn–Teller stabilization energy, if the following circumstance is taken into account: if only interaction with the fully symmetric a1g mode is considered, then the 4 T2g potential surface will be only shifted (with equal changes of 1X , 1Y , 1Z shown in Table 1), without any splitting (the splitting is actually produced by the eg vibrational mode, which effectively lowers the symmetry). The point of this hypothetical minimum is shown in Fig. 3 by an open circle, and the value of the potential energy at this point (which lies between 400 and 500 cm−1 in the Figure) is exactly the Jahn–Teller stabilization energy. Numerical estimations with Eq. (6) and all calculated force constants and displacements yield the value of 469 cm−1 for K2 LiAlF6 . 5. Conclusions The vibronic interaction between the 4 T2g excited state of Cr3+ and the a1g , eg normal vibration modes of the host matrix modifies the geometry of the 4 T2g state relative to the ground state. Using the harmonic approximation for vibrations and linear approximation for the electron-vibrational interaction, we have calculated the equilibrium displacements of the 4 T2g potential surface minimum from the ground state along the a1g and eg Jahn–Teller active modes The Ki force constants for the a1g and eg modes are calculated with the FG matrix method and are equal to 255.3 and 180.3 N/m. The Huang–Rhys factors Sa1g and Seg are obtained from the experimental value of the total Huang–Rhys factor and Stokes shift. Using these data and frequencies of the normal modes, we
2073
estimated 1Qa1g = 0.10 Å and 1Q eg θ = 0.10 Å. These eq eq values have been converted into the changes of the metal–ligand bond lengths 1xeq , 1yeq = 0.072 Å and 1zeq = −0.015 Å. The Jahn–Teller stabilization energy EJT = 469 cm−1 has been estimated from the contour plot of potential surface of the 4 T2g electronic state. The obtained results are compared with those for other fluorides [9]; a few common trends are discussed and related to the corresponding changes of the lattice constants of the considered hosts. Acknowledgement N.M. Avram appreciates the financial support from Romanian National Authority for Research in the frame of CEEX Program (Project Nr. 51/2006/MATNANTECH). References [1] P.A. Tanner, Chem. Phys. Lett. 388 (2004) 488. [2] M.A.F.M. da Silva, R.B. Barthem, L.P. Sosman, J. Solid State Chem. 179 (2006) 3718. [3] J. Graulich, S. Drueeke, D. Babel, Z. Anorg. Allg. Chem. 624 (9) (1998) 1460. [4] S. Sugano, Y. Tanabe, H. Kamimura, Multiplets of Transition-Metal Ions in Crystals, Academic Press, New York, 1970. [5] M.D. Sturge, Advanced in Solid State Physics, vol. 20, Academic Press, New York, 1967, 91–211. [6] K. Wissing, J. Degen, Mol. Phys. 95 (1998) 51. [7] T. Schönherr, Topics Current Chem. 191 (1997) 87. [8] T.C. Brunold, H.U. Güdel, Inorganic Electronic Structure and Spectroscopy, in: Methodology, vol. I, Wiley, New York, 1997. [9] K. Venkateswarlu, S. Sundaram, Z. Phys. Chem. Neue Folge 9 (1956) 174. [10] R.B. Wilson, E.I. Solomon, Inorg. Chem. 17 (1978) 1729–1736. [11] N.M. Avram, M.G. Brik, J. Mol. Struct. 838 (2007) 198. [12] M.G. Brik, N.M. Avram, J. Mol. Struct. 838 (2007) 193–197. [13] A. Vedrine, J.P. Besse, G. Baud, M. Capstan, Rev. Chim. Miner. 7 (1990) 593. [14] B.F. Aull, H.P. Jenssen, Phys. Rev. B 34 (1986) 6647.