Electronic structure of CeF3 and TbF3 by valence-band XPS and theory

ARTICLE IN PRESS Journal of Physics and Chemistry of Solids 70 (2009) 1302–1311

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Electronic structure of CeF3 and TbF3 by valence-band XPS and theory K. Klier a,, P. Nova k b, A.C. Miller c, J.A. Spirko a, M.K. Hatalis d a

Department of Chemistry, Lehigh University, Bethlehem, PA 18015, USA Institute of Physics of ASCR, Cukrovarnicka 10, 162 53 Prague 6, Czech Republic The Scienta Laboratory of the Materials, Research Center of Lehigh University, USA d Department of Electrical Engineering and Computer Science, Lehigh University, Bethlehem, PA 18015, USA b c

a r t i c l e in f o

a b s t r a c t

Article history: Received 12 November 2008 Received in revised form 29 May 2009 Accepted 21 July 2009

Electronic structures of the rare earth trifluorides CeF3 (P3 c1) and TbF3 (Pnma) were examined by highresolution valence-band X-ray photoelectron spectroscopy (VB-XPS) and all-electron periodic-crystal DFT theory including the spin-polarization (SP) combined with spin–orbit (SO) coupling using a secondvariational treatment. Calculations using the Perdew–Burke–Ernzerhof (PBE) functional and the LDA+U method were carried out and compared. The results show that a complete analysis does require a full DFT–SP–SO treatment to obtain a quantitative account for the observed VB-XPS spectra, with an additional insight of the theory with regard to the nature of the topmost orbitals, and the bonding–antibonding character of orbitals within the VB and sub-VB levels. The band structure at the bottom of the conduction band (BCB) shows a strong dispersion in TbF3 but not in CeF3, predicting photoconductivity in TbF3. & 2009 Elsevier Ltd. All rights reserved.

Keywords: A. Inorganic compounds C. Ab initio calculations C. Photoelectron spectroscopy D. Electronic structure D. Magnetic properties

1. Introduction Trivalent rare earth (RE) phosphors are widely used in practical applications involving lighting, lasers, and detectors of many forms of radiation. The present research is motivated by the potential of various RE-containing inorganic and organic complexes and compounds as viable components of LED displays, including our continuing research effort in fabrication of flexible displays [1]. Recently we have reported on the synthesis of a mesoporous silica derivatized with propyl-sulfonic groups exchanged with Ce(III) ions to form novel intraporous complexes (Si–CH 2 CH 2 CH 2 SO 3 ) 2 Ce(III) OH(H 2 O) x [Ce(III)SBA and Ce(III)SBA–(H 2 O) 14 ], which displayed intense purple luminescence [2]. The 4f-electron of Ce(III) was located at the top of the valence band by XPS, and a theoretical account was provided for this observation using the all-electron periodic DFT FP-LAPW method. During this research, it became apparent that the theoretical approach would probe into far greater detail of the electronic structure if well-defined simple compounds were subject to a similar analysis. Presently we chose CeF 3 , a single 4f-electron compound, and TbF 3 , an eight 4f-electron compound, to examine whether the theory accounts correctly and quantitatively for exchange-correlation (XC) effects, spin-polarization (SP),

 Corresponding author.

E-mail address: [email protected] (K. Klier). 0022-3697/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2009.07.024

spin–orbit (SO) coupling in the valence band (VB), and the crystalfield (CF) effects. A comparison of the theoretical results with the VB-XPS spectra has now led to the conclusion that a satisfactory interpretation of the VB-XPS spectra is obtained only if all four of these effects are taken into play, as they are of comparable magnitude. In addition, a combination of SP and SO allows the prediction of magnetic moments, and analysis of the band structure in the k-space yields information on effective masses of the current carriers in the VB and the conduction band (CB).

2. Experimental 2.1. Materials and procedures Commercial materials used in this work were CeF3 99.9% Alfa Aesar CAS #7758-88-5, stock #21118, Lot #F16R032, and TbF3 99.9% Alfa Aesar CAS #13708-63-9, stock #13657, Lot #B08M33. Both these compounds are highly stable, insoluble in water, and display sharp X-ray powder patterns consistent with hexagonal CeF3 (Tysonite, space group 165 P3 c1) and orthorhombic TbF3 (space group 62 Pnma). The trivalent Ln(III) (Ln ¼ Ce or Tb) have electronic configurations [Xe]4f1 (Ce) with a single 4f-electron on top of the closed [Xe] shell and [Xe]4f8 (Tb) with one electron on top of the half-filled [Xe]4f7 shell. Characteristic optical properties of these white materials include optical bandgaps in the UV (4.13 eV for CeF3 and 5.4 eV for TbF3) and luminescence in the

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visible region (purple emission at 364 nm for CeF3 and green emission with maximum at 544 nm for TbF3). 2.2. XPS analysis High-resolution XPS analyses, recording the Ln3d, Ln4d, Ln5p, F1s and the valence band (VB) spectral regions, and possible impurities indicated by C1s and O1s emissions were carried out using the Scienta ESCA-300 spectrometer [3]. An electron flood gun was used to neutralize charging of the insulating samples, as in our earlier work. The XPS data were analyzed using the CasaXPS processing software [4]. 2.3. Visible and UV diffuse reflectance spectroscopy (VIS–UV-DRS) for the band-to-band transitions The Cary 500 spectrophotometer equipped with a diffuse reflectance (DRS) attachment with a PbSe NIR detector at the periphery of the Halon-coated integration sphere was used for a rapid determination of the band-to-band transitions. The Halon standard was also used as a reference, and corrections to the Schuster–Kubelka–Munk (SKM) for the ratio of absorption and scattering coefficients based on the albedo of isotropic scattering in turbid media were made following the procedure outlined by one of the authors [5]. The range 10,000–45,000 cm1 was covered in the same experimental setup reported earlier [2]. 2.4. Theory The computational platform used for all-electron calculations of the RE crystals was the all-electron periodic DFT-FP-LAPW theory imbedded in the Wien2k-9.1 package [6]. In this set of codes, the crystal space is divided into atomic spheres in which the eigenfunctions are described by spherical harmonics, and interstitial space with planewaves that match amplitudes and derivatives at the interface with the atomic spheres. Reciprocal space is characterized by a user-defined set of k-vectors, presently 16 k-points. The currently used options included the following: optimization of atomic positions within the unit cells using the PORT routine, the (on/off) LDA+U method, spin-polarization (SP), and spin–orbit (SO) coupling using the second-variational method [7]. Calculations with inclusion of the SP and SO options are in Wien carried out step-wise: First, in the lapw1c code, the SP eigenvalues and eigenvectors are generated by solving the Kohn–Sham equations by diagonalizing the matrix of the system Hamiltonian in the basis of eigenvectors based on the density functionals for the spin-up and spin-down manifolds, with the exchange-correlation potential generated by lapw0. Presently we used (A) the PBE functional [8], which entails the generalized gradient approximation, and (B) the LDA functional with the Coulomb and exchange parameters U and J as user-defined in the Wien2k-9.1 package. In the subsequent step, SO coupling is added in the lapwso code, which computes new eigenvalues and eigenvectors for the following steps, which entail the generation of valence charge densities, core states, and re-entering the result into the next scf cycle till convergence is reached. The Hamiltonian for the SO step has the form ! ~~ ‘ 1 dV s l 0 ^ ; H SO ¼ 2Mc2 r dr 0 0 where sx, sy, sz are the Pauli-spin matrices, l is the angular momentum vector, M the relativistically enhanced mass of electron, and V the potential within the atomic sphere. Applica^ SO couples the spin-up and spin-down eigenvectors tion of H calculated in the lapw1 step. The limitations are in that the SO

1303

procedure applies only to the space inside atomic spheres, not the interstitial space, and that only ‘‘small’’ spin–orbit interactions are calculated. However, none of these limitations has been encountered in the present study, primarily for two reasons: the 4f orbitals of the rare-earth ions have a small radius and spin–orbit coupling of 2–3 eV, comparable to experimental findings, is shown to be well accounted for by the present method.

3. Results and discussion 3.1. Structures of CeF3 and TbF3 These two trifluorides are representative of two basic structures in the entire LnF3 series [9]: the trigonal P3 c1, here represented by CeF3, dominating in the early Ln series, and the orthorhombic Pnma, represented by TbF3, in the later part of the series. The two structures are shown in Fig. 1 alongside the valence-band XPS spectra. The unit cell dimensions were taken from crystallographic data, a ¼ b ¼ 0.7110 nm, c ¼ 0.7270 nm, a ¼ b ¼ 901, g ¼ 1201 for CeF3 (P3 c1), and a ¼ 0.6513 nm, b ¼ 0.6949 nm, c ¼ 0.4384 nm, a ¼ b ¼ g ¼ 901 for TbF3 (Pnma). In the unit cell of the CeF3 P3 c1 structure, there are 6 equivalent Ce atoms (Ce(1)) and 3 sets of unique F atoms with multiplicities 12 (F(2)), 4 (F(3)), and 2 (F(4)). Of these, the smallest Ce–F distances are with fluorine atoms from the least populated subsets F(3) and F(4) and the largest Ce–F distances are with the higher populated atoms F(2). This has consequences for observed XPS binding energies, the electrostatic contributions for which are the largest for fluorine emissions (F2s and F2p here) for the F atoms closest to the Ce(III) cation and smaller for the more distal F atoms. The unit cell of the TbF3 Pnma structure contains 4 equivalent Tb atoms (Tb(1)) and 2 sets of unique F atoms with multiplicities 4 (F(2)) and 8 (F(3)). The Tb–F distances are not very different for F(2) (0.2330 nm) and 6 of the 8 atoms F(3) (0.2327 nm), which then gives rise to nearly equal electrostatic ‘‘bond’’ energies. The longrange Madelung interactions are also evaluated in the calculations below, but the above qualitative electrostatic argument provides a good guideline for rationalizing the observed XPS level shifts due to electrostatic interactions. 3.2. XPS spectra of the valence and sub-valence band regions of CeF3 and TbF3 The VB and sub-VB XPS spectra of the two trifluorides are shown in Fig. 1 alongside the structures. The narrow peaks reflect the ionic character of these compounds, but they display a fine structure, which is a consequence of the combined SP, SO, and CF effects. The Ce4f emission appears some 2.4 eV above the top of the valence band (TVB), unlike in the Ce-SBA material in which the separation was smaller and overlap with the main VB was significant [2]. The ‘‘main’’ VB in CeF3 is F2p, which includes the SO and SP splittings for each of the 3 sets of the F atoms. The Ce5p emissions are clearly split into the 5p3/2 and 5p1/2 bands with intensity ratio 2:1. The F2s band shows an asymmetric structure, which again is due to the set of unique, non-equivalent fluoride ions. These assignments are confirmed by the theory below. The Tb4f spectrum is seemingly simpler but actually involves overlaps between parts of the spin-polarized Tb4f manifold, one that is separated above the ‘‘main’’ TVB and another that falls within the ‘‘main’’ F2p VB. Furthermore, the Tb5p emissions overlap with the F2s emissions, giving rise to an apparent ‘‘doublet’’ at 20–30 eV. An earlier paper involving XPS analysis of rare-earth trihalides [10] did report the valence band spectra of CeF3 but not of TbF3, but a theoretical analysis of the observed spin–orbit split

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<00.1> 500 Ce5p3/2

400 F2s

CPS

300

Ce5p1/2 Ce5s F2p

200

Ce4f

100

0 -30

<11.0>

-20

-10

0

Binding Energy, eV

<100> 400 7 F2p, Tb4 f↑

300 F2s and Tb5p

200 Tb4f↓

100

0 -30 <010>

-20

-10

0

Binding Energy, eV

Fig. 1. Structures of the hexagonal P3 c1 CeF3 and orthorhombic Pnma TbF3 showing the different motifs of an 8-coordinated Ce (top left) and Tb (bottom left) and the corresponding VB XPS spectra referenced to the top of the VB at the Fermi level (right). Ce is labeled red and Tb green. The structure files of these Ln(III) trifluorides are given in the appendix (for interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

emissions was not provided and the spin-polarization effects were not included in the qualitative discussion of the VB spectra.

3.3. Theoretical calculations and results

agrees with the theoretical calculations for P3 c1 but not with P63cm. Another important property is the magnetic moment, yielding the relative theoretical stabilities of ferromagnetic and antiferromagnetic phases. And further, the theory decides between local and delocalized opto-electronic phenomena.

The purpose of the present calculations is to provide a framework for a quantitative interpretation of the XPS spectra in the valence band region, using the two well-defined compounds, and apply the same level of the theory to more complex systems, e.g. exemplified in our earlier study of the Ce-SBA material. Theory, in combination with spectroscopy, can discriminate between stable and metastable structures, predict magnetic properties, and assess the degree of delocalization of orbitals involved in spectroscopic transitions such as PL and EL. One example is the issue of the trigonal P3 c1 vs. P63cm structures of CeF3 and LaF3, which both had been proposed in the crystallographic literature based on very similar sets of data [11]. The present calculations yielded the total energy per unit cell 109984.858602 Ry for P3 c1 and 109984.534557 Ry for P63cm, favoring P3 c1 by a significant energy difference of 0.735 eV per Ce(III) ion, even though small displacements of atoms from crystallographic positions relaxed back to either of the two of these structures. Consistent with this, the XPS VB spectrum

3.3.1. Calculations using the PBE functional The DOS of the VB of the two trifluorides calculated by method (A) are shown in Figs. 2, 3 (CeF3) and Fig. 4 (TbF3). The spin-up, spin-down, and total DOS are represented in Figs. 2 and 4 in a Gaussian-broadened fashion in order to match the line broadening in the XPS experiment. Spin–orbit (SO) coupling in the VB is included using the second-variational treatment [7]. Without SO, the theory fails to account for the split in the Ce5plevels in CeF3 and for the more complex interplay of SP and SO in TbF3. Inclusion of SO corrects this deficiency. This results in a 2.55 eV separation and 2:1 ratio of the intensities of the two Ce5plevels in CeF3, compared with the observed XPS separation of 2.97 eV—a satisfactory agreement, as well as an overall agreement, including that for TbF3. The computational parameters specific to the Wien code are listed in Appendix I—Supplemental Information. Since the SO calculations often require inclusion of highly excited eigenstates, we have also examined the effects of increasing the EMAX value from 2 to 10 Ry

ARTICLE IN PRESS

-20

-10 Energy, eV

15 10

F2p*0.0478

Ce4f*0.0781

Ce5*0.439

-20

-10

0

10

Fig. 4. Theoretical DOS of CeF3 modified by RSF from Ref. [4].

10

20

15 Tb4f7/2,F2p Tb4f5/2

EF

10

SO and SP mixed Tb5p, F2s

5 (-19.46) -19.86

(-24.67) -23.70

20 DOS

-30

BE, eV

DOS

and spin-orbit coupling Red: spin-up Blue: spin-down Black: total dos Dotted: spin-up minus spin-down

25

EF

0

5

-30

0

-20

-10

0

10

Energy, eV

-5

Fig. 5. Theoretical DOS of TbF3 modified by RSF from Ref. [4]. In particular, the Tb5p and F2s DOS at 18 to 28 eV that shows three peaks at a higher resolution are smeared to nearly equal intensity doublet when the theoretical partial DOS are modified by the relative sensitivity factors given in text. The energy and relative intensities of this doublet are in an excellent agreement with experiment, cf. Fig. 1.

-10 -15 -20

F2s*0.21

-40

0

(-4.98) -4.23

VB structure of TbF3 with spin polarization

30

2

Ce4f

Fig. 2. Gaussian-broadened theoretical DOS of CeF3. The Fermi level EF is marked by a dashed vertical line, showing partial occupancy of the Ce orbitals with one 4felectron occupying the spin-up levels while the spin-polarization shifts the spindown levels to higher energies by 0.75 eV, leaving them empty. The individual bands are labeled with the emission levels, which are well separated from one another. Numerical values of theoretical energies are given as vertical labels, accompanied by experimental XPS energies in parentheses.

35

4

0

Ce5p1/2-5p3/2 F2p

-30

40

6

EF

Ce5s*0.23

DOS, RSF-modified

-5.89,40.98 (-4.89,19)

Ce5s F2s

1305

8

EF

-16.91,18.47 (-15.92,45)

(-18.80,28) -19.52,10.92

(-25.83.33) -24.98,24.49

55 50 45 40 35 30 25 20 15 10 5 0 -5 -10 -15 -20 -25 -30 -35

VB structure of CeF3 with spin polarization and spin-orbit coupling Red: spin-up Blue: spin-down Black: total dos Dotted: spin-up minus spin-down

-35.03,9.88 (-34.55,23)

DOS

K. Klier et al. / Journal of Physics and Chemistry of Solids 70 (2009) 1302–1311

XPS peak maxima in parentheses

-25 -30

-20

-10 Energy, eV

0

10

Fig. 3. Gaussian-broadened theoretical DOS of TbF3. The Fermi level EF is marked by a dashed vertical line, showing partial occupancy of the Tb orbitals with one 4felectron occupying the spin-down levels while the spin-polarization shifts the spin-up levels to lower energies, where they overlap with the F2p ‘‘main’’ valence band. Spin-polarization is large because of exchange stabilization in the sevenelectron half-filled shell. The band just below EF is Tb4f1 spin-down, that at 4.23 eV is F2p overlapping with Tb4f7 spin-up, and the bands at 19.86 and 23.70 eV ensue from an overlap between F2s and Tb5p levels.

and found no further increase of the SO splitting exemplified by the separation of the Ce5p3/2–5p1/2 levels, 2.55 eV. The theoretical treatment deals with the initial states only and assumes common final states as in Ref. [12]. Therefore, the DOS maxima and structure are not absolute, nor are the XPS DOS, which are referenced to the Fermi level of the XPS instrument. However, an excellent agreement of the relative DOS from the

present theory and experiment is shown in Fig. 10, yielding an overall linear regression coefficient R ¼ 0.997 and slope 0.986 for all the data from both compounds. 3.3.1.1. Bandgaps. While the theory underpredicts the magnitude of the bandgap, as is the well-known feature of the DFT calculations at this level, the difference in magnitudes between CeF3 (3.22 eV) and TbF3 (4.22 eV), 1.0 eV, is in good agreement with the experimental values, 4.13 eV [13] and 5.4 eV [14], with the difference of 1.27 eV. 3.3.1.2. Intensities in XPS spectra and theoretical DOS. The observed XPS intensities are related to the population of levels in partial DOS through the relative sensitivity factors (RFS), which involve ionization cross-sections of atomic levels that constitute the crystal orbitals in the VB. Multiplication of the partial DOS by the RFS of the particular atomic orbitals gives rise to modified orbital population, which yields

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F

F

Ce↑ ↑

Ce↑

0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3

Fig. 6. Valence band electron density in CeF3 in the energy range from 0.5 eV to the top of the valence band in the zx plane: Top left, the Ce(III) ions showing a di-trigonal shape are surrounded by the spherical fluoride ions; top right, net spin density is located at the Ce ions; bottom, 3-dimensional plot of net electron density showing nonspherical symmetry of the 4f orbitals on the Ce ions.

modified DOS suited for comparison of peak intensities with experimental spectra. We have used RFS values from the CasaXPS file system tables [4], here given in parentheses for each emission: Ce5s (0.23), Ce5p (0.66), Ce5p3/2 (0.439), Ce5p1/2 (0.221), F2s (0.21), F2p (0.0478), F2p3/2 (0.0317), F2p 1/2 (0.0161), Ce4f (0.139), Ce4f 7/2 (0.0781), Ce4f5/2 (0.0609): Tb5s (0.23), Tb5p (0.804), Tb5p3/2 (0.539), Tb5p1/2 (0.265), Tb4f (1.97), Tb4f7/2 (1.10), Tb4f5/2, and (0.867). A particular feature worth mentioning is the ratio of sensitivity factors for the Tb5p and F2s emissions, roughly 4-fold in favor of Tb5p, which is responsible for the apparently equal XPS intensities in the overlapping region of the two emissions. An overall satisfactory account for relative intensities was obtained, as is evident from Figs. 4 and 5.

3.3.1.3. Character of the VB 4f orbitals from electron density plots. A full analysis of the 4f orbitals is based on methodology presented in Refs. [15] and [16]. Here we summarize the computational results in graphical forms, which exemplify the salient features of the 4f orbitals in the two fluorides, namely their localization, non-spherical shape, and antibonding character. The VB 4f orbitals are the source

for excitations resulting in photoluminescence of the rare-earth ions planted in various inorganic, organic and hybrid materials. In many such materials, the 4f-electron density falls within the VB or appears as a barely discernible shoulder at the top of the VB. In the present trifluorides, some of the 4f levels are well separated from the VB while others fall, in the case of Tb(III), within the F2p band (cf. Fig. 3). While the effects of SP and SO in energy spectra are well accounted for by the present theory, it is also useful to display the spatial distribution of the 4f orbitals in a way that reveals their localization and directionality in relation to the positions of the ligands, the fluoride ions. Fig. 6 displays the valence-band electron density plots with spin-up, spin-down, and net spin, showing lack of overlap of densities between the Ln(III) cations and the fluoride ligands, and the localization of the net spin at the Ln(III) cations. The Ce 4f orbital in CeF3 avoids the fluoride ligands, hence displaying an antibonding character. The closed-shell fluoride ligands are spherically symmetric, as expected from their ionic character. The Tb 4f orbitals in TbF3 are split into two spin-polarized manifolds: the spin-down 4f-electron occupies the highest energy

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Tb↓ ↓

Tb↑

F

F

F

F

1307

F

F

F

F

Tb↑

Tb↓

Fig. 7. Valence band electron density in TbF3 in the energy range from 10.9 eV to the Fermi level in the yx plane consistent with coordinates of Fig. 1: Left, spin-down density showing non-spherical symmetry of the Tb 4f supra-valence band electron; right, spin-up density showing the spherical symmetry of the half-filled shell Tb 4f7. The spin-up density contour values are reduced 7  compared with the spin-down density contour values. Tb 4f densities from the next plane below are shown as asymmetric features in the left-hand spin-down plot.

20

Ce4f

10 0 -10

Ce5s F2s

Ce5p1/2-5p3/2

-20

F2p

VB structure of TbF3 with spin polarization and spin-orbit coupling Red: spin-up Blue: spin-down Black: total dos Dotted: spin-up minus spin-down

-30

-30 -30

-20

-10 Energy, eV

0

EF (-4.98) -1.67

45 40 35 30 25 20 15 10 5 0 -5 -10 -15 -20 -25

(-24.67) -20.00 (-19.46) -16.62

(-4.89) -3.36

DOS

DOS

30

-32.48 (-34.55)

40

EF

-14.26 (-15.92)

50

(-18.80) -16.92

60

(-25.83) -21.90

70

VB structure of CeF3 with spin polarization and spin-orbit coupling Red: spin-up Blue: spin-down Black: total dos Dotted: spin-up minus spin-down

-20

-10 Energy, eV

0

10

10

Fig. 8. Gaussian-broadened theoretical DOS of CeF3 calculated with LDA+U. The Fermi level EF is marked by a dashed vertical line, showing a single occupancy of the Ce orbitals with one 4f-electron occupying the spin-up level while the unoccupied 6 the spin-up levels are shifted to higher energies by 4.9 eV. The individual bands are labeled with the emission levels, which are well separated from one another. Numerical values of theoretical energies are given as vertical labels, accompanied by experimental XPS energies in parentheses.

level with shape so as to avoid the fluoride ligands, hence displaying an antibonding character, and the spherically symmetric spin-down half-filled shell 4f7, which are in non-bonding relationship with the fluoride ligands. 3.3.1.4. Magnetic moments. Both fluorides are ferromagnetic at absolute zero temperature. The combined effects of SP and SO result in a slight quenching of the spin-only magnetic moment of Ce(III)F3 from 1 to 0.95 Bohr magnetons per Ce(III) ion and of Tb(III)F3 from 6 to 5.94 Bohr magnetons per Tb(III) ion. These

Fig. 9. Gaussian-broadened theoretical DOS of TbF3 calculated with LDA+U. The Fermi level EF is marked by a dashed vertical line, showing occupancy of the Tb orbitals with one spin-down 4f-electron at EF while the remaining 6 unoccupied spin-down levels are shifted to higher energies by 5.0 eV above EF. The 7 occupied spin-up levels are shifted to the bottom of the F2p ‘‘main’’ valence band peaking at 1.7 eV. Spin-polarization is large because of exchange stabilization in the sevenelectron half-filled shell. The band at EF is Tb4f1 spin-down, that with maximum at 1.67 eV is F2p, and the bands at 16.62 and 20.0 eV ensue from an overlap between F2s and Tb5p levels.

predicted moments need to be compared with experiments, for which data are currently unavailable. 3.3.1.5. Critique of the method. Notwithstanding the excellent correlation of the occupied VB levels for both CeF3 and TbF3 with the XPS data shown in Fig. 10, the GGA method using the PBE functional suffers from deficiencies in describing the partially occupied and empty states. The present PBE calculations have resulted in near-degeneracy of the 4f orbitals at the top of the VB: in the case of CeF3, the singly occupied 4f level, even though

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correctly positioned in the energy scale, is nearly degenerate with the remaining 6 unoccupied 4f levels of the same spin; and in TbF3, similar near-degeneracy occurs for the minority spin manifold occupied with one electron at the top of the VB with the 6 empty minority-spin levels, while the 7 occupied majority spinup levels are, again, correctly positioned at the bottom of the F2p VB. The near-degeneracies of the partially occupied levels might be viewed as a ‘‘metallic band’’, although the flat energy dispersion indicates zero mobility, and hence agrees with the insulating 0

EF

-5 Theoretical Energy, eV

LDA+U TbF3

-10 -15 -20

LDA+U CeF3 PBE CeF3 and TbF3

-25

TVB

-30 -35 -35

-30

-25 -20 -15 -10 Experimental XPS Energy, eV

-5

0

Fig. 10. A comparison of binding energies obtained using the DFT-FP-LAPW-SP–SO theory using the PBE functional and the LDA+U method with experimental XPS values for CeF3 and TbF3 showing an excellent linear correlation between the calculated PBE orbital energies (square symbols for both CeF3 and TbF3) and the XPS spectra. Two other separate sets of LDA+U energies indicate a systematic upward shift of ca 2.6 eV for CeF3 (down-triangles) and ca. 3.6 eV for TbF3 (uptriangles), indicating that lower values of U would bring the LDA+U calculations to a closer agreement with experiment, specifically for the occupied states.

character of both trifluorides. This is also evident from the highly localized, spatially and energy resolved electron density plots in Figs. 6 and 7. An alternative method for describing highly localized orbitals, the LSDA+U theory [17], has provided a new physical insight especially into the physics and chemistry of rare-earth compounds. An application of this method to the present trifluorides is investigated in the next section, and both approaches are compared thereafter.

3.3.2. Calculations using the LDA+U method This method uses the local density approximation (LDA), but for selected atomic states the DFT exchange-correlation potential is replaced by its Hartree–Fock-like form. To correct for the double counting we used the ‘‘Fully Localized Limit’’ version of the LDA+U [18]. For rare earth compounds with partially occupied 4f-shells, U and J are related to Slater’s integrals U ¼ F0 and J is a linear combination of the F2, F4, and F6 integrals [19]. Because of screening of nuclear charge by electrons of closed inner shells and partial screening by other electrons in the open 4f-shell, the subintegral radial functions in the F-integrals are influenced by a number of factors including electron configuration, long-range electrostatic interactions, the nature of counterions to the rareearth element, and ionicity/covalency of the interatomic bonds. These factors make an ab initio determination of U and J a very challenging task, and therefore it has become more expedient to empirically modify these integrals to fit experimental data such as energy levels and bandgaps in compounds. Thus the LDA+U method becomes a semiempirical theory and depends on availability of the experimental data, which can be provided by various spectroscopies, including XPS. In the present work, we have used U ¼ 0.493 Ry and J ¼ 0.051 for CeF3 from Ref. [20], while for TbF3 U ¼ 0.563 Ry and J ¼ 0.051 Ry, appropriate for Gd [21], were used as a starting point. This will be seen from the results as requiring a rescaling particularly of U to lower values to reach a quantitative agreement with both the XPS experiment and the PBE theory.

Fig. 11. Band structures of the hexagonal P3 c1 CeF3 (left) and orthorhombic Pnma TbF3 (right) showing differences in the bandgap and character at the bottom of the conduction band, in particular the larger bandgap and strong dispersion at the BCB of TbF3 that extends into the interstitial space. Heavier ‘‘band character’’ marks of CeF3 emphasize the 4f spin-up levels and those of TbF3 emphasize the spin-down levels near the Fermi level EF. Special points of the Brillouin zones in the hexagonal CeF3 are G(0 0 0), Mð12 0 0Þ, Kð13 13 0Þ, Að0 0 12Þ, and those in the orthorhombic TbF3 are Rð12 12 12Þ, G(0 0 0), Xð12 0 0Þ, Sð12 12 0Þ, and G(0 0 0). Calculations using the PBE finctional.

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The main results are summarized as follows. The LDA+ U-calculated DOS of CeF3 and TbF3 (Figs. 8 and 9) strongly resemble those of the PBE DOS in Figs. 2 and 3 with regard to the fully occupied VB levels. A significant difference appears in the partially occupied 4f states in that, as expected, degeneracies between occupied and unoccupied levels are removed with a separation on the order of the magnitude of U. 3.3.2.1. Magnetic moments. The LDA+U method also yields a stable ferromagnetic state at absolute zero temperature for both fluorides, with the spin-only magnetic moment of Ce(III)F3 of 1.00 Bohr magnetons per Ce(III) ion for Ce(III)F3 and 5.99 Bohr magnetons per Tb(III) ion for TbF3. These values are comparable to albeit a little higher than those obtained with the PBE functional. 3.3.2.2. Electron densities and relative intensities. These properties are in a qualitative agreement between LDA+U with those obtained with the PBE functional, cf. Figs. 4–7. 3.3.3. Correlation of the PBE and LDA+U calculations with the XPS valence band spectra A comparison of theory with experiment shown in Fig. 10 demonstrates an excellent correlation for the occupied states using both methods. However, the present choice of the U and J parameters in the LDA+U calculations has led to an underprediction of the gap between the Ln4f occupied state at EF and the ‘‘main’’ valence band F2p and, concurrently, to systematically higher values of all orbital energies than the PBE calculation and experiment. Also, the TbF3 correlation line lies higher than the CeF3 correlation line. Both could be brought into a better agreement with experiment by a lowering of U, but with a consequence of shifting the unoccupied 4f levels to lower energies, thus reducing the bandgaps below experimental values of 5.40 eV for TbF3 and 4.13 eV for CeF3 and creating a new gap between the unoccupied 4f states and the bottom of the conduction band. We have also calculated the electronic structures of the Ce and Tb trifluorides using the values of U

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and J given by Larson et al. for the cubic rare-earth nitrides [22], U ¼ 0.549 Ry and J ¼ 0.073 Ry for Ce and Uf ¼ 0.696 Ry and Jf ¼ 0.092 Ry for Tb, with added Ud ¼ 0.47 Ry. These values, while applicable to the more covalent nitrides, have further increased the undesirable trends for the fluorides in that the topmost occupied Ce4f and Tb4f levels were buried deep within the F2p valence band and the bandgap energy increased even more into the far-UV region of optical absorption spectra, contrary to the experimental values mentioned above. Thus the parameters for rare earth nitrides do not appear applicable to the presently investigated fluorides. In summary, the PBE method yields a quantitative account for the observed energies of the occupied valence states, and an adjustment of the U parameters in LDA+U calculations holds promise for obtaining a similar result. Both methods have deficiencies in the treatment of partially occupied and unoccupied spin manifolds, herein mainly in that PBE correctly accounts for the gap between the highest occupied Ln(4f) orbital and the ‘‘main valence-band’’ F2p orbital, while underpredicting the bandgap between the highest occupied Ln(4f) level and the bottom of the conduction band; and LDA+U (using values of U that successfully interpreted physical properties of other compounds) underpredicts the gap between this Ln(4f) level and the occupied F2p states while overpredicting the bandgap between the highest occupied Ln(4f) level and the bottom of the conduction band. It is noted that both methods deal with orbitals that are not eigenfunctions of L2 and S2, as aptly put among others by Liechtenstein et al. [18], yet the open-shell eigenstates of the Ce(III) and Tb(III) compounds lie close to the free-ion levels, which do possess good total spin and orbital angular momentum, as evidenced by e.g. luminescence spectra in ionic compounds of these rare earths.

3.4. Energy dispersion curves The band dispersions of the two LnF3 trifluorides calculated by the use of the PBE and the LDA+U functionals reflect in greater

Fig. 12. Band structures of the hexagonal P3 c1 CeF3 with heavier ‘‘band character’’ marks emphasizing the Ce4f spin-up (left) and spin-down (right) levels. Special points of the Brillouin zones are as described in Fig. 11. Calculations using the LDA+U method. The single spin-up 4f occupied level at EF is clearly separated from the main F2p valence band and from other empty 4f spin-up and spin-down levels forming flat bands close to the bottom of the conduction band.

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Fig. 13. Band structures of the orthorhombic P3nma TbF3 with heavier ‘‘band character’’ marks emphasizing the Tb4f spin-up (left) and spin-down (right) levels and those of TbF3 emphasize the spin-down levels near the Fermi level EF. Special points of the Brillouin Zones are as described in Fig. 11. Calculations were made using the LDA+U method. The majority spin-up 4f occupied states are below the bottom of the main F2p valence band. The single spin-down 4f occupied level at EF is above the main F2p valence band and is clearly separated from other empty spin-down levels, forming flat bands close to the bottom of the conduction band. A strong dispersion at the bottom of the conduction band at the G-point is consistent with that found by the PBE calculation, cf. Fig. 11.

detail salient features of their electronic structures, as shown in Figs. 11–13. Among the features that agree well between the two methods, the dispersion of the 4f orbitals is flat in accordance with their highly localized nature, one of the 4f states appears as the highest occupied orbital at EF, and in the case of TbF3 the occupied majority 4f levels are located at the bottom of the F2p band. There are also differences, already evident from the DOS plots in Figs. 2, 3, 8, and 9. Aside from the gaps between EF and the F2p ‘‘main valence band’’ and between the bottom of the conduction band (BCB) and EF, the LDA+U method with the presently used U and J parameters places empty 4f states close to the bottom of the conduction band, while the PBE method has these empty 4f states only slightly above EF. This is an expected effect imparted by LDA+U, which however yields a poorer agreement with experiment than the PBE method. It is noted however that the near-degeneracy of the occupied and unoccupied 4f levels in the PBE treatment is not inconsistent with the insulating nature of these fluorides, as the dispersions at EF are flat, yielding practically zero mobility of the highly localized 4f-electrons. Near the bottom of the conduction band, there is a significant difference between the two trifluorides, which predicts localized orbital character for CeF3 but a highly delocalized character for TbF3 with effective mass comparable to that of a free electron, determined 2 (for the PBE calculation) as meff ¼ ‘ =ð@2 EBCB =@k2 Þwhere EBCB is the orbital energy at the bottom of the conduction band of TbF3 at the G-point, yielding meff ¼ 0.76me. The dispersion at BCB of TbF3 also results from the LDA+U calculation, even though in this case the BCB is overlapped by flat unoccupied 4f states. Thus both methods predict photoconductivity in TbF3 and the lack thereof in CeF3 by excitation across the bandgap.





4. Conclusions

  The DFT-FP-LAPW theory correctly interprets the VB structure of the RE trifluorides only if both spin-polarization and

spin–orbit coupling are taken into account, based on the analysis of the two structures with partially filled 4f-shells: a single 4f-electron in CeF3 and a highly correlated 4f8 configuration in TbF3. A linear relation is obtained between the theory using both the PBE functional and the LDA+U method and the XPS experiment, although LDA+U requires further optimization of the Coulomb and exchange parameters to yield a quantitative agreement. This sets the stage for interpretation of more complex solids such as the partially substituted, ion conducting trifluorides, and the anchored molecular materials such as the Ce(III)SBA previously published [2]. The net spin is localized in antibonding 4f orbitals in the ground states of both the analyzed trifluorides, but there is a significant difference between CeF3 and TbF3 with regard to delocalized character at the bottom of the main conduction band (CB): while the flat CB of CeF3 indicates no electron mobility, the CB of TbF3 displays a large dispersion associated with the effective electron mass equal to a fraction of that of free electron, predicting fast photoconductivity and an efficient excitation energy transfer in TbF3. The PBE method underpredicts the bandgaps, but the the differences between the observed and theoretical bandgaps of the CeF3 and TbF3 show a reasonable trend in that the bandgap of CeF3 is by 1.2–1.9 eV smaller than that of TbF3. The LDA+U method overpredicts the bandgaps with the use of the present parameters, but could be brought to a better agreement with experiment by lowering the value of the U parameter from 0.6 to 0.3 Ry. The present implementation of the secondvariational method for calculation of spin–orbit interactions has led to an excellent agreement with experiment, e.g. 2.88 eV for Ce5p3/2–Ce5p1/2 separation) by both the PBE (2.61 eV) and LDA+U methods (2.66 eV), as well as the correct ratio of intensities (2:1 for 5p3/2:5p1/2). Magnetic moments are close to spin-only values, being only slightly reduced by 5% (Ce) and 1% (Tb) using the PBE method, and closer to spin-only using the LDA+U method.

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Acknowledgements We gratefully acknowledge the support of this work by the Displays Lab of the Center for Optical Technologies of Lehigh University. Work of one of the authors (P.N.) was supported by the Czech project AVOZ10100521.

Appendix A. Supplementary materials The online version of this article contains additional supplementary data. Please visit doi:10.1016/j.jpcs.2009.07.024.

Appendix B. Supplementary materials

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