Infrared and Raman spectra and the band structure of yttrium trifluoride YF3

Computational Materials Science 50 (2011) 2391–2396

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Infrared and Raman spectra and the band structure of yttrium trifluoride YF3 R. Vali ⇑ School of Physics, Damghan University, P.O. Box 36715/364, Damghan, Iran

a r t i c l e

i n f o

Article history: Received 2 September 2010 Received in revised form 11 March 2011 Accepted 13 March 2011 Available online 5 April 2011 Keywords: YF3 Infrared reflectivity Raman intensity Electronic band structure

a b s t r a c t The infrared and nonresonant Raman spectra of YF3 have been studied within the framework of density functional perturbation theory. We report the calculated frequencies of three Raman active modes and one IR active mode that could not be detected experimentally. The valence and conduction band structure of YF3 have been calculated, using density functional theory. A good agreement between the calculated valence band width and experimental result was obtained, and an indirect energy gap of 7.58 eV is estimated in the local density approximation. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction Fluoride crystalline materials are suitable for use as the active media in solid state lasers and scintillators, due to their good optical properties. Among these crystals, LiYF4 is the most studied and used fluoride host crystal with several emission bands, depending on its active center [1]. Yttrium trifluoride (YF3) is another fluoride crystal that can be used as solid state lasers and scintillators, since it can easily be doped with rare earth ions due to its wide band gap and suitable Y3+ sites where other trivalent rare earth or transition metal ions can be substituted without additional charge compensation [2]. Lanthanide trifluorides (LnF3) with Ln = Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb and Lu also emerged as good candidates for use as active media for tunable solid state laser or as fast scintillator materials [3]. At room temperature these lanthanide trifluorides and YF3 crystallize in an orthorhombic Pnma structure with four formula units per unit cell, known as b-YF3 structure [4]. The knowledge of the electronic properties and optical phonon modes is crucial for development of tunable solid state lasers. The excitation [2], Raman [4] and infrared [5] spectra of b-YF3-type crystals have been studied experimentally. Also, the scintillation properties of Ce doped YF3 have been studied theoretically and experimentally by Boutchko et al. [6]. They have found that YF3 exhibits cerium scintillation. However, to the best of our knowledge there is no theoretical work on the infrared (IR) and Raman spectra of theses crystals, using ab initio methods. In this paper, we present the results of a density functional theory study of the electronic band structure, Raman and IR active spectra of the prototype compound YF3 crystal. The paper is organized as follows. ⇑ Tel.: +98 0232 523 30 54; fax: +98 0232 524 47 87. E-mail address: [email protected] 0927-0256/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2011.03.017

Section 2 describes theoretical methods and technical details of the calculations. Section 3 presents the results including the electronic band structure, IR reflectivity spectra, and Raman spectra. Section 4 concludes the paper. 2. Theoretical methods and technical details The calculations were performed using the linear response approach together with an iterative minimization pseudopotential plane-wave method [7–9] and within the framework of the density functional theory [10] as implemented in the ABINIT code [11]. In the density functional theory, the ground state energy of the electronic system is given by the following minimum principle [7]

Eel fwa g ¼

occ X hwa jT þ V ext jwa i þ EHxc ½n

ð1Þ

a

where the wa’s are the Kohn–Sham orbitals, to be varied until the minimum is found, T is the kinetic energy operator, Vext is the potential external to the electronic system, EHxc is the Hartree and exchange-correlation energy functional of the electronic density P  nðrÞ ¼ occ a wa ðrÞwa ðrÞ, and the summation runs over the occupied states a. The phonon modes, susceptibilities and the Born effective charge tensors describing the LO–TO splitting are evaluated by the density functional perturbation theory [7–9], which is the basis of the linear response approach. Linear response method provides an efficient means for computing quantities that can be expressed as derivatives of the total energy with respect to a perturbation, including the force constant matrix elements, Born effective charges and dielectric tensors. In this method the perturbed external potential is expanded in terms of a small parameter k as ð1Þ 2 ð2Þ follows: V ext ðkÞ ¼ V ð0Þ ext þ kV ext þ k V ext þ . . . Since Eel satisfies a

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variational principle under constraints, it is possible to derive a constrained variational principle for the 2nth order derivative of Eel with respect to the nth order derivative of wa. The second-order derivative of the density functional theory total energy E2el is the minimum of the following expression [7] ð2Þ

Eel fwð0Þ ; wð1Þ g ¼

occ X ð0Þ ð1Þ ð1Þ ð1Þ ð0Þ ð0Þ ½hwð1Þ a jH  ea jwa i þ hwa jV ext jwa i

a ð1Þ ð2Þ ð1Þ ð0Þ ð0Þ þ hwð0Þ a jV ext jwa i þ hwa jV ext jwa i  Z Z  ! ! ! 1 d2 EHxc  ! þ  nð1Þ ð r Þnð1Þ ð r 0 Þd r d r0 ! ! 0  2 dnð r Þdnð r Þ nð0Þ   Z 2 d dEHxc  1 d EHxc  ð1Þ ! ! ð2Þ þ   n ð r Þd r þ dk dnð! 2 dk2  ð0Þ r Þnð0Þ n

Table 1 Calculated structural parameters of YF compared to experimental values. 0 3 The lattice constants a, b and c are in A Å and the atomic coordinates x, y and z are in fractional units of cell parameters. Calculated

Experimental [19]

a b c

6.2740 6.7649 4.3383

6.3537 6.8545 4.3953

Y(4c) x y z

0.3694 0.2500 0.0623

0.3673 0.2500 0.0591

F1(4c) x y z

0.5211 0.2500 0.5858

0.5227 0.2500 0.5910

F2(8d) x y z

0.1665 0.0626 0.3805

0.1652 0.0643 0.3755

where the first order wave functions wð1Þ a are varied under conD E ð1Þ ð0Þ straints wa j wb ¼ 0 for all occupied states a and b, while the P ! ð1Þ ! first order density n(1)(r) is given by nð1Þ ð r Þ ¼ occ a wa ð r Þ ! ð0Þ ! ð1Þ ! wð0Þ a ð r Þ þ wa ð r Þwa ð r Þ. Eq. (2) can be specialized to a phonon perturbation with wave vector q, in which atom s is moved in direction i, or an electric field perturbation along direction j. Once the first order perturbed wave functions wð1Þ a for all phonon polarizations i at a given q or all electric field perturbation along j have been obtained, the force constant matrix elements, the Born effective charge tensor, and the electronic dielectric susceptibility tensor are easily computed. The force constant matrix elements can be calculated as

@us

@2 E @us

1 i1

2 i2

,

where us1 i1 is the displacement of atom s1 in direction i1 from its position in the equilibrium crystal structure. The Born effective charge tensor can be computed as a mixed second-order derivative of the energy with respect to an electric field and to an atomic dis2

placement: Z sij ¼  @d@ @Ee , where dsi is the uniform displacement of si j the atomic sublattice s in direction i from its position in the equilibrium unit cell and ej is the electric field in direction j. The electronic dielectric susceptibility tensor is given by

vij ¼  V1

@2 E , @ ei @ ej

where V is the volume of the primitive unit cell. The calculation of the Raman tensor requires the calculation of a third rank tensor @ vij/@rsc, where rsc is the Cartesian coordinate in direction k of atom s in the unit cell. Thus, it requires third derivatives of the total energy. This can be achieved using first order change of the wave functions using the 2n + 1 theorem. Further details on the calculation of the Raman tensors can be found in Ref. [12]. The calculations were first done with the Hartwigsen–Goedecker–Hutter (HGH) pseudopotential [13] for the Y and F atoms, where the Y (4s, 4p, 4d, 5s), and F (2s, 2p) orbitals were considered as valence states. Convergency was reached for an energy cutoff of 150 Ha for the plane-wave expansion and a 3  3  4 k-point mesh for the Brillouin zone integration. Although HGH pseudopotentials demand a higher energy cutoff than for example Troullier–Martins

Wyckoff positions of the reported atoms are mentioned in parentheses.

10

8

Energy (eV)

6 T o ta l Y 4d F 2p

4

2

0

-2

-4

Γ

Y

T

Z

Γ

X

Wave vector

U

R

S

X

DOS

Fig. 1. The electronic band structure of YF3 along high symmetry lines of the Brillouin zone (left) and the corresponding density of states (right) in an energy window from 4 to +10 eV, where the Fermi energy set to zero.

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(TM) pseudopotentials [14], they give better and more reliable results. Then, we have tested the TM pseudopotential for the Y and F atoms. The use of TM pseudopotential for the F atom gives nearly no change in phonon frequencies, whereas deviations up to 30% appear when a TM pseudopotential for Y atom is used, independent of the pseudopotential used for F atom. In HGH pseudopotential of Y atom the semicore Y 4s and Y 4p orbitals were considered as valence states. In the case of Y the use of semicore states is necessary for a good description of phonon modes. The inclusion of semicore states in the TM pseudopotentials is possible but can generate unphysical ‘‘ghost states’’. In the following we present results obtained with the use of HGH pseudopotential for Y and TM pseudopotential for F atom and with an energy cutoff of 80 Ha. The exchange-correlation energy is evaluated within the local density approximation [15] by using Ceperley–Alder homogeneous electron gas data [16]. The conjugate gradient method [17] was employed to minimize the energy functional.

1.0

[001] B1u modes

(a) 0.8

0.6

0.4

0.2

0.0 1.0

[010] B2u modes

(b) 0.8

At room temperature, yttrium trifluoride exhibits the b-YF3 structure, an orthorhombic pnma structure with four formula units per unit cell. In this structure the fluorine atoms occupy two sites with different symmetries. They are called F1 and F2 and are lo  cated at x; 14 ; z and ðx; y; zÞ on the 4c and 8d Wyckoff sites, respectively. The yttrium atoms occupy also sites with 4c symmetry. Both the lattice constants and atomic positions have been relaxed under constraints of the Pnma space group. The structural relaxation was conducted using the Broyden–Fletcher–Goldfarb–Shanno minimization [18], modified to take into account the total energy in addition to the gradients. The relaxation was terminated when the 0 residual forces on each atom were less than 0.01 eV/Å A. The calculated as well as experimental structural parameters are given in Table 1. The calculated lattice constants a, b and c as well as the internal parameters x, y and z are found to be in very good agreement with their corresponding experimental values [19]. Fig. 1 displays the valence and conduction band structures of YF3 along the high symmetry lines of the Brillouin zone of pnma orthorhombic structure, together with the corresponding density of states in the 4 to 10 eV range. The analysis of local density of states identifies that the valence band is mainly dominated by the 2p orbitals of the fluorine atom weakly hybridized with the 4d orbitals of the yttrium atom, whereas the conduction band is dominated by the 4d orbitals of yttrium atom with some mixing with the 2p orbitals of the fluorine. The top of the valence band was fixed at 0 eV. The top of the valence band and the bottom of the conduction band are respectively located at the Z and C high symmetry points. Therefore YF3 is an insulator material with an indirect band gap of 7.58 eV. However the energy of the top of the valence band at Z is only 0.04 eV higher than the energy at the C point so that the direct band gap at C is only slightly higher, 7.62 eV. The experimental value of the band gap of YF3 was reported as 10.53 eV in Ref. [20] and 12.5 eV in Ref. [2]. Our underestimate for the band gap is a common feature of the density functional theory calculations, as is well known from work on a wide variety of other materials. The calculated valence band width is 3.84 eV which is in reasonable agreement with result of the X-ray photoelectron study of the Ref. [21], where the value of the full width at half maximum of the valence band was found to be 3.5 eV. From the factor group analysis, the zone center optical phonon modes are classified as 7Ag + 5B1g + 7B2g + 5B3g + 6B1u + 4B2u + 6B3u + 5Au. The B1u B2u and B3u modes polarized along c, b and a, respectively, are IR active. The Ag, B1g, B2g and B3g even modes are Raman active while Au modes are silent. The computed frequencies

Reflectivity

3. Results 0.6

0.4

0.2

0.0 1.0

[100] B3u modes

(c) 0.8

0.6

0.4

0.2

0.0

0

100

200

300

400

500

600

700

-1

ω(cm ) Fig. 2. Calculated IR reflectivity spectra (without damping) at normal incidence on the (a) [0 0 1], (b) [0 1 0] and (c) [1 0 0] surface of a YF3 single crystal.

of Au modes are, respectively, 163.1, 197.9, 212.0, 263.3 and 324.6 cm1. These frequencies can not be compared to experimental values since no experimental inelastic scattering data are presently available. Panels (a), (b) and (c) of Fig. 2 display the calculated IR reflectivity spectra (without damping) at normal incidence, respectively, on the [0 0 1], [0 1 0] and [1 0 0] surfaces of a YF3 single crystal, calcu 1=2 2  Þ1 lated as RðxÞ ¼ ee1=2 ððx xÞþ1 , where e(x) is frequency dependent dielectric constant. As can be seen from this figure, there are 6 phonon peaks corresponding to 6B1u modes, 4 phonon peaks corresponding to 4B2u modes and 6 phonon peaks corresponding to 6B3u modes, as expected from group theory. Table 2 summarizes the calculated frequencies of IR active phonons for longitudinal

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Table 2 Calculated and experimental frequencies (in cm1) of IR active modes of YF3. The symbol TO (LO) denotes transverse (longitudinal) optical mode. The calculated oscillator strengths Sm (in 104 a.u.) are also provided for each mode. Symmetry

Calculated

5

(a) 4

Experimental [5]

TO

LO

Sm

TO

LO

126.7 254.7 269.2 314.2 388.9 424.8

181.1 268.3 292.6 388.4 409.8 484.8

3.26 3.27 0.10 2.03 0.01 0.39

132 234 267 333 373 434

191 237 311 373 419 514

3

B2u

210.6 232.6 294.5 348.9

230.2 255.2 311.9 513.7

4.00 0.18 1.65 4.17

216 243 307 352

242 273 331 525

1

B3u

170.6 210.9 247.1 310.1 375.0 470.6

172.0 215.6 296.4 355.0 463.7 509.5

0.01 0.96 6.21 1.49 1.41 0.11

– 227 262 322 392 486

– 237 272 376 481 520

Table 3 Calculated and experimental frequencies (in cm1) of the Raman active modes of YF3. Symmetry

Calculated

Experimental [4]

Ag

121.9 148.2 171.4 248.2 356.3 380.2 455.6

118 145 171 244 351 367 444

B1g

133.8 223.3 303.2 373.3 402.8

138 221 295 367 391

B2g

128.2 201.7 281.7 326.4 343.3 373.5 524.4

120 188 – – 350 373 514

149.4 199.2 271.5 344.8 431.6

149 190 262 341 –

B3g

optic (LO) and transverse optic (TO) modes in comparison with available experimental values [5]. The overall agreement is good; however two possible reassignments are suggested. We find one weak IR active B3u mode at 170.6 cm1 that is not seen in experiment. As can be seen from Table 2, this mode has weak oscillator strength. Due to its weak oscillator strength this mode is difficult to detect experimentally and it does not give rise to a considerable LO–TO splitting (see figure 2). Instead, our calculations do not give any B3u mode with frequency close to the experimental value [5] of 275 cm1. We have considered this reassignment in Table 2. The peak intensity of the nonresonant Stokes Raman spectrum can be calculated within the Placzek approximation as I / j^ei Rm ^es j2 x1 ðnB ðxÞ þ 1Þ, where nB(x) = [exp ( hx/KBT)  1]1 [22]. Here ^ei and ^es are the polarization of incident and scattered radiation and T is the temperature. The Raman susceptibility tensor Rm associated with the mth phonon is given by Rm ij ¼

2

0 5

(b) Raman intensity (arb. units)

B1u

4

3

2

1

0 5

(c) 4

3

2

1

0

0

100

200

300

400

500

600

-1

Wavenumber (cm ) Fig. 3. The Raman spectra of YF3 for the Ag modes in the (aa) polarization (a), (bb) polarization (b) and (cc) polarization (c).

pffiffiffiffiP @vij V sc @rsc um ðscÞ, where V is the unit cell volume, vij is the first order dielectric susceptibility tensor, rsc is the Cartesian coordinate in direction k of atom s in the unit cell and um(sc) is the eigenvector corresponding to mode m. The Raman tensor for the Raman active irreducible representations of an orthorhombic crystal has the following form [23]

0

a

B Ag @ 0

0 0 b

1

C 0 A;

0

0

B B1g @ d

0 0 c 0 1 0 0 0 B C and B3g @ 0 0 f A 0 f 0

d

0

1

C 0 0 A; 0 0 0

0

0 0

e

1

B C B2g @ 0 0 0 A e

0 0

R. Vali / Computational Materials Science 50 (2011) 2391–2396

agreement is very good however, we find three Raman active modes at 281.7 cm1[B2g], 326.4 cm1[B2g] and at 431.6 cm1[B3g], that could not be detected experimentally. Figs. 3 and 4 display the calculated Raman spectra of a YF3 single crystal, where the Raman line shape is assumed to be Lorentzian, and the line width is fixed at 4 cm1 . Figs. 3a–c represent Raman spectra in the three parallel polarizations. As can be seen from these figures, the intensity of totally symmetric Ag modes depends on the direction of the polarization. A similar behavior was observed experimentally for lanthanide fluorides LnF3 and was attributed to the special arrangement of F1 atoms in b-YF3 structure [4]. In this structure, around each Ln (or Y) ion a coordination polyhedron of three F1 and six F2 ions forms, where the three F1 ions lie in planes parallel to the crystallographic ac plane. Thus the intensity of the totally symmetric Raman bands is expected to depend on the direction of polarization. Fig. 4a represents Raman spectra in (ab) crossed polarization. It can be found that there are two relatively weak B1g bands at 133.8 and 223.3 cm1 and three strong B1g bands at 303.2, 373.3 and 402.8 cm1. Fig. 4b represents Raman spectra in (ac) crossed polarization. There should be seven bands at frequencies listed in Table 3, however the mode at 326.2 cm1 is too weak to be distinguished and the mode at 281.2 cm1 is very weak. Hence, it is not surprising that these two B2g modes were not observed in experiment. Fig. 4c represents Raman spectra in (bc) crossed polarization. There should be five bands at frequencies listed in Table 3; however the mode at 431.6 cm1 is too weak to be distinguished. This is the main reason for the absence of this B3g mode in experimental results.

(a)

3

2

1

0

(b)

Raman Intensity (arb. units)

4

3

2

1

0

4. Conclusions

(c)

2.0

Based on density functional perturbation theory, the IR reflectivity and Raman spectra of YF3 have been calculated. The overall agreement between the calculated and experimental frequencies is good. However, we find one IR active B3u mode at 170.6 cm1 and three Raman active modes at 281.7, 326.4 and 431.6 cm1 that could not be detected experimentally. Our calculation confirms the experimental result that for trifluoride crystals with b-YF3 structure the intensity of the totally symmetric Ag modes depends on the direction of polarization. The calculated electronic band structure revealed that the top of valence band and the bottom of the conduction band are decided by F 2p and Y 4d states, respectively, and that YF3 presented and indirect band gap of 7.58 eV.

1.5

1.0

0.5

0.0

2395

0

100

200

300

400

500

600

References

-1

Wavenumber (cm ) Fig. 4. The Raman spectra of YF3 for the B1g modes (in the ab crossed polarization) (a), B2g modes (in the ac crossed polarization) (b) and B3g modes (in the bc crossed polarization) (c).

For Ag symmetry, the Raman tensor is diagonal which means that incident and scattered light must have parallel polarization. As pointed in Ref. [4], since the structure of YF3 has an inversion center, none of the Raman modes can be polar and thus the Raman bands do not depend on the light propagation directions. Thus, only the polarization directions of the incident and scattered lights, within Porto notation, are required to distinguish the modes. The Ag modes are observable in (aa), (bb) or (cc) polarizations, B1g modes in (ab) polarization, B2g modes in (ac) polarization and B3g modes in (bc) polarization. In Table 3, the calculated frequencies of the Raman active modes are compared with experimental values [4]. Overall, the

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