Explanation of the EPR g factors for Co2+ impurities in trigonal Cd2P2S6 crystal

Physica B 307 (2001) 28–33

Explanation of the EPR g factors for Co2+ impurities in trigonal Cd2P2S6 crystal Wen-Chen Zhenga,b,c,*, Wu Shao-Yia,b a Department of Material Science, Sichuan University, Chengdu 610064, People’s Republic of China International Centre for Materials Physics, Chinese Academy of Sciences, Shenyang 110015, People’s Republic of China c Key Laboratory for Radiation Physics and Technology of Ministry of Education, in Sichuan University, People’s Republic of China b

Received 4 October 2000; received in revised form 2 July 2001

Abstract By extending the theory of Abragam and Pryce, the formulas of EPR parameters g8 and g> for 3d7 ion in trigonal octahedral crystals are established from a cluster approach. In these formulas, not only the configuration interaction effect, but also the covalency effect are considered and the parameters related to both effects and the trigonal distortion are determined from the optical spectra and the structural data of the crystal under study. Based on these formulas, the g8 and g> of Cd2P2S6 : Co2+ crystal are reasonably explained from its structural data. The relationship between the sign of Dg ð¼ g8  g> Þ and the trigonal distortion (elongated or compressed) of ligand octahedron, which is opposite to that obtained in the previous paper, is given and the results are discussed. r 2001 Elsevier Science B.V. All rights reserved. PACS: 76.30Fc; 71.70Ch; 75.10Dg Keywords: Electron paramagnetic resonance; Crystal- and ligand-field theory; Trigonal distortion; Co2+; Cd2P2S6

1. Introduction The M2P2S6 lamellar materials are currently enjoying considerable attention because of the ability to undergo intercalation reactions. The layers in these materials are composed of arrange of [P2S6]4 units coordinated to M2+ cations through M–S bonds, so that the M2+ cations are in a trigonally distorted octahedral environment [1]. The EPR spectra of Co2+ replacing Cd2+ in *Corresponding author. Department of Material Science, Sichuan University, Chengdu 610064, People’s Republic of China. Fax: +86-28-540-5541. E-mail address: [email protected] (W.-C. Zheng).

Cd2P2S6 crystals were measured and the anisotropy g-factors g8 and g> were reported [2,3]. In order to explain these EPR parameters, Hefni et al. [2] diagonalized the matrix of Hamiltonian H ð¼ Hcryst þ HZeeman Þ of 3d7 ion in trigonal symmetry and found that the sign of the g-value anisotropy Dg ð¼ g8  g> Þ depends upon the sign of the trigonal distortion Dy (¼ y  y0 ; where y is the angle between the M–S bond and C3 axis, y0 is the same angle in cubic symmetry), i.e., when Dyo0; Dgo0 and when Dy > 0; Dg > 0: Thus, from the Dg (E1, [2,3]) of Cd2P2S6 : Co2+, they concluded that Co2+ is in a trigonally compressed octahedral site. The conclusion is in disagreement with the fact obtained by X-ray diffraction where the metal

0921-4526/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 9 5 9 - 0

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coordination in Cd2P2S6 lattice is trigonally elongated by about 2–31 [1]. Hefni et al. [2] attributed the disagreement to the local deformation of the Co2+ coordination environment in Cd2P2S6. However, considering that Co2+ in pure Co2P2S6 crystal is also trigonally elongated [1], the above opinion and hence the relationship are doubtful. In order to obtain the correct relationship between the signs of Dg and Dy; in this paper, by extending the Abragam and Pryce theory [4], the formulas of g8 and g> for 3d7 ion in trigonal octahedral crystals are established from a cluster approach. In these formulas, not only the configuration interaction (CI) effect, but also the covalency (CO) effect are considered and the parameters related to both effects and the trigonal distortion can be calculated from the optical spectra and the structural data of the studied system. From these formulas, the relationship between the signs of Dg and Dy; which is opposite to that in Ref. [2], is obtained and so the EPR parameters g8 and g> of Cd2P2S6 : Co2+ crystal are reasonably explained from the structural data. The reasonableness of this relationship is discussed.

2. Calculation formulas From the cluster approach for 3dn ion in octahedra, the LCAO orbital [5,6] jgS ¼ Ng1=2 ðjdg S  lg jpg SÞ;

ð1Þ

is taken as the one-electron orbital, where g=t2g or eg, is the irreducible representation of Oh group. Ng and lg are, respectively, the normalization and mixing coefficients. |dgS is the d orbital of 3dn ion and |pgS is a linear combination of valence p orbitals of six ligands transforming like |dgS. In principle, one has also to add in Eq. (1) (but only for the eg orbital) the corresponding linear combination involving the valence s orbitals of ligands. However, numerical calculations [5,7,8] show that the contribution due to the s orbital is small, and so we neglect the s orbital in Eq. (1). Thus, the spin–orbit (SO) coupling coefficients z, z0 and the orbital reduction factors k, k0 due to the CO effect can be

written as [5,6] z ¼ Nt ðz0d þ l2t z0p =2Þ; z0 ¼ ðNt Ne Þ1=2 ðz0d  lt le z0p =2Þ; k ¼ Nt ð1 þ l2t =2Þ;

k0 ¼ ðNt Ne Þ1=2 ð1  lt le =2Þ; ð2Þ

where z0d and z0p are the SO coupling coefficients of 3dn ion and ligand in free ions. The CO parameters Ng and lg can be determined from a semiempirical LCAO method, which yields the normalization correlation [5,6] Ng ð1  2lg Sdp ðgÞ þ l2g Þ ¼ 1

ð3Þ

and approximate relation [5] fg ¼ /g2 je2 =r12 jg2 S=/d2 je2 =r12 jd2 S E fNg ½1  lg Sdp ðgÞ g2 2 ¼ Ng2 ½1 þ l2g Sdp ðgÞ  2lg Sdp ðgÞ ;

ð4Þ

where fg ½EðB=B0 þ C=C0 Þ=2 is the ratio of Racah parameters for a 3dn ion in crystal to those of free ion [5]. Sdp ðgÞ are the group overlap integrals calculated from Slater-type SCF functions and the average metal–ligand distance R. Thus, from the optical spectral parameters B, C and the structural parameter R of the studied system, the parameters z, z0 , k and k0 can be determined. When a free 3d7 (Co2+) ion is placed in an octahedral crystal field, the 4F ground state is split into 4A2, 4T2 and 4T1. The lowest 4T1 is further split by SO coupling and trigonal field into six Krammers doublets and the lowest Krammers doublet has an effective spin S0 =1/2. The Hamiltonian for the system can be written as H ¼ H0 þ H 0 ; a ðDq; V; V 0 Þ; H0 ¼ HCoul ðB; CÞ þ Vcrys b ðDq; V; V 0 Þ; H 0 ¼ Hz ðk; k0 ; a; a0 Þ þ HSO ðz; z0 Þ þ Vcrys

ð5Þ where HCoul denotes the Coulomb repulsion a b interaction. Vcrys and Vcrys are, respectively, diagonal and off-diagonal parts of crystal field with the cubic field parameter Dq and trigonal field parameters V and V0 . Hz is the Zeeman term with the effective lande factors a and a0 in the axial and perpendicular directions. HSO is the SO coupling

30

W.-C. Zheng, S.-Y. Wu / Physica B 307 (2001) 28–33

term (note: because of the large SO coupling coefficient of Co2+, the role of Jahn–Teller effect is not important, i.e., HJT 5HSO for Co2+ octahedra [9–11]. So, HJT is not considered here, as made in Ref. [2,4,9,10]). By using the perturbation method similar to that in Ref. [4], the second-order perturbation formulas of g8 and g> for 3d7 ions in trigonal symmetry based on the cluster approach are derived as follows [6]:

where the energy denominators E1X ; E1Z ; E2X ; E2Z and E3 and the splitting D can be determined from the d–d transition energy matrices of 3d7 ion in trigonal symmetry. The parameters a; a0 ; fi and qi can be calculated from the perturbation method [6], i.e., pffiffiffi 5er þ 12r2  t2 ;

a ¼ 32e2 



 
  a 
 a  3 4 9 4 3 4  4ðka þ 2Þ 2  v3 v2  2 þ2 2 v1 þ 0 x x a a0 x xþ2 ðx þ 2Þ2 ðx þ 2Þ2
   g8 ¼ 2 þ ; a 2 6 8 þ þ a0 x2 ðx þ 2Þ2
     a 4 a 2 2ka 12 a 2 8 12 4 þ v þ v þ  þ þ v v7 4 5 6 a0 x þ 2 xðx þ 2Þ a0 xðx þ 2Þ a0 ðx þ 2Þ ðx þ 2Þ2
   ; g> ¼ a 2 6 8 þ 2þ a0 x ðx þ 2Þ2


where x can be obtained from the energy splitting D ½¼ Eð4 A2 Þ  Eð4 EÞ of 4T1 ground orbital state in trigonal crystal field, the CI coefficients a, a0 and the spin–orbit coupling coefficient z by solving the equation
 za02 3 4 za þ ð8Þ D¼  ðx þ 3Þ: 6 3a x x þ 2 The parameters vi are defined as
 k0 z0 15f12 2q21 10k0 z0 f42 v1 ¼ þ ; v2 ¼ 3 2E1X E2X 3E2Z
 k0 z0 5f1 f2 5f3 f4 2q1 q2 v3 ¼ þ þ ; 3 2E2X E2Z E1X
 k0 z0 5f22 4q2 v4 ¼ þ 2 ; 3 E2X E1X
 k0 z0 5f32 4q23 4q24 v5 ¼ þ þ ; 3 E2Z E3 E1Z
 k0 z0 5f52 5f32 2q23 2q24 v6 ¼ þ þ þ ; 3 E2X E2Z E3 E1Z
 k0 z0 5f1 f2 5f3 f4 q1 q2 þ þ ; v7 ¼ 3 E2X E2Z E1X

a

0

¼ 32ee0

pffiffi

5 0



00

2

e r þ 2rs  tt0 ;




f1 ¼  ee 1 þ

r2  e

"

# 0 00 4 a 2 t t 3 r f2 ¼ e00 e 1 þ pffiffiffi 0  pffiffiffi 0 00 þ pffiffiffi ; 5e 5ee 5e "

# 1 r f3 ¼ e 1 þ pffiffiffi ; 5e " 00

f5 ¼  ee

" q1 ¼ eN1X

0

q2 ¼ e N1X

"

# 2 s f4 ¼ e 1 þ pffiffiffi 0 ; 5e 0

# 4 r 2 t0 t00 r2 1 þ pffiffiffi  pffiffiffi 0 00  ; e 5e 5ee

pffiffiffi # 5rt 5 t0  0þ ; 2e 2 ee

" ð9Þ

;

pffiffiffi # 5 t00 t0 3 t 2st00 þ 0 00 þ þ ; 0 e 2e ee 2 e00

ð6Þ

ð7Þ

W.-C. Zheng, S.-Y. Wu / Physica B 307 (2001) 28–33

"

pffiffiffi # 5 sr 2r 3s ts þ 0þ þ eN ; q3 ¼ 2 e 2e eN1Z 2 ee0 " q4 ¼ eN1Z

pffiffiffi # 5 rt0 t 3t0 2rs þ 0 þ ; e 2e eN1Z 2 ee0

k ¼ k0 ¼ 1; z ¼ z0 ¼ z0d ; Eqs. (6) and (7) become the formulas in Ref. [4].

ð10Þ

where ei ; ti ; si ; r and Nij are the admixture (or CI) coefficients. According to the normalization relationship, we have e2 þ t2 þ r2 ¼ 1; e0 ½1 þ ðt0 =e0 Þ2 þ ðs=e0 Þ2 1=2 ¼ 1; e00 ½1 þ ðt00 =e00 Þ2 þ ðr=eÞ2 1=2 ¼ 1; N1X ½1 þ ðt00 =e00 Þ2 þ ðt=eÞ2 1=2 ¼ 1;

ð11Þ

N1Z ½1 þ ðt0 =e0 Þ2 þ ðs0 =N1Z Þ2 1=2 ¼ 1; N2 ½1 þ ðs0 =N1Z Þ2 þ ðs=e0 Þ2 1=2 ¼ 1: From a modified perturbation procedure [4] and the d–d transition energy matrices, we obtain [6] pffiffiffi t 40Dq þ ð2V  2V 0 Þ pffiffiffi ; E e 60Dq þ ð150B þ 3V þ 11 2V 0 Þ pffiffiffiffiffi pffiffiffiffiffiffiffi r 5V þ 2 10V 0 pffiffiffi ; E e 80Dq þ 4V þ 8 2V 0 pffiffiffi t0 40Dq þ 8 2V 0 pffiffiffi ; E e0 60Dq þ 150B  13 2V 0

3. Calculations on Cd2P2S6 : Co2+ For Co2+ in Cd2P2S6, since the ionic radius ri ( [12]) of impurity is different from the (E0.72 A ( [12]) of the replaced host ion radius rh (E 0.97 A 2+ 2+ Cd , the Co –S2 distance R is unlike the ( [1]) in the pure Cd2+–S2 distance RH (E2.72 A crystal. We estimate reasonably the distance R using the approximate formula [13] ( RERH þ ðri  rh Þ=2E2:595 A:

ð13Þ

Thus, we obtain the group overlap integrals Sdp(t2g)E0.0097 and Sdp(eg)E0.0296 from the Slater-type SCF functions [14,15] and the distance R. The absorption spectrum of Co2P2S6 shows DqE  837 cm1 [16]. The pressure experiment for NiO [17] and the theoretical studies based on the molecular orbital calculation for 3dn ions in many crystals [18,19] show that DqpR5 is approximately valid. So, from the Co2+–S2 distance R0 ( [1]) in Co2P2S6 and the R value in (E2.509 A Cd2P2S6 : Co2+, we obtain for Cd2P2S6 : Co2+ DqE  700 cm1 :

ð14Þ

Considering that the Racah parameter B in an octahedral site is greater than that in a tetrahedral site, we take BE690 cm1 according to the data in Ref. [20] and CE3.92B as in the case of free ion. For free Co2+ ion, B0E1115 cm1, C0E4366 cm1 and z0d E533 cm1 [21], and for free S2 ions, z0p E365 cm1 [22]. Thus, we have fg E0.6209 and so

pffiffiffiffiffi pffiffiffi t00 2 5V  10V 0 pffiffiffi ; E e00 20Dq þ 150B  V þ 3 2V 0 pffiffiffiffiffi s 2 10V 0 pffiffiffi ; E e0 180Dq þ 2V 0 pffiffiffiffiffi s0 4 10V 0 pffiffiffi : E N1Z 120Dq þ 150B þ 14 2V 0

31

ð12Þ

Obviously, when the optical spectral parameters B, C, Dq, V and V 0 (note: the V and V 0 are often calculated from the structural data of the studied system because it is difficult to measure them ) are obtained, the CI and CO coefficients and hence the factors g8 and g> can be calculated. Noteworthily, if the CO effect is neglected, i.e., lg ¼ 0; Ng ¼ 1;

zE462 cm1 ; kE0:900;

z0 E384 cm1 ;

k0 E0:686:

ð15Þ

From the superposition model [23], the trigonal field parameters can be expressed as V ¼ ð18=7ÞA% 2 ðRÞð3 cos2 y  1Þ þ ð40=21ÞA% 4 ðRÞð35 cos4 y  30 cos2 y þ 3Þ þ ð40O2=3ÞA% 4 ðRÞ sin3 y cos y;

ð16Þ

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W.-C. Zheng, S.-Y. Wu / Physica B 307 (2001) 28–33

Table 1 EPR g factors for Co2+ in Cd2P2S6 crystal

g8 g>

Calculationa

Calculationb

Experiment

5.18 4.12

5.01 3.90

4.99–5.00 [2], 4.86 [3] 3.94–3.99 [2], 4.01 [3]

a

Calculated neglecting the CO effect, i.e., lg ¼ 0; Ng ¼ 1; k ¼ k ¼ 1; z ¼ z0 ¼ z0d : b Calculated using the formulas in this paper, i.e., considering the CO effect. 0

V 0 ¼ ð6O2=7ÞA% 2 ðRÞð3 cos2 y  1Þ þ ð10O2=21ÞA% 4 ðRÞð35 cos4 y  30 cos2 y þ 3Þ þ ð20=3ÞA% 4 ðRÞsin3 y cos y; where A% 2 ðRÞ and A% 4 ðRÞ are the intrinsic parameters. For 3dn ions in octahedra, A% 4 ðRÞE ð3=4ÞDq [23]. The ratio A% 2 ðRÞ=A% 4 ðRÞ is in the range of 9–12 for 3dn ions in many crystals [24–25] and we take A% 2 ðRÞE9A% 4 ðRÞ here. Since in Cd2P2S6 lattice the metal coordination is trigonal elongated by about 2–31 from X-ray diffraction measurements [1], we take y E52.741 here. Substituting all these parameters into the above formulas, the parameters g8 and g> of Cd2P2S6 : Co2+ are calculated. The results show good agreement with the observed values (see Table 1). For comparison, the calculated values by neglecting the CO effect are also shown in Table 1.

4. Discussions From Table 1, one can find that if the contribution of CO effect is neglected, the calculated g8 and g> are larger than the observed values. So, in the exact theoretical calculation, the contribution of CO effect should be taken into account. The group overlap integrals Sdp ðgÞ calculated by using the Slater-type SCF wavefunctions in this paper are smaller than those reported for transition-metal complexes calculated with functions of Hartree–Fock quality [8,26–28]. This may indicate that the right tail of the 3d function is not well reproduced by the approximated wavefunctions

employed in the present paper. However, according to the calculations, the changes in Sdp ðgÞ result in only very small changes in the calculated values of g factors, e.g., no more than 2% and 4% errors are found for g8 and g> ; respectively, in despite of the increases for Sdp ðgÞ by about 10 times. So, we employ the approximated wavefunctions here. For dn ions in crystals, the contributions to g factors come from the crystal-field (CF) mechanism related to the influence of CF excitations and the charge-transfer (CT) mechanism related to influence of CT excitations [8,29]. Since the CT bands in energy are higher than the CF bands [30], their contribution to the g factors of ground state may be small and is not considered in the above calculation. However, the contribution of CT mechanism increases with the increasing covalency of the system under study and so it is more important when the ligand is S2 ion than when the ligand is O2 ion. So, the neglecting of the contribution due to the CT mechanism may result in the calculated errors. Even so, considering that by using the above formulas based on the CF mechanism, the g8 and g> for Cd2P2S6 : Co2+ can be satisfactorily explained from the structural data, these formulas can be regarded as reasonable. In the calculation, we find that when Dyo0; Dg > 0 and when Dy > 0; Dgo0: The relationship is opposite to that in Ref. [2]. The following points support our relationship: (i) For 3dn ion in trigonal octahedra, we have [31,32]

pffiffiffi qDg DgEðy  y0 Þ E  3 2ðy  y0 ÞF44 ; ð17Þ qy 0 where F44 is the spin-lattice coupling coefficient in cubic symmetry. For 3d7 ions in octahedra, we have F44>0, e.g., F44E10 for MgO : Co2+ [33]. Thus, we obtain when Dyo0; Dg > 0: (ii) For giving the evidence in favour of their relationship between the signs of Dg and Dy; Hefni et al. [2] suggested that the metal coordination in cobalt fluorosilicate (CoSiF6 6H2O) is trigonal compressed and that in CdCl2 : Co2+ is trigonally elongated. However, these suggestions are opposite to the experimental data obtained from the Xray diffraction for CoSiF6 6H2O [34] and from

W.-C. Zheng, S.-Y. Wu / Physica B 307 (2001) 28–33

EPR studies for CdCl2 by using 3dn ions as probes [35]. In fact, the values of Dy and also Dg for CoSiF6 6H2O and CdCl2 : Co2+ support the relationship in the present paper rather than that in Ref. [2]. (iii) Nagasundaram and Francis, two of the authors of Ref. [2], thought that for Cd2P2S6 : Co2+ the ground state is 4A2 [16]. However, the calculations based on the crystal-field theory related to the structural data show that for Cd2P2S6 : Co2+ (elongated octahedra), the ground state is 4E rather than 4A2. The misassignment of the ground state may result in the incorrect relationship between the signs of Dg and Dy obtained in Ref. [2].

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