Theoretical calculations of spin-Hamiltonian parameters for the rhombic-like Mo5+ centers in KTiOPO4 crystal

Physica B 430 (2013) 27–30

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Theoretical calculations of spin-Hamiltonian parameters for the rhombic-like Mo5 þ centers in KTiOPO4 crystal Mei Yang a,c, Zheng Wen-Chen b,n, Liu Hong-Gang b a

School of Physics & Electronic Engineering, Mianyang Normal University, Mianyang 621000, PR China Department of Material Science, Sichuan University, Chengdu 610064, PR China c Research Center of Laser Fusion, CAEP, Mianyang 621900, PR China b

art ic l e i nf o

a b s t r a c t

Article history: Received 15 May 2013 Received in revised form 13 August 2013 Accepted 14 August 2013 Available online 24 August 2013

The spin-Hamiltonian parameters (g factors gi and hyperfine structure constants Ai, were i ¼ x, y and z) for Mo5 þ ion occupying the Ti(1) site with approximately rhombic symmetry in KTiOPO4 crystal are calculated from the high-order perturbation formulas based on the two-mechanism model. In the model, not only the contribution due to the conventional crystal-field (CF) mechanism, but also those due to the charge-transfer (CT) mechanism are included. The six calculated spin-Hamiltonian parameters with four adjustable parameters are in reasonable agreement with the experimental values. The calculations show that for more accurate calculations of spin-Hamiltonian parameters of the high valence dn ions (e.g., Mo5 þ considered here) in crystals, the contribution from CT mechanism, which is ignored in the conventional crystal field theory, should be taken into account. The reasonable crystal field energy levels of Mo5 þ in KTiOPO4 are also predicted from calculations. & 2013 Elsevier B.V. All rights reserved.

Keywords: Spin-Hamiltonian parameters Electron paramagnetic resonance Crystal- and ligand-field theory Charge-transfer mechanism KTiOPO4:Mo5 þ

1. Introduction Potassium titanyl phosphate KTiOPO4 (KTP) is an important optical crystal for a wide range of nonlinear optical applications, such as optical parametric oscillation (OPO), optical parametric amplifier (OPA) and second harmonic generation (SHG) because of its many favorable properties like large nonlinear optical coefficients, wide transparency, high optical damage threshold and good mechanical properties [1–5]. Since the transition metal (dn) impurities can influence the optical, electric and magnetic properties of crystals and can be used as a probe to study structural properties of the host crystals, the spectroscopic studies of KTP doped with transition metal ions have aroused interest [6–12]. The electron paramagnetic resonance (EPR) spectra of Mo5 þ -doped KTP crystal were measured at TE 77 K [11]. From the measurement [11], a Mo5 þ center due to the substitutional Mo5 þ at the octahedral Ti4 þ (1) position was found and its g factors gi (i¼ x, y and z, the principal values of g factor and corresponding direction cosines relating the principal axes to the crystallographic axes are given in Ref. [11], this shows that the rhombic system exhibits approximately the first-kind rhombic distortion named in Ref. [13]) were given. In addition, the hyperfine structure constants |A ? | (E|Ax| E|Ay|)| and |A//| (E |Az|) of this Mo5 þ center were also obtained from the EPR experiment for the powder spectra [11].

n

Corresponding author. Tel.: þ 862 885 412 371; fax: þ862 885 416 050. E-mail address: [email protected] (Z. Wen-Chen).

0921-4526/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2013.08.021

The site symmetry of Ti4 þ (1) site is triclinic (C1) [12–14]. However, if the triclinic distortion may be considered as small, the Mo5 þ center at Ti4 þ (1) site may be regarded as rhombic-like. Since the observed gz 4gx (and gy), it can be expected that the ground state of rhombic-like Mo5 þ center is |dxy〉 [11–15]. On the basis of the rhombic symmetry approximation and in terms of the following simple second-order perturbation formulas of g factors for d1 ions in rhombic symmetry with the ground state |dxy〉 2λ gx  ge  ; E1

2λ gy  ge  ; E2

8λ gz  ge  E3

ð1Þ

(where ge E2.0023, the g factor of free electron, λ ( ¼ζ for d1 ions [16]) is the spin–orbit parameter), Chandrasekharan et al. [11] estimated the crystal field energy levels E1 [ ¼E(|dxz〉)  E(|dxy〉)], E2[ ¼E(|dyz〉)  E(|dxy〉)] and E3 ½ ¼ Eðjdx2 y2 〉ÞEðjdxy 〉Þ by matching the observed g factors using the spin–orbit parameter for the free Mo5 þ ion, λ E1013 cm  1 and assumed reduction to 900 cm  1 due to the covalence effect. The estimated values are E1 E12679 cm  1, E2 E 15,679 cm  1 and E3 E 19,823 cm  1 (note that there is an obvious misprint in Ref. [11], the values of E1 and E3 should be exchanged). However, when we apply these values to Eq. (1), the calculated gx (E1.860), gy (E 1.887) and gz (E 1.639), in particular, gz, do not agree with the observed gi E1.868(2), 1.885(2) and 1.905 (2) for i ¼x, y and z, respectively. In addition, the calculation for the hyperfine structure constants Ai was not carried out in Ref. [11]. So far the further theoretical investigations for all these spinHamiltonian parameters (gi and Ai) have not been made. In fact, the spin-Hamiltonian parameters of dn ions in crystals come from

28

M. Yang et al. / Physica B 430 (2013) 27–30

two mechanisms or contributions, one is the crystal-field (CF) mechanism connected with CF excited states and another is the charge-transfer (CT) mechanism related to CT excited states [15,17–19]. In the extensively-applied CF theory [20–22], the contribution to spin-Hamiltonian parameters of ground state from the CT mechanism is not considered because the CT energy levels are often much higher than the CF energy levels [23]. However, the CT energy levels lower with the increase of valence state of dn ions [23]. So in the investigations of spin-Hamiltonian parameters for dn ions with high valence state (e.g., Mo5 þ considered here) in crystals, the contribution due to CT mechanism should not be ignored and a two (CF and CT)-mechanism model should be considered. In this paper, we calculate theoretically the spinHamiltonian parameters gi and Ai for the rhombic-like Mo5 þ center in KTP crystal from the high-order perturbation formulas based on the two-mechanism model.

ð2ÞCF Ax ¼ Að1Þ þ Axð2ÞCT x þ Ax 2 Að1Þ x ¼ P CF ðκ þ 7 Þ;

Axð2ÞCT ¼ P ′CT

In consideration of the CF and CT mechanisms, the one-electron basis functions |ψγ〉 for dn octahedral clusters MX6 can be represented as the molecular orbitals (MO) being the combination of d orbitals |dγ〉 of dn ions and p orbitals |pγ〉 of ligands [15–24] jψ γ 〉 ¼ N xγ ðjdγ 〉 þ λxγ jpγ 〉Þ

ð2Þ

where the subscript γ¼ t or e denotes the irreducible representation of Oh group, and the superscript x¼a or b indicates the anti-bonding orbitals (for CF mechanism) and bonding orbitals for CT mechanism). Nγx (the normalization coefficient) and λγx (orbital mixing coefficient) are MO coefficients. By adding the spin–orbit interaction term HSOCT, the Zeeman interaction term HZeCT and hyperfine interaction term HhfCT related to CT mechanism to the conventional Hamiltonian in the CF theory, for d1 ion with the ground state |dxy〉 in a first-kind [13] rhombically distorted octahedron, the high-order perturbation formulas based on the two-mechanism model for g factors are derived in Ref. [15] and those for hyperfine constants Ai are derived here. They are CT g x ¼ g e þΔg CF x þ Δg x

ECF 1

Δg CT x ¼

þ

2 Að1Þ y ¼ P CF ðκ þ 7Þ;  3 ðg CF 14 x g e Þ

Ayð2ÞCT ¼ P CT0

2ζ CF ζ CF0 kCF0

CT g y ¼ g e þ Δg CF y þ Δg y

Δg CF y ¼

2kCF ζ CF

þ

ζ 2CF kCF 2ζ CF0 2 kCF  CF CF CF ECF E2 E3 1 E2

ζ2 g 2ζ 0 2 g  CFCF e2  CFCF 2 e CF CF E1 E3 2ðE1 Þ ðE3 Þ 2ζ CF0 kCF0 3ζ CT0 2 kCT0  CT CT ECT E1 E2 2 þ

Δg CT y ¼

ECF 2

2ζ CF ζ CF0 kCF0

8ζ CT kCT ECT 1

2ζ CT0 kCT0 ECT 2

Az ¼ Azð1Þ þ Azð2ÞCF þ Azð2ÞCT 4 Að1Þ z ¼ P CF ðκ Þ; h 7

þ

6ζ CT ζ CT0 kCT 2 ðECT 2 Þ

0

i

8ζ CT kCT ECT 1

ð4Þ

in which the superscripts and subscripts CF and CT stand for the parameters connected with CF and CT mechanisms. κ is the core polarization constant. EJCF and EJCT represent the CF and CT energy 0 levels. The spin–orbit parameters ζ, ζ , the orbit reduction factors k, k′ and the dipolar hyperfine structure constants P, P′ concerning the CF and CT mechanisms are expressed as [15] h i ζ CF ¼ ðNat Þ2 ζ 0d þ ðλat Þ2 ζ 0p =2 ζ CF0 ¼ N ae N at ðζ 0d λat λae ζ 0p =2Þ ζ CT ¼ N at N bt ðζ 0d þ λat λbt ζ 0p =2Þ; ζ CT0 ¼ N be N at ðζ 0d λat λbe ζ 0p =2Þ h i kCF ¼ ðN at Þ2 1 þ 2λat Sdp ðtÞ þ ðλat Þ2 =2   ′ kCF ¼ N at N ae 1 þ λat Sdp ðtÞ þ λae Sdp ðeÞλat λae =2 h i ′ kCT ¼ Nat Nbt 1 þ ðλat þ λbt ÞSdp ðtÞ þ λat λbt =2 h i ′ kCT ¼ Nat Nbe 1 þ λbe Sdp ðeÞ þ λat Sdp ðtÞλat λbe =2 P CF ¼ ðN at Þ2 P 0 ; P 0CF ¼ N at N ae P 0 ; P CT ¼ N at N bt P 0 ; P 0CT ¼ N at N ae P 0 ;

NXγ ½1 þ 2λXγ Sdp ðγÞ þ ðλXγ Þ2 1=2 ¼ 1

ð5Þ

ð6Þ

and the orthogonality connection [15] 1 þλaγ Sdp ðγÞ λbγ ¼  a λγ þ Sdp ðγÞ

CT g z ¼ g e þ Δg CF z þΔg z 0 0 0 0 0 0 8k ζ CF CF ζ CF2 kCF 2ζ CF ζ CF kCF Δg CF  CF CF  z ¼ CF CF ECF E E E E 3 1 2 1 3 2ζ CF ζ CF0 kCF0 ζ CF0 2 g e ζ CF0 2 g e    CF 2 2 ECF 2ðECF 2ðECF 2 E3 1 Þ 2 Þ

Δg CT z ¼

h Ayð2ÞCF ¼ P ′CF ðg CF y g e Þ

where the superscript or subscript ‘0’ represents the corresponding parameter in the free state. For the (MoO6)7  cluster under consideration, we have ζ 0d (Mo5 þ )E 1030 cm  1 [21], ζ 0p (O2  ) E150 cm  1 [25] and P0(Mo5 þ )E  66.7  10  4 cm  1 [26]. Sdp(γ) are the group overlap integrals which can be calculated from the Slater-type self-consistent field (SCF) functions [27,28] with the average metal–ligand distance R. For Mo5 þ at the Ti4 þ (1) site of KTP crystal, from the average metal–ligand distance RE0.1974 nm [14], we obtain Sdp(t) E0.0409 and Sdp(e)E 0.1097. From Eq. (2), the MO coefficients Nγx and λγx required in the calculations of parameters in Eq. (5) can be related by the normalization relation [15–24]

ζ 2CF kCF 2ζ CF0 2 kCF  CF CF CF ECF E1 E3 1 E2

ζ2 g 2ζ 0 2 g  CFCF e2  CFCF 2 e CF CF E2 E3 2ðE2 Þ ðE3 Þ 2ζ CT0 kCT0 3ζ CT0 2 kCT0  CT CT ECT E1 E2 2 þ

ECT 2

ð2ÞCF Ay ¼ Að1Þ þ Ayð2ÞCT y þ Ay

Azð2ÞCT ¼ P

2kCF ζ CF

′

2ζ ′CT kCT

CF CF 3 Azð2ÞCF ¼ P CF0 ðg CF z g e Þ þ 14ðg x þ g y 2g e Þ

2. Calculation

Δg CF x ¼

h i CF 3 Axð2ÞCF ¼ P ′CF ðg CF x g e Þ14ðg y g e Þ

ð3Þ

ð7Þ

Thus, the MO coefficients λγb and Nγx can be determined when the coefficients λγb are known. Since many studies of the g factors in dn octahedral clusters [15,29–31] suggested that the coefficients λat and λae in the CF mechanism are close to each other, to reduce the number of adjustable parameter, we assume λta E λea Eλγa and let λγa be an adjustable parameter. The CT energy levels EJCT in Eqs. (3) and (4) may be obtained from the CT optical spectra, however, for (MoO6)7  (Mo5 þ  O2 

M. Yang et al. / Physica B 430 (2013) 27–30

3. Discussion

combination) clusters in crystals, no CT optical spectra were reported so far. So we estimate them approximately by comparing with the values of E1CT E 26,400 cm  1 and E2CT E30,200 cm  1 [23] of (MoCl6)  (Mo5 þ  Cl  combination) octahedral clusters in crystals. It is known that for d1 octahedral cluster, the CT energy levels EJCT depend upon the relation [χ(L)  χ(M)]  30000 cm  1 [23], where χ(L) and χ(M) denote the optical electronegativities of ligand and central metal ion. From the values of χ(Cl  )E3.0 and χ(O2  )E 3.2 [23] and the above EJCT of (MoCl6)  cluster, we have E1CT E 32400 cm  1 and E2CT E36,200 cm  1 for (MoO6)7  clusters. The CF energy levels EJCF (i.e., EJ in Ref. [11]) may be estimated from the optical spectra (d–d transitions). Since no the optical spectra of CF KTP:Mo5þ were reported, we take E1p (¼E(|d 〉)E(|d xzp xy〉)¼ ffiffiffiffiffi ffi ffiffiffi 3Ds þ 5Dt þ 3D 4D ¼3=7B 5=21B þ 70B44 þ 6=7B22 þ ξ η 20 40 pffiffiffiffiffiffi 2 10=21B42 , where Bkl are the crystal-field parameters in Wybourne notation [32,33]) and E2CFp(¼E(|d þ ffi5Dt 3Dξ þ ffiffiffiffiffiffi yz〉) pffiffiffiE(|dxy〉)¼3Ds pffiffiffiffiffi 4Dη ¼3=7B20 5=21B40 þ 70B44 þ 6=7B22 þ 2 10=21B42 ) as the adjustable parameters (note: if the above four crystal field parameters are treated as the adjustable parameters, which can result in large uncertainty because only two optical band positions are required in the calculations of spin-Hamiltonian parameters, so we take E1CF and E2CF as the adjustable parameters for decreasing the number of adjustable parameter) and estimate E3 CF ¼ Eðjdx2 y2 〉Þ Eðjdxy 〉Þ ¼ 10Dq [15] as follows. By studying the spectrochemical series for metal ion and ligand in dn clusters, Jørgensen [34] factored the Dq value into a ligand factor f(L) and a metal ion factor g(M) as Δ ¼ 10Dq  f ðLÞg ðMÞ

The absolute values of hyperfine structure constants |A//|E88(2)G E79(2)  10  4 cm  1 (with the observed g// ¼ gz E1.905) and |A ? | E43(2)G E38(2)  10  4 cm  1 (with the observed g ? E(gx þ gy)/ 2E1.877) were only given in Ref. [11]. Our calculations suggest that all the hyperfine structure constants Ai in KTP:Mo5 þ are positive (see Table 3). This is consistent with the data in Refs. [26,36–38] and can be regarded as reasonable. From Table 3, one can find that the difference Ax Ay (E1.1  10  4 cm  1) is within the experimental error (E2  10  4 cm  1 [11]), this explains the observed values A ? EAx EAy in the powder KTP:Mo5 þ [11]. The small difference between Ax and Ay (and also between gx and gy) suggests that the rhombic-like distortion of the (MoO6)7  cluster in KTP:Mo5 þ is small. From the CF theory [15,23], the rhombic crystal field splits the energy level E(dxz;yz 〉) in  tetragonal   symmetry into two energy levels E1CF(dxz ) and E2CF (dyz ) and so the rhombic distortion can be pffiffifficharacterized pffiffiffiffiffiffiby the parameter δð ¼ E2 CF E1 CF ¼ 6Dξ 8Dη ¼ ð2 6=7ÞB22 þ ð4 10=21ÞB42 , if there is no rhombic distortion, the parameter δ¼0). From Eq. (9), our value of δ used in the calculation is 1250 cm  1, which is smaller than that (¼ E2  E1 E3000 cm  1) applied in Ref. [11]. So our CF energy levels E1CF, E2CF (which correspond to the smaller rhombic distortion) and E3CF ( ¼10Dq, which is estimated reasonably from the other Mo5 þ octahedral cluster, as mentioned above) can be regarded as more rational than those given in Ref. [11]. The core polarization constant κ (E0.99) used in the above calculation is close to those for Mo5þ ions in others octahedral clusters (e.g., κE0.91 and 0.99 for (MoCl6)  clusters in Cs2ZrCl6 and Cs2HfCl6 crystals [18]) and seems to be proper. Thus, in terms of four suitable adjustable parameters, the six spin-Hamiltonian parameters gi and Ai (i¼ x, y and z) are explained satisfactorily (see Table. 3). In Table. 3, one can see that for KTP:Mo5 þ , the signs of Δg CT and i Aið2ÞCT due to CT mechanism are opposite to those of the corresponding ð2ÞCF Δg CF due to the CF mechanism. The relative importance of i and Ai CT mechanism can be characterized by |QCT/QCF|. From the values shown in Table. 3, we have for KTP:Mo5þ , |QCT/QCF|E12%, 14%, 20%, 10%, 12% and 9% for Q¼Δgx, Δgy, Δgz, Ax(2), Ay(2) and Az(2), respectively. So, in order to calculate more accurately the spin-Hamiltonian

ð8Þ

According to the values of f(Cl  ) E0.78, f(O2  ) E1 [22–31] and DqE 2170 cm  1 [35] for (MoCl6)  octahedral cluster, we obtain DqE 2800 cm  1 for (MoO6)7  octahedral cluster. Thus, in the above formulas, four unknown parameters λγa, E1CF and E2CF and κ are left as the adjustable parameters to fit the experimental spinHamiltonian parameters. By calculating the six spin-Hamiltonian parameters gi and Ai of KTP:Mo5 þ through the above high-order perturbation formulas based on the two-mechanism model, we find that the parameters 1 1 λaλ  0:64; ECF ; ECF ; κ  0:99 1  8500 cm 2  9750 cm

ð9Þ

Table. 3 The spin-Hamiltonian parameters (g factor gi and hyperfine structure constants Ai, where iEx, y and z; the principal axis system of spin-Hamiltonian parameters is given in [11], which exhibits approximately the first-kind rhombic distortion named in [13]; Ai are in units of 10  4 cm  1) for the rhombic (MoO6)7  cluster in KTP:Mo5 þ .

can lead the calculated spin-Hamiltonian parameters to be in reasonable agreement with the experimental values. From the value of λγa, the MO coefficients λγb and Nγx obtained from Eqs. (6) and (7) are listed in Table. 1. The parameters in Eq. (5) used for the calculation of spin-Hamiltonian parameters are given in Table. 2, whereas the resulting spin-Hamiltonian parameters and the experimental ones are listed in Table. 3.

Table. 1 The molecular orbital coefficients of (MoO6)7  cluster in KTP:Mo5 þ crystal. λbt

λbe

N at

N ae

N bt

N be

1.5212

1.6226

0.8431

0.8726

0.5393

0.5007

29

ΔgxCF

ΔgxCT

gx (Calc.)

gx (Expt. [11])

 0.1536 ΔgyCF  0.1356 ΔgzCF  0.1231 Ax(1) 33.4 Ay(1) 33.4 Ay(1) 74.0

0.0186 ΔgyCT 0.0186 ΔgzCT 0.0251 Ax(2)CF 6.1 Ay(2)CF 5.0 Ay(2)CF 9.1

1.8673 gy (Calc.) 1.8853 gz (Calc.) 1.9043 Ax(2)CT  0.6 Ay(2)CT  0.6 Ay(2)CT  0.8

1.868(2) gy (Expt. [11]) 1.885(2) gz (Expt. [11]) 1.905(2) Ax (Calc.) 38.9 Ay (Calc.) 37.8 Ay (Calc.) 82.3

Ax (Expt. [11]) 38(2) Ay (Expt. [11]) 38(2) Ay (Expt. [11]) 79(2)

Table. 2 The spin–orbit parameters ζ, ζ′ (in cm  1), the orbital reduction factors k, k′ and the polar hyperfine structure constants P, P′ (in 10  4 cm  1) connected with CF and CT mechanisms for (MoO6)7  cluster in KTP:Mo5 þ crystal. ζCF

ζ′CF

ζCT

ζ′CT

kCF

k′CF

kCT

k′CT

PCF

P′CF

PCT

P′CT

757

732

433

470

0.8357

0.4903

0.2352

0.7184

 47.4

 49.1

 30.3

 31.4

30

M. Yang et al. / Physica B 430 (2013) 27–30

parameters of high valence state dn ions in crystals, the contribution due to CT mechanism should not be ignored and a two-mechanism model should be applied.

[12] [13] [14] [15] [16]

Acknowledgments

[17] [18] [19] [20]

This project is supported by the Key Foundation of Mianyang Normal University (Grant no. 2011A16), the Initial Foundation of Mianyang Normal University (Grant no. MQD2011A05) and the Science and Technology Development Foundation of China Academy of Engineering Physics (Grant nos. 2011B0302059, 2012A0302015 and 2012B0302050). References [1] W. Koechner, Solid-State Laser Engineering, 5th ed., Springer, Berlin, 1999. [2] M. Scheldt, B. Beler, R. Knappe, K.J. Boller, R. Wallensteln, J. Opt. Soc. Am. B12 (1995) 2087. [3] L. Zhang, P.J. Chandler, P.D. Townsend, Z.T. Alwahabi, S.L. Pltyana, A. J. McCaffery, J. Appl. Phys. 73 (1993) 2695. [4] F. Laurell, T. Calmano, S. Muller, P. Zeil, C. Canalias, G. Huber, Opt. Express 20 (2012) 22308. [5] H.H. Li, S.G. Li, X.H. Ma, J.T. Wang, X.L. Zhu, Chin. Phys. Lett. 29 (2012) 114215. [6] S.D. Setzler, K.T. Stevens, N.C. Fernelius, M.P. Scripsick, G.J. Edwards, L.E. Halliburton, J. Phys. Condens. Matter. 15 (2003) 3969. [7] M.G. Rolofs, J. Appl. Phys. 65 (1989) 4976. [8] S. Han, J. Wang, Y. Xu, Y. Liu, J. Wei, J. Phys. Condens. Matter 4 (1992) 6009. [9] C. Rudowicz, S.B. Madhu, I. Akhmadoulline, Appl. Magn. Reson. 16 (1999) 457. [10] S.W. Ahn, S.H. Choh, J. Phys. Condens. Matter 11 (1999) 3193. [11] K. Chandrasekharan, V.S. Murty, F.J. Kumar, P. Ramaswamy, P.S. Rao, J. Mater. Sci. 35 (2000) 1561.

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