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PHYSICAL REVIEW B 93, 245125 (2016)

Nonlinear electronic polarization and optical response in borophosphate BPO4 Zhi Li,1 Qiong Liu,1,2 Shujuan Han,1 Toshiaki Iitaka,3 Haibin Su,4,5 Takami Tohyama,6 Huaidong Jiang,7 Yongjun Dong,8 Bin Yang,9 Fangfang Zhang,1 Zhihua Yang,1,* and Shilie Pan1,† 1

Key Laboratory of Functional Materials and Devices for Special Environments of CAS, Xinjiang Technical Institute of Physics and Chemistry of CAS, Xinjiang Key Laboratory of Electronic Information Materials and Devices, 40-1 South Beijing Road, Urumqi 830011, China 2 University of Chinese Academy of Sciences, Beijing 100049, China 3 Computational Astrophysics Laboratory, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan 4 Division of Materials Science, Nanyang Technological University, 50 Nanyang Avenue, 639798, Singapore 5 Institute of Advanced Studies, Nanyang Technological University, 60 Nanyang View, 639673, Singapore 6 Department of Applied Physics, Tokyo University of Science, Katsushika, Tokyo 125-8585, Japan 7 State Key Laboratory of Crystal Materials, Shandong University, Jinan 250100, China 8 Shanghai Institute of Optics and Fine Mechanics, CAS, Shanghai 201800, China 9 Department of Physics, Harbin Institute of Technology, Harbin 150001, China (Received 29 March 2016; revised manuscript received 13 May 2016; published 13 June 2016) The electronic structure, nonlinear electronic polarization induced by a static external electric field, and frequency dependent second-harmonic susceptibility tensor of the borophosphate BPO4 are studied by a firstprinciples calculation based on density-functional theory. Our calculated results show that the borophosphate BPO4 has a large band gap ∼10.4 eV, which is larger than the band gap of the widely used nonlinear optical crystal KBe2 BO3 F2 . However, BPO4 also has a nonlinear coefficient d36 = 0.92 pm/V at static limit, which also is larger than the nonlinear coefficient d11 = 0.47 pm/V of KBe2 BO3 F2 . The unexpected larger nonlinear coefficient of BPO4 can be interpreted by the relatively strong s-p hybridization in BPO4 , which can enhance the inter-band Berry connections, while the O 2p orbitals dominating valence bands in KBe2 BO3 F2 are very flat, resulting from weak s-p hybridization. DOI: 10.1103/PhysRevB.93.245125 I. INTRODUCTION

Borates with the BO4 tetrahedron usually have a large band gap and potential applications as nonlinear optical (NLO) crystals. NLO crystals play an important role in the frequency conversion of laser light [1]. With the help of NLO crystals, the production of laser light unattainable by conventional lasers becomes possible. Borate crystals LiB3 O5 , β-BaB2 O4 , and KBe2 BO3 F2 (KBBF) are widely used as NLO crystals because of their moderate birefringence and excellent NLO effect [2–4]. Especially, KBBF has a short ultraviolet (UV) cutoff wavelength ∼150 nm, a relatively large second-order harmonic generation (SHG) d11 ∼ 0.47 pm/V, and a moderate birefringence of 0.077 [3], and it is widely used to produce laser light with wavelengths in deep UV region. Borophosphate BPO4 , as another typical deep UV NLO crystal, also has attracted lots of research interest because of its simple crystal structure, which facilitates theoretical research into the mechanism of its large NLO coefficient. The noncentrosymmetric BPO4 crystalizes into tetragonal structure with I 4 (S4 ) space (point) group. In 2011, Zhang et al. reported the refractive indices and second-order NLO coefficients of BPO4 [5,6]. They measured one of the nonvanishing NLO coefficients, d36 , which is about twice that of KH2 PO4 (KDP), while the d15 coefficient is too weak to be measured. As the archetype of many NLO crystals, BPO4 also has a relatively large second-harmonic generation (SHG) among borates with the NLO effect [7]. BPO4 has a short UV

* †

[email protected] [email protected]

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cutoff wavelength ∼134 nm, which is shorter than that of the well-known NLO crystal KBBF [3]. The shorter UV cutoff wavelength of BPO4 means that BPO4 has a larger band gap. Since the possibility of quantum transition between valence and conducting bands is proportional to the inverse of energy difference at each momentum point, BPO4 should have a weaker NLO response resulting from the larger band gap. However, BPO4 has an even larger NLO coefficient than that of KBBF. The electronic structure of BPO4 has always been studied by first-principles calculation before [6]. However, the origin of the large NLO coefficient of d36 of BPO4 and the external field dependent NLO response are still absent. The NLO response of large band gap insulators is essentially connected with the nonlinear electronic polarization [8–10]. Currently, it is widely accepted that the Berry phase of occupied Bloch wave functions, i.e., the accumulated adiabatic flow of current occurring as the wave functions are deformed or perturbed, is the theoretical cornerstone of electronic polarization. The Berry phase and Berry curvature afford new interpretations of many interesting phenomena in condensed matter physics, such as ferroelectricity, piezoelectric effects, the (anomalous) quantum Hall effect, and the circular photogalvanic effect [11–16]. The nonlinear electronic polarization induced by an external laser light field also can be interpreted with the concept of the Berry phase. Additionally, the optical response can also be adopted to measure the dynamical Berry curvature of metallic materials with time-reversal invariant symmetry [17–22]. In this work, we studied the electronic structure, external field induced nonlinear electronic polarization, and the frequency dependent NLO property by first-principles calculation. The calculated band gap of BPO4 is about 10.4 eV, which

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is consistent with the experimental UV cutoff edge of BPO4 , ∼134 nm. By the comparison of band structures between BPO4 and KBBF, the large SHG of BPO4 can be interpreted by the relatively strong s-p hybridization in BPO4 , which can enhance the interband Berry connections, while the O 2p orbitals dominating valence bands in KBe2 BO3 F2 are very flat, resulting from weak s-p hybridization. The Berry phase calculation also predicts a parabolic electronic polarization along the [001] direction under static external electric field along the [110] direction. By the Berry phase method with static external electronic field along the [110] direction, the second-order optical susceptibility χ2 is about 4.0 pm/V, while the NLO coefficient 2d36 = χ312 (−2ω; ω,ω) is about 1.84 pm/V by sum-over-states (SOS) approximation at the static limit. Extracting the rectified polarization, χ2 contributed by the double frequency polarization is about 2.0 pm/V. The calculated results by both methods are qualitatively consistent with the experimental one, i.e., 1.52 pm/V.

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Ef

II. COMPUTATIONAL METHOD

The first-principles calculations for electronic structure employed the all electron, full-potential linearized augment plane wave (FPLAPW) method with modified Becke-Johnson local density approximation (MBJLDA) implemented in WIEN2K code [23–26]. The spin-orbital coupling (SOC) is not included in the self-consistent calculations since it should be very weak for light elements, and only the orbital effect is important in BPO4 . The NLO coefficient was calculated by ABINIT code with norm-conserving pseudopotentials and an energy cutoff of 45.0 hartree [27–32]. A dense k-point sample, 24 × 24 × 24, was adopted for the nonlinear electronic polarization with the Berry phase method and frequency dependent SHG with the SOS approximation. III. RESULTS AND DISCUSSION

Electronic structure. The calculated band structure by MBJLDA is shown in Fig. 1, in which the weights of O 2p orbitals and s orbitals from B and P are dyed by blue and red colors, respectively. The MBJLDA calculation renders a band gap ∼10.6 eV, while the band gap by the first-principles calculation with the general gradient approximation (GGA) is about 8.0 eV. The MBJLDA band gap is slightly larger than the optical band gap ∼9.2 eV from optical transmittance. Because of the exciton effect, the optical band gap is smaller than the fundamental band gap, indicating that the MBJLDA band gap is reasonable, while the GGA underestimates the band gap of BPO4 . The band structure shown in Fig. 1 reveals that the valence bands near the Fermi level are mainly from the O 2p orbitals (in blue), and the lowest conducting band is mainly from the s orbitals of B and P(in red). The lowest conducting band is only dispersive around the point. Around the point, one of the valence bands (at −1.6 eV) is obviously dispersive, resulting from the hybridization between the O 2p orbitals and the s orbitals from P and B. This band character near the Fermi level is ubiquitous in borates, as in the band structure of KBBF [3]. However, in the band structure of KBBF, the valence bands are extremely flat because of minute hybridization with other orbitals. The electronic structures of

FIG. 1. The band structures of BPO4 calculated by MBJLDA. The O p orbitals are in blue, while both B and P s orbitals are in red. The unit cell of BPO4 and the Brillouin zone of a primitive cell are shown in the inset.

BPO4 and KBBF can be described by a model Hamiltonian, ds (k) V (k) . (1) H0 (k) = V † (k) dp (k) Here, ds (k) > 0 [dp (k)] is a diagonal matrix describing the pure s orbitals (p orbitals), and complex V (k) = dx (k) − idy (k) is the hybridization between s orbitals and p orbitals. Here, we take the p as constant and ignore the direct hybridization between O 2p orbitals because of large O-O bond length. For simplification, we assume that there is only one p orbital dominating valence band and one s orbital dominating conducting band. However, the physics picture is no different if multiple p orbitals dominating valence bands are adopted. If the hybridization term V (k) is vanishing, the model Hamiltonian has eigenvalues ds (k), dp (k) and eigenvectors e1 = (1,0)T , e2 = (0,1)T , respectively. With these eigenvectors, both the interband and intraband Berry connections Amn (k) = iem |∇k |en are vanishing over the whole Brillouin zone (BZ). In realistic borate materials, the hybridization term is not vanishing, and it will mix the low energy p orbitals with high energy s orbitals into the p orbitals dominating the valence band. The hybridization term also populates the low energy p orbitals into the s-dominating conducting band. With broken spatial inversion symmetry, the hybridization term V (k) is complex. With a nonvanishing hybridization term, the eigenvalues ε± (k) and eigenvectors

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u± (k) for the two-band model Hamiltonian read ds (k) 2 ds (k) ± (k) = + V + (k)V (k), ± 2 2 u+ (k) = (cos α(k), sin α(k)eiβ(k) )T , u− (k) = ( − sin α(k)e−iβ(k) , cos α(k))T , + (k) cos α(k) = , +2 (k) + V + (k)V (k) tan β(k) =

dy (k) , dx (k)

respectively. In BPO4 , |V (k)| ds (k), so α(k) ∼ 0. The hybridization V (k) between s orbitals and p orbitals will create an electron-hole pair with parabolic dispersion. From the band structure of BPO4 , we can see that this electron-hole pair with parabolic dispersion is around the point. Now, the intraband and interband Berry connections read [33] A++ (k) = − sin2 α∇k β, A−− (k) = sin2 α∇k β, A+− (k) = − exp(−iβ)(i∇k α + sin α cos α∇k β), A−+ (k) = − exp(iβ)(−i∇k α + sin α cos α∇k β), respectively. With time-reversal invariant symmetry, the intraband connection always is real and even with respect to while the real part and imaginary part of the momentum k, interband connection are even and odd, respectively. If the spatial inversion symmetry is preserved, then β(k) is constant, and the expression for the Berry connection becomes much simpler. With |u− (k) as the occupied band, the ith component of spontaneous electronic polarization P0i is i d 3 kAi−− (k) e, (2) 0 P0 = e BZ

where e is the charge of electron. Also, there is no spontaneous electronic polarization if spatial inversion is preserved. From the above discussion, we know that the interband Berry connection is determined by the hybridization term. Strong hybridization will enhance the interband Berry connection. In the calculated band structure of borates, strong hybridization also means dispersive p orbitals dominating the valence band. The band structures of both BPO4 and KBBF have dispersive s orbitals dominating the conducting band around the . However, the p-dominating valence bands in KBBF are very flat near the Fermi level. In contrast, BPO4 has relatively dispersive valence bands resulting from relatively strong s-p hybridization which can enhance the interband Berry connection and contribute a relatively large SHG effect in BPO4 , though the band gap of BPO4 is larger. Polarization under static field. With dipole approximation the model Hamiltonian including the static light-matter interaction reads Hint = − Pmac · E,

(3)

where Pmac is macroscopic polarization including contributions from electrons and ions, and is the volume of unit cell. Since the NLO effect is essentially related to the

FIG. 2. Calculated electronic polarization under a static external electric field along the [111] direction by the Berry phase approach (a), and calculated refractive indices by the derivation of polarization with respect to field strength (b).

nonlinear polarization of material, it is significant to study the electronic polarization of BPO4 under an external static electronic field. In principle, we can construct ting-binding (TB) model as above to calculate the external dependent electronic polarization. However, in realistic NLO crystals like BPO4 and KBBF, there are many bands near the Fermi level, and the two-band model is never enough to fully describe the real borates. Here, we calculate the static external field induced electronic polarization by the Berry phase of occupied Bloch states which minimize the electric enthalpy functional F [34–36]. The functional F is expressed as = EKS − Pmac · E. F[un (k); E]

(4)

The field dependent wave functions un (k) are solved by the variational method. The electronic polarization is expressed as the Berry phase of occupied Bloch states, i.e., u (k)|i∇ n k |un (k). Although a different external field is BZ applied, the charge polarization from ions is constant, so we ignore the charge polarization from ions. With the applied static external electric field along [111] direction, the calculated electronic polarization is shown in Fig. 2(a). Because of the dominating linear response, which is of order 10−4 a.u., the nonlinear electronic polarization of order 10−8 a.u. is overwhelmed by the linear polarization. The calculated polarizations reveal that the field induced electronic polarizations Px and Py along [100] and [010], respectively, are identical because of the S4 point group, while the polarization

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FIG. 4. Calculated incident photon energy dependent SHG by SOS approximation. FIG. 3. Calculated electronic polarization with different external static field along [110], [1 − 1 0], [101] directions (a), respectively, and along the [100] direction by the Berry phase approach (b).

Pz along the [001] direction is slightly larger than Px and Py . By the first-order derivative of polarization with respect to the field strength, the dielectric susceptibility χ1 and the refractive indices nx (= ny ) and nz can be obtained. The field strength dependent refractive indices are shown in Fig. 2(b), indicating minute birefringence ∼0.008, which is very close to the experimental value 0.005 at the static limit. With the applied external electric field along the [110] and [101] directions of the BPO4 crystal, as shown in Fig. 3(a), the calculated electronic polarizations Pz and Py demonstrate nonlinear polarization. The calculated field strength dependent polarizations are roughly parabolic, though a third-order term also is not ignorable. By the difference of polarization at E = ±8 × 10−4 a.u., the third-order polarization is estimated to be about 2.3 × 10−8 a.u., which is about 1/4 of the secondorder polarization. Especially, the spontaneous electronic polarizations along the [010] and [001] directions are not exactly vanishing because of the absence of spatial inversion symmetry. Of interest, if we apply the external field along the [1 − 1 0] direction, the direction of field induced polarization is inverted compared with the direction of polarization induced by a field along the [110] direction. This inverted direction of polarization implies that the dominating second-order y y z z polarization P(2) (P(2) ) is in the form of P(2) = χ2 Ex Ey (P(2) = χ2 Ez Ex ). In Fig. 3(b), we show the calculated electronic polarizations Py and Pz with applied electric field along the [100] direction. However, the second-order polarization

is not obvious compared with the transverse polarization induced by external fields along the [110] and [101] directions. With Ey = 0, this absence of second-order polarization can z be interpreted by P(2) = χ2 Ex Ey = 0. By the second-order derivative of polarization with respect to the external field strength, the calculated second-order susceptibility χ2 is about 4.0 pm/V, when the external field is applied along [110] direction. Second-harmonic generation. In a NLO crystal, we are interested in the frequency dependent SHG, which is an important index to evaluate the application of the crystal. Since the calculated χ2 includes both polarizations with the frequency sum and difference at the static limit, it is not identical to the SHG susceptibility. With P 2,z = 14 χ2 (Ex eiωt + Ex e−iωt )(Ey eiωt + Ey e−iωt ), we can subtract the rectified polarization with zero frequency. The SHG χ312 (−2ω; ω,ω) (= 12 χ2 ) is about 2.0 pm/V with ω → 0. Usually, the frequency dependent SHG can be calculated by the SOS approximation with a scissor shift to deal with the underestimated band gap by GGA. With a scissor shift of 3.0 eV (approximately the band gap difference between GGA and MBJLDA), the calculated SHG by the SOS approximation is shown in Fig. 4. The calculated absolute value of the NLO coefficient d36 = 1 χ (−2ω; ω,ω) = 0.92 pm/V at the static limit, which 2 312 is very close to the result calculated by the Berry phase approach. The slightly larger SHG by the Berry phase approach results from the underestimated band gap from GGA, while a scissor shift is used to reproduce the correct band gap in the SOS approximation [37]. Both the SOS and Berry phase approaches render a similar nonlinear coefficient d36 , which is slightly larger than the experimental value, 0.76 pm/V. The

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calculated d15 = 12 χ113 (−2ω; ω,ω) by the SOS approximation is about 0.21 pm/V, which is consistent with the minute field induced nonlinear electronic polarization calculated by the Berry phase method. Since the intensity of generated light with doubled frequency is proportional to the square of nonlinear coefficients, the small nonlinear coefficient d15 at the static limit may be unobservable [5].


[110] ([101]). With a static external electronic field along the [110] direction, the second-order optical susceptibility χ2 is about 4.0 pm/V, while the SHG χ312 (−2ω; ω,ω) is about 1.84 pm/V by the SOS approximation at the static limit. Both the calculated results are qualitatively consistent with the experimental results, i.e., 1.52 pm/V, and larger than the SHG of KBBF, χ111 (−2ω; ω,ω) = 2d11 = 0.94 pm/V at the static limit, although BPO4 has a large band gap.

IV. SUMMARY

In summary, we studied the band structure and NLO property of the large band gap insulator BPO4 by first-principles calculation. The calculated band structure reveals that the O 2p orbitals dominating the valence bands are relatively dispersive by comparison with the band structure of KBBF. The dispersive valence bands imply relatively strong s-p hybridization which can enhance the interband Berry connection in BPO4 . With the Berry phase approach, our calculation reveals that a relatively strong nonlinear electronic polarization along [001] ([010]) can be induced by a static external electric field along

This work is supported by the Recruitment Program of Global Experts (1000 Talent Plan, Xinjiang Special Program), the Director Foundation of XTIPC, CAS (Grant No. 2016RC001), the Funds for Creative Cross and Cooperation Teams of CAS, the National Basic Research Program of China (Grant No. 2014CB648400), and the National Natural Science Foundation of China (Grants No. 11104344, No. 51425206, No. U1129301, and No. 51172277).

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ACKNOWLEDGMENTS

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