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First-principles calculations of electronic and optical properties of F, C-codoped cubic HfO2
Journal of Magnetism and Magnetic Materials 375 (2015) 61–64
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
First-principles calculations of electronic and optical properties of F, C-codoped cubic HfO2 Yu-Fen Zhang a,n, Hao Ren a, Zhi-Tao Hou b a b
School of Chemistry and Chemical Engineering, University of Jinan, Jinan 250022, PR China Cooperation Development Department, Shandong University, Jinan 250100, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 12 July 2014 Received in revised form 21 September 2014 Available online 23 September 2014
First-principles calculations based on DFTþ U were performed on electronic and optical properties of F, C-codoped cubic HfO2. The calculations show that strong 2p–2p/5d admixtures result in half-metallic ferromagnetism behaviors of F, C-codoped cubic HfO2. Both the direct 2p–2p interaction and the indirect 2p–5d/2p–2p coupling interactions can be expected to contribute to the long-range magnetic coupling. Meanwhile, F and C codoping induces obvious increase of refractive index and new steep absorption peaks at lower energy region ∼2.8 eV, which can be used for photoabsorption applications. & Elsevier B.V. All rights reserved.
Keywords: DFT Electronic and optical properties Codoping Cubic HfO2
1. Introduction As a transition metal oxide, hafnium dioxide (HfO2) is widely studied due to its excellent dielectric properties, wide band gap, high bulk modulus, high melting point and good thermal stability [1–3]. At ambient pressure, HfO2 exists in three polymorphs: monoclinic (P21/c ) at low temperature, tetragonal (P42/nmc) at around 2000 K, and cubic fluorite (Fm3m) at about 2900 K [4]. However, at lower temperatures the cubic phase of HfO2 can be stabilized by element doping. Thus, this phase is in fact the stable structure and can be used in many industrial applications [5–7]. With the addition of small amounts of impurities, various structures of HfO2 has been used as the high-k dielectric films [8], optical and protective coatings [9]. More recently, the resistive switching phenomena were observed in the HfO2 thin film [10] and the doped HfO2 films [11]. Compared to traditional charge-based semiconductor devices, spintronics devices have attracted great interest due to their unique advantages such as being more powerful, endurable and efficient. For use in applications, materials that can both generate and manipulate electronic spin at room temperature (RT) or above are essential. Recently, unexpected ferromagnetism called d° ferromagnetism (FM) has been successively observed in undoped HfO2 and ZrO2 [12] , TiO2 [13] . Meanwhile, C (or N) substitution has also received much attention as potential ways to realize n
Corresponding author. E-mail address: [email protected] (Y.-F. Zhang).
http://dx.doi.org/10.1016/j.jmmm.2014.09.043 0304-8853/& Elsevier B.V. All rights reserved.
ferromagnetism in ZrO2 and TiO2 [14–16] , which challenge the traditional understanding for the origin of ferromagnetism. However, substitutional C (or N) dopants generate oxygen vacancies, and induce band gap narrowing [14,17]. It was repoted that F has a large beneficial effect on charge trapping, and can fill up oxygen vacancies by creating a shallow donor state [17]. So it can be speculated that codoping of low concentrations of F and C impurities in cubic HfO2 could induce a different electronic structure and physical properties that may lead to new applications. It is therefore timely to investigate the electronic structures of these systems. To date, few works have been focused on the electronic and optical properties of F, C-codoped HfO2. For recent years, HfO2 based materials have attracted many experimental and theoretical investigations, in which, the method of density functional theory (DFT) has been successfully used in predicting crystal structures and properties of undoped and doped cubic HfO 2 [2,17–19]. In this paper, we carried out first-principles calculations based on DFT to investigate the electronic structures of the F, C-doped cubic HfO2 (denoted c-HfO2:F, C) and to find out the probable relations between electronic structures and optical properties.
2. Computational method and details The first-principles electronic structure calculations based on DFT [20] within CASTEP code [21] were carried out to determine the stability and electronic structures of undoped and F,
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Y.-F. Zhang et al. / Journal of Magnetism and Magnetic Materials 375 (2015) 61–64
C-codoped cubic HfO2. In the calculations, the electron–ionic core interaction is represented by the Vanderbilt ultrasoft pseudopotential [22]. The Hf (5d, 6s), O (2s, 2p), F (2s, 2p) and C (2s, 2p) levels are treated as valence states. To treat electron exchange and correlation, we chose the Perdew–Burke–Ernzerhof formulation of the generalized gradient approximation (GGA-PBE) [23] taking into account the on-site Coulomb repulsive interaction. A cutoff energy of 550 eV for undoped and F, C-codoped cubic HfO2 in the plane wave expansion was employed in the calculations. A 2 2 2 (or 5 5 5) Monkhorst–Pack grid [24] was used for integration over the irreducible part of the Brillouin zone of the F, C-codoped cubic HfO2 (or undoped system). Good convergence was achieved with the cutoff energy and number of k points. The default convergence criteria of CASTEP was applied within energy tolerance 5.0e–6 eV/atom, max force tolerance 0.03 eV/Å, max displacement tolerance 5.0e–4 Å and max stress tolerance 0.02 GPa. To explore the electronic and optical properties of these solid solutions, we employed a 96-atoms supercell (containing 2 2 2 full cubic cells) with the starting configuration of cubic HfO2 suggested in Ref. [18]. We substituted O atoms by C and F atoms to model substitutional C and F impurities in cubic HfO2, the resulting supercell is Hf32F4C4O56. For the F, C-codoped system, the C concentration is 6.25%, which results from evenly substituting four O atoms with four C atoms in a 96-atoms supercell. There is only one anion site in cubic HfO2, so subsitituting O atom with C does not need to consider the site preference. In this system, the F concentration is also 6.25%, which results from evenly substituting four O atoms with four F atoms in the 96-atoms C-doped supercell. There are sereval O sites in C-doped cubic HfO2, so subsitituting O atom with F needs to consider the site preference. We modeled two structures of F, C-codoped cubic HfO2, including structure (I) (F and C share a Hf atom at a distance of ∼4.59 Å) and structure (II) (F and C share no Hf atom at a distance of ∼5.30 Å). Meanwhile, the calculations show that the structure (I) energetically favors over the structure (II) by 182 meV per F/C. This implies that the admixture of F and C atoms will obviously stablize the F, C-codoped cubic HfO2. Within the Broyden–Fletcher–Goldfarb–Shanno (BFGS) scheme [25], geometry optimization was performed allowing both cell parameter and internal coordinates relaxation. In the optical property calculation, the underestimation of the band gap (inherent in DFT calculations) was corrected by introducing a “scissors operator” (2.64 eV), by which the conduction bands positions were raised in energy prior to the interband transition strength calculation to match the general features of the measured imaginary part of dielectric function [19]. The standard DFT formulation usually fails to describe strongly correlated electrons behavior. This limitation can be corrected using the DFTþ U method, which introduces a Hubbard parameter “U” for the description of the on-site interactions of those electrons [26]. Considering that the strength of the effective onsite Coulomb repulsion interaction between Hf 5d electrons is comparable with the valence bandwidth and the screening of Hf 6s electrons, we used the DFTþU methodology with an U value of 2.0 eV to describe the interactions of Hf 5d electrons [27,28].
3. Results and discussions 3.1. Structural stability and population analysis The calculated equilibrium structural parameters and the results of population analysis of undoped and F, C-codoped cubic HfO2 structures are summarized in Table 1. The calculated equilibrium structural parameter a for cubic HfO2 is 5.274 Å, which is in agreement with the experimental value of 5.08 Å [29]. This can
Table 1 Calculated equilibrium structural parameters and the results of population analysis of undoped and F, C-codoped cubic HfO2 structures. Structure
a (Å)
Charge (e)/μB
Bond order (bond length, Å)
c-HfO2
5.274
O: 0.76/0 Hf: 1.52/0
Hf–O: 0.32 (2.284) In total: 10.24
c-HfO2: F, C (I)
5.297 C: 0.80/0.56 F: 0.51/0.02 O1NN: 0.74/0.04 O2NN: 0.77/0 O3NN: 0.77/0 O4NN: 0.74/0.04 HfI: 1.47–1.48/7 0.04 HfII: 1.49–1.50/0
hc-HfO2: F, C (II) 5.299 C: 0.83/0.64 F: 0.50/0 O1NN: 0.75/0.02– 0.04 O2NN: 0.76/0 O3NN-1: 0.74/0.14 O3NN-2: 0.74/0 HfI: 1.48/0∼ 0.02 HfII: 1.49/0
Hf–O: 0.28–0.32 (2.261– 2.319) Hf–C: 0.43–0.44 (2.183– 2.212) Hf–F: 0.10–0.12 (2.376– 2.404) In total: 9.49
Hf–O: 0.24–0.34 (2.226– 2.373) Hf–C: 0.46–0.48 (2.254– 2.281) Hf–F: 0.11 (2.361–2.368) In total: 9.54
demonstrate the applicability of our theoretical model in geometry optimization for HfO2. For F, C-codoped cubic HfO2, the equilibrium structural parameter a is about 0.4–0.5% larger than cubic HfO2, which can be explained by the larger radius and lower electronegativity of C compared with O atoms. From the results of population analysis, it is clearly found that substitutional codoping of F and C will greatly decrease the Mulliken charge of C atoms, slightly increase the charge of O atoms, and obviously decrease the charge of Zr atoms. This implies that F and C codoping will obviously enhance the covalent character of Zr–C bond. Spin-polarized calculations reveal the nonmagnetic character of the undoped cubic HfO2. By F and C codoping, a total magnetic moment of ∼1.0 μB per F/C is mostly contributed by the C dopant (∼0.60 μB) with some contributions from its first nearest-neighboring (NN) O atoms (∼0.2 μB). Meanwhile, the calculations show that the magnetic solutions energetically favor over the nonmagnetic solutions by 49–57 meV per F/C, which are even larger than that of Cu-doped ZnO (42 meV), which is known to be RTFM [30]. Therefore, RTFM in F, C-codoped cubic HfO2 is possible. In both F, C-codoped structures, there are two kinds of Hf atoms, the C-connected Hf (HfI) and the C-unconnected Hf (HfII). In structure (I), the neighboring O atoms of C dopant can be numbered based on their distances from C dopant, including the first-NN to the fourth-NN O atoms (denoted O1NN, O2NN, O3NN and O4NN) at distances of about 2.69, 3.75, 4.59 and 5.30 Å, respectively. It is found that both O1NN and O2NN share Hf with C dopant, while O3NN and O4NN share no Hf with C dopant. In structure (II), the neighboring O atoms of C dopant can be numbered based on their distances from C dopant, including the O1NN, O2NN, O3NN (O3NN-1 and O3NN-2) at distances of about 2.77, 3.75 and 4.59 Å, respectively. It is found that O1NN, O2NN and O3NN-1 share Hf with C dopant, while O3NN-2 does not share Hf with C dopant. From the calculations, it is found that the obvious magnetic moments of the O1NN in both F, C-codoped structures suggest that the wave function of the C dopant extends to its O1NN, which are polarized in the same direction with that of the C dopant, hinting
Y.-F. Zhang et al. / Journal of Magnetism and Magnetic Materials 375 (2015) 61–64
strong magnetic coupling occurs between the C dopant and its O1NN. The local magnetic moment may result from the interplay between the strong effective Hund's rule coupling and hybridization [31,32]. In HfO2, an O vacancy (Ov) as a donor can introduce two electrons to the system leaving cation dangling bonds. When the C and F dopants simultaneously occupy the Ov sites, the C dopant accepts two electrons to fulfill the C 2p orbitals, while the F dopant will donate one electron to the system. According to Hund′ s rules, the substitutional F and C dopants were both left with four 2p electrons, which may create 1.0 μB per F/C with a high-spin configuration. Additionally, the O4NN (sharing no Hf atom with C) in structure (I) have the obvious polarization in the same direction with that of the C dopant. This may suggest that an indirect magnetic coupling interaction is induced by the hybridization among the 2p orbitals of the C dopant, O1NN and the nearest-neighboring O atom of the O1NN (at a distance from O1NN ∼2.60 Å), and originates from the delocalized nature of 2p orbitals, and it is strong when the angle (V) of C‒O1NN‒O chain is 180° (VC-Hf–O3NN-1 ¼ 180°). It is also found that the O3NN-1 in structure (II) also have obvious polarization in the same direction with that of the C dopant. This suggests another indirect magnetic coupling interaction mediated by the holes along the C–Hf–O chain. This kind of interaction is induced by the hybridization among the C 2p, Hf 5d and O 2p orbitals, and originates from the delocalized nature of the 2p orbitals compared to the d orbitals of transtion metals, and it is strong when the angle of C–Hf–O chain is 180° (VC–Hf–O3NN–1 ¼ 180°). Taking into account the above results, both the direct 2p–2p interaction and the indirect 2p–5d/2p–2p coupling interaction can be expected to contribute to the long-range magnetic coupling in F, C-codoped cubic HfO2. The bond order between a pair of atoms is a measure of the strength of the bond. It can be calculated from the overlap integrals between the Bloch functions with an atomic basis expansion [33]. The total bond order of a crystal can be obtained by multiplying the individual bond order by the number of bonds per molecular unit. In F, C-codoped cubic HfO2 structures (shown in Table 1), it is clearly found that the Mulliken bond order for Hf– C about 0.43–0.48 (bond length ¼2.183–2.345 Å) is obviously larger than Hf–O ∼0.32 (bond length ¼2.284 Å), the bond order for Hf–F about 0.10–0.12 (bond length ¼2.361–2.404 Å) is much smaller, and the total bond order (9.49–9.54) is smaller (by ∼7%), compared with undoped system (10.24). So it can be predicted that the covalent character of F, C-codoped cubic HfO2 is lower than that of cubic HfO2.
63
Fig. 1. TDOS and PDOS of cubic HfO2. Fermi level (EF) is set to zero.
Fig. 2. Spin polarized TDOS and PDOS of F, C-codoped (I) cubic HfO2 near the Fermi level (EF). Fermi level is set to zero. Solid and dash line refer to majority and minority spin channels, respectively.
3.2. Electronic properties In Figs. 1–3, we plot the spin polarized total density of states (TDOS) and partial density of states (PDOS) spectra for both undoped and F, C-codoped cubic HfO2, in which, the Fermi level (EF) locates at 0 eV. The TDOS and the PDOS of cubic HfO2 are presented in Fig. 1 and show a nonmagnetic character. For cubic HfO2, the band gap is calculated to be about 3.06 eV, which is smaller than the experimental data of 5.7 eV [34] due to the wellknown underestimation of conduction band state energies in ab initio calculations [35]. In Fig. 1, it is found that the bandwidth of the lower valence bands is about 3.07 eV with the peak at ∼ 16.72 eV, and the bandwidth of the upper valence bands is 6.02 eV with three main peaks at ∼ 4.55, ∼ 3.84 and ∼ 1.42 eV. It is also found that the lower valence bands originate from O 2s and the upper valence bands consist of O 2p states, which shows a hybridization character with Hf 5d states. With some contributions from Hf 5p and Hf 6s states, the conduction bands are mostly contributed by Hf 5d states, which show a hybridization character with O 2p states.
Fig. 3. Spin polarized TDOS and PDOS of F, C-codoped (II) cubic HfO2 near the Fermi level (EF). Fermi level is set to zero. Solid and dash line refer to majority and minority spin channels, respectively.
Figs. 2 and 3 show the calculated spin polarized TDOS and PDOS for both F, C-codoped cubic HfO2 structures. By F and C codoping, a slight band gap reduction is observed and the impurity bands both at the top and at the bottom of upper valence bands make the width of upper valence bands obviously broaden. In Figs. 2 and 3, it is clearly found that the impurity bands near the Fermi level are attributed to C 2p states with a strong admixture of O 2p and Hf 5d states, and the impurity bands at the bottom of upper valence bands are attributed to F 2p states with a admixture
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Fig. 4. Real and imaginary part of dielectric functions of (a) undoped, (b) F, C-codoped (I) and (c) F, C-codoped (II) cubic HfO2.
of O 2p and Hf 5d states. At the top of valence bands, the majority spin states are fully occupied while the minority spin states are partially filled and cross the Fermi level, which results in halfmetallic ferromagnetism (HMF) [14] behaviors of the systems and may be the reason to cause the band gap reduction. The finite density at the Fermi level implies that the magnetic interaction in F, C-codoped cubic HfO2 can be mediated by the holes induced by F and C codoping. It is also found that the O 2p states at the top of valence band increase, while Hf 5d states at the bottom of conduction bands decrease due to the reduced interaction between Hf 5d–O 2p states compared to that in pure HfO2. This may be another reason to cause the band gap reduction. With some contributions from O 2p and C 2p states, the bottom of the conduction bands is mainly contributed by the Hf 5d states, which contribute equally for both spin channels. Figs. 2 and 3 clearly show a substantial obvious mixing among dopants 2p, O 2p and Hf 5d states. 3.3. Dielectric constants The optical properties of undoped and F, C-codoped cubic HfO2 were also calculated. For these systems, the calculated dielectric functions, including the real part ε1 and imaginary part ε2, were calculated up to 25 eV and shown in Fig. 4. Because the calculated band gap of pure cubic HfO2 (3.06 eV) is smaller than the experimental value (5.7 eV) [34], the scissor operator with the value of 2.64 eV is used for the dielectric function. The static dielectric constant ε(0) is obtained as the zero frequency limit of the real part of the frequency-dependent dielectric function. The calculated ε(0) is 4.13 for undoped, 6.52 and 5.81 for F, C-codoped cubic HfO2 structure (I) and (II), which are obviously larger than that of undoped system. A larger ε(0) means a larger refractive index [33]. A higher optical density for the F, C-codoped cubic HfO2 may have special applications in certain optical components [36]. In Fig. 4a, the ε2 plot for undoped cubic HfO2 exhibits five major absorption peaks (at 7.06, 8.24, 9.51, 10.22 and 10.88 eV, respectively), which are the absorptive transitions from the valence bands to the conduction bands. The lower energy peaks at around 7 eV are associated with the the electronic transition between the O 2p states in the upper valence bands and the Hf 5d states in the conduction bands, while the peaks at around 10 eV may be due to the transition between the O 2s and the Hf 5d states in the valence bands [2]. For both F, C-codoped structures (shown in Fig. 4b and c), the ε2 plots exhibit new major absorption peaks at ∼2.8 eV, which should be attributed to the strong orbitals mixing between C 2p and Hf 5d states (shown in Figs. 2 and 3). And the large position changes of the other absorption peaks correspond to the delocalized effects of 2p orbitals in the supercell. From the results
of Fig. 4, it can be concluded that the F, C-codoped cubic HfO2, with steep absorption peaks at lower energy region ∼2.8 eV, can be used for photoabsorption applications [37]. In summary, first-principles calculations based on DFTþ U were performed on electronic and optical properties of F, C-codoped cubic HfO2. The calculations show that the magnetic solutions energetically favor over the non-magnetic solutions by 49–57 meV per F/C in F, C-codoped systems. In the minority spin, the states near the Fermi level are attributed to C 2p states with a strong admixture of O 2p and Hf 5d states, which results in half-metallic ferromagnetism behaviors of F, C-codoped cubic HfO2. Both the direct 2p–2p interaction and the indirect 2p–5d/2p–2p coupling interactions can be expected to contribute to the long-range magnetic coupling. Meanwhile, F and C codoping induces obvious increase of refractive index and new steep absorption peaks at lower energy region ∼2.8 eV, which can be used for photoabsorption applications. Acknowledgments This work was supported by the Natural Science Foundation of Shandong Province, China (No. ZR2010BM028), the National Natural Science Foundation of China (No. 51074078), the Research Foundation of university of Jinan, China (No. XBS0922).
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Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
First-principles calculations of electronic and optical properties of F, C-codoped cubic HfO2 Yu-Fen Zhang a,n, Hao Ren a, Zhi-Tao Hou b a b
School of Chemistry and Chemical Engineering, University of Jinan, Jinan 250022, PR China Cooperation Development Department, Shandong University, Jinan 250100, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 12 July 2014 Received in revised form 21 September 2014 Available online 23 September 2014
First-principles calculations based on DFTþ U were performed on electronic and optical properties of F, C-codoped cubic HfO2. The calculations show that strong 2p–2p/5d admixtures result in half-metallic ferromagnetism behaviors of F, C-codoped cubic HfO2. Both the direct 2p–2p interaction and the indirect 2p–5d/2p–2p coupling interactions can be expected to contribute to the long-range magnetic coupling. Meanwhile, F and C codoping induces obvious increase of refractive index and new steep absorption peaks at lower energy region ∼2.8 eV, which can be used for photoabsorption applications. & Elsevier B.V. All rights reserved.
Keywords: DFT Electronic and optical properties Codoping Cubic HfO2
1. Introduction As a transition metal oxide, hafnium dioxide (HfO2) is widely studied due to its excellent dielectric properties, wide band gap, high bulk modulus, high melting point and good thermal stability [1–3]. At ambient pressure, HfO2 exists in three polymorphs: monoclinic (P21/c ) at low temperature, tetragonal (P42/nmc) at around 2000 K, and cubic fluorite (Fm3m) at about 2900 K [4]. However, at lower temperatures the cubic phase of HfO2 can be stabilized by element doping. Thus, this phase is in fact the stable structure and can be used in many industrial applications [5–7]. With the addition of small amounts of impurities, various structures of HfO2 has been used as the high-k dielectric films [8], optical and protective coatings [9]. More recently, the resistive switching phenomena were observed in the HfO2 thin film [10] and the doped HfO2 films [11]. Compared to traditional charge-based semiconductor devices, spintronics devices have attracted great interest due to their unique advantages such as being more powerful, endurable and efficient. For use in applications, materials that can both generate and manipulate electronic spin at room temperature (RT) or above are essential. Recently, unexpected ferromagnetism called d° ferromagnetism (FM) has been successively observed in undoped HfO2 and ZrO2 [12] , TiO2 [13] . Meanwhile, C (or N) substitution has also received much attention as potential ways to realize n
Corresponding author. E-mail address: [email protected] (Y.-F. Zhang).
http://dx.doi.org/10.1016/j.jmmm.2014.09.043 0304-8853/& Elsevier B.V. All rights reserved.
ferromagnetism in ZrO2 and TiO2 [14–16] , which challenge the traditional understanding for the origin of ferromagnetism. However, substitutional C (or N) dopants generate oxygen vacancies, and induce band gap narrowing [14,17]. It was repoted that F has a large beneficial effect on charge trapping, and can fill up oxygen vacancies by creating a shallow donor state [17]. So it can be speculated that codoping of low concentrations of F and C impurities in cubic HfO2 could induce a different electronic structure and physical properties that may lead to new applications. It is therefore timely to investigate the electronic structures of these systems. To date, few works have been focused on the electronic and optical properties of F, C-codoped HfO2. For recent years, HfO2 based materials have attracted many experimental and theoretical investigations, in which, the method of density functional theory (DFT) has been successfully used in predicting crystal structures and properties of undoped and doped cubic HfO 2 [2,17–19]. In this paper, we carried out first-principles calculations based on DFT to investigate the electronic structures of the F, C-doped cubic HfO2 (denoted c-HfO2:F, C) and to find out the probable relations between electronic structures and optical properties.
2. Computational method and details The first-principles electronic structure calculations based on DFT [20] within CASTEP code [21] were carried out to determine the stability and electronic structures of undoped and F,
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C-codoped cubic HfO2. In the calculations, the electron–ionic core interaction is represented by the Vanderbilt ultrasoft pseudopotential [22]. The Hf (5d, 6s), O (2s, 2p), F (2s, 2p) and C (2s, 2p) levels are treated as valence states. To treat electron exchange and correlation, we chose the Perdew–Burke–Ernzerhof formulation of the generalized gradient approximation (GGA-PBE) [23] taking into account the on-site Coulomb repulsive interaction. A cutoff energy of 550 eV for undoped and F, C-codoped cubic HfO2 in the plane wave expansion was employed in the calculations. A 2 2 2 (or 5 5 5) Monkhorst–Pack grid [24] was used for integration over the irreducible part of the Brillouin zone of the F, C-codoped cubic HfO2 (or undoped system). Good convergence was achieved with the cutoff energy and number of k points. The default convergence criteria of CASTEP was applied within energy tolerance 5.0e–6 eV/atom, max force tolerance 0.03 eV/Å, max displacement tolerance 5.0e–4 Å and max stress tolerance 0.02 GPa. To explore the electronic and optical properties of these solid solutions, we employed a 96-atoms supercell (containing 2 2 2 full cubic cells) with the starting configuration of cubic HfO2 suggested in Ref. [18]. We substituted O atoms by C and F atoms to model substitutional C and F impurities in cubic HfO2, the resulting supercell is Hf32F4C4O56. For the F, C-codoped system, the C concentration is 6.25%, which results from evenly substituting four O atoms with four C atoms in a 96-atoms supercell. There is only one anion site in cubic HfO2, so subsitituting O atom with C does not need to consider the site preference. In this system, the F concentration is also 6.25%, which results from evenly substituting four O atoms with four F atoms in the 96-atoms C-doped supercell. There are sereval O sites in C-doped cubic HfO2, so subsitituting O atom with F needs to consider the site preference. We modeled two structures of F, C-codoped cubic HfO2, including structure (I) (F and C share a Hf atom at a distance of ∼4.59 Å) and structure (II) (F and C share no Hf atom at a distance of ∼5.30 Å). Meanwhile, the calculations show that the structure (I) energetically favors over the structure (II) by 182 meV per F/C. This implies that the admixture of F and C atoms will obviously stablize the F, C-codoped cubic HfO2. Within the Broyden–Fletcher–Goldfarb–Shanno (BFGS) scheme [25], geometry optimization was performed allowing both cell parameter and internal coordinates relaxation. In the optical property calculation, the underestimation of the band gap (inherent in DFT calculations) was corrected by introducing a “scissors operator” (2.64 eV), by which the conduction bands positions were raised in energy prior to the interband transition strength calculation to match the general features of the measured imaginary part of dielectric function [19]. The standard DFT formulation usually fails to describe strongly correlated electrons behavior. This limitation can be corrected using the DFTþ U method, which introduces a Hubbard parameter “U” for the description of the on-site interactions of those electrons [26]. Considering that the strength of the effective onsite Coulomb repulsion interaction between Hf 5d electrons is comparable with the valence bandwidth and the screening of Hf 6s electrons, we used the DFTþU methodology with an U value of 2.0 eV to describe the interactions of Hf 5d electrons [27,28].
3. Results and discussions 3.1. Structural stability and population analysis The calculated equilibrium structural parameters and the results of population analysis of undoped and F, C-codoped cubic HfO2 structures are summarized in Table 1. The calculated equilibrium structural parameter a for cubic HfO2 is 5.274 Å, which is in agreement with the experimental value of 5.08 Å [29]. This can
Table 1 Calculated equilibrium structural parameters and the results of population analysis of undoped and F, C-codoped cubic HfO2 structures. Structure
a (Å)
Charge (e)/μB
Bond order (bond length, Å)
c-HfO2
5.274
O: 0.76/0 Hf: 1.52/0
Hf–O: 0.32 (2.284) In total: 10.24
c-HfO2: F, C (I)
5.297 C: 0.80/0.56 F: 0.51/0.02 O1NN: 0.74/0.04 O2NN: 0.77/0 O3NN: 0.77/0 O4NN: 0.74/0.04 HfI: 1.47–1.48/7 0.04 HfII: 1.49–1.50/0
hc-HfO2: F, C (II) 5.299 C: 0.83/0.64 F: 0.50/0 O1NN: 0.75/0.02– 0.04 O2NN: 0.76/0 O3NN-1: 0.74/0.14 O3NN-2: 0.74/0 HfI: 1.48/0∼ 0.02 HfII: 1.49/0
Hf–O: 0.28–0.32 (2.261– 2.319) Hf–C: 0.43–0.44 (2.183– 2.212) Hf–F: 0.10–0.12 (2.376– 2.404) In total: 9.49
Hf–O: 0.24–0.34 (2.226– 2.373) Hf–C: 0.46–0.48 (2.254– 2.281) Hf–F: 0.11 (2.361–2.368) In total: 9.54
demonstrate the applicability of our theoretical model in geometry optimization for HfO2. For F, C-codoped cubic HfO2, the equilibrium structural parameter a is about 0.4–0.5% larger than cubic HfO2, which can be explained by the larger radius and lower electronegativity of C compared with O atoms. From the results of population analysis, it is clearly found that substitutional codoping of F and C will greatly decrease the Mulliken charge of C atoms, slightly increase the charge of O atoms, and obviously decrease the charge of Zr atoms. This implies that F and C codoping will obviously enhance the covalent character of Zr–C bond. Spin-polarized calculations reveal the nonmagnetic character of the undoped cubic HfO2. By F and C codoping, a total magnetic moment of ∼1.0 μB per F/C is mostly contributed by the C dopant (∼0.60 μB) with some contributions from its first nearest-neighboring (NN) O atoms (∼0.2 μB). Meanwhile, the calculations show that the magnetic solutions energetically favor over the nonmagnetic solutions by 49–57 meV per F/C, which are even larger than that of Cu-doped ZnO (42 meV), which is known to be RTFM [30]. Therefore, RTFM in F, C-codoped cubic HfO2 is possible. In both F, C-codoped structures, there are two kinds of Hf atoms, the C-connected Hf (HfI) and the C-unconnected Hf (HfII). In structure (I), the neighboring O atoms of C dopant can be numbered based on their distances from C dopant, including the first-NN to the fourth-NN O atoms (denoted O1NN, O2NN, O3NN and O4NN) at distances of about 2.69, 3.75, 4.59 and 5.30 Å, respectively. It is found that both O1NN and O2NN share Hf with C dopant, while O3NN and O4NN share no Hf with C dopant. In structure (II), the neighboring O atoms of C dopant can be numbered based on their distances from C dopant, including the O1NN, O2NN, O3NN (O3NN-1 and O3NN-2) at distances of about 2.77, 3.75 and 4.59 Å, respectively. It is found that O1NN, O2NN and O3NN-1 share Hf with C dopant, while O3NN-2 does not share Hf with C dopant. From the calculations, it is found that the obvious magnetic moments of the O1NN in both F, C-codoped structures suggest that the wave function of the C dopant extends to its O1NN, which are polarized in the same direction with that of the C dopant, hinting
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strong magnetic coupling occurs between the C dopant and its O1NN. The local magnetic moment may result from the interplay between the strong effective Hund's rule coupling and hybridization [31,32]. In HfO2, an O vacancy (Ov) as a donor can introduce two electrons to the system leaving cation dangling bonds. When the C and F dopants simultaneously occupy the Ov sites, the C dopant accepts two electrons to fulfill the C 2p orbitals, while the F dopant will donate one electron to the system. According to Hund′ s rules, the substitutional F and C dopants were both left with four 2p electrons, which may create 1.0 μB per F/C with a high-spin configuration. Additionally, the O4NN (sharing no Hf atom with C) in structure (I) have the obvious polarization in the same direction with that of the C dopant. This may suggest that an indirect magnetic coupling interaction is induced by the hybridization among the 2p orbitals of the C dopant, O1NN and the nearest-neighboring O atom of the O1NN (at a distance from O1NN ∼2.60 Å), and originates from the delocalized nature of 2p orbitals, and it is strong when the angle (V) of C‒O1NN‒O chain is 180° (VC-Hf–O3NN-1 ¼ 180°). It is also found that the O3NN-1 in structure (II) also have obvious polarization in the same direction with that of the C dopant. This suggests another indirect magnetic coupling interaction mediated by the holes along the C–Hf–O chain. This kind of interaction is induced by the hybridization among the C 2p, Hf 5d and O 2p orbitals, and originates from the delocalized nature of the 2p orbitals compared to the d orbitals of transtion metals, and it is strong when the angle of C–Hf–O chain is 180° (VC–Hf–O3NN–1 ¼ 180°). Taking into account the above results, both the direct 2p–2p interaction and the indirect 2p–5d/2p–2p coupling interaction can be expected to contribute to the long-range magnetic coupling in F, C-codoped cubic HfO2. The bond order between a pair of atoms is a measure of the strength of the bond. It can be calculated from the overlap integrals between the Bloch functions with an atomic basis expansion [33]. The total bond order of a crystal can be obtained by multiplying the individual bond order by the number of bonds per molecular unit. In F, C-codoped cubic HfO2 structures (shown in Table 1), it is clearly found that the Mulliken bond order for Hf– C about 0.43–0.48 (bond length ¼2.183–2.345 Å) is obviously larger than Hf–O ∼0.32 (bond length ¼2.284 Å), the bond order for Hf–F about 0.10–0.12 (bond length ¼2.361–2.404 Å) is much smaller, and the total bond order (9.49–9.54) is smaller (by ∼7%), compared with undoped system (10.24). So it can be predicted that the covalent character of F, C-codoped cubic HfO2 is lower than that of cubic HfO2.
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Fig. 1. TDOS and PDOS of cubic HfO2. Fermi level (EF) is set to zero.
Fig. 2. Spin polarized TDOS and PDOS of F, C-codoped (I) cubic HfO2 near the Fermi level (EF). Fermi level is set to zero. Solid and dash line refer to majority and minority spin channels, respectively.
3.2. Electronic properties In Figs. 1–3, we plot the spin polarized total density of states (TDOS) and partial density of states (PDOS) spectra for both undoped and F, C-codoped cubic HfO2, in which, the Fermi level (EF) locates at 0 eV. The TDOS and the PDOS of cubic HfO2 are presented in Fig. 1 and show a nonmagnetic character. For cubic HfO2, the band gap is calculated to be about 3.06 eV, which is smaller than the experimental data of 5.7 eV [34] due to the wellknown underestimation of conduction band state energies in ab initio calculations [35]. In Fig. 1, it is found that the bandwidth of the lower valence bands is about 3.07 eV with the peak at ∼ 16.72 eV, and the bandwidth of the upper valence bands is 6.02 eV with three main peaks at ∼ 4.55, ∼ 3.84 and ∼ 1.42 eV. It is also found that the lower valence bands originate from O 2s and the upper valence bands consist of O 2p states, which shows a hybridization character with Hf 5d states. With some contributions from Hf 5p and Hf 6s states, the conduction bands are mostly contributed by Hf 5d states, which show a hybridization character with O 2p states.
Fig. 3. Spin polarized TDOS and PDOS of F, C-codoped (II) cubic HfO2 near the Fermi level (EF). Fermi level is set to zero. Solid and dash line refer to majority and minority spin channels, respectively.
Figs. 2 and 3 show the calculated spin polarized TDOS and PDOS for both F, C-codoped cubic HfO2 structures. By F and C codoping, a slight band gap reduction is observed and the impurity bands both at the top and at the bottom of upper valence bands make the width of upper valence bands obviously broaden. In Figs. 2 and 3, it is clearly found that the impurity bands near the Fermi level are attributed to C 2p states with a strong admixture of O 2p and Hf 5d states, and the impurity bands at the bottom of upper valence bands are attributed to F 2p states with a admixture
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Fig. 4. Real and imaginary part of dielectric functions of (a) undoped, (b) F, C-codoped (I) and (c) F, C-codoped (II) cubic HfO2.
of O 2p and Hf 5d states. At the top of valence bands, the majority spin states are fully occupied while the minority spin states are partially filled and cross the Fermi level, which results in halfmetallic ferromagnetism (HMF) [14] behaviors of the systems and may be the reason to cause the band gap reduction. The finite density at the Fermi level implies that the magnetic interaction in F, C-codoped cubic HfO2 can be mediated by the holes induced by F and C codoping. It is also found that the O 2p states at the top of valence band increase, while Hf 5d states at the bottom of conduction bands decrease due to the reduced interaction between Hf 5d–O 2p states compared to that in pure HfO2. This may be another reason to cause the band gap reduction. With some contributions from O 2p and C 2p states, the bottom of the conduction bands is mainly contributed by the Hf 5d states, which contribute equally for both spin channels. Figs. 2 and 3 clearly show a substantial obvious mixing among dopants 2p, O 2p and Hf 5d states. 3.3. Dielectric constants The optical properties of undoped and F, C-codoped cubic HfO2 were also calculated. For these systems, the calculated dielectric functions, including the real part ε1 and imaginary part ε2, were calculated up to 25 eV and shown in Fig. 4. Because the calculated band gap of pure cubic HfO2 (3.06 eV) is smaller than the experimental value (5.7 eV) [34], the scissor operator with the value of 2.64 eV is used for the dielectric function. The static dielectric constant ε(0) is obtained as the zero frequency limit of the real part of the frequency-dependent dielectric function. The calculated ε(0) is 4.13 for undoped, 6.52 and 5.81 for F, C-codoped cubic HfO2 structure (I) and (II), which are obviously larger than that of undoped system. A larger ε(0) means a larger refractive index [33]. A higher optical density for the F, C-codoped cubic HfO2 may have special applications in certain optical components [36]. In Fig. 4a, the ε2 plot for undoped cubic HfO2 exhibits five major absorption peaks (at 7.06, 8.24, 9.51, 10.22 and 10.88 eV, respectively), which are the absorptive transitions from the valence bands to the conduction bands. The lower energy peaks at around 7 eV are associated with the the electronic transition between the O 2p states in the upper valence bands and the Hf 5d states in the conduction bands, while the peaks at around 10 eV may be due to the transition between the O 2s and the Hf 5d states in the valence bands [2]. For both F, C-codoped structures (shown in Fig. 4b and c), the ε2 plots exhibit new major absorption peaks at ∼2.8 eV, which should be attributed to the strong orbitals mixing between C 2p and Hf 5d states (shown in Figs. 2 and 3). And the large position changes of the other absorption peaks correspond to the delocalized effects of 2p orbitals in the supercell. From the results
of Fig. 4, it can be concluded that the F, C-codoped cubic HfO2, with steep absorption peaks at lower energy region ∼2.8 eV, can be used for photoabsorption applications [37]. In summary, first-principles calculations based on DFTþ U were performed on electronic and optical properties of F, C-codoped cubic HfO2. The calculations show that the magnetic solutions energetically favor over the non-magnetic solutions by 49–57 meV per F/C in F, C-codoped systems. In the minority spin, the states near the Fermi level are attributed to C 2p states with a strong admixture of O 2p and Hf 5d states, which results in half-metallic ferromagnetism behaviors of F, C-codoped cubic HfO2. Both the direct 2p–2p interaction and the indirect 2p–5d/2p–2p coupling interactions can be expected to contribute to the long-range magnetic coupling. Meanwhile, F and C codoping induces obvious increase of refractive index and new steep absorption peaks at lower energy region ∼2.8 eV, which can be used for photoabsorption applications. Acknowledgments This work was supported by the Natural Science Foundation of Shandong Province, China (No. ZR2010BM028), the National Natural Science Foundation of China (No. 51074078), the Research Foundation of university of Jinan, China (No. XBS0922).
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