Abstract
Techniques for rapid design of dielectric materials with appropriate permittivity for many important technological applications are urgently needed. It is found that functional structure blocks (FSBs) are helpful in rational design of inorganic dielectrics with expected permittivity. To achieve this, coordination polyhedra are parameterized as FSBs and a simple empirical model to evaluate permittivity based on these FSB parameters is proposed. Using this model, a wide range of examples including ferroelectric, high/low permittivity materials are discussed, resulting in several candidate materials for experimental follow-up.
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Introduction
Dielectric materials are essential for many technological applications in optical, electronic and micro-electronic devices. For instance, high-permittivity materials are required for gate dielectrics and high-energy storage capacitors and low-permittivity dielectrics are necessary for transparent windows and miniaturized integrated circuits. The search for these dielectric materials over a wide range of compounds is time-consuming. A major reason is the lack of a clear and intuitive data set to give an idea about which materials should be focused on1. Fortunately, we now have computational tools such as codes based on density functional theory2,3 (DFT), capable of accurately predicting many important materials properties. With the help of computations, materials discovery can be accelerated4,5,6.
Up to now, high-throughput computational approach have been employed to screen thousands of compounds for new materials7,8,9,10,11,12,13,14. Structure prediction methods15, such as USPEX16,17, have also been developed to optimize certain properties of materials with only the chemical composition given18,19,20,21. However, the efficiency of these theoretical methods requires a fast and accurate evaluation of the properties of interest, while dielectric properties are relatively time-consuming. Therefore, it would be desirable to find a way to compute them from crystal structure, most transparently using functional structure blocks (FSBs), which are directly linked to the materials properties. The application of this FSB method mainly depends on: (1) the determination of a suitable FSB for a certain property of materials; and (2) the establishment of an explicit relationship between this property and its FSB. With such structure-property relations, one can quantitatively or qualitatively evaluate properties for a material in seconds. In this paper, we will demonstrate that the idea of FSBs could be very useful for rational design of materials with expected permittivity.
Inspired by Rignanese et al.22 and our previous studies21,23, we choose the coordination polyhedron as FSB for permittivity due to its major and easy to rationalize effect on permittivities of materials. Coordination polyhedron to a very large extent determines many aspects of lattice dynamics and thus can be used to determine permittivity21,22. Rignanese et al.22 proposed an empirical model to calculate permittivity, for each coordination polyhedron using three characteristic parameters (electronic polarizability 
, charge 
and force constant 
). In this present study, we suggest a simplified empirical model with each type of coordination polyhedra characterized by two parameters: electronic polarizability 
and ionic oscillator strength 
. Furthermore, by introducing the volume 
of each type of polyhedron, we can extend our model to estimate permittivity of a crystal structure provided that the type of coordination polyhedron is known. This means that dielectric materials with expected permittivity could be constructed by selecting appropriate coordination polyhedra.
Results and Discussions
Description of the model
According to Rignanese’s model22, it is possible to evaluate the electronic24, lattice and static permittivities of a given structure based on its electronic polarizability 
, charge 
and force constant 
:
where 
is the electronic permittivity; 
is the lattice permittivity; 
is the static permittivity; and 
is the volume of the structure. They define 
, 
and 
values for each type of coordination polyhedron i and assuming that:
where 
is the number of type-i coordination polyhedron contained in a structure. Summation is done over all types of coordination polyhedra. The optimal 
, 
and 
values for each type of coordination polyhedron i can be determined using least-squares method based on the 
, 
and 
values calculated from first principles for a set of materials. However, 
obtained by their model is sometimes very different from that calculated from first principles. This may be due to the fact that 
and 
are considered as two independent variables in their model, which, however, may be correlated to each other. Therefore, we suggest defining a single parameter of ionic oscillator strength 
:
Then, the lattice permittivity 
can be calculated as:
By analogy with 
, we define 
for each type of coordination polyhedron i such that:
The optimal values 
can be determined in the same way as for 
. As shown in the following part of this paper, 
obtained from our simplified model improve upon those calculated from Rignanese’s model in most cases.
Test of the model
We have calculated permittivity of various inorganic compounds constructed from three binary oxide systems (MgO, Al2O3 and SiO2). With the crystal structures of these compounds obtained from Materials Project1, we performed full structure relaxation before calculating permittivity using the density functional perturbation theory (DFPT25) approach. Structural information and DFPT permittivities of these compounds can be found as Supplementary Table Is. The optimal 
and 
values of seven coordination polyhedra, MgO4, MgO6, AlO4, AlO5, AlO6, SiO4 and SiO6 obtained in our model are listed in Table 1.
in Å3), ionic oscillator strengths (
in Å3), effective volumes (
in Å3), electronic polarizabilities per volume (
) and ionic oscillator strengths per volume (
) of 26 coordination polyhedra.In Fig. 1, 
and 
values of MgO, Al2O3 and SiO2 compounds given by our model are compared to those calculated from DFPT approach, with quite good agreement for most of the structures. In particular, 
values obtained in our model agree very well with those computed by the DFPT approach, with an average relative error as low as 1.5%. Although a few 
values have error higher than 10%, it can be concluded that our 
values of MgO4, MgO6, AlO4, AlO5, AlO6, SiO4 and SiO6 coordination polyhedra are reliable.
Characteristic parameters α and η.
Comparison between characteristic parameters α (in Å3) and η (in Å3) of many MgO, Al2O3 and SiO2 phases calculated from DFPT and those derived from optimal αi and ηi values reported for coordination polyhedron i.
To test the applicability of our model, we evaluated permittivities of many ternary and quaternary oxides in (MgO)x(Al2O3)y(SiO2)z system (see Table 2). DFPT results obtained by us and some experimentally or theoretically reported values are also listed in Table 2 for comparison. We can see that our model with optimized 
and 
is really helpful to evaluate materials permittivity. Moreover, our model may provide a way to obtain permittivity for very complex systems where DFPT approach is not feasible, e.g., enstatite MgSiO3 (80 atoms/cell) listed in Table 2.
and static−
) of some ternary and quaternary oxides in the (MgO)x(Al2O3)y(SiO2)z system.However, one must keep in mind the limitations of the model (see 
values shown in Fig. 1). We conclude that our simplified model is not suitable for materials with low-frequency polar modes having large contributions (due to large 
values) to the lattice permittivity. We return to this point later in this paper.
Our model can also be extended to evaluate permittivity of a hypothetical structure, for which only the types of coordination polyhedra are given. To achieve this, we define volume 
for each type of coordination polyhedron i and determine optimal 
values in the same way as for 
and 
(as listed in Table 1). The addition of 
of coordination polyhedron i can reproduce volume of a structure well (as shown in Fig. 2). Then the 
(
) values of a structure can be obtained from:
Volume V.
Comparison between volume V (in Å3) of many MgO, Al2O3 and SiO2 compounds calculated from DFPT and those derived from optimal Vi values reported for coordination polyhedron i.
The corresponding 
(
) values are comparable to those calculated from DFPT approach (see Fig. 3). In this way, permittivity of a hypothetical structure can be reasonably evaluated.
Parameters α/V and η/V.
Comparison between parameters (α/V and η/V) of many MgO, Al2O3 and SiO2 phases calculated from DFPT and those estimated by using αi, ηi and Vi values of coordination polyhedron i.
Application of the model
The 
, 
and 
values of each type of coordination polyhedra obtained from our model are helpful to design dielectric materials with expected permittivity. First, we extended our model to study some other oxides, nitrides and fluorides (see Supplementary Table IIs). We obtained 
, 
and 
values for another 19 coordination polyhedra (see Table 1). With the 
, 
and 
values of 26 coordination polyhedra listed in Table 1, we illustrated how to rationally design ferroelectric and high/low permittivity materials.
We have calculated 
of 95 compounds using 
values of these 26 coordination polyhedra. Some of these compounds are listed in Table 2. The complete list of compounds can be found as Supplementary Tables Is and IIs. We compare 
values of these 95 compounds with those calculated from DFPT approach (see Fig. 4). The agreement between the two data sets is good. However, there are two deviating structures, P42/nmc HfO2 and Pbnm MgSiO3, for which the actual 
is much higher than that from our model. We found that the “unusual” enhancement of 
is related to large 
values. This may originate from low-frequency polar phonon modes, which means that these two structures can be close to a ferroelectric instability.
Lattice permittivity εL.
Comparison between lattice permittivity εL of 95 compounds obtained by using the present simplified semi-empirical model and those calculated from DFPT.
In fact, the P42/nmc HfO2 is a well-known ferroelectric material. Another structure, Pbnm MgSiO3, possesses a perovskite structure adopted by many ferroelectric materials. We calculated the contributions to 
from each polar phonon mode of Pbnm MgSiO3 (as listed Table 3). The Pbnm MgSiO3 indeed possesses a low-frequency polar phonon mode (at 175 cm−1) contributing to 
much more than other phonon modes. In other words, our model underestimates permittivities of ferroelectrics and crystals with softened polar modes. This can actually be used for rapid screening of potential ferroelectric materials.
[cm−1]) and their contributions to the permittivity (
) computed for Pbnm MgSiO346.Our model is also helpful in the design of materials with high/low permittivity. Our results show, quite intuitively, that coordination polyhedra with high 
(
) and low 
are favorable for high dielectric permittivity.
At a glance at Table I, we can find that HfO8 has much higher 
and 
values than others among the 26 coordination polyhedra. Indeed, Hf oxides are excellent high-permittivity oxides (ref. 20). On the other hand, SiO4 tetrahedron possesses the lowest 
and 
values among O-based coordination polyhedra. Indeed, SiO2 (quartz and silica glass) with SiO4 tetrahedra is a well-known low-permittivity material in micro-electronics industry.
Noticeably, 
and 
values of N-based coordination polyhedra are higher than those of O-based coordination polyhedra. For instance, AlN6 coordination polyhedron has much higher 
and 
values than AlO6. We may expect high-permittivity in nitrides, e.g., Hf3N4 with HfN8 coordination polyhedron. As listed in Table 1, 
and 
values of HfN8 coordination polyhedron are higher than those of the HfO8 polyhedron. Therefore, 
Hf3N4 with HfN8 coordination polyhedron has higher permittivities than most of hafnium oxides (see Supplementary Table IIs).
For the design of low-permittivity materials, we can immediately expect that permittivity of an oxide can be decreased by replacing O with F (see Table 1). Experimentally, SiF4 material with SiF4 tetrahedra has much lower permittivity than quartz26,27. In a similar way, we can expect that 
and 
values of MgF4 coordination polyhedron may be much lower than those of MgO4 polyhedron. Therefore, we try to design low-permittivity MgF2 material with MgF4 coordination polyhedron. We constructed a new 
MgF2 phase (Fig. 5(a)) with very low permittivity using 
SiO2 structure (cristobalite) with SiO4 tetrahedra (detailed structural information can be found as Supplementary Table IIIs). The static permittivity 
of 
MgF2 (2.5) is much lower than that of quartz (3.927) and comparable to most low-permittivity polymers. The dynamical and mechanical stability of 
MgF2 was verified by phonon and elastic constants calculations (see Supplementary Fig. 1s and Table IVs). The enthalpy of 
MgF2 phase is only 0.1 eV/atom higher than that of the most stable MgF2 structure (P42/mnm phase). Moreover, this inorganic material may have a better mechanical strength than polymers (see Supplementary Table IVs). This suggests that 
MgF2 may be synthesized and tested as a potential low-permittivity material.
Crystal structures of MgF2 and BeF2.
(a) 
MgF2 constructed from MgF4 coordination polyhedra; (b) 
BeF2 constructed from BeF4 coordination polyhedra. Blue spheres denote F atoms, brown spheres denote Mg atoms and green spheres denote Be atoms.
From the Materials Project, we also found a near-ground-state BeF2 structure (
) with BeF4 coordination polyhedra, as shown in Fig. 5(b). The static permittivity 
of 
BeF2 is 2.5, indicating that BeF2 is also a good low-permittivity material. We suggest that compounds constructed from LiF4, BF4, NaF4 and AlF4 coordination polyhedra may also have low permittivities, e.g., 
of P3121 LiBF4 with LiF4 and BF4 coordination polyhedra can be as low as 3.6.
We have to mention that coordination number is an important factor to design high/low-permittivity materials. There is a trend21,23: low coordination number, low permittivity. Our present study agrees with this trend well; coordination polyhedra with low coordination number have low 
and 
values. For example, our study shows that the 
SiC2N4 structure, with 1/3 SiN4 and 2/3 CN2 coordination polyhedra, has much lower permittivity (4.6) than P63/m Si3N4 (8.3) containing SiN4 coordination polyhedra.
To summarize, we have presented a method for designing new inorganic dielectrics with expected permittivity is discussed. Coordination polyhedron is adopted as the functional structural block (FSB) of permittivity. Three parameters (electronic polarizability 
, ionic oscillator strength 
and volume 
) are chosen to characterize each coordination polyhedron. We show applications of this model evaluate materials permittivity. Results derived from this model agree well with those from density-functional perturbation theory. Moreover, 
, 
and 
values assigned to coordination polyhedra may be helpful to make intuitive choices of materials to focus on. Successful applications include ferroelectric, high- and low-permittivity materials.
Methods
Before calculating the properties, we perform full structure relaxation using density functional theory (DFT2,3) as implemented in the Vienna ab intio Simulation Package (VASP28) with the PBEsol-GGA29,30 exchange-correlation functional. The all-electron projector-augmented wave (PAW) method31 is used, with a plane-wave energy cutoff of 900 eV and k-point meshes with reciprocal-space resolution of 
. These settings enable excellent convergence for the energy differences, stress tensors and structural parameters. With fully relaxed structures, dielectric25 and mechanical32 properties (e.g. the elastic constants) were computed. Permittivities and phonon dispersion curves are calculated using density functional perturbation theory (DFPT25). Phonon dispersion curves were obtained by PHONOPY33.
Additional Information
How to cite this article: Xie, C. et al. Rational design of inorganic dielectric materials with expected permittivity. Sci. Rep. 5, 16769; doi: 10.1038/srep16769 (2015).
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Acknowledgements
This work was supported by the Foreign Talents Introduction and Academic Exchange Program of China (No. B08040), the National Science Foundation (EAR-1114313, DMR-1231586), DARPA (Grant No. W31P4Q1210008) and the Government of Russian Federation (grant 14.A21.31.0003). The authors also acknowledge the High Performance Computing Center of NWPU for the allocation of computing time on their machines.
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C.-W.X. and A.R.O. designed the project and performed the calculations. C.-W.X., A.R.O. and D.D. analyzed the data and wrote the paper. N.L. got the structural information of the structures studied in this paper. D.L. and T.T.D. helped to plot the figures.
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Xie, C., Oganov, A., Dong, D. et al. Rational design of inorganic dielectric materials with expected permittivity. Sci Rep 5, 16769 (2015). https://doi.org/10.1038/srep16769
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DOI: https://doi.org/10.1038/srep16769
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