Rational design of inorganic dielectric materials with expected permittivity

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.

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.

Table 1. Electronic polarizabilities ( in ų), ionic oscillator strengths ( in ų), effective volumes ( in ų), electronic polarizabilities per volume ( ), and ionic oscillator strengths per volume ( ) of 26 coordination polyhedra.

Coordinationpolyhedron	α	η	V	α/V	η/V
LiO4	1.16	4.79	12.42	0.093	0.386
LiF6	1.03	11.54	16.75	0.061	0.689
BeO4	1.39	4.54	14.03	0.099	0.323
BeF4	1.83	4.53	44.99	0.041	0.101
BO3	2.17	4.25	24.47	0.089	0.173
BO4	1.84	5.09	19.16	0.096	0.266
NaO4	2.3	7.22	21.21	0.108	0.341
NaF6	1.24	7.16	24.66	0.050	0.291
MgN4	3.08	8.76	20.81	0.148	0.421
MgO4	2.29	6.07	23.82	0.096	0.255
MgO6	1.91	10.89	18.92	0.101	0.575
MgF6	2.00	9.06	33.58	0.060	0.270
AlN4	2.77	6.94	21.30	0.130	0.326
AlN6	2.34	19.57	16.85	0.139	1.161
AlO4	2.72	8.10	31.71	0.086	0.255
AlO5	2.45	15.35	23.91	0.102	0.642
AlO6	2.27	13.44	22.06	0.103	0.609
AlF6	2.69	11.21	47.17	0.057	0.238
SiN4	3.13	7.71	24.86	0.126	0.310
SiN6	2.56	12.45	17.15	0.149	0.726
SiO4	3.21	6.61	49.33	0.065	0.134
SiO6	2.66	17.15	23.61	0.112	0.726
HfO6	5.17	31.84	32.22	0.120	0.737
HfO7	4.61	40.24	34.48	0.134	1.167
HfO8	4.49	53.40	32.36	0.139	1.650
HfN8	4.63	52.39	24.99	0.185	2.096

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.

Figure 1. Characteristic parameters α and η.

Comparison between characteristic parameters α (in ų) and η (in ų) of many MgO, Al₂O₃, and SiO₂ phases calculated from DFPT and those derived from optimal α_i and η_i 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.

Table 2. Space group (SG), and permittivities (electronic − , and static− ) of some ternary and quaternary oxides in the (MgO) _x (Al₂O₃) _y (SiO₂) _z system.

Compound	SG
		model	DFPT	reported	model	DFPT	reported
MgAl2O4 (Spinel)		3.18	3.06	2.8934	9.27	8.51	8.4035,8.7536
MgAl2O4 (CaFe2O4-type)	Pbnm	3.46	3.31		11.36	15.13
MgAl2O4 (CaTi2O4-type)	Cmcm	3.36	3.30		11.07	14.46
MgSiO3 (Enstatite)	Pbca	3.11	–		7.35	–	8.2337
MgSiO3 (Clinoenstatite)	P21/c	3.09	2.82		7.30	9.25
MgSiO3 (Protoenstatite)	Pnab	2.88	2.78		6.84	7.10	6.7038
MgSiO3 (Clinoenstatite)	C2/c	2.88	2.78		6.83	7.31
MgSiO3 (Corundum)		3.20	3.15		11.00	10.07
MgSiO3 (Perovskite)	Pbnm	3.52	3.38		11.94	16.80
Mg2SiO4 (Forsterite)	Pbnm	2.96	2.84	2.7839	7.76	7.52	6.8040,7.3041
Mg2SiO4 (Wadsleyite)	Imma	3.21	3.01		8.39	8.45
Mg2SiO4 (Ringwoodite)		3.33	3.03		8.64	8.14
Al2SiO5 (Andalusite)	Pmnn	2.78	2.83	2.7842	7.51	7.79	8.2837,8.043
Al2SiO5(Sillimanite)	Pmcn	2.97	2.88	2.8542	7.16	7.47	9.2937,6.244
Al2SiO5 (Kyanite)		3.24	3.09	3.1442	8.78	8.78
Mg2Al4Si5O18(Cordierite)	Cccm	2.42	2.39		5.34	4.97	5.045,6.1446

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:

Figure 2. Volume V.

Comparison between volume V (in ų) of many MgO, Al₂O₃, and SiO₂ compounds calculated from DFPT and those derived from optimal V_i 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.

Figure 3. 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 α_i, η_i, and V_i 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.

Figure 4. Lattice permittivity ε_L.

Comparison between lattice permittivity ε_L 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.

Table 3. Frequencies of polar phonon modes ( [cm⁻¹]) and their contributions to the permittivity ( ) computed for Pbnm MgSiO₃ ⁴⁶.

Mode	[cm−1]		Mode	[cm−1]		Mode	[cm−1]
B2u	175	6.14	B2u	430	1.83	B3u	662	0.11
B3u	239	0.61	B1u	449	0.14	B1u	688	~0
B1u	253	0.24	B2u	464	0.64	B2u	690	0.02
B2u	293	0.83	B3u	474	2.02	B2u	715	0.20
B1u	307	1.80	B1u	486	1.80	B3u	737	0.22
B3u	332	1.04	B3u	514	0.07	B3u	749	~0
B3u	367	0.23	B1u	541	~0	B1u	760	0.16
B3u	405	0.63	B1u	582	0.37
B1u	416	0.30	B2u	586	0.23

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 AlO_6. 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.

Figure 5. 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).