Dielectric Properties of Selected Metal–Organic ... - ACS Publications

Article pubs.acs.org/JPCC

Dielectric Properties of Selected Metal−Organic Frameworks Robert Warmbier,*,†,‡ Alexander Quandt,†,‡ and Gotthard Seifert†,§ †

Materials for Energy Research Group, University of the Witwatersrand, Private Bag 3, 2050 Johannesburg, Gauteng, South Africa School of Physics & DST-NRF Centre of Excellence in Strong Materials, University of the Witwatersrand, Private Bag 3, 2050 Johannesburg, Gauteng, South Africa § Physikalische Chemie, Technische Universität Dresden, Bergstraße 66b, 01062 Dresden, Sachsen, Germany ‡

S Supporting Information *

ABSTRACT: The electronic structure of a class of [Zn4O(CO2)6] based metal−organic frameworks (MOFs) is theoretically analyzed by means of density functional perturbation theory. The calculated static dielectric constants vary in a range between 1.33 and 1.54, characterizing the structures as ultralow-k dielectric materials and confirming earlier estimates qualitatively. We also present the results of first-principle calculations of the real and imaginary parts of the dielectric function and give the frequency-dependent dielectric constant up to the near-ultraviolet, which is important for high frequency semiconductor and optical applications of MOFs. The dielectric and electronic properties are governed by the linker molecules, so that the band gap and the dielectric constant can be engineered.

■

INTRODUCTION Metal−organic frameworks (MOFs) represent a subsystem of nanoporous crystals characterized by metal oxygen central cage clusters interconnected by organic linkers. Due to their controlled porosity, extremely large specific surface area, and structural variety, they have been of great interest for a series of applications such as gas storage, gas separation, and catalysis.1,2 MOFs could also show very small values of the dielectric constant ε (alternatively also κ or k in technological applications), due to their low density. Materials with extremely low dielectric constants are, on the other hand, highly interesting for applications as interlayer dielectrics for semiconducting devices. With the persistent downscaling of device features in integrated circuits, serious problems with the resistance−capacitance interconnect delay time, cross-talk noise, and power dissipation became a major bottleneck. Therefore, as a next technological step, ultralow-k dielectrics have to be introduced as insulating material. The structural, energetic, electronic, and mechanical properties of MOFs have been studied in some detail (see, e.g., ref 3), but a thorough investigation of dielectric properties of MOFs is still missing. With the exception of the IRMOF-10, whose dielectric properties have been studied with ab initio techniques,4 there exists only a theoretical work, where the static dielectric properties of MOFs were determined using a semiempirical Clausius−Mossotti model.5 The dielectric constant of a material is in general frequency dependent. The operating frequency of modern microprocessors is of the order of 109 Hz, which lies clearly below phonon excitations (about 1013 Hz) and electronic ones (about 1015 Hz). Therefore, the © XXXX American Chemical Society

frequency dependence of the dielectric constant can be neglected, and the static dielectric constant is a good approximation for alternating fields of frequencies acting inside a microprocessor. However, emerging communication technologies up to the THz region and applications in the optical range, like optical sensors,6 require also the knowledge of the dielectric properties beyond the static limit. An understanding of optical properties in MOFs and of their relation to structure is needed. Here we present first-principle studies for a subset of isoreticular metal−organic frameworks (IRMOF), which had previously been surveyed using density functional tight binding (DFTB) in refs 3 and 5. All of the IRMOFs studied below have fcc unit cells, which consist of two Zn4O(CO2)6 units and six hydrogen-terminated carbon linkers, as shown in Figure 1 for the simplest IRMOF with benzene linkers. MOFs are commonly named after a numerical scheme (e.g., IRMOF-1), but for easier reading, we are going to refer to the IRMOFs according to the chemical name of their linkers (e.g., benzene). The eight IRMOFs in this paper can be grouped according to the geometry of their linkers, which are printed in ref 3. Benzene (IRMOF-1) and biphenyl (IRMOF-10) linkers consist of a single line of one and two tip-to-tip connected benzene rings, respectively. The (poly)-acenic linkers anthracene (IRMOF-M1b) and naphtacene (IRMOF-M2c) consist of a single line of three and four edge connected benzene rings, Received: March 25, 2014 Revised: May 9, 2014

A

dx.doi.org/10.1021/jp5029646 | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

Article

sampling of the k-space, because their calculation involves an integral over the k-space. However, the size of the reciprocal cell is inverse proportional to the size of the unit cell in real space. Therefore, less k-points are needed for larger cells. For the smallest systems (benzene, cubane, biphenyl), we compared 1 × 1 × 1 and 2 × 2 × 2 k-point grids and found no significant difference in the optical properties, because the band structure is essentially flat. For all other systems, 1 × 1 × 1 (Γ-point only) was used. The energy limit for the plane wave basis was set to 500 eV for most calculations and up to 650 eV for geometry relaxation.

■

RESULTS AND DISCUSSION Geometry. The [Zn4O(CO2)6] IRMOF series has face centered cubic symmetry, space group Fm3̅m (no. 225), but symmetry can be lowered by the linker. The calculated cell parameters are summarized in Table 1. All DFT cell parameters are consistently (∼2−4%) smaller than the DFTB ones reported by Kuc et al.3 They agree very well (within ±1%) with available experimental data.16,17 The symmetric position of the six linkers within a unit cell potentially allows for a high internal symmetry of the MOFs. The linkers are only connected with a single carbon atom to the cores. Phonon modes, which correspond to rotations of the linkers, have very low energies of about 10 meV. This means that at room temperature many linker types can rotate freely. While this has little consequence for the physical properties, varying angles between the linkers reduce the internal symmetry of the system, which makes numerical simulations more cumbersome. Static Dielectric Constant. The dielectric tensor, or relative permittivity, relates external and induced electric fields. In the case of cubic systems, the tensor reduces to identical diagonal elements, effectively making the dielectric constant a scalar. The static, or low frequency, dielectric constant ε0 = ε∞ + εion for static electric fields contains electronic (ε∞) and ionic (εion) contributions. The static ionic contributions are the low energy limit of the phonon contributions to the infrared dielectric function, which is calculated using a damped Lorentz oscillator, as described in detail in ref 18. As such, the ionic part is calculated from the optical phonon modes, which represent the 3N − 3 internal degrees of freedom. For systems with many atoms, like MOFs, and for low point group symmetries, this becomes computationally very expensive, so that we concentrate here on two examples. For the benzene linked IRMOF-1, we find 315 optical phonon energies in a wide range between 11 and 391 meV, which yield a static ionic contribution of εion = 0.24. The biphenyl linked IRMOF-10 has 495 optical phonons in an

Figure 1. Face-centered cubic unit cell of IRMOF with benzene linkers (IRMOF-1). Carbon atoms are illustrated by small brown spheres, hydrogen atoms by small white spheres, oxygen atoms by small red spheres, and zinc atoms by large gray spheres. Created using VESTA.7

respectively. Those two groups are clearest to identify, as they follow the idea of linker extensions. The other two groups are less rigorously defined. Pyrene (IRMOF-14) and corannulene (IRMOF-M8) linkers are polyaromatic hydrocarbons. Pyrene is planar with benzene connected in a 2 × 2 honeycomb matrix. Corannulene is a polycyclic molecule with five edge connected hexagons around a pentagon forming a bowl-like structure. Cubane (IRMOFM11) and dodecahedrane (IRMOF-M13) are hypothetical cage-like linkers. Cubane is cubic, while dodecahedrane is the smallest fullerenic cage.

■

COMPUTATIONAL DETAILS We performed density functional theory (DFT)8,9 calculations to determine the dielectric properties of the MOFs. Initial guesses for the optimization of the IRMOF geometries were taken from previous tight binding studies.3 The calculations were performed using the Vienna Ab initio Simulation Package (VASP).10 In order to ensure reliability of the results, we used both the local density approximation (LDA) in the Ceperley− Alder parametrization 11,12 and the generalized gradient approximation (GGA) in the Perdew−Burke−Ernzerhof (PBE) parametrization13 to describe the exchange-correlation part of the electron-density-dependent energy functional. VASP uses projector augmented wave method14,15 based pseudopotentials for the treatment of the core electrons and a plane wave basis for the valence and conduction band electrons. Calculations of extended systems require a sampling of the band structure in the reciprocal k-space. The number of kpoints is crucial for the quality of DFT calculations. In particular, the optical properties are very sensitive to the Table 1. Basic Properties of Selected IRMOFsa

a

linker

MOF

N

aLDA

aPBE

aDFTB

aExp

benzene biphenyl anthracene naphtacene pyrene corannulene cubane dodecahedrane

IRMOF-1 IRMOF-10 IRMOF-M1b IRMOF-M2c IRMOF-14 IRMOF-M8 IRMOF-M11 IRMOF-M13

106 166 178 214 190 214 130 274

25.78 34.29 34.33 38.98 34.20 32.82 25.68 29.30

26.07 34.79 34.80 39.35 34.71 33.25 26.09 29.45

26.88b 35.56b 35.52b 39.95b 35.39b 33.67b 26.67b 30.16b

25.83c 34.28d

34.48d

The table shows the number of atoms N per unit cell and the lattice parameter a (Å). bReference 3. cReference 16. dReference 17. B

dx.doi.org/10.1021/jp5029646 | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

Article

considered here. There is however experimental data on the frequency-dependent refractive index within the visible spectrum of the [Cu3(BTC)2] MOF,6 which has a similar linker as the IRMOF-1. This allows our results to be compared to experimental data on a level of plausibility and consistency arguments. The experimental data of the [Cu3(BTC)2] MOF suggest a band gap in the near-ultraviolet. The dielectric constant at the red limit of the visible spectrum is ε(1.65 eV) = 1.93. The static limit can be estimated by the gradient of the dielectric function. The gradient in the dielectric function below the band gap comes from the tail of the Lorentzians, which characterize the interband transitions and therewith the dielectric function above the band gap. The experimental data can be fitted to a Lorentzian model, which yields a static limit of ε∞ ≈ 1.85. Zagorodniy et al. estimated ε∞ = 1.7 for that particular MOF using Clausius−Mossotti, which is roughly 8% less than our estimate.5 This is consistent with the finding that Clausius−Mossotti underestimates the dielectric constant by about 10% for the IRMOF-1 and similar IRMOFs. Band Gap. The fundamental electronic band gap is an important material property, because it has direct influence on the usability of a material for electronic and optical applications. It is known that standard DFT strongly underestimates the electronic band gap of semiconductors, often by 50%. It has been found, however, that GGA-PBE predicts astonishingly good band gaps for different IRMOFs.4,20,21 Indeed, the DFT band gaps for the benzene linked IRMOF-1 are very close, and slightly larger, to the experimental gap of 3.4 eV.22 Hybrid functionals are often used to predict better band gaps than possible with pure DFT. In this case, however, hybrid functionals will overestimate the electronic band gap; i.e., we calculated the HSE0623 band gap of the IRMOF-1 to be 4.66 eV. Table 2 lists the electronic band gaps as estimated by LDA and GGA-PBE and as calculated by DFTB from ref 3. It is unknown if the DFT band gaps are well approximated for all IRMOFs. Assuming though that the trend holds, it is clear that the hypothetical IRMOFs with the cage linkers cubane and dodecahedrane have the largest band gaps of about 4.4 eV. The smallest band gap can be found for the polyacenic IRMOFs with anthracene and naphtacene linkers, which have estimated band gaps of about 2.0 and 1.4 eV, respectively. These two IRMOFs are the only ones examined here, which potentially have a band gap within the optical spectral range. Linear linkers seem to have similar dielectric constants, independent of whether the linkers appear as chains of benzene, like biphenyl, or as polyacenic ones, like anthracene and naphtacene. On the other hand, this difference in structure influences the band gap, so that it can used to select low or high band gap regimes as desired without changing the dielectric constant. Projected Density of States and Partial Charge Analysis. The analysis of atomic projected density of states (pDOS) and the partial charges of bands around the band gap helps to understand the nature of the electronic, dielectric, and optical properties we found. Projected Density of States. The IRMOFs are constructed such that the same [Zn4O(CO2)6] cores are connected via different hydrocarbon linkers. We would therefore expect the zinc and oxygen pDOS to be more or less invariant, while the carbon and hydrogen pDOS should change strongly. Bands which are localized at, or associated with, the core should show contributions from oxygen, zinc, and to a smaller extent carbon. Bands belonging to the linkers should show contributions from

energy range of 8−391 meV, which yield a static ionic contribution of εion = 0.12. These ionic contributions are very small, which is the result of the low densities of the MOFs and the mainly covalent character of the bonding. Other MOFs should have similar ionic contributions due to structural similarities. The decrease in the ion contribution from the benzene to the biphenyl linked IRMOF is roughly proportional with the doubling in linker length as well as with the slightly more than doubled volume. In order for a phonon mode to contribute to the dielectric constant, it needs to be infraredactive, the ionic motion must induce dipole moments. Depending on the environment, vibrating carbon−carbon bonds can only create small or no dipole moments. This implies that the linkers have only a weak infrared signature and that the major contributions to εion must come from the cores. In that case, a reasonable estimate for the ionic contributions to the static dielectric constants for the different IRMOFs can be obtained from the benzene and biphenyl linked IRMOF values scaled by the volume quotients. The electronic static dielectric constant ε∞ can be computed to high accuracy within density functional perturbation theory (DFPT).19 The values of ε∞ with full response including local field effects (εDFPT) are shown in Table 2. For comparison, the Table 2. Calculated Material Properties of Selected IRMOFsa ELDA g benzene biphenyl anthracene naphtacene pyrene corannulene cubane dodecahedrane

3.54 3.01 1.97 1.40 2.47 2.49 4.41 4.31

EPBE g

EDFTB g

εind ∞

εDFPT ∞

εCM ∞

3.56 2.98 1.96 1.39 2.48 2.51 4.41 4.34

b

1.67 1.48 1.61 1.57 1.68 1.89 1.63 1.77

1.47 1.33 1.39 1.35 1.41 1.52 1.46 1.54

1.37c 1.23c 1.25c 1.21c 1.28c 1.39c 1.45c 1.50c

3.73 3.07b 2.13b 1.61b 2.63b 2.66b 4.91b 5.49b

a

The table shows band gaps Eg (eV), electronic static dielectric constants ε∞ from GGA-PBE in the independent particle approxDFPT and Clausius−Mossotti imation εind ∞ and in full response ε∞ c CM b estimates ε∞ . Reference 3. Reference 5.

responses in the independent particle approximation and without local field effects (εind) are given as well. LDA and GGA-PBE results do not differ by more than 0.01, so that only GGA-PBE results are given. The independent particle picture is a simpler, less accurate model. The DFPT dielectric constant on the other hand is an accurate description for the linear response regime (aka for moderate field strengths) within the limits of DFT. The calculated static dielectric constants agree quite well with previous semiempirical and estimated dielectric constants using a Clausius−Mossotti approach,5 which underestimates the dielectric constants by about 10% for all IRMOFs but the ones with cage linkers. The Clausius−Mossotti estimates for the cage linked cubane and dodecahedrane IRMOFs are fairly close to the DFT data. The smallest dielectric constants can be obtained for linear linkers, like biphenyl, anthracene, and naphtacene. This is logical, as IRMOFs with those linkers show the largest vacuum in any line of sight. IRMOFs with short or space-filling linkers, like corannulene, cubane, or dodecahedrane, have a larger dielectric constant. To our knowledge, there are no experimental data available of the dielectric and optical properties for the IRMOFs C

dx.doi.org/10.1021/jp5029646 | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

Article

carbon and hydrogen. Figures for the partial density of states can be found in the Supporting Information due to the number and size of those figures. As a representative example, we show the partial density of states for carbon and oxygen + zinc of the anthracene linked IRMOF-M1b around the band gap in Figure 2. The band energies are given relative to the Fermi level, which is defined by convention to be the valence band maximum. The DOS is artificially broadened for better visibility.

contribution can be found about 5 eV below the Fermi level, as can be seen in Figure 2. Those contributions decrease toward the Fermi level. Oxygen also contributes strongly to the DOS with most contributions located closely below the Fermi level (p-states). Hydrogen does have contributions to numerous valence bands but always in a minor role. Carbon dominates the DOS for most of the valence bands having more or less equally distributed contributions. Carbon is also dominating the bands at the Fermi level, with the exception of benzene and the hypothetical cage linked cubane and dodecahedrane. Here zinc, oxygen, and carbon contribute to the Fermi level band. The valence band pDOS here agree very well with the pDOS of the IRMOF-10 calculated by Yang et al.4 Above the Fermi level, most states near the band gap energy are dominated by carbon, with a few core dominated bands in between. There are a few distinct conduction bands before the DOS goes over into the continuum at less than 10 eV above the band gap. Yang et al. find a strong increase in the DOS about 5−10 eV above the Fermi level for the IRMOF-10, which we do not affirm. In the anthracene IRMOF DOS, one can clearly see that the valence band maximum and the conduction band minimum are linker localized states, which are separated considerably from the rest of the valence and conduction bands, respectively. All carbon atoms are sp2 hybridized with π states around the band gap. It is apparent that the linkers are controlling the band gap and most dielectric and optical properties. Longer linkers add more and more carbon dominated bands around the band gap. Going from biphenyl over anthracene to naphtacene IRMOFs, we find that the highest occupied levels are more and more separated from the rest of the valence bands and that the additional conduction bands reduce the band gap. All of these bands come in pairs of six, two sets of degenerate sp2/π bands separated by about 0.1 eV.

Figure 2. Density of states (DOS) for anthracene linked IRMOF-M1b around the band gap. The carbon DOS is given as a black line, the oxygen + zinc contribution, as a light green dashed line. The band energies are relative to the Fermi level EF.

The calculated pDOS for zinc and oxygen indeed do not change much over the different IRMOFs, with the small exception of bands near the Fermi level, which are more influenced by changes in the environment. Zinc has a very strong pDOS from d-states a few eV below the Fermi level. For the anthracene linked IRMOF-M1b, the strongest zinc

Figure 3. Partial charges of the highest occupied bands of biphenyl, anthracene, and naphtacene linked IRMOFs (from left to right). Created using VESTA.7 D

dx.doi.org/10.1021/jp5029646 | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

Article

Figure 4. Partial charges of the lowest unoccupied bands of benzene, biphenyl, anthracene, and naphtacene linked IRMOFs (from left top to right bottom). Created using VESTA.7

Partial Charge Analysis. The partial charge ϱ, defined as ϱl ′ l ″(r) = ∑ll ″′ |Ψ(r)|2, gives the contribution to the charge density of certain bands, where l′ and l″ are the lower and upper band boundaries for the summation. We performed a partial charge analysis for the five IRMOFs in this paper with planar linkers based on GGA-PBE calculations. A collection of charge density images for the most important low energy transitions is given in the Supporting Information. The states at the upper valence band and lower conduction band edges determine essentially the static and frequencydependent dielectric properties. Therefore, their analysis gives a better understanding of how the electronic structure determines these properties. All IRMOFs with planar linkers in this work show a clear separation of the first six quasidegenerate unoccupied bands from the rest of the conduction bands. As mentioned before, those consist of two sets of each three degenerate π bands formed by the sp2-hybridized carbon atoms. Most of those IRMOFs also show a separation of the six quasi-degenerate highest occupied bands from the rest of the valence bands. Partial charge density plots for the highest occupied bands of biphenyl, anthracene, and naphtacene linked IRMOFs are presented in Figure 3. Correspondingly, the lowest unoccupied bands of benzene, biphenyl, anthracene, and naphtacene linked IRMOFs are shown in Figure 4. The valence band maximum of the benzene linked IRMOF is dominated by zinc and oxygen and does not contribute to strong low energy transitions and is therefore not of interest here. The origin of the low energy transitions for this IRMOF comes from a number of carbon dominated bands below the valence band maximum. The valence band maximum partial charges (see Figure 3) of the anthracene and naphtacene based IRMOFs are almost identical. The charge density is localized at the outer ridge of the linkers almost like a zigzag pattern. The biphenyl linked IRMOF has a different valence band maximum

partial charge. Benzene and biphenyl IRMOFs show a similar partial charge for the conduction band minimum (see Figure 4) with parallel diatomic connections along the linker line. Again, anthracene and naphtacene IRMOFs have the same behavior compared to each other but a different one compared to the benzene and biphenyl IRMOFs. This suggests that for linkers of the same type but different length the bands around the band gap are similar. We would therefore expect similarities in the dielectric function for linkers of the same type, and more pronounced differences between different types of linkers. Frequency-Dependent Dielectric Function. The frequency-dependent dielectric function, covering the nearinfrared, visible, and ultraviolet spectral ranges, does not have ionic contributions, because ions are too heavy to react to those fast-changing electric fields. In contrast to the static dielectric constant, the frequency-dependent dielectric function is a complex property. It is more complicated to calculate than the static constant as well. The quality is limited by the accuracy of the band gap and whether the computationally demanding excitonic and local field effects can be taken into account. The large unit cells of the MOFs limit us to use the independent particle approximation applied to the LDA and GGA band energies and wave functions. Results are therefore in principle of qualitative rather than quantitative nature. However, the uncommon accurate DFT band gaps of the IRMOFs allow a better dielectric function to be calculated than for many semiconductors. Nevertheless, the low energy spectrum may suffer from underestimated oscillator strengths. LDA and GGAPBE results are nearly identical, so only PBE-GGA results are shown in Figure 5. The dielectric function data can be found in the Supporting Information. A comparison of the static E = 0 eV limit of the frequencydependent dielectric function and independent particle static E

dx.doi.org/10.1021/jp5029646 | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

Article

Figure 5. Complex dielectric function of MOFs within the independent particle model from GGA-PBE calculations. The real part ℜ[ε] is shown as a dashed red line, while the imaginary part ℑ[ε] is shown as a black line.

also come from the zinc d-states. The biphenyl linked IRMOF10 has been subjected to a similar numerical analysis in ref 4. The position and number of the major transitions are identical, which is natural, as similar numerical methods were used, but the present work shows a higher resolution. The cage linker IRMOFs show a completely different behavior. They do not have aromatic ring systems that the other IRMOFs exhibit. The bands around the band gap of cubane and dodecahedrane linked IRMOFs are not dominated by the carbon linkers but also have strong contributions for oxygen and zinc. That seems to be insufficient to produce strong low energy transitions. The overall magnitude of the dielectric functions and their features is rather small, as expected for a system with a lot of vacuum. Most semiconductors possess a negative real part of the dielectric function right after an absorption resonance,

dielectric constant from DFPT shows that they are identical to within 0.02 eV, which indicates that enough empty bands were used. The IRMOFs do not have strong absorption features from valence electrons beyond 20 eV, which many other materials have. Therefore, it was sufficient to add empty bands up to about 50 eV above the Fermi level. The overall structure of the dielectric functions is very similar in the six IRMOFs without cage linkers. They have one or a few strong transitions just above the band gap energy and a number of weak transitions. Atom projected density of states and partial charge analysis show that these strong transitions are dominated by the carbon linkers and π to π* transitions. Hydrogen and zinc play no role in the strong low energy transitions. Furthermore, these six IRMOFs show a broader absorption peak around 13−14 eV, which can also be found in the dielectric function of graphite/graphene24 but which may F

dx.doi.org/10.1021/jp5029646 | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

Article

(4) Yang, L.-M.; Ravindran, P.; Vajeeston, P.; Tilset, M. Ab Initio Investigations on the Crystal Structure, Formation Enthalpy, Electronic Structure, Chemical Bonding, and Optical Properties of Experimentally Synthesized Isoreticular Metal- Organic Framework-10 and Its Analogues: M-IRMOF-10 (M = Zn, Cd, Be, Mg, Ca, Sr and Ba). RSC Adv. 2012, 2, 1618−1631. (5) Zagorodniy, K.; Seifert, G.; Hermann, H. Metal-Organic Frameworks as Promising Candidates for Future Ultralow-k Dielectrics. Appl. Phys. Lett. 2010, 97, 251905. (6) Redel, E.; Wang, Z.; Walheim, S.; Liu, J.; Gliemann, H.; Wöll, C. On the Dielectric and Optical Properties of Surface-Anchored MetalOrganic Frameworks: A Study on Epitaxially Grown Thin Films. Appl. Phys. Lett. 2013, 103, 091903. (7) Momma, K.; Izumi, F. VESTA3 for Three-Dimensional Visualization of Crystal, Volumetric and Morphology Data. J. Appl. Crystallogr. 2011, 44, 1272−1276. (8) Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas. Phys. Rev. 1964, 136, B864−B871. (9) Parr, R. G.; Yang, W. Density-Functional Theory of Atoms and Molecules; International Series of Monographs on Chemistry; Oxford University Press: Oxford, U.K., 1989; Vol. 16. (10) Kresse, G.; Furthmüller, J. Efficient Iterative Schemes for Ab Initio Total-Energy Calculations Using a Plane-Wave Basis Set. Phys. Rev. B 1996, 54, 11169−11186. (11) Ceperley, D. M.; Alder, B. J. Ground State of the Electron Gas by a Stochastic Method. Phys. Rev. Lett. 1980, 45, 566−569. (12) Perdew, J. P.; Zunger, A. Self-interaction Correction to DensityFunctional Approximations for Many-Electron Systems. Phys. Rev. B 1981, 23, 5048−5079. (13) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865−3868. (14) Blöchl, P. E. Projector Augmented-Wave Method. Phys. Rev. B 1994, 50, 17953−17979. (15) Kresse, G.; Joubert, D. From Ultrasoft Pseudopotentials to the Projector Augmented-Wave Method. Phys. Rev. B 1999, 59, 1758− 1775. (16) Li, H.; Eddaoudi, M.; O’Keeffe, M.; Yaghi, O. Design and Synthesis of an Exceptionally Stable and Highly Porous Metal-Organic Framework. Nature 1999, 402, 276−279. (17) Eddaoudi, M.; Kim, J.; Rosi, N.; Vodak, D.; Wachter, J.; O’Keeffe, M.; Yaghi, O. M. Systematic Design of Pore Size and Functionality in Isoreticular MOFs and Their Application in Methane Storage. Science 2002, 295, 469−472. (18) Warmbier, R.; Mohammed, F.; Quandt, A. Optical and Other Material Properties of SiO2 from Ab Initio Dtudies. Opt. Eng. 2014, 53. (19) Wu, X.; Vanderbilt, D.; Hamann, D. R. Systematic Treatment of Displacements, Strains, and Electric Fields in Density- Functional Perturbation Theory. Phys. Rev. B 2005, 72, 035105. (20) Yang, L.-M.; Vajeeston, P.; Ravindran, P.; Fjellvag, H.; Tilset, M. Theoretical Investigations on the Chemical Bonding, Electronic Structure, And Optical Properties of the Metal-Organic Framework MOF-5. Inorg. Chem. 2010, 49, 10283−10290. (21) Yang, L.-M.; Ravindran, P.; Vajeeston, P.; Tilset, M. Properties of IRMOF-14 and its Analogues M-IRMOF-14 (M = Cd, Alkaline Earth Metals): Electronic Structure, Structural Stability, Chemical Bonding, and Optical Properties. Phys. Chem. Chem. Phys. 2012, 14, 4713−4723. (22) Alvaro, M.; Carbonell, E.; Ferrer, B.; Llabres i Xamena, F. X.; Garcia, H. Semiconductor behavior of a metal-organic framework (MOF). Chem.Eur. J. 2007, 13, 5106−5112. (23) Krukau, A. V.; Vydrov, O. A.; Izmaylov, A. F.; Scuseria, G. E. Influence of the Exchange Screening Parameter on the Performance of Screened Hybrid Functionals. J. Chem. Phys. 2006, 125, 224106. (24) Marinopoulos, A. G.; Reining, L.; Olevano, V.; Rubio, A.; Pichler, T.; Liu, X.; Knupfer, M.; Fink, J. Anisotropy and Interplane Interactions in the Dielectric Response of Graphite. Phys. Rev. Lett. 2002, 89, 076402.

which is sometimes referred to as optical metallicity. Most of the IRMOFs here do not exhibit this behavior, with the exception of anthracene and naphtacene, which have a very short and only slightly negative energy interval. The large vacuum partitions of the unit cells also lead to very low reflectivity. Even at the absorption resonances, the reflectivity does not exceed 20% for most IRMOFs and 30% for anthracene and naphtacene linked IRMOFs.

■

CONCLUSIONS We presented the results of first-principle DFT calculations for eight different [Zn4O(CO2)6] based metal−organic frameworks (IRMOFs). The calculated structures agree very well with available experimental data and are in good agreement with results from earlier DFTB calculations. The calculated static constants vary in a range between 1.33 and 1.54, characterizing the structures as ultralow-k dielectric materials. The results confirm earlier estimations using a semiempirical Clausius− Mossotti model qualitatively but show also that these estimates are generally too small by about 10%. We present also the results of first-principle calculations of the real and imaginary parts of the dielectric function and give the frequencydependent dielectric constant up to the near-ultraviolet, that are important for high frequency semiconductor and optical applications of MOFs. The dielectric and electronic properties are governed by the linker molecules, so that the band gap and the dielectric constant can be engineered.

■

ASSOCIATED CONTENT

S Supporting Information *

The raw data for the frequency-dependent function and additional figures for the partial density of states and partial charges. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors thank Cem Ozdogan from the Cankaya University for the computational resources. R.W. and A.Q. would like to thank the DST-NRF Centre of Excellence in Strong Materials at the University of the Witwatersrand for support. A.Q. acknowledges financial support by the National Research Foundation (NRF) (Grant specific unique reference number (UID) 85972) and the Department of Science and Technology (DST) of South Africa, as well as by the Gauteng node of the National Institute of Theoretical Physics (NITheP). G.S. acknowledges also support by the DFG within the priority program “Poröse metallorganische Gerüstverbindungen”.

■

REFERENCES

(1) Davis, M. Ordered Porous Materials for Emerging Applications. Nature 2002, 417, 813−821. (2) Kreno, L. E.; Leong, K.; Farha, O. K.; Allendorf, M.; Van Duyne, R. P.; Hupp, J. T. Metal-Organic Framework Materials as Chemical Sensors. Chem. Rev. 2012, 112, 1105−1125. (3) Kuc, A.; Enyashin, A.; Seifert, G. Metal- Organic Frameworks: Structural, Energetic, Electronic, and Mechanical Properties. J. Phys. Chem. B 2007, 111, 8179−8186. G

dx.doi.org/10.1021/jp5029646 | J. Phys. Chem. C XXXX, XXX, XXX−XXX