Computation of Dielectric Response in Molecular Solids for High

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Computation of Dielectric Response in Molecular Solids for High Capacitance Organic Dielectrics Henry M. Heitzer, Tobin J. Marks,* and Mark A. Ratner* Department of Chemistry and the Materials Research Center, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, United States CONSPECTUS: The dielectric response of a material is central to numerous processes spanning the fields of chemistry, materials science, biology, and physics. Despite this broad importance across these disciplines, describing the dielectric environment of a molecular system at the level of first-principles theory and computation remains a great challenge and is of importance to understand the behavior of existing systems as well as to guide the design and synthetic realization of new ones. Furthermore, with recent advances in molecular electronics, nanotechnology, and molecular biology, it has become necessary to predict the dielectric properties of molecular systems that are often difficult or impossible to measure experimentally. In these scenarios, it is would be highly desirable to be able to determine dielectric response through efficient, accurate, and chemically informative calculations. A good example of where theoretical modeling of dielectric response would be valuable is in the development of high-capacitance organic gate dielectrics for unconventional electronics such as those that could be fabricated by high-throughput printing techniques. Gate dielectrics are fundamental components of all transistor-based logic circuitry, and the combination high dielectric constant and nanoscopic thickness (i.e., high capacitance) is essential to achieving high switching speeds and low power consumption. Molecule-based dielectrics offer the promise of cheap, flexible, and mass producible electronics when used in conjunction with unconventional organic or inorganic semiconducting materials to fabricate organic field effect transistors (OFETs). The molecular dielectrics developed to date typically have limited dielectric response, which results in low capacitances, translating into poor performance of the resulting OFETs. Furthermore, the development of better performing dielectric materials has been hindered by the current highly empirical and labor-intensive pace of synthetic progress. An accurate and efficient theoretical computational approach could drastically decrease this time by screening potential dielectric materials and providing reliable design rules for future molecular dielectrics. Until recently, accurate calculation of dielectric responses in molecular materials was difficult and highly approximate. Most previous modeling efforts relied on classical formalisms to relate molecular polarizability to macroscopic dielectric properties. These efforts often vastly overestimated polarizability in the subject materials and ignored crucial material properties that can affect dielectric response. Recent advances in first-principles calculations via density functional theory (DFT) with periodic boundary conditions have allowed accurate computation of dielectric properties in molecular materials. In this Account, we outline the methodology used to calculate dielectric properties of molecular materials. We demonstrate the validity of this approach on model systems, capturing the frequency dependence of the dielectric response and achieving quantitative accuracy compared with experiment. This method is then used as a guide to new high-capacitance molecular dielectrics by determining what materials and chemical properties are important in maximizing dielectric response in selfassembled monolayers (SAMs). It will be seen that this technique is a powerful tool for understanding and designing new molecular dielectric systems, the properties of which are fundamental to many scientific areas.

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field-effect transistors) and other electronics.11−14 However, a traditional limitation of such materials is the low dielectric constants of those that might serve as gate dielectrics, typically 2.0−4.0, smaller than many inorganics (e.g., SiO2, HfO2). Gate dielectrics are fundamental components of all transistor-based logic circuitry, and the combination of high dielectric constant

INTRODUCTION The dielectric properties of liquids and solids have central importance in many fields, including biological function,1,2 materials science,3−5 molecular electronics,6−8 and solubility.9,10 Since the 1800s, researchers have determined the dielectric constants of materials, yet little progress has been made in accurately computing dielectric properties. Advances in performance and processing of organic electronic materials makes them attractive targets for next-generation OFETs (organic © XXXX American Chemical Society

Received: April 8, 2016

A

DOI: 10.1021/acs.accounts.6b00173 Acc. Chem. Res. XXXX, XXX, XXX−XXX

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Accounts of Chemical Research and nanoscopic thickness (i.e., high capacitance) are essential for achieving high switching speeds and low power consumption. Furthermore, developing better-performing dielectric materials has been hindered by the current highly empirical and laborintensive pace of synthetic progress. An accurate and efficient theoretical computational approach should accelerate the discovery process by screening potential organic dielectric materials and providing reliable design rules for future materials. Early modeling efforts related polarizability, a property of an isolated molecule in vacuum, to a materials property, dielectric constant, via classical formalisms such as the Clausius−Mossotti relation.15 While sufficient for modeling dilute systems, these classical relationships are often inadequate for modeling dielectric response in solids and liquids.16 Polarizability in solids is greatly overestimated in standard, small-molecule quantum calculations, leading to overestimations in dielectric response; these approaches often neglect important materials’ properties such as molecular surface coverage and orientation.17,18 As chemistry becomes more interrelated to biology and materials science, it has become increasingly important to incorporate environmental effects surrounding a chemical process, such as dielectric response, into modeling efforts. Fortunately, recent advances in computational power and methodology have enabled increasingly accurate calculations of solids.19−22 In the materials science community, the dielectric response of oxides,23,24 piezoelectrics,20 ferroelectrics,25,26 and interfaces21,27,28 has been successfully modeled, but few studies have dealt with organic solids.18,22 Liquids and gases require additional translational and rotational terms to capture their entire dielectric response. While we focus primarily on solid materials here, we believe that extending this technique to calculating dielectric responses in liquids and gases is realistic. In this Account, we focus on modeling and optimizing dielectric response in molecular dielectric materials using firstprinciples methodology. We describe an approach using periodic boundary conditions in conjunction with density-functional theory (DFT). Subsequently, we demonstrate how both materials properties (molecular density, molecular tilt angle, intermolecular disorder) and chemical design (conjugation length, polarizable substituents, donor−acceptor motifs) can significantly alter dielectric response in molecular dielectrics. Lastly, we discuss how these results can be used to design new molecular gate dielectrics for organic electronics and potential trade-offs between dielectric response and device performance.

Figure 1. Coordinate system used for first-principles DFT calculation of the optical and static dielectric response of aromatic molecular arrays. Diagram of a 2 × 2 × 2 simulation cell for a benzene monolayer (a) and orientations of the benzene molecules looking down the x, y, and z axes (b). The z axis is parallel to the applied electric field and perpendicular to the xy plane. The simulation cell is periodic in all three directions, but the large imposed vacuum along the z axis insures that the monolayers do not interact.

a parallel plate capacitor model,29,30 the experimentally observable dielectric constant ε can be computed using eq 1. Here, ηi is the calculated local dielectric constant at a given index i along the a−b = ε

b

∑ i=a

1 ηi

(1)

z coordinate, the a and b indices correspond to the origin and terminus of the molecule, respectively, and ε is the dielectric constant averaged over the length of the molecule. Here the origin and terminus of the molecule are defined as the positions along the z-axis of the lowermost and uppermost constituent atoms, respectively. To calculate the local dielectric response, the system is first relaxed in the absence of an electric field. After the system is relaxed, two calculations are performed. In the first, an electric field is applied, Eapplied = 0.001 au, parallel to the z axis, while in the second, Eapplied = −0.001 au. The 3D charge density profile, ρind(x,y,z), is then calculated as the difference in charge density between the two different applied electric fields. Subsequently a planar average is taken for each along the z axis. ρ̅ind(z), the difference in the charge density profiles at the different electric field strengths, is related to the induced polarization, P̅(z), using eq 2. The induced polarization is then used to calculate the local dielectric constant, η(z) (eq 3), with ε0 the vacuum permittivity divided by 4π, and Eext = 2Eapplied. The calculation of charge density change gives the equivalent of the relative voltage drop for each segment, so the boundary condition set by the total voltage applied to the molecule

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COMPUTATIONAL APPROACH Computing the dielectric response of a molecular material measures the change in charge density as a function of applied electric field. The electric field is applied along the molecular axis perpendicular to the substrate surface on which molecular monolayers lie; this axis will be referred to as the z axis, as shown in Figure 1. The local dielectric constant computed at each z coordinate is thus an average over the xy plane, providing a planar-averaged local dielectric response that varies as a function of molecular architecture along the z-axis. Note that a smoothing function is applied to the electron density at each z coordinate as described in other articles.21,27 The planar-averaged local dielectric constant, η(z), provides two informative quantities in monolayer systems. First, once the local dielectric response along a particular molecular direction (i.e., at a specific z coordinate) is known, a direct comparison between specific structural features and the corresponding dielectric response can be made. Second, while η(z) is a nonmeasurable quantity, by applying B


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Accounts of Chemical Research d P ̅(z) = −ρind ̅ (z ) dz η(z ) =

ε0Eext ε0Eext − P ̅(z)

electrons respond to the rapidly oscillating electric field. It is therefore important that a theoretical description be able to differentiate between these frequency regimes. In the high frequency limit, ω → ∞, described by the optical dielectric constant, only the electrons are allowed to relax in the presence of the electric field while the nuclear coordinates are held fixed. In the low frequency limit, ω → 0 described by the static dielectric constant, internal geometric reorganization is allowed by permitting displacements of the nuclei. To verify that the present approach accurately captures the dielectric response at high and low frequencies in molecular systems, sparsely packed (1 molecule/nm2) periodically spaced monolayers of benzene, fluorobenzene, and chlorobenzene molecules are examined. In general, high-frequency dielectric response is primarily determined by the atomic radii and the density of the atoms comprising a molecule.36,37 Correspondingly, it might be expected that the optical dielectric responses of benzene and fluorobenzene would be similar due to the similar atomic radii of H and F atoms, while chlorobenzene might have a significantly greater response. Figure 2A details the computed optical dielectric response of

(2)

(3)

enables calculation of the series of voltages and thus the total dielectric constant. The optical dielectric constant where ω → ∞ is computed assuming that only the electrons respond to the electric field (i.e., optical regime). The static dielectric constant where ω → 0 is calculated by allowing the internal molecular geometry to relax in the presence of the electric field. In the optical and static response, no translational or rotational motion is allowed. The observable dielectric constant of a monolayer can also be computed using the induced polarization of the entire monolayer, as shown by Natan31 and others. The relationship between monolayer polarization, P, and the dielectric constant, ε, is as in eq 4 where ε0 and Eext are the same terms as used above. The polarization for the monolayer is then defined as ε=

P=

ε0Eext ε0Eext − P

(4)

Δμ VML

(5)

where Δμ is the change in monolayer dipole moment induced by the applied field and VML is the volume of the monolayer. VML is the area of the unit cell times the monolayer thickness. The calculated ε is theoretically identical in eqs 1 and 4; however there are numerical discrepancies between the two values for highly polarizable molecules. This is caused by a locally large polarization that causes eq 3 to diverge. If the local polarization is properly averaged,21 then the two values agree. By implementing periodic boundary conditions, the medium of the surrounding molecules is accounted for. If the x and y dimensions are set to large enough values, the molecules are essentially treated in vacuum. Conversely, if the x and y dimensions are set to smaller values then the calculation models a molecule within a dielectric medium. The local dielectric method is desirable when analyzing local dielectric effects in materials, while the dipole method is more robust for highly polarizable materials and is easier to compute. For specific simulation parameters used in these calculations, please see our recent articles.22,32−34

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MODEL VALIDATION To demonstrate the validity of the present computational technique, examples are analyzed that would reveal whether important dielectric characteristics were captured. The goal is to determine whether this technique qualitatively captures the dielectric response frequency dependence and quantitatively calculates the static dielectric response in ordered molecular solids.

Figure 2. Computed local optical (a) and static (b) dielectric constant of a monolayer of benzene (solid), fluorobenzene (dashed), and chlorobenzene (dotted). There is an imposed 10 Å separation in x and y directions between molecules. The applied electric field is parallel to the z axis and has a strength of ±0.001 au, and the response is averaged over the xy plane.

three monolayers composed of these three molecules. The optical dielectric response computed with this technique is in good agreement with basic chemical intuition. Thus, the local dielectric constant is similar across the three molecules until the z coordinate reaches the point of differing substitution. At this site, benzene and fluorobenzene exhibit nearly identical optical dielectric responses while chlorobenzene exhibits a markedly greater response.

Dielectric Response at High and Low Frequencies

Different contributions to the dielectric response of molecular materials are evident at distinct frequencies of the applied electric field.35 In the low frequency regime, geometric rearrangements/ oscillations contribute to large dielectric responses, while at high frequencies, only the electronic response is important since only C


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calculated from eq 4 with the theoretical bulk dielectric constant, εtheory, representing the averaged dielectric value from the x, y, and z axes. The computed (εtheory = 2.46) and the experimental bulk dielectric constant (εexp = 2.34) are in excellent agreement.39 The crystalline bulk is defined as the region in which the dielectric response repeats periodically and is indicated in Figure 3a by the dashed line. These results demonstrate the excellent accuracy of the present computational approach. Benzene is relatively nonpolar; to demonstrate validity in polar materials, the dielectric constant of o-dinitrobenzene was calculated and found to be in similar agreement.22 The dielectric response reaches periodicity within one layer in benzene (Figure 3b), indicating rapid convergence of the local dielectric constant to the bulk response.

Since benzene has minimal polar bonding, the static dielectric response should be nearly identical to the optical response. However, for fluorobenzene and chlorobenzene, there should be variations caused by oscillations of the polar carbon−halogen bonds. Figure 2B details the static dielectric response of the three materials. As anticipated, for benzene the static and optical dielectric constants are indistinguishable. In contrast, the halogenated benzenes contain polar bonds and consequently have different static and optical dielectric responses, primarily differing in the vicinity of the carbon−halogen bond, as evident in Figure 2B. In modeling the optical and static dielectric responses of these molecular layers, note that the present approach is capable of (a) correlating chemical functional groups to specific dielectric responses and (b) differentiating dielectric responses based on the frequency regime of the applied field.

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MATERIALS PROPERTIES Depending on the film growth and assembly process, materials may display significantly different electronic properties.11,14,40,41 In recent work on high-capacitance organic dielectrics, selfassembled monolayers (SAMs) were used to create well-ordered, ultrathin dielectrics.7,8,42,43 SAMs exhibit variable surface coverage densities, molecular tilt angles, and intermolecular packing geometries, depending on details of substrate−molecule and molecule−molecule interactions, which in turn are influenced by both constituent molecule chemical properties and the film growth technique.11,14,40,41,44 Therefore, it is important to understand and predict in advance how these variables affect the film dielectric response. Additionally, computational results for model SAMs can be applied to other molecular materials where similar variations in bulk properties, such as crystalline versus amorphous, can be obtained through variations in processing conditions or alterations to chemical architecture. Here we examine how surface coverage, molecular tilt angle, and intermolecular geometry govern SAM dielectric response and offer design rules for maximizing dielectric response.

Dielectric Response in Crystalline Solids

To verify the accuracy of the technique in computing the dielectric response of ordered multimolecular materials, a seven layer slab of benzene molecules, with molecular coordinates taken from the benzene crystal structure,38 was analyzed. Figure 3A

Surface Coverage Effects

To examine surface density effects on SAM dielectric response, coverage was varied from 0.1 adsorbed chains/nm2 to 5.0 adsorbed chains/nm2, arrayed in a square lattice with uniform x and y direction molecular spacing (c.f., Figure 1). Polyethylene and polyyne-based SAMs were chosen to assess differences between nonconjugated and conjugated constituents. Molecular long axes are assumed to be oriented parallel to the electric field, and tilt angle effects are addressed in the next section. Figure 4 shows trends in computed dielectric constant versus surface coverage for the two different hydrocarbon SAMs. In both, the dielectric constant scales linearly with surface coverage. This behavior can be explained by the incremental replacement of the vacuum, ε = 1, with hydrocarbon chains as the surface coverage increases. Replacing vacuum with an electron-rich molecular material increases the density of electrons that can respond to the applied field, thereby increasing the dielectric response. At high surface coverage, the simple polyyne hydrocarbon has a computed dielectric constant >8.0, very high for an organic material.18 Because PBE and other simple functionals are known to overestimate polarizability of conjugated organics,30 the Coulomb potential screening hybrid functional of Heyd, Scuseria, and Ernzerhof (HSE) 0645 was used to test the validity of the results. Using the HSE 06 functional as implemented in QUANTUMESPRESSO gave similar results for polyyne at 5.0 molecules/nm2 with ε = 6.76. These results highlight an important design guideline for high capacitance molecular gate dielectrics: all other factors being equal, maximizing the surface

Figure 3. Local optical dielectric constant (a) of seven benzene layers and (b) at different thicknesses, one (blue), three (red), five (green), and seven (purple) benzene layers, along the z axis. Images next to the dielectric response are xz cross sections of the benzene crystal structure slabs looking down the y axis. Note that the benzene is infinitely repeated in the x and y directions using periodic boundary conditions. The dashed line in panel a represents the region considered to be bulk, that is, with minimal surface effects. The electric field is applied parallel to the z axis and has a strength of ±0.001 au, and the response is averaged over the xy plane.

shows the benzene slab and the computed optical dielectric response. Only the optical dielectric constant is examined here because, as discussed above, the benzene optical and static dielectric constants are essentially identical. The molecular orientation and density are different viewed down each axis, requiring the dielectric response along the remaining two axes to also be computed.22 The dielectric constant along each axis is D


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absent in the polyethylene SAM without a π system. Clearly to achieve large dielectric responses, conjugated π-electron SAMs must be properly oriented in the electric field direction.

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CHEMICAL DESIGN In the previous section, we examined how materials properties alter the dielectric response in molecule-based dielectrics. Next, modification of chemical structures is analyzed using two different approaches. We first examine how introducing large, polarizable substituents alters the dielectric response of simple π-hydrocarbons. Then we introduce donor and acceptor substituents to determine whether dielectric response can be enhanced in a similar fashion for donor−bridge−acceptor (DBA) molecules. Lastly, we explore how large dielectric constants can be realized while retaining high capacitances.

Figure 4. First-principles DFT calculation of the optical dielectric constant for polyethylene and polyacetylene monolayers as a function of surface coverage in molecules/nm2. Computed optical dielectric constant for 10-carbon polyethylene and polyyne oligomers as a function of surface coverage. Lines are fit with a linear regression. At each surface coverage, molecules are arrayed such that the x and y distances between neighboring molecules are identical. Molecules are assumed to be standing perpendicular to the surface, parallel to the applied electric field. Applied electric fields are parallel to the z axis with a magnitude of ±0.001 au. A k-point scheme of 2 × 2 × 2 is used for all simulations.

Polarizable Substituents

The next logical step beyond conjugated hydrocarbons is to inquire how different backbone substituents alter the dielectric response. Figure 6A shows the model conjugated backbone,

coverage of an oriented molecular material should maximize dielectric response. Dense surface coverage imbues SAMs composed of relatively simple conjugated molecules with noteworthy dielectric response. Molecular Orientation Effects

The alignment angles of the polyethylene and polyyne long axes were next varied from 0° to 60° with respect to the surface normal (Figure 5). Beyond 60°, the hydrocarbon molecules

Figure 5. First-principles DFT calculation of the optical dielectric constants of C-10 polyethylene and polyacetylene monolayers as a function of molecular tilt angle θ. The molecules are separated by 5 Å in the x and y directions. Applied electric fields are parallel to the z axis and have a magnitude of ±0.001 au. A k-point scheme of 2 × 2 × 2 is used for all simulations.

Figure 6. First-principles DFT calculation of the static dielectric constants of SAMs having various backbone substituent(s). (A) (left) Monosubstituted trans-polyacetylene with replacement of a terminal H atom; (right) di- and trisubstituted polyacetylenes with the 2nd and 3rd substitutions at the backbone positions indicated. (B) Static dielectric constants of the various monolayers. Each unit cell contains one molecule and has dimensions 5.47 × 5.47 × 25.0 Å3. The applied electric fields are parallel to z with a magnitude of ±0.001 au. A k-point sampling scheme of 2 × 2 × 2 is used for each SAM.

come into unphysical contact. Figure 5 shows the computed dielectric constants of C-10 polyethylene and polyyne SAMs as a function of tilt angle. While the dielectric constant of polyethylene SAM remains essentially unaffected by increased tilting (Δε < 0.2), the polyyne SAM dielectric constant falls significantly. The tilt angle dependence of the dielectric response in polyyne SAMs reflects changes in the interaction of the π system with the electric field. When the molecular long axis is parallel to the electric field, the π electrons can respond over the entirety of the π pathway. As the molecule is brought normal to the electric field, there is less efficient coupling along the molecular length, attenuating dielectric response. This phenomenon is

trans-PA, with single substituent group R. In principle, replacing H with more polarizable substituents might increase the dielectric response, so substituents of varying atomic numbers and polarizabilities (F, CH3, OH, SH, Br, and I) are introduced, and the computed static dielectric responses at a constant 3.33 molecules/nm2 coverage are shown in Figure 6B. Note that E


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atomic polarizations that occur on longer time scales due to atomic rearrangement, than electronic polarization that occurs instantaneously.

the dielectric response increases in all cases except when H is replaced with F, which has similar polarizability.18 In contrast, incorporating SH, Br, or I significantly increases the dielectric response in accord with their larger atomic polarizabilities. To investigate further how substituents modulate dielectric response, additional H atoms are next replaced with iodine atoms (Figure 6A), and the results are plotted in Figure 6B. Note that 2-I and 3-I exhibit progressively larger dielectric constants, with 3-I achieving a remarkable computed static dielectric constant of 12.98. This corresponds to a thin film capacitance exceeding 9.0 μF/cm2. As shown above, at high surface coverages and with proper molecular alignment, even simple molecular structural changes can dramatically increase the dielectric response of a thin molecular film.

Dielectric Response versus Capacitance

Squaraines have large optical responses and can be configured in donor−acceptor−donor (DAD) architectures.48−50 Figure 8A

Donor−Bridge−Acceptor Structures

To ascertain whether DBA systems can exhibit large dielectric constants, five different structures based on six-carbon polyacetylene fragments were evaluated (Figure 7A). Substituents

Figure 8. First-principles DFT computation of optical and static dielectric constants of squaraine SAMs: (A) structures of substituted squaraines; (B) optical and static dielectric constants of substituted squaraines. The unit cell contains one molecule and has dimensions of 6.2 × 5.1 × 40.0 Å3. Applied electric fields are in the z direction with a magnitude of ±5.14 × 108 V/m. A k-point sampling scheme of 2 × 2 × 1 is used for each material.

Figure 7. First-principles DFT calculation of optical and static dielectric constant of six-carbon polyacetylene SAMs with the indicated substituents: (A) substitutions examined; (B) optical and static dielectric constants of the various polyacetylene SAMs. The polyacetylene unit cell contains one molecule and has dimensions 5.00 × 5.00 × 30.0 Å3.

shows squaraines with three possible substituents. Squaric acid acts as a powerful acceptor and the two phenyl rings can act as donors, depending on substituents. Squaraine with one squaric acid unit and two H substituents is designated 1-Sq-HH. Similarly, squaraines with NO2 (acceptor) and NH2 (donor) substituents are designated 1-Sq-AA and 1-Sq-DD, respectively. In Figure 8B, optical and static dielectric responses are presented for three substituted squarine SAMs. Note that 1-Sq-HH and 1-Sq-AA have substantially smaller static dielectric constants than 1-Sq-DD, likely reflecting smaller donor characteristics since H does not exhibit donor/acceptor behavior and NO2 acts as an acceptor. 1-Sq-AA’s larger dielectric constant versus 1-Sq-HH may reflect a larger electron count, similar to the AH pattern above. 1-Sq-DD has a remarkable static dielectric constant that is essentially twice its optical dielectric constant. The small length and large dielectric constant means that 1-Sq-DD has a predicted capacitance of C = 6.99 μF/cm2, 5× larger than that of the current organic gate dielectric champion.51 Proceeding via C4−O linking from D−A to D−A−D achieves further enhancement in dielectric performance. Further catenation by joining one (2-Sq-HH) or two (3-Sq-HH) squarate units terminated by H (HH) or NH2 (DD) to a squarate core further increases the dielectric constants (Table 1), with 3-Sq-DD having the largest calculated ε. Nevertheless the capacitance is slightly lower than 1-Sq-DD due to the increased layer thickness (C = 5.99 μF/cm2), highlighting possible capacitance limits

H, NH2, and NO2 were used to represent neutral, donor, and acceptor moieties, respectively. Figure 7B shows the optical/high frequency and static/low frequency dielectric constants for the SAMs at 4.0 molecules/nm2 coverage. Note that molecule HH with the least polarizable substituents and minimal polar bonding has the smallest response at both high and low frequencies (εs = εo= 4.26). In marked contrast, molecules DH and AH exhibit far larger optical and static dielectric responses, reflecting the differing electron densities of NH2 and NO2,46 with the introduction of polar bonds creating additional polarization. Polarization induced by polar bonding includes dipolar and atomic polarization due to reorientation of dipoles and atomic displacements.46 Similarly, DD’s dielectric constant is greater than that of DH due to the additional electron-rich substituent. Furthermore, DA has significantly greater computed dielectric response (εs = 7.62) than DD (εs = 5.29). If dielectric response were strictly additive, that of DD should approximate that of DA since the dielectric constant of DH ≈ AH. That introducing donor and acceptor groups at opposite ends of the molecule yields a larger dielectric response indicates cooperative enhancement and agrees qualitatively with Clausius−Mossotti in that donor−bridge−acceptor molecules typically have large polarizabilites.47 That the static dielectric constant is enhanced more than the optical argues that donor− acceptor interactions more strongly influence dipolar and F


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Accounts of Chemical Research Table 1. Static and Optical Dielectric Constants of 1-Sq-XX, 2-Sq-XX, and 3-Sq-XX dielectric constant material

ε0

1-Sq-HH 1-Sq-DD 2-Sq-HH 2-Sq-DD 3-Sq-HH 3-Sq-DD

3.78 5.03 5.82 7.33 7.91 9.68

a

εs

capacitance (μF/cm2)

b

3.91 10.91 6.10 14.59 8.34 19.72

a

Coc

Csd

2.68 3.22 2.55 3.00 2.52 2.94

2.77 6.98 2.67 5.97 2.66 5.99

Optical dielectric constant. bStatic dielectric constant. frequency capacitance. dStatic frequency capacitance.

c

Optical

for organic dielectrics. To obtain higher capacitance, organic dielectrics must remain thin (∼5.0 nm) while simultaneously increasing ε.

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DESIGN CONSIDERATIONS FOR HIGH-CAPACITANCE DIELECTRICS

Multilayers

In self-assembled nanodielectrics (SANDs), an oxide nanolayer is introduced between organic π-layers and finally capped with another oxide nanolayer to minimize leakage, ensure a smooth interface with the overlying TFT semiconductor, minimize I−Vgate hysteresis, and preserve or enhance capacitance.52,53 The growth procedure and a representative SAND structure are shown in Figure 9. SANDs are an example of a complex multicomponent dielectric, and computational studies have not yet been completed. Nevertheless, the parallel plate capacitor model provides insight into such multilayer systems (eq 6),29,30 where di and εi are the thickness and dielectric constant, respectively, of component material i, d is the total thickness of the multilayer, and ε is the multilayer dielectric constant. Equation 6 shows d = ε

x

∑ i=1

di εi

Figure 10. Total dielectric constants of bilayer dielectrics. (A) Bilayer in each slab having a different dielectric constant; d1 and d2 are the thicknesses of slabs 1 and 2, respectively, while ε1 and ε2 are the corresponding dielectric constants of each slab, and εT is dielectric constant of the multilayer of thickness d. (B) Multilayer dielectric constant as a function of ε1 and ε2. Each slab has a thickness of 5 nm. (C) Multilayer dielectric constant as a function of slab 1 dielectric constant and thickness. Slab 2 is assumed to be a 5 nm SiO2 layer with ε2 = 3.9. Note that actual εT values will change gradually rather than abruptly as shown here as a guide to the eye.

(6)

that the dielectric response of a multilayer is dominated by the lowest ε constituent. For example, in the bilayer of Figure 10A, consisting of two dielectric layers where d1 = 5.0 nm and ε1 = 20.0 and d2 = 5.0 nm with ε2 = 2.0, the computed bilayer dielectric constant is only 3.64, significantly closer to 2.0 than to 20.0. In fact, if ε1 is doubled to 40.0, εT only increases to 3.80. Figure 10B shows εT’s of bilayers consisting of two different dielectric

materials. Note that the multilayer dielectric constant cannot reach values near ε ≈ 20.0 unless both layers have a dielectric

Figure 9. Growth procedure for a SAND (self-assembled nanodielectric) multilayer gate dielectric. Inset: cross-sectional transmission electron micrograph. G


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Figure 11. Calculated dielectric constants εs and ε0 and single-molecule conductance, dI/dV, of anthracene-based (AC) and anthraquinone-based (AQ) molecular monolayer materials with Au electrodes. In the vicinity of the Au Fermi energy, EF (red vertical line), the conductance of the AQ and AC junctions are 0.21 and 11.4 μS, respectively. The origin of the ∼1.7 order of magnitude drop in AQ conductance, a value in close agreement with experiment,17 is destructive quantum interference. In contrast, the optical and static dielectric constants of these materials are nearly identical ε0 ≈ εs ≈ 2.2. All calculations are for junctions operating in linear response at 300 K.

that this approach is well-suited for molecular dielectrics, able to accurately capture the frequency dependence of the dielectric response as well as reproduce experimental dielectric constants in thin films. This approach provides useful insights for designing next-generation molecular dielectrics as well as showing how chemical/structural alterations can dramatically increase dielectric response. Lastly, we show that dielectric matching is crucial in multicomponent materials and how molecular conductance can be tuned while achieving large dielectric responses. In this Account, we focused on solid-state materials primarily because of interest in altering dielectric properties for transistors; however it should be evident that similar approaches are applicable to liquids.

constant >15.0. In Figure 10C, the total dielectric constant of a bilayer consisting of 5 nm SiO2 (ε = 3.90), and a layer of variable thickness and ε is shown. Note that a 40 nm layer with ε = 40.0 is insufficient to raise the total dielectric constant above 20.0 as long as the second layer is 5 nm SiO2. Thus, introducing a low ε layer, even if far thinner than the accompanying layers, significantly diminishes the overall dielectric response. Thus, all constituent layers of a multilayer must have similar dielectric constants to achieve appreciable overall stack dielectric constants. Leakage Current in Ultrathin Dielectrics

In designing FET gate dielectric materials, it is important to minimize leakage currents while maintaining large capacitance. It is of course well-known that highly polarizable molecules tend to have smaller band gaps54,55 and that classical inorganic materials have a well-established relationship between decreasing band gap and increasing dielectric constant.56,57 Note that delocalized organic π system materials (with smaller band gaps) can also transport charge,58−61 potentially increasing leakage currents between the source/drain electrodes and the gate. Thus, the trade-off between dielectric constant and charge transport capacity must be considered in designing highcapacitance organic dielectrics. Recent work demonstrates that molecular dielectrics offer a unique opportunity in this regard.62 Computing the conductance and dielectric properties of linearly conjugated single-molecule Au−AC−Au and Au−AQ−Au junctions with anthracenyl (AC) or cross-conjugated anthraquinoid (AQ) blocks reveals dramatic differences in transport capacity with minor structural changes, attributable to quantum interference effects (Figure 11), 11.4 μS and 0.21 μS, for AC and AQ, respectively. In contrast, the optical and static dielectric constants of these materials are virtually identical at ∼2.21 for AC and ∼2.24 for AQ.33,62 Furthermore, creating DBA structures with donor and acceptor substituents more than doubles the dielectric constants without changing the conductance.33,34 These results demonstrate the possibility of realizing high dielectric response organics with minimal conductance by utilizing quantum interference and highly polarizable structures.

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AUTHOR INFORMATION

Corresponding Authors

*T.J.M. E-mail: [email protected]. *M.A.R. E-mail: [email protected]. Notes

The authors declare no competing financial interest. Biographies Henry Heitzer was born in Waukesha, Wisconsin. He received B.S. degrees in Chemistry and Mathematics from Carlton College in 2010 and a Ph.D. in Chemistry from Northwestern University in 2014 where he was a National Defense Science and Engineering Graduate Fellow. He is currently a Consultant for Boston Consulting Group. Tobin J. Marks was born in Washington, D.C. He is currently Ipatieff Professor of Chemistry and Professor of Materials Science and Engineering at Northwestern University. He received a B.S. degree from the University of Maryland and Ph.D. from MIT in Chemistry. He is a member of the U.S. National Academy of Sciences, a Fellow of the American Academy of Arts and Sciences, a Member of the German National Academy of Sciences, and a Fellow of the Royal Society of Chemistry. Among other recognitions, he received the U.S. National Medal of Science and the American Chemistry Society Priestley Medal.

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Mark A. Ratner was born in Cleveland, Ohio. He is currently Dumas Emeritus Professor of Chemistry and Professor of Materials Science and Engineering at Northwestern University. He received a B.A. degree from Harvard University and Ph.D. from Northwestern University in Chemistry. He is a member of the U.S. National Academy of Sciences

CONCLUSIONS AND OUTLOOK Here we described the application of a first-principles method to calculate dielectric response in molecular materials. We show H


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and a Fellow of the American Academy of Arts and Sciences. Among other recognitions, he received the Willard Gibbs Medal from the ACS and honorary Doctor of Science Degrees from the University of Copenhagen and the Hebrew University of Jerusalem.

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ACKNOWLEDGMENTS This research was supported by the NSF MRSEC program (Grant DMR-1121262) through the Northwestern University Materials Research Center and by AFOSR (Grant FA9550-15-10044). H.M.H. was supported by an NDSEG Graduate Fellowship.

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Computation of Dielectric Response in Molecular Solids for High

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