Chain Length Dependence of the Dielectric ... - ACS Publications

Jun 2, 2017 - molecules and the film dielectric constant, using periodic density functional theory (DFT) calculations, for polyyne and saturated alkan...
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Chain Length Dependence of the Dielectric Constant and Polarizability in Conjugated Organic Thin Films Colin Van Dyck,* 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 S Supporting Information *

ABSTRACT: Dielectric materials are ubiquitous in optics, electronics, and materials science. Recently, there have been new efforts to characterize the dielectric performance of thin films composed of molecular assemblies. In this context, we investigate here the relationship between the polarizability of the constituent molecules and the film dielectric constant, using periodic density functional theory (DFT) calculations, for polyyne and saturated alkane chains. In particular, we explore the implication of the superlinear chain length dependence of the polarizability, a specific feature of conjugated molecules. We show and explain from DFT calculations and a simple depolarization model that this superlinearity is attenuated by the collective polarization. However, it is not completely suppressed. This confers a very high sensitivity of the dielectric constant to the thin film thickness. This latter can increase by a factor of 3−4 at reasonable coverages, by extending the molecular length. This significantly limits the decline of the thin film capacitance with the film thickness. Therefore, the conventional fit of the capacitance versus thickness is not appropriate to determine the dielectric constant of the film. Finally, we show that the failures of semilocal approximations of the exchange-correlation functional lead to a very significant overestimation of this effect. KEYWORDS: dielectric constant, thin film, conjugation, thickness, chain length, permittivity, depolarization radiation-hard,31,32 and are readily functionalized in comparison to metal oxide dielectrics.21,33 In the context of molecular electronics, the dielectric properties of molecular thin films have also been characterized to investigate their suitability as capacitors,34 gate dielectrics,16−22 and to relate their dielectric properties to their transport properties.23,35−37 This relationship can be traced back to the Simmons barrier tunneling model,23,38 but is also evidenced by the important role played by the voltage drop across molecular junctions.39−43 A figure of merit to characterize the performance of these organic dielectric layers is given by C, the capacitance per unit area (eq 1), where ε is the dielectric constant of the dielectric

O

ver the past few decades, the scientific community has witnessed a large and growing interest in carbonbased electronic materials.1 Since the discovery of the conducting properties of conjugated polymers,2 the properties of the delocalized π-electrons have been exploited to create organic electronic devices, such as light-emitting diodes,3−6 solar cells,7,8 and transistors.9−11 Note that these conjugated materials not only exhibit useful conductance properties, but their remarkable response to applied electric fields makes them attractive candidates for nonlinear optical materials.12−15 More recently, a number of research groups proposed to take advantage of this response to create organic thin film transistors.16−22 The idea is not only to use an organic material as an active layer between a source and drain electrodes but also to include an organic dielectric layer to insulate the gate electrode.16−22 One particular embodiment takes the form of a self-assembled monolayer (SAM) deposited on the gate electrode. A related form uses multilayers, such as selfassembled nanodielectric (SAND) materials, which offer highly promising performance, with measured dielectric constants up to ∼13 and low leakage currents.23−28 These have the advantage of being flexible, low cost, suppressing charge trapped between the dielectric and the semiconductor,29,30 © 2017 American Chemical Society

C=

ε 4πd

(1)

medium and d is the thickness of the layer, both in atomic units (which are used throughout this work). To enhance the Received: March 15, 2017 Accepted: June 2, 2017 Published: June 2, 2017 5970

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RESULTS/DISCUSSION Polarizability of the Isolated Molecules. Polyyne chains, i.e., C2iH2 as represented in Figure 1, possess a very simple

performance of organic dielectric layers, we then have two main possibilities. First, we can increase the dielectric constant by using highly conjugated and polarizable materials. Second, we can make the organic layer as thin as possible. Unfortunately, these two strategies lead to increasing leakage currents through the film which destroys the dielectric functionality, and creates a dilemma for the design of organic gate dielectrics with enhanced performance. Recently, we showed that the introduction of a quantum interference unit in the molecular backbone is a promising means to escape from this dilemma.44 In this contribution, we investigate the actual thickness dependence of the capacitance for a conjugated monolayer, using periodic density functional theory (DFT).45−47 For simplicity, we focus on the one-dimensional conjugated polyyne chains48 and compare the dielectric response to that of one-dimensional saturated alkane chains. This investigation is motivated by the superlinear dependence of the polarizability with the chain length in conjugated molecules.49−51 The impact of this behavior on the dielectric properties of thin films has not been reported or characterized yet in the literature. Strikingly, we show here that the delocalization of the electrons leads to a strong thickness dependence of the dielectric constant. These results are rationalized by a simple qualitative model that relates the polarizability of the chain to the dielectric constant of the film, taking the depolarization effect into account. Importantly, we show that this behavior is largely overestimated by a nonhybrid DFT functional, such as PBE. Only the inclusion of a significant amount of Hartree−Fock exchange provides computed results that agree with the most accurate coupledcluster methods, the current “gold standard” of quantum chemistry, used in this context. The results of the present study argue that a very simple strategy to significantly tune the dielectric constant of a thin film is to modify the conjugated chain length. Also, because of the thickness dependence, the classical linear inverse proportionality relation between the capacitance of the dielectric film and its thickness, as suggested by eq 1, no longer applies. This leads to a dampened dependence of the capacitance on the film thickness that permits reducing the leakage current through the monolayer exponentially without significantly affecting the capacitance of the organic film. This article is structured as follows. First, we compute the polarizability of the isolated molecules and highlight the importance of the choice of the exchange-correlation functional. We then compare these isolated polarizabilities with those of the molecules when embedded in a monolayer. The results illustrate the significant impact of the depolarization effects on the embedded molecule polarizability. In this section, we also introduce a simple predictive model that essentially captures the electrostatic depolarization in a thin film, in a way that is similar to the Clausius−Mossotti relation for bulk systems. This model performs well at predicting the qualitative trends observed in the periodic DFT calculations. Finally, we relate the molecular polarizability to the dielectric constant of the corresponding thin films. The results demonstrate that, because the conjugation of a molecular chain leads to a superlinear response in polarizability, the dielectric constant does not rapidly converge but rather increases significantly with the film thickness. In the last section describing our methodology, we formally define the dielectric constant of a thin film and explain how this can be evaluated at the theoretical level, using periodic DFT.

Figure 1. Two types of constituent thin film molecules, with varying lengths, investigated in this study. Above, a conjugated polyyne chain with alternating triple and single bonds, characterized by different carbon−carbon distances dt (triple) and ds (single). This provides substantial electron delocalization in the carbon chain. Below, a fully saturated alkane chain which does not have significant electron delocalization in the carbon chain.

molecular structure and have been widely studied.48,52−54 These are textbook examples of conjugated chains, characterized by frontier orbitals that are delocalized along the conjugated backbone. Furthermore, placing the molecule in an electric field induces a large reorganization of the electronic from one extremity of the chain to the other. This confers a large longitudinal polarizability on conjugated molecular chains.55 This delocalized polarization effect can be seen, at a quantitative level, in Figure 2. There, we show the electron density reorganization profile (panel a) that is induced by an applied electric field in isolated C8H2 and C20H2 polyyne chains, at the DFT/HSE06 level of theory and defined as: δρ(z) =

∫ ∫Cell [ρ(x , y , z , E ≠ 0) − ρ(x , y , z , E = 0)]dxdy

(2)

where ρ is the electron density. We observe that the charge profile increases from the left terminal carbon atom toward the right terminal carbon atom. This corresponds to a general polarization with negative and positive net charges on each side of the molecule. As the chain becomes longer (2i increases from 8 to 20), the induced dipole moment increases from 0.71 to 4.15 D. The corresponding charge reorganization profile is amplified, which means that a larger net charge is induced and accumulated at each side of the longer molecule, thereby increasing the dipole moment. This understanding is particularly noticeable as we integrate the reorganization profile of eq 2 (panel c), from the left to the right: Q (z ) =

∫0

z

δρ(z′)dz′

(3)

Q(z) denotes the electronic charge accumulated from the left to the right up to the z position along the backbone. One can see a general well-shaped profile indicating a net charge at each side of the molecule. The minimum of the profile goes down as the chain length increases, indicating a larger amount of electron charge on each side of the molecule. 5971

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Figure 2. (Top) Charge reorganization profiles induced by an applied electric field computed for the two indicated conjugated polyyne chains (a) and two saturated alkane chains (b) having different lengths, at the DFT/HSE06 + 0.45 and DFT/HSE06 levels of theory, respectively, in 132 Å2 unit cells. (Bottom) Integrated charge reorganization profiles along the molecular backbones, from left to right, corresponding to the top profiles. The delocalized polarization taking place in the conjugated molecules in comparison with saturated molecules is evident, as an accumulation effect can be seen in panel (c).

Figure 3. Evolution of single molecule polarizabilities with chain length for conjugated (a) and saturated (b) chains. For the conjugated chain (a), a superlinear increase is observed that is overestimated at the PBE level of theory. For the saturated chain (b), a linear increase is observed that is consistent for both hybrid and nonhybrid functionals.

significant delocalization related charge transfer along the saturated chain. We observe here that the induced dipole increases with chain length. Thus, the total polarizability of the molecule increases with LCC, the distance between the two terminal carbon atoms in eq 4. In general, it is well established that the polarizability follows a power law dependence,55

For comparison, the profile of a saturated carbon chain is also shown in Figure 2b and d. Note that a generally increasing trend is observed but much less pronounced than for the polyynes, and that, after a similar chain length extension from C8H18 to C20H42, the induced dipole moment increases from only 0.33 to 0.96 D. The dipole moment increase is different from the conjugated molecules. The profile is not amplified, and the net charge at each terminus remains constant (panel b). This result indicates that the increase in induced dipole is here due to the superposition of aligned dipole moments at each saturated C−C bond. This is particularly noticeable as the reorganization profile is integrated (panel d), revealing a less marked and less delocalized, well-shaped profile. That may be due to inductive charge transfer at terminal C−H bonds. Nevertheless, and in clear opposition to the conjugated profile, the minimum of the saturated alkane profile is almost insensitive to the length extension, verifying that there is no

n α(LCC) = αHC + α1LCC = αHC + αCC

(4)

Here, αHC, α1, and n are parameters that can be determined by a fitting procedure. For a saturated molecule, we reasonably expect a linear increase of the polarizability, i.e., n = 1. A small superlinear increase (n > 1) may also be seen in nonconjugated materials because of a repolarization effect due to adjacent polarized units.45 This effect may lead to a higher polarization at the center of the molecule. In the following, we associate the y-intercept, αHC, with the polarizability of the two terminal C− 5972

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Figure 4. Evolution of embedded molecule polarizabilities with chain length for conjugated polyyne (a) and saturated alkane chains (b) packed in a surface area of 33 Å2 per molecule. As discussed in the text, the evolution appears linear in panel (a) but a limited superlinear dependence persists.

H bonds. The second contribution is defined as αCC, and associates with the conjugated or saturated carbon chain. In Figure 3 the evolution of the polarizability with chain length extension for the isolated conjugated and saturated molecules is shown. To be consistent throughout this work and use the same methodology for each of the calculations, we compute the isolated molecule with a periodic DFT implementation. This naturally generates an error, due to the induced dipole interactions. This error has been accounted for in this work. Specifically, we place the molecules in large unit cells, apply an electric field and then apply a dipolar correction scheme to limit the error. This is explained in detail in the Supporting Information. The polarizabilities are computed directly from eq 17. In Figure 3a, note that the longitudinal polarizabilities of the conjugated chains increase more than linearly with the chain length. This specific feature of conjugated chains has been established by numerous theoretical models,49−51,55−58 and has also been confirmed in the experimental literature.59−64 The polarizability is very sensitive to the conjugated chain geometry and more particularly to the degree of bond length alternation (BLA).15,58,65,66 This latter term is defined as the averaged length difference between triple and single bonds (see Figure 1), along the conjugated backbone. It generally characterizes the electron delocalization, and the lower the BLA, the greater the delocalization.67,68 In the present PBE calculations, we observe a BLA that decreases by more than 50%, from 0.142 Å for C4H2 down to 0.067 Å for C32H2 (see Supporting Information). This shows enhanced conjugation as the conjugated chain extends in length. This is accompanied by the superlinear increase of the polarizability. After a fit of eq 4 to the PBE calculations, we obtain n = 2.30 ± 0.03 (see Supporting Information). We now compare the results of our computations to more sophisticated ones reported in the literature. For the polyyne chain, state-of-the-art coupled-cluster and highly accurate DFT calculations predict a BLA that goes from 0.164 Å58 for C4H2 down to 0.130 Å58,69 for C32H2, a decrease of only ∼21%. Moreover, coupled-cluster calculations, usually used as benchmark references in quantum chemistry,70,71 predict that n = 1.64,58 a value that is much lower than the value of 2.30 obtained here at the DFT/PBE level of theory. This disparity with respect to these highly reliable calculations is due to wellknown limitations of semilocal XC functionals. Indeed, this functional type tends to predict BLAs that are too small and polarizabilities that are too large.67,71−73 Moreover, not only are

the BLAs too short, but also the convergence of the BLA with the conjugation length is generally too slow,67,71 leading to overestimation of the superlinearity in Figure 3a. In the literature, it has been suggested that addition of a significant amount of Hartree−Fock exchange in the XC functional can largely correct for this misbehavior.57,65,67,69,71,72,74 Specifically, about 45% of Hartree−Fock exchange has been shown to reproduce the correct BLA convergence with the chain length for polyenes.71,75 This motivated us to use the HSE06 + 0.45 functional, a hybrid functional available in the VASP package, to study the chain length dependence of the polarizability. The corresponding results are also presented in Figure 3. A significant improvement in the computed values is immediately evident. The fitting procedure of eq 4 shows that n = 1.75 ± 0.02, in good agreement with the coupled cluster results. Moreover, this hybrid functional predicts a BLA that goes from 0.168 Å for C4H2 down to 0.126 Å for C32H2. This decrease of about 25% is in much better agreement with state-of-the-art calculations. It is then reasonable to consider that the results computed at the DFT/HSE06 + 0.45 level are a reliable reference, while the results computed at the DFT/PBE level can be tainted with a pretty large error. We will discuss the direct consequences on the dielectric constant evaluation in the final section. For saturated chains, we observe the expected linear increase for both nonhybrid PBE and hybrid HSE06 calculations. Indeed, the same fitting procedure gives n = 1.08 ± 0.02 and n = 1.08 ± 0.02 for HSE06 and PBE calculations, respectively. This is expected, as there is no delocalization in the saturated chains and the polarizabilities of the sigma-type C−C bonds simply add up. Consequently, as the separated fragment polarizabilities are not largely affected by the XC functional, there is no large difference between hybrid and nonhybrid functionals. We attribute the very limited superlinearity to the interfragment electrostatic polarization.45 Polarizability of the Molecules Embedded in a Monolayer. In Figure 4, we report the evolution of the polarizability with chain length for saturated and conjugated molecules, embedded in a small periodic unit cell. The chosen unit cell represents a significant coverage, with an area per molecule of 33 Å2 and an intermolecular distance of 5.74 Å. This evolution must be compared with the isolated counterparts, in Figure 3. Interestingly, we observe that the coverage significantly reduces the superlinearity of the polarizability in the conjugated chain case. This is especially true for longer chains, which tend to approach a linear evolution. An 5973

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ACS Nano apparently similar behavior was previously reported for oligophenyl chains.45 However, as we point out in the previous section, we are focusing here on the case of a strongly conjugated system, with a large charge transfer across the chain upon polarization (Figure 2c). Therefore, the superlinearity does not originate from an interfragment polarization and the model and argument discussed in this previous work do not apply in the context of this analysis. In the previous section, we reported that the BLA decreases as the chain is extended. Note that this observation remains valid at high packing densities. Actually, the coverage does not affect the BLA in the computations. We computed the BLA of the C8H2 chain with different coverages, at 33 Å2 and 264 Å2 per molecule. The associated change of BLA between these two coverages is less than 10−4 Å. This means that the chain intrinsically remains highly polarizable, independent of the coverage. Therefore, for these simple carbon chains, the only difference induced by the coverage originates in the depolarization. By depolarization, we mean the interaction used in the organic thin film context to describe the reduction of an isolated molecule dipole moment upon embedding in a monolayer, due to dipole−dipole interactions.76−78 The loss of superlinearity indicates that the depolarization increases together with the polarizability and the chain length. If the induced dipole in the embedded molecule is proportional to the applied field times the embedded molecule polarizability, α′, we obtain eq 5: μ ind = α′Eapp = α(Eapp + Enear)

Figure 5. Evolution of the depolarization factor, D, with chain length for the conjugated monolayers. In the dashed line the comparison is made with a simplified model for the near field, which is illustrated above the graph.

To explain the origin of this superlinearity loss at high coverages, or equivalently, the increase of the depolarization factor, we propose here a simple qualitative model. First, we assume that a long polarized polyyne molecule is a dipolar stick, made of two point charges, Q and −Q, separated by the conjugated chain length, LCC, with a dipole moment equal to μind in eq 5. This can represent an ideally delocalized polarization with charges accumulated at each end of the molecule. This charge is determined by the polarizability, the chain length, and the applied field (eq 7):

(5)

which can be reformulated as ⎛ E ⎞ α′ = α⎜⎜1 + near ⎟⎟ = α(1 − D) Eapp ⎠ ⎝

(6)

where we introduce D, a depolarization factor, in terms of the applied and response near fields. This factor can be evaluated by comparing the embedded and isolated molecule polarizabilities shown in Figures 3 and 4. Note that this factor and the near field fall to zero in three-dimensional cubic crystals and films having a cubic structure composed of polarizable point dipoles, in accord with the Clausius−Mossotti relation.79,80 As defined by eq 6, it has a microscopic nature. We stress that it is different from the macroscopic shape-dependent depolarization factor, as defined in some textbooks,81 in the general context of dielectric materials. For the following development, we focus only on the depolarization in the carbon chain. To do so, we extrapolate the first three data points in Figures 1, 2, 3, and 4 to a power law. The y-intercept then determines the terminal C−H bond polarizabilities (see Supporting Information), and ultimately the conjugated backbone polarizability, αCC, given in eq 4. This quantity has the advantage of being directly proportional to the chain length, which removes the y-intercept in the following discussion. Moreover, it allows the development of a simplified depolarization model. Finally, it will allow separation of the total dielectric constant into carbon backbone and terminal bond contributions, as seen in the next section. The depolarization factors associated with the carbon backbone contribution are shown in Figure 5. As expected, we see that D increases with the chain length. The same figure is reported for the saturated chain case in Supporting Information. In that case the depolarization rapidly converges toward 43% and 44% at the HSE and PBE levels of theory, respectively.

Q=

′ Eapp αCC LCC

(7)

Second, we assume that the near field generated by this induced charge distribution is uniform along the conjugated backbone. Third, we assume the charge distributions at the top and the bottom of the film are uniform and form two parallel plates at the edge of the conjugated backbones. The near field is then given by a well-known result of electrostatics (eq 8): Enear =

′ Eapp −4παCC −4πQ = A LCCA

(8)

with A the area per molecule. After insertion of this term into eq 5, we obtain: αCC ′ = αCC 4πα 1 + L CC (9) CCA and,

D=

4παCC LCCA + 4παCC

(10)

From this simple model, we can analytically compute the depolarization factor and compare with the actual depolarization extracted from the extensive and accurate periodic DFT calculation. This is shown in Figure 5. Note that we observe a 5974

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ACS Nano very good qualitative agreement. Moreover, as the chain is extended, the model gains in accuracy. The origin of this is a self-depolarization error discussed in the Supporting Information. A simple correction scheme can be introduced and can largely correct for this behavior. However, it complicates the mathematical description without any influence for longer chains, in which case the correction does not play any role. As a result, this simplified model captures the essential electrostatic depolarization effects in a thick conjugated monolayer. We can now estimate the effect of the depolarization on the chain length dependence, on the basis of our simple model. From eqs 4 and 9, we find eq 11 which predicts that the ′ (LCC) = αCC

n α1LCC

1+

n−1 4πα1LCC

A

Table 1. Polarizability Evolution with Chain Length for a Polyyne Chain, Embedded in a Monolayer at 33 Å2 Coverage, at the HSE06 + 0.45 Levela index (j)

LCC (Å)

αCC (a.u.)

ratios αCC : j+1/j

ratios LCC: j+1/j

n

1 2 3 4 5 6 7

1.188 3.753 8.868 16.521 24.251 31.823 39.482

10.95 42.83 118.72 239.99 363.09 486.27 609.81

3.91 2.77 2.02 1.51 1.34 1.25

3.16 2.36 1.86 1.47 1.31 1.24

1.19 1.19 1.13 1.08 1.07 1.05

a

Successive polarizability ratios are compared with the successive length ratios to highlight the degree of superlinearity in the molecule embedded in a monolayer. Superlinear behavior is characterized by a polarizability ratio higher than the corresponding conjugation length ratio. Superlinearity in between two successive lengths is then estimated in the last column, from eq S.29 in the Supporting Information.

1 ∼ LCC

L →∞

(11)

embedded molecule polarizability converges toward a linear dependence for sufficiently long chains, independent of n and in agreement with both the periodic DFT/PBE (n = 2.3) and DFT/HSE06 + 0.45 (n = 1.75) calculations. This shows that the embedded molecule polarizability does not follow a power law with chain length, contrary to the isolated molecule case. Moreover, if the evolution of the polarizability is strictly linear, i.e., n = 1, we obtain from eqs 4 and 10 that the depolarization through the conjugated material remains constant. The reason is that if the increase in polarizability is linear, the surface charge density on each charged plane is constant, according to eq 7. Therefore, the depolarization field given by eq 8 also remains constant. On the other hand, if the increase in polarizability is superlinear, a larger charge density will accumulate on each plane by extending the chain length, according to eq 7. This will increase the depolarization field, according to eq 8, and progressively eliminate the superlinearity. Very importantly, if it is true that a linear increase in polarizability tends to be recovered in a dense monolayer, the convergence toward this regime is rather slow and, strictly speaking, n > 1 for each of the embedded conjugated chains considered here. This is true for both the simple model and the actual DFT results, as established in Table 1. In this table, we compare the ratios of embedded polarizabilities with chain length ratios. From these data we clearly see that the polarizability ratios are always larger, especially for shorter chains. We show in the next section that this has major consequences for the dielectric constant and capacitance. In contrast, the polarizability evolution with chain length is very close to linear for the saturated chains, as shown in Table 2. This is expected, since there is no accumulation effect increasing the depolarization factor, D, if the isolated molecule polarizability is linear. Consequences for the Dielectric Constant and Thin Film Capacitance. As we noted in the previous section, we separate the induced dipole into three layer contributions. The central layer lies in between two shorter layers, constituted by the external C−H bonds. This central layer is characterized by a length that is equal to LCC (see Figure 5) and an embedded polarizability that is equal to α′CC, as defined in eq 4 for its isolated counterpart. The external layers are characterized by the y-intercept in eq 4 which is a constant for each chain length. Following this procedure, we can define a dielectric constant for the central layer from eqs 16, 17, and 18. If we express the induced dipole in terms of the embedded molecular polar-

Table 2. Polarizability Evolution with Chain Length for a Saturated Chain, Embedded in a Monolayer with a 33 Å2 Coverage, at the HSE06 Levela index (j)

LCC (Å)

αCC (a.u)

ratios αCC : j+1/j

ratios LCC: j+1/j

n

1 2 3 4 5 6 7

1.150 3.766 8.936 16.655 24.374 29.516 37.227

9.11 30.51 73.59 137.43 201.63 244.24 307.75

3.35 2.41 1.87 1.47 1.21 1.26

3.27 2.37 1.86 1.46 1.21 1.26

1.02 1.02 1.00 1.01 1.00 1.00

a

This Table contains the same data as Table 1 for the saturated molecule case.

izability, we obtain eq 12. It can be seen that if the embedded molecule polarizability evolves linearly with the chain length, εCC =

1 1−

′ 4παCC ALCC

(12)

the dielectric constant remains constant. In contrast, if by extending the chain the polarizability increases more than linearly, as can be observed in Table 1, the LCC factor in the denominator cannot compensate for the polarizability increase, and the dielectric constant is expected to grow with the chain length. This chain length dependence can be estimated by the simple model we introduced in the previous section. After insertion of the embedded polarizability estimated by this model (see eq 9) and the power law in eq 4, we obtain an estimate for the dielectric constant (eq 13). εCC(LCC) ≈ 1 +

n−1 4παCC 4πα1LCC =1+ LCCA A

(13)

This simple model predicts that the dielectric constant of the thin film grows with the chain length, following a power law of Ln−1 CC . As this simplified model overestimates the depolarization (see Figure 5), we expect the dielectric constant defined in eq 13 to be lower than that from the periodic DFT calculation. Moreover, as the overestimation of the depolarization is more 5975

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Figure 6. Evolution of the dielectric constant with the chain length for conjugated (a) and saturated (b) chains at different levels of theory. For the conjugated chain (a), the dielectric constant increases with chain length while it remains constant for the saturated chain (b).

consequence of the fact that the increase of the embedded molecule polarizability is still greater than linear, even though depolarization limits this effect to a large extent. Since conforming to the power law in eq 4 with n > 1 is a generic feature of conjugated oligomers, we expect this behavior to be observable for any kind of conjugated molecule. We note that a similar behavior has been observed in a previous theoretical study on acene oligomers.84 Also, this precise behavior has been observed experimentally for thin films of conjugated azobenzene oligomers.23 In that case, the measured dielectric constant evolved from 2.7 to 12.7 for a film thickness going from 2.2 to 5.2 nm. The data extracted from another recent experimental study can also be interpreted as a dielectric constant that evolves from 5.4 ± 0.3 to 9.9 ± 0.5 for a film thickness going from 1.4 to 7.5 nm (see Supporting Information).35 We hope that this study will motivate more experimental efforts to characterize this important behavior of conjugated chains. The striking difference between the HSE06 + 0.45 and PBE level of theory is directly explained by our simplified model and eq 13. Indeed, this model predicts a power law evolution with n = 0.75 and n = 1.3 at the HSE06 + 0.45 and PBE levels of theory, respectively. We clearly notice in Figure 6 that n < 1 and n ≈ 1 for the HSE06 + 0.45 and PBE calculations, respectively. A power law fitting procedure, included in the Supporting Information, estimates that the exponents are about n = 0.53 ± 0.08 and n = 0.93 ± 0.04 at the HSE06 + 0.45 and PBE levels, respectively. As expected, the exponents are overestimated by the simplistic model by about 42% and 40%, respectively. However, the conjugated chain dielectric constant evolution is reasonably well characterized by a power law with coefficients of determination R2 = 0.995 and R2 = 0.999, respectively. Note that the above exponent differences are a direct consequence of the notorious limitation of semilocal exchangecorrelation approximations in predicting the correct parameters for the power law in eq 4. The large overestimation of the exponent directly leads to an evolution of the dielectric constant with chain length that is qualitatively incorrect. As a result, the error in the dielectric constant increases with the chain length and culminates in a factor of 2.6 for the longest chain examined, taking the HSE06 + 0.45 results as a reference. Actually, in eq 13 a direct proportionality between the dielectric constant and the isolated molecule polarizability is seen. Consequently, the simple model predicts that the relative error in the dielectric constant will be directly proportional to the relative error in the isolated molecule polarizability. Eq 13 also

marked at shorter chain lengths, the exponent of the power law, n − 1, may be viewed as an upper bound estimation. Note that this formula predicts a direct proportionality of the dielectric constant with the molecular density in the monolayer. This agrees with previous findings from periodic DFT calculations.82,83 Consequently, the increasing rate of the dielectric constant change with chain length is predicted to be proportional to the packing density, for a fixed exponent n. The total dielectric constant, εtot, is also defined according to eqs 16, 17, and 18. The difference is that it includes the terminal C−H bond polarization and length. Equivalently, Natan et al.45 showed that εtot is expected to follow the capacitance in a series law, in the case of a dense thin film (eq 14).

L tot 2L HC L = + CC εtot εHC εCC

(14)

This means that the total dielectric constant may also increase with chain length for the simple reason that the conjugated portion of the molecule increases with a larger expected dielectric constant than that of the C−H bonds. This result shows the importance of separating the dielectric system into three layers. Indeed, it separates conjugation effects from this proportion effect to unambiguously demonstrate that the increase of the dielectric constant is mainly due to increased conjugation with the chain length. We note that a series model cannot be applied to the conjugated layer because of the charge delocalization in this part. This implies that a “local dielectric constant” in such a layer would be ill-defined.47 The evolution of the carbon chain and total dielectric constants, computed at the periodic DFT level, are shown in Figure 6. The increase of both dielectric constants is very large for conjugated species. At the HSE06 + 0.45 level of theory, the carbon chain dielectric constant evolves from 2.08 to 7.79. At the PBE level of theory, the same dielectric constant grows even more, starting from 2.21 to culminate at 20.36 for the longest chain. The effect on the saturated chain dielectric constant is very small as expected, evolving from 1.79 to 1.83 at the HSE06 level of theory. This means that the dielectric constant can reasonably be assumed to be constant with the chain length, in the absence of conjugation effects. As the saturated chains are well represented by a fragment superposition, there is no quantitative difference between the PBE and HSE06 levels of theory. This large increase of the dielectric constant is expected from the discussion in the previous sections. It is a direct 5976

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Figure 7. Evolution of the thin film capacitances (a) and carbon chain dielectric constants (b) with chain length, at different coverages, at the HSE06 + 0.45 level of theory. The dashed line in (a) is the decay expected if the dielectric constant does not have length dependence, according to eq 1, by extrapolating the value of the dielectric constant obtained for the shortest chain. The fact that the real curve lies above shows the maintenance of the capacitance due to conjugation effects.

dielectric constant. For the data reported in Figure 7a at a 33 Å2 coverage, a linear fit yields a dielectric constant of 1.3 versus the values reported in Figure 6a, where the actual total dielectric constant lies in between 1.8−6.5, see Supporting Information. Note that this length dependence is not expected if the molecule is a superposition of nonconjugated fragments, see Figure 6b. This indicates that one can directly fit eq 1 in the case of saturated chains (see Supporting Information). Very interestingly, this behavior for conjugated chains permits exponential minimization of the tunneling current through the layer (leakage current), while maintaining a large capacitance as required for a transistor gate dielectric layer. Note also that this maintenance of capacitance increases with molecular coverage. Furthermore, this effect may be combined with the introduction of a quantum interference unit to design high-performance gate dielectrics.44 Finally, we consider that the increase of the dielectric constant is too smooth to be directly noticeable as a transition in the tunneling decay constants. The evolution of the carbon layer dielectric constant at different coverages is depicted in Figure 7b. The rate of dielectric constant increase with chain length increases with the coverage. This is well-explained by the simple model and eq 13, which predicts that the dielectric constant increases linearly with coverage. We further discuss the dependence of the dielectric constant on coverage in the Supporting Information. Note that this increase is slightly superlinear for the shortest conjugated chains, while it is essentially linear for longer conjugated chains and for saturated chains. Interestingly, the choice of the XC functional has no influence on this behavior. We suspect that the collective hybridization of the π orbitals may be at the origin of this behavior, with only a minor impact on the polarizability, making it noticeable only for shortest chain lengths.

indicates that the absolute error will increase with the molecular density. Consequently, the computed dielectric constants of densely packed thin films composed of long conjugated chains, calculated at the semilocal level of density functional approximation, must be regarded with reservation. Dielectric materials have many important technology applications. We now discuss some consequences of this chain length dependence in this context. First, the present results show that one can tune the dielectric constant of a thin film by simple modification of the number of units in an oligomer chain. Moreover, the dielectric constant is very controllable since the increase is monotonic. Also, it may not substantially modify the packing density, and be easier to implement in comparison to the introduction of electronically active substituents on the molecular backbone.82,85 Finally and very importantly, chain length dependence has a major consequence for eq 1 that characterizes the capacitance of a thin film sandwiched between electrodes, the figure-of-merit to minimize the power consumption of thin film transistors. Figure 7 shows the evolution of the capacitance of conjugated thin films at different molecular surface coverages. The capacitance is computed from the total dielectric constant and total molecular lengths at the HSE06 + 0.45 level (d = LCC + 2LHC) by inserting these parameters in eq 1. We also include the evolution of the dielectric constants with molecular length in the same figure. Note that the length dependence of the dielectric constant confers a specific property on conjugated organic thin films. Their capacitance does not fall, as commonly expected, linearly with the film thickness, which in this case varies from 3.308 to 41.602 Å. Instead of decreasing by a factor of 41.602/3.308 = 12.6 from the shortest to the longest oligomer, it only decreases by factors of 4.4, 3.4, and 3.1 for coverages of 66, 33, and 16.5 Å2, respectively. This is clearly evidenced by the extrapolated linear decay shown in Figure 7a (dashed lines). An area per molecule of 16.5 Å2 is still a reasonable value for thiol-based carbon chain monolayers.36 The increase of conjugation limits the reduction of the capacitance with film thickness, since it always remains above the dashed lines. For this reason, one cannot directly fit eq 1 on a capacitance versus thickness plot to deduce the dielectric constant, even if the evolution appears to be linear, see Supporting Information. This procedure would severely underestimate the actual

CONCLUSIONS In this study, we investigated the chain length dependence of the polarizability, dielectric constant, and capacitance of thin films composed of conjugated oligomers. The specific effect of the conjugation on these properties was highlighted by comparing conjugated polyyne chains to saturated alkane chains. The results provide three important conclusions for the fields of organic and molecular electronics: 5977

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induced dipoles are normal to the film plane. This focus on the nonperiodic direction avoids complications related to the definition of the full dielectric tensor, overcome by the modern theory of polarization.88,89 Importantly, we note that it is the relevant direction for capacitor applications. In the thin film case, it is straightforward to demonstrate that the polarization field is given by EP = −4πP. After insertion of this polarization field in eq 15, we obtain the following expression for the dielectric constant of a thin film:

(1) The superlinear power law dependence of the polarizability on conjugated chain length does not hold for molecules embedded in a monolayer. The superlinear dependence is largely attenuated, and a linear dependence is partially recovered, especially for longer chains. A simple depolarization model rationalizes this behavior. Indeed, a superlinear dependence leads to an accumulation of charges at both extremities of the monolayer with increasing chain length. This leads to a depolarization field that also increases with chain length and significantly limits this behavior. However, its strength is not sufficient to completely eliminate the superlinear dependence. (2) If the embedded molecule polarizability maintains a superlinear dependence, the dielectric constant of the corresponding thin film acquires a thickness dependence, significantly increasing the dielectric constant by a factor of 3−4. Thus, simple modification of the conjugated chain length will tune the film dielectric constant. Also, because the capacitance decline with film thickness is strongly attenuated, determining the thin film dielectric constant by the conventional fit of a capacitance versus thickness plot is not appropriate. (3) The semilocal approximations for the exchange-correlation functionals, such as PBE, are not suitable for the evaluation of thin films composed of long conjugated oligomers. This is due to the well-known overdelocalization limitation which significantly overestimates polarizabilities. The consequence is that the thickness dependence of the dielectric constant with chain length is largely overestimated. Quantitatively, this leads to an overestimation of the dielectric constant by a factor of 2−3 for polyyne chains. Including a significant amount of Hartree−Fock exchange largely corrects for this behavior in isolated molecules. As these corrections can then compare well with more accurate wave function methods, the corresponding computed dielectric constants should be reasonably accurate.

ε=

μind = α(Eapplied + Enear)

(16)

(17)

film and Enear the field generated by the other polarized molecules. This latter term actually corresponds to an infinite sum over dipolar fields. Once this induced dipole moment is known as a function of the applied field, taking the near field into account, the uniform and macroscopic polarization density of the thin film can be computed from eq 18 where A is the area per molecule and d is the thickness of the organic layer. μ P = ind (18) A. d Several approaches have been used to compute the induced dipole defined in eq 18, for a given applied electric field. First, each molecule can be approximated as single polarizable point dipole. Then, the infinite sum can be estimated analytically or numerically,80,86,90 to relate the polarizability with the dielectric constant in the spirit of the Clausius−Mossotti relation. Alternatively, this approximation can be refined by distributing the total polarizability along each molecular backbone in a group of polarizable point dipoles.80,84 This approach better accounts for the inhomogeneity of the near field and the polarizability along the embedded molecule backbone. Finally, we can rely on the periodic implementation of density functional theory (DFT) under an applied electric field. This last approach has one great advantage: it explicitly treats the molecules as polarizable electron densities. The corresponding near field is embedded in the Hartree potential computed from Poisson’s equation under periodic conditions. The expected accuracy is very high since it is directly determined by the grid resolution. This can also account for any intermolecular electronic overlap. This periodic treatment has been shown to be very accurate for organic thin-films,82 organic crystals,47 or solid bulk materials,91,92 when compared to experimental measurement data. Still and very important, the electron density and its response are limited in accuracy by the approximate treatment of the exchange-correlation functional. In this work, we adopted the periodic DFT approach and gained physical insight in our results from a simple depolarization model, assuming a physically motivated dependence of the near field with the isolated molecule polarizability. All our DFT calculations were carried out with the VASP package using the same procedure. The molecules are optimized geometrically and placed in squared periodic boxes with fixed lateral dimensions, which generate an infinite monolayer. We generally do not allow the geometry of the molecule to relax under the electric field, since we focus on the optical dielectric response in this work. The position of

The dielectric constant of a thin film has been the subject of several definitions in the literature.45,46,83,86 In this section, we follow the textbook definitions79,81,87 that are used for the common Clausius− Mossotti relationship derivation. We stress here that we do not use this common relationship in this work. We show in Supporting Information that the definition of the dielectric constant can also be deduced directly from eq 1, in a heuristic way. The dielectric constant is a macroscopic quantity, intrinsically related to the susceptibility of a polarizable medium, χ, as ε = 1 + χ. The susceptibility is defined as the constant of proportionality between the macroscopic polarization per volume unit, P, and the macroscopic electric field in the medium, χE . This leads to a general relation between the Emacro: P = 4macro π dielectric constant, the macroscopic field, and the polarization density, given by eq 15: Emacro + 4πP Emacro

Eapplied − 4πP

Equation 16 gives the definition of the dielectric constant for a thin film that is used in this work. It is in agreement with other existing studies.45,84 Note that this relation has a divergent nature as emphasized in a previous study.84 This means that a small increase of polarization density can lead to a large amplification of the dielectric constant. This relation enables establishing the dielectric constant of a medium if the macroscopic polarization density created in response to the applied field is known. This is a challenging problem since the macroscopic polarization has a microscopic origin. At this microscopic level, a molecule in a well-ordered periodic film encounters two electric field contributions: (i) the applied electric field and (ii) the field generated by the other surrounding polarized molecules. This creates an induced dipole moment in the molecule according to eq 17 where α is the molecular polarizability in the direction normal to the

METHODS/EXPERIMENTAL

ε=

Eapplied

(15)

The macroscopic field is the sum of two contributions: (i) the applied electric field and (ii) the field that originates from the uniform and macroscopic (space-averaged) polarization density, in response to the applied electric field. In general, as the polarization field approaches the applied field, the macroscopic field approaches zero and the dielectric constant increases, according to eq 15. We assume in the following discussion that the applied field and the corresponding 5978

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the lowest carbon atom is constrained during the optimization procedure. A vacuum layer of 15 Å plus half the height of the subject molecule is introduced above the molecule to prevent any interaction between superposed layers. An electric field of 0.001 au is applied in the direction normal to the monolayer plane to obtain the induced dipole moment of eq 18 after convergence of the DFT procedure. The polarization density is then computed from eq 18, using the normal distance between the center of the external carbon or hydrogen atoms as the definition of thickness. Finally, the dielectric constant of the monolayer is determined from eq 16. More computational details are provided in the Supporting Information. In this study, three different levels of approximation were used for evaluating the exchange-correlation (XC) functional: PBE,93 HSE06,94 and HSE06 + 0.45. HSE06 is a hybrid functional which includes an explicit Hartree−Fock exchange contribution. We use two different percentages of Hartree−Fock exchange in this work: (i) the default, 25%, and (ii) a tuned HSE functional with 45% of Hartree−Fock exchange. This latter is referred as HSE06 + 0.45 in this article. We discuss the reason why we decided to carry out Hybrid DFT calculations in the previous sections. At a purely practical level, note that evaluating the Hartree−Fock exchange for these systems increases the computational time considerably.

ASSOCIATED CONTENT S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.7b01807. Details about the definition of the dielectric constant and the computational procedures, discussion of the simple depolarization model, and more detailed analysis about the dielectric constant of the thin-films (PDF)

AUTHOR INFORMATION Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. *E-mail: [email protected]. ORCID

Colin Van Dyck: 0000-0003-2853-3821 Tobin J. Marks: 0000-0001-8771-0141 Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS The authors thank R. Gieseking for discussions of bond length alternation and conjugation concepts and P. Darancet for insightful discussions. This work was supported by the National Science Foundation MRSEC program (DMR-1121262) at the Materials Research Center of Northwestern University. Initial numerical simulations were carried out at the Center for Nanoscale Materials (Argonne National Lab), an Office of Science user facility, supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. A second part of the numerical simulations were carried out on the XSEDE computational facilities, supported by National Science Foundation Grant ACI-1053575. REFERENCES (1) Jariwala, D.; Sangwan, V. K.; Lauhon, L. J.; Marks, T. J.; Hersam, M. C. Carbon Nanomaterials for Electronics, Optoelectronics, Photovoltaics, and Sensing. Chem. Soc. Rev. 2013, 42, 2824−2860. (2) Shirakawa, H.; Louis, E. J.; MacDiarmid, A. G.; Chiang, C. K.; Heeger, A. J. Synthesis of Electrically Conducting Organic Polymers: 5979

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DOI: 10.1021/acsnano.7b01807 ACS Nano 2017, 11, 5970−5981