Vibronic Interactions in Fractionally Charged Hydrocarbons

Article pubs.acs.org/JPCC

Vibronic Interactions in Fractionally Charged Hydrocarbons Takashi Kato* Institute for Innovative Science and Technology, Graduate School of Engineering, Nagasaki Institute of Applied Science, 3-1, Shuku-machi, Nagasaki 851-0121, Japan S Supporting Information *

ABSTRACT: We investigate the vibronic stabilization energy (Δvib,−x) in aromatic 1,2:8,9-dibenzopentacene (Kx30dba) by comparison with electronic properties of aromatic hydrocarbons such as picene (K322ph). Approximately twice larger value of the density of states at the Fermi level (N(εF)−x) for K3.1730dba than that for K330dba is the main reason why the experimental Δvib,−x value of 28.2 K for K3.1730dba is much larger than that of 7.4 K for K330dba. Approximately 5.1−5.6 and 4.3−4.8 times larger N(εF)−x values reproducing the Δvib,−3.45 values of 33.1 and 21.3 K, respectively, for K3.4530dba than that reproducing the Δvib,−3.00 value of 7.4 K for K330dba are the main reasons why the Δvib,−3.45 values of 33.1 and 21.3 K for K3.4530dba are much larger than the Δvib,−3.00 value of 7.4 K for K330dba. The molecular crystals of possible minimum sized molecules, with small charges, in which the stable fractionally charged (±0.50∼ ± 1.00 (in particular, ±0.57∼ ±0.86)) states with respect to the closed-shell electronic states can be realized, are the best candidates for large vibronic stabilization energies.

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INTRODUCTION The effect of vibronic interactions and electron−phonon interactions1−6 in molecules and crystals is an important topic of discussion in modern chemistry and physics. The vibronic and electron−phonon interactions play an essential role in various research fields such as the characterization of molecular structures, electrical conductivity, superconductivity, Peierls distortions, and spectroscopy.1−12 We have investigated the vibronic interactions and electron−phonon interactions in various charged molecular crystals for more than 10 years.13−18 Modification of the electronic structure of materials with πelectron networks by metal doping is a very important way to bring out novel physical properties.19−24 In 2002, we theoretically analyzed the electron−phonon interactions1−4 and estimated possible vibronic stabilization energies (Δvib,±x) of the negatively charged various molecular systems such as polyacenes and polyphenacenes, including picene (22ph) and phenanthrene (14ph). Specifically, we predicted the Δvib,±x values of around 10 K in the negatively charged 22ph.16 Eight years later, in 2010, it was experimentally established that intercalating alkali metal atoms into 22ph produces metallic behavior and superconductivity.25 Solid Kx22ph has a Δvib,±x of 7 or 18 K, depending on the metal content. This is the first aromatic hydrocarbon superconductor. Superconductivity in the trianions of polyphenacenes were recently investigated theoretically.18,26,27 We found that the electron−phonon coupling constant (l−x) increases with a decrease in molecular size from 22ph to 14ph and thus the Δvib,±x value for K314ph was expected to be larger than that for K322ph.18 In 2011, the discovery of superconductivity in K314ph was reported.28 However, the experimental results show that the Δvib,±x of 5 K for K314ph is © 2013 American Chemical Society

much smaller than that of 18 K for K322ph. Furthermore, it was recently found that K-doped 1,2:8,9-dibenzopentacene (Kx30dba) such as K330dba and K3.1730dba exhibit superconductivity at 7.4 and 28.2 K, respectively, and K3.4530dba exhibits superconductivity at 33.1, 21.3, and 4.5 K.29 That is, the Δvib,±x values for various sized and charged molecules are now measured, and thus we can systematically assign many parameters which were unable to be assigned. Recently,30 we discussed the dimensionless electron−phonon coupling constants (λ±x), which set the Δvib,±x values with charge ±x. The λ±x values are set by the electron−phonon coupling constants (l±x) and the density of states at the Fermi level (N(εF)±) (λ±x (= N(εF)±x l±x)). In previous research,30 we investigated how l±x, N(εF)±, and λ±x depend on the amount of doped carriers, and we suggested a guiding principle for the synthesis of molecular crystals with large λ±x values. In this article, as a good example, on the basis of the hypothesis suggested in the previous research,30 we will show that the Δvib,±x values for K3.1730dba and K3.4530dba can be qualitatively explained in view of the relationships between the N(εF)± and ± x values in the fractionally charged materials. We first estimate the l−x and vln,−x values in K330dba, K3.1730dba, and K3.4530dba. We next estimate the Coulomb pseudopotential (μ−x*) for K330dba, K3.1730dba, and K3.4530dba from the μ−3.00* value for K322ph in view of the relative relationships between K330dba and K322ph. Furthermore, we estimate the N(εF)−x values for K330dba, K3.1730dba, and K3.4530dba, on the basis of the Received: September 24, 2012 Revised: July 2, 2013 Published: July 2, 2013 17211

dx.doi.org/10.1021/jp402725t | J. Phys. Chem. C 2013, 117, 17211−17224

The Journal of Physical Chemistry C

Article

inversion of the McMillan’s formula (eq 1), from the knowledge of the experimental Δvib,−x values and an educated guess of the l−x, vln,−x, and μ−x* values for K330dba, K3.1730dba, and K3.4530dba. We suggest the reason why the Δvib,−x value of 28.2 K in the fractionally charged K3.1730dba, and those of 33.1 and 21.3 K for two of three transition phases in the fractionally charged K3.4530dba, are much larger than that of 7.4 K in the integrally charged K330dba, in view of the hypothesis suggested in the previous research.30 Furthermore, we also suggest a guiding principle toward large Δvib,−x values in view of the physical parameters such as l−x, μ−x*, and N(εF)−x values. In this research, we discuss the electron−phonon interactions by mainly considering that the reported nominal numbers x are equal to the actual x values in Kx30dba and Kx22ph.

Table 1. List of Physical Values for Hydrocarbons Considered in This Study molecule (unit)

0.00−1.00 K314ph K322ph (phase I) 1.00 K322ph (phase II) 1.00 K330dba 1.00 K3.1730dba 0.83 K3.4530dba (phase I) 0.55 K3.4530dba (phase II) 0.55 K3.4530dba (phase III) 0.55 l−x (eV) l−x (eV/carrier·atom)

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0.000−0.336 0.206 0.206 0.137 0.112 0.049 0.049 0.049

ELECTRON−PHONON INTERACTIONS AND VIBRONIC STABILIZATION ENERGIES We investigate the experimentally determined Δvib,±x by using an approximate solution (McMillan’s formula)31 of the Eliashberg equation from the calculated electron−phonon coupling constants, Δ vib, ±x =

⎡ ⎤ 1.04(1 + λ±x) ⎥ exp⎢− ⎢⎣ λ±x − μ±x *(1 + 0.62λ±x) ⎥⎦ 1.2

νln , ±x

⎡ ⎤ 1.04(1 + N (εF)±x l±x) ⎥ = exp⎢− ⎢⎣ N (εF)±x l±x − μ±x *(1 + 0.62N (εF)±x l±x) ⎥⎦ 1.2

0.000−4.704 4.532 4.532 4.110 4.877 4.860 4.860 4.860 vln,−x (cm−1)

λ−x − 0.494 0.403 0.356 (0.301−0.411) 0.588 (0.497−0.677) 0.683 (0.556−0.816) 0.589 (0.474−0.710) 0.416 (0.321−0.514)

νln , ±x

(1)

where μ±x* is the Coulomb pseudopotential describing the electron−electron repulsion, and vln,±x is the logarithmically averaged phonon frequencies (see Supporting Information SI1−5).32 The μ−x* values as a function of the distances between carriers and the nearest environmental charges (rc) for polyphenacene and 30dba molecular crystals13−15 can be defined as q q x(x − n) μ−x *∝ U −x = env vib = rc rc (2)

− 1001 1001 1040 775 775 775 775

number of carriers (/atom) 0.000−0.071 0.045 0.045 0.033 0.028 0.018 0.018 0.018 N(εF)−x (states/eV unit·spin)  1.2 (expt) (ref 25) 1.0 1.3 (1.1−1.5) 2.6 (2.2−3.0) 7.0 (5.7−8.3) 6.0 (4.8−7.2) 4.2 (3.3−5.2) Δvib,−x (K) (expt) μ−x*

− 0.095 0.095 0.066 (0.030−0.100) 0.082 (0.037−0.124) 0.110 (0.050−0.167) 0.110 (0.050−0.167) 0.110 (0.050−0.167)

5 (ref 28) 18 (ref 25) 7 (ref 25) 7.4 (ref 29) 28.2 (ref 29) 33.1 (ref 29) 21.3 (ref 29) 4.5 (ref 29)

essential role in the electron−phonon interactions in 30dbax− (3.00 < x ≤ 4.00) (Figure 1b,c). Total Electron−Phonon Coupling Constants. The l−x values for various charges of 22ph and 30dba are shown in Figure 2. The l−3.00 value for 22ph was estimated to be 0.206 eV.18 The l−3.00, l−3.17, and l−3.45 values for 30dba are estimated to be 0.137, 0.112, and 0.049 eV, respectively. The l−3.00 value for larger sized 30dba is estimated to be much smaller than that for smaller sized 22ph, as expected. The l−x value decreases with an increase in the x value from 30dba3− to 30dba3.45−, as shown in Figure 2a,b. This is because the carrier density decreases with an increase in the x value from 30dba3− (1.00) to 30dba3.45− (0.55). Logarithmically Averaged Phonon Frequencies. The vln,±x values for 22ph and 30dba are listed in Table 1. The vln,−x values for 2.00 ≤ x ≤ 3.00 and 3.00 < x ≤ 4.00 in Kx30dba are estimated to be 1040 and 775 cm−1, respectively. Therefore, the vln,−x value for 2.00 ≤ x ≤ 3.00 is larger than that for 3.00 < x ≤ 4.00 in Kx30dba. Furthermore, the vln,−x value for 2.00 ≤ x ≤ 3.00 in Kx22ph was estimated to be 1001 cm−1.18 Coulomb Pseudopotential. The μ−x* values as a function of various charge in 22phx− and 30dbax−, estimated on the basis of the assumption that the μ−x* values are approximately proportional to the U−x values, are shown in Figure 3. The μ−x* value increases with an increase in the negative charge −x, as expected. Furthermore, the μ−x* value decreases with an increase in the molecular size, as expected.

n = 0 for 0.00 ≤ x < 2.00 = 2 for 2.00 ≤ x < 4.00

number of carriers (/unit)

(3)

where the U−x value denotes a typical Coulomb interaction, qvib(= n−x) denotes the charge of the carrier directly related to the vibronic interactions, and qenv(= −x) is the environmental charges of the nearest molecules (Figure SI3, Supporting Information). In eq 2, we assume that the μ−x* values are approximately proportional to the U−x values. Justification of qualitative discussions on the estimation of the μ−x* values will be shown later. For these calculations, we used the hybrid Hartree−Fock (HF)/density-functional-theory (DFT) method of Becke,33 and Lee, Yang, and Parr34 (B3LYP), and the 6-31G* basis set.35 We used the Gaussian 03 program package36 for theoretical analyses. Physical values for hydrocarbon materials considered in this study are listed in Table 1. Electron−Phonon Coupling Constants. The electron− phonon coupling constants versus frequency in 30dba are shown in Figure 1a. The low-frequency modes of 229 and 503 cm−1 as well as the high-frequency C−C stretching modes of 1392, 1571, and 1635 cm−1 play an essential role in the electron−phonon interactions in 30dbax− (2.00 ≤ x ≤ 3.00) (Figure 1b,c), and the high-frequency C−C stretching modes of 1343, 1553, and 1600 cm−1 as well as the low-frequency mode of 228 cm−1 play an 17212



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Figure 1. Electron−phonon interactions in the negatively charged 1,2:8,9-dibenzopentacene (30dbax−). (a) Electron−phonon coupling constants (l−x,m) (meV) versus frequency (ν−x,m) plots for the LUMO of dianion and the HOMO of tetraanion in 30dba. (b) Phase patterns of the frontier orbitals in the neutral 30dba. Both the LUMO of 30dba2− and the HOMO of 30dba4− correspond to the LUMO+1 of the neutral 30dba. (c) Vibronically active modes in 30dba2− and 30dba4−.

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18 K for K322ph can be reproduced by using μ−3.00* = 0.095 for the experimental N(εF)−3.00 value of 1.2 states per (eV molecule spin),25 and the calculated l−3.00 value of 0.206 eV and vln,−3.00 value of 1001 cm−1.18 And the N(εF)−3.00 value reproducing the experimental Δvib,−x value of 7 K can be estimated to be 1.0 states per (eV molecule spin) by using μ−3.00* = 0.095, in eq 1. K3-Dibenzopentacene. Let us look into vibronic stabilization energies in K330dba. The Δvib,−x for K330dba is reported to be 7.4 K.29 We estimate the μ−3.00* values as a function of the

DISCUSSION ON VIBRONIC STABILIZATION ENERGIES K3-Picene. Let us first look into vibronic stabilization energies in K322ph. The Δvib,−x values for K322ph are reported to be 18 and 7 K.25 We estimate the μ−3.00* values as a function of the N(εF)−3.00 values reproducing the Δvib,−x values of 18 and 7 K by using the theoretical values of vln,−3.00 and l−3.00, as shown in Figure 4a. On the other hand, Mitsuhashi et al. reported an experimental estimate for N(εF)−3.00 of K322ph (Δvib,−x = 18 K) to be 1.2 states per (eV molecule spin).25 Therefore, the Δvib,−x of 17213



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Figure 2. Total electron−phonon coupling constants. (a) The l−x value as a function of the −x value in Kx30dba. (b) The l−x/l−3.00 value as a function of the −x value in Kx30dba. (c) The l−x value as a function of the −x value in Kx22ph.

Figure 3. Coulomb pseudopotential versus charge. (a) The μ−x* value as a function of the −x value in Kx30dba. The opened-circles, opened-triangles, opened-squares, crosses, closed-circles, closed-triangles, and closed-squares denote the μ−x* values for the N(εF)−x values of 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, and 1.6 states per (eV molecule spin), respectively. (b) μ−x*/μ−3.00* value as a function of the −x value in Kx30dba. (c) The μ−x* value as a function of the −x value for the N(εF)−x value of 1.2 states per (eV molecule spin) in Kx22ph.

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Figure 4. Coulomb pseudopotential versus the density of states at the Fermi level reproducing the vibronic stabilization energies. (a) The μ−3.00* value versus the N(εF)−3.00 value reproducing the Δvib,−3.00 values of 18 and 7 K in K322ph. The circles and triangles denote the μ−3.00* values for the Δvib,−3.00 values of 18 and 7 K, respectively. A cross denotes the μ−3.00* for the Δvib,−3.00 value of 18 K estimated from the experimental results. (b) The μ−3.00* value versus the N(εF)−3.00 value reproducing the Δvib,−3.00 value of 7.4 K in K330dba. (c) The μ−3.17* value versus the N(εF)−3.17 value reproducing the Δvib,−3.17 value of 28.2 K in K3.1730dba. (d) The μ−3.45* value versus the N(εF)−3.45 value reproducing the Δvib,−3.45 values of 33.1, 21.3, and 4.5 K in K3.4522ph. The circles, triangles, and squares denote the μ−3.45* values for the Δvib,−3.45 values of 33.1, 21.3, and 4.5 K, respectively.

N(εF)−3.00 values reproducing the Δvib,−x value of 7.4 K by using the theoretical values of vln,−3.00 and l−3.00, as shown in Figure 4b. Considering the ratio of the rc value for 30dba (rc ≈ 16.3 Å) to that for 22ph (rc ≈ 11.4 Å) (mainly, the molecular size difference between 30dba and 22ph), the μ−3.00* value for K330dba can be estimated to be about 0.066. In this estimation, we assume that the largest distance between two carbon atoms in a molecule is related to the rc value in each molecule in eqs 2 and 3. The Δvib,−x of 7.4 K for K330dba can be reproduced by using N(εF)−3.00 = 1.3 states per (eV molecule spin) for μ−3.00* = 0.066. Therefore, in view of the reasonable relationships between the μ−3.00* values of K322ph and K330dba, the most probable N(εF)−3.00 values are estimated to be about 1.3 states per (eV molecule spin) for K330dba, which are similar to that of 1.2 states per (eV molecule spin) for K322ph. Let us next compare the Δvib,−x values for K330dba with those for K322ph. The Δvib,−x value of 7.4 K for larger sized K330dba is smaller than that of 18 K for smaller sized K322ph. This can be understood as follows. The v−3.00 value of 1040 cm−1 for 30dba is only slightly larger than that of 1001 cm−1 for 22ph. The estimated μ−3.00* value of 0.066 for larger sized 30dba3− with N(εF)−3.00 = 1.3 states per (eV molecule spin) is smaller than that of 0.095 for smaller sized 22ph3− with N(εF)−3.00 = 1.2 states per (eV molecule spin) (Figure 3). However, the estimated l−3.00 value of 0.137 eV for 30dba3− is much smaller than that of 0.206 eV for 22ph3− (Figure 2a,c). In general, both the l−3.00 and μ−3.00* values decrease with an increase in the molecular size, as shown

in Figures 2 and 3. Therefore, the competition between these decreasing values determines whether smaller or larger sized molecules have larger Δvib,−x values. In the medium or relatively large sized materials such as 22ph and 30dba, the decrease of the l−3.00 value rather than the decrease of the μ−3.00* value with an increase in molecular size can be expected to be dominant factor for determination of the magnitudes of the Δvib,−x values. Therefore, we can conclude that much smaller l−3.00 value for larger sized 30dba than that for smaller sized 22ph is the main reason why the Δvib,−x value for K330dba is smaller than that for K322ph, even though the μ−3.00* value for larger sized 30dba is smaller than that for smaller sized 22ph. It should be noted that the μ−3.00* value of 0.066 for K330dba is estimated by considering only molecular size differences between K322ph and K330dba, on the basis of the assumption that the molecular crystal structures for K330dba are not significantly different from those for K322ph and K314ph. On the other hand, because the μ−3.00* values would be slightly changed by orientation between the two neighboring molecules, the μ−3.00* value of 0.066 for K330dba would be slightly different from the actual μ−3.00* value. In general, the μ−x* value is very difficult to be estimated and has been usually used as a fitting parameter for qualitative discussions.1−12 However, considering the large molecular size differences between 22ph and 30dba, the μ−3.00* values for K330dba are at least expected to be smaller than that of 0.095 for K322ph, and thus we can expect that the μ−3.00* value of 0.066 is at least available for qualitative discussions made 17215



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Figure 5. Density of states at the Fermi level. (a) The N(εF)−3.17 value reproducing the Δvib,−3.17 value of 28.2 K in K3.1730dba versus the μ−3.00* value in K330dba. (b) The N(εF)−3.45 value reproducing the Δvib,−3.45 value of 33.1 K in K3.4530dba versus the μ−3.00* value in K330dba. (c) The N(εF)−3.45 value reproducing the Δvib,−3.45 value of 21.3 K in K3.4530dba versus the μ−3.00* value in K330dba. (d) The N(εF)−3.45 value reproducing the Δvib,−3.45 value of 4.5 K in K3.4530dba versus the μ−3.00* value in K330dba. Circles indicate the N(εF)−x values for Kx30dba estimated on the basis of the μ−3.00* values for K330dba, and crosses indicate the N(εF)−x values directly estimated from the experimental Δvib,−x value for Kx30dba.

in this research. For security, the possible various μ−3.00* values as a function of the N(εF)−3.00 value reproducing the experimental Δvib,−x value of 7.4 K for K330dba are shown in Figure 4b. Even though the N(εF)−3.00 and μ−3.00* values for K330dba may not be very clearly determined, we can at least conclude that a much smaller l−3.00 value for larger sized 30dba than that for smaller sized 22ph is the main reason why the Δvib,−x value for K330dba is smaller than that for K322ph. However, it should be also noted that the Δvib,−x value of 7 K for one of two transition phases in smaller sized K322ph is similar to that of 7.4 K in larger sized K330dba. This is because the estimated N(εF)−x value of 1.0 states per (eV molecule spin) for K322ph with Δvib,−x = 7 K is much smaller than that of 1.3 states per (eV molecule spin) for K330dba even though the l−3.00 value of 0.206 eV for smaller sized K322ph is much larger than that of 0.137 eV for larger sized K330dba. Therefore, we can expect that in general, in the medium- and large-sized materials, the Δvib,−x value for the same electronic states and the similar N(εF)−x values decreases with an increase in molecular size if the stabilization of the electronic structure occurs as a consequence of the intramolecular vibronic interactions, as suggested in our previous research.14,18 K3.17-Dibenzopentacene. Let us next compare the vibronic stabilization energy in the fractionally charged K3.1730dba with that in the integrally charged K330dba. As described in the previous section, precise estimation of the μ−3.00* value for K330dba on the basis of the μ−3.00* value of 0.095 for the N(εF)−3.00 value of 1.2 states per (eV molecule spin) in K322ph may be difficult only in view of the molecular size difference (rc values difference) between these molecules. Once the μ−3.00*

value for K330dba is fixed, we estimate the μ−3.00* values for K3.1730dba and K3.4530dba on the basis of the μ−3.00* values for K330dba, by assuming that the μ−x* values are proportional to the U−x values, as described previously. At least, relative relationships between the μ−3.00*, μ−3.17*, and μ−3.45* values analyzed in this study can be expected to be qualitatively precise, as will be discussed later. The μ−x* values as a function of the x value for various N(εF)−3.00 values estimated on the basis of the charge difference between Kx30dba and K330dba are shown in Figure 3a. This figure shows that the μ−x* value increases with an increase in doped charge x, as expected. For example, considering the most probable μ−3.00* value of 0.066 and the N(εF)−3.00 value of 1.3 states per (eV molecule spin), the μ−3.17* and μ−3.45* values are estimated to be 0.082 and 0.110, respectively (Figure 3a, Table 1). Even though, because of difficulty of the precise estimation of the μ−3.00* value for K330dba from that for K322ph, the μ−x* values for K3.1730dba and K330dba may slightly deviate from 0.082 and 0.066, respectively, we can at least expect that the relative relationships between the μ−3.17* and μ−3.00* values (i.e., the ratio of the μ−3.17* value to the μ−3.00* value (μ−3.17*/μ−3.00*) do not significantly depend on the estimated μ−3.00* value of K330dba. In view of the N(εF)−3.00 values for K322ph and the molecular size difference between 22ph and 30dba, we can expect that for the μ−3.00* values of 0.030−0.100 in K330dba, which are generally smaller than that of 0.095 in K322ph, the N(εF)−3.00 values are estimated to be around 1.1−1.5 states per (eV molecule spin) in K330dba, as shown in Figure 4b. The l−3.17 value of 0.112 eV for K3.1730dba is slightly smaller than the l−3.00 value of 0.137 eV for K330dba (Figure 2a), and the 17216



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Figure 6. Relative relationships between the density of states at the Fermi level in various charges. (a) The N(εF)−x/N(εF)−3.00 value for Kx30dba versus the μ−3.00* value for K330dba. Circles denote the N(εF)−x/N(εF)−3.00 values for the Δvib,−3.17 value of 28.2 K in K3.1730dba, and triangles, squares, and diamonds denote the N(εF)−x/N(εF)−3.00 values for the Δvib,−3.45 values of 33.1, 21.3, and 4.5 K, respectively, in K3.4530dba. (b) The most probable N(εF)−x value versus the −x value in Kx30dba. (c) The most probable N(εF)−x/N(εF)−3.00 value versus the −x value in Kx30dba.

vln,−3.17 value of 775 cm−1 for K3.1730dba is smaller than the vln,−3.00 value of 1040 cm−1 for K330dba (Table 1). Furthermore, the μ−3.00* value of 0.082 for K3.1730dba is larger than the μ−3.00* value of 0.066 for K330dba (Figure 3a). Therefore, the Δvib,−x value for K3.1730dba would be expected to be smaller than that for K330dba in view of the l−x, vln,−x, and μ−x* values. For example, by using the N(εF)−x value of 1.3 states per (eV molecule spin), the Δvib,−3.17 value for μ−3.17* = 0.082 in K3.1730dba is estimated to be 0.94 K while the Δvib,−3.00 value for μ−3.00* = 0.066 in K330dba is estimated to be 7.4 K. However, the experimental Δvib,−x value of 28.2 K for the fractionally charged K3.1730dba is much larger than that of 7.4 K for the integrally charged K330dba. Therefore, let us next discuss the reason why the Δvib,−x value of 28.2 K for the fractionally charged K3.1730dba is much larger than that of 7.4 K for the integrally charged K330dba. For the most probable μ−3.17* value of 0.082, the experimental Δvib,−3.17 value of 28.2 K can be reproduced by using the N(εF)−3.17 value of 2.6 states per (eV molecule spin) (Figure 4c). Therefore, we can expect that the estimated approximately twice larger N(εF)−3.17 value of 2.6 states per (eV molecule spin) than that of 1.3 states per (eV molecule spin) reproducing the experimental results is the main reason why the Δvib,−x value of 28.2 K for the fractionally charged K3.1730dba is much larger than that of 7.4 K for the integrally charged K330dba. However, because of the difficulty of the precise determination only in view of the molecular size difference between 22ph and 30dba, the μ−3.00* value may slightly deviate from 0.066, and furthermore the μ−3.17* value may slightly deviate from 0.082. Therefore, for security, let us next investigate the relationships between the various N(εF)−3.17 and μ−3.17* values reproducing

the Δvib,−3.17 value of 28.2 K. The μ−3.17* value as a function of the N(εF)−3.17 value reproducing the Δvib,−3.17 value of 28.2 K is shown in Figure 4c. Furthermore, the estimated N(εF)−x values as a function of the μ−3.00* value in Kx30dba are shown in Figure 5. On the basis of the N(εF)−x and μ−3.17* values reproducing the Δvib,−x value of 7.4 K for K330dba, the Δvib,−x value of 28.2 K can be well reproduced by using the various N(εF)−x values of 2.2− 3.0 states per (eV molecule spin) and the μ−x* values of 0.030− 0.100 in K330dba, as shown in Figure 5a. In particular, most probably the Δvib,−3.17 value of 28.2 K can be reproduced if we consider the N(εF)−3.17 value of 2.6 states per (eV molecule spin) for the μ−3.17* value of 0.082. Figure 5a shows that the N(εF)−3.17 values estimated from the unified parameters of K330dba and K3.1730dba are in excellent agreement with those directly estimated from the experimental values of K3.1730dba. Furthermore, let us next estimate the ratio of the N(εF)−3.17 values to the N(εF)−3.00 values (N(εF)−3.17/N(εF)−3.00) for 30dba estimated on the basis of the μ−3.00* values of K330dba. The N(εF)−x/N(εF)−3.00 values as a function of the μ−3.00* value in Kx30dba are shown in Figure 6a. The N(εF)−3.17/N(εF)−3.00 values for μ−3.00* = 0.030−0.100 are estimated to be around 2.0. Therefore, an approximately twice larger N(εF)−x value (the most probable, 2.6 states per (eV molecule spin)) for K3.1730dba than that (most probably 1.3 states per (eV molecule spin)) for K330dba (Figure 6a) is the main reason why the experimental Δvib,−x value of 28.2 K for K3.1730dba is much larger than that of 7.4 K for K330dba. In a similar way, an approximately twice larger N(εF)−x value for K3.1730dba than that for K322ph is the main reason why the experimental Δvib,−x value of 28.2 K for the fractionally charged K3.1730dba is larger than that of 18 K for the integrally charged K322ph, even though the l−3.17 value of 0.112 17217



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eV for larger sized 30dba is much smaller than the l−3.00 value of 0.206 eV for smaller sized 22ph. K3.45-Dibenzopentacene. Let us next look into vibronic stabilization energies in K3.4530dba.29 There are three transition phases in K3.4530dba. The experimental Δvib,−x values for three transition phases of K3.4530dba are 33.1, 21.3, and 4.5 K. Let us next compare the vibronic stabilization energies in the fractionally charged K3.4530dba with that in the integrally charged K330dba. The l−3.45 value of 0.049 eV for K3.4530dba is much smaller than the l−3.00 value of 0.137 eV for K330dba (Figure 2a), and the vln,−3.45 value of 775 cm−1 for K3.4530dba is smaller than the vln,−3.00 value of 1040 cm−1 for K330dba (Table 1). Furthermore, the μ−3.45* value of 0.110 for more negatively charged K3.4530dba is much larger than the μ−3.00* value of 0.066 for less negatively charged K330dba (Figure 3a). Therefore, the Δvib,−x value for K3.4530dba would be expected to be smaller than that for K330dba in view of the l−x, vln,−x, and μ−x* values. For example, by using the N(εF)−x value of 1.3 states per (eV molecule spin), the Δvib,−3.45 value for μ−3.45* = 0.110 in K3.4530dba is estimated to be 3.27 × 10−56 K while the Δvib,−3.00 value for μ−3.00* = 0.066 in K330dba is estimated to be 7.4 K. However, the experimental Δvib,−x values of 33.1 and 21.3 K for two of three transition phases in the fractionally charged K3.4530dba are larger than that of 7.4 K in the integrally charged K330dba, and that of 4.5 K for only one of three transition phases in the fractionally charged K3.4530dba is smaller than that of 7.4 K in the integrally charged K330dba. Therefore, let us next discuss the reason why the experimental Δvib,−x values of 33.1 and 21.3 K for two of three transition phases in the fractionally charged K3.4530dba are larger than that of 7.4 K in the integrally charged K330dba. In view of Figure 3, the μ−3.45* value for the most probable N(εF)−3.00 value of 1.3 states per (eV molecule spin) is estimated to be 0.110. For the most probable μ−3.45* value of 0.110, the experimental Δvib,−x values of 33.1, 21.3, and 4.5 K for K3.4530dba can be reproduced by using the N(εF)−3.45 values of 7.0, 6.0, and 4.2 states per (eV molecule spin), respectively, as shown in Figure 4d. Furthermore, let us next estimate the (N(εF)−3.45/N(εF)−3.00) values for 30dba estimated on the basis of the μ−3.00* values of K330dba. The (N(εF)−3.45/N(εF)−3.00) values for the most probable μ−3.00* value of 0.066 are estimated to be 5.4, 4.6, and 3.3, respectively (Figure 6a). However, the actual μ−3.45* value may slightly deviate from 0.110 because of difficulty of precise estimation of the μ−3.00* value of 30dba from that of 22ph, and thus, for security, let us investigate the relationships between the possible various N(εF)−3.45 and μ−3.45* values reproducing the Δvib,−3.45 values of 33.1, 21.3, and 4.5 K. The μ−3.45* values as a function of the N(εF)−3.45 value reproducing the Δvib,−3.45 values of 33.1, 21.3, and 4.5 K are shown in Figure 4d. Furthermore, the N(εF)−3.45 values as a function of the μ−3.00* value in 30dba are shown in Figure 5b−d. Considering the possible μ−3.00* values of around 0.030−0.100, the N(εF)−3.45 values reproducing the Δvib,−x values of 33.1, 21.3, and 4.5 K are estimated to be 5.7−8.3 (the most probable, 7.0), 4.8−7.2 (the most probable 6.0), and 3.3−5.2 (the most probable, 4.2) states per (eV molecule spin), respectively. Figure 5b−d shows that the N(εF)−3.45 values estimated from the unified parameters of K330dba and K3.4530dba are in excellent agreement with those directly estimated from the experimental values of K3.4530dba at various μ−3.00* values which range around 0.030−0.100. Furthermore, let us next look into the (N(εF)−3.45/N(εF)−3.00) values for 30dba estimated on the basis of various μ−3.00* values

of K330dba. Figure 6a shows that the (N(εF)−3.45/N(εF)−3.00) values for the μ−3.00* values of 0.030−0.100 are estimated to be approximately 5.1−5.6, 4.3−4.8, and 2.8−3.5 in K3.4530dba with the Δvib,−3.45 values of 33.1, 21.3, and 4.5 K, respectively. Therefore, we can at least conclude that 5.1−5.6 and 4.3−4.8 (the most probable, 5.4 and 4.6) times larger N(εF)−3.45 values of 5.7−8.3 and 4.8−7.2 (the most probable, 7.0 and 6.0) states per (eV molecule spin) than the N(εF)−3.00 value of 1.1−1.5 (the most probable, 1.3) states per (eV molecule spin), are the main reason why the Δvib,−3.45 values of 33.1 and 21.3 K for two of three transition phases in K3.4530dba are larger than the Δvib,−3.00 value of 7.4 K in K330dba, respectively. On the other hand, the much smaller l−3.45 value of 0.049 eV for K3.4530dba than the l−3.00 value of 0.137 eV for K330dba is the main reason why the experimental Δvib,−3.45 value of 4.5 K for one of three transition phases in K3.4530dba is smaller than the experimental Δvib,−3.00 value of 7.4 K in K330dba even though the N(εF)−3.45 value of 3.3−5.2 (the most probable, 4.2) states per (eV molecule spin) is 2.8−3.5 (the most probable, 3.3) times larger than the N(εF)−3.00 value of 1.1− 1.5 (the most probable, 1.3) states per (eV molecule spin). Possible Origin of the Three Transition Phases in K3.45Dibenzopentacene and the Two Transition Phases in K3Picene. There are three transition phases at 33.1, 21.3, and 4.5 K in K3.4530dba, and two transition phases at 18 and 7 K in K322ph. In the previous section, we estimated the various N(εF)−3.45 values for K3.4530dba and the N(εF)−3.00 values for K322ph, reproducing the experimental Δvib,−x values by considering that various transition phases originate from molecular crystal structure differences between K3.4530dba and K322ph. That is, we considered that the nominal x values are equal to the actual x values in Kx30dba and Kx22ph. However, there is also a possibility that the intramolecular electronic structures are unstable, and the nominal numbers of K atoms would slightly deviate from the actual number of carriers located on each molecule. For example, there is a possibility that the actual intramolecular electronic states in 30dba molecules in K3.4530dba are in the range x = 3.45−3.00, from the point of view of vibronic interactions. Similar discussions can be made in the two transition phases at 7 and 18 K in Kx22ph. According to recent experimental work,37 the electron−phonon coupling constant estimated from the specific heat jump suggests that the role of the intramolecular vibrational modes in the stabilization of the electronic structures in Ba1.514ph could not be observed (linter ≠ 0) for some reason. One of the most plausible possibilities is that because of the strong electron−electron repulsion in small intramolecular ranges, 14ph molecules in Ba1.514ph cannot accept three electrons for long enough time, and are in the dianionic (14ph2−) from the point of view of the vibronic interactions (lintra ≈ l−2.00 = 0), and one carrier electron goes only through the intermolecular and intercalants regions. This is the main reason why the Δvib,−x values of 5 K for Ba1.514ph and K314ph are much smaller than that of 18 K for K322ph even though the l−3.00 value of 0.336 eV for 14ph3− is much larger than that of 0.206 eV for 22ph3−. In a similar way, because the numbers of carriers of 22ph3− and 30dba3.45− are near the upper limits of their abilities as acceptors, the electronic states of 22ph3− and 30dba3.45− are not always stable. Therefore, let us next look into the possibility that the nominal numbers of K atoms slightly deviate from the actual numbers of carriers located on each molecule. The estimated N(εF)−x values as a function of the x (3.05−3.50) value reproducing the Δvib,−x values of 33.1, 21.3, and 4.5 K by using the estimated l−x and μ−x* values are shown in Figure 7a and listed in Table 2. Let us first 17218



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Figure 7. Density of states at the Fermi level versus charge. (a) The N(εF)−x values reproducing the Δvib,−x values of 33.1, 21.3, and 4.5 K in Kx30dba. The circles, triangles, and squares denote the N(εF)−x values reproducing the Δvib,−x values of 33.1, 21.3, and 4.5 K, respectively. (b) The N(εF)−x values reproducing the Δvib,−x values of 18 and 7 K in Kx22ph. The circles and triangles denote the N(εF)−x values reproducing the Δvib,−x values of 18 and 7 K, respectively.

ranges around 2.57 ≤ x ≤ 2.87 and 2.4 ≥ N(εF)−x ≥ 1.2 states per (eV molecule spin) and most probably originates from an electronic state which ranges near from x = 2.87 and N(εF)−x = 1.2 states per (eV molecule spin) in K322ph, as shown in Figure 7b and Table 3.

Table 2. x and μ−x* Values versus the N(εF)−x Values (States/ eV·Unit·Spin) Reproducing the Δvib,−x Values of 33.1 and 21.3 K in K3.4530dba N(εF)−x x (33.1 K) μ−x* x (21.3 K) μ−x*

2.0

2.5

3.0

3.5

4.0

4.2

3.05 0.070 3.11 0.076

3.13 0.078 3.18 0.083

3.20 0.084 3.25 0.089

3.25 0.089 3.29 0.093

3.29 0.093 3.34 0.098

3.32 0.096 3.35 0.099

Table 3. x and μ−x Values versus the N(εF)−x Values (States/ eV·Unit·Spin) Reproducing the Δvib,−x Value of 7 K in K322ph N(εF)−x

consider that the molecular crystal structures are not significantly changed by actual charge decrease and the N(εF)−x value does not significantly change with a decrease in the actual charge. That is, by considering the same N(εF)−x values, the Δvib,−x value significantly increases with a decrease in the x value from 3.50 to 3.00 in Kx30dba. For example, considering the N(εF)−x value of 4.2 states per (eV molecule spin) reproducing the Δvib,−x value of 4.5 K for x = 3.45, the Δvib,−x value of 21.3 K is reproduced by x = 3.35, and that of 33.1 K is reproduced by x = 3.32 (Figure 7a, Table 2). Therefore, the Δvib,−x value for N(εF)−x = 4.2 states per (eV molecule spin) significantly increases (by 28.6 K) with an only slight decrease in the x value from 3.45 to 3.32 (by 0.13). On the other hand, there is also a possibility that the molecular crystal structures are slightly changed and the N(εF)−x value slightly deviates from N(εF)−x = 4.2 states per (eV molecule spin) with a decrease in the x value. Therefore, let us next look into the x values reproducing the experimental Δvib,−x values for various N(εF)−x values. Table 2 shows that the x values reproducing the Δvib,−x value of 33.1 K for the N(εF)−x values of 4.2−2.0 states per (eV molecule spin) are estimated to be 3.32−3.05. Therefore, there is a possibility that the transition phase at 33.1 K originates from an electronic state ranging around 3.05 ≤ x ≤ 3.32 and 2.0 ≤ N(εF)−x ≤ 4.2 states per (eV molecule spin) and most probably originates from an electronic state which ranges near from x = 3.32 and N(εF)−x = 4.2 states per (eV molecule spin). In a similar way, there is a possibility that the transition phase at 21.3 K originates from an electronic state ranging around 3.11 ≤ x ≤ 3.35 and 2.0 ≤ N(εF)−x ≤ 4.2 states per (eV molecule spin) and most probably originates from an electronic state which ranges near from x = 3.35 and N(εF)−x = 4.2 states per (eV molecule spin). Furthermore, in a similar way, there is a possibility that the transition phase at 7 K originates from an electronic state which

x (7 K) μ−x*

x (7 K) μ−x*

1.2

1.3

2.87 0.079

2.83 0.074

1.4

1.5

2.80 2.75 0.071 0.065 N(εF)−x

1.6

1.7

2.73 0.063

2.70 0.060

1.8

1.9

2.0

2.1

2.2

2.3

2.4

2.68 0.058

2.66 0.056

2.64 0.054

2.62 0.051

2.60 0.049

2.59 0.048

2.57 0.046

It should be noted that the Δvib,−x value significantly changes with an only slight change in the x value, and thus we can expect possible various Δvib,−x values even within the small ranges of the x values (Figure 8c−f). Even though we cannot very clearly determine the x and N(εF)−x values for the electronic states around which the phase transitions occur, the most probable x and N(εF)−x values for such electronic states are estimated to be x = 3.32 and N(εF)−x = 4.2 states per (eV molecule spin) in K3.4530dba (Δvib,−x = 33.1 K), x = 3.35 and N(εF)−x = 4.2 states per (eV molecule spin) in K3.4530dba (Δvib,−x = 21.3 K), and x = 2.87 and N(εF)−x = 1.2 states per (eV molecule spin) in K322ph (Δvib,−x = 7 K), according to our calculated results. We can rationalize the three transition phases in K3.4530dba and the two transition phases in K322ph in view of the difference between the nominal and actual x values as well as the molecular crystal structures differences. However, we cannot clearly determine whether three transition phases in K3.4530dba and two transition phases in K322ph originate from the molecular crystal structure differences or the differences between the nominal and actual x values. The experimental works which elucidate the molecular crystal structures and the actual electronic structures in more detail are yet to be performed. 17219



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Figure 8. Dimensionless electron−phonon coupling constants and the vibronic stabilization energies. (a) The most probable λ−x value versus the −x value in Kx30dba. (b) The most probable λ−x/λ−3.00 value versus the −x value in Kx30dba. (c) The Δvib,−x (

Vibronic Interactions in Fractionally Charged Hydrocarbons

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