BANDLIKE HOLE MOBILITY IN NAPHTHALENE CRYSTAL
2.6 Supplementary Materials
Band structure and phonon dispersion
-400 -300 -200 -100 0 100
Y Γ K Z Γ B A Γ D
HOMO and HOMO-1 electronic bands (meV)
DFT GW-corrected
(a)
0 20 40 60 80 100 120 140
B Γ Y A Γ Z
Frequency ω(q) (cm-1 )
(b) (c)
1
Figure 2.5: Naphthalene band structures and phonon dispersions. Band struc- tures and phonon dispersions of naphthalene crystal, for the structure used at 300 K. (a) The HOMO and HOMO−1 electronic bands, where black is used for the DFT bands and red for the bands with the GW correction. (b) Dispersion of the 12 intermolecular phonon modes. (c) Sketch of the first Brillouin zone.
Results with different van der Waals correction
100 140 180 220 260 300
Temperature (K)
10-1 100 101
Mobility (cm2/ V s) n = -2.68 n = -2.61
µa experiment
µa ab initio
(a)
100 140 180 220 260 300
Temperature (K)
10-1 100 101
Mobility (cm2/ V s) n = -2.41 n = -2.34
µb experiment
µb ab initio
(b)
100 140 180 220 260 300
Temperature (K)
10-2 10-1 100
Mobility (cm2/ V s) n = -2.63 n = -2.88
µc∗ experiment
µc∗ ab initio
(c)
Figure 2.6: Calculated hole mobility using the TS-vdW correction.1 The mobility at 300 K obtained using a structure relaxed with the TS-vdW correction is shown with black crosses. The values fall within the error bars.
Comparison of calculated phonon dispersions with experiment
B
ΓY A
ΓZ
0 20 40 60 80 100 120 140
Fre qu en cy
ω(q) (c m
−1) Experiment DFPT
1
Figure 2.7: Comparison of calculated phonon dispersions with experiment.
Calculated dispersion of the 12 intermolecular phonon modes for perdeuterated naphthalene, with lattice constants taken from Refs. [52, 63]. The markers are the experimental data at 6 K from Ref. [64].
Refined interpolation and its impact on the calculated hole mobility
Figure 2.8: Coupling constants from refined interpolation. Comparison of the coupling constants used in Chapter 2 and those from a refined interpolation scheme developed in our recent work. The original interpolation overestimates the coupling strength for low-frequency phonons, and hence exaggerates the scattering rates from intermolecular phonons.
100 140 180 220 260 300 Temperature (K)
100 101
μa (cm2/ V s) n = 2.68 n = 2.61
n = 2.70 experiment PRB 2018 refined
(a)
100 140 180 220 260 300 Temperature (K) 10−1
100 101 102
μb (cm2/ V s)
n = 2.41 n = 2.34
n = 2.40 e periment PRB 2018 refined
(b)
100 140 180 220 260 300 Temperature (K) 10−2
10−1 100 101
μc* (cm2/ V s)
n = 2.63 n = 2.88 n = 3.14
experiment PRB 2018 refined
(c)
Figure 2.9: Hole mobility calculated with refined coupling constants.1 Calcu- lated hole mobility using the refined coupling constants. The result is in excellent agreement with experiment, indicating that the discrepancy is mainly due to the improperly-interpolated𝑔for low-frequency phonons.
In our recent work, the open problem that the calculated phonon-limited hole mobil- ity is lower than experiment by a factor of 3−4 is resolved. Upon refining our results, we found that the commonly-adopted interpolation scheme cannot properly handle aq-point grid that is as coarse as 2×4×2, leading to a large overestimation on the coupling strength with low-frequency phonons (see Fig. 2.8). Improvements in the acoustic sum rules and Wigner-Seitz cell sampling used for𝑒-ph interpolation [55]
have allowed us to refine the results and obtain accurate interpolated𝑒-ph coupling constants that match the DFPT benchmark at all grid points. The overestimation of 𝑒-ph coupling with low-frequency modes that affected our results in Ref. [65]
artificially increases the scattering rate due to intermolecular phonons, resulting in a lower mobility. Using a refined interpolation scheme [55], we can bring the calcu- lated hole mobility to a value just slightly higher than experiment (see Fig. 2.9). This result is consistent with our expectation, and shows that transport of hole carriers in naphthalene can be fully explained by band theory.
References
[1] M. Muccini,Nat. Mater.5, 605–613 (2006).
[2] V. A. Dediu, L. E. Hueso, I. Bergenti, and C. Taliani,Nat. Mater.8, 707–716 (2009).
[3] A. L. Briseno, S. C. B. Mannsfeld, M. M. Ling, S. Liu, R. J. Tseng, C. Reese, M. E. Roberts, Y. Yang, F. Wudl, and Z. Bao,Nature444, 913–917 (2006).
[4] T. Sekitani, U. Zschieschang, H. Klauk, and T. Someya,Nat. Mater.9, 1015–
1022 (2010).
[5] M. Kitamura, T. Imada, and Y. Arakawa, Appl. Phys. Lett. 83, 3410–3412 (2003).
[6] N. Karl, "Organic semiconductors, Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology New Series - Group III Crystal and Solid State Physics", Vol. 17i (Springer-Verlag Berlin-Heidelberg, 1985).
[7] L. B. Schein, C. B. Duke, and A. R. McGhie, Phys. Rev. Lett.40, 197–200 (1978).
[8] H. Oberhofer, K. Reuter, and J. Blumberger,Chem. Rev.117, 10319–10357 (2017).
[9] S. Canola and F. Negri,Phys. Chem. Chem. Phys.16, 21550–21558 (2014).
[10] A. Troisi,Adv. Mater.19, 2000–2004 (2007).
[11] J. C. Tully,J. Chem. Phys.93, 1061–1071 (1990).
[12] L. Wang and D. Beljonne,J. Phys. Chem. Lett.4, 1888–1894 (2013).
[13] A. Troisi and G. Orlandi,Phys. Rev. Lett.96, 086601 (2006).
[14] S. Fratini, D. Mayou, and S. Ciuchi,Adv. Funct. Mater.26, 2292–2315 (2016).
[15] S. Fratini, S. Ciuchi, D. Mayou, G. T. de Laissardiére, and A. Troisi, Nat.
Mater.16, 998–1002 (2017).
[16] R. Silbey and R. W. Munn,J. Chem. Phys.72, 2763–2773 (1980).
[17] V. M. Kenkre, J. D. Andersen, D. H. Dunlap, and C. B. Duke,Phys. Rev. Lett.
62, 1165–1168 (1989).
[18] K. Hannewald and P. A. Bobbert,Phys. Rev. B69, 075212 (2004).
[19] K. Hannewald and P. A. Bobbert,Appl. Phys. Lett.85, 1535–1537 (2004).
[20] F. Ortmann, F. Bechstedt, and K. Hannewald,Phys. Rev. B79, 235206 (2009).
[21] F. Ortmann, F. Bechstedt, and K. Hannewald,New J. Phys.12, 023011 (2010).
[22] L. J. Wang, Q. Peng, Q. K. Li, and Z. Shuai, J. Chem. Phys. 127, 044506 (2007).
[23] L. Tang, M. Long, D. Wang, and Z. Shuai,Sci. China Ser. B-Chem.52, 1646–
1652 (2009).
[24] Y. C. Cheng, R. J. Silbey, D. A. da Silva Filho, J. P. Calbert, J. Cornil, and J. L. Brédas,J. Chem. Phys.118, 3764–3774 (2003).
[25] L. Giuggioli, J. D. Andersen, and V. M. Kenkre, Phys. Rev. B 67, 045110 (2003).
[26] O. Ostroverkhova, D. G. Cooke, F. A. Hegmann, J. E. Anthony, V. Podzorov, M. E. Gershenson, O. D. Jurchescu, and T. T. M. Palstra,Appl. Phys. Lett.
88, 162101 (2006).
[27] D. Moses, C. Soci, X. Chi, and A. P. Ramirez,Phys. Rev. Lett. 97, 067401 (2006).
[28] Z. Q. Li, V. Podzorov, N. Sai, M. C. Martin, M. E. Gershenson, M. Di Ventra, and D. N. Basov,Phys. Rev. Lett.99, 016403 (2007).
[29] S. Machida, Y. Nakayama, S. Duhm, Q. Xin, A. Funakoshi, N. Ogawa, S.
Kera, N. Ueno, and H. Ishii,Phys. Rev. Lett.104, 156401 (2010).
[30] H. Kobayashi, N. Kobayashi, S. Hosoi, N. Koshitani, D. Murakami, R. Shi- rasawa, Y. Kudo, D. Hobara, Y. Tokita, and M. Itabashi,J. Chem. Phys.139, 014707 (2013).
[31] J. I. Mustafa, M. Bernardi, J. B. Neaton, and S. G. Louie, Phys. Rev. B 94, 155105 (2016).
[32] J.-J. Zhou and M. Bernardi,Phys. Rev. B94, 201201 (2016).
[33] V. A. Jhalani, J.-J. Zhou, and M. Bernardi,Nano Lett.17, 5012–5019 (2017).
[34] M. Bernardi,Eur. Phys. J. B89, 239 (2016).
[35] M. Casula, M. Calandra, and F. Mauri,Phys. Rev. B86, 075445 (2012).
[36] A. Girlando, L. Grisanti, M. Masino, I. Bilotti, A. Brillante, R. G. Della Valle, and E. Venuti,Phys. Rev. B82, 035208 (2010).
[37] N. Vukmirović, C. Bruder, and V. M. Stojanović,Phys. Rev. Lett.109, 126407 (2012).
[38] P. Giannozzi et al.,J. Phys. Condens. Matter21, 395502 (2009).
[39] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865–3868 (1996).
[40] N. Troullier and J. L. Martins,Phys. Rev. B43, 1993–2006 (1991).
[41] M. van Setten, M. Giantomassi, E. Bousquet, M. Verstraete, D. Hamann, X.
Gonze, and G.-M. Rignanese,Comput. Phys. Commun.226, 39–54 (2018).
[42] C. P. Brock and J. D. Dunitz,Acta Cryst. B38, 2218–2228 (1982).
[43] S. Gražulis, A. Daškevič, A. Merkys, D. Chateigner, L. Lutterotti, M. Quirós, N. R. Serebryanaya, P. Moeck, R. T. Downs, and A. Le Bail,Nucleic Acids Res.40, D420–D427 (2012).
[44] S. Gražulis, D. Chateigner, R. T. Downs, A. F. T. Yokochi, M. Quirós, L.
Lutterotti, E. Manakova, J. Butkus, P. Moeck, and A. Le Bail,J. Appl. Cryst.
42, 726–729 (2009).
[45] S. Grimme,J. Comput. Chem.27, 1787–1799 (2006).
[46] V. Barone, M. Casarin, D. Forrer, M. Pavone, M. Sambi, and A. Vittadini,J.
Comput. Chem.30, 934–939 (2009).
[47] T. Rangel, K. Berland, S. Sharifzadeh, F. Brown-Altvater, K. Lee, P. Hyldgaard, L. Kronik, and J. B. Neaton,Phys. Rev. B93, 115206 (2016).
[48] A. Marini, C. Hogan, M. Gr𝑢¥ning, and D. Varsano,Comput. Phys. Commun.
180, 1392–1403 (2009).
[49] A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari,Comput. Phys. Commun.185, 2309–2310 (2014).
[50] L. A. Agapito, M. Fornari, D. Ceresoli, A. Ferretti, S. Curtarolo, and M. B.
Nardelli,Phys. Rev. B93, 125137 (2016).
[51] S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi,Rev. Mod. Phys.
73, 515–562 (2001).
[52] F. Brown-Altvater, T. Rangel, and J. B. Neaton, Phys. Rev. B 93, 195206 (2016).
[53] S. Poncé, E. Margine, C. Verdi, and F. Giustino, Comput. Phys. Commun.
209, 116–133 (2016).
[54] F. Giustino, M. L. Cohen, and S. G. Louie,Phys. Rev. B76, 165108 (2007).
[55] J.-J. Zhou, J. Park, I.-T. Lu, I. Maliyov, X. Tong, and M. Bernardi, Preprint at arXiv:2002.02045 (2020).
[56] G. Pizzi, D. Volja, B. Kozinsky, M. Fornari, and N. Marzari,Comput. Phys.
Commun.185, 422–429 (2014).
[57] P. E. Blöchl, O. Jepsen, and O. K. Andersen,Phys. Rev. B49, 16223–16233 (1994).
[58] J. Hoja, A. M. Reilly, and A. Tkatchenko, Wiley Interdiscip. Rev. Comput.
Mol. Sci.7, 1294 (2017).
[59] A. M. Reilly and A. Tkatchenko,J. Chem. Phys.139, 024705 (2013).
[60] M. Bernardi, D. Vigil-Fowler, J. Lischner, J. B. Neaton, and S. G. Louie, Phys. Rev. Lett.112, 257402 (2014).
[61] B. F. E. Curchod, U. Rothlisberger, and I. Tavernelli, ChemPhysChem 14, 1314–1340 (2013).
[62] G. D. Mahan,"Many-Particle Physics", 3rd ed. (Springer, 2000).
[63] I. R. Thomas, I. J. Bruno, J. C. Cole, C. F. Macrae, E. Pidcock, and P. A.
Wood,J. Appl. Crystallogr.43, 362–366 (2010).
[64] I. Natkaniec, E. L. Bokhenkov, B. Dorner, J. Kalus, G. A. Mackenzie, G. S.
Pawley, U. Schmelzer, and E. F. Sheka,J. Phys. C13, 4265 (1980).
[65] N.-E. Lee, J.-J. Zhou, L. A. Agapito, and M. Bernardi, Phys. Rev. B 97, 115203 (2018).
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