Huge Trionic Effects in Graphene Nanoribbons - Nano Letters (ACS

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Huge trionic effects in graphene nanoribbons Thorsten Deilmann, and Michael Rohlfing Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b03111 • Publication Date (Web): 25 Oct 2017 Downloaded from http://pubs.acs.org on October 26, 2017

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Huge trionic effects in graphene nanoribbons Thorsten Deilmann∗,† and Michael Rohlfing‡ †Center for Atomic-Scale Materials Design (CAMD), Department of Physics, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark ‡Institut f¨ ur Festkörpertheorie, Westf¨ alische Wilhelms-Universit¨ at M¨ unster, 48149 M¨ unster, Germany. E-mail: [email protected] Abstract One- and two-dimensional materials are presently being intensively investigated due to their interesting properties for next-generation opto-electronic devices. Among these, armchair-edged graphene nanoribbons are very promising candidates with optical properties which are dominated by excitons. In the presence of additional charges, trions (i.e. charged excitons) can occur in the optical spectrum. With our recently developed first-principle many-body approach (Phys. Rev. Lett. 116, 196804), we predict strongly bound trions in free-standing nanoribbons with large binding energies of 140 to 660 meV for widths of 14.6 to 3.6 ˚ A. Both for the trions and for the excitons we observe an almost linear dependency of their binding energies on the band gap. We observe several trion states with different character derived from the corresponding excitons. Due to the large bindings energies, this opens a route to applications which optical properties are easily manipulated, e.g., by electrical fields.

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Keywords graphene nanoribbons, 1D materials, optional spectra, many-body perturbation theory, excitons, trions Graphene nanoribbons (GNRs) are quasi one-dimensional stripes of graphene. 1 In contrast to two-dimensionally extended graphene, a band gap is found in GNRs 2–4 which allows its usage for build opto-electronic devices. Depending on width and type (i.e., armchair (AGNR), or zigzag (ZGNR) nanoribbons), distinctly different band gaps as well as excitonic properties have been observed. 5,6 Recently several experiments were able to build atomically precise stripes of different widths, e.g., AGNR(N = 7, 13), 7–9 AGNR(N = 5), 10 and ZGNR(N = 6). 11 In a combined experimental and theoretical first-principle study Denk et al. 9 have confirmed large excitonic effects in their optical spectra, which underlines their potential for applications. On the other hand, experimental and theoretical studies for transition metal dichalcogenides (TMDCs) 12–15 and carbon nanotubes (CNTs) 16–26 have shown significant changes if the systems are doped: Trions, i.e. correlated states of two electrons and one hole (or two holes and one electron) appear, which results in a distinctly different optical spectrum, in particular in photoluminescence. Over the years trions have been identified in a number of semiconducting materials. In Tab. 1 we compare the binding energy of some of them. While in 2d (or 3d) materials small binding energies of only < 50 meV are observed, 1d or quasi-1d materials (including phosphorus, which is at the border between 2d and 1d behavior) reveal much larger energy splitting between the trion and exciton line due to the enhanced confinement. Table 1: Comparison of trion binding energies in different materials. Material CdTe quantum well MoS2 (and similar TMDCs) Black phosphorus CNT (different (n, m)) GNR

Dimension 3d/2d 2d 2d/1d 1d 1d

2

Trion binding energy 2.1 meV 27 18–43 meV 12,28,29 ∼100 meV, 30,31 ∼50 meV 32 100–220 meV, 16,17 130 meV 33 140–660 meV (this work)


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to strong contribution to luminescence. Furthermore the large splitting makes it possible to separate trions from excitons even at room temperature, which might become particular interesting for applications. For the understanding of the detailed results like the occurrence of three different AGNR families (N = 3p, N = 3p + 1, N = 3p + 2) we have to move one step back and reconsider the ground state properties. AGNR(N ) are built up by 2N carbon atoms per unit cell and four hydrogen atoms (two at each side of the ribbon, see Fig. 1(b)). Structure optimization within the local density approximation (LDA) 37 of density-functional theory (DFT)

1

leads

to a lattice constant of a = 4.24 ˚ A along the ribbon axis and widths (from carbon atom to carbon atom) of 3.62 to 14.61 ˚ A for AGNR(N = 4) to AGNR(N = 13). The band structure of all ribbons reveals a direct gap at Γ. Due to the back folding on the graphene band structure the nanoribbons split into three families with N = 3p, 3p + 1, and 3p + 2 (p being an integer number). 3,40 The largest gaps show up for N = 3p + 1, while the family of N = 3p + 2 reveals the smallest. In DFT(LDA) we find values of 0.20 to 2.56 eV (for N =4-13) in good agreement to Ref., 4 with a decrease of the gap for increasing ribbon width (within all three families). By employing many-body perturbation theory within the GW approximation 34,35 we find a distinct opening of the gaps to 0.58 to 5.11 eV 2 . This finding is in excellent agreement with previous results. 5,6 The excited states and the resulting optical properties for neutral excitations (electronhole pairs) result from the Bethe-Salpeter equation (BSE) with the Hamiltonian H (eh) = ˆ BS + H ˆ eh commonly used within MBPT. 41,42 Herein H ˆ BS denotes the band-structure conH ˆ eh is the direct and exchange interaction between electron and hole. The cortributions and H responding wave function (using the Tamm-Dancoff approximation) for the S-th excitation 1 Norm-conserving pseudopotentials 38 in Kleinman-Bylander form 39 are used. The DFT wave functions are expanded in a basis of atom-centered Gaussian orbitals of s, p, d, and s∗ symmetry with decay constants 0.22, 0.84, and 3.52 for carbon and 0.15, 0.68, and 3.53 1/a2B for hydrogen atoms. The structural optimization is performed using 12 k points in irreducible part of Brillouin zone. 2 For the calculation of the GW self-energy operator we employ at least 24 q points in the irreducible part of the Brillouin zone. A cutoff energy of 5 Ry (3757 plane waves) is used for the auxiliary basis set in which two-point quantities (e.g., the dielectric function, the screened Coulomb interaction etc.) are represented.

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with a total momentum Q = −kv + kc is given by Φ(S,Q) (xh , xe ) =

P

vc

(S,Q) ∗ φv (xh )φc (xe ).

Bvc

Additional charges (from doping or gating) can form a (bound) three-particle state. We concentrate on negatively charged trions composed from two electrons and one hole (for details see Ref. 33 ) and set up their wave function as Φ(T,K) (xh , x1 , x2 ) =

X

∗ A(T,K) v,c1 ,c2 φv (xh )A{φc1 (x1 )φc2 (x2 )},

(1)

v,c1 ,c2

where K = k1 + k2 − kv denotes the momentum of the trion, v = (v, kv ) band index and wave number of a hole (c1 and c2 for two electrons analogously), xh , x1 , x2 denote their position and spin, and A is the antisymmetrizer. The corresponding Hamiltonian is set up as the sum of band-structure part and the two-particle interactions between both electrons, and between the first and second electron and the hole ˆ (eeh) = H ˆ BS + H ˆ ee + H ˆ eh,1 + H ˆ eh,2 . H

(2)

Although the resulting Hamilton matrix is vast (about 0.5 to 14 million, depending on the exact numerical details), it is possible to handle it using parallel architecture and suitable algorithms. 43,44 In Fig. 1(a) the band gaps are compared to the transition energies Ω. For excitons Ω(S,Q) are directly given by the eigenvalues of the BSE. For a trion, Ω denotes the energy difference (i.e. excitation energy) between the charged exciton and a single-electron state. Assuming that the latter is given by an electron in the conduction-band minimum (CBM), the excitation ˆ (eeh) . Since energy is given by Ω(T,K) = E (T,K) − ECBM with E (T,K) being an eigenvalue of H an optical excitation leaves the total momentum nearly unchanged, the trion momentum K (for Eq. (1)) is set by the CBM, which (for the AGNR) occurs at Γ. Both transition energies (of excitons and trions) follow the same trend as the band gaps for the three families 3p, 3p + 1, and 3p + 2. The increase for the exciton binding energy EbS = Egap − Ω(S,Q) with decreasing ribbon width is given, e.g. for 3p + 1, by 1.03 eV for N = 13 to 3.05 eV for N = 4. 5


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of the exciton and trion spectrum cannot be compared with one another because the trion spectrum scales with the density of these donated electrons. If we consider the luminescence process instead of absorption, we assume that the trions (and excitons as well) achieve thermal equilibrium after the excitation process, with energy dependent occupation probabilities given by a room-temperature Boltzmann distribution (see 33 ). This will enhance the trion peaks distinctly. To gain deeper insight the real-space wave function of the trions is analysed. To visualize this quantity we reduce the six- or nine-dimensional nature of the wave functions by fixing a hole at rh and integrating perpendicular to the ribbon axis, leading to electron distributions along the ribbon: (S,Q)

Z

Z

dze |Φ(S,Q) (rh , xe , ye , ze )|2 Z Z Z (T,K) ρ (x1 ) = dy1 dz1 dr2 |Φ(T,K) (rh , x1 , y1 , z1 , r2 )|2 . ρ

(xe ) =

dye

(3) (4)

For trions (Eq. (4)) the second electron is integrated out which allows for a direct comparison of the trion with the exciton. The electron distributions of the excitons for AGNR of the 3p + 1 family are shown in Fig. 4(a). In all ribbons the electron probability of the first bright exciton state is centered around the fixed hole (at x = 0). With increasing width the extent along the ribbon axis increases. The analogous trions are shown in Fig. 4(b). In comparison to excitons the trions are much more extended, because the electron-electron repulsion drives the electrons somewhat apart from each other, and thus weakening the attraction of both electrons to the hole. The trend of increased extent along the ribbon axis for increased width shows up for trions analogously. For any given AGNR, the trions exhibit a large variety of their spatial structures. As a particular rich example we discuss the N = 10 AGNR. The spectrum is shown in Fig. 5(a) with two exciton peaks E11 and E22 at 1.38 and 1.67 eV. For the nine most prominent trions (indicated by labels in Fig. 5(a)) the wave functions are plotted in Figs. 5(b) and (c). The

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valence band and both electrons in the spin degenerated second lowest conduction band) (1)

has a localized wave function but a vanishing optical activity. T22 directly corresponds to (1)

the E22 exciton with a binding energy of 240 meV. Similar to T11 , it is spatially extended in comparison with its exciton counterpart. Two examples of linear combinations with a free (2,3)

electron and scattering states are labeled as T22 . Similar analyses for AGNR of different width (including different families) lead to very similar characteristics of the trions. Thus we conclude that the eigenstates of Eq. (2) include (i.) the original excitons accompanied by a free conduction-band electron (in the limit of negligible doping, i.e. large unit cells). Additionally, it leads to (ii.) new localized threeparticle states with decreased optical transitions, and (iii.) binding and antibinding linear combinations with scattering states appear. In this letter we have evaluated the optical properties of free-standing AGNRs of different width. Trions with huge binding energy of up to 0.7 eV occur, which should be measurable in experiment. With increasing width we find a reduction of band gap, exciton and trion binding energy. A detailed investigation of the corresponding wave functions reveals a distinctly more extended shape of localized trions (in comparison to the exciton counterparts). Furthermore, a large set of trions has been identified as linear combinations of a localized exciton state and delocalized electron states, with characteristics of free or scattered electrons. Notes The authors declare no competing financial interests.

Acknowledgement T.D. acknowledges the financial support from the Villum foundation. The authors gratefully acknowledge the computing time granted by the John von Neumann Institute for Computing (NIC) and provided on the super-computer JURECA at J¨ ulich Supercomputing Centre (JSC).

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References (1) Novoselov, K. S. Science 2004, 306, 666–669. (2) Barone, V.; Hod, O.; Scuseria, G. E. Nano Letters 2006, 6, 2748–2754. (3) Son, Y.-W.; Cohen, M. L.; Louie, S. G. Phys. Rev. Lett. 2006, 97, 216803. (4) Yang, L.; Park, C.-H.; Son, Y.-W.; Cohen, M. L.; Louie, S. G. Physical Review Letters 2007, 99, 186801. (5) Yang, L.; Cohen, M. L.; Louie, S. G. Nano Letters 2007, 7, 3112–3115. (6) Prezzi, D.; Varsano, D.; Ruini, A.; Marini, A.; Molinari, E. Physical Review B 2008, 77, 041404. (7) Cai, J.; Ruffieux, P.; Jaafar, R.; Bieri, M.; Braun, T.; Blankenburg, S.; Muoth, M.; Seitsonen, A. P.; Saleh, M.; Feng, X.; M¨ ullen, K.; Fasel, R. Nature 2010, 466, 470–473. (8) Linden, S.; Zhong, D.; Timmer, A.; Aghdassi, N.; Franke, J. H.; Zhang, H.; Feng, X.; M¨ ullen, K.; Fuchs, H.; Chi, L.; Zacharias, H. Physical Review Letters 2012, 108, 216801. (9) Denk, R.; Hohage, M.; Zeppenfeld, P.; Cai, J.; Pignedoli, C. A.; Söde, H.; Fasel, R.; Feng, X.; M¨ ullen, K.; Wang, S.; Prezzi, D.; Ferretti, A.; Ruini, A.; Molinari, E.; Ruffieux, P. Nature Communications 2014, 5, 4253. (10) Kimouche, A.; Ervasti, M. M.; Drost, R.; Halonen, S.; Harju, A.; Joensuu, P. M.; Sainio, J.; Liljeroth, P. Nature Communications 2015, 6, 10177. (11) Ruffieux, P.; Wang, S.; Yang, B.; Sánchez-Sánchez, C.; Liu, J.; Dienel, T.; Talirz, L.; Shinde, P.; Pignedoli, C. A.; Passerone, D.; Dumslaff, T.; Feng, X.; M¨ ullen, K.; Fasel, R. Nature 2016, 531, 489–492. (12) Mak, K. F.; He, K.; Lee, C.; Lee, G. H.; Hone, J.; Heinz, T. F.; Shan, J. Nat Mater 2013, 12, 207–211. 12


Page 12 of 16

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(13) Ross, J. S.; Wu, S.; Yu, H.; Ghimire, N. J.; Jones, A. M.; Aivazian, G.; Yan, J.; Mandrus, D. G.; Xiao, D.; Yao, W.; Xu, X. Nat Commun 2013, 4, 1474. (14) Mouri, S.; Miyauchi, Y.; Matsuda, K. Nano Letters 2013, 13, 5944–5948. (15) Berkelbach, T. C.; Hybertsen, M. S.; Reichman, D. R. Phys. Rev. B 2013, 88, 045318. (16) Matsunaga, R.; Matsuda, K.; Kanemitsu, Y. Phys. Rev. Lett. 2011, 106, 037404. (17) Santos, S. M.; Yuma, B.; Berciaud, S.; Shaver, J.; Gallart, M.; Gilliot, P.; Cognet, L.; Lounis, B. Phys. Rev. Lett. 2011, 107, 187401. (18) Park, J. S.; Hirana, Y.; Mouri, S.; Miyauchi, Y.; Nakashima, N.; Matsuda, K. Journal of the American Chemical Society 2012, 134, 14461–14466. (19) Jakubka, F.; Grimm, S. B.; Zakharko, Y.; Gannott, F.; Zaumseil, J. ACS Nano 2014, 8, 8477–8486. (20) Rnnow, T. F.; Pedersen, T. G.; Cornean, H. D. Physics Letters A 2009, 373, 1478 – 1481. (21) Rønnow, T. F.; Pedersen, T. G.; Cornean, H. D. Phys. Rev. B 2010, 81, 205446. (22) Watanabe, K.; Asano, K. Phys. Rev. B 2012, 85, 035416. (23) Nishihara, T.; Yamada, Y.; Okano, M.; Kanemitsu, Y. Applied Physics Letters 2013, 103 . (24) Yuma, B.; Berciaud, S.; Besbas, J.; Shaver, J.; Santos, S.; Ghosh, S.; Weisman, R. B.; Cognet, L.; Gallart, M.; Ziegler, M.; Hönerlage, B.; Lounis, B.; Gilliot, P. Phys. Rev. B 2013, 87, 205412. (25) Mouri, S.; Miyauchi, Y.; Iwamura, M.; Matsuda, K. Phys. Rev. B 2013, 87, 045408.

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(26) Koyama, T.; Shimizu, S.; Miyata, Y.; Shinohara, H.; Nakamura, A. Phys. Rev. B 2013, 87, 165430. (27) Huard, V.; Cox, R. T.; Saminadayar, K.; Arnoult, A.; Tatarenko, S. Physical Review Letters 2000, 84, 187–190. (28) Zhang, Y.; Li, H.; Wang, H.; Liu, R.; Zhang, S.-L.; Qiu, Z.-J. ACS Nano 2015, 9, 8514–8519. (29) Low, T.; Chaves, A.; Caldwell, J. D.; Kumar, A.; Fang, N. X.; Avouris, P.; Heinz, T. F.; Guinea, F.; Martin-Moreno, L.; Koppens, F. Nature Materials 2016, 16, 182–194. (30) Yang, J.; Xu, R.; Pei, J.; Myint, Y. W.; Wang, F.; Wang, Z.; Zhang, S.; Yu, Z.; Lu, Y. Light: Science & Applications 2015, 4, e312. (31) Xu, R.; Zhang, S.; Wang, F.; Yang, J.; Wang, Z.; Pei, J.; Myint, Y. W.; Xing, B.; Yu, Z.; Fu, L.; Qin, Q.; Lu, Y. ACS Nano 2016, 10, 2046–2053. (32) Chaves, A.; Mayers, M. Z.; Peeters, F. M.; Reichman, D. R. Physical Review B 201603-25, 93, 115314. (33) Deilmann, T.; Dr¨ uppel, M.; Rohlfing, M. Phys. Rev. Lett. 2016, 116, 196804. (34) Rohlfing, M.; Louie, S. G. Phys. Rev. B 2000, 62, 4927–4944. (35) Onida, G.; Reining, L.; Rubio, A. Rev. Mod. Phys. 2002, 74, 601–659. (36) Soavi, G.; Dal Conte, S.; Manzoni, C.; Viola, D.; Narita, A.; Hu, Y.; Feng, X.; Hohenester, U.; Molinari, E.; Prezzi, D.; M¨ ullen, K.; Cerullo, G. Nature Communications 2016, 7, 11010. (37) Perdew, J. P.; Zunger, A. Phys. Rev. B 1981, 23, 5048–5079. (38) Hamann, D. R. Phys. Rev. B 1989, 40, 2980–2987.

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(39) Kleinman, L.; Bylander, D. M. Phys. Rev. Lett. 1982, 48, 1425–1428. (40) Nakada, K.; Fujita, M.; Dresselhaus, G.; Dresselhaus, M. S. Physical Review B 1996, 54, 17954–17961. (41) Strinati, G. Phys. Rev. Lett. 1982, 49, 1519–1522. (42) Strinati, G. Phys. Rev. B 1984, 29, 5718–5726. (43) Haydock, R.; Heine, V.; Kelly, M. J. Journal of Physics C: Solid State Physics 1972, 5, 2845. (44) Hernandez, V.; Roman, J. E.; Vidal, V. ACM Trans. Math. Software 2005, 31, 351–362. (45) Garcia-Lastra, J. M.; Thygesen, K. S. Phys. Rev. Lett. 2011, 106, 187402. (46) Choi, J.-H.; Cui, P.; Lan, H.; Zhang, Z. Physical Review Letters 2015, 115, 066403. (47) Olsen, T.; Latini, S.; Rasmussen, F.; Thygesen, K. S. Physical Review Letters 2016, 116, 056401.

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