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Abstract
ABSTRAK
Perhitungan rapat keadaan (density of states/DOS) material graphene bilayer model ikatan kuat atau Tight-Binding (TB) melalui penyelesaian persamaan Schrӧdinger Gayut Waktu (PSGW) telah dilakukan dengan menggunakan metode rambatan waktu Trotter-Suzuki. Fungsi gelombang awal dibangkitkan dengan menggunakan bilangan acak kompleks sebagai fungsi superposisi dari basis tiap-tiap site atom berstruktur heksagonal untuk memperoleh fungsi korelasi. Nilai rapat keadaan dihitung dengan mentransformasi Fourier fungsi korelasi yang sudah didapatkan. Penelitian ini terfokus pada perhitungan rapat keadaan dari graphene bilayer model AA-Stacking. Hasil penelitian menunjukkan bahwa metode rambatan waktu Trotter-Suzuki sangat efisien digunakan untuk menghitung rapat keadaan dari sistem atom yang besar. Model TB dapat menjelaskan rapat keadaan material graphene bilayer dengan variasi ukuran sistem dan hopping parameter. Banyaknya jumlah atom (site) yang ditinjau mempengaruhi keteraturan hasil perhitungan rapat keadaan. Semakin banyak atom yang ditinjau, semakin teratur perhitungan rapat keadaan yang dihasilkan. Variasi nilai hopping interlayer mengakibatkan munculnya singularitas van Hove di level tenaga Fermi untuk t1=t2 dan bertambahnya sifat metallicity akibat bertambahnya nilai DOS pada level tenaga Fermi untuk t1<t2.
Kata kunci: graphene bilayer, rapat keadaan, dan metode Trotter-Suzuki,
ABSTRACT
The density of states (DOS) of the bilayer graphene material with Tight-Binding (TB) model has been calculated by solving the Time Dependent Schrӧdinger Equation (TDSE) with Trotter-Suzuki formula. The generation of the initial wave function is carried out by using a random complex number as a superposition function of the base of each hexagonal structure atomic site to obtain the correlation function. The results of the calculation of the correlation function are then Fourier transformed to calculate the density of the state. This research focuses on calculating the state density of the graphene bilayer AA-Stacking. The results showed that the Trotter-Suzuki method was very efficient in calculating the state density of large atomic systems. The TB model can explain the density of states of the bilayer graphene material with variations in system size and hopping parameters. The number of atoms (sites) reviewed affects the regularity of the DOS calculation results. The more atoms that are reviewed, the more orderly the DOS calculations will be generated. The variation of the hopping interlayer value would be appearing van Hove singularity at the Fermi level for t1=t2 and increase in the metallicity properties with the increase in the DOS value at the Fermi level for t1<t2.
Keywords: bilayer graphene, density of states, Trotter-Suzuki formula
Keywords
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References
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References
Zhu YW, Murali S, Cai WW, Li XS, Suk JW, Potts JR, et al. Graphene and graphene oxide: synthesis, properties and applications. Adv Mater. 2010 Jan 1;20:1–19.
Castro Neto AH, Guinea F, Peres NMR, Novoselov KS, Geim AK. The electronic properties of graphene. Rev Mod Phys [Internet]. 2009 Jan 14;81(1):109–62. Available from: https://link.aps.org/doi/10.1103/RevModPhys.81.109
Bonaccorso F, Sun Z, Hasan T, Ferrari AC. Graphene photonics and optoelectronics. Nat Photonics [Internet]. 2010;4(9):611–22. Available from: https://doi.org/10.1038/nphoton.2010.186
McCann E, Koshino M. The electronic properties of bilayer graphene. Reports on Progress in Physics [Internet]. 2013;76(5):056503. Available from: https://dx.doi.org/10.1088/0034-4885/76/5/056503
Rozhkov A V, Sboychakov AO, Rakhmanov AL, Nori F. Electronic properties of graphene-based bilayer systems. Phys Rep [Internet]. 2016;648:1–104. Available from: https://www.sciencedirect.com/science/article/pii/S0370157316301612
Uppstu A. Electronic properties of graphene from tight-binding simulations [Doctoral thesis]. Aalto University; 2014.
Alben R, Blume M, Krakauer H, Schwartz L. Exact results for a three-dimensional alloy with site diagonal disorder: comparison with the coherent potential approximation. Phys Rev B [Internet]. 1975 Nov 15;12(10):4090–4. Available from: https://link.aps.org/doi/10.1103/PhysRevB.12.4090
De Raedt H. Product formula algorithms for solving the time dependent Schrödinger equation. Computer Physics Reports [Internet]. 1987;7(1):1–72. Available from: https://www.sciencedirect.com/science/article/pii/0167797787900025
Yuan S, De Raedt H, Katsnelson MI. Modeling electronic structure and transport properties of graphene with resonant scattering centers. Phys Rev B [Internet]. 2010 Sep 28;82(11):115448. Available from: https://link.aps.org/doi/10.1103/PhysRevB.82.115448
Häfner V, Evers Korreferent F, Busch Prüfexemplar K. Large scale simulation of wave-packet propagation via Krylov subspace methods and application to graphene Simulation von Wellenpaketen mit Krylov-Methoden und Anwendung auf Graphen. 2011.
Dhand I, Sanders BC. Stability of the Trotter–Suzuki decomposition. J Phys A Math Theor [Internet]. 2014;47(26):265206. Available from: https://dx.doi.org/10.1088/1751-8113/47/26/265206
Ariasoca TA, Sholihun, Santoso I. Trotter-Suzuki-time propagation method for calculating the density of states of disordered graphene. Comput Mater Sci. 2019;
Yuan S, De Raedt H, Katsnelson MI. Electronic transport in disordered bilayer and trilayer graphene. Phys Rev B [Internet]. 2010 Dec 6;82(23):235409. Available from: https://link.aps.org/doi/10.1103/PhysRevB.82.235409
Dietl P. Numerical studies of electronic transport through Graphene nanoribbons with disorder. Karlsuhe Institute of Technology. 2009;
De Raedt H, De Raedt B. Applications of the generalized Trotter formula. Phys Rev A (Coll Park) [Internet]. 1983 Dec 1;28(6):3575–80. Available from: https://link.aps.org/doi/10.1103/PhysRevA.28.3575
Suzuki M. Generalized Trotter’s formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems. Commun Math Phys. 1976;51(2):183–90.