About the time evolution of coherent electron states in monolayers of boron allotropes


  • Erik Díaz-Bautista Instituto Politécnico Nacional, Unidad Profesional Interdisciplinaria de Ingeniería Campus Hidalgo, Departamento de Formación Básica Disciplinaria, Ciudad del Conocimiento y la Cultura, Carretera Pachuca-Actopan km 1+500, San Agustín Tlaxiaca, 42162 Hidalgo, México




Tilted Dirac cones, anisotropic Dirac materials, borophene, coherent states, Wigner function


In this paper, we theoretically analyze the massless Dirac fermion dynamics in twodimensional monolayers of boron allotropes, 8B and 2BH − pmmn borophene, interacting with external electric and magnetic fields. We study the effect of the Dirac cone tilt in these materials, which is known as valley index, through the time evolution of probability density of coherent electron states as well as their phase-space representation obtained via the Wigner function. Our results show that the time evolution of the coherent electron states in these materials is valley dependent, which is reinforced in the presence of external electric fields.


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E. Schrödinger. Der stetige Übergang von der Mikrozur Makromechanik. Naturwissenschaften 14(28):664–666, 1926. https://doi.org/10.1007/BF01507634.

R. J. Glauber. Coherent and incoherent states of the radiation field. Physical Review 131:2766–2788, 1963. https://doi.org/10.1103/PhysRev.131.2766.

J. Weinbub, D. K. Ferry. Recent advances in Wigner function approaches. Applied Physics Reviews 5(4):041104, 2018. https://doi.org/10.1063/1.5046663.

C. Gerry, P. Knight, C. C. Gerry. Introductory Quantum Optics. Cambridge University Press, 2010. ISBN 052152735X.

E. Wigner. On the quantum correction for thermodynamic equilibrium. Physical Review 40(5):749–759, 1932. https://doi.org/10.1103/physrev.40.749.

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner. Distribution functions in physics: Fundamentals. Physics Reports 106(3):121–167, 1984. https://doi.org/10.1016/0370-1573(84)90160-1.

A. Kenfack, K. Zyczkowski. Negativity of the Wigner function as an indicator of non-classicality. Journal of Optics B: Quantum and Semiclassical Optics 6(10):396–404, 2004. https://doi.org/10.1088/1464-4266/6/10/003.

D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani. Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum. Physical Review Letters 70(9):1244–1247, 1993. https://doi.org/10.1103/physrevlett.70.1244.

C. Baune, J. Fiurášek, R. Schnabel. Negative Wigner function at telecommunication wavelength from homodyne detection. Physical Review A 95(6):061802, 2017. https://doi.org/10.1103/physreva.95.061802.

H.-W. Lee. Theory and application of the quantum phase-space distribution functions. Physics Reports 259(3):147–211, 1995. https://doi.org/10.1016/0370-1573(95)00007-4.

W. B. Case. Wigner functions and Weyl transforms for pedestrians. American Journal of Physics 76(10):937–946, 2008. https://doi.org/10.1119/1.2957889.

R. P. Rundle, P. W. Mills, T. Tilma, et al. Simple procedure for phase-space measurement and entanglement validation. Physical Review A 96(2):022117, 2017. https://doi.org/10.1103/physreva.96.022117.

M. V. Berry. Semi-classical mechanics in phase space: A study of Wigner’s function. Philosophical Transactions of the Royal Society of London Series A, Mathematical and Physical Sciences 287(1343):237–271, 1977. https://doi.org/10.1098/rsta.1977.0145.

C. Jacoboni, P. Bordone. The Wigner-function approach to non-equilibrium electron transport. Reports on Progress in Physics 67(7):1033, 2004. https://doi.org/10.1088/0034-4885/67/7/r01.

C. Bäuerle, D. C. Glattli, T. Meunier, et al. Coherent control of single electrons: a review of current progress. Reports on Progress in Physics 81(5):056503, 2018. https://doi.org/10.1088/1361-6633/aaa98a.

E. Colomés, Z. Zhan, X. Oriols. Comparing Wigner, Husimi and Bohmian distributions: which one is a true probability distribution in phase space? Journal of Computational Electronics 14(4):894–906, 2015. https://doi.org/10.1007/s10825-015-0737-6.

D. J. Mason, M. F. Borunda, E. J. Heller. Revealing the flux: Using processed Husimi maps to visualize dynamics of bound systems and mesoscopic transport. Physical Review B 91(16):165405, 2015. https://doi.org/10.1103/physrevb.91.165405.

C. M. Carmesin, P. Kling, E. Giese, et al. Quantum and classical phase-space dynamics of a free-electron laser. Physical Review Research 2(2):023027, 2020. https://doi.org/10.1103/physrevresearch.2.023027.

O. Morandi, F. Schürrer. Wigner model for quantum transport in graphene. Journal of Physics A: Mathematical and Theoretical 44(26):265301, 2011. https://doi.org/10.1088/1751-8113/44/26/265301.

E. Díaz-Bautista, Y. Betancur-Ocampo. Phase-space representation of Landau and electron coherent states for uniaxially strained graphene. Physical Review B 101(12):125402, 2020. https://doi.org/10.1103/physrevb.101.125402.

E. Díaz-Bautista, Y. Concha-Sánchez, A. Raya. Barut–Girardello coherent states for anisotropic 2D-Dirac materials. Journal of Physics: Condensed Matter 31(43):435702, 2019. https://doi.org/10.1088/1361-648x/ab2d18.

E. Díaz-Bautista, D. J. Fernández. Graphene coherent states. The European Physical Journal Plus 132(11):499, 2017. https://doi.org/10.1140/epjp/i2017-11794-y.

D. J. Mason, M. F. Borunda, E. J. Heller. Semiclassical deconstruction of quantum states in graphene. Physical Review B 88(16):165421, 2013. https://doi.org/10.1103/physrevb.88.165421.

G. J. Iafrate, V. N. Sokolov, J. B. Krieger. Quantum transport and the Wigner distribution function for Bloch electrons in spatially homogeneous electric and magnetic fields. Physical Review B 96(14):144303, 2017. https://doi.org/10.1103/physrevb.96.144303.

P. Ghosh, P. Roy. Quasi coherent state of the Dirac oscillator. Journal of Modern Optics 68(1):56–62, 2021. https://doi.org/10.1080/09500340.2021.1876261.

D. K. Ferry, I. Welland. Relativistic Wigner functions in transition metal dichalcogenides. Journal of Computational Electronics 17(1):110–117, 2017. https://doi.org/10.1007/s10825-017-1094-4.

E. Díaz-Bautista, M. Oliva-Leyva. Coherent states for dispersive pseudo-Landau-levels in strained honeycomb lattices. The European Physical Journal Plus 136(7):765, 2021. https://doi.org/10.1140/epjp/s13360-021-01753-w.

J. R. Schaibley, H. Yu, G. Clark, et al. Valleytronics in 2D materials. Nature Reviews Materials 1(11):16055, 2016. https://doi.org/10.1038/natrevmats.2016.55.

A. Kundu, H. A. Fertig, B. Seradjeh. Floquet-engineered valleytronics in Dirac systems. Physical Review Letters 116:016802, 2016. https://doi.org/10.1103/PhysRevLett.116.016802.

Y. S. Ang, S. A. Yang, C. Zhang, et al. Valleytronics in merging Dirac cones: All-electric-controlled valley filter, valve, and universal reversible logic gate. Physical Review B 96:245410, 2017. https://doi.org/10.1103/PhysRevB.96.245410.

A. Lopez-Bezanilla, P. B. Littlewood. Electronic properties of 8−Pmmn borophene. Physical Review B 93(24):241405, 2016. https://doi.org/10.1103/physrevb.93.241405.

A. D. Zabolotskiy, Y. E. Lozovik. Strain-induced pseudomagnetic field in the Dirac semimetal borophene. Physical Review B 94(16):165403, 2016. https://doi.org/10.1103/physrevb.94.165403.

SK Firoz Islam, A. M. Jayannavar. Signature of tilted Dirac cones in weiss oscillations of 8 − Pmmn borophene. Physical Review B 96(23):235405, 2017. https://doi.org/10.1103/physrevb.96.235405.

SK Firoz Islam. Magnetotransport properties of 8-Pmmn borophene: effects of Hall field and strain. Journal of Physics: Condensed Matter 30(27):275301, 2018. https://doi.org/10.1088/1361-648x/aac8b3.

Y. Betancur-Ocampo, E. Díaz-Bautista, T. Stegmann. Valley-dependent time evolution of coherent electron states in tilted anisotropic Dirac materials. Physical Review B 105(4):045401, 2022. https://doi.org/10.1103/PhysRevB.105.045401.

M. Assili, S. Haddad, W. Kang. Electric field-induced valley degeneracy lifting in uniaxial strained graphene: Evidence from magnetophonon resonance. Physical Review B 91(11):115422, 2015. https://doi.org/10.1103/physrevb.91.115422.

D. Sabsovich, T. Meng, D. I. Pikulin, et al. Pseudo field effects in type II Weyl semimetals: new probes for over tilted cones. Journal of Physics: Condensed Matter 32(48):484002, 2020. https://doi.org/10.1088/1361-648x/abaa7e.

K. Das, A. Agarwal. Linear magnetochiral transport in tilted type-I and type-II Weyl semimetals. Physical Review B 99(8):085405, 2019. https://doi.org/10.1103/physrevb.99.085405.

A. Menon, B. Basu. Anomalous Hall transport in tilted multi-Weyl semimetals. Journal of Physics: Condensed Matter 33(4):045602, 2020. https://doi.org/10.1088/1361-648x/abb9b8.

P. P. Ferreira, A. L. R. Manesco, T. T. Dorini, et al. Strain engineering the topological type-II Dirac semimetal NiTe2. Physical Review B 103(12):125134, 2021. https://doi.org/10.1103/physrevb.103.125134.

J. Sári, M. O. Goerbig, C. Tőke. Magneto-optics of quasirelativistic electrons in graphene with an inplane electric field and in tilted Dirac cones in α-(BEDT TTF)2I3. Physical Review B 92(3):035306, 2015. https://doi.org/10.1103/physrevb.92.035306.

M. O. Goerbig, J.-N. Fuchs, G. Montambaux, F. Piéchon. Tilted anisotropic Dirac cones in quinoid-type graphene and α-(BEDT-TTF)2I3. Physical Review B 78(4):045415, 2008. https://doi.org/10.1103/physrevb.78.045415.

M. O. Goerbig, J.-N. Fuchs, G. Montambaux, F. Piéchon. Electric-field–induced lifting of the valley degeneracy in α-(BEDT-TTF)2I3 Dirac-like Landau levels. EPL Europhysics Letters 85(5):57005, 2009. https://doi.org/10.1209/0295-5075/85/57005.

T. Morinari, T. Himura, T. Tohyama. Possible verification of tilted anisotropic Dirac cone in α-(BEDT-TTF)2I3 using interlayer magnetoresistance. Journal of the Physical Society of Japan 78(2):023704, 2009. https://doi.org/10.1143/jpsj.78.023704.

X.-F. Zhou, X. Dong, A. R. Oganov, et al. Semimetallic two-dimensional boron allotrope with massless Dirac fermions. Physical Review Letters 112(8):085502, 2014. https://doi.org/10.1103/physrevlett.112.085502.

A. J. Mannix, X.-F. Zhou, B. Kiraly, et al. Synthesis of borophenes: Anisotropic, two-dimensional boron polymorphs. Science 350(6267):1513–1516, 2015. https://doi.org/10.1126/science.aad1080.

B. Feng, J. Zhang, Q. Zhong, et al. Experimental realization of two-dimensional boron sheets. Nature Chemistry 8(6):563–568, 2016. https://doi.org/10.1038/nchem.2491.

W. Li, L. Kong, C. Chen, et al. Experimental realization of honeycomb borophene. Science Bulletin 63(5):282–286, 2018. https://doi.org/10.1016/j.scib.2018.02.006.

Z.-Q. Wang, T.-Y. Lü, H.-Q. Wang, et al. Review of borophene and its potential applications. Frontiers of Physics 14(3):33403, 2019. https://doi.org/10.1007/s11467-019-0884-5.

S.-H. Zhang, W. Yang. Oblique Klein tunneling in 8-Pmmn borophene p − n junctions. Physical Review B 97(23):235440, 2018. https://doi.org/10.1103/physrevb.97.235440.

S.-H. Zhang, W. Yang. Anomalous caustics and Veselago focusing in 8-pmmn borophene p–n junctions with arbitrary junction directions. New Journal of Physics 21(10):103052, 2019. https://doi.org/10.1088/1367-2630/ab4d8f.

N. Tajima, K. Kajita. Experimental study of organic zero-gap conductor α-(BEDT-TTF)2I3. Science and Technology of Advanced Materials 10(2):024308, 2009. https://doi.org/10.1088/1468-6996/10/2/024308.

N. P. Armitage, E. J. Mele, A. Vishwanath. Weyl and Dirac semimetals in three-dimensional solids. Reviews of Modern Physics 90(1):015001, 2018. https://doi.org/10.1103/revmodphys.90.015001.

Y. Betancur-Ocampo, F. Leyvraz, T. Stegmann. Electron optics in phosphorene pn junctions: Negative reflection and anti-super-Klein tunneling. Nano Lett 19(11):7760–7769, 2019. https://doi.org/10.1021/acs.nanolett.9b02720.

Y. Betancur-Ocampo, E. Paredes-Rocha, T. Stegmann. Phosphorene pnp junctions as perfect electron waveguides. Journal of Applied Physics 128(11):114303, 2020. https://doi.org/10.1063/5.0019215.

N. Levy, S. A. Burke, K. L. Meaker, et al. Straininduced pseudo-magnetic fields greater than 300 tesla in graphene nanobubbles. Science 329(5991):544–547, 2010. https://doi.org/10.1126/science.1191700. 11

F. Guinea, M. I. Katsnelson, A. K. Geim. Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering. Nature Physics 6(1):30–33, 2009. https://doi.org/10.1038/nphys1420.

S. H. R. Sena, J. M. Pereira Jr, G. A. Farias, et al. The electronic properties of graphene and graphene ribbons under simple shear strain. Journal of Physics: Condensed Matter 24(37):375301, 2012. https://doi.org/10.1088/0953-8984/24/37/375301.

P. Ghosh, P. Roy. Collapse of Landau levels in graphene under uniaxial strain. Materials Research Express 6(12):125603, 2019. https://doi.org/10.1088/2053-1591/ab52ad.

V. M. Pereira, A. H. Castro Neto. Strain engineering of graphene’s electronic structure. Physical Review Letters 103(4):046801, 2009. https://doi.org/10.1103/physrevlett.103.046801.

V. M. Pereira, A. H. Castro Neto, N. M. R. Peres. Tight-binding approach to uniaxial strain in graphene. Physical Review B 80(4):045401, 2009. https://doi.org/10.1103/physrevb.80.045401.

G. Cocco, E. Cadelano, L. Colombo. Gap opening in graphene by shear strain. Physical Review B 81(24):241412, 2010. https://doi.org/10.1103/physrevb.81.241412.

F. M. D. Pellegrino, G. G. N. Angilella, R. Pucci. Strain effect on the optical conductivity of graphene. Physical Review B 81(3):035411, 2010. https://doi.org/10.1103/physrevb.81.035411.

H. Rostami, R. Asgari. Electronic ground-state properties of strained graphene. Physical Review B 86(15):155435, 2012. https://doi.org/10.1103/physrevb.86.155435.

S. Barraza-Lopez, A. A. Pacheco Sanjuan, Z. Wang, M. Vanević. Strain-engineering of graphene’s electronic structure beyond continuum elasticity. Solid State Communications 166:70–75, 2013. https://doi.org/10.1016/j.ssc.2013.05.002.

G. G. Naumis, S. Barraza-Lopez, M. Oliva-Leyva, H. Terrones. Electronic and optical properties of strained graphene and other strained 2D materials: a review. Reports on Progress in Physics 80(9):096501, 2017. https://doi.org/10.1088/1361-6633/aa74ef.

D.-N. Le, V.-H. Le, P. Roy. Graphene under uniaxial inhomogeneous strain and an external electric field: Landau levels, electronic, magnetic and optical properties. The European Physical Journal B 93(8):158, 2020. https://doi.org/10.1140/epjb/e2020-10222-3.

S. M. Cunha, D. R. da Costa, L. C. Felix, et al. Electronic and transport properties of anisotropic semiconductor quantum wires. Physical Review B 102(4):045427, 2020. https://doi.org/10.1103/physrevb.102.045427.

T. Stegmann, N. Szpak. Current flow paths in deformed graphene: from quantum transport to classical trajectories in curved space. New Journal of Physics 18(5):053016, 2016. https://doi.org/10.1088/1367-2630/18/5/053016.

T. Stegmann, N. Szpak. Current splitting and valley polarization in elastically deformed graphene. 2D Materials 6(1):015024, 2019. https://doi.org/10.1088/2053-1583/aaea8d.

Y. Betancur-Ocampo. Partial positive refraction in asymmetric Veselago lenses of uniaxially strained graphene. Physical Review B 98(20):205421, 2018. https://doi.org/10.1103/physrevb.98.205421.

Y. Betancur-Ocampo, P. Majari, D. Espitia, et al. Anomalous floquet tunneling in uniaxially strained graphene. Physical Review B 103(15):155433, 2021. https://doi.org/10.1103/physrevb.103.155433.

T. Cheng, H. Lang, Z. Li, et al. Anisotropic carrier mobility in two-dimensional materials with tilted Dirac cones: theory and application. Physical Chemistry Chemical Physics 19:23942–23950, 2017. https://doi.org/10.1039/C7CP03736H.

B. Feng, J. Zhang, R.-Y. Liu, et al. Direct evidence of metallic bands in a monolayer boron sheet. Physical Review B 94:041408, 2016. https://doi.org/10.1103/PhysRevB.94.041408.

Y. Zhao, X. Li, J. Liu, et al. A new anisotropic Dirac cone material: A B2S honeycomb monolayer. The Journal of Physical Chemistry Letters 9(7):1815–1820, 2018. https://doi.org/10.1021/acs.jpclett.8b00616.

D. Ferraro, B. Roussel, C. Cabart, et al. Real-time decoherence of Landau and Levitov quasiparticles in quantum hall edge channels. Physical Review Letters 113(16):166403, 2014. https://doi.org/10.1103/physrevlett.113.166403.

T. Jullien, P. Roulleau, B. Roche, et al. Quantum tomography of an electron. Nature 514(7524):603–607, 2014. https://doi.org/10.1038/nature13821.

J. Oertel. Solutions of the Dirac equation in spacetime-dependent electric fields. Master’s thesis, University of Duisburg-Essen, Freiberg, Germany, 2014.

M. Castillo-Celeita, E. Díaz-Bautista, M. Oliva-Leyva. Coherent states for graphene under the interaction of crossed electric and magnetic fields. Annals of Physics 421:168287, 2020. https://doi.org/10.1016/j.aop.2020.168287.

V. Lukose, R. Shankar, G. Baskaran. Novel electric field effects on Landau levels in graphene. Physical Review Letters 98(11):116802, 2007. https://doi.org/10.1103/physrevlett.98.116802.

N. Gu, M. Rudner, A. Young, et al. Collapse of Landau levels in gated graphene structures. Physical Review Letters 106(6):066601, 2011. https://doi.org/10.1103/physrevlett.106.066601.

J. M. Ziman. Principles of the Theory of Solids. Cambridge University Press, 2nd edn., 1972. https://doi.org/10.1017/CBO9781139644075.

T. Huang, R. Chen, T. Ma, et al. Electronic Bloch oscillation in a pristine monolayer graphene. Physics Letters A 382(42-43):3086–3089, 2018. https://doi.org/10.1016/j.physleta.2018.07.035.

M. Kai, W. Jian-Hua, Y. Yi. Wigner function for the Dirac oscillator in spinor space. Chinese Physics C 35(1):11–15, 2011. https://doi.org/10.1088/1674-1137/35/1/003.

D. J. Fernández, D. I. Martínez-Moreno. Bilayer graphene coherent states. The European Physical Journal Plus volume 135(9):739, 2020. https://doi.org/10.1140/epjp/s13360-020-00746-5.




How to Cite

Díaz-Bautista, E. (2022). About the time evolution of coherent electron states in monolayers of boron allotropes. Acta Polytechnica, 62(1), 38–49. https://doi.org/10.14311/AP.2022.62.0038



Analytic and Algebraic Methods in Physics