Graphene under uniaxial inhomogeneous strain and an external electric field: Landau levels, electronic, magnetic and opt
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THE EUROPEAN PHYSICAL JOURNAL B
Regular Article
Graphene under uniaxial inhomogeneous strain and an external electric field: Landau levels, electronic, magnetic and optical properties Dai-Nam Le 1,2,a , Van-Hoang Le 3 , and Pinaki Roy 1,2,b 1
2 3
Atomic Molecular and Optical Physics Research Group, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam Department of Physics, Ho Chi Minh City University of Education, 280 An Duong Vuong St., Dist. 5, Ho Chi Minh City, Vietnam Received 30 April 2020 / Received in final form 27 June 2020 Published online 24 August 2020 c EDP Sciences / Societ`
a Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 2020 Abstract. We investigate graphene under an inhomogeneous uniaxial strain and an in-plane electric field. We examine in detail the effect of strain and the electric field on relativistic Landau levels, Hall conductivity, de Haas-van Alphen oscillation and optical conductivity. Using Lorentz transformation in combination with supersymmetric quantum mechanics, we examine three different structures of Landau levels induced by three different profiles of inhomogeneous uniaxial strain and external electric fields. It is shown that straininduced pseudomagnetic field forms Landau levels while electric field opposes formation of these levels. Besides the collapse of strain induced Landau levels, the influences of electric field on the quantization of strain dependent valley Hall conductivity, de Haas-van Alphen quantum oscillation of magnetization as well as optical conductivity have been investigated.
1 Introduction Graphene, during the first two decades of 21st century, has been one of the hotspots in the condensed matter as well as high energy physics communities because of the special nature of its charge carriers. Due to the honeycomb lattice structure of graphene monolayer, its 2pz -orbital electrons mimic massless fermions obeying the Dirac-Weyl equation (instead of Schr¨ odinger equation) with the velocity of light c replaced by the Fermi velocity vF ≈ c/300 [1–3]. One of the challenges in this field is related to confinement or controlling the motion of electrons in graphene or other 2D materials. Such confinement is possible, for example, by using various types of magnetic fields. In the case of magnetic confinement relativistic √Landau levels are formed with energy proportional to B instead of B as in conventional Landau levels [4]. There have been many investigations regarding the magnetic confinement of Dirac electron [5–11], quantum Hall effect [3,12–15] or de Haas-van Alphen (dHvA) oscillation [16–18]. An interesting phenomena takes place when in addition to a magnetic field, graphene is subjected to an electric field perpendicular to the magnetic field. In this case the a b
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electric field opposes the formation of Landau levels and beyond a critical electric
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