Chiral phase transition and quantum revivals in graphene

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Mater. Res. Express 2 (2015) 095602 doi:10.1088/2053-1591/2/9/095602 PAPER Chiral phase transition and quantum revivals in graphene Rfaqat Ali1 and Farhan Saif2 1 Department of Physics, Quaid-i-Azam University, Islamabad, Pakistan 2 Department of Electronics, Quaid-i-Azam University, Islamabad, Pakistan E-mail: [email protected] and [email protected] Keywords: graphene, quantum revivals, chirality Abstract We explain the dynamics of charge carriers in graphene using a two dimensional Dirac oscillator in the presence of an external magnetic field. The energy dispersion relation with linear behavior corresponds to monolayer graphene in a relativistic regime, whereas parabolic behavior appears in the case of bilayer graphene in a non-relativistic regime. We show that in the bilayer graphene model, a magnetic field-dependent energy gap exists, whereas a changing external magnetic field leads to a chiral phase transition. Our model explains the phenomenon of collapse and revivals and chiral phase transition in the presence of a magnetic field in monolayer and bilayer graphene. The displayed collapses and revivals occur due to Zitterbewegung and classical cyclotron motion. 1. Introduction Flexibility of bonding makes carbon as an important material for life and a fundamental stone of organic chemistry. Carbon based materials show a large number of stable structures with an equally large variety of physical properties. Among these systems with only carbon atoms, graphene is a material made up of layers of carbon atoms arranged in a planer hexagonal structure, which is isolated from graphite. Graphite is a three dimensional isotropic form of carbon which has become widely known after the invention of the pencil. Graphite is made up of the stocking of graphene layers, which are weakly coupled by van der Waals forces [1]. Graphene has many interesting features, and, due to its planar structure it has two atoms per unit cell. It has attracted enormous attention due to its unique electronic properties. The Velocity of an electron in the graphene lattice is much higher (108 cm s−1) than that in any other material. The charge carriers in graphene are called quasi-particles due to their effectively mass-less interactions and relativistic nature. In the quasi-particle scheme, the carriers are represented as single particles that scatter from, and are surrounded by, a cloud of other particles, such as phonons [2]. These quasi-particles follow from the formal equivalence of the Schrödinger equation to the relativistic Dirac equation for graphene. In the relativistic domain graphene has a linear dispersion relation and gap-less edge states. These edge states are also known as Dirac points; at these points the conduction and valence bands almost touch each other and form a Dirac cone. In the tight bonding approximation, the Hamiltonian derived at the Dirac points is independent of mass and is written as H v f . q =  s , here, vf is the Fermi velocity, q is the quasi momentum, and σ is the Pauli spin matrix. In order to study the transport of spin- half quasi-particles in graphene, the mass-less Dirac equation is used instead of the Schrödinger equation. Graphene exhibits unconventional two-dimensional electronic properties, resulting from the symmetry of its quasi-particles, which leads to the concepts of pseudo-spin and electronic chirality [4, 5]. In graphene, the direction of propagation of a quasi-particle and the amplitude of its wave function are not independent, so the quasi-particle possesses the property of chirality. This contribution shows the chiral phase transition in graphene due to the external magnetic field. The projection of the pseudo-spin on the direction of the momentum defines the chirality of the quasi-particles in graphene. If the direction of momentum of the quasi- particle and pseudo-spin is parallel, the quasi-particle has right-handed chirality or it it is antiparallel the quasi- particle has left-handed chirality [8, 9]. Within one single Dirac cone, electrons (or equivalently holes) of opposite direction have opposite pseudo-spin, but the same chirality. RECEIVED 1 June 2015 REVISED 31 July 2015 ACCEPTED FOR PUBLICATION 4 August 2015 PUBLISHED 7 September 2015 © 2015 IOP Publishing Ltd.

In this paper we explain a two dimensional Dirac oscillator model that effectively describes the dynamics of a quasi-particle in two dimensional graphene in the presence of a magnetic field. In our analytical work we explain the energy eigenstates of a chiral quasi-particle in graphene. Moreover, we explain the time evolution of a wave packet and the temporal behavior of a wave packet, as it is exposed to an external magnetic field. This leads us to derive analytic results for the semi-classical expressions for the auto-correlation function. Our analytical results are in very good agreement with the numerical results. In section 2, we formulate the Dirac equation in the presence of a magnetic field and linear potential, and provide a discussion on the energy spectrum of quasi- particles in relativistic and non-relativistic regimes. In section 3, we discuss the wave packet dynamics and construction of the auto-correlation function that defines the temporal behavior of the wave packet in graphene. In the evolution of a wave packet, certain collapses and revivals in the spin degree of freedom appear as a consequence of Zitterbewegung [7, 27]. In other words, we obtain the energy spectrum and formulate the wave function, which leads us to derive analytic results for the semi-classical expression for the auto-correlation function. 2. Chiral phase transition using the Dirac oscillator model in graphene The three-dimensional system defined by the Dirac equation in the presence of a linear vector potential is called a Dirac oscillator. The Dirac oscillator model explains the dynamics of quasi-particles in a two dimensional graphene sheet in the presence of a linear vector potential of the form m r i . , wba  which originates due to presence of a magnetic field [3, 10]. The system behaves as a harmonic oscillator with a strong spin–orbit coupling in the non-relativistic limit [6]. Here m is the mass of a Fermion, ω the oscillator frequency, r is the vector distance of the fermium from the origin of the potential. In addition, α and β are the four Dirac matrices. Recently, an experimental setup has been developed to analyze the characteristics of the Dirac oscillator model [12]. The dynamics of a quasi-particle for the system is described by the Dirac equation as t v P eB r eA mv i 1 4 i . 1 j j j z j j z f 1 2 I f ∣ ∣ ( ) ⎜ ⎟ ⎡ ⎣ ⎢ ⎢ ⎛ ⎝ ⎞ ⎠ ⎤ ⎦ ⎥ ⎥  å y s s s y ¶ ¶ ñ = - - + ñ = In order to discuss the chiral properties of electrons, we study the Dirac equation in the presence of weak and strong magnetic fields. 2.1. Dirac equation in the presence of a weak magnetic field The model is based on the understating that the dynamics of quasi-particles in graphene are effectively described by a two-dimensional Dirac oscillator. The Dirac equation in (2+1) dimensions is written as t v P eB r mv i 1 4 i , 2 j j j z j z f 1 2 I f ∣ ∣ ( ) ⎜ ⎟ ⎡ ⎣ ⎢ ⎢ ⎛ ⎝ ⎞ ⎠ ⎤ ⎦ ⎥ ⎥  å y s s s y ¶ ¶ ñ = - + ñ = where, BI is the internal magnetic field, js and zs are Pauli matrices. The internal uniform effective magnetic field arises due the motion of the electrons relative to the planar hexagonal arrangement of carbon atoms. It is analogous to the internal magnetic field of a hydrogen atom, which occurs from the relative motion of the electron relative to the proton, which determines the spin–orbit coupling of the hydrogen atom. The value of the internal magnetic field for a hydrogen atom is approximately 0.3 T. The estimated value for graphene is less than 1 T. We express the dynamics of charge carriers in the presence of a perpendicular magnetic field by means of a Hamiltonian as H v P eB x P eB y i i , ml x x z y y z f 1 4 I 1 4 I ( ) ( ) ⎡ ⎣ ⎤ ⎦ ts s s s = - + - which leads to single particle energy eigenstates around one of the two equivalent corners of the first Brillouin zone [13]. Here, the dimensionless parameter τ has two possible values 1, which determine the corners of the Brillouin zone that is considered [14]. For monolayer graphene, we consider that the quasi-particle is mass-less and at the corner 1 t = + . The Dirac spinor ∣yñis described by two components such that , T 1 2 ∣ (∣ ∣ ) y y y ñ = ñ ñ , where 1 ∣y ñand 2 ∣y ñhave spin- up and spin-down, respectively, with positive and negative energies. By using euation (2), we obtain a set of coupled equations, i.e. E mv v P w v x P w v y i i i , 3 x y f 2 1 f 2 f 2 2 f 2 2 ( ) ( ) ⎡ ⎣ ⎢⎢ ⎛ ⎝⎜ ⎞ ⎠⎟ ⎛ ⎝⎜ ⎞ ⎠⎟ ⎤ ⎦ ⎥⎥   y y - = + - + E mv v P w v x P w v y i i i . 4 x y f 2 1 f 2 f 2 2 f 2 2 ( ) ( ) ⎡ ⎣ ⎢⎢ ⎛ ⎝⎜ ⎞ ⎠⎟ ⎛ ⎝⎜ ⎞ ⎠⎟ ⎤ ⎦ ⎥⎥   y y + = - + - Here, w v l2 f B = is the cyclotron frequency and l eB B I  = represents the magnetic length scale in the problem. In order to find the energy dispersion relation, we express the above equations in terms of annihilation and creation 2 Mater. Res. Express 2 (2015) 095602 R Ali and F Saif.

operators and define as a r pi j j y 1 2 1( )  = + x x a r p i j j j 1 2 1( ) † † †  = - x x . Here, aj and aj † are the annihilation and creation operators of the harmonic oscillator and v w x = f represents the ground state oscillator width. It is convenient to introduce chiral creation and annihilation operators in terms of aj and aj †. The right-handed chiral annihilation and creation operators are expressed as a a ai x y r 1 2 ( ) = - and a a ai , x y r 1 2 ( ) † † † = + respectively, and the left handed chiral annihilation and creation phase operators as a a ai x y l 1 2 ( ) = + and a a ai x y l 1 2 ( ) † † † = - . The orbital angular momentum Lz is expressed in terms of the chiral operators as L a a a a , z r r l l ( ) † † =  - which leads to a physical description of ar † and al †; here, ar † and al † are the operators that contribute, respectively, a right-handed and left-handed angular momentum [10]. Equations (3) and (4) are expressed in terms of chiral operators as wa E mv 2 i , 5 1 l f 2 2 ( ) ( ) †  y y = - wa E mv 2 i . 6 2 l f 2 1 ( ) ( )  y y = - + Here, an important point is that only left-handed chiral operators appear in the coupled equations. In order to check the stability of the system, the commutation relation is established, which obeys the canonical commutation relation a a , 1 l l [ ] † = . The number operator is simply defined as n a a l l † l = . On substitution of equations (5) and (6) in equation (2), the governing Hamiltonian controlling the dynamics of the left handed quasi-particle is expressed as H g a g a mv z l l l l l f 2 ( ) † *  s s s = + + - . Here, s+ and s- represent the spin raising and lowering operators, respectively, and g vi eB l f I  = . Equations (5) and (6) lead to the energy equation as E w n m v 1 2 1 2 , 7 n 2 2 l 2 f 4 l ( ) ⎜ ⎟ ⎛ ⎝ ⎞ ⎠  =  + +  where n l = 0, 1, 2, 3 ... is the eigenvalue of the number operator, which defines the nth Landau level. The energy spectrum in momentum coordinates with k eB 2 I =  is E k v n m v 1 2 1 2 . 8 n 2 2 f 2 l 2 f 4 l ( ) ⎜ ⎟ ⎛ ⎝ ⎞ ⎠  =  + +  When the effective mass is zero the energy spectrum is written as E kv n 1 2 1 2 , 9 n f l l ( ) ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ =  +  which relates the energy spectrum of graphene to the relativistic energy dispersion relation. It depends linearly on the momentum k,  which is consistent with the well known result for quasi-particles in monolayer graphene [13]. On the other hand, when the effective mass of the quasi-particle is not zero m 0 ( ) ¹ , the system is in the non-relativistic regime. Here equation (9) becomes E k v n m v mv 1 2 1 2 10 n 2 2 f 2 l 2 f 4 f 2 l ( ) ⎜ ⎟ ⎛ ⎝ ⎞ ⎠  =  + + -  In bilayer graphene, it is valid to write mv k f   and the non-relativistic dispersion relation is written as E k n 2m , n 2 2 l 1 2 1 2 l ( )  = + +  which represents the non-relativistic energy dispersion relation depending parabolically on the momentum coordinate for a quasi-particle [15]. In the above discussion, it is clear that, the quasi-particle is left-handed in the presence of a weak external magnetic field.If the external magnetic field is strong then the pseudo spin changes its direction which leads to another phase, namely the right handed chiral phase. Now we discuss the case when the external magnetic field is strong and obtain the energy expression for the right handed chiral electron. 2.2. Dirac equation in the presence of a strong magnetic field We consider equation (1) to discuss the chirality of an electron in the presence of a strong external magnetic field. The dynamics of a quasi particle in the presence of an external vector potential A, is effectively described by the Dirac equation in (2+1) dimensions as 3 Mater. Res. Express 2 (2015) 095602 R Ali and F Saif.

t v P eA mv i 11 j j j j z f 1 2 f 2 ( ) ∣ ∣ ( ) ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥  å y s s y ¶ ¶ ñ = - + ñ = Here, we consider the the corner of the first Brillouin zone is defined by 1 t = . In order to analyze the dynamics of the quasi-particle in the presence of an external magnetic field, we assume the external potential is stronger than the linear potential. Here, A A A x, y ( ) = is the external vector potential and e is the charge of the quasi-particle. The external vector potential A r A x y 0 , , ( ) ( ) = is connected to the magnetic field B r A r ( ) ( ) =  ´ and contributes to the Hamiltonian as A A A r i x y ( ) = - . By using the Dirac spin or , T , 1 2 ∣ (∣ ∣ ) y y y ñ = ñ ñ equation (11) simplifies in the form of two coupled equations, E mv v P w x v P w y v i i i , 12 x y f 2 1 f 2 f 2 2 f 2 2 ( ) ˜ ˜ ( ) ⎡ ⎣ ⎢ ⎢ ⎛ ⎝⎜ ⎞ ⎠⎟ ⎛ ⎝ ⎜⎜ ⎤ ⎦ ⎥ ⎥   y y - = - - - E mv v P w x v P w x v i i i . 13 x y f 2 2 f 2 f 2 2 f 2 1 ( ) ˜ ˜ ( ) ⎡ ⎣ ⎢⎢ ⎛ ⎝⎜ ⎞ ⎠⎟ ⎛ ⎝⎜ ⎞ ⎠⎟ ⎤ ⎦ ⎥⎥   y y + = + + + Using the definition of the creation and annihilation operators, a set of coupled equations are obtained. Furthermore, the definition of the right-handed chiral annihilation a a ai x y r 1 2 ( ) = - and the creation a a ai x y r 1 2 ( ) † † † = + operators [11], that leads to a set of coupled equations wa E mv 2 i , 14 1 r f 2 2 ( ) ( )  y y = - - wa E mv 2 i , 15 2 r f 2 1 ( ) ( ) †  y y = + Hence, on solving these coupled equations, it is possible to write the effective Hamiltonian of the system as H g a g a mv z r r r r r f 2 ( ) ˜ ˜ ˜ ˜ † *  s s s = - + + + - with g vi eB r f 2 ˜  = effectively coupling between the charge carrier and the magnetic field. Here, it is clear from the above coupled equations that the quasi-particle shows right-handed chiral properties. The Hamiltonian Hr describes the right-handed chirality of a massive electron. So the energy dispersion relation is expressed in terms of the external magnetic field, as E eB v n m v 2 1 2 1 2 , 16 n f 2 r 2 f 4 r ( ) ⎜ ⎟ ⎛ ⎝ ⎞ ⎠  =  + +  where, n r = 0, 1, 2, 3 ... . For a mass-less quasi-particle E eB v n 2 1 2 1 2 , n f 2 r r ⎜ ⎟ ⎛ ⎝ ⎞ ⎠  =  +  which corresponds to the well known Landau energy level spectrum of monolayer graphene [17–19]. The Landau energy level for bilayer graphene in the non-relativistic limit, when the mass of the quasi-particle is non- zero is expressed as E eB v n m v mv 2 1 2 1 2 . n f 2 r 2 f 4 f 2 r ⎜ ⎟ ⎛ ⎝ ⎞ ⎠  =  + + -  This Landau energy level spectrum, which implies zero energy level for n r = 0, has a field dependent gap which is expressed as E B , nr 0 2 d g d = - + + here, 2mev f 2 d = and 8ev f 2 g = [19]. This section of the article shows that the chiral property of the quasi- particle depends on the strength of the external magnetic field. The strength of the external magnetic field leads to the phase transition from left-handed chiral phase to right-handed chiral phase. For a full description of the problem, equation (1) can be solved simultaneously. The Hamiltonian that gives a complete description of the quasi-particle in graphene in the presence of a magnetic field is written as H H H mv z, l r f 2s = - + where the left-handed Hamiltonian is written as H g a g a l l l l l ( ) † *  s s = + + - and the right-handed Hamiltonian is H g a g a , r r r r r ( ) † *  s s = + + - with s+ and s- representing the spin raising and lowering operators and g w v 2i i eB l f I  = = and g w v 2i i eB r f I  = = ; these two Hamiltonians depend on the strength of the external magnetic field that is applied to the system. 4 Mater. Res. Express 2 (2015) 095602 R Ali and F Saif.

3. Dynamics of electronic wave packet in graphene Here, we discuss the time evolution of the electronic wave packet in graphene that comprises both positive and negative Landau levels. The study of the wave packet in monolayer graphene is discussed in the literature [23–26] in the presence of a magnetic field, where the phenomenon of quantum revival in monolayer graphene is discussed [25, 26, 28]. In this contribution, a dynamic approach is developed to discuss the time evolution of the wave packet in monolayer as well as bilayer graphene. In order to discuss the temporal behavior of the wave packet, a time-correlation function is an effective and intuitive way to describe the dynamics of a quantum system [22, 27]. An auto-correlation function refers to the correlation of a time series with its own past and future values. Revivals in the time evolution of a wave packet in graphene manifest beyond Zitterbewegung and classical cyclotron motion [7, 22]. In order to establish the auto-correlation function for the system, we require the normalized energy eigenstates of the system. Consider the energy eigenvalue of the system when an external magnetic field is dominant as discussed in section 2. In order to find the corresponding eigenstates, we solve equations (5) and (6), which are not linearly independent. The normalized energy eigenstates are E E mv E n E mv E n i 2 1 2 , f 2 r f 2 r ( ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ = -      with n r = 1, 2, 3 ... quantum number. These energy eigenstates are transparently expressed in components of Pauli spinors with ∣c ñ  spin up and ∣c ñ  spin down, such that, E n n i 1 , n r r r a c b c + = - - +   E n n i 1 , n r r r b c a c - = - +   where E mv E 2 n n r f 2 r a = - and E mv E 2 n n r f 2 r b = + are probability amplitudes. In order to establish the ground state wave function of the quasi particle, we need to solve the above expressions. Here the energy states display entanglement between orbital and spin degrees of freedom. The initial state of the wave packet is the standard Gaussian distribution of eigenstates a n 0 1 n n r r r ∣ ( ) ∣ ∣ å y c ñ - ñ ñ  and superposition of positive energy and negative energy components as E E 0 i i n n r r ∣ ( ) ∣ ∣ y b a ñ = + ñ - - ñ[10], therefore, the time evolved state is formed as t a E e E i i e . 17 n n n w t n w t i i n n r r r r r r ∣ ( ) ( ) ⎡⎣ ⎤⎦ å y b a ñ = + - - - We assume that the expansion coefficients are given by a e n n 1 2 n n n r r r 0 2 r ( )   = p - - . By substituting the values of ∣ + E ñand ∣ - E ñequation (17) becomes t a w t mv E w t n eB n v E w t n cos i sin 1 sin . 18 n n n n n n n f 2 r r f 2 r r r r r r r r ∣ ( ) [ ( ) ⎛ ⎝⎜ ⎞ ⎠⎟ ⎤ ⎦⎥  å y c c ñ = + - +   Here, wn En r r =  is the frequency of oscillation between nr ∣ ñand n 1 ∣ r - ñstates. Hence the auto-correlation function A t t 0 , ( ) ( )∣ ( ) y y = á ñ is written as A t a w t mv E w t cos i sin . 19 n n n n n 2 f 2 r r r r r ( ) ( ) ⎡ ⎣⎢ ⎤ ⎦⎥ = å + This expression gives the temporal behavior of the wave packet in graphene. The occurrence of revivals is a purely quantum mechanical phenomenon, which emerges in graphene due to quantum interference of Landau energy levels. The time evolution of wave packets in relativistic and non-relativistic regimes is very complex. There are many types of periodicities that may emerge depending upon the nature of the energy spectrum. The system shows non-linearity in the energy spectrum. In order to find the periodicities in the auto-correlation function subjected to the evolution of time, the energy given in equation (16) is expanded by using a Taylor expansion around n , 0 as the mean quantum number [21], such that, 5 Mater. Res. Express 2 (2015) 095602 R Ali and F Saif.

E E E n n E n n 2 ...... 20 n n n n r 0 r 0 2 r 0 0 0 ( ) ( ) ( ) » + ¢ - +  - with the assumption n n 1 0   . Here, En0¢ and En0 stand for the first and second derivatives of En with respect to n, respectively, at n = n0 . We shall make use of equation (20) to identify several types of periodicities in the temporal development of the wave packet, namely, Zitterbewegung, classical motion and quantum revivals. When the effective mass of the quasi-particle is zero m 0 ( ) = then equation (19) gives the temporal behavior of the wave packet in monolayer graphene (see figure 1), when m ¹ 0 then it gives the behavior of a wave packet in bilayer graphene a is shown in figures 2 and 3. The first term of the energy expression provides simple phase oscillation. The amplitude of these oscillations decreases due to quantum interference of positive and negative energy components [10, 22]. The time period of these oscillations is very small, of the order of femto seconds. This time is known as Zitterbewegung time T , ZB and is expressed as Figure 1. Periodicities that appear in monolayer graphene at short evolution time; at magnetic field B = 1 T, s = 1 and n 0 = 10 , classical period Tcl is 1 ps. (a) Re A ( t ) vs time up to Tcl, which shows that after 40 TZB one classical period occurs; the time between the two arrows shows one TZB. (b) At ∣ ∣vs time for a few classical periods is plotted. During the time in panel (c) the dispersion of the initially localized wave packet grows, leading to a uniform distribution which recovers to a full revival Trev after 80 TCl. Figure 2. Periodicities that appear in bilayer graphene at short evolution time; at magnetic field B = 20 T and n 0 = 10 , classical period is 1.8 ps. (a) Re A ( t ) vs time up to Tcl, which shows that after 2500 phase oscillations one classical period occurs. (b) At ∣ ∣vs time for a few classical periods is plotted. 6 Mater. Res. Express 2 (2015) 095602 R Ali and F Saif.

T E 2 . n ZB 0 p = This is a short time scale effect, however as time increases up to the order of pico seconds then this effect disappears and it cannot be observed experimentally [10]. It behaves like a time dependent phase in a single stationary state. The phase oscillations between n 1 r ∣ -  ñand nr ∣  ñshow close resemblance to Rabi oscillations which occur in the Jaynes–Cummings model of a two level atomic system. Furthermore, when time increases another periodicity emerges with periodic time of the order of pico-seconds which is known as the classical cyclotron time Tcl for the wave packet. The second term of the Taylor expansion gives the exact mapping of this periodic time. This expression is similar to the correspondence principle which is associated with the classical periodicity of cyclotron motion in a bound system [27]. It is expressed on an energy time scale as T eB v n m v ev B 2 2 . cl f 2 0 1 2 1 2 2 f 4 f 2 ( )  p = + +  When a wave packet is allowed to expand it collides with the boundary of the system and bounces back, at time Tcl. At this time the wave packet comes back to its initial position and the auto-correlation function regains its initial value. The key point here is that during this motion the wave packet does not go into a collapsed state, but it produces a very small phase difference. By the repetition of this cycle, the phase difference becomes prominent and the wave packet goes into a collapsed state, which leads to minimum amplitude of the auto-correlation function. With the passage of time the wave packet attains the same initial phase, which was at initial time; this time; is called the revival time Trev. The third term of the Taylor expansion leads to a periodicity at the quantum revival time, which is expressed as T ev B eB v n m v 4 2 1 2 1 2 . rev f 2 2 f 2 0 2 f 4 3 ( ) ⎜ ⎟ ⎛ ⎝ ⎞ ⎠    p = + +  It is associated with the quantum revival time on an energy scale, and shows the importance of the term n n0 ( ) - in the exponent of e t i Enr -  . When time increases as t 0 , ( ) > it is possible to have a long time as compared to the classical time t T , cl  when the wave packet attains the same distribution as it was at the initial time. This time is the revival time for the wave packet, and depends on the external magnetic field. Conclusions In this article, we discussed the Dirac equation under the two conditions to solve the Dirac oscillator model in graphene; first, when the applied external magnetic field B is weaker than the internal magnetic field BI, second, when the external magnetic field is greater than the internal magnetic field. In the first case, we lead to the first approximation that the correction of order O B BI( )approaches zero. The dynamics of the system are described by a left-handed chiral Hamiltonian, which leads to the energy dispersion relation of a left-handed chiral electron. In the second case, corrections of order O B B I( ) are negligible, the right-handed chiral Hamiltonian describes the dynamics of the quasi-particle and the right-handed energy dispersion relation appears. This Figure 3. Quantum revival of wave packet appearing in bilayer graphene at time T rev = 8.8 ns , when magnetic field B = 20 T , and initially wave packet is at n 0 = 10 . It appears after a large number of classical periods. 7 Mater. Res. Express 2 (2015) 095602 R Ali and F Saif.

means that when the magnetic field is weak, the quasi-particle has left-handed chirality and when the magnetic field is strong, it has right-handed chirality. Hence, changing the external magnetic field’s strength, a chiral phase transition appears. In the case of bilayer graphene, a magnetic field dependent energy gap exists at zero mode energy. Furthermore, we have studied the regeneration of a wave packet built as a superposition of eigenstates sharply peaked around some energy level for bilayer graphene in a perpendicular magnetic field. In the wave- packet evolution, three different regeneration times like Zitterbewegung, classical, and revival, corresponding to three different time scales have been shown. This type of periodicity depends on the energy eigenvalue spectrum. Assuming an initial wave packet as a superposition of eigenstates sharply peaked around some level n , 0 this wave packet initially will evolve with a semi-classical periodicity T , cl and then it will spread and delocalize. At later times integer multiples of the time T , rev it will recover its initial shape. These time-dependent effects are very conspicuous in graphene. These phenomena of quantum revivals in graphene occur due to the special structure of the Landau levels, cyclotron motion and interference of positive energy and negative energy components. Acknowledgments We gratefully acknowledge Shahid Iqbal for useful discussions on the phenomena of quantum revivals and fractional revivals. References [1] Castro Neto A H, Guinea F, Peres N M R, Novoselov K S and Geim A K 2009 Rev. Mod. Phys. 81 109 [2] Bostwick A and Otha T 2007 Nat. Phys. 3 36 [3] Moshinsky M and Szczepaniak A 1989 J. Phys. A 22 L817 [4] Brihuega I and Mallet P 2008 Phys. Rev. Lett. 101 206802 [5] Pichler T 2011 Physics 4 79 [6] Strange P 1998 Relativistic Quantum Mechanics (Cambridge, England: Cambridge University Press) [7] Romera E and de los Santos F 2009 Phys. Rev. B 80 165416 [8] Katsnelson M I, Novoselov K S and Geim A K 2006 Nat. Phys. 2 620 [9] Geim A K and Novoselov K S 2007 Nat. Mater. 6 183 [10] Bermudez A, Martin-Delgado M A and Solano E 2007 Phys. Rev. A 76 041801(R) [11] Bermudez A, Martin-Delgado M A and Solano E 2007 Phys. Rev. Lett. 99 123602 [12] Franco-Villafane J A, Sadurni E, Barkhofen S, Kuhl U, Mortessagne F and Selig-man T H 2013 Phys. Rev. Lett. 111 170405 [13] Castro Neto A H, Guinea F, Peres N M R, Novoselov K S and Geim A K 2009 Rev. Mod. Phys. 81 109 [14] Schliemann J 2008 New J. Phys. 10 043024 [15] McCann E and Koshino M 2013 Rep. Prog. Phys. 76 056503 [16] Novoselov K S, McCann E, Morozov S V, Falko V I, Katsnelson M I, Geim A K, Schedin F and Jiang D 2006 Nat. Phys. 2 177 [17] Gusynin V P and Sharapov S G 2005 Phys. Rev. B. 71 125124 [18] Peres N M R, Guinea F and Neto A H Castro 2006 Phys. Rev. B 73 125411 [19] Velasco J Jr et al 2012 Nat. Nano-tech 7 156 [20] Klimov A B and Chumakov S M 2010 A group-theoretical approach to quantum optics Contemp. Phys. 51 376 [21] Iqbal S, Qurat-ul-Ann and Saif F 2006 Phys. Lett. A 356 231 [22] Garc T, Cordero N A and Romera E 2014 Phys. Rev. B 89 075416 [23] Rakhimov K Y, Chaves A, Farias G A and Peeters F M 2011 J. Phys.: Condens. Matter 23 275801 [24] Chaves A, Covaci L, Rakhimov K Y, Farias G A and Peeters F M 2010 Phys. Rev. B 82 205430 [25] Torres J J and Romera E 2010 Phys. Rev. B 82 155419 [26] Maksimova G M, Demikhovskii V Y and Frolova E V 2008 Phys. Rev. B 78 235321 [27] Robinett R W 2004 Phys. Rep. 392 1 [28] Krueckl V and Kramer T 2009 New J. Phys. 11 093010 8 Mater. Res. Express 2 (2015) 095602 R Ali and F Saif.