Magnetic Excitations of Graphene in 8-Spinor Realization of Chiral Model

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Introduction. Scalar Chiral Model Since the very discovery of mono-atomic carbon layers called graphenes [1, 2] this material attracted deep interest of researchers due to its extraordinary properties concerning magnetism, stiffness and high electric and thermal conductivity [3-5]. The interesting connection of graphene was revealed with nano-tubes and fullerenes [6]. A very simple explanation of these unusual properties of graphene was suggested in [7], where the idea of massless Dirac-like excitations of honeycomb carbon lattice was discussed, the latter one being considered as a superposition of two triangular sub-lattices. Some phenomeno-logical development of this idea was realized in [8, 9]. As is well known, the carbon atom possesses of four valence electrons in the socalled hybridized 2-states, the one of them being “free” in graphene lattice and all others forming -bonds with the neighbors. It appears natural to introduce scalar 0 and 3-vector a fields corresponding to the -orbital and the -orbital states of the “free” electron respectively. These two fields can be combined into the unitary matrix (2) considered as the order parameter of the model in question, the long-wave approximation being adopted, i. e. = 0 0 + a · , (1) where 0 is the unit 2 2-matrix and are the three Pauli matrices, with the (2)-condition 2 + a2 = 1 (2) being imposed. It is convenient to construct via the differentiation of the chiral field (1) the so-called left chiral current = +∂, (3) the index running 0, 1, 2, 3 and denoting the derivatives with respect to the time 0 = and the space coordinates , = 1, 2, 3. Then the simplest Lagrangian density reads (4) and corresponds to the sigma-model approach in the field theory with the mass term. Here the constant model parameters and are introduced. Comparing the Lagrangian density (4) with that of the Landau-Lifshits theory corresponding to the quasiclassical long-wave approximation to the Heisenberg magnetic model [10], one can interpret the parameter in (4) as the exchange energy between the atoms (per spacing). Inserting (1) into (3) and (4) and taking into account the condition (2), one easily finds the following Lagrangian density: . (5) For the case of small a-excitations the equations of motion generated by (5) read as D a - (2/)a = 0 and correspond to the dispersion law = 0, 2 = k2 + 2/, which in the high-frequency approximation has the linear photon-like form. First we begin with the static 1 configuration corresponding to the ideal graphene plane, the normal being oriented along the -axis. In this case the order parameter has the form = exp(Θ3), Θ = Θ(), with the Lagrangian density being . (6) The Lagrangian (6) yields the equation of motion (7) The solution to (7) satisfying the natural boundary conditions has the well-known kink-like (or domain-wall) form: Θ0() = 2 arctan exp(-/ℓ), (8) with the characteristic thickness (length parameter) (9) and the energy per unit area Spinor Chiral Model of Graphene Now we intend to include in the model the interaction with the electromagnetic field for the description of conductivity and magnetic properties. To this end, we suggest 8-spinor generalization of the scalar chiral model and use the gauge invariance principle for introducing the electromagnetic interaction. The motivation for such a generalization is the following. For the description of spin and quasi-spin excitations in graphene, the latter ones corresponding to independent excitation modes of the two triangular sub-lattices of graphene, we introduce the two Dirac spinors 1, 2 and consider the combined spinor field Ψ as a new order parameter: Ψ = ⊗ (1 ⊕ 2) , (10) where stands for the first column of the unitary matrix (1). The Lagrangian density of the model (11) contains the projector = on the positive energy states, where = Ψ, = 0, 1, 2, 3, designates the Dirac current, Ψ = Ψ+0 and stands for the Dirac matrix. The model contains the two constant parameters of the previous scalar √model: the exchange energy per lattice spacing and some characteristic inverse length . The interaction with the electromagnetic field is realized through the extension of the derivative: = ∂ - 0Γ, with 0 > 0 being the coupling constant and Γ = (1 3)/2 being the charge operator chosen in accordance with the natural boundary condition at infinity: 0( ) = 1. However, the additional interaction term of the Pauli type should be added to take into account the proper magnetic moments of the electrons. Here = [, ]/4, = ∂ - ∂ , and 0 > 0 denotes the Bohr magneton per lattice spacing cubed. Let us consider as an illustration the interaction of the mono-atomic carbon layer = 0 with the static uniform magnetic field B0 oriented along the axis. We introduce first the vector potential = (), with the intensity of the magnetic field being = () = -′() and the natural boundary condition at infinity: → -0 . The model in question admits the evident symmetry 1 ⇔ 2 , 0 - invariance Ψ ⇒ 0Ψ and also the discrete symmetry: ⇔ *; 2,3 ⇒ -2,3. Therefore, one can introduce the chiral angle Θ(): 0 = cos Θ, 1 = sin Θ and the real 2-spinor () = col(, ), where 1 = 2 = col(, ). As a result the new Lagrangian density takes the form: = -2 ′2 - 8 2(Θ′2 + 22 sin2 Θ) - 4 sin2 Θ(22 + 0′) - ′2/(8), (12) where the new variable is introduced: = 2 = 22. Taking into account that 2 = 16 2, one can deduce from (12) and the boundary conditions at infinity: 2(∞) = 1, Θ(∞) = 0, ′(∞) = -0 the following “energy” integral: = -2 ′2 - 8 2(Θ′2 - 22 sin2 Θ) + 82 2 sin2 Θ - ′2/(8) = -2/(8), that implies the Hamilton-Jacobi equation for the “action” : . (13) Here the following definitions of the Jacobi momentums are used: . (14) Let us study the behavior of solution to the equations (13) and (14) in the asymptotic domain → ∞, where ≈ -0. In the first approximation one gets: . (15) Inserting (15) into (14), one derives the differential equation ′ = 4 Θ′ tan Θ with the evident integral 4 = cos-4 Θ corresponding to the boundary condition ( ) = 1/4. In view of (14) this fact permits one to obtain the equation for Θ(): with the solution of the form: , (16) where Θ0 stands for the integration constant. Finally, combining (16) and the last relation in (14), one can find the magnetic field intensity in the asymptotic domain → ∞: (17) As can be seen from (17), the effect of weakening of the magnetic field is revealed for the positive value of the constant 0 20 , this effect being similar to that of London “screening” caused by the second term in the electromagnetic current: (18) The current (18) contains beyond the standard conduction term, the diamagnetic current and the Pauli magnetization-polarization one. As follows from (17), for the negative value of the constant 0 20 the paramagnetic behavior of the material takes place. 10. Interaction with Magnetic Field Orthogonal to Graphene Plane Let us now study the case with the orientation of the magnetic field B0 along the -axis. Using the cylindrical coordinates , , , we introduce the vector potential = , with the intensity of the magnetic field being = ∂ ( )/, = -∂ , and the natural boundary condition at infinity being imposed: ( → ∞) = 0/2. The model in question admits the evident symmetry 1 ⇔ 2 and 0 - invariance Ψ ⇒ 0Ψ, that permits one to introduce 2-spinor by putting 1 = 2 = col(, ), = col(, ). To simplify the calculations, let us suppose the smallness of the radial magnetic field: ≪ . In this approximation the new discrete symmetry holds: ⇒ -3, ⇒ -, ⇒ *, 2,3 ⇒ -2,3, that permits one to introduce the chiral angle Θ: 0 = cos Θ, 1 = sin Θ and consider the axially-symmetric configuration: = (, ), Θ = Θ(, ). As a result the new Lagrangian density takes the form: , (19) where the new variable is introduced: = 2 and ∂ signifies the differentiation with respect to and . The equations of motion corresponding to (19) read: , (20) , (21) . (22) Let us now search for solutions to the equations (20), (21), (22) in the asymptotic domain → ∞, where Θ → 0; = 1/4 + , → 0; = 0/2 + , → 0. Thus, the equation (21) takes the form: and its solution can be found by separating variables: Θ = Θ0 exp -2 - , Θ0 = const, (23) with the following constant parameters: . (24) Inserting (23) into (20) and (22), one gets the inhomogeneous equations for and : , (25) (26) with the solutions of the form: , (27) where the radial functions () and () satisfy the following equations: (28) . (29) Let us now estimate the magnetic intensity: Taking into account that due to (29) ≈ (22)-1 as → ∞, one gets from (27): (30) (31) However, at small 0 one finds from (29) that 3/8, and therefore the intensity of the magnetic field reads: (32) (33) As can be seen from (30)-(33) , according to the sign of the multiplier 0 40 our graphene material reveals diamagnetic or paramagnetic behavior. Therefore, it would be interesting to obtain numerical estimates for the parameters of the model. In view of definitions adopted one has where the exchange energy is usually adopted as exch = 2.9 eV and the lattice spacing as = 3.56 10-8 cm, with being the absolute value of the electron charge. Finally, one can find the following numerical values: It means that the parameter 0 40 is positive and the weakening of the magnetic field inside the graphene is predicted in accordance with (17), (30) and (31) for large and its strengthening for small in accordance with (32) and (33). In view of the importance of the latter conclusion it would be desirable to investigate the magnetic field behavior in the central domain of the graphene material, i. e. at small but arbitrary . To this end, we consider the extrapolation of the configuration (23) to the domain wall structure of the form: . (34) Later it will be shown that this approximation is valid in the small field limit 0 0. To start with, we insert (34) and = 0/2 + , 1/4 into (23), this amounting to the equation: . (35) Solution to the equation (35) satisfying boundary condition ( = 0) = 0 can be found by Green’s function method: , (36) where 1 and 1 stand for the modified Bessel functions of the imaginary argument. Taking into account their asymptotic behavior as → 0: one finds from (35) and (36) that at small : , (37) that confirms the paramagnetic behavior of the graphene in the central domain. Finally, inserting (37) and = 1/4 + , ≈ 0/2 into (20), one gets the equation: . (38) Solution to (38) can be found also by Green’s function method along similar lines as for (35): . (39) Taking into account the asymptotic behavior of the Bessel functions as → 0: 0() ≈ 1 + , 0() ≈ log[2/], one finds from (38) and (39) that in the central domain . (40) Using (40), one can verify the validity of the approximation (34) as solution to (21) for the central domain in the small field limit, that is if the following strong inequalities hold: 2/ ≫ 00, 002 ≪ 1, 22 ≪ 1. Conclusions We analyzed the two phenomenological approches to the description of the graphene: the simplest scalar chiral model and its 8-spinor generalization. The scalar model admits very simple domain-wall solution describing one layer graphene configuration. On the opposite, the 8-spinor chiral model contains all previous results of the scalar model and also permits one to describe graphene interaction with the electromagnetic field. Magnetic excitations in graphene, for the case of the external magnetic field parallel to the graphene plane, reveal the evident diamagnetic effect: the weakening of the magnetic field within the graphene sample. As for the case of the magnetic field orthogonal to the graphene plane, the strengthening of the magnetic intensity inside the material is revealed in the central domain (at small ).

About the authors

Yu P Rybakov

Peoples’ Friendship University of Russia (RUDN University)

Author for correspondence.
Email: soliton4@mail.ru

Department of Theoretical Physics and Mechanics

6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation

M Iskandar

Peoples’ Friendship University of Russia (RUDN University)

Email: iskaandanr@gmail.com

Department of Theoretical Physics and Mechanics

6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation

A B Ahmed

Peoples’ Friendship University of Russia (RUDN University)

Email: garkuwaz@yahoo.com

Department of Theoretical Physics and Mechanics

6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation

References