Magnetic excitations of graphene in 8-spinor realization of chiral model Текст научной статьи по специальности «Физика»

Физика

UDC 538.9

DOI: 10.22363/2312-9735-2017-25-3-266-275

Magnetic Excitations of Graphene in 8-Spinor Realization

of Chiral Model Yu. P. Rybakov, M. Iskandar, A. B. Ahmed

Department of Theoretical Physics and Mechanics Peoples' Friendship University of Russia (RUDN University) 6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation

The simplest scalar chiral model of graphene suggested earlier and based on the SU(2) order parameter is generalized by including 8-spinor field as an additional order parameter for the description of spin (magnetic) excitations in graphene. As an illustration we study the interaction of the graphene layer with the external magnetic field. In the case of the magnetic field parallel to the graphene plane the diamagnetic effect is predicted, that is the weakening of the magnetic intensity in the volume of the material. However, for the case of the magnetic field orthogonal to the graphene plane the strengthening of the magnetic intensity is revealed in the central domain (at small r). Thus, the magnetic properties of the graphene prove to be strongly anisotropic.

Key words and phrases: graphene, spin excitations, chiral model, 8-spinor

1. 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 so-called hybridized sp2-states, the one of them being "free" in graphene lattice and all others forming sp-bonds with the neighbors. It appears natural to introduce scalar a0 and 3-vector a fields corresponding to the s-orbital and the p-orbital states of the "free" electron respectively. These two fields can be combined into the unitary matrix U G SU(2) considered as the order parameter of the model in question, the long-wave approximation being adopted, i. e.

U = a0 To + i a ■ r, (1)

where r0 is the unit 2 x 2-matrix and t are the three Pauli matrices, with the SU(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

= u+d^u, (3)

Received 12th January, 2017.

the index ^ running 0,1, 2, 3 and denoting the derivatives with respect to the time x° = ct and the space coordinates xl, i = 1, 2, 3. Then the simplest Lagrangian density reads

£ = -1ISp(l,I») - 2A2a2 (4)

and corresponds to the sigma-model approach in the field theory with the mass term. Here the constant model parameters I and A 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 I 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:

£ =21 (^«0 d11 ao + d^a ■ ^a) - 1 A2a2. (5)

For the case of small a-excitations the equations of motion generated by (5) read as

□ a - (A2//)a = 0 and correspond to the dispersion law

w = k0c, k2 = k2 + A2//,

which in the high-frequency approximation has the linear photon-like form.

First we begin with the static configuration corresponding to the ideal graphene plane, the normal being oriented along the z-axis. In this case the order parameter has the form

U = exp^Grs), e = 0(z),

with the Lagrangian density being

T \2

£ = -2e - Tsin2e. (6)

The Lagrangian (6) yields the equation of motion

21 e'' - A2sin2e = 0. (7)

The solution to (7) satisfying the natural boundary conditions

e(-rc) = e(+œ) = 0

has the well-known kink-like (or domain-wall) form:

e0 (z) = 2 arctanexp(-z/l), (8)

with the characteristic thickness (length parameter)

l = V7/A (9)

and the energy per unit area

1 ^ ' '2 ,

E = 2Jdz (/eo2 + A2sin2 e^ = 2AV7.

2. 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 and consider the combined spinor

field $ as a new order parameter:

$ = £ ® © ^2), (10)

where £ stands for the first column of the unitary matrix (1). The Lagrangian density of the model

T__\2 _

L = ^ + — a2j^j" + F$ (11)

contains the projector P = 7^ on the positive energy states, where = $7^^ = 0,1,2, 3, designates the Dirac current, $ = $+70 and 7^ stands for the Dirac matrix. The model contains the two constant parameters of the previous scalar model: the exchange energy I per lattice spacing and some characteristic inverse length \/A . The interaction with the electromagnetic field is realized through the extension of the derivative:

D^ = dM - ieoA^Te,

with e0 > 0 being the coupling constant and Te = (1 — r3)/2 being the charge operator chosen in accordance with the natural boundary condition at infinity: a0(<x>) = 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

= [7^, ]/4 = — ^ A^,

and ¡i0 > 0 denotes the Bohr magneton per lattice spacing cubed.

Let us consider as an illustration the interaction of the mono-atomic carbon layer z = 0 with the static uniform magnetic field B0 oriented along the x axis. We introduce first the vector potential Ay = A(z), with the intensity of the magnetic field being

Bx = B(z) = —A' (z)

and the natural boundary condition at infinity: A ^ —B0 z.

The model in question admits the evident symmetry ip2 , j0 — invariance

$ ^ 70$ and also the discrete symmetry:

^ vp*', a.2,3 ^—02,3-

Therefore, one can introduce the chiral angle 0(z):

a0 = cos 0, ai = sin 0

and the real 2-spinor ip(z) = col(w, —u), where = = col(^, —<p). As a result the new Lagrangian density takes the form:

L = — 2IU'2 — 8IU2(6/2 + e20A2 sin2 6) — 4U sin2 6(2A2U + ^0A') — A'2/(8k), (12)

where the new variable is introduced: U = |^|2 = 2u2. Taking into account that j2 = 16U2, one can deduce from (12) and the boundary conditions at infinity:

j 2(to) = 1, 6(to) = 0, A' (to) = -Bo the following "energy" integral:

E = -2IU'2 - 8IU2(6'2 - e20A2 sin2 6) + 8X2U2 sin2 6 - A'2/(8k) = -B%/(8n), that implies the Hamilton-Jacobi equation for the "action" S:

1 ( 9S\2 1 ( dS\2 ( dS t TT 2\2

8i\dû) + 332ÏÛ2 U) +2*{a + sin2eJ

R2

= +8 v2 sin2 6 (A2 + /e2^2) . (13)

Here the following definitions of the Jacobi momentums are used:

^ = -4IU'; |6 = -161U 26'2; ^ = sin2 6 --A'/(4«). (14)

Let us study the behavior of solution to the equations (13) and (14) in the asymptotic domain z ^ to, where A ~ -B^z. In the first approximation one gets:

5 « - 8 e0IU2 sin2 6) A. (15)

e^ a.

Inserting (15) into (14), one derives the differential equation

U' = 4 Ue' tan e

with the evident integral 4 U = cos-4 e corresponding to the boundary condition U(to) 1/4. In view of (14) this fact permits one to obtain the equation for e(z):

2e'

= e0A k -e0B0z

sin 26

with the solution of the form:

tan 6 = tan 60 exp (-eoBoz2/2) , (16)

where 60 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 z ^ to:

B = -A ~ B0 - 2v(e0I - 2^0) tan2 60 exp {-e0B0z2) . (17)

As can be seen from (17), the effect of weakening of the magnetic field is revealed for the positive value of the constant e0I - 2^0 , this effect being similar to that of London "screening" caused by the second term in the electromagnetic current:

= eoIIm (WP- e0Ij2(a21 + a&A^ + 2i^dv (a2fff,,W) . (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 e0I — 2^0 the paramagnetic behavior of the material takes place.

3. Interaction with Magnetic Field Orthogonal to Graphene

Plane

Let us now study the case with the orientation of the magnetic field B0 along the z-axis. Using the cylindrical coordinates r, <fi, z, we introduce the vector potential A$ = A, with the intensity of the magnetic field being

Bz = dr (rA)/r, Br = —dz A,

and the natural boundary condition at infinity being imposed: A(z ^ œ) = B0r/2.

The model in question admits the evident symmetry ^ ip2 and 70 — invariance ^ ^ 70^, that permits one to introduce 2-spinor by putting

= = col(^, ifi), = col(w, u).

To simplify the calculations, let us suppose the smallness of the radial magnetic field:

Br ^ Bz.

In this approximation the new discrete symmetry holds:

p ^ w ^ —w, U ^ U*, 0,2,3 ^ —0-2,3,

that permits one to introduce the chiral angle 0:

a0 = cos 0, ai = sin 0 and consider the axially-symmetric configuration:

u = u(r,z), 0 = 0(r,z). As a result the new Lagrangian density takes the form:

£ = -81

R2(d±0)2 + J(^)2 + e2R2A2 sin2 0

— 8\2R2 sin2 0 +

+ 8 u0Rsin2 0-dr(rA) — — r 8k

-2 (dr (rA))2 + (dz A)2

, (19)

where the new variable is introduced: R = u2 and signifies the differentiation with respect to r and z. The equations of motion corresponding to (19) read:

1 dr(rdrR) + d2zR — 4 R(d±0)2 — 4 e20R A2 sin2 0

= 2 sin2 0

2X2R — ^o1 dr (r A)

(20)

2

-dr(rR2dr0) + 2dz (R2dz0) — e20R2A2 sin 20

= R sin 26

X2R - u0-dr(r A) r

, (21)

1

1 A'

- dr (rdr A) + d\A - —

161 e0R2A sin2 6 + 8ß0dr(R sin2 6).

(22)

Let us now search for solutions to the equations (20), (21), (22) in the asymptotic domain z ^ to, where

e ^ 0; R =1/4 + C, C ^ 0; A = Bor/2 + a, a ^ 0. Thus, the equation (21) takes the form:

I

- dr (rdr 6) + d2z6 - - e2B2r26 r 4

6(A2 - 4/j,0Bo),

and its solution can be found by separating variables:

6 = 6q exp(-vr2 - kz), 6q = const

with the following constant parameters:

v = e050/4; Ik2 = B0 (e0I - 4№) + A2

(23)

(24)

Inserting (23) into (20) and (22), one gets the inhomogeneous equations for ( and a:

- dr (rdrC) + d2z C = (ô±)2 + r

U , „ . „Q a

4egB2r2 + -(A2 - 2^oB0)

62

-dr(rdra) + d2za - = 2neoBo(egI - 4№)r62 = 6r62

Jt rp2

with the solutions of the form:

c = 60 exp(-2^r2 - 2kz)N(r); a = 5620 exp(-2^r2 - 2kz)K(r) where the radial functions N(r) and K(r) satisfy the following equations:

(25)

(26)

(27)

N" + N' - + N 2 Bo (eg - 8^) +4-Ç + eg^V

I J I 1 2

2

ono

2

2 e2oB2r2 + eoBo + - (A2 - 3ßoBo), (28)

K" + K' - + K [ak2 - 8v + 16v2r2 - 4 ) = r.

(29)

Let us now estimate the magnetic intensity:

Bz = B0 + bz, bz = —dr (ra), Br = br = -dza. Taking into account that due to (29) K & (r)-1 as r ^ to, one gets from (27):

bz = -2^( eoI - 4^0)62 exp(-2 vr2 - 2kz) ,

(30)

br = ^^(eol - 4^o)e0 exp(-2ur2 - 2kz), (31)

eoB0r

However, at small r ^ 0 one finds from (29) that K & r3/8, and therefore the intensity of the magnetic field reads:

bz = neoBo(eoI - 4^o)e2r2 exp(-2^r2 - 2kz), (32)

br = ™eoBo(eoI - 4^o)e2or3 exp(-2^r2 - 2kz). (33)

As can be seen from (30)-(33) , according to the sign of the multiplier eoI - 4^o 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

_ e eh Eexch

eo = T- , Vo = ~-3, i =-,

he 2 mec a3 a

where the exchange energy is usually adopted as Eexch = 2.9 eV and the lattice spacing as a = 3.56 ■ 10-8 cm, with e being the absolute value of the electron charge. Finally, one can find the following numerical values:

eoI = 2 ■ 103 Gauss, = 2 ■ 102 Gauss.

It means that the parameter eoI - 4^o is positive and the weakening of the magnetic field inside the graphene is predicted in accordance with (17), (30) and (31) for large r and its strengthening for small r 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 r but arbitrary z. To this end, we consider the extrapolation of the configuration (23) to the domain wall structure of the form:

e = 2 arctan [exp(-2^r2 - 2kz)] . (34)

Later it will be shown that this approximation is valid in the small field limit Bo ^ 0. To start with, we insert (34) and A = Bor/2 + a, R & 1/4 into (23), this amounting to the equation:

-dr(rdra) + d2za - = 2nr eoBo sin2 e[eoI - 4/j,o tanh(Vr2 + kz)] = 2nr j. (35)

Solution to the equation (35) satisfying boundary condition a(r = 0) = 0 can be found by Green's function method:

CO CO r

a = J dz' J ds exp[«s(z - z')} J dr' r' j' [I1(sr)K1(sr') - Ki(sr)Ii(sr')}, (36)

-C -C o

where I1 and K1 stand for the modified Bessel functions of the imaginary argument. Taking into account their asymptotic behavior as x ^ 0:

I1(x) & x/2, K1(x) & x

-1

one finds from (35) and (36) that at small r:

nr 3eoBo

a(r, z)

4 cosh2(vr2 + kz)

[e oI — 4^o tanh( v r2 + kz)]

(37)

that confirms the paramagnetic behavior of the graphene in the central domain.

Finally, inserting (37) and R = 1/4 + A & B0r/2 into (20), one gets the equation:

-dr(rdr0 + d2zÇ = sin2 0 r

2 /1 j(X2 — 3/ioBo) + eoBol 1 + 2 eoBor2

= 3i-

(38)

Solution to (38) can be found also by Green's function method along similar lines as for (35):

œ CO

c=Udz' Ids eMis(z—[Ko( sr) /o( s/)—/o( sr^Ko( s/)] •(39)

— oo — oo

Taking into account the asymptotic behavior of the Bessel functions as x ^ 0:

Io(x) & 1 + x, Ko(x) & log[2/x], one finds from (38) and (39) that in the central domain

C(r, z) « —

4 cosh (vr2 + kz)

2

j(\2 — 3^oBo) + eoBo ( 1 + 2eoBor

(l + 2 eoBor2^)

(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:

A2/1 > eoBo, eoBor2 < 1, k2r2 < 1.

r

2

4. 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 ).

References

1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, A. A. Firsov, Electric Field Effect in Atomically Thin Carbon Films, Science 306 (2004) 666-669.

2. A. K. Geim, Graphene: Status and Prospects, Science 324 (2009) 1530-1534.

3. C. Lee, X. Wei, J. W. Kysar, J. Hone, Measurement of Elastic Properties and Intrinsic Strength of Monolayer Graphene, Science 321 (2008) 385-388.

4. A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, C. N. Lau, Superior Thermal Conductivity of Single-Layer Graphene, Nano Lett. (8) (2008) 902-907.

5. K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, H. L. Stormer, Ultrahigh Electron Mobility in Suspended Graphene, Solid State Commun. 146 (2008) 351-355.

6. D. Yu, L. Dai, Self-Assembled Graphene/Carbon Nanotube Hybrid Films for Super-Capacitors, J. Phys. Chem. Lett. 1 (2010) 467-470.

7. G. W. Semenoff, Condensed-Matter Simulation of a Three-Dimensional Anomaly, Phys. Rev. Lett. 53 (1984) 2449-2452.

8. Yu. P. Rybakov, On Chiral Model of Graphene, Solid State Phenomena 190 (2012) 59-62.

9. Yu. P. Rybakov, Spin Excitations in Chiral Model of Graphene, Solid State Phenomena 233-234 (2015) 16-19.

10. A. M. Kosevich, B. A. Ivanov, A. S. Kovalev, Nonlinear Magnetization Waves. Dynamical and Topological Solitons, Naukova Dumka, Kiev, 1983, in Russian.

УДК 538.9

Б01: 10.22363/2312-9735-2017-25-3-266-275

Магнитные возбуждения графена в рамках 8-спинорной реализации киральной модели Ю. П. Рыбаков, М. Искандар, А. Б. Ахмед

Кафедра теоретической физики и механики Российский университет дружбы народов ул. Миклухо-Маклая, д. 6, Москва, Россия, 117198

Простейшая киральная модель графена, предложенная ранее и основанная на Яи(2) параметре порядка, обобщается путем введения 8-спинорного поля как дополнительного параметра порядка для описания спиновых (магнитных) возбуждений в графене. В качестве иллюстрации мы изучаем взаимодействие графенового слоя с внешним магнитным полем. В случае магнитного поля, параллельного графеновой плоскости, предсказывается диамагнитный эффект, т. е. ослабление магнитной индукции внутри образца. Однако в случае магнитного поля, ортогонального графеновой плоскости, обнаруживается усиление магнитной индукции в центральной области (при малых г). Таким образом, магнитные свойства графена оказываются сильно анизотропными.

Ключевые слова: графен, спиновые возбуждения, киральная модель, 8-спинор

Литература

1. Electric Field Effect in Atomically Thin Carbon Films / K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, A. A. Firsov // Science. — 2004. — Vol. 306. — Pp. 666-669.

2. Geim A. K. Graphene: Status and Prospects // Science. — 2009. — Vol. 324. — Pp. 1530-1534.

3. Measurement of Elastic Properties and Intrinsic Strength of Monolayer Graphene / C. Lee, X. Wei, J. W. Kysar, J. Hone // Science. — 2008. — Vol. 321. — Pp. 385-388.

4. Superior Thermal Conductivity of Single-Layer Graphene / A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, C. N. Lau // Nano Lett. — 2008. — No 8. — Pp. 902-907.

5. Ultrahigh Electron Mobility in Suspended Graphene / K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, H. L. Stormer // Solid State Commun. — 2008. — Vol. 146. — Pp. 351-355.

6. Yu D., Dai L. Self-Assembled Graphene/Carbon Nanotube Hybrid Films for Super-Capacitors // J. Phys. Chem. Lett. — 2010. — Vol. 1. — Pp. 467-470.

7. Semenoff G. W. Condensed-Matter Simulation of a Three-Dimensional Anomaly // Phys. Rev. Lett. — 1984. — Vol. 53. — Pp. 2449-2452.

8. Rybakov Yu. P. On Chiral Model of Graphene // Solid State Phenomena. — 2012. — Vol. 190. — Pp. 59-62.

9. Rybakov Yu. P. Spin Excitations in Chiral Model of Graphene // Solid State Phenomena. — 2015. — Vol. 233-234. — Pp. 16-19.

10. Косевич А. М., Иванов Б. А., Ковалев А. С. Нелинейные волны намагниченности. Динамические и топологические солитоны. — Киев: Наукова думка, 1983. — 190 с.

© Rybakov Yu.P., IskandarM., Ahmed A.B., 2017