Study of magnetism of two-dimensional ferromagnetic graphene

Bin-Zhou Mi1,*, Yong-Hong Xue2, Huai-Yu Wang3, Yun-Song Zhou4, and Xiao-Lan Zhong5

1,2Department of Basic Curriculum, North China Institute of Science and Technology, Beijing- East 101601, China

3Department of Physics, Tsinghua University, Beijing 100084, China

4Department of Physics, Capital Normal University, Beijing 100048, China

5 Laboratory of Optical Physics, Institute of Physics, Chinese Academy of Science, P.O. Box 603,

Beijing 100190, China

1,*Corresponding author: email: mbzfjerry2008@126.com

2email: aristotle@ncist.edu.cn 3email: wanghuaiyu@mail.tsinghua.edu.cn

4email: 263zys@263.net 5email: zhongxl@iphy.ac.cn

Keywords: Heisenberg model; Ferromagnetic graphene; Magnetic properties; Many-body Green’s function method.

Abstract In this paper, the magnetic properties of ferromagnetic graphene nanostructures,

especially the dependence of the magnetism on finite temperature, are investigated by use of the

many-body Green’s function method of quantum statistical theory. The spontaneous magnetization

increases with spin quantum number, and decreases with temperature. Curie temperature increases

with exchange parameter J or the strength K2 of single-ion anisotropy and spin quantum number.

The Curie temperature TC is directly proportional to the exchange parameter J. The spin-wave

energy drops with temperature rising, and becomes zero as temperature reaches Curie temperature.

As J(p,q)=0, ω1=ω2, the spin wave energy is degenerate, and the corresponding vector k=(p, q) is

called the Dirac point. This study contributes to theoretical analysis for pristine two-dimensional

magnetic nanomaterials that may occur in advanced experiments.

Introduction

More than 70 years ago, Landau and Peierls argued that strictly 2D crystals were

thermodynamically unstable and could not exist [1,2]. Until 2004, the isolation of a single layer of

graphene, which is a two-dimensional honeycomb lattice of carbon atoms, was first successfully

fabricated in experiment by Micromechanical cleavage technique [3]. During the following years,

electronic properties of graphene nanostructures and graphene nanoribbons have been studied

extensively [4,5]. Recently, ferromagnetic (FM) graphene is being investigated although ideal

graphene sheets are known to be non-magnetic. In order to obtain FM graphene, scientists have

proposed a variety of theoretical or experimental ways [6-16], including cut it into zigzag

configuration with the zero-dimensional or one-dimensional nano-ribbon structure [6-8], or

introduce various defects [9-13], and consciously doped transition metal atoms [14,15], etc.

Particularly, J. Zhou et al. had simulated theoretically that semi-hydrogenated graphene [16] (which

is called graphone) becomes a ferromagnetic semiconductor with a small indirect gap by use of the

density functional theory. Compared with other methods, this method can giving rise to an infinite

magnetic sheet with structural integrity and magnetic homogeneity. Stable and precision magnetic

spin ordering make the graphene sheet very appealing for further experimental study and the actual

application in nanoelectronic devices. Furthermore, the electronic and magnetic properties of VS2

and VSe2 monolayers were studied by a first-principles density functional theory calculation [17]. It

was pointed out that the pristine VS2 and VSe2 monolayers surprisingly exhibit ferromagnetic

ordering, and offering evidence of the existence of magnetic behavior for pristine 2D monolayers.

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In such research background, we think that theoretical investigations of magnetic properties of

graphene nanostructures in the case that they are FM are also desirable and inevitable, especially the

dependence of the magnetism on temperature. Under this motivation, the present paper is devoted to

the research of the spontaneous magnetization, Curie point, and the magnon dispersion of FM

graphene. As we all know, the first-principles density functional theory calculation is not involved

in the case of finite temperature. So this study contributes to theoretical analysis for

two-dimensional magnetic nanomaterials that may occur in advanced experiments.

Model and method

The calculation model of two-dimensional magnetic system is the hexagonal honeycomb lattice

structure with spins situated at the lattice sites. The distance between the nearest neighbor sites is a.

The lattice has to be divided into two sublattices and the lattice constants of the z-direction and

y-direction in the plane are 1 23 , 3 ,b a b a= = respectively.

The Heisenberg exchange Hamiltonian is 2 2

1 2 2 1 2 2 1 2

(1 ,2 ) 1 2 1 2

( ) ( ) .z z z z

i j i j z i z j

i j i j i j

H J K S K S B S B S= − ⋅ − − − −∑ ∑ ∑ ∑ ∑S S (1)

In Eq. (1), the first term represents the Heisenberg FM exchange, with exchange parameter J,

where the nearest-neighbor interaction is considered. The subscripts 1i and 2j denote the

corresponding sublattice sites. The second and the third term describe the single-ion anisotropy of

sublattice 1 and 2, respectively, which causes the uniaxial anisotropy of the system and thus is

responsible for appearance of spontaneous magnetization. The strength of K2 is usually believed to

be less than J by two orders of magnitude. The easy axis is the z-axis in the graphene plane. The

last two term stand for Zeeman energy of sublattice 1 and 2, respectively, when an external

magnetic field is applied along the z-direction. In calculation, we set J/ K2=100. Then all the

parameters and temperature are taken as dimensionless quantities.

Here, we point out that our Hamiltonian is mainly an isotropic Heisenberg exchange model plus a

single-ion anisotropy term, if the external field is absent. According to the Mermin–Wagner

theorem [18], for one- or two-dimensional (2D) isotropic Heisenberg systems, there is no

spontaneous magnetization. However, the spontaneous magnetization can occur when a single-ion

anisotropy is introduced no matter how small it is. This is verified by us [19,20].

The many-body Green’s function method (MBGFM) is a powerful means [19,20] to calculate

magnetization of the magnetic nanostructures since this method takes into account the spin

fluctuation, and is valid in the whole temperature range. The retarded Green’s function is defined as

follows: T T T( ) ; ( ) ,R RG t t A B i t t A B BAθ′ ′− = ⟨⟨ ⟩⟩ = − − ⟨ − ⟩ (2)

where the operators A and B are actually operator vectors in the following form:

)( )(1 2 1 1 2 2, , exp( ) ,exp( )z zA S S B S S S Sα α+ + − −= = (3)

The approach is similar to that in Ref. [19,20]. Here we do not write the detailed process because of

the limitation of article length, readers can refer to Ref. [19,20]. Finally, the magnetization zSµ⟨ ⟩

and correlation functions z zS Sµ µ⟨ ⟩ of an arbitrary S are expressed generally [21,22] by

2 1 2 1

2 1 2 1

( 1 ) ( )( 1),

( 1)

S S

z

S S

S SS µ

µ µ µ µ

µ µ

+ +

+ +

Φ + + Φ − Φ − Φ +⟨ ⟩ =

Φ + −Φ (4)

and

( 1) (1 2 )z z zS S S S Sµ µ µµ⟨ ⟩ = + − + Φ ⟨ ⟩ , (5)

90 Management, Manufacturing and Materials Engineering II

where 1

1 2

1.

1

U U

N N e τ

µτ τµµ βω

τ

−

Φ =−∑∑

k

(6)

In Eq. (6), N1 and N2 label the site numbers along the z-axis and y-axis, respectively. The vector k is:

k=(p, q), and the wave vector component p, q is within the first Brillouin zone. Note that we use zS µ⟨ ⟩ to denote the magnetization of the sublattice 1 and 2 and S to denote spin quantum number.

We here consider the case when the external field is absent. Note that in the present case the

magnetizations of the two sublattices are the same, namely, 1 2 .z z zS S S⟨ ⟩ = ⟨ ⟩ = ⟨ ⟩ Since the lattice is

divided into two sublattices, there are two branches of dispersion relationships:

[ ]1,2 0 22 ( , ) ,zJ K C J p q Sω = + ± ⟨ ⟩ (7)

where

0 3 ,J J= (8)

1 1 2( , ) 1 4cos cos cos ,2 2 2

pb pb qbJ p q J

= + +

(9)

and

2

11 [ ( 1) ].

2

z zC S S S SS

= − + − ⟨ ⟩ (10)

After calculating

1 21 2

1 2

1 1 1 1(coth coth ) .

4 2 2 2p qN N

βω βωΦ = Φ = + −∑ ∑ (11)

Equations (4), (5) and (7-11) are the transcendental equations for calculating magnetization zS⟨ ⟩ .

For S=1/2, the anisotropy term in Eq. (1) does not play a role. Therefore, in the present paper, we

study the cases of S=1, 3/2, 2, 5/2.

Numerical results and discussion

Spontaneous magnetization and Curie point

The spontaneous magnetization zS⟨ ⟩ of FM graphene as a function of temperature is calculated

numerically. Figure 1 plots the magnetization versus temperature for four spin quantum numbers.

The zS⟨ ⟩ decreases with temperature rising. The temperature at which zS⟨ ⟩ reaches zero is Curie

point TC, and which increases with spin quantum number rising.

0 50 100 150 200 2500.00

0.25

0.50

0.75

1.00J=100, K

2=1.

5/223/2S=1

T

<S

z >/S

Figure1. The spontaneous magnetization of FM graphene as a function of temperature for four S

values.

Advanced Materials Research Vol. 601 91

0 50 100 150 200 250 300 3500.00

0.25

0.50

0.75

1.00

100 200 300 400 500 6000

50

100

150

200

250

S=1, J/K2=100.

J

TC

T<

S z >

600500400300200J=100

S=1, J/K2=100.

Figure2. The spontaneous magnetization of FM graphene as a function of temperature for several

values of J, J/ K2=100 and the spin quantum number is S=1. The inset: The Curie point as a

function of J for S=1.

Figure 2 illustrates the spontaneous magnetization zS⟨ ⟩ of FM graphene as a function of

temperature for several values of J with S=1. When the exchange parameter J or the strength K2 of

single-ion anisotropy rises, TC increases. The Curie point TC versus exchange parameter J of the FM

graphene for S=1 and J/ K2=100 is plotted in the inset of Fig. 2. The Curie temperature TC is

directly proportional to the exchange parameter J.

The magnon dispersive relation

The spin wave spectra of FM graphene are expressed by ( 1, 2)τω τ = . Here we only calculate the

spin wave spectrum when the external magnetic field is absent. Since the two dimensional

honeycomb-lattice is divided into two sublattices, there are two branches of dispersion relationships,

ω1andω2, see for Eq. (7). It shows that the spin-wave energy ω(p,q) is proportional to the

magnetization, while the zS µ⟨ ⟩ decreases with temperature rising, thus the elementary excitation

energies also decrease with temperature. At Curie point the magnetization becomes zero, so that the

spin-wave energies ω(p,q) reach zero, and this can be so-called mode softening.

Recently, the Dirac point have been observed and studied in graphene nanostructures or other

materials, which is characterized by the linear splitting of energy bands in its vicinity. As can be

seen from Eq. (7), as J(p,q)=0, then ω1=ω2, the spin wave energy spectra of FM graphene is

degenerate, and the corresponding vector k=(p, q) is called the Dirac point.

Concluding remarks

In this paper, the magnetic behaviors of FM graphene are investigated by means of the MBGFM.

We carry out numerical calculation of spontaneous magnetization and Curie point. Furthermore, the

spin wave spectrum and the Dirac point are analyzed qualitatively. The spontaneous magnetization

increases with spin quantum number, and decreases with temperature. Curie temperature increases

with exchange parameter J or the strength K2 of single-ion anisotropy and spin quantum number.

The Curie temperature TC is directly proportional to the exchange parameter J. A higher temperature

leads to lower energy spectra. At Curie point, the spin-wave energies ω(p,q) reach zero, and this can

be so-called mode softening. As J(p,q)=0, ω1=ω2, the spin wave energy is degenerate, and the

corresponding vector k=(p, q) is called the Dirac point.

This study contributes to theoretical analysis for two-dimensional magnetic materials that may

occur in advanced experiments. We hope that the present study will stimulate further experimental

effort in pristine two-dimensional magnetic nanomaterials.

Acknowledgements This work was supported by the Special Fund for Basic Scientific Research of

Central Colleges, North China Institute of Science and Technology for Nationalities (No.

JCB1202B).

92 Management, Manufacturing and Materials Engineering II

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