Tight-binding model study of substrate induced pseudo-spin polarization and magnetism in mono-layer graphene

Journal of Magnetism and Magnetic Materials ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Tight-binding model study of substrate induced pseudo-spin polarization and magnetism in mono-layer graphene Sivabrata Sahu a, G.C. Rout b,n a

School of Applied Sciences (Physics), KIIT University, Bhubaneswar 751024, Odisha, India Condensed Matter Physics Group, Physics Enclave, Plot No. – 664/4825, Lane-4A, Shree Vihar, Chandrasekharpur, Po-Patia, Bhubaneswar 751031, Odisha, India

b

art ic l e i nf o

a b s t r a c t

Article history: Received 19 April 2015 Received in revised form 25 December 2015 Accepted 20 January 2016

We present here a tight-binding model study of generation of magnetism and pseudo-spin polarization in monolayer graphene arising due to substrate, impurity and Coulomb correlation effects. The model Hamiltonian contains the first-, second- and third-nearest-neighbor hopping integrals for π electrons of graphene besides substrate induced gap, impurity interactions and Coulomb correlation of electrons. The Hubbard type Coulomb interactions present in both the sub-lattices A and B are treated within the meanfield approximation. The electronic Green's functions are calculated by using Zubarev's technique and hence the electron occupancies of both sub-lattices are calculated for up and down spins separately. These four temperature dependent occupancies are calculated numerically and self-consistently. Then we have calculated the temperature dependent pseudo-spin polarization, ferromagnetic and anti-ferromagnetic magnetizations. We observe that there exists pseudo-spin polarization for lower Coulomb energy, u < 2.2t1 and pseudo-spin polarization is enhanced with substrate induced gap and impurity effect. For larger Coulomb energy u > 2.5t1, there exists pseudo-spin polarization (p); while ferromagnetic (m) and antiferromagnetic (pm) magnetizations exhibit oscillatory behavior. With increase of the substrate induced gap, the ferromagnetic and antiferromagnetic transition temperatures are enhanced with increase of the substrate induced gap; while polarization (p) is enhanced in magnitude only. & 2016 Elsevier B.V. All rights reserved.

Keywords: Graphene Coulomb interaction Pseudo-spin polarization FM Magnetization AFM Magnetization

1. Introduction Ferromagnetic instability due to exchange interaction in the three dimensional (3D) electron gas has been studied in great detail [1,2]. Similar studies have suggested the existence of ferromagnetic phase in the diluted two-dimensional (2D) electron gas with transition from paramagnetic to ferromagnetic phase [3]. There has been strong experimental indications on the existence of ferromagnetism in highly disordered graphites [4,5]. Though a number of different mechanisms have been proposed [6,7], the origin of the magnetic phase is still unclear. The recent experiments in true 2D-graphene systems [8,9] show that electronelectron interaction and disorder effect have to be taken in order effect to obtain a fully consistent picture of magnetism in graphene [10]. The electronic structure of perfect graphene plane exhibits the linear relationship between momentum and electronic energy. This singular dispersion relation in graphene is a direct n

Corresponding author. E-mail addresses: [email protected] (S. Sahu), [email protected] (G.C. Rout).

consequence of its honey-comb lattice structure. The effective mass of electron in graphene is zero at the Dirac point similar to the universal electrodynamics. In perfect graphene sheet, the chemical potential crosses the Dirac point and electronic density of states vanishes at the Fermi-energy. The vanishing of the effective mass or density of states has profound consequences. The Coulomb interaction remains unscreened and gives rise to an inverse quasi-particle life time that increases linearly with energy or temperature. It is well known that direct exchange interaction between electrons can lead to a ferromagnetic instability in dilute electron gas [1,11]. Similar exchange instability of electron gas can be generalized for pure and doped 2D-graphene. It has been reported by Peres et al. [12] that the extended defects such as dislocation, disinclination edges and micro cracks can leads to self-doping where charge is transformed to/from defect to graphene. The extended defects can also lead to self-doping and hence localized disorders such as vacancies and adatoms are introduced. Thus Peres et al. [12] have considered the influence of disorder by introducing Coulomb interaction for the generation of ferromagnetism. The possibilities of other instabilities in a graphene plane reflecting the Coulomb interaction have been studied in the

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literature [13,14]. In spite of this, the study of instability due to Coulomb interaction in graphene is extensive. It is worth mentioning that the simple analysis of Stoner criterion for ferromagnetism fails in graphene, as the density of states of un-doped graphene vanishes at the Fermi level. The Coulomb interaction between massless fermions in pristine graphene remains long ranged and unscreened. It is not clear whether this would lead to strongly correlated insulating phase or weakly correlated semi-conducting phase of graphene [15,16]. The local Coulomb interaction is important for the theory to understand the defect induced magnetism [17] and Mott insulator-metal transition in graphene on the surface of Si:X (X ¼Si, C, Sn, Pb ) [18]. The two dimensional graphene [19,20] and polymers [21,22] display strong local as well as non-local Coulomb interactions. The bare on-site Coulomb interaction in benzene was estimated long back to be ∼16.93 eV [17]. The weak coupling perturbation theory gives an effective repulsive on-site Coulomb interaction energy of ∼10 eV to explain the optical spectroscopy experiments of poly -acetylene [23,24]. Wehling et al. [19] have estimated that the onsite Coulomb interaction is U ∼ 3.3t1 and the nearest-neighbor Coulomb interaction is U ∼ 2t1 in graphene, where t1 = − 2.8 eV is the nearest-neighbor hopping integral. The mean-field Hubbard model calculation for bipartite graphene lattice yields antiferromagnetic ground state for UAFM ≥ 2.2t1 [25,26], while the quantum Monte-Carlo and finite size scaling for the Hubbard model give the evidence of a zero temperature transition between non-magnetic semimetal and antiferromagnetic insulator for UAFM ≥ 4.5t1 [25–27]. The local magnetic moments can be introduced to graphene in a varieties of forms i.e along the edges of the nano-ribbons [28], around vacancies [29] and adatoms [30]. However, long range ferro-magnetic order in graphene does not occur without exchange coupling between the local moments. By coupling the graphene to an atomically flat Yttrium iron garnet (YIG) [31], ferromagnetic insulator film, ferromagnetism in graphene can be introduced without sacrificing its excellent transport properties. The hybridization between the π orbitals in graphene and the near-by spin polarized d-orbitals in magnetic insulator gives rise to the exchange interaction required for long range ferromagnetic ordering. It is known that point defects induce localized magnetic moments in graphene [32–34]. Lieb [35] has predicted the presence of bi-bipartite lattice in graphene in case of half filling and repulsive local electron-electron repulsive interaction‘U’. The ground state is characterized by the total spin 2S = |NA − NB |, where NA and NB are the number of sites in sub-lattices A and B respectively. Point defects like adsorbed atoms or single vacancies unbalance the number of atoms in both the sub-lattices giving rise to localized magnetic moments in graphene. The peculiar characteristics of different magnetic states in graphene have been studied and characterized by model Hamiltonians [36–39]. Many first principle calculations have been performed for hydrogen vacancies by Yazyev [34,40] and others [33,41] and fluorine by Sofo et al. [42]. The recent experiments on fluorinated graphene and irradiated graphene indicate a magnetic behavior competitive with localized spin one-half defects [43–46]. The electron–electron interaction in graphene can lead to other instabilities at low temperatures like ferromagnetic phase [47]. A local on-site repulsive interaction can lead to an anti-ferromagnetic phase, when its value exceeds a critical threshold [25,48]. In the following, we will concentrate on the on-site Coulomb interaction in low and relatively large limits in order to study the pseudo-spin polarization, ferromagnetism (FM) and antiferromagnetism (AFM) in doped graphene on substrate. Earlier we have reported the study of band gap opening in graphene by single impurity taking up to third nearest hoppings in a tight-binding

model in absence of Coulomb interaction [49]. We find that Coulomb interaction (U) in bipartite lattice enhances pseudo-spin effect leading to band gap opening [50]. Now we investigate here the charge and spin polarizations leading to oscillatory magnetic phases due to Coulomb interaction in doped graphene on substrate. We present the tight-binding Hamiltonian in Section 2. We calculate the electron Green's functions by Zubarev technique [51] and calculate the electron occupancies of both sub-lattices for two spin orientations in Section 3. The temperature dependent electron occupancies are solved numerically and self-consistently. We present the results and discussion in Section 4 and conclusion in Section 5.

2. Theoretical model Based upon our earlier model [49,50], the tight-binding model Hamiltonian including π electron hoppings up to third-nearestneighbor hoppings with hopping integrals t1, t2 and t3 respectively is written as

H0 =

∑ ϵa ai†, σ ai, σ + ∑ ϵbb†j, σ bj, σ i, σ

J, σ

∑

− t2

〈〈i, j〉〉, σ

∑

− t3

〈〈〈i, j〉〉〉, σ

where

− t1

ai,†σ

∑ 〈i, j〉, σ

⎛ ⎜ a † aj, σ + b† bi, σ + H . j, σ ⎜ i, σ ⎝

⎞ ⎛ ⎜ a † bj, σ + b† ai, σ ⎟ j, σ ⎟ ⎜ i, σ ⎠ ⎝

⎞ C ⎟⎟ ⎠

⎞ ⎛ ⎜ a † bj, σ + b† ai, σ ⎟ j, σ ⎟ ⎜ i, σ ⎠ ⎝

( ai, σ ) and

(1)

bi,†σ

( bi, σ ) create (annihilate) respectively an ⎯→ ⎯ electron with spin s on site Ri on sub-lattices A and B. The Fourier transformed band dispersions for first-, second- and third-nearestneighbor hoppings of π electrons are found to be ϵ1k = − t1γ1 (k ), ϵ2k = − t2 γ2 (k ) and ϵ3k = − t3 γ3 (k ), where γ1 (k ) for the nearestneighbor hopping is

γ1 (k ) = eik x a0 + 2e−i (1/2) k x a0 . cos

3 k a 2 y 0

(2)

→ where a0 is the lattice constant and k (k x , k y ) is the electron mo→→

mentum. The other dispersions are γ2 (k ) = ∑δ ei k . δ2 and 2 →→ ⎯→ ⎯ ⎯→ ⎯ γ3 (k ) ¼ ∑δ ei k . δ3, where δ2 and δ3 are corresponding lattice vec3 tors. Gharekhanl et al. [52] have reported the electronic band structure of patterned graphene taking tight-binding approximation for the π electrons including upto five nearest-neighbor hopping integrals and found opening of gap near Dirac point in graphene by this symmetry breaking. The localized defects such as vacancies and impurities are included in the tight-binding description of the model by adding a local energy term [53]

Himp =

⎛

⎞

⎝

⎠

∑ Vi ⎜⎜ ai†, σ ai, σ + bi†+ δ, σ bi + δ, σ ⎟⎟ i, σ

(3)

→ where Vi is the random potential at sites Ri . In momentum space the Hamiltonian appears as

Himp =

⎛

⎞

⎝

⎠

∑ V ⎜⎜ x a ak†, σ ak, σ + xb bk†, σ bk, σ ⎟⎟ k, σ

(4)

where V is the interaction potential at impurity sites with impurity concentrations xa and xb at sites A and B respectively. On growing epitaxial graphene on silicon carbide substrates by annealing 4H– and CH–SiC surfaces [54,55], magnetism is

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generated in graphene. Both theoretical and experimental studies [56,57] indicate that the first layer of epitaxial graphene (buffer layer)is strongly coupled to the substrate and has electronic properties that differ significantly from pristine graphene. Subsequent layers are nearly decoupled from the substrate and then display graphene like behavior. While early theoretical studies [57–59] focus on graphene on perfect bulk terminated SiC (0001), recent STM observations [60] and x-ray reflecting measurements [56] clearly indicate the presence of defects, in particular Si adatoms at the graphene-SiC (0001) interface. The substrate induced potential can break the symmetry of the honeycomb lattice and generate gap in the electronic spectra of graphene. As a result, a modulated potential leads to the breaking of the symmetry between A and B sites, where A site has energy +Δ and B site with energy −Δ and gives rise to a gap of 2Δ. Such symmetry breaking Hamiltonian is written as

H1 = Δ ∑

a†j, σ aj, σ

j, σ

− Δ∑

b†j, σ bj, σ

j, σ

(5)

It is seen, in our earlier work [49,50], that the band dispersion touches near the Dirac point and the electron density of states vanishes near the Dirac point in absence of substrate induced gap. In this case, there is no electronic screening [61] and the electrons interact with long range Coulomb forces. In presence of impurity and substrate effects, the Coulomb correlation energy is strongly screened. The on-site electron–electron interactions at A and B sites can be written as

HU = U ∑ [na, j ↑ na, j ↓ + nb, j ↑ nb, j ↓ ] j

(6)

where nα, j ↑ (nα, j ↓ ) with α ∈ A , B sub-lattices, represents the occupation number operator of up(down) spin and U as an effective Coulomb energy. The Coulomb interaction induces electron scattering in inter-bands and also intra-bands and provides an electron–electron scattering which allows for the possibilities of a ferromagnetic transition at weak coupling. The spin polarization is also preferred, when long range interactions are present. Since kinetic energy and Coulomb energy are minimized simultaneously by the Pauli's exclusion principle, the Hubbard type short range interaction also benefits antiferromagnetic coupling via a kinetic exchange mechanism. The Hamiltonian given in Eq. (4) is decoupled by Hartree–Fock decoupling scheme i.e. Unα, j, σ nα, j, −σ ≈ U 〈nα, j, σ 〉nα, j, −σ + U 〈nα, j, −σ 〉nα, j, σ − U 〈nα, j, σ 〉〈nα, j, −σ 〉, where s stands for up and down spins. After Fourier transformation in momentum space, the total Hamiltonian is written as

H = H0 + Himp + H1 + HU

(7)

B1 (k, ω) = ⪡bk, σ ; bk†, σ ⪢ω =

3

1 ⎛ ω − εa, σ (k ) ⎞ ⎜ ⎟ 2π ⎝ |Dσ (ω)| ⎠

(9)

where |Dσ (ω)| is written as

|Dσ (ω)| = (ω − ϵa, σ (k ))(ω − ϵb, σ (k )) − 4|ϵ13 (k )|2

(10)

Here

ϵa, σ (k ) = ϵa + ϵ2k + x a V + Δ + U 〈n−aσ 〉 ϵb, σ (k ) = ϵb + ϵ2k + xb V − Δ + U 〈n−bσ 〉

(11)

The combined band dispersion is ϵ13 (k ) = ϵ1k + ϵ3k . The impurity concentrations at A and B sites are xa and xb respectively with impurity potential V. Setting |Dσ (ω)| = 0, we find two quasi- particle band energies i.e. ωαk, σ , for α = 1, 2 1

ωαk, σ = 2 [ϵa, σ (k ) + ϵb, σ (k ) − ( − 1)α pk, σ ]

(12)

where pk, σ = (ϵa, σ (k ) − ϵb, σ (k ))2 + 4|ϵ13 (k ) |2. The electron occupancy for sub lattice index α = a, b and spin s is defined as

〈nα, σ 〉 =

1 Ns

∑k 〈cα†, k, σ cα, k, σ 〉, where Ns is the number of unit cells per

area. According to Zubarev's technique [51], the electron correlation is written as

〈cα†, k, σ cα, k, σ 〉 = i lim η→0

∞

∫−∞ f (βω)[G (ω + iη) − G (ω − iη)] dω

(13)

Here the Fermi function is written as f (βω) = [1 + exp (βω)]−1, with β = (kB T )−1, where kB, T and ω are respectively Boltzmann constant, temperature in degree Kelvin and energy. Further G (ω + iη) is the electron Green's function in general with η as a small spectral width. The temperature dependent electron occupancies associated with A and B sub-lattices are defined as

〈na, σ 〉 =

1 N

∑k 〈ak†, σ ak, σ 〉 and 〈nb, σ 〉 =

1 N

∑k 〈bk†, σ bk, σ 〉. They are calcu-

lated from Green's functions A1 (k, σ ) and B1 (k, σ ) given in Eqs. (8) and (9) respectively and are written as

〈na, σ 〉 =

1 Ns

∑ k

1 [(ω1k, σ − ϵb, σ ) f (βω1k, σ ) pk, σ

− (ω2k, σ − ϵb, σ ) f (βω2k, σ )]

(14)

〈nb, σ 〉 =

1 Ns

∑ k

1 [(ω1k, σ − ϵa, σ ) f (βω1k, σ ) pk, σ

− (ω2k, σ − ϵa, σ ) f (βω2k, σ )] 3. Calculation In order to calculate temperature dependent pseudo-spin potential, ferro magnetism and antiferromagnetism of graphene-onsubstrate, we calculate the electron Green's functions for A and B sub-lattices by equations of motion by using Zubareve's Green's function technique. The electron Green's functions associated with A sub-lattice are related by two coupled equations. The Fourier transformed electron Green's function associated with A sub-lattice is written as

A1 (k, ω) = ⪡ak, σ ; ak†, σ ⪢ω =

1 ⎛ ω − εb, σ (k ) ⎞ ⎜ ⎟ 2π ⎝ |Dσ (ω)| ⎠

(8)

Similarly the electron Green's function associated with B sites is defined as

(15)

The electron occupancies associated with numbers for A and B site electrons for different spin orientation are computed numerically and self-consistently for total Brillouin zone taking 100  100 grid points in X–Y momentum plane. To facilitate numerical computation, the physical parameters for graphene are made dimensionless with respect to suitable energy scale. Here physical parameters involved in the calculation are scaled by first-nearestneighbor hopping integral ( t1 = − 2.78 eV ). The temperature T in degree Kelvin appears as t =

kB T t1

in dimensionless form. Then other

physical parameters involved in the calculations are scaled by first-nearest-neighbor hopping integral t1 = − 2.78 eV . The scaled t t V parameters are t˜1 = 1, t˜2 = 2 , t˜3 = 3 , impurity potential v = , t1

t1

Coulomb repulsive energy u = site energy at A site ea =

ϵa t1

U , t1

substrate induced gap d1 =

and site energy at B site eb =

t1 Δ , t1

ϵb . t1

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4. Results and discussion In the model tight-binding calculations, we have considered the nearest-neighbor hopping integral t1 = − 2.78 eV . The Coulomb repulsive energy is treated within a Hartree–Fock type mean-field approximation where same on-site Coulomb energy U among the A and B sub-lattices of carbon atoms of graphene is taken. The electron density operators 〈nα, ↑ 〉 and 〈nα, ↓ 〉 of up and down spin electrons given in Eqs. (12) and (13) for sub-lattices α ∈ A , B are solved self-consistently. Based on earlier report [62] we have introduced electron density nα and magnetization mα represented as

nα = 〈nα, ↑ 〉 + 〈nα, ↓ 〉,

mα = 〈nα, ↑ 〉 − 〈nα, ↓ 〉

(16)

The total electron density (n) and total ferromagnetic (FM) magnetization (m) are defined as

n = n a + nb ,

m = ma + mb

(17)

where n and m are symmetric with respect to the sub-lattices. Here, we introduce pseudo-spin and antiferromagnetic (AFM) magnetizations as

p = n a − nb ,

pm = ma − mb

(18)

which are anti-symmetric with respect to the sub-lattices. Here p (pm ) represents charge polarization (spin polarization) for the hexagonal unit cell of the graphene system. We have computed ferromagnetic magnetization (m), pseudo-spin polarization (p), spin polarization or antiferromagnetic sub-lattice magnetization (pm) from the self-consistent values of the temperature dependent electron density operators. The temperature dependence of these parameters are shown below in Figs. 1–10. We consider below the effect of Coulomb repulsive interaction (u) on charge and spin polarizations under two limits. i.e. Case-I: for lower u and Case-II: for larger u. 4.1. Case I: For lower u The temperature dependence of electron occupancies na, ↑ and nb, ↑ is shown in Fig. 1. For given Coulomb potential U = uc t1 = 1.7t1 and substrate induced gap d1 =

Δ t1

= 0.1, the electron occupancies

0.38

Occupation number

0.36

0.09

0.38 0.36 0.34

P

Occupation number

are na, ↑ = na, ↓ for A site and nb, ↑ = nb, ↓ for B site. Here ferromagnetic and anti-ferromagnetic spin orders vanish for this low values of Coulomb energy u¼ 1.7. The electron occupancy at B site is larger

0.32 0.3 0

0.34

0.1 0.2 0.3 0.4 0.5 Temperature (t)

0.085 0.08 0.075 0.07 0.065 0.06

0

0.1 0.2 0.3 0.4 Temperature (t)

0.5

0.32

na nb

0.3 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Temperature (t)

Fig. 1. The variation of occupation number of electrons for both A and B sites vs. temperature (t) for fixed Coulomb potential u¼ 1.7 and substrate induced gap d1 ¼ 0.1. The inset figure shows the variation of pseudo-spin polarisation (p) with temperature (t) for fixed Coulomb potential.

than that of A site. For both the cases, electron occupancies gradually decrease and attain flat minima at temperature t ≈ 0.6 and then slowly increase towards higher temperatures. Due to the difference in electron occupancies for given spin, there exists pseudo-spin polarization (p) which decreases with temperature as shown in inset Fig. 1. The cause of pseudo-spin polarization is explained in Fig. 2. The experimental measurements have shown that the graphene-on-substrate develops gap of Δ ≃ 250 meV for substrate SiC [63], Δ ≃ 200 meV for substrate gold on ruthenium [64] and Δ ≃ 100 meV for substrate boron nitride (BN) [65]. Accordingly, we have considered the dimensionless substrate induced gap Δ d1 = t = 0.036 –0.10. The pseudo-spin polarization (p) increases 1

with increase of substrate induced gap as shown in Fig. 2a. This indicates that the difference in sub-lattice occupancy is enhanced with increase of substrate induced gap. Fig. 2b shows that, with the decrease of substrate induced gap, the occupancy nb, ↑ is suppressed, while the occupancy na, ↑ is enhanced. Ultimately na, ↑ will be equal to nb, ↑ when d1 = 0. Min et al. [66] have reported the study of pseudo-spin version of ferromagnetism due to a competition between band gap and kinetic energies leading to symmetric breaking in bilayer graphene. Thus, it is concluded that the substrate induced gap is the cause of pseudo-spin polarization [66]. The effect of Coulomb repulsive interaction on pseudo-spin polarization or charge polarization is shown in Fig. 3 for given value of substrate induced gap d1 = 0.10. With increase of Coulomb interaction, the polarization (p) is gradually suppressed throughout the temperature range. This indicates that the difference in electron occupancies nb, ↑ and na, ↑ is reduced, indicating that the Coulomb interaction suppresses the substrate induced gap present in the graphene system. This type of behavior is observed for lower values of the Coulomb interaction. Rout et al. [50] reported earlier that the gap opening in graphene-on-substrate attains a maximum value for Uc = 1.7t1 and the magnitude of the gap is suppressed with the increase of Coulomb potential. The DFT calculations by Yazyev and co-workers [34,40] have reported the exchange-correlation functions for U ≈ 1.3t1 using generalized gradient approximation. We have introduced a localized impurity interaction given by Hamiltonian in Eqs. (3) and (4), where xa and xb are the impurity concentrations at A and B sub-lattices respectively. The impurity concentrations may be due to electrons or holes. The impurity concentrations xa and xb may be taken as positive for electrons and negative for holes. The impurities at real space lattice points will appear in the vicinity of K/K′ points (Dirac points) in the Brillouin zone. The localized electron impurity states will appear just above the Dirac point in the conduction band and hole impurity states will appear just below the Dirac point in the valency band. The effect of impurity at A sub-lattice on temperature dependent electron occupancies na, ↑ and nb, ↑ is shown in Fig. 4(a) and (b). In the present calculation, we have considered the impurity concentration xa with impurity potential v at sub-lattice A. In absence of impurity (xa ¼0), the occupancy na, ↑ gradually decreases with increase of temperature. When the hole doping at A sub-lattice is increased i.e. xa ¼0, 0.01, 0.03, the electron occupancy na, ↑ is suppressed throughout the temperature range as shown in Fig. 4(a), Similarly when electron doping at A sub lattice is increased i.e. xa ¼ 0,  0.01,  0.03, the electron occupancy na, ↑ is enhanced throughout the temperature range (see Fig. 4(a)). With hole or electron doping at A sub-lattice, the electron occupancy nb, ↑ exhibits reverse effect. In other words the electron occupancy nb, ↑ is enhanced with increasing of hole doping, while it is suppressed with increase of electron doping as shown in Fig. 4b. With electron doping or hole doping at A site, we have considered the net effect of doping on the temperature dependence of charge or pseudospin polarization as shown in Fig. 4c. With increase of hole doping

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(b) 0.09 d1 = 0.10 d1 = 0.090 d1 = 0.070 d1 = 0.036

0.08 0.07

nb

0.06 0.05 P

Occupation number

(a) 0.39 0.385 0.38 0.375 0.37 0.365 0.36 0.355 0.35 0.345 0.34 0.335 0.33 0.325 0.32 0.315 0.31 0.305 0.3 0.295 0.29

5

0.04

na

0.03 0.02 0.01

0

0.1

0.2 0.3 0.4 Temperature (t)

0.5

0

0

0.1

0.2 0.3 0.4 Temperature (t)

0.5

Fig. 2. (a) shows the variation of occupation numbers of electrons for both A and B sub-lattices vs. temperature (t) for different substrate induced gap d1 = 0.1, 0.09, 0.070, 0.036 and for fixed Coulomb potential u¼ 1.7. Fig. 2(b) shows the variation of pseudo-spin polarization (p) with temperature (t) for different substrate induced gaps and fixed Coulomb potential u ¼1.7. 0.09 u = 1.5 u = 1.7 u = 2.0 u = 2.2 u = 2.5

0.085 0.08

P

0.075 0.07 0.065 0.06 0.055 0.05

0

0.05

0.1

0.15

0.2 0.3 0.25 Temperature (t)

0.35

0.4

0.45

0.5

Fig. 3. The variation of pseudo-spin polarisation (p) with temperature (t) for different Coulomb potential u¼ 1.5,1.7, 2.0, 2.2, 2.5 and fixed substrate induced gap d1 ¼ 0.1.

the pseudo-spin polarization is enhanced throughout the temperature range; while the polarization (p) is suppressed with electron doping throughout the temperature range. It is concluded that the pseudo-spin polarization can be induced in the pristine graphene by introducing either hole or electron doping. The effect of impurity at B sub-lattice on temperature dependent electron occupancies na, ↑ and nb, ↑ is shown in Fig. 5a and b. In the present calculation, we have considered the impurity concentrations xb with impurity potential v at sub-lattice B. In absence of impurity xb ¼ 0, the occupancy na, ↑ gradually decreases with increase of temperature. When hole doping at B sub-lattice is increased, i.e. xb = 0, 0.01–0.03 the electron occupancy na, ↑ is enhanced throughout the temperature range as shown in Fig. 5(a). When electron doping at B sub-lattice is enhanced i.e

xb = 0, − 0.01, − 0.03, the electron occupancy na, ↑ is suppressed through out temperature as shown in Fig. 5(a). With hole or electron doping at B sub-lattices, the electron occupancy nb, ↑ exhibits a reverse effect. In other words the electron occupancy nb, ↑ is enhanced with increase of electron doping at B site, while it is suppressed with increase of hole doping as shown in Fig. 5(b). With electron doping or hole doping at B site, we have considered the net effect of doping on the temperature dependent charge or pseudo-spin polarization as shown in Fig. 5(c). With increase of electron doping, the pseudo-spin polarization is enhanced throughout the temperature range; while the pseudo-spin polarization is suppressed with hole doping throughout the temperature range. It is concluded that the pseudo-spin polarization can be induced in the pristine graphene by introducing either electron or hole doping. From the temperature dependence of polarization (p) shown in Figs. 4c and 5c, we infer that the hole doping at A site sub-lattice enhances the pseudo spin-polarization, while electron doping at B site sub-lattice enhances the pseudo-spin polarization. Hence, in a bi-bipartite lattice like graphene, pseudo-spin or charge polarization can be induced in the system experimentally by hole (electron) doping in A (B) sub-lattice. Peres and coworkers [12] have reported the study of the ferromagnetic zero temperature phase diagram of the two dimensional honey-comb lattice as a function of the strength of the Coulomb interaction and doping with two types of carriers (electrons and holes). 4.2. Case-II: For larger u We consider here the larger Coulomb interactions required to investigate charge polarization as well as spin-polarization. The occupancies at A and B sub-lattices for up and down spin orientations are computed self-consistently and their temperature dependences are shown is shown in Fig. 6a. Fig. 6a shows that the electron occupancy (nb ) at B sub-lattice is higher than electron occupancy (na ) at A sub-lattice for lower Coulomb energy u ¼2.0 giving rise to only charge or pseudo-spin polarization. Fig. 6b

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6

(b)

(a) 0.39

0.39 xa = 0 xa = 0.01 xa = - 0.01 xa = 0.03 xa = -0.03

0.38 0.37 0.36

0.12 0.11

0.38

0.1 0.37

na

0.34 0.33 0.32

nb

0.09

0.36 P

0.35

Occupation number

Occupation number

(c)

0.08

0.35

0.07

0.34

0.06

0.31 0.3

0.05

0.29 0.28

0.33 0

0.1 0.2 0.3 0.4 0.5 Temperature (t)

0

0.1 0.2 0.3 0.4 0.5 Temperature (t)

0.04

0

0.1 0.2 0.3 0.4 0.5 Temperature (t)

Fig. 4. (a) and (b) shows the variation of occupation number of electrons for both A and B sites vs. temperature (t) for different impurity doping concentrations at A site, for xa ¼ 0.01,  0.01, 0.03,  0.03 having impurity potential v¼2 and Coulomb potential u ¼1.7 and substrate induced gap d1 ¼ 0.1. Fig. 4(c) shows the variation of pseudo-spin polarization (p) vs. temperature (t) for different doping concentration at A site.

shows both the electron occupancies na, ↑ (na, ↓ ) and nb, ↑ (nb, ↓ ) which just split for a critical Coulomb interaction Uc = uc t1 = 2.2t1 leading to pseudo-spin polarization as well as spin polarization leading to magnetization. For higher Coulomb energy u = 2.5 > uc , Fig. 6c shows that spin polarization is higher upto the magnetic transition temperature tm = 0.05 ( Tm ≈ 1390 K ) corresponding to nearestneighbor hopping t1 = 2.78 eV ≃ 27800 K . It is observed that the spin polarization reverses its direction for higher temperatures t > tm . This indicates that there exists magnetization in one direction up to temperatures tm and reverses its direction for higher temperature t > tm , for given higher Coulomb energy u ¼2.5 and given substrate induced gap d1 ¼ 0.1. Thus, we observe a complex temperature dependent magnetic effect in graphene. This is more clearly discussed in the following figures. To study how ferromagnetism and antiferromagnetism evolve from electron occupancies for different spin orientations, we show, in Fig. 7, the variation of electron occupancies of A and B sublattices with Coulomb potential (u). We observe that na, ↑ = na, ↓ for sub-lattice A and nb, ↑ = nb, ↓ for sub lattice B up to a critical Coulomb interaction uc ≃ 2.2. This indicates that there is no signature of the existence of magnetization up to uc ≃ 2.2 in graphene. Since electron occupancies for A and B sub lattices are different, there exists only pseudo-spin polarization up to uc ≃ 2.2. For u > uc , it is observed that the electron occupancies for both the sub-lattices are different for different spin orientations and they exhibit oscillatory behavior. The magnetic states exist for u > uc . The FM and

AFM magnetizations are clearly shown below in Fig. 8. Wehling and co-workers [19] have considered local and non-local Coulomb interactions with Hubbard approach and found an effective local Coulomb interaction of U⁎ = 1.6t1 in graphene. The Hubbard model within Hartree–Fock mean-field calculation by Sorella [25] has shown a transition from nonmagnetic semi-metal to an antiferromagnetic insulator in 2D honey-comb lattice for U = 2.23t1. The present critical Coulomb potential of uc ≈ 2.2 agrees with the above results. The quantum Monte-Carlo calculation yields antiferromagnetic transition for higher Coulomb energy UAFM ≥ 4.5t1 [25–27]. We show, in Fig. 8, how Coulomb interaction brings about the change of strengths of FM magnetization (m) and AFM magnetization (pm ). It is observed that both magnetizations are m ¼0 and pm = 0 up to Coulomb energy uc ≃ 2.2. Then both the magnetizations exhibit oscillatory behavior for higher Coulomb interactions i.e. u > uc . It is to note further that, the oscillatory FM and AFM states are oppositely directed for u > uc . The temperature dependent ferromagnetic (FM) magnetization (m) and antiferromagnetic (AFM) sub-lattice magnetization (pm) are shown in Fig. 9 for a higher fixed Coulomb potentials u = 2.5 > uc . The FM phase exists upto Curie temperature tc1 ≃ 0.05 (Tc1 ≃ 1390 K) corresponding to nearest-neighbor hopping integral t1 = − 2.78 eV ≃ 27800 K . For t > tc1, the magnetization reverses its direction up to another transition at tc2 ≃ 0.3 and then the FM phase reverses its direction for still higher

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(b)

(a) 0.39 xb = 0 xb = 0.01 xb = -0.01 xb = 0.03 xb = -0.03

Occupation number

0.36 0.35

na

0.34 0.33 0.32

(c)

0.39

0.12

0.38

0.11 nb

0.1

0.37

0.09 0.36 P

0.37

Occupation number

0.38

7

0.08

0.35 0.07 0.34

0.06

0.31 0.33

0.05

0.3 0.29

0

0.1 0.2 0.3 0.4 0.5 Temperature (t)

0.32

0

0.1 0.2 0.3 0.4 0.5 Temperature (t)

0.04

0

0.1 0.2 0.3 0.4 0.5 Temperature (t)

Fig. 5. (a) and (b) shows the variation of occupation number of electrons for both A and B sites vs. temperature (t) for different impurity doping concentration at B site for xb ¼ 0.01,  0.01, 0.03,  0.03 having impurity potential v¼2 and Coulomb potential u ¼1.7 on substrate induced gap d1 ¼ 0.1. Fig. 5(c) shows the variation of pseudo-spin polarization (p) vs. temperature (t) for different doping concentration at B site.

temperature. In other words, for higher Coulomb interactions, the ferromagnetic state exhibits oscillatory behavior for Coulomb interaction u ¼2.5 for a fixed substrate induced gap d1 ¼ 0.1. The AFM magnetization (pm ) decreases with temperature up to the Ne´els temperature tN1 ≃ 0.055 ( TN1 = 1529 K ) and then reverses its direction up to another Ne´els temperature tN2 = 0.3. Beyond this temperature, the oscillatory behavior of AFM magnetization pm becomes smaller and then becomes zero. It is concluded that oscillatory behavior of spin polarization with temperature exists for given higher value of Coulomb interaction u > uc and then it subsides and becomes steady at higher temperatures. The low temperature dependence of m and pm is shown in inset of Fig. 9. which shows that the anti-ferromagnetic order exists in opposite direction to that of ferromagnetic state in graphene for u ¼2.5. The origin of oscillatory behavior FM(m) and AFM (pm) magnetizations for higher Coulomb potentials u > uc is not obvious. However, it may be associated with the gap (d1) arising due to the substrate effect. The magnetization (m and pm) are oriented in one direction for 2.2uc1 ≤ u ≤ uc2 = 4 . For high Coulomb potentials, the spin orders in A and B sub-lattices in one temperature range reverses direction in the next higher temperature range leading to such oscillatory nature. The upper critical Coulomb energy uc2 = 4 for anti-ferromagnetic transition approximately agrees with UAFM ≥ 4.5t1 obtained from Monte-Carlo simulation results [25–27]. The effect of different substrate induced gaps on ferromagnetic

magnetization (m) antiferromagnetic magnetization (pm ) is shown in Fig. 10. It is experimentally observed that the substrate induced gaps in graphene are Δ ≃ 250 meV for substrate SiC [63], Δ = 200 meV for gold on ruthenium [64] and Δ = 100 meV for BN [65]. To study the substrate induced magnetization in graphene for Coulomb interaction u ¼2.5, we consider here the effect of reduced gaps d1 ¼0.1 and d1 = 0.12 on magnetization in graphene. For the given gap d1 ¼0.1, the AFM magnetization exhibits nearly meanfield temperature dependence up to Ne´els temperature tN ≃ 0.05. Similarly the FM magnetization (m) also exhibits meanfield behavior upto Curie temperature tc ≃ 0.05 for the same u ¼2.5. When the substrate induced gap is enhanced to d1 = 0.12, the Ne´els temperature is enhanced to tN = 0.125 and Curie temperature to tc = 0.13. Under this condition, the magnetization ‘m’ and ‘pm’ do not exhibit meanfield behavior, but they are directed in opposite directions. Thus, it is concluded that the substrate induced gap in graphene enhances the magnetic orders. The effect of substrate induced gap on temperature dependent charge polarization or pseudo-spin polarization (p) is shown in inset of Fig. 10. The polarization (p) slowly decreases with temperature for a given Coulomb interaction u ¼2.5 and given gap d1. The magnitude of the pseudo-spin polarization is enhanced with increase of the substrate induced gap leading to the more asymmetry in charge densities of two sub-lattices A and B. Most of the calculations in graphene physics involve nearestneighbor-hopping integral (t1 = − 2.78 eV to 3 eV) with dispersion

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(b)

(a) 0.37

(c)

0.37

0.37 u = 2.2

u = 2.0 0.365

na na nb nb

0.365

0.365 0.36

0.355

0.355

0.355

0.35

0.35 0.345 0.34

Occupation number

0.36

Occupation number

Occupation number

0.36

0.35 0.345 0.34

u = 2.5

0.345 0.34 0.335 0.33 0.325

0.335

0.335

0.32

0.33

0.33

0.315

0.325

0.325

0.31

0.32

0

0.32

0.04 0.08 0.12 Temperature (t)

0.305 0

0.3

0.04 0.08 0.12 Temperature (t)

0

0.04 0.08 0.12 Temperature (t)

Fig. 6. The variation of occupation numbers of electrons for both spins at A and B sites vs. temperature (t) for different Coulomb potentials u ¼2.0, 2.2, and 2.5 and fixed substrate induced gap d1 ¼ 0.1.

0.3

0.48 na

0.2

nb

0.4

0.15

nb 0.36

Magnetization

Occupation number

m Pm

0.25

na 0.44

0.32 0.28

0.1 0.05 0 -0.05

0.24

-0.1

0.2 -0.15

0.16 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

-0.2

Coulomb potential (u)

Fig. 7. The variation of occupation numbers of electrons for both spins at A and B sites vs. Coulomb potential (u) for temperature t ¼0 and fixed substrate induced gap d1 ¼0.1.

near the Dirac point only. The value of second-nearest-neighbor hopping integral (t2) is not well known, but ab-initio calculations by Reich et al. [67] show 0.02t1 ≤ t2 ≤ 0.2t1 depending on the tightbinding parameters. These calculations also include the effect of a third-nearest-neighbor-hopping integral which has value t3 ≃ 0.07 eV . A tight-binding fit to cyclotron resonance experiment

1

1.5

2

3

2.5

3.5

4

4.5

5

Coulomb potential (u)

Fig. 8. The variation of ferromagnetism (m) and antiferromagnetism (pm) vs. Coulomb potential (u) for fixed temperature t ¼0 and fixed substrate induced gap d1 ¼0.1.

by Deacon et al. [68] find the value t2 = 0.1 eV . In present calculation, we have employed the following values of the hopping integrals i.e t1 = − 2.78 eV , t2 = − 0.12 eV and t3 = − 0.066 eV . The temperature dependent pseudo-spin polarization is shown in Fig. 11 for the dimensionless hopping parameters t˜1 = − 1, t t t˜3 = 3 = − 0.024 . The polarization p t˜2 = 2 = − 0.043 and t1

t1

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magnetization (m, Pm)

0.008 Pm 0.006

Magnetization (m, Pm)

0.004 0.002

0.008 Pm 0.004 0 m

-0.004 0

0.01 0.02 0.03 0.04 0.05 0.06 t

0 -0.002 m -0.004

9

decreases with temperature for given set of hopping parameters. The polarization p is enhanced throughout the temperature range with the increase of different hopping integrals. The magnitudes of polarization at t¼ 0 are p ¼0.078 for t˜1 = − 1; p ¼0.081 for t˜1 = − 1, t˜2 = − 0.043 and p¼ 0.0825 for hopping integral upto third-nearest-neighbors. This indicates that the pseudo-spin polarization (here charge gap) is enhanced with inclusion of higher order hopping integrals. Sahoo and Rout [50] have reported that the substrate induced gap is enhanced with higher order hopping integrals in graphene-substrate. Castro Neto et al. [15] have observed no gap in electronic dispersion in the honey-comb lattice at Dirac point for t1 = − 2.7 eV and a band gap for t1 = − 2.7 eV and t2 = − 0.02t1 eV .

-0.006 0

0.05

0.1

0.15

0.2

0.3

0.25

0.4

0.35

0.45

0.5

Temperature (t)

Fig. 9. The variation of ferro-magnetics magnetization (m) and anti-ferromagnetic magnetizations (pm) vs. temperature (t) for fixed Coulomb potential u ¼2.5 and fixed substrate-induced gap d1 ¼ 0.1. The inset figure shows the cross over transition of both magnetism.

0.008 0.075 0.006

0.07 P

Pm

magnetization (m, Pm)

0.065 0.004

0.06 0

0.002

0.1

0.2

0.3

0.4

0.5

t

0

m

-0.002

d1 = 0.1 d1 = 0.12 -0.004 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Temperature (t)

P

Fig. 10. The variation of ferromagnetic magnetization (m) and sub lattice magnetization (pm) vs. temperature (t) for fixed Coulomb potential u ¼2.5 and different substrate-induced gap d1 = 0.10, 0.12. The inset figure shows the variation of pseudo-spin polarization (p) vs. temperature (t) for different substrate induced gaps.

0.083 0.082 0.081 0.08 0.079 0.078 0.077 0.076 0.075 0.074 0.073 0.072 0.071 0.07 0.069 0.068 0.067 0.066

~ t1 = -1 ~ ~ t1 = -1, t2 = - 0.043 ~ ~ ~ t1 = -1, t2 = - 0.043 , t3 = - 0.024

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Temperature (t)

Fig. 11. The plot off pseudo-spin polarization (p) vs. temperature (t) for Coulomb interaction u ¼1.7, substrate induced gap d1 ¼0.1 for different values of hopping integrals.

5. Conclusion In the present report, we propose a tight-binding model calculation for doped graphene-on-substrate consisting of the hopping of pz electrons up to third-nearest-neighbors. For the graphene-on-substrate, we introduce a symmetry breaking interaction such that the A sub-lattice acquires an energy +Δ and B sublattice acquires energy −Δ giving raising to the total gap of 2Δ. Further, we have introduced on-site Coulomb repulsive interaction at A and B sub-lattices with same effective Coulomb potential U. Here the Coulomb interaction in the Hamiltonian is considered within Hartree–Fock mean-field formalism. The total Hamiltonian is solved by using the electron Green's function technique of Zubarev. The temperature dependent electron occupancies at A and B sub-lattices are calculated and computed numerically and selfconsistently. Finally the temperature dependent charge- and spinpolarizations for antiferromagnetic and ferromagnetic orders are computed for different values of Coulomb potentials, substrate induced gaps and impurities. Our earlier calculation [49,50] estimates a maximum gap in substrate induced gap in graphene at 0 K for critical Coulomb interaction Uc = uc t1 = 1.7t1. For the same u ¼1.7 in the lower limits of Coulomb potential in the present calculation, we find that the electron occupancy at B site becomes higher than that of occupancy at A site leading to a nonmagnetic temperature dependent pseudo-spin polarization or a charge gap. For a given u = 1.7, the decrease of substrate induced gap suppresses B site electron occupancy (nb) and enhances the A site electron occupancy (na)leading to the condition na = nb for which the pseudo-spin polarization gap vanishes. The pseudospin polarization (or a gap) is enhanced with the increase of the substrate induced gap. It is further noted that the temperature dependent substrate induced pseudo-spin polarization is suppressed with the increase of Coulomb interaction. This indicates that Coulomb interaction leads to the charge neutrality of both the sub-lattices of the bi-bipartite lattice of the graphene system. For low u, the impurity in graphene exhibits interestingly effects. For given Coulomb potential and substrate induced gap, the A site electron occupancy is enhanced with A site electron doping, while it is suppressed with A site hole doping. Similar effect is observed for B site occupancy with the B site electron and hole doping. Finally we conclude that the pseudo-spin polarization can be enhanced in pristine graphene by introducing electron and hole dopings in graphene system. Further, we observe that A and B electron occupancies tend to split for different spin-orientations for a of a critical coulomb interaction uc = 2.2 leading to the generation of magnetism in graphene. For u = 2.25 > uc , A and B site electron occupancies reverse their magnitudes for up and down spin orientations at temperature t ≃ 0.05. This leads to the occurrence of interesting magnetism in graphene-on-substrate. The variation of Coulomb potential

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shows that the electron occupancies exhibit no magnetization up to uc = 2.2 and then the electron occupancies exhibit oscillatory variation for two spin orientations of two sub-lattices for u > uc . Similarly there is no occurrence of magnetization up to u = uc = 2.2. However the ferromagnetic magnetization (m) and anti-ferromagnetic magnetization (pm) exhibit oscillatory behavior for u > uc . The temperature dependent FM magnetization (m) and AFM magnetization (pm) exhibit oscillatory behavior which dies down at higher temperatures. We observe low temperature magnetic transition at temperature tc1 = tN1 ≃ 0.05 and another transition temperature tN2 ≃ 0.3. To the best of our knowledge, this type of oscillatory magnetic order is reported for the first time in the substrate induced graphene with Coulomb interaction of (with nearest-neighbor-hopping integral U ≃ 2.5t1 = 6.95 eV t1 = − 2.78 eV ). It is to note further that the substrate induced gap also enhances the AFM and FM transition temperatures to higher values.

Acknowledgments The authors gracefully acknowledge the research facilities offered by the Institute of Physics, Bhubaneswar, India. The author (Sivabrata Sahu) also acknowledges the leave granted by the authorities of Synergy Institute of Engineering and Technology, Dhenkanal, Odisha to continue research work.

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Please cite this article as: S. Sahu, G.C. Rout, Journal of Magnetism and Magnetic Materials (2016), http://dx.doi.org/10.1016/j. jmmm.2016.01.062i