Magnetization of Epitaxial Graphene Induced by Magnetic Metallic Substrate

Abstract

A model of the energy diagram is proposed for graphene (Gr) monolayer formed on a magnetic transition metal (MTM). The regimes of the strong and weak Gr–MTM coupling are considered; the analytical expressions have been derived for the magnetization induced by a substrate in epitaxial graphene in these regimes. It is shown that, in the strong coupling regime, a quasi-local state gives the main contribution to the magnetization; in this case, the coupling has an antiferromagnetic character. In the weak coupling regime, the main contribution to the magnetization is given by states of the d band of MTM, which leads to the ferromagnetic character of the coupling. The numerical estimations are performed for the Gr/Ni(111) and Gr/Co(poly) and they are compared with the experimental data for Gr/Ni(111) and the calculated results for Cr/Co/Ni(111). In the first case, the results agree quantitatively and, in the second case, qualitatively.

APPENDIX

Consider the problem of adsorption of an individual atom on MTM. In this case, for the density of states, we obtain, instead of Eq. (7), expression

$${{\bar {\rho }}_{\sigma }}(\omega ) = \frac{1}{\pi }\frac{{{{\Gamma }_{\sigma }}(\omega )}}{{[{{{(\omega - {{\varepsilon }_{0}} - {{\Lambda }_{\sigma }}(\omega )]}}^{2}} + \Gamma _{\sigma }^{2}(\omega )}},$$

((A1))

where ε₀ is the adatom quasi-level energy, Γ_σ(ω) and Λ_σ(ω) are given by Eqs. (4) and (5). Assume that the adatom orbital energy ε₀ = 0, which corresponds to the Dirac point. The matter is that here (in the strong coupling regime Γ/t$\gg$ 1) we consider a pseudo-isolated epigraphene atom, rather than a solitary carbon adatom. The maxima of density of states (A1) correspond to the roots of equation

$$\omega - {{\Lambda }_{\sigma }}(\omega ) = 0.$$

((A2))

If inequality 1 < ${{(d{{\Lambda }_{\sigma }}(\omega ){\text{/}}d\omega )}_{{{{\omega }_{{0\sigma }}}}}}$, there is only one root $\omega _{\sigma }^{*}$ in the region of the continuous spectrum (Fig. 1b). Since dΛ_σ(ω)/dω = W_dΓ/π[(W_d/2)² – (ω – ω_0σ)²], we obtain πW_d/4Γ < 1, which is valid for the Gr/Ni(111) system, according to the estimations presented in the text. Assuming that |$\omega _{\sigma }^{*}$ – ω_0σ| $\ll$W_d/2, we have Λ_σ(ω) ≈ –4(ω – ω_0σ)Γ/πW_d, and, instead of Eq. (A2), we have $\omega _{\sigma }^{*}$ + 20(V_d/W_d)²($\omega _{\sigma }^{*}$ – ω_0σ) = 0. Because, according to the abovementioned estimations, 20(V_d/W_d)²$\gg$ 1, we have $\omega _{\sigma }^{*}$ ≈ ω_0σ. Then, in the vicinity of the maximum, the density of states can be represented as ${{\bar {\rho }}_{\sigma }}(\omega )$ ≈ Γ/π[(ω – ω_0σ)² + Γ²]. Since E_F = 0, we obtain the contribution of band states (${{\bar {n}}_{\sigma }}$)_band to the occupation number ${{\bar {n}}_{\sigma }}$ from Eq. (10) as

$${{({{\bar {n}}_{\sigma }})}_{{{\text{band}}}}} \approx \frac{1}{\pi }\left( {\arctan \frac{{{{W}_{d}}}}{{2\Gamma }} - \frac{{{{\omega }_{{0\sigma }}}}}{\Gamma }} \right).$$

((A3))

In the case that |$\omega _{\sigma }^{*}$ – ω_0σ| $\gg$W_d/2, we have two solutions of Eq. (A2) corresponding to local levels with energies $\omega _{\sigma }^{ + }$ ≈ $\pm {{V}_{d}}\sqrt 5$ + ω_0σ/2. We refine: these levels can be taken to be local only with respect to MTM, since they lie off the dσ subband of a metal (Fig. 1b). Actually, these levels are quasi-local or resonant, since they are overlapping with the continuous spectrum of graphene. However, we will perform estimations assuming their local, since the density of states of MTM is much higher than the graphene density of states (Fig. 1). It is easy to see that this simplification is of the same order as the neglect of the s band of MTM.

The occupation number of level $\omega _{\sigma }^{ - }$ calculated by formula (${{\bar {n}}_{\sigma }}$)_loc = $\left| {1 - d{{\Lambda }_{\sigma }}(\omega ){\text{/}}d\omega } \right|_{{\omega _{\sigma }^{ - }}}^{{ - 1}}$ [29] is

$${{({{\bar {n}}_{\sigma }})}_{{{\text{loc}}}}} \approx \frac{1}{2}\left( {1 + \frac{{{{\omega }_{{0\sigma }}}}}{{{{V}_{d}}\sqrt 5 }}} \right).$$

((A4))

The resulting magnetization counted per one epigraphene atom is $\bar {m}$ = ${{\bar {n}}_{ \uparrow }}$ – ${{\bar {n}}_{ \downarrow }}$, where ${{\bar {n}}_{\sigma }}$ = (${{\bar {n}}_{\sigma }}$)_band + (${{\bar {n}}_{\sigma }}$)_loc. Using Eqs. (A3) and (A4), we obtain Eq. (13) of this report. Since the summary occupation number of epigraphene atom is $\bar {n}$ = $\sum\nolimits_\sigma {[{{{({{{\bar {n}}}_{\sigma }})}}_{{{\text{band}}}}}}$ + ${{({{\bar {n}}_{\sigma }})}_{{{\text{loc}}}}}]$, then

$$\bar {n} \approx 1 + \frac{2}{\pi }\arctan \frac{{{{\eta }^{2}}}}{{2\pi }},$$

((A5))

where η = W_d/V_d$\sqrt 5$. In the framework of the Harrison theory, we have η ≈ 0.63, so that $\bar {n}$ ≈ 1.04. We emphasize that the local state gives the main contribution to $\bar {n}$. Since the model energy diagram considered in this work for the Gr/Ni(111) is symmetric with respect to E_F = 0 (Fig. 1); the exact value of the occupation number must be 1. Thus, the error is only 4%.