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Dynamical and Static Spin Susceptibilities of Doped Gapped Graphene Nanoribbon Due to Local Electronic Interaction

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Abstract

We present the behaviors of both dynamical and static spin susceptibilities of doped gapped armchair graphene nanoribbon using the Green’s function approach in the context of Hubbard model Hamiltonian. Specially, the effects of spin polarization and gap parameter on the spin excitation modes of armchair graphene nanoribbon are investigated via calculating correlation function of spin density operators. Our results show the increase of electron concentration leads to disappear low frequency spin excitation mode for gapless case. We also show that low frequency excitation mode for both gapped and gapless nanoribbon disappears under spin full polarization condition. Finally, the electron density dependence of static charge structure factor of armchair graphene nanoribbon is studied. The effects of both gap parameter and magnetic ordering on the static structure factor are discussed in details.

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Authors and Affiliations

  1. Department of Physics, Razi University, Kermanshah, Iran

    H. Rezania & A. Abdi

Authors
  1. H. Rezania
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  2. A. Abdi
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Correspondence to H. Rezania.

Appendix: The Explicit Expressions of Factors in Eq. 18

Appendix: The Explicit Expressions of Factors in Eq. 18

In this Appendix, we present the full expression of factors which has been mentioned in Eq. 18.

$$\begin{array}{@{}rcl@{}} \mathcal{F}&=&{f^{2}_{1}}(k_{x},p){f^{2}_{1}}(k_{x}+q_{x},p^{\prime})+ f_{1}(k_{x},p)f_{3}(k_{x},p)f_{1}(k_{x}+q_{x},p^{\prime})f^{*}_{3}(k_{x}+q_{x},p^{\prime}) \\&&+c.c.+ |f_{3}(k_{x},p)|^{2}|f_{3}(k_{x}+q_{x},p^{\prime})|^{2}|\\ \mathcal{I}&=&{f^{2}_{1}}(k_{x},p){f^{2}_{2}}(k_{x}+q_{x},p^{\prime})+ f_{1}(k_{x},p)f_{3}(k_{x},p)f^{*}_{4}(k_{x}+q_{x},p^{\prime})f_{2}(k_{x}+q_{x},p^{\prime})\\&&+c.c.+ |f_{3}(k_{x},p)|^{2}|f_{4}(k_{x}+q_{x},p^{\prime})|^{2}|\\ \mathcal{J}&=&{f^{2}_{1}}(k_{x},p){f^{2}_{1}}(k_{x}+q_{x},p^{\prime})+ f_{1}(k_{x},p)f_{3}(k_{x},p)f^{*}_{3}(k_{x}+q_{x},p^{\prime})f_{1}(k_{x}+q_{x},p^{\prime})\\&&+c.c.+ |f_{3}(k_{x},p)|^{2}|f_{3}(k_{x}+q_{x},p^{\prime})|^{2}|\\ \mathcal{K}&=&{f^{2}_{1}}(k_{x},p){f^{2}_{1}}(k_{x}+q_{x},p^{\prime})+ f_{1}(k_{x},p)f_{3}(k_{x},p)f_{4}^{*}(k_{x}+q_{x},p^{\prime})f_{2}(k_{x}+q_{x},p^{\prime})\\&&+c.c.+ |f_{3}(k_{x},p)|^{2}|f_{4}(k_{x}+q_{x},p^{\prime})|^{2}|\\ \mathcal{L}&=&{f^{2}_{2}}(k_{x},p){f^{2}_{1}}(k_{x}+q_{x},p^{\prime})+ f_{2}(k_{x},p)f_{4}(k_{x},p)f^{*}_{3}(k_{x}+q_{x},p^{\prime})f_{1}(k_{x}+q_{x},p^{\prime})\\&&+c.c.+ |f_{4}(k_{x},p)|^{2}|f_{3}(k_{x}+q_{x},p^{\prime})|^{2}|\\ \mathcal{M}&=&{f^{2}_{2}}(k_{x},p){f^{2}_{2}}(k_{x}+q_{x},p^{\prime})+ f_{2}(k_{x},p)f_{4}(k_{x},p)f^{*}_{4}(k_{x}+q_{x},p^{\prime})f_{2}(k_{x}+q_{x},p^{\prime})\\&&+c.c.+ |f_{4}(k_{x},p)|^{2}|f_{4}(k_{x}+q_{x},p^{\prime})|^{2}|\\ \mathcal{N}&=&{f^{2}_{2}}(k_{x},p){f^{2}_{1}}(k_{x}+q_{x},p^{\prime})+ f_{2}(k_{x},p)f_{4}(k_{x},p)f_{3}(k_{x}+q_{x},p^{\prime})f_{1}(k_{x}+q_{x},p^{\prime})\\&&+c.c.+ |f_{4}(k_{x},p)|^{2}|f_{3}(k_{x}+q_{x},p^{\prime})|^{2}|\\ \mathcal{N}&=&{f^{2}_{2}}(k_{x},p){f^{2}_{2}}(k_{x}+q_{x},p^{\prime})+ f_{2}(k_{x},p)f_{4}(k_{x},p)f^{*}_{4}(k_{x}+q_{x},p^{\prime})f_{2}(k_{x}+q_{x},p^{\prime})\\&&+c.c.+ |f_{4}(k_{x},p)|^{2}|f_{4}(k_{x}+q_{x},p^{\prime})|^{2}| \end{array} $$
(21)

The functions f 1,f 2,f 3,and f 4 have been introduced in Eq. 9.

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Rezania, H., Abdi, A. Dynamical and Static Spin Susceptibilities of Doped Gapped Graphene Nanoribbon Due to Local Electronic Interaction. Plasmonics 13, 845–856 (2018). https://doi.org/10.1007/s11468-017-0580-7

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