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Numerical modelling of doped graphane superconducting and normal properties To cite this article: N A Kudryashov and A A Kutukov 2017 J. Phys.: Conf. Ser. 937 012024
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MPMM IOP Conf. Series: Journal of Physics: Conf. Series 1234567890 937 (2017) 012024
IOP Publishing doi:10.1088/1742-6596/937/1/012024
Numerical modelling of doped graphane superconducting and normal properties N A Kudryashov, A A Kutukov National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), 31 Kashirskoye shosse, 115409, Moscow, Russia E-mail:
[email protected] Abstract. The normal properties of graphane with various degrees of doping are calculated with the help of the generalized Eliashberg theory. Within the theory of strong electron-phonon interaction, a superconducting order parameter of the doped graphane has been found. The critical temperature of superconductivity of graphane has been calculated as a function of the doping degree taking into account the renormalization of the electron mass, the chemical potential, and the density of electronic states.
1. Introduction Graphene is the two-dimensional hexagonal lattice of carbon atoms. This material was discovered and obtained in 2004 [1]. As a result of the interaction of graphene with atomic hydrogen, a new substance called graphane with the chemical formula CH can be formed. The existence of graphane was theoretically predicted in 2006 [2] and the work devoted to experimental obtained graphane was published in 2009 [3]. Graphane is obtained by reversible hydrogenation of graphene and retains two-dimensional hexagonal lattice. In order to the graphane has a metallic conductivity and can transform into a superconducting state it must be doped. It is proposed in [4] to dope graphane similarly to B-doped diamond and study the electronic structure of the substance based on the density functional theory. In [4] using the Eliashberg function the electron-phonon interaction constant was calculated and the critical temperature of the doped graphane was estimated from the McMillan formula. In this paper we calculate the critical temperature of the doped graphane taking into account the renormalization of the density of electronic states, the electron mass, and the chemical potential. 2. Numerical study of normal and superconducting properties of doped graphane To determine the critical temperature of superconductivity for a strong electron-phonon interaction, the generalized Eliashberg theory [5] is used. To calculate the normal state, a system of nonlinear integral equations (1)-(3) is used for the real and imaginary parts of the selfenergy part of the electron Green’s function and the renormalization of the density of electronic
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MPMM IOP Conf. Series: Journal of Physics: Conf. Series 1234567890 937 (2017) 012024
IOP Publishing doi:10.1088/1742-6596/937/1/012024
states [6]. Real part of the self-energy part of the electron Green’s function +∞ +∞ Nn (−z ) Nn (z ) 2 dzα (z)F (z) dz f (−z ) − ReΣ(ω) = −P + z + z + ω z + z − ω 0 0 (−z ) N Nn (z ) n , +f (z ) − + z − z + ω z − z − ω
(1)
where f (z ) = ez /T1 +1 is the Fermi distribution. Imaginary part of the self-energy part of the electron Green’s function +∞ ImΣ(ω) = −π dzα2 (z)F (z) [Nn (ω − z) + Nn (ω + z)] nB (z) (2) 0 +Nn (ω − z)f (z − ω) + Nn (ω + z)f (z + ω) , nB (z) = ez /T1 −1 is the Bose distribution. The renormalized by the electron-phonon interaction electronic density of states ∞ 1 ImΣ(z ) Nn (z ) = − dξ N0 (ξ ) . (3) π [z − ξ − ReΣ(z )]2 + [ImΣ(z )]2 −μ
Figure 1 shows the results of solving the system of equations (1)-(3) for normal state of 1% doped graphane. The frequency is expressed in dimensionless units corresponding to the limiting frequency of the phonon spectrum 0.158 eV.
Figure 1. The real and the imaginary part of the self-energy part of the electron Green’s function for 1% doped graphane and T=60K The superconducting properties of doped graphane are described by a system of nonlinear integral equations for the superconducting order parameter [7], which are solved numerically by the iteration method. Figure 2 shows the results of solving the system of equations for superconducting state of 8% doped graphane for 25 and 30 iterations. The frequency is expressed in dimensionless units corresponding to the limiting frequency of the phonon spectrum 0.164 eV. 2
MPMM IOP Conf. Series: Journal of Physics: Conf. Series 1234567890 937 (2017) 012024
IOP Publishing doi:10.1088/1742-6596/937/1/012024
Figure 2. The real and the imaginary part of the order parameter for 8% doped graphane and T=90K 3. Conclusion The critical temperature of superconductivity is found by determining the transition to the zero value of the order parameter. For 1% and 8% of the doped graphane the critical temperature of superconductivity T = 70 K and T = 90 K respectively was found. The calculations for intermediate degrees of doping indicate that the superconducting transition temperature increases with increasing degree of doping. When performing the calculations, we used the Eliashberg function and the density of electronic states found in [4]. Acknowledgments This work was supported by the Ministry of Education and Science of the Russian Federation (base part of state task, project no. 1.9746.2017/BCh) References [1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsov A A 2004 Science 306 (5696) 666-9. [2] Sofo J O, Chaudhari A S, Barber G D 2006 Graphane: a two-dimensional hydrocarbon arXiv:condmat/0606704. [3] Elias D C, Nair R R, Mohiuddin T M G, Morozov S V, Blake P, Halsall M P, Ferrari A C, Boukhvalov D W, Katsnelson M I, Geim A K, Novoselov K S 2009 Science 323 (5914) 610-3. [4] Savini G, Ferrari A C, Giustino F 2010 Phys. Rev. Lett. 105 037002-1-4. [5] Mazur E A 2010 Europhys. Lett. 90 47005-16. [6] Kudryashov N A, Kutukov A A and Mazur E A 2016 J. Exp. Theor. Phys 123 (3) 481-8. [7] Kudryashov N A, Kutukov A A and Mazur E A 2016 JETP Letters 104 (7) 460-5.
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