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Dispersion equation, electromagnetic wave slowing down and electron beam instability

Dispersion equation, electromagnetic wave slowing down and electron beam instability

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310



Figure 1. The instability increment (1/cm) versus wave vector k for graphene bilayer.



4. Conclusion

Dispersion equation for wave propagation in a graphene layer results in the

possibility of the Cherenkov instability stimulated by the action of electron

beam.



References

1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang,

S. V. Dubonos, I. V. Grigorieva, A. A. Firsov, Science 306, 666 (2004).

2. G. Ya. Slepyan,

S. A. Maksimenko, A. Lakhtakia, O. Yevtushenko,

A. V. Gusakov, Phys. Rev. B 60, 17136 (1999).

3. K. Batrakov, P. Kuzhir, S. Maksimenko, C. Tomsen, Phys. Rev. B 79,

125408 (2009).



PHYSICS, CHEMISTRY AND APPLICATION OF NANOSTRUCTURES, 2011



FIRST PRINCIPLES STUDY OF HYDROGEN-INDUCED

DECOUPLING OF EPITAXIAL GRAPHENE

FROM SiC SUBSTRATES

A. MARKEVICH, R. JONES

School of Physics, University of Exeter, Stocker Road

Exeter EX4 4QL, United Kingdom

S. ÖBERG

Department of Mathematics, Luleå University of Technology

Luleå S-97187, Sweden

P. R. BRIDDON

School of Electrical, Electronic and Computer Engineering, Newcastle University

Newcastle upon Tyne NE1 7RU, United Kingdom

Density functional theory calculations have been used to investigate possible

mechanisms of hydrogen-induced decoupling of graphene from SiC(0001). The results

suggest that hydrogen atoms reach SiC surface through extended defects in graphene

layers and then migrate and passivate bonds on the SiC surface.



1. Introduction

Graphene has a great potential for applications such as nanoelectronics,

medicine, gas detection, hydrogen storage, etc. At the moment, the only feasible

way for the production of large scale homogeneous graphene samples is the

epitaxial growth on silicon carbide (SiC) substrates [1-3]. However, electronic

properties of epitaxial graphene on SiC are significantly affected by the

interaction with the substrate [4-6]. The first graphene layer grown on SiC(0001)

(Si terminated SiC surface) is partially sp3-hybridized by covalent bonds with the

substrate and, thus, no linear dispersion of π-bands characteristic for free

standing graphene is observed. This interface graphene layer is non-conductive

and is often called buffer layer (BL). The next carbon layer above the buffer

layer behaves electronically as a monolayer graphene. However, this layer was

found to be electron doped (n≈1013 cm-2) [4] due to the charge transfer from the

substrate. It was also suggested that the strong substrate influence is responsible

for the considerable reduction of carrier mobility in this layer, which does not

exceed 2000 cm2/Vs at low temperatures [3]. The subsequently grown graphene

layers have the AB stacking symmetry when grown on Si face [4,7]. It was

shown recently that annealing in molecular or atomic hydrogen ambient at

311



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temperatures above 600 °C resulted in decoupling the BL from the SiC substrate,

and it exhibited the electronic properties characteristic for an isolated graphene

sheet [7,8]. It was suggested that hydrogen atoms penetrate below the graphene

layer, break graphene/SiC covalent bonds and saturate surface dangling bonds.

This process was found to be reversible with the reverse transformation started at

about 700 °C and associated with hydrogen desorption from the SiC surface.

However, the mechanism of hydrogen penetration between SiC(0001) and BL is

not clear. Moreover, in the recent experimental study of atomic hydrogen

adsorbates on epitaxial graphene on SiC(0001) no evidence of hydrogen

penetration below the BL was observed even at 800 °C [9].

In order to elucidate the mechanism of hydrogen-induced decoupling of

graphene buffer layer from the SiC substrate, interaction of hydrogen atoms with

the BL and possible ways of hydrogen penetration below the BL have been

investigated with the use of density functional theory (DFT). For comparison,

similar calculations have been made for a separate graphene layer.

2. Details of calculations

All calculations were performed using AIMPRO DFT code [10]. We use local

density approximation (LDA) and pseudopotentials by Hartwigsen et al. [11].

All structures were modeled with periodic boundary conditions. The 4H-SiC

substrate was represented by 4 bilayers of SiC. A vacuum space of 25 Å was

included above the slab. The buffer layer was modeled by placing a flat

graphene layer on top of the SiC(0001) surface and subsequent optimization of

positions of all atoms. In our calculations we used two different supercells: 4×4

SiC with 5×5 graphene and 3 × 3R30 (R3) SiC with 2×2 graphene. Although,

the R3 model was shown to reproduce qualitatively well the nature of

graphene/SiC(0001) interface [5], it requires 8 % extension of the graphene

lattice constant for SiC and graphene cells to commensurate. Our calculations

show that this results in big errors in modeling of the hydrogen migration

through the BL. However, the R3 model is more appropriate for calculations of

hydrogen migration on the SiC surface below the BL. For modeling of separate

graphene layers we used 4×4 graphene cell.

Migration paths for a hydrogen atom were modeled using nudged elastic

band (NEB) method. The energy barrier for migration was determined as the

total energy difference between a saddle point and the most stable initial

configurations. The binding energy was calculated as the total energy difference

between the structure with a hydrogen atom far from the slab and when

chemisorbed on the required site.



313



3. Results and discussion

The following requirements have to be met for an occurrence of hydrogen atom

migration through the buffer layer and saturation of the SiC surface: 1) H atoms

have enough energy to overcome the energy barrier for migration and 2) the

process is energetically favorable. For the initial and final positions of hydrogen

atom diffusion path we adopted the most favorable chemisorption site of H atom

on the BL and on the SiC surface below the BL, respectively. The binding

energies for hydrogen atom in the initial and final configurations were calculated

to be 2.1 eV and 2.7 eV, respectively. This result indicates that diffusion of H

atom through the BL is energetically favorable. The energy barrier for H atom

diffusion through the BL was calculated to be 4.7 eV. This value seems too high

to be overcome at 600 °C. The results obtained suggest that the mechanism of

hydrogen-induced decoupling of the BL from SiC(0001) is not related to the

direct diffusion of hydrogen through the BL.

Other possible mechanisms of hydrogen penetration below the BL consist of

different ways of hydrogen migration through some extended defects in the BL,

such as holes, discontinuity of the layer, grain boundaries, sample edges, etc. For

these mechanisms to be responsible for graphene decoupling from SiC surface, it

is essential that hydrogen atoms can migrate on the SiC surface and break Si-C

bonds between SiC surface and BL. The energy barrier for hydrogen migration

between two Si atoms, accompanied by breaking a Si-C bond, was calculated to

be 1.3 eV. At temperatures of about 600 °C one can then expect hydrogen atoms

on SiC surface below the BL to be very mobile.

For a free standing graphene layer binding energy of a hydrogen atom and

the barrier for its diffusion through the layer were calculated to be 1.2 eV and

3.8 eV, respectively. Both these values are lower by 0.9 eV than those calculated

for the BL. It should be noted, that the energy of initial configuration Einit and

thus the diffusion barrier depend on the binding energy Ebind of H atom:

Q = ESP - Einit = ESP - (Esub + EH - Ebind), where Q is the diffusion barrier, Esp is

the energy of the saddle point configuration, Esub is the energy of the SiC

substrate with the BL and EH is the energy of a distant hydrogen atom. The

difference in calculated energy barriers for H atom diffusion through the BL and

free standing graphene can be explained by the difference in binding energies of

H atom on the BL and free graphene.

For the higher value of binding energy of H atom on the BL compared to a

separate graphene layer, we propose the following explanation. While the BL on

SiC is diamagnetic, the adsorption of a hydrogen atom can result in occurrence

of an unpaired pz electron on a neighboring C atom. This unpaired electron can



314



interact with a Si dangling bond from the substrate that gives the gain in the

electronic energy.

4. Conclusion

The calculated energy barrier for hydrogen migration through a graphene buffer

layer on SiC(0001) is too high to be overcome at temperatures close to 600 °C.

However, it is found that hydrogen atoms can easily migrate on SiC surface

below the graphene layer in this temperature range. It is suggested that hydrogen

atoms reach SiC surface through extended defects in the graphene layer and then

migrate and saturate SiC surface resulting in decoupling of the graphene layer

from the substrate. Hydrogen atoms are more stable on the buffer layer than on a

free standing graphene layer.

References

1. I. Forbeaux, J.-M. Themlin, J.-M. Debever Phys. Rev. B 58, 16396 (1998).

2. C. Berger, Z. Song, T. Li, X. Li, A. Y. Ogbazghi, R. Feng, Z. Dai,

A. N. Marchenkov, E. H. Conrad, P. N. First, W. A. de Heer, J. Phys.

Chem. B 108, 19912 (2004).

3. K. V. Emtsev, A. Bostwick, K. Horn, J. Jobst, G. L. Kellogg, L. Ley,

J. L. McChesney, T. Ohta, S. A. Reshanov, J. Röhrl, E. Rotenberg,

A. K. Schmind, D. Waldmann, H. B. Weber, T. Seyller, Nature Mater. 8,

203 (2009).

4. K. V. Emtsev, F. Speck, Th. Seyller, L. Ley, J. D. Riley, Phys. Rev. B 77,

155303 (2008).

5. A. Mattausch, O. Pankratov, Phys. Rev. Let. 99, 076802 (2007).

6. F. Varchon, P. Mallet, J.-Y. Veuillen, L. Magaud, Phys. Rev. B 77, 235412

(2008).

7. C. Riedl, C. Coletti, T. Iwasaki, A. A. Zakhanov, U. Starke, Phys. Rev. Let.

103, 246804 (2009).

8. C. Virojanadara, A. A. Zakharov, R. Yakimova, L. I. Johanson, Surf. Sci.

604, L4 (2010).

9. N. P. Guisinger, G. M. Rutter, J. N. Crain, P. N. First, J. A. Stroscio, Nano

Lett. 9, 1462 (2009).

10. R. Jones, P. R. Briddon, Semicond. Semimet. 51, 287 (1998).

11. C. Hartwigsen, S. Goedecker, J. Hutter, Phys. Rev. B 58, 3641 (1998).



PHYSICS, CHEMISTRY AND APPLICATION OF NANOSTRUCTURES, 2011



SCATTERING OF THE ELECTROMAGNETIC FIELD

BY A DIELECTRIC NANOTUBE COVERED

BY A THIN METAL LAYER

D. USHAKOU, A. M. NEMILENTSAU, G. Ya. SLEPYAN

Institute for Nuclear Problems, Belarus State University

Bobruiskaya 11, 220030 Minsk, Belarus

V. V. SERGENTU

Institute of Applied Physics, Academy of Sciences of Moldova

Academy Str. 5, 2028 Chisinau, Moldova

Scattering of electromagnetic fields by a finite-length dielectric nanotube covered by a

thin metal layer was theoretically investigated. The dispersion characteristics of the

surface plasmons in the nanotube were obtained. The axial polarizability of the nanotube

was calculated and the pronounced resonance lines in the polarizability spectra were

demonstrated. The resonance frequencies depend both on the nanotube length and on the

thickness of the metal layer.



1. Introduction

Devices allowing effective light manipulation at the nanoscale are the subject of

the intensive researches nowadays. Among them nanoantennas are of a great

interest. Though prospects of the nanoparticles made from plasmonic metals and

carbon nanotubes as nanoantennas have already been demonstrated [1,2] the

search for new systems allowing effective conversion of the freely propagating

electromagnetic radiation to the near-fields is continuing. In this paper we

present the potentiality of a dielectric nanotube covered by a thin metal layer as a

nanoantenna in the near-infrared and optical ranges.

2. Theory

Let an isolated dielectric nanotube of length L and radius Rn, covered by a thin

metal layer (see Fig. 1), and aligned along the z-axis of the cylindrical coordinate

system, be exposed to an external electromagnetic field E0eiky-iωt, where k=ω/c

and ω is the angular frequency. The electric field in such system is a solution of

the homogeneous Helmholtz equation



( ∇ × ∇ × −k ε ( r ) ) E(r ) = 0,

2



where ε(r) is equal to εD inside the dielectric and ε(r)=1 outside of it.



315



(1)



316



un



εM



d

Rn



εD



VD

SM

Figure 1. A dielectric nanotube of radius Rn covered by a thin metal layer of thickness d.

Dielectric constants of dielectric and metal are εD and εM, respectively.

un is the unit vector normal to the nanotube surface.



Equation (1) should be supplemented by the effective boundary conditions

on the nanotube surface

u n × [ H ( ze z + R n + 0) − H ( ze z + R n − 0) ] = ( 4π / c ) e z σ Meff E z ( ze z + R n ),

u n × [ E( ze z + R n + 0) − E( ze z + R n − 0)] = 0,



(2)



where z ∈ [ − L / 2, L / 2] , σ Meff = −iω d (ε M − 1) / 4π is the effective conductivity

of a thin metal layer (d « λ, λ is the free-space wavelength of the electromagnetic

field).

In order to solve system of Eqs. (1), (2) we assume that in the nanotube

exposed to the external electromagnetic field the polarization of the dielectric is

small compared to the polarization of the metal layer. In this case, by analogy

with Ref. [2], we obtain the following approximate expression for the density of

the surface axial current induced in the metal layer:

j( z ) = e z j ( z ) = e z



σ Meff (h 2 − k 2 )

2ih



L/2

 eihL / 2 (1 − e −ihL ) 2 ihz



ih z − z ′

E0 

(e + e −ihz ) + ∫ e

dz ′ ,

−2 ihL

ih

(

e



1)

−L/2







(3)



where the waveguide number h of the surface wave in the nanotube is the

solution of the characteristic equation



ik [κ BK1 (κ Rn ) I 0 (κ D Rn ) + ε Dκ K 0 (κ Rn ) I1 (κ D Rn ) ] =

= ( 4π / c ) σ Meff κκ D K0 (κ Rn ) I 0 (κ D Rn ),

where κ = h 2 − k 2 , κ D = h 2 − k 2ε D .



(4)



317



3. Numerical results

Spectra of the waveguide number h of the surface waves in the dielectric

nanotube with εD=10 covered by a gold layer are presented in Fig. 2. The

dielectric

function

of

gold

was

assumed

to

be

equal

−1

to ε M (ω ) ≅ 1 − ω p2 [ω (ω + iΓ) ] , where ωp=13.8×1015 rad/s, Γ=1.075×1014 s-1. As

one can see, both the imaginary and real parts of h depend strongly on the

thickness of the gold layer d.



Re h, cm-1



Im h, cm-1



ω, rad/s



ω, rad/s



Figure 2. Spectra of the waveguide number h of the surface plasmon in a dielectric nanotube

(Rn=25 nm, εD=10) covered by a thin gold layer of thickness d.



In the long wavelength limit (λ » L) the far-field electromagnetic response of

the nanotube can be described by the axial polarizability



α=



2π iRn

ω E0



L/2







j ( z )dz =



− L/ 2





2π Rnσ Meff (h 2 − k 2 )  eihL (1 − e − ihL )3

+ ihL − (eihL − 1)  . (5)



−2 ihL

3



 1− e





The polarizability spectra of the nanotubes of length L = 300 nm are presented in

Fig. 3. Both the imaginary and real parts of the polarizability spectra contain

pronounced resonance lines, the resonance frequencies being dependent on the

thickness of the gold layer d. Particularly, the increase of the thickness d leads to

the blue shift of the resonance frequency.



4. Conclusion

We studied theoretically interaction of the plane electromagnetic wave with the

finite-length dielectric nanotube covered by a thin metal layer. For this purpose,



318



the expression for the current density induced on the nanotube surface by the

external field was derived. The structure of the surface electromagnetic modes in

the system was investigated and the characteristic equation for the waveguide

number of the surface wave was obtained. The nanotube polarizability was

calculated and the pronounced resonance lines in the polarizability spectra were

demonstrated.



Re α, 10-15 cm3



Im α, 10-15 cm3



ω, rad/s



ω, rad/s



Figure 3. The spectra of the real and imaginary part of a polarizability α of a dielectric nanotube

(Rn=25 nm, εD=10) covered by a thin gold layer of thickness d. L=300 nm.



Acknowledgments

This research was partially supported by (i) the Belarus Republican Foundation

for Fundamental Research under Project No. F10Mld-003; (ii) EU FP7 under

Projects No. FP7-230778 TERACAN and No. FP7-266529 BY-NanoERA



References

1. P. Bharadwaj, B. Deutsch, L. Novotny, Adv. Opt. Photon. 1, 438 (2009).

2. G. Ya. Slepyan, M. V. Shuba, S. A. Maksimenko, A. Lakhtakia, Phys. Rev.

B 73, 195416 (2006).



PHYSICS, CHEMISTRY AND APPLICATION OF NANOSTRUCTURES, 2011



ANISOTROPY AND ELECTROMAGNETIC PROPERTIES OF

CARBON NANOTUBE BASED DEFORMED POLYMER

COMPOSITES IN MICROWAVES

D. BYCHANOK, M. SHUBA, A. PODDUBSKAYA, A. PLIUSHCH,

A. NEMILENTSAU

Institute for Nuclear Problems, Belarus State University

Bobruiskaya 11, 220030 Minsk, Belarus

M. KANYGIN, A. KURENYA, A. OKOTRUB

Nikolaev Institute of Inorganic Chemistry SB RAS

Acad. Lavrentiev Ave. 3, 630090 Novosibirsk, Russia

The deformation influence on electromagnetic response of carbon nanotube based

polymer composites was theoretically modeled and experimentally investigated in

Ka-band.



1. Introduction

Cylindrical structure of carbon nanotubes (CNT) causes anisotropy of their

electromagnetic and mechanical properties. Particularly, polarizability of an

isolated CNT in the axial direction is much bigger than polarizability in the

perpendicular direction. By homogeneous dispersion of CNT in a polymer

matrix composite material is isotropic. Effects of anisotropy can be observed in

the case of oriented CNTs in polymer matrix. For the orientation of CNTs in a

composite, the stretching deformation may be used.

2. Sample preparation

CNTs were fabricated on a silicon substrate (Fig. 1) using chemical vapor

deposition (CVD) at ~800 °C [1]. CNTs were then dispersed in the solution of

PMMA in toluene. The solution was subjected to sonication for 30 min in order

to make the CNTs distribution homogeneous. Composite films containing

1 wt. % of CNTs were finally produced. The samples were deformed with the

stretch velocity of 5 μm/s. The relative deformation of all samples were χ = 1.5.

Rectangles of 9×3.4 mm were cut from the stretched films perpendicular and

parallel to the tension direction.



319



320



Figure 1. (left) SEM image of carbon nanotubes. (center) Polymer composite film with 1 wt.% of

CNTs. (right) Optical microscope image of polymer composite film contained 1 wt.% of CNTs.



3. Experiment

Electromagnetic response of the stretched films was investigated in frequency

range 26-37 GHz (Ka-band), using scalar network analyzer P2-408P. Samples

cut perpendicular and parallel to the tension direction were placed into the

waveguide system of the analyzer, perpendicular and parallel to the electric field

vector of the scattered electromagnetic wave, respectively.

Re Eps (no deformation)

Re Eps (parallel deformation)

Re Eps (perpendicular deformation)

Im Eps (no deformation)

Im Eps (parallel deformation)

Im Eps (perpendicular deformation)



7

6



Epsilon



5

4

3

2

1

0

26



28



30



32



34



36



38



Frequency, GHz



Figure 2. Frequency dependence of dielectric permittivity of stretched composite materials.



As we have found (Fig. 2), a pure PMMA matrix is practically transparent

and does not absorb electromagnetic radiation. Because of homogeneous

dispersion of CNTs in the polymer matrix, the undistorted composite material is

isotropic. By the stretching deformation, CNTs are primarily oriented along the

tension direction. The polarizability of CNTs in the axial direction is much



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