NEAR-FIELD ELECTRODYNAMICS OF ATOMICALLY DOPED CARBON NANOTUBES

Transcription

1 NER-FIELD ELECTRODYNMICS OF TOMICLLY DOPED CRBON NNOTUBES Igor Bondarev The Institute for Nuclear Problems The Belarusian State University Minsk, BELRUS

2 Collaborators: Prof. Philippe Lambin Laboratoire de Physique du Solide Facultes Universitaires Notre-Dame de la Pai Namur, BELGIUM cknowledgements: Belgian Office for Scientific, Technical and Cultural ffairs

3 OUTLINE Basic physical properties of singlewalled carbon nanotubes (CNs) Motivation Near-field electrodynamics. The model tomic spontaneous decay dynamics near a carbon nanotube van der Waals energy of an atom near a carbon nanotube Conclusions

4 BSIC PHYSICL PROPERTIES OF SINGLE-WLLED CNs () classification of single-walled CNs Graphene single sheet y ma + na a 3 a Single-walled carbon nanotube of (m,n) type

5 BSIC PHYSICL PROPERTIES OF SINGLE-WLLED CNs () Brillouin zone structure of achiral CNs p z s p φ = s R cn, =,,, m p z p φ p φ Zigzag: (m,) m = 3q metallic (q =,,3, ) rmchair: (m,m) metallic for all m Conductivity normalized Typical frequency dependence of the conductivity Zigzag (9,) Re σzz Im σzz -,,5,,5,, ω =, γ =.7eV -8 γ -

6 MOTIVTION Eperimental observations TEM image of a sample partially filled by Cs atoms. The inset sets off the encapsulated atoms by dotting with black circles [after J.-H.Jeong et al. PRB68,754(3)] Recent eperiments on single atom encapsulation into single-wall carbon nanotubes and atom intercalation into single-wall carbon nanotube bundles: J.-H.Jeong et al. PRB68,754(3) H.Shimoda et al. PRL88,55 () L.Duclau Carbon4,75 (). Nanotube/atomic physical properties are substantially modified in a controllable way when doping nanotubes with impurity atoms. New challenging nanophotonics applications of atomically doped carbon nanotube systems as various sources of coherent light emitted by dopant atoms

7 NER-FIELD ELECTRODYNMICS. THE MODEL () the geometry of the problem I.V.Bondarev & Ph.Lambin, Phys. Lett., to be published I.V.Bondarev & Ph.Lambin, PRB69,35(4) I.V.Bondarev & Ph.Lambin, cond-mat/433 I.V.Bondarev & Ph.Lambin, cond-mat/44 [submitted to SSC] I.V.Bondarev et al., Carbon4,997(4) I.V.Bondarev et al., PRB68,733(3) I.V.Bondarev et al., PRL89,554() R = ( R, φ, Z) cn

8 THE MODEL () Hamiltonian Field quantization in the presence of dispersing and absorbing bodies: b L.Knöll, S.Scheel, D-G.Welsch, D in: Coherence and Statistics of Photons and toms, New York,Wiley, Hamiltonian: r ˆ E ˆ r r E ˆ r () = dω( iω) (, ω) + hc.., ˆ ϕ() = dω (, ω) + hc.. ˆ Hˆ = Hˆ + Hˆ + Hˆ + Hˆ (, ω) = d ( ) ˆ(, ω), = {, r ϕ,} z () () F F F ˆ ˆ+ Field: H = dω ω d f (, ω) fˆ R R ( R, ω) tom: Interaction I: (electric dipole) Interaction II: (diamagnetic) Hˆ Hˆ F ( ) ( ) E r rδ r r E r r pˆ = + qq i j i mi i, j ri rj () qi ˆ ˆ ˆ ˆ F = pi r + d ϕ r i mc i () qi ˆ F = r i mc i Hˆ ( ) ( ) ( ) Mawell equations with an eternal source (CN) ˆ ω (, ) ˆ (, ), ˆ ω (, ) ˆ 4π Erω = i Hrω Hrω = i E ( r, ω) + Iˆ (, r ω ) c c c Ir ˆ(, ω) = drδ( r RJR )( ˆ, ω) = Jˆ ( R, ϕ, z, ω ) δ ( r R ) JR ˆ (, ω ) = ω Re σ ( R, ω )/ π fˆ ( R, ω ) e, R = { R, φ, Z } zz cn cn z cn cn ˆ ˆ (,, ) ( ˆ (, r ω) = i d GrR ω J R, ω), = i( ω/ c) ˆ E 4πω c R H E GrRω (,, ) is the classical field Green tensor

9 THE MODEL (3) two-level approimation ω ˆ σ z + dω d g (,, ω) σ g (,, ω) σ fˆ (, ω) + hc.. ( + ) + ( ) ˆ ˆ R r R r R R Renormalized atomic frequency (due to diamagnetic interaction) ω ( r ) = ω dω ω d g (,, ) ( ) ω ω R r R Electric dipole coupling (electric dipole interaction) ω g ( r, R, ω) = g ( r, R, ω) ± g ( r, R, ω) ( ± ) ω () 4ω ( ) r = e ( ) zz αgαβ g (, R, ω) i π ωr σ ω d ( r, R, ω) c ξ ˆ ˆ+ H = dω ω d f (, ω ) fˆ (, ω ) + R R R Local photonic DOS ( ( ) ω ) ω ( ) (,, ω) ξ Γ dr g r R = π ω ( ) ( ) ( ) αβ ( ) r ω ξ Im Gαβ ( ω) 8π Γ = dd G G r r = c ( ω) Im ( ), Im (,, ) α β αβ ω αβ ω δαβ ( r, ω) Im G ( r, r, ω) (, ) = = + ( r, ω) ω 6π c

10 Local photonic DOS for an atom near a carbon nanotube I.V.Bondarev & Ph.Lambin, PRB69 69,35(4) [cond-mat/433] r > R cn : R cn 4 βω ( ) vip( vrcn) Kp( vr) cn 3 C 4 k p= + Rcnβ ω v Ip vrcn Kp vrcn 3R ξ( r, ω) = + Im π ( ) ( ) ( ) dh ω v= h k ; k = ; c k σ zz ( Rcn, ω) βω ( ) = 4πi k ω h r < R cn : R cn I p K p in the numerator of the integrand

11 Eample of local photonic DOS for an atom near a carbon nanotube I.V.Bondarev & Ph.Lambin, PRB69 69,35(4) [cond-mat/433] Local photonic DOS for an atom outside the (9,) CN at different distances from the wall 8 (9,) R cn 7 6 r =.5R cn ; r = 3.5R cn r =.5R cn ; ξ() ω =, γ =.7eV γ

12 (I) SPONTNEOUS DECY DYNMICS I.V.Bondarev & Ph.Lambin, PRB69 69,35(4) [cond-mat/433] Γ ( ω) ω ω ω ξ π ω ( r) = d ( r, ) ω ω u u { } ω ω l r R cn ω l l { ( R, ω) } { } ψ i( ω /) t () t = Cu () t e u {} + R R R Wave function of the system dω d C ω t e ω } i( ω ω /) t l (,, ) l {(, ) Evolution equation of the upper state t C () t = + K( t t ) C ( t ) dt u i( ω ω )( t t ) e ( + ) ( ) = ω (,, ) ω i( ω ω ) R r R Kt t d d g u, ~ ξ ( r, ω)

13 SPONTNEOUS DECY DYNMICS Solution of evolution equation I.V.Bondarev and Ph.Lambin, PRB69 69,35(4) [cond-mat/433] Γ ( ) ω γ Γ ( ) =, =, τ = t, γ =.7eV γ γ C ( τ) = + K( τ τ ) C ( τ ) dτ Γ K( τ τ ) = u τ ( ) i( )( τ τ ) 3 d ξ ( ) 3 π i( ) u e e i( )( τ τ ) Markovian approimation πδ ( ) + ip i ( ) NUMERICLLY!!! Γ( ) K( τ τ ) + i ( ) Cu( τ) = e 4γ Γ ( ) = ξ ( ) Γ ( ) Γ( ) τ γ

14 SPONTNEOUS DECY DYNMICS Numerical results I.V.Bondarev & Ph.Lambin, PRB69 69,35(4) [cond-mat/433] Local photonic DOS for an atom outside the (9,) CN at different distances from the wall 8 (9,) R cn 7 6 r =.5R cn ; r = 3.5R cn r =.5R cn ; ξ() ω =, γ =.7eV γ

15 SPONTNEOUS DECY DYNMICS Numerical results I.V.Bondarev & Ph.Lambin, PRB69 69,35(4) [cond-mat/433] Spontaneous decay dynamics for an atom outside the (9,) CN at different distances from the wall R cn,,8 C u (τ),6,4 (9,), =.33 r =.5R cn ; r =.5R cn ; r = 3.5R cn - eact, - ep. decay,, τ τ γ t,.7ev = γ =

16 SPONTNEOUS DECY DYNMICS Numerical results I.V.Bondarev & Ph.Lambin, PRB69 69,35(4) [cond-mat/433] tom outside the (9,) CN at different distances from the wall (a) local photonic DOS 8 (9,) 7 r =.5R cn ; r =.5R cn ; 6 r = 3.5R cn 5 ξ ( ) 4 ω =, γ =.7eV γ 3 C u (τ) 3.5 (b) upper-level spontaneous decay dynamics compared with eponential decay (Markovian approimation),,8,6,4, (a) (9,), =.5 r =.5R cn ; r =.5R cn ; r = 3.5R cn - eact, - ep. decay, τ γ τ = t R cn

17 SPONTNEOUS DECY DYNMICS Numerical results I.V.Bondarev & Ph.Lambin, PRB69 69,35(4) [cond-mat/433] tom outside the (9,) CN at different distances from the wall (a) local photonic DOS 8 (9,) 7 r =.5R cn ; r =.5R cn ; 6 r = 3.5R cn 5 ξ ( ) 4 ω =, γ =.7eV γ 3 C u (τ) 3.58 (b) upper-level spontaneous decay dynamics compared with eponential decay (Markovian approimation),,8,6,4, (b) (9,), =.58 r =.5R cn ; r =.5R cn ; r = 3.5R cn - eact, - ep. decay, τ γ τ = t R cn

18 C u (τ) SPONTNEOUS DECY DYNMICS Numerical results I.V.Bondarev & Ph.Lambin, PRB69 69,35(4) [cond-mat/433] tom in the center of different CNs: 4 ξ ( ) ω =, γ =.7eV 3 γ.45 3 (b) upper-level spontaneous decay dynamics compared with eponential decay (Markovian approimation),,8,6,4, (a) local photonic DOS (5,5); (,); (3,) r =, =.45 (5,5); (,); (3,) - eact, - ep. decay, τ R cn γ τ = t

19 SPONTNEOUS DECY DYNMICS Numerical results I.V.Bondarev & Ph.Lambin, PRB69 69,35(4) [cond-mat/433] tom inside the (,) CNs at 3 from the wall [PRB68,754]: (a) local photonic DOS 7 (,); r =.56R cn 6 R cn ξ ( ) 5 4 ω =, γ =.7eV γ 3,,,,3,4,5,6,7,8,9,, (b) upper-level spontaneous decay dynamics compared with eponential decay (Markovian approimation) C u (τ),,8,6,4, (,); r =.56R cn ; =.; =.3; =.5; =.3; =.45; - eact; - ep.decay, τ γ τ = t

20 How to observe eperimentally? Optical absorption eperiments Weak coupling Strong coupling ω u { } ω ω Rabi ( {} { ( R, )} ) u l ω ( {} + { ( R, )} ) u l ω ω l l { ( R, ω) } { } ω l { } Optical absorbance profile in the neighborhood of the atomic transition energy ω Rabi ~ ω ω ~ ω ω

21 SPONTNEOUS DECY DYNMICS CONCLUSIONS (I.) Spontaneous decay dynamics of an ecited atom near a carbon nanotube is in general NON-EXPONENTIL because of the strong non-markovian memory effects arising from the rapid variation of the photonic density of states with frequency near the nanotube (I.) Spontaneous decay may proceed via damped Rabi oscillations a principal signature of STRONG atom-vacuum-field coupling in the vicinity of the nanotube (I.3) tom-field coupling and spontaneous decay dynamics may be controlled eperimentally by changing the distance between an atom and a nanotube surface by means of a proper preparation of atomically doped carbon nanotube systems This opens routes for possible new challenging applications of atomically doped carbon nanotube systems in: cavity QED nanophotonics quantum information processing

22 (II) GROUND-STTE van der WLS ENERGY I.V.Bondarev & Ph.Lambin, cond-mat/44 [submitted to SSC] Γ ( ω) ω ω ω ξ π ω ω ( r) = d ( r, ) ω u ω u u { ( R, ω) } { } ω r R cn ω l l { } ψ = C l {} +dωd R C ( R, ω) u {( R, ω)} l Wave function of the system u van der Waals energy ω dω E = d g ω ω + E ω E = + E vw ( r ), ~ ξ ( r, ω) ( ) R r R (,, ω)

23 GROUND-STTE van der WLS ENERGY Integral equation I.V.Bondarev and Ph.Lambin, cond-mat/44 [submitted to SSC] Evw( r ) Γ ( ) ω εvw( r) =, Γ ( ) =, =, γ =.7eV γ γ γ ε vw( r ) = π Γ ( ) d ξ ( r, ) ξ (, ) + ξ (, ) ( ) Γ d π Γ ( ) + d ξ ( r, ) r ε vw( r) r vw r d ε ( ) Γ ( ) ξ ( r, ) Weak coupling approimation Γ ( ) εvw( r ) d ξ (, ) π r r ξ (, ) ξ (, ) Γ ( ) d π Γ ( ) + d ξ r (, ) + r Strong coupling approimation Γ ( ) εvw( r ) d ξ (, ) π r ξ ( r, ) (, ) + ξ r Γ ( ) d π εvw( r)

24 GROUND-STTE van der WLS ENERGY Numerical results I.V.Bondarev & Ph.Lambin, cond-mat/44 [submitted to SSC] tom outside the (9,) CN: eact solution for the vdw energy compared with weak/strong coupling approimation. r /R cn,4,6,8,,,4, -, E vw /γ -, -,3 -,4 (9,), =.33 eact solution strong coupling appro. weak coupling appro. ω =, γ =.7eV γ

25 GROUND-STTE van der WLS ENERGY Numerical results I.V.Bondarev & Ph.Lambin, cond-mat/44 [submitted to SSC] tom outside the (9,) CN: eact solution for the vdw energy compared with weak/strong coupling approimation for different atomic transition frequencies r /R cn,4,6,8,,,4, -, 3 E vw /γ -, 3 -,3 -,4 3 (b) (9,) eact solution strong coupling appro. weak coupling appro. - =.33; - =.5; 3 - =.58 ω =, γ =.7eV γ

26 GROUND-STTE van der WLS ENERGY Numerical results I.V.Bondarev & Ph.Lambin, cond-mat/44 [submitted to SSC] tom in/outside the (,) CN: van der Waals energy in/outside the nanotube for a few atomic transition frequencies r /R cn,,,4,6,4,6,8, -, E vw /γ -, -,3 (,) eact solution =.; =.3; =.45 ω =, γ =.7eV γ

27 GROUND-STTE van der WLS ENERGY CONCLUSIONS (II.) Due to STRONG atom-vacuum field coupling in a close vicinity of the nanotube surface, one has to be careful when calculating the van der Waals energy of an atom near a carbon nanotube (II.) n integral equation for the van der Waals energy has been derived which is valid for both strong and weak atom-vacuum-field coupling (II.3) The inapplicability has been demonstrated of weak-couplingbased van der Waals interaction models in a close vicinity of the nanotube surface (II.4) Encapsulation of doped atoms into the nanotube has been shown to be energetically more favorable than their outside adsorption by the nanotube surface

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