Modélisation et étude des propriétés optiques des nanotubes de carbone Modelling and study of
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Mode´lisation et e´tude des proprie´te´s optiques
des nanotubes de carbone
Modelling and study of optical properties of
carbon nanotubes
Benjamin Ricaud
´Universite du Sud Toulon Var
Centre de Physique The´orique, 22 Octobre 20076
6
Carbon nanotubes
Tubes made of carbon atoms with = radiuses, = chiralities.
◮ Different properties.
◮ How to sort nanotubes.
Benjamin Ricaud -2/26- 22 Octobre 2007 ,
Optical spectrum
◮ A way to sort nanotubes. Monochromatic light sent on CN, low
temp.
I ω0
I ω1
◮ Optical absorption spectrum /ω
◮ Optical absorption spectrum related to radius & chirality.
◮ explain a part of the optical absorption spectrum (Infrared).
−(Absorption by e )
Benjamin Ricaud -3/26- 22 Octobre 2007 ,Optical response, physical explanation
Sending light and looking at the absorption of semiconductor CN.
− −
◮ Without e -e interaction:
E
◮ periodic lattice⇒ bands& gaps Conduction band −e
◮ Semiconductor: G. state=valence bands full, E(k)c
conduction bands empty.
Eg◮ Absorption↔ Energy.
− − E(k)◮ vWith e -e interaction (weak):
+Valence band h k◮ “excitons”. Wannier (1937), Elliot(1957), Mahan.
− + E◮ exciton, coupled e -h : H φ = E φexc n n n
Conduction band
◮ Absorption spectrum at 0K with Eg=gap, η=control
E(k)cadiabaticity:
En2X |ψ (0)|n
α(ω)∼
2 2(E + E +~ω)[(E + E −~ω) +(~η) ] E(k)n g n g vn
+Valence band h k
[Haug, Koch]
◮ Nanotubes: Exciton eigenstates depend on r.
Benjamin Ricaud -4/26- 22 Octobre 2007 ,Benjamin Ricaud -5/26- 22 Octobre 2007 ,Model for exciton
◮ Pedersen 2003, Kostov et al. 2002
◮ Exciton: 2 particles on a tube.
2∂◮ Exciton model Hamiltonian: H =− −22m ∂x1 1
2 2 2∂ ∂ ∂ r− − − V (x − x , y − y )2 2 2 1 2 1 22m ∂y 2m ∂x 2m ∂y1 2 21 2 2
◮ Hamiltonian with Coulomb pot. on a cylinder
1rV (x, y) = q
2 y2 2x + 4r sin 2r
◮ ”center of mass” separation: X = (m x + m x )/(m + m )1 1 2 2 1 2
x = x − x , Y = y , y = y − y1 2 2 1 2
2 2 2 2∂ ∂ ∂ ∂ reH =− − − − − V (x, y)
2 2 22∂x 2∂y 2m ∂Y m ∂Y∂y2 2
Benjamin Ricaud -6/26- 22 Octobre 2007 ,r rFrom H to H , one-dimensional effective Hamiltonianeff
2 2 2 2∂ ∂ ∂ ∂ r◮ H =− − − − − V (x, y)2 2 22μ∂x 2μ∂y 2m ∂Y m ∂Y∂y2 2P 2Δ nY◮ Projection over modes of− = P (p.b.c.)2 n2 2r
n
2n
◮ Transverses modes with energy and22r
n1 i Y√ rχ (Y) = e , P =|χ ><χ |n n n n2πr
◮ Only low lying spectrum is interesting.
◮ Projection on the ground transverse mode n = 0:
ΔΔx yr r 2 1(1⊗P )H (1⊗P ) =− − −V , in L (R×rS )0 0
2 2
◮ same for y:
Δxr r 2H =− − V in L (R)eff eff
2Z πr1 1r qV (x) = dyeff 2πr 2 y−πr 2 2x + 4r sin
2r
Benjamin Ricaud -7/26- 22 Octobre 2007 ,6
r rFrom H to H , one-dimensional effective Hamiltonianeff rH Vn,meff
r 1 H + 2eff 2r f r r 2H = = H + H , indiag offdiagH + 2eff r
...Vm,n
2 2ℓ (Z; L (R)). Z π i(m−n)y1 eqV (x) = dy x = 0m,n
y2π 2 2−π x + 4r sin 2
f r◮ Perturbation theory: comparison H and Hdiag
r
◮ z∈ρ(H ), r smalleff
r→0f r −1 r −1◮ k(H − z) −(H − z) k → 0, informations on the spectrumeff
◮ H is H -form bounded.offdiag diag
r
◮ Difficulty: H depend on r, choice of z depend on r.eff
Benjamin Ricaud -8/26- 22 Octobre 2007 ,Unperturbed model
Z πr
1 1rV (x) = q dyeff 2πr 2 y2 2−πr x + 4r sin
2r
r
◮ Using perturbation theory for small r+ quadratic forms, Heff
approximated by:
2d 1
◮ H ψ = ψ− ψ = Eψ with conditions at zero:C 2dx |x| ′ ′ψ (ε)−ψ (−ε) r
lim + 2 ln ψ(0) = 0
ε→0 2 2ε
′
◮ Oddψ not continuous.
◮ final Hamiltonian depend on r, spectrum is asymptotic.
1◮ Similar to Loudon,1959: V(x) = , a = r/2.|x|+a
Benjamin Ricaud -9/26- 22 Octobre 2007 ,results
1
◮ even states (S), energies: E =− and functions: Whittakern 2α(r,n)
ψ (x) = W 1(|x|).n α(r,n), 2
◮ α(r, n) such that f(α(r, n), r, n) = 0.
1
◮ odd states (P), energies: E =− and functions (P): Laguerren 2n
1− |x| 1
2ψ (x) = e xL (|x|).n n−1
2
◮ fundamental energy: E ∼−4(ln r) for small r.n
0
-4
Second p state E
3p◮ Stronger bound in 1D. -8s state E
2s
◮ excited states: energies & First p state E
2p
Ground state Eeigenvectors. -12
1s
Variational E
1s
-16
0 0,1 0,2 0,3 0,4 0,5 0,6
Nanotube radius r [ a* ]
B
Benjamin Ricaud -10/26- 22 Octobre 2007 ,
Exciton binding energy E [Ry*]
