Ground State Absorption Spectroscopy and Dephasing

Absorption spectra are acquired using the Varian Cary 5000, which an accessible wavelength range from 175-3300 nm. Absorption spectra recorded with this instrument simply reflect the wavelength λ dependence of the absorbance A

( )

λ . Note, that the term optical density OD

( )

λ is often used synonymously for absorbance. According to the Beer-Lambert law the absorbance is defined as ^A

( ) ( )

λ =α λ ^lC, where α is the absorption coefficient defined in chapter 1, lis the optical path length, and is the concentration of the absorbing species. In order to obtain the optical density

C

( )

λ OD of a

sample, one measures the intensity of the light transmitted sample through the sample,

( )

λ

Is , and the intensity of the incident light, I₀

( )

λ . Then by using the relation,

( ) ( )

( )

^⎟⎟_⎠^⎞

⎜⎜⎝

= ⎛

λ λ λ

IS

OD log I⁰ , (1)

the optical density of the sample is found. If the material of interest is dissolved in solution then a reference spectrum of the solvent must be taken and its absorbance be subtracted from the solution spectrum.

A discussion focused on a simple two-level model under the dipole approximation is instructive to explain the concept of the absorption of electromagnetic radiation and emission from a bound entity such as an exciton. The system is described by a ground state, S₀ , and an excited state given by, S₁ , where as in chapter 1 these are singlet states. The excited state is dipole coupled to the ground state and the quantum yield of fluorescence is unity. Following the formalism outlined by Loudon [1983], a time- dependent electromagnetic field with a frequency ω, ^Eω

( )

^t

v will induce a polarization on an ensemble of systems such that,

( )

^{t E}

{ ( )

^{e i t}

( )

^{ei t}

}

Pv_ω =12ε₀v₀ χ ω ^−ω +χ −ω ^ω

, (2) where ε₀ is the permittivity of free space, Ev₀

is the field strength, and χ

( )

ω is the frequency depended susceptibility which is related to the absorption coefficient by

( )

ω

[

ω

( )

ω

]

χ

( )

ω

α = cn Im [Lou83]. Here the index of refraction, ⁿ

( )

ω , is usually assumed to vary slowly with ω and can be treated as a constant. The time dependent polarization determines the susceptibility and the dynamics of the system determine the time dependent dipole moment dv

which is proportional to the time dependent

polarization. The dynamics of the system are governed by the optical Bloch equations [Lou83]. Including radiative γ and environmental damping γ_env, an incident electromagnetic field Ev₀

with a frequency ω, the Bloch equations are given by,

( ) ( )

( )

(

00 11

) ( [

01

) ]

01

01

11 10

11 01

01

01 01

2 1

2 2

* 1 2 1

ρ γ ω ω ρ

ρ ρ ζ

γρ ρ

ζ ρ

ρ ζ

ω ω

ω ω ω

ω

− ′

− +

−

=

− +

−

=

−

−

−

−

−

i e

dt i d

e i e

dt i d

t i

t i t

i

, (3)

where ρ₁₁is the population in S₁ , ρ₀₁is the coherence between S₁ and S₀ , _hω₀₁ is the transition energy between from S₀ and S₁ , e S₀ Ev₀ dv S₁

h ⋅

ζ = is the dimensionless transition dipole moment, and γ′=γ +γ_env is the dephasing rate.

Figure 1 a. Schematic of a two level system absorbing a photon, which takes the system from the ground state (S₀) to an excited state (S₁). b. Absorption spectra of the hypothetical system in a. If the system is ideal, the only broadening would be due to the finite lifetime of the photoexcitation and this would be reflected in the full width half maximum of a Lorentzian profile. Such a system is said to be homogeneously broadened. c. The same two level system under the presence of static disorder that smears out the discrete energy level S₀ by a Gaussian of width 2δE. d. The effect of the disorder is said to inhomogeneously broaden the spectra and the resulting absorption spectra is a convolution of both a Lorentzian and Gaussian profile which is called a Voigt profile.

The dephasing rate determines the time over in which the polarization decays. If the system is initially in its ground state S₀ , ρ11

( )

⁰ =⁰ and ρ₀₁

( )

0 =0, then the solutions to equations 3 to the lowest order in ζ are,

( )

[ ] [ ( ) ]

{ }

( ) {

^{t i( )t}

}

t t

e i e

i

e t C

t C

e C C

ω ω γ

γ γ

γ ω ω ρ ζ

ω ω ω

ω ζ

ρ

−

−

− ′

− ′

−

′ −

−

= −

−

−

−

− +

=

01

01 01

01 4

01 3

2 2 1 2 11

2

sin 4 cos

1

, (4)

where,

( ) ( ) ( )

(

₀₁

) (

²

)

²

2 2

2

γ γ ω

ω

γ γ γ

− ′ +

−

− ′

=

2 C

2 01

1 ω ω γ

γ γ

+ ′

−

= ′

C , ,

( ) ( )

[ ]

( ) ( )

[ ] [ ⁽

01

^{) (}

²

⁾

²

]

2 2 01

2 01

3 2

2 2

γ γ ω

ω γ ω ω

γ γ γ ω ω

− ′ +

′ − +

−

− ′ + ′

= −

C , and

( )( )

[ ]

( ) ( )

[ ] [ ⁽

01

^{) (}

²

⁾

²

]

2 2 01

01

4 2

4

γ γ ω

ω γ ω ω

γ γ ω ω

− ′ +

′ − +

−

′−

= −

C .

The time dependence of the dipole moment is given by,

( )

^{t e}

{

^{S d S e i t S d S ei t}

}

d^v =− ρ_{10 0} ^v ₁ ^−ω01 +ρ_{01 1} ^v ₀ ^ω01 , (5) and by comparing with the time dependent polarization the susceptibility is found and the absorption coefficient can be written as,

( )

( )

γ ζ γ γ

ω ω

γ ω ω

α

2

2 2 2

01 2 1

0 ′

′ + +

−

=C S dv S ′

, (6)

where C is a constant describing the concentration of the ensemble of two level systems, the second term is the oscillator strength, and the last term the line profile homogeneously broadened by the saturation of the transitions and by both radiative and environmental damping [Van77]. The last term determines the lineshape of the

absorption spectrum and is denoted by F_L

( )

ω , where L stands for a Lorentzian lineshape.

The full width half maximum (FWHM) of the absorption spectrum in this model is given by 2 γ′+γ′ζ ² 2γ , and in the limit that the transitions are not saturated 2γ +2γ_env. The radiative lifetime expressed in terms of the radiative rate is T₁=1 2γ and dephasing time as T₂=1

(

γ +γenv

)

. In the absorption experiment one measures the FWHM of a specific transition which – in the case of a homogeneously broadened transition can be used to determine the pure dephasing time τ if the radiative lifetime is known from

(

1 2−12 1 ⁻¹

= T T

τ

)

, Fig. 1 a and b. In carbon nanotubes each subband will have a series of excitons with only one of them being dipole allowed [And97, Spa04, Per04, Cha04].

Therefore, the absorption spectra will contain a series of peaks like in figure 1, one associated with each subband. Dephasing and life times of the optically allowed states will be discussed in chapter 4.

In almost all circumstances the interaction of excitons with the environment will shift and/or broaden the excitonic energy levels.. This can be considered due to a disordered dielectric environment and will introduce inhomogeneous broadening into the spectra. Broadening of this type is considered to cause a smearing of the energy levels δ over a normal distribution, F_G

( )

ω , centered at the resonance, Fig. 1 c. A convolution of the inhomogeneous and homogeneous linewidths can be calculated by,

( )

( )

(

ω ν

)

γ ^ν π

δ

ω γ ^δ

ν ω

e d F

2

2 2 3

2

2

∫

^∞

∞

−

− −

+

= − , (7)

which is referred to as the Voigt lineshape and can be used to estimate the amount of

contribution of inhomogeneous broadening to the linewidth will be important for a determination of the pure and total dephasing times.