Theoretical analysis - ePrints@IISc

Supplementary Information for manuscript titled "Spontaneous light emission from molecular junctions: Theoretical analysis of

up-conversion signal"

Hari Kumar Yadalam, Souvik Mitra, and Upendra Harbola

^∗

Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore, 560012 India

E-mail: uharbola@iisc.ac.in Phone: +91 (080)229 33410

The Liouville space algebraic expression for the current-induced fluorescence can be writ- ten down readily¹ from the double-sided diagram shown in Fig. (1a). We get,

S⁽¹⁾(ω_s) = 1

¯

h⁴ReX

iα

|T_iα|² Z t

−∞

dτe^{−iωs(t−τ)} Z τ

−∞

dτ₁ Z τ

−∞

dτ₂e^{¯hiα(τ2−τ1)}f_α(_α)hTˆµˆ^†_sR(t)ˆµ_sL(τ)ˆc^†_iL(τ₁)ˆc_iR(τ₂)i (1)

where the electron is created due to the interaction with αth lead. Note that the relative time-ordering in the first two operators at τ₁ and τ₂ is irrelevant as they act on different branches. It is straightforward to write down the corresponding Hilbert space expression.

After inserting the molecular many-body states and performing the time integrals, we obtain,

S⁽¹⁾(ωs) = γ

¯ h

X

iα

X

a⁰e⁰

|Tiα|² |ha|ci|e⁰ihe⁰|µ^†_s|a⁰i|²fα(α)

[(¯hω_s−E_e⁰_a⁰)²+γ²][(_α−¯hω_s−E_a⁰_a)²+ Γ²] (2)

1

whereΓ and γ are the life-times of the a→e⁰ and e⁰ →a⁰ excitations, and E_e⁰_a⁰ =E_e⁰−E_a⁰ is the energy difference between the anionic excited and (lower energy) ground states, |e⁰i and |a⁰i, respectively, and the first Lorentzian with width γ in (2) is to make sure that the correct a⁰ and e⁰ states are picked up from the sum. This fact has been used above to replace E_e⁰_a⁰ by¯hω_s in the second Lorentzian. Sum over the lead states α can be converted to integration by introducing lead density of states ρ_α(_α). Taking small temperature limit such that fα(α) = Θ(α−EF −eV) and performing the integral over the lead energies, α, we obtain the results given in Eq. (5).

Similarly, for the EL process depicted in Fig. (1b), we obtain the following expression in terms of the Liouville space correlation function.

S⁽²⁾(ω_s) = − 1

12¯h⁶ReX

ii⁰

X

αβ

|T_iα|²|T_i⁰_β|² Z t

−∞

dτ Z τ

−∞

dτ₁ Z τ

−∞

dτ₂ Z τ2

−∞

dτ₃ Z τ1

−∞

dτ₄

e^iωs(τ−t)e^{¯hiα(τ2−τ1)}e^{¯hiβ(τ4−τ3)}f_α(_α)(1−f_β(_β))hTˆµˆ^†_sR(t)ˆµ_sL(τ)ˆc_iR(τ₂)ˆc^†_iL(τ₁)ˆc_i⁰_L(τ₄)ˆc^†_i0R(τ(3)₃)i

where α 6= β. Again, it is straightforward to write down the corresponding Hilbert space expression and then inserting the molecular many-body states as depicted in the diagram and leads to

S⁽²⁾(ω_s) =− 1

12¯h⁵ReX

ii⁰

X

αβ

|T_iα|²|T_i⁰_β|²f_α(_α)(1−f_β(_β))|ha|c^†_i0|e⁰⁰ihe⁰⁰|c_i|eihe|µ^†_s|ai|²δ(¯hω_s−E_ea)

Z ∞ 0

dτ₁ Z ∞

0

dτ₂ Z ∞

0

dτ₃ Z ∞

0

dτ₄e^{¯hi(α−β−Eea)(τ1−τ2)}e^{¯hi(β−Eae00(τ3−τ4)} (4)

Straightforward integration over times and replacing sum over lead energies we obtain,

S⁽²⁾(ω_s) = γ 12¯h

X

ii⁰ee⁰⁰

ρ_αρ_β|T_iαT_i⁰_β|² (¯hωs−Eea)²+γ²

Z d

Z d⁰

ha|c_i⁰|e⁰⁰ihe⁰⁰|c_i|eihe|µ^†|ai

(−⁰−hω¯ s−iΓae)(⁰+Ee⁰⁰a−iΓe⁰⁰a)

2

f_α()(1−f_β(⁰)). (5)

In order to perform energy integrals, we assume small Γ_ae limit and then taking small temperature limit, such that the Fermi functions become heavy-side function and µα = eV

2

and µ_β = 0, we obtain the result given in Eq. (6).

Green function result

When the self-energy matrix, Γ, is not diagonal, that is, when the coherences between the molecular orbitals induced by the leads are not negligible, Green functions are not diagonal in the molecular orbital basis. For this case, G^RL(ω) and G^LR(ω) for a non-interacting system at steady-state and within the wide band approximation can be expressed in matrix form as,

G^RL(ω) = −i X

α=L,R

[1−f_α(ω)]G^r(ω)S_αG^a(ω) (6) G^LR(ω) = i X

α=L,R

f_α(ω)G^r(ω)S_αG^a(ω) (7)

where f_α(ω) = e^{β(ω−µα)}+ 1−1

are the Fermi functions of the reservoirs and S_α is the self- energy matrix due to coupling to the αth lead. G^r/a(ω) are the retarded/advanced Green functions of the molecule under the influence of the leads given as,

G^r(ω) = G^a(ω)^†=

"

ωI −H_sys+ i 2

X

α=L,R

S_α

#−1

(8)

with H_sys

ij =ξ_iδ_ij and[S_α]_ij = 2πρ_αT_iα^∗T_jα. G^r(ω)can be expressed in quasi-diagonal form using the right (|R_ni) and the left (hL_n|) eigenstates corresponding to the nth eigenvalue, γ_n, of the non-hermitian operatorH_sys− ₂ⁱ P

α=L,RS_α. This gives, G^r(ω) =X

n

|R_nihL_n|

ω−γ_n . (9)

3

Substituting this in (6), we obtain,

G^RL(ω) = −i X

α=L,R

X

mn

(1−f_α(ω))|R_ni hL_n|S_α|L_mi

(ω−γ_n)(ω−γ_m^∗)hR_m| (10) G^LR(ω) = i X

α=L,R

X

mn

f_α(ω)|R_ni hLn|Sα|Lmi

(ω−γ_n)(ω−γ_m^∗)hR_m| (11)

Using the above expressions for G^LR(ω) and G^RL(ω) in the equation for signal given in Eq. (11) and performing ω integral using Cauchy’s residue method² gives,

S(ωs) = 2 π

X

α,β=L,R

nαβ(ωs)<

( X

n,m,k,l

"

hL_n|S_α|L_mihR_m|µ^†|R_kihL_k|S_β|L_lihR_l|µ|R_ni (ω_s+γ_n−γ_k)

#

×







 Ψh

1

2 +i^β(γn_2π^−µα)i

−Ψ

1

2 +i^β(^γn−µβ+ωs)

2π

(γ_n−γ_m^∗) (ω_s+γ_n−γ_l^∗) + Ψh

1

2 +i^β(γk−µ_2π^α−ωs)i

−Ψ

1

2 +i^β(^γk−µ_β)

2π

(γ_k−γ_l^∗) (ω_s−γ_k+γ_m^∗)

















 (12)

wheren_αβ(ω_s) = e^{β(µα−µβ+¯hω)}−1−1

is the Bose-Einstein distribution function, Ψ[Z] is the digamma function³ and [µ]_ij = Θ[_i−_j]V_ij.

References References

(1) C. Marx, U. Harbola and S. Mukamel, Nonlinear optical spectroscopy of single, few, and many molecules: Nonequilibrium Green function QED approach, Phys. Rev. A77, 022110 (2008).

(2) J. E. Marsden and M. J. Hoffman, Basic Complex Analysis, 3^rdEdition, W. H. Freeman Comp., New York (1999).

(3) Handbook of Mathematical Functions, Edited by M. Abramowitz and I. A. Stegun, Dover Publications, New York (1972).

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