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Raman Scattering from Nonequilibrium Molecular Conduction Junctions Michael Galperin, Mark A Subscriber access provided by TEL AVIV UNIV (AO) Letter Raman Scattering from Nonequilibrium Molecular Conduction Junctions Michael Galperin, Mark A. Ratner, and Abraham Nitzan Nano Lett., 2009, 9 (2), 758-762• DOI: 10.1021/nl803313f • Publication Date (Web): 21 January 2009 Downloaded from http://pubs.acs.org on February 18, 2009 More About This Article Additional resources and features associated with this article are available within the HTML version: • Supporting Information • Access to high resolution figures • Links to articles and content related to this article • Copyright permission to reproduce figures and/or text from this article Nano Letters is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 NANO LETTERS 2009 Raman Scattering from Nonequilibrium Vol. 9, No. 2 Molecular Conduction Junctions 758-762 Michael Galperin,*,† Mark A. Ratner,‡ and Abraham Nitzan§ Department of Chemistry & Biochemistry, UniVersity of California at San Diego, La Jolla, California 92093, Department of Chemistry, Northwestern UniVersity, EVanston Illinois 60208, and School of Chemistry, The Sackler Faculty of Science, Tel AViV UniVersity, Tel AViV 69978, Israel Received November 3, 2008; Revised Manuscript Received December 26, 2008 ABSTRACT Raman scattering is a potentially important probe of structure, dynamics, and thermal properties of single-molecule conduction junctions. We combine a nonequilibrium Green’s function description of the junction with a generalized scattering theory of the Raman process, which provides the first theoretical description of Raman scattering from such systems. The voltage dependence of the Raman flux shows a characteristic behavior at the conductance threshold resulting from (a) partial populations in the ground and excited molecular levels that give rise to two scattering pathways as well as interference between them and (b) junction heating that affects the Raman intensities. Comparing “effective temperatures” obtained from Raman scattering and heat balance serves to establish the integrity of this concept for nonequilibrium junctions. The interaction of molecular conduction junctions with the The molecule is modeled as a two-electronic-levels system radiation field has been explored as a way for characterizing interacting with a molecular vibration, with the continuum and controlling their operation.1,2 In particular, optical electronic manifolds of the electrodes (with an applied - switching of such junctions was demonstrated3 5 and studied voltage across them) and with the radiation field. The 6,7 theoretically Laser control of junction operation and their theoretical treatment needs to account for nonequilibrium 8-14 noise properties was studied theoretically. The effect of nature of the electronic transport together with the scattering illumination on the conduction properties of atomic size of photons off of the molecule. contacts was studied experimentally,15 and implications for molecular junctions were considered theoretically.16 We focus on response to the local electromagnetic field, Surface-enhanced Raman and resonance Raman spec- which is assumed independent of the imposed potential bias troscopies (SERS and SERRS)17,18 have become important and therefore amenable to independent calculation. We diagnostic tools for many science applications. The observa- employ a generalized spinless model,28 comprising a mol- tion of single-molecule SERS and SERRS19,20 indicates that ecule coupled to two metal leads (L and R) each in its own the signal is often dominated by molecules adsorbed at thermal equilibrium. The molecule is represented by its particular “hot spots”. Single-molecule transport junctions, highest occupied and lowest unoccupied molecular orbitals, molecules connecting two metal leads, whose electrical HOMO ) 1 and LUMO ) 2, with populations (n1, n2), transport properties are under intensive study,21 have struc- respectively, that describe the ground (1,0) and lowest excited tures similar (e.g., ref 22) to models used for such hot spots. (0,1) molecular states, as well as positive (0,0) and negative Monitoring SERS and SERRS together with electronic ˆ ˆ ˆ (e-v) ˆ (et) (1,1) ion states. The Hamiltonian is H ) H0 + V + V 23-26 transport in such junctions will provide new diagnostic + ˆ (v-b) + ˆ (e-h) + ˆ (e-p) + ˆ (e-p) V V VM VCT with tools for molecular electronics and requires a suitable theoretical description of the optical response of nonequi- ˆ ) ˆ† ˆ + ˆ†ˆ + † + H0 ∑ εmdmdm ωVbVbV ∑ εkcˆkcˆk librium open nanosystems. Here we outline a first theoretical m)1,2 k∈L,R treatment of Raman scattering from nonequilibrium molec- ˆ† ˆ + † ∑ ωbb ∑ νRaˆRaˆR (1) ular conduction junctions (for the full theoretical treatment R∈{i,{f}} see ref 27) and describe its main experimental implications. ˆ (e-v) ) (e-v) ˆ ˆ† ˆ V ∑ V m QVdmdm (2) * Corresponding author, [email protected]. m)1,2 † University of California at San Diego. (et) (et) † (et) † ‡ ˆ ) ˆ + ˆ Northwestern University. V ∑ ∑ (V km cˆkdm V mk dmcˆk) (3) § Tel Aviv University. K)L,R k∈K;m 10.1021/nl803313f CCC: $40.75 2009 American Chemical Society Published on Web 01/21/2009 - - Vˆ (v b) ) ∑ U (v b)Qˆ Qˆ (4) energy currents, and their balance at steady-state determines V 30 the junction “effective temperature” according to (e-h) (e-h) † † + Vˆ ) ∑ (V Dˆ cˆ cˆ + H.c) (5) -1 ΩV NBE(ωV) I- k1k2 k1 k2 ≡ - ) * NV exp[ωV ⁄ TV 1] (7) k1 k2 ΩV + (I+ - I-) / ˆ (e-p) ) (e-p) ˆ † + (e-p) † ˆ where V M ∑ (U R D aˆR U R aˆRD) (6a) R∈{i,{f}} 2 ΩV ) 2π∑ |U | δ(ωV - ω ) ˆ (e-p) ) + † (e-p) ˆ + (e-p) ˆ V CT ∑ ∑ ∑ (aˆR aˆR)[V km,R Dkm V mk,R Dmk] R∈ ∈ ) {i,{f}} k {L,R} m 1,2 ) - -1 (6b) NBE(T, ωV) exp[ω ⁄ T 1] ˆ and H0 includes additively the Hamiltonians for the molecular +∞ electronic and vibrational subsystems, the metal electrons, (e-v) 2 dE < > ˆ† ˆ † I( ≡ ∑ |V | ∫ G (E)G (E ( ωV) and the radiation field. dm (dm) and cˆk (cˆk) create (annihilate) m -∞ 2π m m an electron in the molecular state m and in the lead state k m)1,2 ˆ† ˆ ˆ† ˆ of energies εm and εk, respectively. bV (bV) and b (b) create The single electron GFs of the biased junction given in terms (annihilate) vibrational quanta in the molecular mode,V, and of spectral functions associated with the molecule-leads † (K) ) ) the thermal bath mode, , respectively. aˆR (aˆR) stands for coupling Γm (E), m 1, 2, K L, R, and the leads Fermi creation (annihilation) operators of radiation field quanta. ω functions fK(E)by and ν denote frequencies of phonon and photon modes, ˆ ≡ ˆ + ˆ† )V Γ(L)(E)f (E) + Γ(R)(E)f (E) respectively. Also Qj bj bj , j , are displacement < ) m L m R V G m(E) i operators for the molecular ( ) and thermal bath () (E - ε )2 + (Γ (E)⁄2)2 vibrations, respectively. The corresponding momentum op- m m > ˆ ≡- ˆ - ˆ† )V ˆ et ˆ (e-h) ˆ (e-v) G E if E -i - f E erators are Pj i(bj bj ), j , . V , V , V , and (for m ( ) with K( ) replaced by (1 K( )). It should Vˆ (v-b) are the couplings associated with electron transfer be noted, however, that the definition of “effective temper- between molecule and leads, coupling of molecular excitation ature” in a nonequilibrium system is not unique. In particular, to electron-hole pairs in the leads, electron-molecular how the effective temperatures obtained from different vibration interaction and coupling between the molecular observables compare with each other in nonequilibrium ˆ (e-p) (e-p) vibration and the thermal bath, respectively. VM and VCT situations is an open question. are radiative coupling terms. The former is associated with Our focus in the present work is the inelastically scattered the molecular transition, while the latter accounts for - ˆ ≡ ˆ†ˆ photon flux. Its calculation requires substantial departure metal molecule charge transfer (CT) transitions. D d1d2, ˆ † ≡ ˆ†ˆ from former methodologies. First, while the electronic D d2d1 are the molecular polarization operators and ˆ ≡ ˆ† ˆ ≡ †ˆ ) nonequilibrium state of the junction is best described within similarly, Dmk dmcˆk, Dkm cˆkdm (m 1, 2). Here and below we put e ) 1, p ) 1, and kB ) 1 for the electron charge and the NEGF formalism, the Raman process with energy- the Planck and Boltzmann constants, respectively. Note that resolved initial and final fluxes is naturally described by in our formulation we use zero and one photon occupations scattering theory. A unified description is achieved by of the relevant radiation field modes. Therefore the coupling considering the molecule as connecting between two “photon (e-p) (e-p) amplitudes UR and Vkm, R reflect the intensity of the local baths”: an incoming bath with only the incident mode i electromagnetic field, including effects of plasmon excitations populated and an outgoing bath that initially has all modes in the leads. Also, the vibronic coupling (2), a parallel shift vacant but which can be populated by the scattering off of in harmonic nuclear potential surfaces between different electronic states, is common in molecular spectroscopy but the molecule. its applicability to Raman scattering is limited to near Another complication arises from the presence of two resonance situations. This makes our theory valid for types of molecule-radiation fields couplings, eqs 6a and 6b. resonance Raman scattering, although it can provide qualita- Here we assume that the molecular transition energy ∆ε ) tive insight also for the off-resonance case. The model εj - εj and the CT resonances εj - E , E - jε are far Hamiltonian Hˆ describes inelastic light scattering superim- 2 1 2 F F 1 j ) - V(e-v) posed on inelastic electron tunneling in a biased molecular apart (εm εm λm m are phonon-renormalized electronic junction.
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