Subscriber access provided by 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

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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, , EVanston Illinois 60208, and School of Chemistry, The Sackler Faculty of Science, , Tel AViV 69978,

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 jε - jε and the CT resonances jε - 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. The former will be treated in the lowest (fourth) energies), and treat their contributions separately. Finally, order appropriate for Raman scattering. The latter has been because Raman scattering is a coherent process, standard under intense study in recent years,29 mostly in the context approximations used in the analysis of electron-vibration of inelastic tunneling spectroscopy. Its handling follows our dynamics in inelastic tunneling become inadequate. In previous work29,30 and is described in detail in ref 27. Briefly, particular, the decoupling approximation usually made at the following a small polaron transformation,31 Hˆ f H¯ˆ , where single particle GF level is replaced here by decoupling at ˆ each D operator is renormalized by a corresponding phonon what is essentially a two particle GF, yielding the structures ˆ f ˆ ˆ ˆ ) ˆ † ˆ ˆ ≡ ˆ shift operator, e.g., D DX with X X1X2; Xm exp[iλmPV]; - (e-v) of eqs 9 11 below. The procedure is described in detail in λˆ ≡ V /ωV, elastic and inelastic electron currents are m m ref 27. For the molecular mechanism associated with obtained using nonequilibrium Green function (NEGF) theory,32-34 solving using standard approximations coupled coupling (6a) it leads to the light scattering flux in the form equations for electron and phonon Green functions (GFs) (nR) (iR) (intR) J ) J + J + J (8) and self-energies (SEs), which are needed for current iff iff iff iff evaluation. The same formalism yields electron and phonon where

Nano Lett., Vol. 9, No. 2, 2009 759 +∞  (nR) ) 2 2 -  t t × J iff |Ui| |Uf| ∫-∞ d(t t )∫-∞ dt1∫-∞ dt2 - - -   iνi(t1 t2) iνf(t t ) ˆ ˆ † ˆ ˆ † × e e 〈X(t2)X (t )X(t)X (t1)〉 ˆ ˆ †  ˆ ˆ † 〈D(t2)D (t )D(t)D (t1)〉 (9) +∞ +∞ +∞ (iR) 2 2  J f ) |U | |U | ∫ d(t - t )∫ dt ∫ dt × i f i f -∞ t 1 t 2 - - -   iνi(t1 t2) iνf(t t ) ˆ † ˆ ˆ † ˆ × e e 〈X (t )X(t2)X (t1)X(t)〉 ˆ †  ˆ ˆ † ˆ 〈D (t )D(t2)D (t1)D(t)〉 (10) +∞ +∞ (intR) ) 2 2 -  t × j J f |U | |U | ∫ d(t t )∫ dt ∫ dt Figure 1. The integrated Raman signal J f (solid line, black, i f i f -∞ -∞ 1 t 2 j νi {νf} - - -   left vertical axis), its Stokes Jν fν -ωV (dashed line, blue, right vertical iνi(t1 t2) iνf (t t ) ˆ † ˆ ˆ ˆ † × j i i 2Re[e e 〈X (t )X(t2)X(t)X (t1)〉 axis) and anti-Stokes Jνifνi+ωV (dotted line, red, right) components, ˆ †  ˆ ˆ ˆ † and the current I (inset) displayed as functions of the applied bias 〈D (t )D(t2)D(t)D (t1)〉] (11) V. The parameters used are νi ) jε2 - jε1, EF ) 0, jε1 )-1 eV, jε2 (nR) (iR) ) ) ) (e-v) ) (e-v) ) Jiff and Jiff describe processes that start and end in the 1 eV, ωV 0.1 eV, ΩV 0.005 eV, V1 0.1 eV, V2 molecular ground and excited states, respectively, “normal” 0.05 eV and T ) 100 K, “a.u.” denotes atomic units. and “inverse” Raman scattering. The third term, eq 11, results from interference between these scattering pathways and is example, the CT analogue of eq 9, the normal Raman discussed below. - scattering associated with metal-to-molecule charge transfer Next, consider the correlation functions in eqs 9 11. transition that dominates for incident radiation in resonance 〈Xˆ † t Xˆ t Xˆ t Xˆ † t 〉 Phonon correlation functions such as ( 1) ( 2) ( 3) ( 4) with this transition, is obtained in the form27 can be evaluated by standard methods,31 that the steady state junction can be associated with an effective thermal distribu- (nR) (e-p) (e-p) (e-p) (e-p) J f ) ∑ V V  V V × i f k22,i 2k ,f k2,f 2k1,i  tion, here taken to correspond to the effective temperature k,k ,k1,k2∈{L,R} +∞  (7). The polarization correlation functions, e.g., ∫ -  ∫t ∫t × ˆ † ˆ ˆ ˆ † -∞ d(t t ) -∞ dt1 -∞ dt2 〈D (t1)D(t2)D(t3)D (t4)〉, are evaluated by invoking two - - -   iνi(t1 t2) iνf(t t ) × ˆ ˆ † ˆ ˆ † × simplifications. First, radiative level broadening is disre- e e 〈X2(t2)X2(t )X2(t)X2(t1)〉 garded, and second, energy relaxation due to electron-hole † ˆ ˆ†   † ˆ ˆ† 〈cˆk (t2)d2(t2)d2(t )cˆk(t )cˆk(t)d2(t)d2(t1)cˆk (t1)〉 (15) pair creation in the metals is accounted for using the 2 1 ansatz35,36 The nuclear correlation term is evaluated as before. The electronic average is approximated as a product ˆ ˆ ˆ ˆ i ¯ t i ¯ t i ¯ (et) i ¯ (et) (e-h) † † † † H - H H t - H t -Γ t⁄2 〈 ′ ′ 〉 × 〈ˆ ˆ ′ ˆ ˆ 〉 Dˆ (t) ≡ e Dˆ e ≈ e Dˆ e e (12) cˆk2(t2)cˆk (t )cˆk(t)cˆk1(t1) d2(t2)d2(t )d2(t)d2(t1) , and evaluated using Wick’s theorem.27 This again leads to a numerically ¯ˆ(et) ≡ ¯ˆ + ¯ˆ(et) 35,36 where H H0 V and where tractable form. (e-h) ) (e-h) 2 - - - The results shown below are based on the molecular Γ (E) 2π ∑ |Vk k | fk [1 fk ]δ(E (εk εk )) 1 2 1 2 2 1 scattering mechanism, eq 6a. (The charge transfer contribu- k1*k2 (13) tion is considered in more detail in ref 27), and demonstrates the main physical results of our theory: the behavior of is the molecular polarization damping rate due to coupling Raman scattering across the conduction threshold, its ap- to electron-hole excitations in the leads. Disregarding the plicability for effective temperature determination, and the energy dependence of Γ(e-h)(E), the mixing of levels 1 and 2 due to their interactions with the leads and the effect of novel aspects in its interference component. electron-phonon interaction on these functions finally leads Figure 1 shows, as functions of biased voltage, the total j to (elastic and inelastic) scattering flux J f , as well as its j j νi {νf}  - (e-h) - + -  Stokes, Jνifνi-ωV, and anti-Stokes, Jνifνi+ωV, components. Here ˆ ˆ † ˆ ˆ † ≈ Γ (t1 t t2 t )⁄2 × 〈D(t2)D (t )D(t)D (t1) 〉 e j )F F F Jνifνf R(νi) R(νf)Jiff, where R is the density of radiation < - >  - - <  - < - × - [G1(t1 t2)G1(t t) G1(t t2)G1(t1 t)] field modes. The inset shows the current voltage charac- > - < -  - > -  > - teristic, obtained using the procedure of ref 28. A symmetric [G2(t2 t1)G2(t t ) G2(t2 t )G2(t t1)] (14) voltage division factor, i.e., µK ) EF + ηKeV (K ) L, R), ηL and analogous expressions for the polarization correlation ) 1 + η ) 0.5, was used. functions that appear in eqs 10 and 11, where G>,< (t)(m ) R m ) 1, 2) are the Fourier transform of the junction GFs defined At the voltage threshold for conduction (here V 2V) above. Equations 8-14 can now be used to evaluate the the molecular levels enter the window between the leads Raman scattering flux for our model. Explicit expressions chemical potentials. The current increases accordingly (note are given in ref 27, and principal results are shown and its above threshold modulation by the molecular vibration), discussed below. while the total scattering signal drops by almost half. This The mechanism discussed above is expected to dominate can be understood by noting that the total scattering intensity the Raman scattering near the molecular resonance. A similar is approximately a sum of the normal and inverse contribu- procedure is used in the evaluation of the Raman flask tions, which, for weak molecule-leads coupling, are pro- associated with the CT interaction (6b) that can be important portional to the level population factors n1(1 - n2) and n2(1 37,38 for incident frequency near jε2 - EF or EF - jε1. For - n1), respectively. Below threshold n1 ) 1 and n2 ) 0,

760 Nano Lett., Vol. 9, No. 2, 2009 Figure 2. Comparison between the effective vibrational temperature Figure 3. The ratio between the interference component and the Teff, calculated from eq 7 (solid line, blue) and the temperature j(intR) j ) v total Raman flux at the Stokes frequency, Jνifνi-ωV/Jνifνi-ωV,atV - (e-h) estimated from the ratio of Stokes and anti-Stokes intensities TS aS, 2.5 V as a function of Γ for Γm ) 0.01 eV and νi ) ∆ε. Other eq 16 (dashed line, red) vs applied bias. Calculation is done for parameters as in Figure 2. the resonant scattering case νi ) ε2 - ε1 ≡ ∆ε. Parameters are as (e-v) ) (e-v) ) in Figure 1 except that V1 0.1 eV and V2 0.08 eV. temperature even under the inherently nonequilibrium situ- ation imposed by the voltage bias. Indeed, a first application while well above it n ≈ n ≈ 0.5. The integrated scattering 1 2 of Raman scattering for such a temperature estimate has been decreases with n (1 - n ) + n (1 - n ). The Stokes intensity 1 2 2 1 recently described.26 One should keep in mind the limitations similarly decreases; however the anti-Stokes signal increases imposed by our simple model: prediction (a) stems from the beyond threshold because of junction heating. assumption that the resonance scattering involves transitions Figure 2 compares the vibrational temperature T of eq 7 between the molecular HOMO and LUMO. Temperature to that estimated from the ratio between the Stokes and anti- estimates should take into account signal frequency depen- Stokes Raman components according to dence associated with possible plasmon excitation in the ωV leads. Indeed, the electromagnetic field that enters our ) TS-aS (16) jJ ( + )2 calculation is the local field that can be obtained for any ν fν -ωV νi ωV ln i i given junction composition and geometry. With this provi- (j 2) J f + (ν - ωV) νi νi ωV i sion, our conclusions should be valid, at least qualitatively, 2 in many realistic situations. The factors (ν ( ωV) correct for the frequency depen- i In addition, we found a new interference component that dence of the outgoing radiation density of modes. The good contributes to the scattered intensity above the conduction correspondence seen in Figure 2 indicates the usefulness of threshold. While this contribution to the overall signal cannot both measures. (The apparent deviation of the two measures be monitored independently, its appearance, resulting from at low bias results from the inaccuracy in the computed anti- interfering Raman pathways, is interesting and deserves Stokes signal which is extremely small in this voltage further study. regime.) Note, however, that additional frequency-dependent Junction spectroscopy, in particular Raman scattering, corrections to the S/AS measure may result from resonance opens new methodological routes for characterization and structure in the scattering cross section.26 control of molecular conduction junctions and at the same Finally, we draw attention to the interesting observation time provides interesting experimental and theoretical chal- of interference, eq 11, between the two Raman pathways lenges. Future theoretical efforts should focus on extending available above the conduction threshold. In a sense, this is the theory to better describe the nonresonant scattering a double slit situation, where one path sees the lower (upper) regime and on applications to realistic models of molecular molecular electronic level occupied (empty), while for the junctions. other these occupations are interchanged. Interference results from the correlated interchange between these pathways, Acknowledgment. A.N. thanks the US-Israel BSF, ISF, affected by the energy transfer interaction (5) (see Figure GIF, and ERC for support. M.G. acknowledges the support 3), which is effective despite the coupling to a thermal of UCSD startup funds, UC Academic Senate research grant, electronic environment. and of a LANL Director’s Postdoctoral Fellowship. M.R. In this paper we have outlined a theory of Raman thanks the Chemistry and MRSEC Divisions of the NSF for scattering from biased and current-carrying molecular junc- support. tions, valid mainly under resonant scattering conditions. The theory makes specific predictions of generic nature that References should be observable at and above the conduction threshold: (1) Flaxer, E.; Sneh, O.; Chesnovsky, O. Science 1993, 262, 2012. (a) The Stokes component in resonance Raman scattering is (2) Wu, S. W.; Ogawa, N.; Ho, W. 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