Surface-assisted carrier excitation in plasmonic nanostructure
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Surface-assisted carrier excitation in plasmonic nanostructures
Tigran V. Shahbazyan
Department of Physics, Jackson State University, Jackson, MS 39217 USA
We present a quantum-mechanical model for surface-assisted carrier excitation by optical fields inplasmonic nanostructures of arbitrary shape. We derive an explicit expression, in terms of local fieldsinside the metal structure, for surface absorbed power and surface scattering rate that determinethe enhancement of carrier excitation efficiency near the metal-dielectric interface. We show thatsurface scattering is highly sensitive to the local field polarization, and can be incorporated intometal dielectric function along with phonon and impurity scattering. We also show that the obtainedsurface scattering rate describes surface-assisted plasmon decay (Landau damping) in nanostructureslarger than the nonlocality scale. Our model can be used for calculations of plasmon-assisted hotcarrier generation rates in photovoltaics and photochemistry applications.
I. INTRODUCTION
Plasmon-assisted hot carrier excitation and transferacross the interfaces [1] has recently attracted intense in-terest due to wide-ranging applications in photovoltaics[2–11] and photochemistry [12–16]. In metal nanostruc-tures with characteristic size L below the diffractionlimit, i.e., L < c/ω , where c and ω are, respectively,the light speed and frequency, the light scattering is rel-atively weak, and the absorption is dominated by reso-nant excitation of surface plasmons, which subsequentlydecay into electron-hole ( e-h ) pairs through several de-cay mechanisms depending on the size and shape of aplasmonic system [17–21]. While in relatively large sys-tems, excitation of e-h pairs with optical energy ~ ω isaccompanied by momentum relaxation due to phononand impurity scattering, for systems with characteristicsize L ∼
20 or smaller, the dominant momentum relax-ation channel is surface scattering that leads, in par-ticular, to size and shape dependence of the plasmonlinewidth [22–32]. Calculations of surface-assisted decayrate have been performed for some simple shapes (mostlyspherical) within random phase approximation (RPA)[33–41] or time-dependent local density approximation(TDDFT) [42–48] approaches, while for general shapesystems, the classical scattering model [49–53] suggestedthe surface scattering rate in the form γ cs = Av F /L s ,where L s is the ballistic electron scattering length, andthe constant A includes the effects of surface potential,electron spillover, and dielectric environment [54]. Note,however, that recent measurements of plasmon linewidthin nanostructures of various shapes revealed significantdiscrepancies with a simple L − dependence[27, 30–32].Here we present a quantum-mechanical model forsurface-assisted excitation of e-h pairs by alternating lo-cal electric field of the form E ( r ) e − iωt created in themetal either by excitation of a plasmon or as a responseto monochromatic external field. We show that surfacecontribution to the absorbed power due to energy trans-fer to the excited carriers is given by Q s = e π ~ E F ( ~ ω ) Z dS | E n | , (1) where integration is taken over the metal surface, E n isthe local field component normal to the interface, and E F is the Fermi energy in the metal. The above expression,which is derived within RPA approach, is valid for sys-tems of arbitrary shape that are significantly larger thanthe nonlocality scale v F /ω [55, 56], i.e., for systems atleast several nm large. We compute the surface enhance-ment of the carrier excitation efficiency for some commonnanostructures and show that it is highly sensitive to thelocal field polarization and system geometry. II. THEORY
We consider a metal nanostructure characterized bycomplex dielectric function ε ( ω ) = ε ′ ( ω ) + iε ′′ ( ω ) in amedium with dielectric constant ε d , and, for simplic-ity, restrict ourselves by systems with a single metal-dielectric interface. The standard expression for ab-sorbed power has the form [57], Q = ωε ′′ ( ω )8 π Z dV | E | . (2)where integration is carried over the metal volume, whilethe local field E ( r ) is determined, in the quasistatic limit,by the Gauss’s law ∇ [ ε ′ ( ω, r ) E ( r )] = 0 [here ε ( ω, r )equals ε ( ω ) and ε d in the metal and dielectric regions,respectively]. The surface contribution to the absorbedpower is Q s = ~ ω/τ , where 1 /τ is the first-order tran-sition probability rate, leading to the standard RPA ex-pression [41] Q s = πω X αβ | M αβ | [ f ( ǫ α ) − f ( ǫ β )] δ ( ǫ α − ǫ β + ~ ω ) . (3)Here, M αβ = R dV ψ ∗ α Φ ψ β is the matrix element of localpotential Φ( r ) defined as e E = − ∇ Φ ( e is the electroncharge), ψ α ( r ) and ψ β ( r ) are wave-functions for electronstates with energies ǫ α and ǫ β , respectively, separatedby ~ ω , and f ( ǫ ) is the Fermi distribution function. Fornanostructures of arbitrary shape, numerical evaluationof M αβ is a highly complicated task due to the complexityof electron wave functions. However, as we show below,for the hard-wall confining potential, Q s can be derivedin a closed form for any system significantly larger than v F /ω (but still smaller than c/ω ).Excitation of an e-h pair with a large, compared tothe electron level spacing, energy ~ ω requires momen-tum transfer to the cavity boundary. We note that theboundary contribution to M αβ can be presented as anintegral over the metal surface [58], M sαβ = − e ~ m ǫ αβ Z dS [ ∇ n ψ α ( s )] ∗ E n ( s ) ∇ n ψ β ( s ) , (4)where ∇ n ψ α ( s ) is the wave function derivative normal tothe surface, E n ( s ) is the corresponding normal field com-ponent, ǫ αβ = ǫ α − ǫ β is the e-h pair excitation energy,and m is the electron mass. Using this matrix element,Eq. (3) takes the form Q s = e ~ πm ω Z Z dSdS ′ E n ( s ) E ∗ n ′ ( s ′ ) F ω ( s , s ′ ) , (5)where F ω ( s , s ′ ) is the e-h surface correlation function, F ω ( s , s ′ ) = Z dǫf ω ( ǫ ) ρ nn ′ ( ǫ ; s , s ′ ) ρ n ′ n ( ǫ + ~ ω ; s ′ , s ) , (6)defined in terms of normal derivative of the electroncross density of states, ρ ( ǫ ; s , s ′ ) = Im G ( ǫ ; s , s ′ ), at sur-face points: ρ nn ′ ( ǫ ; s , s ′ ) = ∇ n ∇ ′ n ′ Im G ( ǫ ; s , s ′ ). Here G ( ǫ ; s , s ′ ) is the confined electron Green function, andthe function f ω ( ǫ ) = f ( ǫ ) − f ( ǫ + ~ ω ) restricts the initialenergy of promoted electron to the interval ~ ω below E F .To evaluate Q s , we note that excitation of an e-h pairwith energy ~ ω is accompanied by momentum transfer ∼ ~ ω/v F and, hence, takes place in a region of size ∼ v F /ω , so that the e-h correlation function F ω ( s , s ′ )peaks in the region | s − s ′ | . v F /ω and rapidly oscil-lates outside of it (see below). At the same time, thelocal fields significantly change on a much larger scale ∼ L . Therefore, for L ≫ v F /ω , the main contributionto the integral in Eq. (5) comes from the regions with E n ( s ) ≈ E n ( s ′ ), so that Q s takes the form Q s = e ~ πm ω Z dS | E n ( s ) | ¯ F ω ( s ) , (7)where ¯ F ω ( s ) = R dS ′ F ω ( s , s ′ ). Evaluation of ¯ F ω is basedupon multiple-reflection expansion for the electron Greenfunction G ( ǫ ; s , s ′ ) in a hard-wall cavity [59]. For sys-tem’s characteristic size L ≫ λ F , where λ F is the Fermiwavelength, the main contribution comes from the di-rect and singly-reflected paths, while the higher-orderreflections are suppressed as powers of λ F /L . In theleading order, we obtain G ( ǫ ; s , s ′ ) = 2 G ( ǫ, s − s ′ ),where G ( ǫ, r ) = ( m/ π ~ ) e ik ǫ r /r , with k ǫ = √ mǫ/ ~ ,is the free electron Green function and factor 2 comesfrom equal contributions of the direct and singly-reflectedpaths at a surface point [58]. It is now easy to see thatthe integrand of Eq. (6) peaks in the region | s − s ′ | . ( k ǫ + ~ ω − k ǫ ) − and rapidly oscillates outside of it. For ǫ ∼ E F and ~ ω/E F ≪
1, this sets the scale | s − s ′ | ∼ v F /ω for the e-h correlation function F ω ( s , s ′ ) in Eq. (5).Finally, for L ≫ v F /ω , after computing normal deriva-tives in ρ nn ′ ( ǫ ; s , s ′ ) = 2 ∇ n ∇ ′ n Im G ( ǫ, s − s ′ ) relativeto the tangent plane at a surface point [58], we obtain¯ F ω = (2 m E F /π ~ ) ~ ω , yielding Eq. (1).The surface contribution Q s to the full absorbed power Q should be considered in conjunction with its bulk coun-terpart Q , i.e., Q = Q + Q s . The bulk contribution isgiven by the standard expression (2), where the metaldielectric function ε ( ω ) includes only the bulk processes.In the following, we adopt the Drude dielectric function ε ( ω ) = ε i ( ω ) − ω p /ω ( ω + iγ ), where ε i ( ω ) describes in-terband transitions, ω p is the plasma frequency, and γ is the scattering rate. In fact, both bulk and surfacecontributions can be combined, in a natural way, withinthe general expression (2). Indeed, using the relation ω p = 16 e E F / π ~ v F , the surface absorbed power (1)can be recast as Q s = 3 v F π ω p ω Z dS | E n | . (8)Then, it is easy to see that the surface contribution can beincorporated into general expression (2) for the absorbedpower by modifying the Drude scattering rate as γ = γ + γ s , where γ is the usual bulk scattering rate and γ s = 3 v F R dS | E n | R dV | E | , (9)is the surface scattering rate. Indeed, after this modifica-tion, Q s can be obtained from Eq. (2) as the first-orderexpansion term in γ s , indicating that the surface scat-tering mechanism should be treated on par with phononand impurity scattering. Note that γ s is independent ofthe local field overall strength but highly sensitive to itspolarization relative to the metal-dielectric interface.Turning to the surface-assisted plasmon decay (Landaudamping), the plasmon decay rate in any metal-dielectricstructure is given by general expression [60] Γ = Q/U ,where U is the plasmon energy [57], U = ω π ∂ε ′ ( ω ) ∂ω Z dV | E | . (10)Using Eq. (2) for the absorbed power, the decay rate hasthe standard form,Γ = 2 ε ′′ ( ω ) (cid:20) ∂ε ′ ( ω ) ∂ω (cid:21) − . (11)Let us show that the full plasmon decay rate that includesboth bulk and surface contributions is given by Eq. (11),but with ε ( ω ) modified according to Eq. (9). Indeed,using Eq. (8), the surface contribution to Γ takes theform Γ s = Q s U = 2 ω p γ s ω (cid:20) ∂ε ′ ( ω ) ∂ω (cid:21) − , (12)where γ s is given by Eq. (9). The same expression isobtained, in the first order in γ s , from Eq. (11) withsurface-modified ε ( ω ). Note that for ω well below theinterband transitions onset, the plasmon decay rate andscattering rate coincide, Γ ≈ γ = γ + γ s . III. APPLICATIONS
Let us now discuss the effect of field polarization andsystem geometry on the carrier excitation efficiency. Thesurface enhancement factor of the absorbed power isgiven by the ratio of full ( Q ) to bulk ( Q ) absorbedpower, M = Q/Q , which, within RPA, takes a simpleform M = 1 + γ s γ . (13)Evaluation of surface scattering rate γ s is made moreconvenient by noting that the Gauss’s law reduces thevolume integral in Eq. (9) to the surface term, so that γ s = A v F R dS |∇ n Φ | R dS Φ ∗ ∇ n Φ , (14)where real part of the denominator is implied. This formof γ s as the ratio of two surface integrals reflects the factthat e-h pairs are excited in a close proximity (within v F /ω ) to the interface. The constant A , which equals A = 3 / a . A straightfor-ward evaluation of Eq. (14) recovers the standard resultfor a sphere γ sp = Av F /a , while in the recent TDLDAcalculations for relatively large (up to a = 10 nm) spher-ical particles [45, 46] the value A ≈ .
32 was obtained.Note that for systems, whose geometry permits separa-tion of variables, the form (14) of γ s is especially use-ful since it leads to analytical results for some commonstructures that so far eluded attempts of any quantum-mechanical evaluation, as we illustrate below for metalnanorods and nanodisks.In Figs. 1 and 2, we show calculated surface scatter-ing rates for nanorodes and nanodisks, which are mod-eled here by prolate and oblate spheroidal nanoparticles,respectively. These structures support longitudinal andtransverse plasmon modes oscillating along the symme-try axis (semi-axis a ) and within the symmetry plane(semi-axis b ). Using Eq. (14), γ s for all modes can befound in an analytical form [58], but here only the re-sults for the dipole mode are presented. For a nanorod(prolate spheroid) with the aspect ratio b/a <
1, we ob-tain γ s = γ sp f L,T , where f L = 32 tan α (cid:20) α sin 2 α − (cid:21) , f T = 34 sin α (cid:20) − α tan 2 α (cid:21) , (15)are the normalized rates for longitudinal and transversemodes relative to the spherical particle rate, and α = s / s p b/a PL PTPTPL
FIG. 1. Normalized rates for longitudinal and transversedipole modes in prolate spheroidal particles (nanorods) rel-ative to the spherical particle rate are shown with changingaspect ratio b/a
Inset: Schematics for the mode polarizations.
2b 2a s / s p b/a OL OTOTOL
FIG. 2. Normalized rates for longitudinal and transversedipole modes in oblate spheroidal particles (nanodisks) rel-ative to the spherical particle rate are shown with changingaspect ratio b/a
Inset: Schematics for the mode polarizations. arccos( b/a ) is the angular eccentricity. For a nanodisk(oblate spheroid) with b/a >
1, the normalized rateshave the same form (15) but with α = i arccosh( b/a ). InFig. 1, we show the normalized rates in a prolate spheroid( b/a <
1) for both longitudinal (PL) and transverse (PT)modes as the system shape evolves from a needle to asphere. With changing aspect ratio b/a , the rates ex-hibit a dramatic difference in behavior depending on thelocal field polarization. As nanorods become thinner, thenormalized rate decreases for the PL mode but increasesfor the PT modes. These trends are reversed for nan-odisks ( b/a >
IV. CONCLUSIONS
In summary, we developed a quantum-mechanicalmodel for surface-assisted carrier excitation in plasmonic nanostructures of arbitrary shape. We derived explicitexpressions for surface absorbed power and scatteringrate that are highly sensitive to the local field polariza-tion relative to the metal-dielectric interface. Our resultscan be used for calculations of hot carrier generation ratesin photovoltaics and photochemistry applications [1].
ACKNOWLEDGMENTS
This work was supported in part by the National Sci-ence Foundation under grants No. DMR-1610427 andNo. HRD-1547754. [1] M. L. Brongersma, N. J. Halas, and P. Nordlander, Nat.Nanotechnol. , 25 (2015).[2] Y. K. Lee, C. H. Jung, J.Park, H. Seo, G. A. Somorjai,and J. Y. Park, Nano Lett. , 4251 (2011).[3] F. Wang and N. A. Melosh, Nano Lett. , 5426 (2011).[4] M. W. Knight, H. Sobhani, P. Nordlander, and N. J.Halas, Science , 702 (2011).[5] A. Sobhani, M. W. Knight, Y. Wang, B. Zheng, N. S.King, L. V. Brown, Z. Fang, P. Nordlander, and N. J.Halas, Nat. Commun. , 1643 (2013).[6] K. Wu, W. E. Rodriguez-Cordoba, Y. Yang, and T. Lian,Nano Lett. , 5255 (2013).[7] M. W. Knight, Y. Wang, A. S. Urban, A. Sobhani, B. Y.Zheng, P. Nordlander, and N. J. Halas, Nano Lett. ,1687 (2013).[8] C. Clavero, Nat. Photonics , 95 (2014).[9] H. Chalabi, D. Schoen, M. L. Brongersma, Nano Lett. , 1374 (2014).[10] R. Sundararaman, P. Narang, A. S. Jermyn, W. A. God-dard III, and H. A. Atwater, Nat. Commun. , 5788(2014).[11] B. Y. Zheng, H. Zhao, A. Manjavacas, M. McClain, P.Nordlander, and N. J. Halas, Nat. Commun. , 7797(2015).[12] I. Thomann, B. A. Pinaud, Z. Chen, B. M. Clemens, T.F. Jaramillo, and M. L. Brongersma, Nano Lett. , 3440(2011).[13] J. Lee, S. Mubeen, X. Ji, G. D. Stucky, and M. Moskovits,Nano Lett. , 5014 (2012).[14] S. Mukherjee, F. Libisch, N. Large, O. Neumann, L. V.Brown, J. Cheng, J. B. Lassiter, E. A. Carter, P. Nord-lander, and N. J. Halas, Nano Lett. , 240 (2013).[15] S. Mubeen, J. Lee, N. Singh, S. Kr¨amer, G. D. Stucky,and M. Moskovits, Nat. Nanotechnol. , 247 (2013).[16] S. Mukherjee, L. Zhou, A. M. Goodman, N. Large, C.Ayala-Orozco, Y. Zhang, P. Nordlander, and N. J. Halas,J. Am. Chem. Soc. , 64 (2014).[17] W. P. Halperin, Quantum size effects in metal particles ,Rev. Mod. Phys. , 533 (1986).[18] V. V. Kresin, Phys. Rep. , 1 (1992).[19] C. Voisin, N. Del Fatti, D. Christofilos, and F. Vall´ee, J.Phys. Chem. B , 2264 (2001).[20] K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz,J. Phys. Chem. B
668 (2003). [21] C. Noguez, J. Phys. Chem. C , 4249 (1998).[23] C. S¨onnichsen, T. Franzl, T. Wilk, G. von Plessen, J.Feldmann, O. V. Wilson, and P. Mulvaney, Phys. Rev.Lett. , 077402 (2002).[24] S. L. Westcott, J. B. Jackson, C. Radloff, and N. J. Halas,Phys. Rev. B , 155431 (2002).[25] G. Raschke, S. Brogl, A. S. Susha, A. L. Rogach, T. AKlar, and J. Feldmann, Nano Lett. , 1853 (2004).[26] A. Arbouet, D. Christofilos, N. Del Fatti, F. Vall¨ee, J. R.Huntzinger, L. Arnaud, P. Billaud, and M. Broyer, Phys.Rev. Lett. , 127401 (2004).[27] C. L. Nehl, N. K. Grady, G. P. Goodrich, F. Tam, N. J.Halas, and J. H. Hafner, Nano Lett. , 2355 (2004).[28] C. Novo, D. Gomez, J. Perez-Juste, Z. Zhang, H.Petrova,M. Reismann, P. Mulvaney, and G. V. Hartland, Phys.Chem. Chem. Phys. , 3540 (2006).[29] H. Baida, P. Billaud, S. Marhaba, D. Christofilos, E. Cot-tancin, A. Crut, J. Lerm´e, P. Maioli, M. Pellarin, M.Broyer, N. Del Fatti, and F. Vall´ee, Nano Lett. , 3463(2009).[30] M. G. Blaber, A.-I. Henry, J. M. Bingham, G. C. Schatz,and R. P. Van Duyne, J. Phys. Chem. C , 393 (2012).[31] V. Juv´e, M. F. Cardinal, A. Lombardi, A. Crut, P.Maioli, J. P¨erez-Juste, L. M. Liz-Marz´an, N. Del Fatti,and F. Vall´ee, Nano Lett. , 2234 (2013).[32] M. N. O’Brien, M. R. Jones, K. L. Kohlstedt, G. C.Schatz, and C. A. Mirkin, Nano Lett. , 1012 (2015).[33] A. Kawabata and R. Kubo, J. Phys. Soc. Jpn. , 1765(1966).[34] A. A. Lushnikov and A. J. Simonov, Z. Physik , 17(1974).[35] W. A. Kraus and G. C. Schatz, J. Chem. Phys. , 6130(1983).[36] M. Barma and V. J. Subrahmanyam, J. Phys.: Cond.Mat. , 7681 (1989).[37] C. Yannouleas and R. A. Broglia, Ann. Phys. , 105(1992).[38] M. Eto and K. Kawamura, Surf. Rev. Lett. , 151 (1996).[39] A. V. Uskov, I. E. Protsenko, N. A. Mortensen, and E.P. O’Reilly, Plasmonics , 185 (2013).[40] J. B. Khurgin and G. Sun, Opt. Exp. , 250905 (2015). [41] A. S. Kirakosyan, M. I. Stockman, and T. V. Shah-bazyan, Phys. Rev. B , 155429 (2016).[42] R. A. Molina, D. Weinmann, and R. A. Jalabert, Phys.Rev. B , 155427 (2002).[43] G. Weick, R. A. Molina, D. Weinmann, and R. A. Jal-abert, Phys. Rev. B , 115410 (2005).[44] Z. Yuan and S. Gao, Surf. Sci. , 440 (2008).[45] J. Lerm´e, H. Baida, C. Bonnet, M. Broyer, E. Cottancin,A. Crut, P. Maioli, N. Del Fatti, F. Vall´ee, and M. Pel-larin, J. Phys. Chem. Lett. , 2922 (2010).[46] J. Lerm´e, J. Phys. Chem. C , 14098 (2011).[47] X. Li, Di Xiao, and Z. Zhang, New J. Phys. , 023011(2013).[48] A. Manjavacas, J. G. Liu, V. Kulkarni, and P. Nordlan-der, ACS Nano , 7630 (2014).[49] L. Genzel, T. P. Martin, and U. Kreibig, Z. Phys. B ,339 (1975).[50] R. Ruppin and H. Yatom, Phys. Status Solidi , 647(1976). [51] W. A. Krauss and G. C. Schatz, Chem. Phys. Lett. ,353 (1983).[52] E. A. Coronado and G. C. Schatz, J. Chem. Phys. ,3926 (2003).[53] A. Moroz, J. Phys. Chem. C , 10641 (2008).[54] U. Kreibig and M. Vollmer, Optical Properties of MetalClusters (Springer, Berlin, 1995).[55] N. A. Mortensen, Photonic. Nanostruct. , 303 (2013).[56] N. A. Mortensen, S. Raza, M. Wubs, T. Sondergaard,and S. I. Bozhevolnyi, Nat. Commun. , 3809 (2014).[57] L. D. Landau and E. M. Lifshitz, Electrodynamics ofContinuous Media (Elsevier, Amsterdam, 2004).[58] T. V. Shahbazyan, Phys. Rev. B , 235431 (2016).[59] R. Balian and C. Bloch, Ann. Phys. , 401 (1970).[60] T. V. Shahbazyan, Phys. Rev. Lett.117