Effects of Quantum Tunneling in Metal Nano-gap on Surface-Enhanced Raman Scattering
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Effects of Quantum Tunneling in Metal Nano-gap on Surface-Enhanced RamanScattering
Li Mao, Zhipeng Li, Biao Wu, ∗ and Hongxing Xu † Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China (Dated: October 29, 2018)The quantum tunneling effects between two silver plates are studied using the time dependentdensity functional theory. Our results show that the tunneling depends mainly on the separation andthe initial local field of the interstice between plates. The smaller separation and larger local field,the easier the electrons tunnels through the interstice. Our numerical calculation shows that whenthe separation is smaller than 0.6 nm the quantum tunneling dramatically reduces the enhancingability of interstice between nanoparticles.
PACS numbers: 33.20.Fb, 03.65.Xp, 78.67.Bf
Metal nano-gaps offering strong surface plasmon cou-plings have very rich physical properties. The relatedstudies have been very hot topics in the field of plamon-ics, e.g., single molecule surface-enhanced Raman spec-troscopy [1, 2], optical nano-antennas [3], high-harmonicgeneration [4]. The electromagnetic (EM) enhancementnear the metal surface, which is caused by the resonantexcitation of surface plasmon [5], is the dominating rea-son for the surface-enhanced Raman scattering (SERS)[1, 6]. Huge SERS with single molecule sensitivity canbe obtained when molecules are located in the nano-gapbetween two metallic nano-structures [1, 2, 7, 8, 9]. Alot of efforts have been made to seek extreme sensitiveSERS substrates [10, 11, 12].Theoretically, people have used many methods basedon the classical electrodynamics [13, 14, 15] to estimatethe SERS enhancement. These classical results indi-cate that the smaller the nano-gap, the higher the en-hancement. However, as the separation decreases to 1nm, the displacive current would partly become electrontunneling current which can reduce the EM enhance-ment substantially [16]. A recent experiment on thefour-wave mixing at coupled gold nanoparticles clearlydemonstrated that the quantum tunneling (QT) effectbecomes significant for the distance smaller than 0.2 nm[17], and a recent study of the plasmon resonance of ananoparticle dimer gave quantum description of such aphenomenon[18]. It is well known that the EM enhance-ment is the main contribution to SERS. Its enhancementfactor is proportional to the fourth power of the local fieldenhancement, i.e. M , where M = | E loc | / | E | with E loc and E being the local enhanced electric field and theincident electric field, respectively. Therefore, even forsmall QT effects on M , after a fourth power, the influenceto SERS could be huge. In this Letter we investigate theeffects of QT on SERS with the time dependent densityfunctional theory [19]. Our studies are able to quantifythese effects and point out at exactly what conditions theQT has to be taken into account.As the “hot spot”, where the SERS is strongest, is lo- ∗ Electronic address: [email protected] † Electronic address: [email protected]
FIG. 1: Schematic drawing of the “hot spot” between twosilver nano-spheres. As the “hot spot” (shaded area) is small,its local field is almost identical to the one computed by re-placing the sphere with a plate. E is the incident laser field. calized in a very small volume in the interstice betweenparticles, it is convenient to investigate the QT effectbetween two closely placed plates instead of two nano-particles. As shown in Fig. 1, in the vicinity of the “hotspot” (shaded area), two plates are not much differentfrom two nano-spheres. Besides we use two approxima-tions for our numerical calculations: (1) In the general-ized Mie theory, the electric current inside nano-sphereis set to be zero [13], so we can regard the silver platesas equipotential bodies at all time in our calculation; (2)The laser field is treated as a static electric field, andthe QT effect in an oscillating field can be described bythe results of static field in one period of laser. Withthese simplifications, when the separation d is not verysmall, the electric tunneling effect can be studied by themethod developed by Simmons [20], which regards elec-trons are tunneling through a voltage barrier. We findthat Simmons’ method is not proper when the distance d < d = 0 . i ∂∂t ψ k = h − ∂ ∂x + V eff ( x, t ) + V ext ( x, t ) i ψ k , (1)where we have used the atomic units and ψ k denotes aquantum state inside the Fermi surface of the silver plate. V ext ( x, t ) is the external potential coming from the laserfield and its induced field. V eff ( x, t ) is the effective po-tential felt by an electron through Coulomb interactionand correlation and exchange; it depends on the elec-tron density. In our approach, we use Crank-Nicholsonmethod [21] to update the wave function. To quantify theQT effects on the SERS, we monitor time evolution of thepotential difference δV between the two silver plates. Wecompute δV with the formula δV ( t ) = V eff ( x lr + x rr , t ) − V eff ( x ll + x rl , t ) , (2)where x lr,l and x rr,l are coordinates of the left and rightsurfaces of the right (left) plate.Let us now turn on the laser field. The electrons insideeach silver plate will start moving instantly to counter-balance the applied electric field so that the total electricfield inside each plate is zero. At the same time, an en-hanced field is induced in the “hot spot”. Afterwards,the electrons will start to tunnel between the two silverplates under the following external potential V ext ( x, t ) = E x x < X ,E loc ( x − X ) + E X X x X ,E ( x − d ) + E loc d x > X . (3)It is clear from the above analysis that the initial elec-tron state for the Schr¨odinger equations in Eq.(1) is thestate where the electrons have moved to counter-balancethe incident laser field. To obtain this initial state, wecompute with the method developed by Schulte [22] theground state of the metallic plate under the followingexternal potential V ( x ) = x < x l E ( x − x l ) x l < x < x r ,E D x > x r (4)where x l , x r are the left and right surfaces of the plate.Figure 2 shows the calculated time evolution of thepotential difference δV between the two plates separatedby d = 0 . ∼ E = 2 . × V/m, corresponding to a laserwith power P = 100 µW and focal spot ∼ µ m. In mostSERS experiments, even for single molecule detection, amuch smaller P ∼ µ W is used [1, 2]. The diameterof nano-particle is D = 6 nm. Note that we have calcu-lated for three different diameters D = 4 , , δV decays while oscillating with afrequency close to the bulk plasma frequency. The decay gets severe as the separation becomes smaller. This kindof decay can be intuitively understood by viewing thesystem as a bad capacitor that leaks current. FIG. 2: Time evolutions of the potential difference δV between the silver plates for different separations d =0 . , . , . , . D = 6 nm; E = 2 . × V/m; M = 1000. Dashed lines are for the averaged δ e V .FIG. 3: The decay rate η of δV as a function of the separation d . D = 6 nm; E = 2 . × V/m; M = 1000; λ = 500 nm. To measure the decay, or the QT suppression of theenhanced field, we introduce a decay rate defined by η = 1 − δ e V ( T ) δV (0) (5)where T is the typical optical period of the incident laser,e.g., T = 1 .
67 fs for a laser wavelength λ = 500 nm. Notethat δ e V ( T ) is not the value of δV but the averaged valueof δV over one oscillation period at t = T /
2. As thedistance d decreases, the potential difference decays withtime dramatically. At d = 0 . ∼ .
6% after half optical period ( λ = 500nm), that is, the SERS enhancement is 3 . × times((1 − η ) − ) smaller than the one obtained from classi-cal theory. By contrast, at d = 1 nm, the reduction ofthe local field enhancement by the QT is only 14%, corre-sponding to one time decrease of SERS enhancement fac-tor. This means that the enhancement can be sustainedif the separation is larger than 1 nm. The decay rates η for these different separations are computed and plottedin Fig.3, where we see η decreases exponentially as d in-creases. Specifically, when the separation is smaller than0.6 nm, the QT can reduce the local field significantly. FIG. 4: (color online) Time evolutions of the potential differ-ence δV between the silver plates for different enhanced localelectric fields E loc . d = 0 . D = 6 nm. The inset showsthe decay rates η as a function of the local electric field E loc . d = 0 . D = 6 nm. It is evident that both the enhancement factor M andthe laser power P can affect the QT via the enhancedlocal field E loc , which is proportional to M √ P . We findthrough numerical calculations that for the range of laserpower commonly used in experiment, the deciding factoris M √ P , not individual values of M and P . For example,we find that the time evolution of δV for P = 10 mW, M = 100 and P = 100 µ W, M = 1000 is almost the same (not shown). This means that we need to consider onlythe enhanced local field E loc . Figure 4 shows the timeevolutions of δV for different E loc at d = 0 . D = 6 nm. We see clearly that larger local field inducelarger QT, which in turn reduces the enhancement. Asshown in the inset in Fig. 4, the decay rate η decreasesslowly when E loc < × V/m, and reaches a non-zeroconstant when E loc goes to 0. This can be explained bythe fact that when the tunneling is small, we still havethe linear current-voltage relation [20], J ( t ) = βδV ( t ).From this relation, we obtain δV ( t ) = δV (0) e − dβt . (6)Therefore, when δV (0) →
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