aa r X i v : . [ phy s i c s . op ti c s ] S e p Plasmon modes of silver nanowire on a silica substrate
C.-L. Zou, F.-W. Sun, Y.-F. Xiao, C.-H.Dong, X.-D. Chen, J.-M. Cui, Q. Gong, Z.-F. Han, and G.-C. Guo Key Lab of Quantum Information, University of Science and Technology of China, Hefei 230026, Anhui, P. R. China State Key Lab for Mesoscopic Physics, School of Physics,Peking University, Beijing 100871, P. R. China
Plasmon mode in a silver nanowire is theoretically studied when the nanowire is placed on or neara silica substrate. It is found that the substrate has much influence on the plasmon mode. For thenanowire on the substrate, the plasmon (hybrid) mode possesses not only a long propagation lengthbut also an ultrasmall mode area. From the experimental point of view, this cavity-free structureholds a great potential to study a strong coherent interaction between the plasmon mode and singlequantum system (for example, quantum dots) embedded in the substrate.
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The silver nanowire is a potential solution to key com-ponents in future ultra-compact electronic and photoniccircuit [1–3] since it can confine the light in nanoscalebeyond the diffraction limit. The strong confinement oflight in a small area produces strong electric field inten-sity and dramatically changes the density of state nearbythe metal surface. Therefore, the silver nanowire hasbeen suggested to realize strong light-matter interaction,where the nanowire plays as the role of a resonator in cav-ity quantum electrodynamics (QED). When an opticalemitter, e.g., a quantum dot (QD), is placed around thesilver nanowire, its spontaneous emission can be signifi-cantly modified[4], known as the Purcell effect. As a re-sult, single optical emitter coupling with plasmon modesof nanowire holds a great potential for broad-band cav-ity QED [4], single photon source [5], and sub-wavelengthsingle photon transistor [6, 7] for quantum informationscience.Theoretically, the plasmon mode in silver nanowirewas studied as an ideal axis-symmetric mode [4], or stilltreated without the external influence even it was placedon a substrate [8–12]. However, in realistic applications,the silver nanowire is always necessary to be located onor near a substrate. For example, the nanowire on asubstrate makes it convenient for manipulating assistedby the scanning near field optical microscope probe [8–16]. To the best of our knowledge, the influence on theplasmon mode from the substrate is still kept un-studied.Actually, the plasmon mode is very sensitive to the sur-rounding environment. In this Letter, we theoreticallystudy the plasmon mode propagating along the silvernanowire near a substrate with a gap g ≥
0. In thissilver-air-substrate structure, most of the mode energy isconfined around the interface between the substrate andsilver nanowire. It is very different from the nanowiresimply embedded in air. The benefit of the nanowire ona substrate is that the large portion of energy in the sub-strate reduces the mode area by a factor of 5, withoutsignificant degradation of the propagation loss.An infinite long silver nanowire with a diameter of d is near an infinite dielectric substrate with a gap of g .The cross section is shown Fig. 1(a). The surface plas-mon mode propagates harmonically along the nanowire, g (a) (b)(c) d FIG. 1: (color online) (a) Schematic diagram of a silvernanowire near the silica substrate with a gap g . The diameterof nanowire is d . (b) and (c), the energy density distributionson the cross section of nanowire ( d = 100 nm ) with g = ∞ , and the electric field varies as exp(i βz − i ωt ), where thepropagation constant β = 2 πn eff /λ is a complex num-ber because of the energy attenuation in the propaga-tion. In this case, the Maxwell equation is reduced totwo-dimensional cross section, which can be expressed as[ ▽ + ( n − n eff )(2 π/λ ) ] ψ = 0, (1)where n and n eff denote the refractive index of materialand the effective index for the mode, respectively. Thepropagation length of the plasmon modes can be definedas L = 1 / { β } . (2)Besides the propagation loss, another important pa-rameter of the plasmon mode is effective mode area A ,which can be defined by the ratio of a mode’s total energydensity per unit length and its peak energy density, A = Z all W ( r )d s/max { W ( r ) } , (3)where W ( r ) represents the effective energy density with W ( r ) = 12 Re { d[ ωε ( r )]d ω } | E ( r ) | + 12 µ | H ( r ) | . (4) (b) (d)(c)(a) FIG. 2: (color online) Effective index (a), propagation loss(b), mode areas (c), and energy confinement in the substrate(d) of surface plasmon modes for the nanowire on the sub-strate (red dashed, g = 0) and in air (blue solid, g = ∞ )versus the diameter of silver nanowire. Here, | E ( r ) | and | H ( r ) | are the intensity of electric andmagnetic fields, respectively. ε ( r ) and µ are the electricand vacuum magnetic permittivities.To describe the impact from the substrate, we alsodefine the confinement factor η as the portion of energyin substrate, η = Z sub W ( r )d s/ Z all W ( r )d s . (5)The confinement factor η can also quantify the interac-tion of SPP and substrate,A finite element method is used to investigate the plas-mon modes with a fixed wavelength at 637 nm. The en-ergy distribution profiles of the plasmon modes in thenanowire are shown in Fig. 1(b) and (c), with g = ∞ , n air = 1 and n sub = 1 .
45. It is not dif-ficult to find a rotation symmetric field distribution whenthe nanowire is in air ( g = ∞ ), while a strongly asym-metric field distribution is presented in the presence ofthe silica substrate ( g = 0). Moreover, the maximum ofthe electric field is distributed in the substrate-nanowiregap. In other words, the substrate can significantly pullthe mode field.We first investigate the impact of the substrate on theplasmon modes for various nanowire sizes. Fig.2 showsthe effective index ( n eff ), propagation length ( L ), effec-tive mode area ( A eff ) and portion of energy ( η ) versus d . As a comparison, the ideal case without substrate( g = ∞ ) is also depicted. It is found that n eff de-creases with the increase of d , while L and A eff shows the opposite behaviors. This tradeoff between confine-ment and absorption loss is well known in the surfaceplasmon mode.For the nanowire on silica substrate, comparing withthe nanowire in air, n eff is larger and A eff is smallerwith all d . These obvious differences come from the pres-ence of large refractive index material near the metalnanowire. In addition, the mode area of the nanowireon silica substrate is only about 1/5 of that in air. Tocharacterize the field pulling by the silica substrate, wepresent the confinement factor for g = 0 in Fig.2(d). The η increases with d to nearly 80% when d = 200 nm .In Fig.(2), A eff simply increases with the increase of d . However, the other three parameters approach satu-ration values. In order to get a physic interpretation, wecan approximately treat this hybrid system as the metalplane-substrate for large d . The analytical solution to Eq.(1) in the plane condition gives n p eff = p ǫ m ǫ s / ( ǫ m + ǫ s ),where ǫ m and ǫ s is the electric permittivity of metal andsubstrate. So we have n p eff = 1 . L p = 53 µm , and η p = 90% in the plane limit.Now, we turn to discuss the experimental condition,where the nanowire is near the substrate with a finitegap g >
0. This is a realistic case for many experimentsbecause the substrate is not perfect smooth, or in spe-cific experimental config, the nanowire is suspended inair[17]. By fixing the diameter d = 100 nm, we study thepropagation loss and effective index (Fig.3) for differentgaps between nanowire and substrate.With the increase of gap g , n eff monotonously de-creases and gradually approaches the in-air case. This isbecause the influence of substrate becomes smaller whenthe nanowire departs from the substrate. When g = 200nm, the substrate has a very minor effect on the plas-mon mode of the silver. However, the dynamics of L ismuch more complicated than that of n eff . When thesilver nanowire moves away from the silica substrate, L increases first, then quickly drops to 2 µm . After that,it slowly increases, and finally approaches to 25 µ m [19].We can divide the gap into three regimes: the near field,dissipation, and far field regimes, which correspond todifferent dynamics of L and different loss mechanisms.(i) Near field regime. In this regime, we have n eff >n sub . Due to the phase mismatching, the dominant lossof the plasmon mode comes from the metal absorption.By increasing the gap, n eff gradually decreases as thenanowire moves away from the substrate. A smaller n eff indicates a smaller energy portion in the metal, so thatthe absorption loss is reduced with increasing g .(ii) Dissipation Regime. When the gap further in-creases, n eff may be smaller than n sub . Meanwhile, theplasmon mode still has much energy in the substrate, ascan be seen in the inset of Fig.3 ( g = 50 nm). FromEq. (1), when n eff < n sub , the solutions are travelingwaves in substrate as ψ ∝ exp(i q n sub − n eff x ). The en-ergy in traveling wave will dissipate into infinite space.Therefore, in this regime, the mode energy quickly dis- FIG. 3: (color online) Effective index (a) and propagationlength (b) of the silver nanowire with d=100 nm versus thegap. Insets: mode field profiles of the surface plasmon modeswith g=2 nm, 50 nm, 200 nm, respectively. sipates by coupling to the dissipative traveling waves inthe substrate. As a result, L has a sudden decrease.(iii) The far field regime. When the nanowire furthermoves away from the substrate, for example, g = 200nm, the coupling between the plasmon mode and the dis-sipative traveling modes in substrate becomes very weakas it is proportional to their field overlap. Thus, even n eff < n sub , the plasmon mode keeps almost unchangedby the substrate, and the mode profile approaches thecase of g = ∞ (in the absence of the substrate). There-fore, L can be slowly recovered in this regime.The above results indicate that the approximation byneglecting the substrate is not applicable in analyzingexperimental results, where the energy confinement andpropagation loss are significantly influenced by the sub-strate. When the gap is very small, most of the energy is confined near nanowire-substrate interface and n eff islarge. It should be more efficient to coupling light to thesurface plasmon mode through the substrate, especiallyfor large d . The efficiently excitation of surface plas-mon mode through substrate have been demonstratedrecently[10, 11]. It is worth noticing that the propa-gation loss strongly depends on the gap. In a recentexperiment, near field surface plasmon mode detectionis demonstrated[17], where the silver nanowire is puton the Ge nanowire, with g = 40nm, d = 100nm and λ = 600nm. From Fig. 3(b), it works in the dissipationregime with L ≈ µm . For more efficiently detection, g should be larger than 200nm or smaller than 10nm .In conclusion, we have numerically studied the modeprofile, propagation loss and mode area of plasmon modesof a silver nanowire near the silica substrate. To reducethe propagation loss, our result suggests that the gapbetween silver nanowire and the substrate should be assmall as possible. When the silver nanowire is attachedto the substrate ( g = 0), the propagation length is lim-ited by metal absorption and the mode area is reducedby a factor of 5. In addition, for nanowire with d = 200nm, the field energy portion of the plasmon mode in thesubstrate is as large as 80%, which is suitable for couplingto QDs or molecules embed in substrate. Accompanyingwith the scanning near field optical microscope, the silvernanowire can be precisely moved on the substrate, andselectively coupled to single QDs in substrate. This ispotential for experimental realization of the single pho-ton transistors, or coupling two distant QDs to generateentanglement for quantum information science.The work was supported by the National funda-mental Research Program of China under Grant No2006CB921900, the Knowledge Innovation Project ofChinese Academy of Sciences, and National NaturalScience Foundation of China (Grant No. 60621064,60537020). F.W. S. is also supported by the new fac-ulty starting funds from USTC and the Fundamental Re-search Funds for the Central Universities. [1] E. Ozbay, Science , 189 (2006).[2] H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M.Rogers, F. Hofer, F. R. Aussenegg, And J. R. Krenn,Phys. Rev. Lett. , 257403 (2005).[3] R. Kolesov, B. Grotz, G. Balasubramanian, R. J. Stohr,A. A. L. Nicolet, P. R. Hemmer, F. Jelezko, and J.Wrochtrup, Nature Phys. , 470 (2009).[4] D. E. Chang, A. S. S φ rensen, P. R. Hemmer, and M. D.Lukin, Phys. Rev. Lett. , 053002 (2006).[5] A. V. Akimov, A. 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