Simulation of Plasmonics Nanodevices with Coupled Maxwell and Schrödinger Equations using the FDTD Method

A nume rical a pproach that cou ples Loren tz-Drude model incorporated Maxwell equations with Schrödinger equation is presented for the simula tion of pla smonics nanodevices. Maxwell equa tions w ith Lore ntz-Drude (LD ) dispersive model are a pplied to l arge size c omponents, w hereas coupled Maxwell and Schrödinger equations are applied to components where quantum effec ts a re nee ded. The finite difference time do main me thod (F DTD) is appl ied to simulate these coupled equations. Numerical results of the coupled ap proach are compared w ith th e conventional approach.


Introduction
The miniaturization of devices and high speed data are main challenges with existing silicon based technologies, and the reasons behind are diffraction l imit and R C time delay respectively.Fo r t he so lution of such challenges different efforts have been done in the past, how ever, an a rea known as plasmonics has been introduced recently to han dle them, and i t has shown prom ising app lications [1 -3].Plasmonics deals w ith co llective osci llation of fre e electr ons a t t he interface of dielectric a nd m etal, w hich r emain bo unded to the surface.Plasmonics is generally categorized into surface plasmon pol aritons (S PP) and l ocalized surface plasmon resonance (LS PR).The f irst one is mor e suitable for information transmission rela ted applications, w hile th e second one is preferable for se nsing ap plications.A w ide range of plasmonic devices have been simulated, fabricated and characterized [1][2][3][4][5][6][7][8][9][10].Some interesting results show that the surface plasmon polariton has strong analogy to Young's double-slit experiment and is discussed in [4].The concept of semiconductor plasmonics using a so lid state model that includes Pauli exculsion principle, state filling effect, Fermi-Dirac t hermalization, an d externa l magnetic fie ld is presented in [5].A n umerical approach that consists of solid state and Lorentz-Drude m odels is presented t o simul ate active plasmonics devices [6]; and it is also used to simulate a plasm onic source , and t hen l ight ex traction from t he source [7].S ome othe r active plasmonic de vices such a s Plas-MOStor (pla smonic based trans istor), ultrafa st a ctive devices, pa ssive a nd ac tive photonics c ircuits using S PP have be en r eported in [8-1 0].The concept of r eplacing the conventional gold and silver with doped semiconductors and intermetallics has been discussed in [11].Plasmonic bas ed phenomena have pote ntial t o ha ndle t he challenges with existing CMOS and photonics technologies, and ca n be u sed to i nterface ph otonics a nd e lectronics devices e ffectively.H owever, m odeling and simulation of such in terfacing do mains bec ome com plex due to di fferent scales of components.These complexities can be solved by implying different techniques.Nonetheless, when the size of a device redu ces to a few nanom eters, quantum effec ts dominate a nd the ir c onsiderations become important to maintain the ac curacy.The refore, to incor porate them i nto modeling and simulation te chniques, modifications i n t he conventional numerical tec hniques ar e neede d, w hereas conventional numerical techniques have performed well for the simulation of bulk materials and devices.For quantum effects there is need to adopt some appropriate approaches from qua ntum m echanics, a nd usually Schrödinger equa tion is c onsidered to incorporate su ch effects.O n t he other han d Maxwell equations ar e use d for electromagnetic effec ts.Therefore, these equa tions are coupled t o simulate t hose a pplications in w hich c ombined effects are needed [12][13].In [12] a hybrid transmission line matrix ( TLM) [14] and FDTD [15], and in [13] a hybr id locally one dimensional (LO D)-FDTD [16] and F DTD methods are applied to coupled non-dispersive Maxwell and Schrödinger equations.In [12] the FDTD method is app lied to Schrödinger eq uation to simu late ca rbon n anotube whil e the TL M metho d is applied to t he conventional n ondispersive Ma xwell e quations to simula te the r est of the structure.Whe reas in [13], the FDTD method is ap plied to Schrödinger equation to simulate a semiconductor nanowire and t he LOD-FDTD m ethod is a pplied to the c onventional non-dispersive Maxw ell e quations t o simulate rest of t he structure effi ciently.In brief, in [1 2-13] h ybrid ap proaches are a pplied t o na notube, nan owire and non d ispersive materials.In this paper, as compared to the [12][13], the LD dispersive model [17] i ncorporated Maxwell equa tions are coup led with S chrödinger equa tion t o sim ulate pl asmonic nanodevices.Schrödinger e quation i ncorporated Maxw ell equations are applied to sim ulate t he components i n w hich quantum effects are needed.The FDTD method is applied to simulate the se coup led equ ations.In sect ion 2, deta iled formulation of the Ma xwell eq uations w ith LD mode l, formulation of the S chrödinger equa tion in t he presence of external elec tromagnetic fie ld, a nd di scretization us ing the FDTD method are presented.The reason of using LD model as compared to the other dispersive models is because of its better accuracy for broader range of wavelength.In section 3, num erical re sults o f the coupled approach ar e com pared with those from the conventional Maxwell approach and at the end conclusion is given.

Formulations
The time de pendent Maxw ell e quations w ith freque ncy dependent permittivity a nd quantum cur rent de nsity are written as where q J is qua ntum curr ent dens ity, and is obt ained from is the frequency dependent permittivity and is obtained from Lorentz-Drude dispersive m odel.In the model, D rude pa rt dea ls with intraband effe cts an d is gen erally used fo r fre e electr ons, whereas, the Lorentz model deals with interband effects and generally dea ls w ith bo unded electrons.The LD model is written as where pD  is plas ma fre quency a nd D  is damping constant assoc iated w ith D rude mode l (intr aband e ffects) , pL  is plasm a fr equency, L  is da mping c onstant, a nd L  is resona nce fre quency o f the first p ole of Lore ntz model ( interband effects).After pu tting equation (3) in to equation (2) and by using the auxiliary differential equation (ADE) approach, and some mathematical simplifications we get following equations q 0 0 J t Where terms with subscript D and term Q denote Drude model and terms with subscript L and term P denote Lorentz model.D uring the simu lation of a struct ure w ith t he proposed a pproach f our different sc enarios can a rise i) a section of th e struc ture in w hich there is no nee d of dispersive model and quantum current density, ii) a region in which quantum current density is required but not dispersive model, iii) a section in which dispersive model is needed but not quantum current density, iv) a re gion where both effects are ne eded.Under all the se sce narios equa tion (2) w ill b e effected.In t his sec tion, as an exam ple form ulation for t he scenario (iv) is presented, however, it can be modified based on the situation.After some mathematical simplifications equation ( 4) can be written as The discritized form of the equations ( 7) to ( 9) is given as Equations ( 5) and ( 6) are discritzed as where where r = x, y and z.
Equations (10) to (1 2) are similar to conventional magnetic field equations in FDTD method and are discritized as Equations ( 13) to ( 20) r epresent d iscritized form of t he Maxwell e quations (eq uation 13 to 1 5 with LD model a nd quantum current density) after applying the FDTD method.For quan tum effects the ti me de pendent S chrödinger equation is con sidered in the pr esence of e xternal electromagnetic field and is written as , h is Planck con stant, q is charge, m is mass of an ele ctron and r repr esents spa tial var iables x, y, and z.The ve ctor and sca lar poten tials ar e obtai ned from following equations The eq uation ( 21) is co mplex, and by using t he relation and a fter some simplifications; it is separated into real and imaginary parts and is given as Temporal discretization of equations ( 22) and (23) i s given as After calc ulating t he re al and im aginary pa rts of th e wavefunction, and then by using the following relation, the quantum current density is obtained.
Spatial d iscretization o f equa tions (24-26) an d t he corresponding term ) r ( J 1 n q  in equations (13)(14)(15) depends on the user how he /she w ant to im plement t hese e quations in one dimensional or three dimensional fashions.The meshing interface be tween Ma xwell and Schrödinger e quations als o depends on o ne or three dimensional patter n of spat ial discretization.We ha ve used bot h pa tterns a nd found the similar r esults.For i nterface be tween Maxw ell and Schrödinger e quations, w ave function, quantum curr ent density and the corresponding electric field are discretized at same poin t.The val ue o f quan tum current dens ity at interface or boundary of b oth d omains is adde d up w ith electric fie ld.In other words, the qua ntum cur rent den sity can also be used as a source for the Maxwell equations i.e. at the b oundary of Schrödinger equat ion, quan tum curr ent density is injected in to Maxwell e quations.The vec tor and scalar potentials are u sed t o i ncorporate the e xternal electromagnetic fiel d i nto S chrödinger equat ion along t he nanowire.In the sim ulation procedure, the m agnetic field is updated first, the n v ector p otential, scalar pot ential, wave function, quan tum cur rent d ensity, and a t the e nd ele ctric field are updated and this sequence continues, until the last iteration.

Numerical Results
For num erical results two different exa mples tha t inc lude dispersive and quan tum effe cts are st udied.A generalized structure i s show n in F ig. 1, in which sem iconductor nanowires are use d as inter connects be tween p lasmonics nanodevices.The size of plasmonics de vices c an be from few nanometer to few hundred nanometers, whereas the size of inter connects ca n be few n anometers.For such applications in the pa per w e use LD dispersive m odel fo r large si ze c omponents, w hereas S chrödinger eq uation is used for quantum effects needed region.Fig. 1: A ge neralized st ructure for c oupled approach, in which plasmonics devices are interconnected via nanowires.
The str uctures stu died in the pa per ha ve operating concept similar to t he generalized struc ture in F ig.1.The structure for the first example is shown in Fig. 2(a).It consists of two gold nanospheres, each with a radius of 20 nm, with a gap of 10 nm in between them, and a 2 nm thick and 70 nm long semiconductor nanowire (NW) is plac ed a t center in between na nospheres.The purpose o f the struct ure i s to study t he qua ntum effec ts and then comparison o f t he coupled and conventional approaches.The cell size in each direction is uniform i.e. 2nm.To maintain the stability of the Schrödinger equation with the FDTD method, the time step should be smaller t han t he C ourant F riedrich Le vy ( CFL) limit of Maxwell equations [18].Therefore, in the coupled approach, the time step of the Schrödinger equation is taken as the time step for whole simulation domain.We take time step 1 00 times sm aller t han t hat o f the CLN limit to accommodate the NW, in other words, accuracy will also be better i f the cell siz e i s sm aller.We have checked method with d ifferent grids o r c ell s izes and it i s f ound t hat the proposed approach converges properly and in addition there is n o sta bility issue , a s long a s the ti me step for t he simulation domain is sam e as of Schrödinger equation.The parameters used for dispersive m odel are sam e as gi ven in [17].The surrounding medium of the structure is free space.A Gaussian pulse is used as a source to get field localization in between nanospheres and a Gaussian pulse at NW is used to excite the w avefunction.F our diffe rent field e xcitation scenarios may arise during the simulation of the structure, I) excitation th at ca n g enerate f ield localization b etween nanospheres, II) source abo ve or below the N W i n the surrounding medium, III) use of qua ntum current density as a sour ce, IV) combination of the abo ve t hree sc enarios.Figure 2 (b) shows snapshot of field localization in between nanospheres without having t he NW in the xy pl ane, whereas Fig. 2 (c) depicts the snapshot of the total electric field inte nsity in the xy p lane with n anowire.Th ese both snapshots are obtained at steady state.In this application the excitation scenario (I) is used.Results show that most of the field is con fined along the NW. Figure 2(d) show s the field intensity with and without Schrödinger equation with respect to number of t ime steps and depicts the difference between both c ircumstances.F igure 2 (e) is plo tted w ith re spect t o energy (eV ) w ith an d w ithout qua ntum effects.The difference of 0.16 e V i s o bserved.The field ob servation point is at 26 nm away from the center of nanospheres and 12 nm l eft from the c enter of the NW .The se re sults illustrate the c lear diffe rence be tween coupled a nd conventional appr oaches.The possible reason of the difference between the resul ts of bo th appr oaches is quantum effec t.Because in the case of co upled approach, the quantum current density takes into account, kinetic and potential energies of elec trons, vector and scalar potentials.Inclusion of these fa ctors i s ca use of shift i n t he fie ld intensity in Fig. 2 (d and e).It is also observed that if the structure is made of b ulk m aterials, then there is no difference in the numerical results of both approaches, and it is validation of the proposed approach.The struc ture of sec ond example is shown i n F ig. 3 (a), in which two pairs of gold nanospheres are placed at both ends of the nanowire.The thickness and length of the NW, radius and distance between nanospheres is same as i n example 1 at first, however, latter on the distance between nanospheres is varied.These struc tures may have num ber of a pplications in different a reas su ch a s b io se nsing, e .g. h eating a nd bl ood sample ana lysis.Beca use of com paratively longer and stronger fie ld osc illations, secon d exa mple ca n be used for blood or liquid analysis more effectively as compared to the example 1. Figure 4 denotes the normalized field pattern for structure 3(a).In this case three different excitation sources are used, localized fields at both ends between nanospheres, and third close to th e center of the NW.The field pattern in Fig. 4 (a) is captured dur ing the trans ient sta te of t he method, where a small value at the center of the NW shows the exc itation of the w avefunction.F igures 4 (b) and 4 (c) show the e lectric an d m agnetic fiel d patterns for sa me structure a t st eady s tate res pectively.These fie ld patterns describe th at at st eady st ate most of t he fi eld c oncentrates along the NW.H owever, the fie ld values bec ome w eaker and weaker with the passage of time due to field radiation in the surrounding media.These patterns are captured when the gap distance between the nanospheres is 10 nm.Nonetheless, the patterns and resul ts a re als o stud ied for variable d istance be tween n anospheres, and o bserved the similar phenomena but with different field intensities.Figure 5 show s the fiel d p lot of t he struc ture with and w ithout incorporating the S chrödinger e quation.F igure 5 (a) indicates el ectric fi eld i ntensity a t NW wi th respect to number of time st eps, dotte d line sh ows the re sult w ithout quantum ef fect a nd the sol id line with qu antum ef fects.It shows that the dominant mode with quantum effects is at 21.96 eV, while without quantum effects is at 22.65 eV, and the d ifference of 0. 69 eV is observe d.The se resul ts are observed a t a po int o n N W, w hen the distance be tween nanospheres is 5 nm.This plot sh ows that the fiel d v alues near to 0 eV with quantum effects is smoother, as compared to the other curve (without quantum effect) that shows some abnormality, a nd the reason i s abse nce of qua ntum effects.The prop osed a pproach may ha ve po tential a pplications in the fie lds of ac tive nan o-plasmonics, optoelectronics, integration of na no-plasmonis and nano-electronics, a nd nano-sensors.

Conclusion
An a pproach th at c ouples t ime d ependent Schrödinger and LD di spersive model inc orporated Ma xwell eq uations is developed and implemented for plasmonics nanodevices and the FD TD method i s ap plied for ana lysis.The appr oach is applied t o str uctures tha t inv olve b oth dis persive a nd quantum effects.Re sults are compared wi th an d w ithout quantum effects a nd cle ar difference is obs erved am ong them.H owever, both a pproaches d id not sh ow an y difference in numerical resul ts for bul k mat erials.T he proposed approach pa ves the way for m odeling and simulation of nanodevices in the w ide spectrum of electromagnetics, and where quantum effects are needed.

Fig. 2 :
Fig. 2: (a) Structure for example 1, (b) Field intensity in the xy plane without nanowire (c) Field intensity in the xy plane with nanowire (d) Normalized magnetic field intensity with and without Schrödinger equation (e) Normalized field with respect to energy (eV).

Figure 3 (
Figure 3(b) show s the n ormalized electric fie ld i ntensity near the center of t he NW, under two different situations, i) the field is e xcited a nd localized betw een nan ospheres at one-end o f t he N W (fr om exa mple 1, green co lor l ine), ii) the field is ex cited and loca lized a t both-ends o f the NW (example 2, red c olor line).It r epresents that in t he case of resonance fie ld at one end (e xample 1), the field b ecomes weaker w ith the pa ssage of time, w hile in the c ase of resonance fie ld a t both ends of the N W ( example 2), amplitude of field remains stronger for longer time along the

Fig. 3 :
Fig. 3: (a) Structure for e xample 2, ( b) Electric field at N W with l ocalized fi eld at o ne en d and l ocalized f ield a t b oth ends of the nanowire.

Figure 5 (
b) ind icates the corresponding va lues of t he electric field with respect to energy (eV).

Fig. 4 :
Fig. 4: Field pattern in the xy plan for structure 3 (a), ( a) Field l ocalization a t t ransient st ate between nanospheres at both ends of NW and a third source is below and close to the center of N W (b) Electric fi eld pa ttern a t stea dy sta te, (c ) Magnetic field pattern at steady state.

Fig. 5 :
Fig. 5 : Field plot for struc ture 3(a ), (a) Ele ctric fiel d w ith respect to number of t ime s teps wi th a nd wi thout Schrödinger equa tion, b) El ectric fiel d w ith respec t t o energy (eV) with and without Schrödinger equation.