Supplementary Information

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1 Supplementary Informaton Plasmonc Glasses and Flms Based on Alternatve Inexpensve Materals for Blockng Infrared Radaton Lucas V. Bestero, 1,2 Xang-Tan Kong, 1,3 Zhmng Wang, 1* Federco Rose, 2* Alexander O. Govorov 3* 1 Insttute of Fundamental and Fronter Scences, Unversty of Electronc Scence and Technology of Chna, Chengdu , Chna 2 Centre Énerge Matéraux et Télécommuncatons, Insttut Natonal de la Recherche Scentfque, 1650 Boul. Lonel Boulet, Varennes, Quebec J3X 1S2, Canada 3 Department of Physcs and Astronomy, Oho Unversty, Athens Oho 45701, Unted States 1

2 S1. Extncton cross sectons from smulaton. Fgures S1, S2 and S3 show the extncton cross sectons of the dfferent NCs used n ths study, as obtaned through electrodynamc smulatons usng the commercal COMSOL package. The results are obtaned for solated NCs mmersed n an nfnte delectrc substrate (glass, wth a refractve ndex of n = 1.5) and llumnated by lnearly polarzed lght. We present averaged data from three orthogonal ncdences of lght, and two orthogonal lnear polarzatons for each drecton of propagaton. The extncton cross sectons contan nformaton about the NC s absorpton and scatterng ( ext abs ). scat S2. Formalsm for glass optmzaton. As descrbed n the man text, the NC composton of the dfferent plasmonc glasses attempts to reproduce an deal transmsson profle (Fgure 1d), fully transparent to vsble lght and fully opaque to both UV and near IR regons of the electromagnetc spectrum: 1, 390 nm 700 nm T deal 0, 390nm or 700nm We can then defne a metrc that characterzes how dfferent a gven transmsson profle s from the deal one, wth the followng beng the one adopted n the present study: IP n n 1700 f f T T 200 deal, d 200 where T n, s ether calculated usng the Beer-Lambert law alone ( T dr ) or also ncludng the dffuson of scattered photons ( dr parameters and the full set of NC denstes dff d T T ), takng cross sectons and optcal path opt L as n as varables. Addtonally, the scalar dmensonless functon f s ncluded here as a tool to preferentally wegh dfferent spectral regons, but the results presented n ths study use only a flat f 1. Havng defned a metrc for the dstance between an arbtrary transmsson profle and the one that we determne as our target, we can fnd sets of NC denstes, n, that mnmze ths 2

3 dstance IP n. Gven the relatvely small set of geometres used n each of the plasmonc glasses, we have frst manually selected prelmnary values for the components of n that smultaneously provde (1) low values of IP n and (2) a comparatvely low total materal volume of NCs. The choces of denstes were nformed by the observaton of the dfferent NCs extncton profles. As a second step, we compared these manually desgned glasses wth those obtaned through an algorthmc search of the space of possble NC denstes. The search was conducted as follows: Usng IP n as the objectve functon, we used the BFGS method 1 to fnd the set of denstes, n, that mnmzes t. Ths functon has a large number of local mnma, and the mnmzaton method wll only fnd the mnmum of the basn of attracton n whch the orgnal ansatz for n les. Therefore, we have used the basn-hoppng algorthm 2 to sample the space defned by the denstes n. As a fnal step, ntended to reduce the total materal nvested n creatng the plasmonc glass, once a strong canddate for a global mnmum has been found a small set of alternatve glasses are consdered: one by one, each of the NC denstes are set to zero; f the new IP n s ether smaller than IP n, or the dfference s below a threshold, the modfed set n replaces the prevous canddate set and becomes the benchmark to whch we compare the next alternatve. Notably, the glasses obtaned wth ths method do not dffer sgnfcantly from the manually chosen ones, as shown n fgures 3 and 4 n the man text. Of course, wth a larger set of possble NCs, the complexty of the optmzaton would ncrease, and one could expect that usng an algorthmc approach such as ths would be the most effcent tactc. Fnally, we note that ths procedure can be used to desgn plasmonc glasses wth completely dfferent transmsson profles, just by adjustng T deal. 3

4 S3. Dffuson of photons n the plasmonc glass. Drect (ballstc) photons strke the glass and can be absorbed and scattered. The scattered photons then dffuse n both drectons, to the rght and to the left (Fgure 1b). The photons dffusng to the rght surface of the pane contrbute to the transmsson and, therefore, should be counted. The photons that dffuse towards the left surface of the pane radate to the outsde. Then, the total transmsson of the plasmonc glass should be wrtten as (see also Eq. 7 n the man text): T Tdr Tdff I e Tdr t 10 OD e OD I The formalsm of photon dffuson s based on the transport equaton: 3 n whch the parameters are defned as follows. q(,) r t 2 q(,) r t k q(,) r t Gscat (,) r t t 3 1) q(,) r t s the local densty of photon energy, havng the unts J/m. Another related parameter s the local fluence rate (,) r t, whch has the unts (,) r t q(,) r t c a 2 W/m. The above parameters are related va 2) c k s the photon dffuson coeffcent, n whch n 3 s the lnear extncton coeffcent, whch can be splt nto two terms related to scatterng and absorpton: n s a,scat,abs 3) The absorpton lfetme of a photon s gven va the absorpton coeffcent: 1 c c n a,abs a 4) The functon G (,) t scat r s the source n the dffuson equaton, comng from the scatterng of the drect lght beam strkng the pane. Ths quantty has unts of volume power densty, and s gven by n 3 W/m, 4

5 G (,) r t I() z I e scat s s 0 where I() z I0e z s the ntensty of the drect beam nsde the glass, gven by the Beer-Lambert law; I 0 s the external flux of ncomng lght. The boundary condtons are such that the local densty of photon energy at the surfaces s small snce, at the nterface wth ar, photons are free to move wthout almost any scatterng. In addton, we consder the CW llumnaton regme wth no tme dependence and a onedmensonal settng (Fgure S4a). Therefore, the resultng smplfed equaton and the correspondng boundary condtons read: 2 qz () 2 G0 z qz () e 2 z k G0 si0 1 3a k a qz ( 0) 0 qz ( L ) 0 Ths equaton has a smple soluton gven by: opt q( z) ae be Be G0 B k 2 2 z z z L L G0 ( e e ) a 2 2 L L k ( e e ) Then, the explct analytcal equaton s z G ( e e ) bab k G L L L L 2 2 ( e e ) k L L L L G 0 ( e e ) ( ) z e e z z qz ( ) e L L ( ) L L ( ) e e. (S1) k e e e e The photon dffusve currents at the surfaces of the plasmonc glass slab are gven by the transport equatons: j 0, Lopt qz ( ) k z z0, Lopt 5

6 and these fluxes n our geometry (Fgure S4a) have the propertes: j0 0 and j 0. Lopt For the dffusve transmsson and for the total transmsson, we have now An analytcal dffusve transmsson reads: jl opt Tdff I0 j T Tdr Tdff Tdr I L L L L ( ) ( ) s e e L e e L L TE e ( ) ( ) e e L L L L e e e e Fgure S4 shows the physcs of photon dffuson n our system. The dffusve photons generaton source comes from the drect lght beam and t s descrbed by the Beer-Lambert Law: z G () z I e. The dstrbuton of photon energy for dfferent wavelengths n our plasmonc scat s 0 glasses (results for the one wth Ag-shells are shown n Fgure S4) are dfferent for the vsble and IR regons (Fgure S4c). To understand ths, we now look at the photon mean free path n ths partcular glass (Fgure S4b). The photonc mean free path n the slab s gven by the averaged extncton lmfp ( ) 1/ n 1/ In the vsble spectrum, the Ag-shell glass has a long photon mean free path, such that l mfp ( ) L opt. Hence ncdent drect lght creates scattered photons nearly unformly through the slab and the dffusve photon densty s a symmetrc functon, as seen n Fgure S4c (the curve for 500nm). In the IR regme, we have the opposte strong nequalty, l ( ) L. Therefore, the dffusve photons are created now only near the left surface (Fgure S4c for 1200nm). The photon energy dstrbuton becomes strongly asymmetrc. Drect photons become preferentally scattered at the left sde of the glass and the energy dffuses towards the left surface. Then these dffusve photons radate back to the ar regon. For the transton regme at 875nm, we see a less strongly asymmetrc functon qz. ( ) These two regmes of dffusve photon transport are determned the relaton between the two transport-related lengths l mfp ( ) and L opt. Fgure S5 now shows the comparson between Lopt 0 mfp opt 6

7 these two lengths for dfferent glasses. As expected, n all glasses, we observe the two regmes of dffuson: l mfp ( ) L opt n the vsble and l mfp ( ) L opt for the IR. The frst graph n each panel of Fgure S5 shows the two contrbutons to the drect transmsson, due to scatterng and absorpton, as well as the total resultng transmsson: T T T dr abs scat ODe,abs abs, ODe,abs opt,abs ODe,scat scat, ODe,scat opt,scat T e L n T e L n The second graph n each panel of Fgure S5 depcts the comparson between the two relevant dmensons controllng the transmsson of dffusve photons, l mfp and L opt. Overall, the dffuson transmsson s not crucal for the performance of the glasses, but gves some correctons to the transmsson spectra and the fgures of mert for the glasses. For example, the fgures of mert become changed by ~6% at most for our best preformng glasses based on Ag, Cu and TN NCs (Fgure S6). The values for VT mprove and the values SHGC get ncreased correspondngly (so that the energy-savng propertes of our glasses become reduced a lttle due to the dffuson). See Fgure S6 below for a few selected glasses, where we also gve computed fgures of mert. S4. Sample polydspersty. Any real system wth a collecton of NCs wll exhbt some amount of dsperson on the NC sze wth respect to the nomnal dmensons. To explore how much the flterng effect of the plasmonc glass depends on the NC polydspersty, we have computed the transmsson profles of two well-performng plasmonc glasses (Ag and Cu shells) ncludng a dsperson of szes for the nanocrystals that are the most relevant to the profle of these IR-blockng glasses (those whch determne the drop on transmsson at the vsble-ir boundary at ~ 700nm). For the Agglass, ths NC has the parameters (a core = 200 nm, w = 5 nm) and, for the Cu-glass, such crucal NC has the parameters (a core = 50 nm, w = 5 nm) (Table S1). The populaton of such crucal NCs 7

8 s now splt nto three dfferent partcle szes dfferng n one of ts geometrcal parameters, to test the effect of polydspersty. The resultng transmsson profles are presented n Fgure S7, accompaned by a table showng the change on the fgures of mert. Overall, we see that the glasses transmsson profles are relatvely robust to these changes. We expected to see ths robustness, because the transmsson wndow n the vsble spectral range n our glasses (Fgures 3 and 4) does not have a very sharp boundary at the vsble-ir nterface and, therefore, some polydspersty cannot destroy the wndow effect and should not alter the fgures of mert sgnfcantly. References: (1) Fletcher, R. Practcal Methods of Optmzaton, 2 ed.; Wley: Chchester, (2) Wales, D. J. Energy Landscapes; Cambrdge Unversty Press: Cambrdge, (3) Bomedcal Photoncs Handbook; Vo-Dnh, T., Ed.; CRC Press: Boca Raton,

9 shape Ag shells Cu shells glass (a, w) (nm): n (m -3 ) (a, w) (nm): n (m -3 ) Monodsperse TO 2 : (200, 5): TO 2 : (50, 5): Polydsperse 1 TO 2 : (180, 5): (200, 5): (220, 5): Polydsperse 2 TO 2 : (200, 4): (200, 5): (200, 6): (100, 5): (200, 5): TO 2 : (40, 5): (50, 5): (60, 5): (100, 5): (200, 5): TO 2 : (50, 4): (50, 5): (50, 6): (100, 5): (200, 5): Table S1. Denstes used to obtan the data depcted n fgure S7. The glass labeled as monodsperse s compared wth two varatons, n whch the densty of a gven shell geometry s symmetrcally spread among shells that dffer from the orgnal n one of ts geometrcal parameters (core dameter and shell thckness). The denstes of the alternatve NCs, whch determne the drop n the transmsson at ~ 700nm, are marked n blue. 9

10 shape nanoshells nanorod nanocup materal (a, w) (nm): n (m -3 ) (d, L) (nm): n (m -3 ) (a, w) (nm): n (m -3 ) Ag TO 2 : (80, 5): (130, 5): (150, 5): (180, 5): (200, 5): Au TO 2 : (50, 5): (80, 5): (100, 5): (130, 5): (200, 5): TO 2 : (10, 38): (10, 45): (10, 59): (10, 81): (10, 102): TO 2 : (10, 24): (10, 29): (10, 38): (10, 45): (10, 59): (10, 81): (10, 102): Al TO 2 : (200, 5): TO 2 : (10, 81): Cu TO 2 : (50, 5): (80, 5): (100, 5): (150, 5): (200, 5): TN TO 2 : (50, 5): (100, 5): (130, 5): (180, 5): (200, 5): (10, 102): TO 2 : (10, 24): (10, 29): (10, 38): (10, 45): (10, 59): (10, 81): (10, 102): TO 2 : (10, 29): (10, 38): (10, 45): (10, 59): (10, 81): TO 2 : (150, 14): (250, 16): TO 2 : (150, 14): (250, 16): TO 2 : (250, 16): TO 2 : (150, 14): (250, 16): TO 2 : (150, 14): Table S2. Summary of concentratons of NCs used to calculate the propertes of IR blockng metaglasses wth a thckness of 4 mm. These are the denstes obtaned by computatonally optmzng the glasses transmsson profles, accountng for both T dr and T dff. 10

11 Fgure S1. Extncton cross sectons of the ensemblee of nanoshell szes consdered n our calculatons, grouped by materal. Note that the proposed glasses (Fgure 3) use dfferent concentratons of each NC sze, as descrbedd n Tables 1 and S2. 11

12 Fgure S2. Extncton cross sectons of the ensemble of nanorod szes consdered n our calculatons, grouped by materal. Note that the proposed glasses (Fgure 4) use dfferent concentratons of each NC sze, as descrbedd n Tables 1 and S2. 12

13 Fgure S3. Extncton cross sectons of the ensemble of nanocup szes consdered n our calculatons, grouped by materal. Note that the proposed glasses (Fgure 4) use dfferent concentratons of each NC sze, as descrbedd n Tables 1 and S2. 13

14 Fgure S4. a) Schematc of the model of lght scatterng n the plasmonc glass. b) Values for the glass of Ag nanoshells. Its transmsson profle (top) hghlghts the wavelengths for whch we show data n panel c of ths fgure. The photon mean free path n ths glass s compared wth ts thcknesss n the panel below. c) Local densty of radatve energy, q(z), as obtaned wth Equaton S1, for three dfferent wavelengths. 14

15 Fgure S5. Analyss of the relatve weght of scatterngg n dfferent plasmonc glasses. a) Data for an Al rod glass. b-e) Data for shell glasses of dfferent materals, namely Au, Ag, Cu, and TN. Two nsets provde the sketch of the NC geometres. The results shown n ths fgure account for drect transmsson only,.e. lght dffuson s not ncluded. Each panel ncludes (1) a graph wth the drect transmsson profle, accompaned by the separate contrbutons from absorpton and scatterng (whchh relate to the total va T T T mean free path at dfferent wavelengths, referenced aganst the total wdth of the glass, or optcal path Lopt. dr abs scat ), and (2) a graph wth the 15

16 Fgure S6. Selecton of plasmonc glasses (Ag and Cu shells and TN cups) desgned so that the total transmsson (TT total Tdr T dff ) s close to the deal step-wse profle T deal. The panels show the total transmssonn profle (black sold curve) and the dffuse transmsson T dff (red dashed curve). The table compares the fgures of mert of these glassess wth the ones obtaned by mnmzng the dstance between drect transmsson, Td dr, and Tdeal. It s mportant to stress that the values n the dfferent columns of the table are not just the result of changng the varables ( T dr or T total ) used n calculatng the fgures of mert for the same glasses. The values shown correspond to dfferent glasses, wth NC denstes obtaned by mnmzng the dstance of ether T dr or Tto otal wth respect to T deal, respectvely. 16

17 Fgure S7. Transmsson profles of plasmonc glasses composed of ensembles of Ag (left panel) and Cu (rght panel) NCs. Each panel compares data for a glass wth the nomnal set of NC denstes, as descrbed n the man text, wth data for two alternatve versons of that ensemble, whch broaden the NC sze dstrbuton (Table S1). Thee transmsson profles are robust to ths sze dsperson, and the fgures of mert show small absolute and relatve varaton n the central glass parameters, VT and SHGC. 17

18 Fgure S8. Values of the spectral dealty parameter, IP, gven as Equaton 8 n the man text. Ths plot reflects ts value for the manually desgnedd plasmonc glasses, wth transmsson profles shown as black curves n Fgures 3,,4 n the man text. Lower values are better, wth an deal profle havng IP 0. 18