Ab Initio Plasmonics of Externally Doped Silicon Nanocrystals | ACS

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Ab initio Plasmonics of Externally Doped Silicon Nanocrystals Fabio Della Sala, Maria Pezzolla, Stefania D'Agostino, and Eduardo Fabiano ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.9b00126 • Publication Date (Web): 25 Apr 2019 Downloaded from http://pubs.acs.org on April 25, 2019

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Ab initio Plasmonics of Externally Doped Silicon Nanocrystals Fabio Della Sala,∗,†,‡ Maria Pezzolla,‡,¶ Stefania D’Agostino,‡ and Eduardo Fabiano†,‡ †Institute for Microelectronic and Microsystems (CNR-IMM), Via Monteroni, Campus Unisalento, 73100 Lecce, Italy ‡Center for Biomolecular Nanotechnologies@UNILE, Istituto Italiano di Tecnologia (IIT), Via Barsanti, I-73010 Arnesano, Italy ¶Department of Engineering, Universit` a del Salento,Via per Monteroni, 73100 Lecce, Italy E-mail: [email protected] Abstract Heavily-doped semiconductor nanocrystals (NCs) represent a novel class of plasmonic materials: their hyper-tunable plasmonic resonances play a key role in different nanotechnology applications. The plasmonic properties of doped semiconductor NCs have been, to date, mainly modeled using (semi-)classical theoretical approaches in contrast to conventional metallic NCs for which ab initio plasmonics based on TimeDependent Density Functional Theory (TD-DFT) calculations have now become the standard reference. In this work, we aim at filling this gap by presenting a TD-DFT study on the optical properties of silicon NCs doped with an increasing number of excess electrons (dynamical doping). We have considered spherical NCs of different sizes (up to a diameter of 2.4 nm) embedded into an external polarizable medium, which turned out to be very important to obtain stable ground-state configurations.

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TD-DFT results show the presence of a plasmon peak at low energy with an intensity increasing with the number of excess electrons. We use the recently proposed Generalized Plasmonicity Index, with a novel implementation and interpretation in terms of transition densities, to verify the plasmonic properties of this peak. Our analysis demonstrates that the low energy peak is a plasmon peak, but it is strongly screened by the valence electrons. A detailed comparison between TD-DFT and classical results shows that the latter can be safely applied only for NCs with a diameter larger than 2 nm. The presented TD-DFT results can be also used as a reference for other theoretical approaches which aim at modeling quantum-effects beyond the classical regime.

Keywords plasmonics, density-functional theory, silicon nanocrystals, doped nanocrystals

Introduction Heavily-doped semiconductor and metal-oxides nanocrystals (NCs) represent a new class of plasmonic nanosystems, with applications in different fields such as nanophotonics, phototherapy, biomedicine, photocatalysis, and photovoltaics. 1–5 In conventional materials used in plasmonics, e.g. gold and silver, the range of optical frequencies which can be reached in applications is mostly fixed by their electron density: thus, gold or silver NCs typically show a Localized Surface Plasmon Resonance (LSPR) in the visible region. In the classical Drude electrostatic model, 6 the LSPR frequency (ωLSP R , i.e. the energy of the main absorption peak) of a spherical nanoparticle is (in Gaussian units):

ωLSP R =

s

ωp2 , ǫ∞ + 2ǫout

2


(1)

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where ωp =

s

4πne2 m∗e

(2)

is the bulk plasma frequency, n is the excess electron density (i.e. the doping concentration), e is the electron charge, ǫ∞ is the high-frequency dielectric constant, m∗e is the electron effective mass, ǫout is the permittivity of the surrounding environment. The LSPR frequency is thus related to the square-root of the electron density which can be tuned (by doping) over a wide interval, allowing applications also in the near-infrared region (NIR). 1–5,7 Doping can be achieved in two different ways: the conventional static intrinsic doping 1–5 or the external dynamical doping. 8,9 In the latter case, external electrons are added to the NCs via reversible capacitive effect, allowing new type of applications such as electrochromic smart windows. 10–12 In this case the LSPR frequency is directly proportional to the squareroot of the added electrons (Ne ), as the excess electron density is simply

n=

Ne , (4π/3)R3

(3)

where R is the radius of the NC. Eqs. (1,2,3) have been successfully applied to model the optical properties of different heavily-doped semiconductor (and metal-oxide) NCs. 7,13–15 However, these equations are valid only for large nanoparticles and neglect all band-structure effects. 16,17 For smallnanoparticles, quantum effects must be taken into account and different simple analytic approaches to extend the Drude theory have been proposed. 18–22 In particular, with the low-density modification of the Drude dielectric constant: ωp2 , ǫ = ǫ∞ − 2 ω(ω + iΓ) − ωQM

(4)

where the quantity ωQM describes quantum effects in the limit of vanishing density, 21,22 the

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LSPR of a spherical nanoparticles modifies into: ′ ωLSP R =

q 2 2 + ωLSP ωQM R .

(5)

Eqs. (4,5) have been already applied to model doped semiconductor NCs, 23–28 and represent a simple classical model with a quantum-correction to the dielectric constant. Other more sophisticated theoretical approaches 29–37 proposed to describe quantum effects in plasmonics have been mostly focused on metallic systems. In recent years, also first-principles Timedependent Density Functional Theory (TD-DFT) 38 calculations for metallic nanosystems have gained increasing popularity see, e.g., Refs. 39–50, which can can be considered the “reference” results for other quantum-classical methods. On the other hand, few quantum calculations for the plasmonics properties of doped semiconductors have been presented, see e.g. Ref. 21 with a jellium-like description of the NCs and Ref. 51 using a tight-binding model. In this work, we report atomistic first-principles TD-DFT calculations of the externally doped silicon semiconductor NCs. Silicon has been chosen not only as a model semiconductor, but also due to recent investigations in which silicon has been proposed for plasmonics. 52–54 Calculations of the LSPR in P-doped silicon NC have been performed only at the tight-binding level. 51 Other TD-DFT studies on doped silicon NCs are focused on staticdoping as a function of the type of the dopant, 55–58 and not on the plasmonic properties as a function of the doping-concentration, which is the topic of this work. Here, we provide results for silicon NCs with different sizes (up to 2.4 nm) and number of excess electrons (up to 30). We investigate the strong influence of the doping on the plasmonic response of the nanoparticles and compare the TD-DFT results with the semi-classical model from Eqs. (1,2,5). Another important issue in small NCs is whether the peak can be considered as a plasmon or not. 22 Different approaches have been presented in literature to define the plasmonic

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character of an excitation in finite systems. 40,59–65 Recently, the Generalized Plasmonicity Index has been introduced. 66 In this work we present a novel implementation and interpretation of the GPI based on the coupling of the transition densities and we apply it to describe the plasmonic character of the peaks in doped silicon NC.

Systems and Methods Systems and Geometries. To construct the NC geometries we started from the Si bulk lattice, cutting spheres of different radii, centered in the middle of the unit cell of the silicon bulk lattice. Thus, we have obtained bare Si clusters of Td symmetry (this symmetry has been kept as a constraint in all subsequent calculations). The Si dangling bonds at the surface of the resulting clusters have been saturated with hydrogen atoms (-SiH or -SiH2 terminations). Four structures, namely Si35 H36 , Si87 H76 , Si163 H116 , Si281 H172 (see Fig. 1) have been considered, with 2, 3, 4 and 5 full shells of silicon atoms, respectively, as shown in Fig. S1 in the Supporting Information. 35Si 36H

87Si 76H

163Si 116H

281Si 172H

Figure 1: Atomic structures of Si35 H36 (S2 ), Si87 H76 (S3 ), Si163 H116 (S4 ), Si281 H172 (S5 ) nanocrystals. The blue larger spheres represent silicon atoms, while the smaller gray ones the hydrogen atoms. Then, all the structures have been relaxed using DFT (see the Computational Details 5


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section). The main structural and electronic properties of all the NCs are summarized in Table 1: the maximum radius is the maximum radial distance of the outermost H atoms and the excess electron density (i.e. the doping concentration) is given in Eq. (3). Table 1: Main structural and electronic properties of neutral NCs in gas phase: label of the system (used throughout the text), chemical composition, number of silicon shells (Sh), maximum radius (R in bohr), diameter (D in nm), doping concentration in cm−3 per added electrons (Ne ). Label S2 S3 S4 S5

Structure Sh Si35 H36 2 Si87 H76 3 Si163 H116 4 Si281 H172 5

R [a.u.] 12.25 16.32 20.82 22.71

D [nm] 1.30 1.73 2.20 2.40

n/Ne [cm−3 ] 8.76·1020 3.71·1020 1.78·1020 1.37·1020

We then considered the different NCs with different excess electrons. Note that the NC geometries have not been relaxed after the addition of electrons: a more appropriate model of counterions is required for this task. Using a fixed geometry for all doping-density, we will investigate the intrinsic electronic effects, whereas structural-induced effects, which do not have a classical counterpart, will be excluded. To keep the Td symmetry and avoid complicated high spin states, we have restricted our study to charged states that correspond to a fully occupied shell in the electronic configuration of each NC. All the considered systems and electronic configurations are reported in Table S1 in the Supporting Information. The doping concentration, see Tab. 1, is the range form 2.75 ·1020 cm−3 (for S5 with Ne = 2) to 1.75 ·1022 cm−3 (for S2 with Ne = 20), which correspond to the range obtainable experimentally. 1–5 The negative electronic charges of the added electrons must be counterbalanced by an equivalent positive charge. In dynamic plasmonics devices 8,10 this is obtained using an ionic solution. Here we added counterions as fixed counter-charges around the NC.These point charges, summing up to the total value of the excess electrons, have been uniformly distributed on an octahedral Lebedev grid 67 with 26 points, centered in the center of mass

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of the NC, and are located at a fixed distance (6 bohr) from the most external hydrogen of the system. The solvent effect is modeled via the continuum model COSMO, 68 with dielectric constant ǫ. We considered ǫ = 1 (i.e. no solvent effects) and ǫ = 65.0, which corresponds to propylene carbonate, 69 commonly used in smart window applications. 11 Generalized Plasmonicity Index. The Generalized Plasmonicity Index (GPI) 66 is given by R ∗ | d3 rδn(r, ω)vind (r, ω)| , η(ω) = R 3 ∗ | d rδn(r, ω)vext (r, ω)|

(6)

∗ (r, ω) is induced potential which in the where δn(r, ω) in the induced electronic density, vind

random-phase-approximation (RPA) simply equals the Hartree potential of δn(r, ω), and vext is the external potential. In the dipole approximation vext ≈ E0 · r where E0 is the external electric field. At plasmon peaks, η ≫ 1. In Ref. 66 it has been shown that at the resonance frequency wI the GPI can be computed as η(ωI ) ≈

Eplas,I , Γ

Z

d 3 r′

(7)

where Eplas,I =

3

dr

Z

ρI (r)ρI (r′ ) , |r − r′ |

(8)

and Γ is the broadening. In Eq. (8), ρI is the transition density for excitation I, which is readily available in TD-DFT codes based on the Casida equations. The transition densities enter in the density response function

χ(r, r′ , ω) = 2

M X ρs (r)ρs (r′ ) ρs (r)ρs (r′ ) − , ω − ω + iΓ ω + ω + iΓ s s s=1

(9)

where M is the number of excitations considered. Note that M ≤ Nocc ∗ Nvirt , where Nocc is the number of occupied orbitals and Nvirt is the number of virtual orbitals (which depends on the basis-set size).

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However, Eq. (7) assumes that the resonance at wI is well separated in energy from other excitations. Such condition never occurs in plasmonic systems (but for the smallest ones) as a plasmon peak is composed by many different transitions very close in energy. Thus, to compute the GPI using a Casida-like approach, the full expression must be used, as in the R following. Using the fact that δn(r, ω) = d3 r′ χ(r, r′ , ω)vext (r′ ) and the definition in Eq. (9) in the numerator and denominator of Eq. (6) we obtain, after some algebra (see Supporting Information):

η M (ω) =

N M (ω) DM (ω)

(10)

with M

N (ω) = 4 DM (ω) =

where

M X M X

µs θss′ µs′ f (ω, ωs , ωs′ ) s=1 s′ =1 M X µ2s g(ω, ωs ) , 2 s=1

,

(11) (12)

4ωs ωs′ (ω 4 − (ω 2 − Γ2 )(ωs2 + ωs2′ ) + 2ω 2 Γ2 + ωs2 ωs2′ + Γ4 ) (,13) ((ω − ωs )2 + Γ2 )((ω + ωs )2 + Γ2 )((ω − ωs′ )2 + Γ2 )((ω + ωs′ )2 + Γ2 ) 2ωs (ω 2 − ωs2 + Γ2 ) − i4ωs ωΓ g(ω, ωs ) = (14) ((ω − ωs )2 + Γ2 )((ω + ωs )2 + Γ2 )

f (ω, ωs , ωs′ ) =

and the transition dipole (along the indicent field direction) is 1 µs = |E0 |

Z

d3 r vext (r)ρs (r) .

(15)

In Eqs. (11) and (12), θss′ is the generalization (including off-diagonal elements) of Eq. (8), namely θss′ =

Z

3

dr

Z

d 3 r′

8

ρs (r)ρs′ (r′ ) . |r − r′ |


(16)

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Using Eqs. (11) and (12), the GPI can be calculated exactly. The calculation of θss′ is quite R ∗ straightforward from the transition densities. Note that d3 rδn(r, ω)vind (r, ω) is purely real-valued, in the dipole approximation: the imaginary contribution (see Supporting InforR ∗ mation) vanishes with the sum in s, s′ . On the other hand d3 rδn(r, ω)vext (r, ω) is complexvalued, but at ω = 0 where it simply equals the static polarizability α.

In Fig. 2 we report the GPI for a jellium sphere with 18 electrons and rs = 4, computed in three different ways: i) computed directly from Eq. (6) by using a linear-response (LR) TD-DFT formalism in the frequency domain and a real-space grid; 35 ii) using the approach in Eq. (7) with a Gaussian broadening for each excitation (the values of Eplas are reported in the bottom panel of Fig. 2); and iii) using the approach presented in this work, i.e. η M (ω) from Eqs. (10,11,12,13,14), considering all dipole-allowed excitations (see also the Computational Details section). Fig. 2 shows that the approach η M (ω) is numerically very accurate as it can exactly reproduce the results of frequency-domain LR-TDDFT. On the other hand the approach in Eq. (7) largely overestimated the exact GPI and it gives reasonable results only near the main plasmon peak: at low energies there are peaks not present in the exact GPI (due to the off-diagonal contributions of the transition densities). Efficient GPI Implementation. For plasmonics systems, we found that η M (ω) is quite sensitive to the number to excitations considered (M ): this is related to the Lorentzian shapes from the denominators of Eqs. (13,14). In fact, in a Casida-like approach M can be varied and it is usually much smaller than Nocc ∗ Nvirt . Thus we propose to compute the GPI as: η(ω) = η(0) +

N M (ω) N M (0) − DM (ω) DM (0)

(17)

where η(0) can be computed as

η(0) =

R

d3 r

R

9

′

) d3 r′ ∆ρ(r)∆ρ(r |r−r′ |

α


(18)

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20

15

LR-TDDFT [Eq. (6)] Eq. (7) M

GPI

η [Eq. (10)]

10

5

Eplas [a.u.]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0 0.02 0.01 0.00

1

2

3

4

Energy [eV] Figure 2: GPI for a jellium sphere with 18 electrons and rs = 4, computed using three different approaches, see text for details. The broadening is Γ=0.05 eV. The approach in Eq. (10) exactly reproduces results from frequency-dependent LR-TDDFT. The bottom panel report the Eplas values (in Hartree) for each excitation energy. where α is the static polarizability and ∆ρ(r) is the static polarization density, and both can be computed exactly. 70 To verify the accuracy of Eq. (17), we investigated (see Fig. S2 in the Supporting Information) the GPI for a standard system in plasmonic application (i.e. Ag20 tetrahedral geometry, see, e.g., Refs. 65,66,71,72) using different approaches. The curves for η M shift up with increasing M . Note that for M = 100, 500, 1000, excitation energies are considered up to 5.3, 9.2, 12.5 eV, respectively, thus above the range of energies considered in the plot. Nevertheless even with M = 1000 the agreement with the exact results is not sufficient. Instead using the approach in Eq. (17) a very good agreement, in the whole energy range, can be obtained, already with M = 500. 10


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The approach in Eq. (17) have been also tested for different systems considered in this work and very good convergence toward the exact results have been obtained. All the GPI spectra reported in the Results section have been obtained with the approach of Eq. (17). Computational Details. The electronic structure and optical properties calculations have been preformed using DFT and TD-DFT 73 modules of the TURBOMOLE package. 74 Cluster geometries have been optimized using the Perdew-Burke-Ernzerhof (PBE) 75 exchange-correlation functional and the def2-SV(P) 76 basis set. Ground-state and TD-DFT calculations have been performed using the PBE functional and a SV(P) 77 basis set, with the standard grid (gridsize 3). In all cases the resolution of identity (RI) approximation 78 has been employed to evaluate the Coulomb matrix elements. The ground-state calculations have been carried out with the default COSMO parameters of the TURBOMOLE implementation: 79 point-charges have been included in the COSMO response but not in the cavity construction. The optimized atomic COSMO radii have been used (rH = 1.3 ˚ A , rSi =2.2 ˚ A). Note that COSMO is used only in ground-state calculation whereas TD-DFT excitation energies have been computed without the COSMO response. The TD-DFT module employs the Casida approach, i.e. it computes the poles of the response function. 73 All the systems considered have Td symmetry, thus only states of the dipole allowed symmetry (t2 ) have been computed. All the optical spectra, if not differently specified, have been computed with a Lorentzian brodening (Γ=0.2 eV). Concerning the results in Fig. 2, the details of the implementation of jellium model in TURBOMOLE will be presented elsewhere; to achieve the real-space accuracy we used a very large even-tempered basis-set.

Results and Discussion Electronic Properties. When nanosystems with externally added electrons are considered, the first question to answer is how many electrons can be added. Adding electrons to an

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electronic nanosystem, will cause a reduction of the ionization potential (IP): when the IP becomes negative the system will loose electrons spontaneously. Clearly, the larger the nanosystem, the higher the number of electrons that can be added. In Fig. 3 we report the highest-occupied molecular orbital (HOMO) energy for the considered different externally doped silicon NCs, and for the two dielectric constants considered. For these systems the HOMO energy is a good approximation to the I.P. computed from

EHOMO [eV]

total energy differences, see Table S2. Without solvent effects (upper panel), the HOMO

ε=1

S2 S3 S4 S5

12

EHOMO [eV]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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6 0 -6 2 0

ε=65

-2 -4 -6

0

4

8

12

16

Ne

20

24

28

32

Figure 3: EHOM O as function of the number of excess electrons (Ne ), without the solvent effect (upper panel) and with the solvent effect (ǫ = 65.0) (bottom panel). The presence of the solvent strongly stabilizes all the systems. is always positive, with the exception of systems with only two excess electrons: thus, the excess electrons cannot be stabilized only by the fixed counter-charges. In the lower panel of Fig. 3, we report the HOMO energy considering a solvent with a dielectric constant of 65.0. Interpolating this data, the maximum number of electrons (i.e.

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with negative EHOM O ) is thus 10, 16, 24 and 30 for S2 , S3 , S4 , and S5 , respectively. These results show that the presence of a polarizable medium is essential to stabilize the NCs with externally added electrons. On the other hand, if we consider only the COSMO solvent, without fixed point-charges, the stabilization of the HOMO decreases (see Table S3 in the Supporting Information). The second main question is where the added electrons are located. The excess electron density (i.e. the difference between the density of the system with Ne added electrons and the neutral one) can be written as

δρ(Ne ; r) = ρ(Ne ; r) − ρ(0; r) = ρadd (Ne ; r) + δρval (Ne ; r) ,

(19) (20)

where ρadd (Ne ; r) is the density only from the orbitals of the added electrons (see Table S1 in the Supporting Information) and δρval (Ne ; r) is the valence density difference between the system with Ne electron and the neutral one, considering only occupied orbitals up to the HOMO of the corresponding neutral system (HOMON, see Table S1 in the Supporting Information). The density δρval (Ne ; r) measures the relaxation of the valence density due to the addition of the excess electrons. Then we consider the spherically-averaged densities:

δ ρ¯(Ne ; r) =

Z

dΩδρ(Ne ; r) = Ω

Z

dΩ (δρadd (Ne ; r) + δρval (Ne ; r))

(21)

Ω

= ρ¯add (Ne ; r) + δ ρ¯val (Ne , r) ,

(22)

where Ω indicates the solid angle. Note that the following relations hold:

Z

Z

Z

dr4πr2 δ ρ¯(Ne ; r) = Ne ,

(23)

dr4πr2 ρ¯add (Ne ; r) = Ne ,

(24)

dr4πr2 δ ρ¯val (Ne ; r) = 0 .

(25)

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Density [a.u.]

6×10 3×10

-3

a) S2 -3

ρadd

δρ 0

-3×10 -6×10 1×10

Density [a.u.]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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5×10

δρval

-3

-3

-4

5

10 R

20

25

30

20 15 Radius [bohr]

R 25

30

15

ρadd

b) S5

δρ

-4

-1×10

δρval Classical density

-3

0 -5×10

δρ ρadd

δρval

-3

5

10

Figure 4: Plot of the spherically-averaged densities for the S2 (upper panel) and S5 nanocrystal (bottom panel), for Ne = 20. R is the radial position of the last hydrogen atom, see Tab. 1. The classical density is given by Eq. (3). The valence density difference (ρval ) screens the added density (ρadd ). The spherically-averaged excess densities for the smaller (S2 ) and for the larger (S5 ) silicon NC for Ne = 20 are reported in Fig. 4 (other systems, not reported, show a similar behavior). Figure 4 shows that the excess electron density ρ¯ is located at the edge of the NC, in both S2 and S5 . However, this density at the NC surface does not originate from the added electrons: in fact, ρ¯add is almost uniformly distributed in the NC and resembles the classical density, Eq. (3). The density peak at the NC surface originates from the valence density difference which screens almost completely ρ¯add inside the NC. Optical Properties. The TD-DFT photo-absorption cross-section for the smallest (S2 ) and the largest (S5 ) nanosystem are reported in Fig. 5. Other systems (not reported) show 14


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a behavior in-between. We first discuss the neutral systems (solid black curves at the bottom). The computed TD-DFT absorption spectra of neutral NCs show a continuum onset, up to the main plasmon peak at about 10 eV, as shown in the inset of Fig. 5a, where we also report (red dashed curve) the electrostatic result in the vacuum, i.e.

σ(ω) =

4π Im[α(ω)] c

(26)

ǫ(ω) − 1 ǫ(ω) + 2

(27)

with α(ω) = R3

where ǫ(ω) = ǫbulk (ω) is the experimental bulk silicon complex dielectric constant. 6 The plasmon peak at about 10 eV for silicon nanoparticles has been already reported in literature 80–84 and the agreement with the classical result is very good, considering the small size of the system considered (the diameter of S2 is only 1.3 nm). The TD-DFT spectrum reported in the inset of Fig. 5a is up to 20 eV and it has been computed including 4000 excited states, which is computationally very expensive. For larger systems much less excited states can be computed (and thus with a smaller energy range): for S5 we could include only 1200 states, ending with a spectrum only up to 3.5 eV (see Fig. 5b). The interesting result in Fig. 5 is the presence of additional absorbance peaks at lower energies in the doped structures. For S2 a prominent peak at about 2.7 eV (see Fig. 5a) is growing with the number of added electrons (see Section for a detailed discussion about the energy position of the peak). In order to understand the origin of this peak, we report in Fig. 6 the electronic redistribution profile (ERP) map 72 for Ne = 20: negative (positive) colormap values represent the occupied (virtual) orbitals involved at each excitation energy. At the bottom of Fig. 6 we also report the density of states (DOS) from occupied (solid-line) and virtual (dashed-line) orbitals. Note that the energy-gap in doped NCs almost vanish. The double peak in the

15


ACS Photonics

S5

S2

20

20

b)

Ne=30

90

1.5

15

30

10

0

10 20 Energy (eV)

10 N =20 e

20 12 8 2 Ne=0

5

1.3 12

2

0

1.4

sqrt(Ne)

5 Ne=12

Ne=2

0 0

1

30

60

18 20 24

15

Energy (eV)

2

σ [bohr ]

a)

σ [bohr ]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 37

2 3 Energy [eV]

0 0

4

1

2 3 Energy [eV]

Ne=0

4

Figure 5: TD-DFT photo-absorption cross section for S2 [panel a)] and S5 [panel b)] for different excess electrons (Ne ). A well defined peak grows with Ne . The inset of panel a) reports the TD-DFT photo-absorption cross section for S2 up to 20 eV and the classical Mie results (red dashed-line). The inset of panel b) shows the evolution of the energy of the peak with the square root of Ne . DOS at about 1.5 eV corresponds to the orbitals occupied by the excess electrons. Thus the peak at 2.7 eV originates from transition from orbitals occupied by the added electrons (see the arrow in the Figure) to virtual orbitals. On the other hand, at higher excitation energies, the contribution of the excess electrons are negligible in comparison with the low lying orbitals (i.e. the occupied orbitals of the neutral system, below -2.0 eV). Increasing the size of the NC, the energy of the peak decreases: for S5 (see Fig. 5b) the peak is at much lower energy (around 1.4 eV). This behavior is consistent with previous findings in heavily-doped semiconductors. 1–5 Moreover the main peak blue-shifts with the number of added electrons (as shown in the inset): a linear relation between the LSPR

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Page 17 of 37

0.7326 6.5

0.5985

6.0

0.4643

5.5

0.3302

5.0

Excitation Energy [eV]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Photonics

0.1960

4.5 0.0619 4.0 -0.0723 3.5 -0.2064

3.0

-0.3406

2.5

-0.4747

2.0 1.5

-0.6089

1.0

-0.7430

0.5

-0.8772

0.0 -8.0 -6.0 -4.0 -2.0 -0.0

2.0

4.0

6.0

8.0

Energy [eV]

Figure 6: Electronic redistribution profile (ERP) map for S2 with Ne =20. The vertical axis is the excitation energy while the horizontal axis is the orbital energies (the DOS is reported at the bottom, solid-line for occupied and dashed-line for virtual orbitals). The arrow indicates the occupied orbitals involved in the main plasmon peak. energy and

√

Ne is consistent with the formulas in Eqs. (1,2). Also the intensity of the

peaks follows the classical theory, in which the integrated area over the peak is

I=

2π 2 e2 Ne 9ǫout , ∗ me c (2ǫout + ǫ∞ )2

(28)

i.e. it is linear with the number of electrons. 21 In order to analyze in detail the nature of the main peak, we have considered the GPI values, see section . The evolution of η for different numbers of added electrons is reported in Fig. 7 for S2 (left panel) and S5 (right panel). For S2 we observe multi-peak structures, as also found in the absorption spectra (see Fig. 5a). The GPI increases with the number

17


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of electrons and toward a unique peak, which is a clear signature of the plasmonic origin. For Ne = 20 we obtain ηmax ≈ 1.4 for S2 and ηmax ≈ 1.0 for S5 . The smaller values of η for the larger system is not surprising: in fact, see Table 1, the doping density per added electron (n/Ne ) is 6.4 times smaller in S5 . Nevertheless the values of η are quite small, if compared to the GPI of a metallic system with the same number of free electrons: for Ag20 we have η ≈ 2.2, see Fig. S2 in the Supporting Information. This traces back to the strong interaction with the valence electrons, as already shown in Fig. 4. Thus, it will be interesting to understand the plasmonic properties of these nanocrystals, without the strong interaction with the valence electrons.

S5

S2

a)

b) Ne=0 2 8 12 20

1.4

1.2

GPI

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 37

Ne=0 2 12 20 30

1.4

1.2

1.0

1.0

0.8

0.8

0.6

0.6 0

0.5

1

1.5

2

2.5

3

3.5

0

0.5

Energy [eV]

1

1.5

2

2.5

3

3.5

Energy [eV]

Figure 7: GPI for S2 (left panel) and S5 (right panel) for different excess electron (Ne ). Intensities in S5 are smaller than in S2 , because the doping density is smaller in the former by a factor of 6.4.

In order to analyze the optical properties of the added electrons only, we need to switch18


Page 19 of 37

off the their interaction with the valence electrons. We thus performed additional TD-DFT calculations in which only the response of the excess electron is considered (i.e. all electrons up to HONON have been excluded from the TD-DFT response). In Fig. 8 we report the corresponding absorption spectra and the GPI for the S2 and S5 systems. Concerning the

S2

S5

250 a)

200

Ne=2 8 12 20

2

2

150

250 c)

σ [bohr ]

200

σ [bohr ]

100

150

50

50

0 6.0 b)

0 6.0 d)

4.0

4.0

2.0 0.0

Ne=2 12 20 30

100

GPI

GPI

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Photonics

2.0

0

2 4 Energy [eV]

0.0

6

0

1

2

3

Energy [eV]

Figure 8: TD-DFT absorption spectra, panel a) and c), and GPI, panel b) and d), considering only the response of the excess-electrons; panel a) and c) refer to for S2 while panel b) and d) report results for S5 . Note how all the intensities are much larger than the ones considering the full electronic response (Figs 5 and 7). absorption spectra, we now observe a much larger blue-shift with Ne : for S5 the shift from Ne = 12 to Ne = 30 is about 0.6 eV in contrast to 0.16 eV from Fig. 5. The intensities are also much larger: in fact, according to Eq. (28) the absorption intensities are inversely proportional to the square of ǫ∞ , which takes into account the dielectric response of the valence electrons. Instead, ǫ∞ can be set to 1 in the case of the response of the added electrons only, as there is no screening. 19


ACS Photonics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The GPI is also much larger, in the range 2-4. Thus, from the data reported in Fig. 7 and Fig. 8 we can conclude that externally doped silicon NCs exhibit a pronounced plasmon peak at low energy, but it is strongly screened by the interaction with the valence electrons. This is an important finding which can be used to engineering novel heavily-doped semiconductors. To analyze in more detail the screening effects, we report in Fig. 9 the plane-averaged transition density for the S2 (upper panel) and S5 (lower panel) system, for Ne = 20. We considered the excited state with the largest oscillator strength, both in the case of the full-response and considering only-excess electrons. Plane-averaged plots, obtained as R ρs (z) = dxdyρs (x, y, z), can well show the electron distribution both at the surface and in the central region of the nanoparticle. On the other hand, when transition densities are

visualized in three-dimensional plots, see, e.g. Refs. 40,42,43,47, then only contributions R R from the surface can be well distinguished. Note that dzρ(z) = 0 and dzzρ(z) = ds . The plots clearly show the screening effects: while in the excess-only electron response the transition densities show a larger contribution near the surface, they are strongly damped in the case of the full-response, where many more (atomic) oscillations are also present. We recall that in the classical model the transition density is just a delta function localized at the NC surface (vertical dotted lines). Nevertheless, the classical model can well describe optical properties in large nanocrystals, as it will be discussed in the next section. Comparison with Classical Models. In Fig. 10 we compare the TD-DFT results with the classical model from Eqs. (1,2,5) using an effective mass for electrons of 0.3 as in Ref. 51 and ǫout = 1. In our model the electron-mass is fixed: in a more refined model me can be varied with the excess electron to take into account the non-paraboliticy of the bands. 16,17 Recall that the model in Eqs. (1,2,5) is classical, but the dielectric constant include a quantum correction (ωQM ). The values of ωQM have been obtained by a graphical extrapolation and are reported in Table 2: they are inversely proportional to the NC radius, as expected. 19,21 Note that to exact prescription exists for the values of ωQM : previous estimations of ωQM are also mostly empirical. 23,24,28

20


Page 20 of 37

Transition Denisty [a.u.]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Transition Dnesity [a.u.]

Page 21 of 37

0.08

S2

0.04 0 Full-response Only-excess electr.

-0.04 -0.08

0.01

S5

0

-0.01

-30

-20

-10

0 Position [bohr]

10

20

30

Figure 9: Plane-averaged transition densities for the S2 (upper panel) and S5 (bottom panel) system, for Ne = 20. For the full-response we considered the excited state at 2.77 (1.35) eV , whereas for the excess-eletron only the excited state at 4.75 (2.64) eV, for S2 (S5 ), respectively. The boundaries of the nanocrystals are indicated by the dotted vertical blue-line. Without screening effects, the transition densities are closer to the surfaces and show less oscillations. We first consider the comparison with the TD-DFT response of the added-electron only (empty symbols and dot-dashed lines in Fig. 10). Recall that in this case ǫ∞ = 1 in the classical model. In Fig. 10a we report the LSPR energy versus the doping density for S2 and S5 (clearly the doping concentration is much larger for S2 due to the smaller volume). For S5 the agreement between TD-DFT results (black empty squares) and the classical model (black dashed-line) is very good. On the other hand, for S2 the classical results (red dashed-line) largely overestimate the TD-DFT data (red empty diamonds), which can be traced back to the neglect of quantum spill-out effects 19,35 in the classical model. Recall that quantumsize effects are instead already included in the quantum correction to the dielectric constant

21


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Page 22 of 37

Table 2: The quantum-correction to the dielectric constant (ωQM ), the NC radius from DFT electron density, the radius-dependent DFT polarizability and the radius-dependent DFT dielectric constant. Label S2 S3 S4 S5

ωQM [eV] 2.7 2.0 1.5 1.0

RDF T [a.u.] 12.43 15.77 18.95 22.22

α(R) [a.u.3 ] 997 2469 4596 7814

ǫ(R) 4.24 6.10 7.13 8.43

(ωQM ). The behavior with system size is also reported in Fig. 10b, where we consider all the systems with a fixed number (Ne = 20) of excess electrons. The agreement with the classical results is very good only for S5 and S4 , whereas the classical model (blue line) overestimates the TD-DFT results (blue-filled diamonds), due to the neglect of quantum spill-out effects. Thus silicon NCs with diameter smaller than 2 nm cannot be accurately describe with a classical model. Then, we consider the full TD-DFT response (i.e. without excluding any orbitals). In this case, in the classical model ǫ∞ represents the static dielectric constant of the neutral silicon NC, which depends on its size. 51,85 This size-dependent static dielectric constant can be obtained directly from the DFT calculations, 80 as

ǫ∞ → ǫ(R) =

1 + 2α(R)/R3 . 1 − α(R)/R3

(29)

In Eq. (29), the geometrical radius reported in Tab. 1 cannot be used because the systems are not perfectly spherical. Thus, we used a radius computed directly from the DFT density: RR RDF T which defined such that 0 DF T dr4πr2 ρ¯(r)(r) = 0.95N where ρ¯(r) is the spherically-

averaged ground-state density for the neutral system and N is the total number of electrons.

With this definition RDF T agree quite well (within 2%) with geometrical radius for the larger (and more spherical) system. Our results for ǫ∞ are reported in Tab. 2. The TD-DFT and classical LSPR energies are reported in Fig. 10a (filled symbols and full

22


Page 23 of 37

lines, respectively). For S5 the agreement between TD-DFT results (black filled diamonds) and the classical model (black full line) is very good. On the other hand, for S2 the TD-DFT LSPR energy even decreases with the doping concentration, a behavior which is opposite to the classical one. We believe that it is related to the failure of a model with an uniform dielectric constant and/or to very large spill-out effects: further studies (e.g. a comparison with the quantum-hydrodynamic model 35 ) are required to investigate this issue. Similar findings are obtained considering a fixed number of excess electron (Ne = 20): Fig. 10b shows that the screening effects can be well described by a classical model only for the largest NC (with a diameter of 2.4 nm).

6

Energy [ev]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Photonics

6

a)

b)

5

5

4

4

3

3

TD-DFT * Clas. * TD-DFT Clas.

S2 TD-DFT

2

S2 Clas.

2

S2 TD-DFT * S2 Clas. * S5 TD-DFT

1

S5 Clas.

1

S5 TD-DFT * S5 Clas. *

0 0

1 2 -3 n [10 a.u]

3

0

1

2 1.5 Diameter [nm]

2.5

Figure 10: Comparison between TD-DFT (symbols) and the classical model (lines) for the LSPR energy. Empty-symbols or dashed-lines are used when the response have been computed considering the response of excess-electron only (see also a ∗ in the legend). Panel a): LSPR energy for S2 (in red) and S5 (in black) as a function of the doping density. Panel b): LSPR energy with Ne =20 as a function of the nanocrystal diameter. The classical model can well reproduce only the NC with diameter larger than 2 nm.

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Page 24 of 37

Finally, we investigate how the GPI computed with TD-DFT deviates from the classical GPI: this is relevant to analyze the quantum effects in the plasmonicity index. The GPI in the classical limit which is 66 η

clas

ǫ(ω) − 1 , (ω) = ǫ(ω) + 2

(30)

where the dielectric constant is obtained using Eqs. (4,29) and Γ is set to 0.75eV . Whereas for the LSPR energy and intensity simple expression exists, see Eqs. (5,28), no explicit formulas exist for the classical GPI peak and intensity. Thus in Fig. 11 we report the full GPI spectra for S2 and S5 . For the largest system the agreement is very good: Figs. 11c and 11d can be directly compared with Figs. 7b and 8d (the same scale is used in the figures): the agreement is excellent in the case of the excess-only electron response, and it is still quite good for the full-response (especially for high doping). Note that for Ne = 0 curve in Fig. 11c (and also Fig. 11a) is frequency independent, and it is simply given by η clas (0) = α(R)/R3 ,

(31)

which is less then 1 and approaches 1 for very large nanoparticles. For S2 the agreement is of course not accurate: comparing Fig. 11b with Fig. 8b (again in the same scale) we see again that quantum effects introduce more peaks and red-shift the peaks. The agreement is also only qualitative in the case of the full-dielectric response (compare Fig. 11a with Fig. 7a). We also note that in the case of the full-response (see Fig. 11a,c) the classical GPI at energies higher than the main peak goes below its value at zero frequency. This is not the case for the the excess-electron only response (see Fig. 11b,d) and it is related to the fact the the denominator in the GPI expression includes both the real and the imaginary part, and the former is very large in the case of large static dielectric screening.

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Page 25 of 37

S5

S2 a)

1.5

c)

1.4 Ne=0 2 8 12 20

Classical GPI

Classical GPI

2.0

1.0

Ne=0 2 12 20 30

1.2 1.0 0.8

0.5 0.6 6.0

b) Classical GPI

0.0 6.0

Classical GPI

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Photonics

4.0

2.0

0.0

0

2

4

4.0

2.0

0.0

6

d)

0

1

2

3

Energy [eV]

Energy [eV]

Figure 11: Classical GPI for S2 , panels a) and b), and S5 , panels c) and d). Panels a) and c) consider the full dielectric response, whereas panels d) and d) consider the response of the excess electron only. These classical GPI plots can be compared with the TD-DFT ones in Figs. 7 and 8.

Conclusions We have carried out a through TD-DFT study of externally doped hydrogen-terminated spherical silicon NCs with diameters in the 1.3-2.4 nm range. Increasing the number of excess electrons increases the instability of the systems (HOMO energy towards positive energy) and thus each NC can accommodate only a radius-dependent number of electrons. A surrounding polarizable solvent is found to be very important to stabilize the system. The excess electrons are found to be allocated inside the nanoparticles, but valence electrons are pushed at the surface, regardless of the system size and the number of excess electrons. The external-doping largely influences the optical features, with new peaks appearing in 25


ACS Photonics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

the NIR. These peaks can be recognized as plasmon peaks: this is the result of our GPI analysis. We presented a novel efficient approach which can be quite easily implemented in any standard quantum-chemistry code using the transition densities. We found that offdiagonal coupling of transition densities are of fundamental importance to describe the GPI spectrum. In externally doped silicon NCs, the GPI increases if the dielectric response of the valence electrons is switched off, showing that the plasmon in these systems is strongly screened by the valence electrons. The shift in the LSPR energy can be attributed to the sum of two effects: the NC size and the number of excess electrons. Increasing the NC size always red-shifts the LSPR, whereas increasing the number of electrons blue-shifts the plasmon resonance only in large systems. We presented a detailed comparison between the TD-DFT results and a classical model with a simple quantum-correction: we found that the TD-DFT results can be explained classically only for NCs with diameter above 2 nm. The TD-DFT results here presented for silicon NCs can be used as a reference for other theoretical approaches which aim at modeling quantum-effects beyond the classical regime, whereas the newly developed tools here presented in this work can be applied to study the plasmonic behavior of semiconductor NCs with different compositions and shapes.

Supporting Information Available Density of silicon atoms for the considered systems, GPI convergence results for Ag20 , electronic configurations for all the considered systems, comparison between the HOMO and the I.P, comparison of the HOMO with and without point-changes, and full derivation of the GPI formula.

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Acknowledgement The authors thank TURBOMOLE GmbH for providing the TURBOMOLE program package and Dr. M. Manca and Prof. S. Corni for discussions.

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(8) Garcia, G.; Buonsanti, R.; Runnerstrom, E. L.; Mendelsberg, R. J.; Llordes, A.; Anders, A.; Richardson, T. J.; Milliron, D. J. Dynamically Modulating the Surface Plasmon Resonance of Doped Semiconductor Nanocrystals. Nano Lett. 2011, 11, 4415– 4420. (9) Schimpf, A. M.; Knowles, K. E.; Carroll, G. M.; Gamelin, D. R. Electronic Doping and Redox-Potential Tuning in Colloidal Semiconductor Nanocrystals. Acc. Chem. Res. 2015, 48, 1929–1937. (10) Runnerstrom, E. L.; Llordes, A.; Lounis, S. D.; Milliron, D. J. Nanostructured electrochromic smart windows: traditional materials and NIR-selective plasmonic nanocrystals. Chem. Comm. 2014, 50, 10555–10572. (11) Pattathil, P.; Giannuzzi, R.; Manca, M. Self-powered NIR-selective dynamic windows based on broad tuning of the localized surface plasmon resonance in mesoporous ITO electrodes. Nano Energy 2016, 30, 242 – 251. (12) Wang, Y.; Runnerstrom, E. L.; Milliron, D. J. Switchable Materials for Smart Windows. Ann. Rev. Chem. Biomol. Eng. 2016, 7, 283–304. (13) Zhao, Y.; Pan, H.; Lou, Y.; Qiu, X.; Zhu, J.; Burda, C. Plasmonic Cu2xS Nanocrystals: Optical and Structural Properties of Copper-Deficient Copper(I) Sulfides. J. Am. Chem. Soc. 2009, 131, 4253–4261. (14) Dorfs, D.; Härtling, T.; Miszta, K.; Bigall, N. C.; Kim, M. R.; Genovese, A.; Falqui, A.; Povia, M.; Manna, L. Reversible Tunability of the Near-Infrared Valence Band Plasmon Resonance in Cu2xSe Nanocrystals. J. Am. Chem. Soc. 2011, 133, 11175–11180. (15) Mendelsberg, R. J.; Garcia, G.; Li, H.; Manna, L.; Milliron, D. J. Understanding the Plasmon Resonance in Ensembles of Degenerately Doped Semiconductor Nanocrystals. J. Phys. Chem. C 2012, 116, 12226–12231.

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(16) Jung, J.; Pedersen, T. G. Analysis of plasmonic properties of heavily doped semiconductors using full band structure calculations. J. Appl. Phys. 2013, 113, 114904. (17) Liu, Z.; Beaulac, R. Nature of the Infrared Transition of Colloidal Indium Nitride Nanocrystals: Nonparabolicity Effects on the Plasmonic Behavior of Doped Semiconductor Nanomaterials. Chem. Mater. 2017, 29, 7507–7514. (18) Scholl, J. A.; Koh, A. L.; Dionne, J. A. Quantum plasmon resonances of individual metallic nanoparticles. Nature 2012, 483, 421. (19) Monreal, R. C.; Antosiewicz, T. J.; Apell, S. P. Competition between surface screening and size quantization for surface plasmons in nanoparticles. New J. Phys. 2013, 15, 083044. (20) Schimpf, A. M.; Thakkar, N.; Gunthardt, C. E.; Masiello, D. J.; Gamelin, D. R. Charge-Tunable Quantum Plasmons in Colloidal Semiconductor Nanocrystals. ACS Nano 2014, 8, 1065–1072. (21) Zhang, H.; Kulkarni, V.; Prodan, E.; Nordlander, P.; Govorov, A. O. Theory of Quantum Plasmon Resonances in Doped Semiconductor Nanocrystals. J. Phys. Chem. C 2014, 118, 16035–16042. (22) Jain, P. K. Plasmon-in-a-Box: On the Physical Nature of Few-Carrier Plasmon Resonances. J. Phys. Chem. Lett. 2014, 5, 3112–3119. (23) Greenberg, B. L.; Ganguly, S.; Held, J. T.; Kramer, N. J.; Mkhoyan, K. A.; Aydil, E. S.; Kortshagen, U. R. Nonequilibrium-Plasma-Synthesized ZnO Nanocrystals with Plasmon Resonance Tunable via Al Doping and Quantum Confinement. Nano Lett. 2015, 15, 8162–8169. (24) Ni, Z.; Pi, X.; Zhou, S.; Nozaki, T.; Grandidier, B.; Yang, D. Size-Dependent Structures

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Ab Initio Plasmonics of Externally Doped Silicon Nanocrystals | ACS

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