Infrared Absorption and Hot Electron Production in Low-Electron

Letter pubs.acs.org/JPCL

Infrared Absorption and Hot Electron Production in Low-ElectronDensity Nanospheres: A Look at Real Systems R. Carmina Monreal,† S. Peter Apell,‡ and Tomasz J. Antosiewicz*,§,‡ †

Departamento de Física Teórica de la Materia Condensada C5 and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-28049 Madrid, Spain ‡ Department of Physics and Gothenburg Physics Centre, Chalmers University of Technology, SE-412 96 Göteborg, Sweden § Centre of New Technologies, University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland S Supporting Information *

ABSTRACT: Doped semiconductor quantum dots are a new class of plasmonic systems exhibiting infrared resonances. At ultralow concentrations of charge carriers that can be achieved by controlled doping, only few carriers occupy each quantum dot; therefore, a spectrum with well-defined atomic-like peaks is expected. Here we investigate theoretically how surface imperfections and inhomogeneities in shape and morphology (surface “roughness”) always present in these nanocrystals, randomize their energy levels, and blur the atomic-like features. We assume a Gaussian distribution of each energy level and use their standard deviation σ as a measure of the nanocrystals’ roughness. For nearly perfect nanospheres with small roughness (σ), the spectrum exhibits well-defined peaks. However, increasing roughness effectively randomizes the energy level distribution, and when σ approaches 15% of the nanoparticle’s Fermi energy, any trace of an atomic-like structure is lost in the spectrum, and a continuous yet few-conduction-electron localized surface plasmon resonance emerges. fficient light−matter interaction in the form of localized surface plasmon resonances (LSPRs) is made possible by the presence of an incompletely filled energy band of the material in question. Such typically occurs in metals, which have been predominantly the material of choice for obtaining resonances in the visible due to their high free carrier densities ne and associated large plasma frequencies ωp.1 The large values of ne in metals are mostly invariable, and for metal nanoparticles, typically their size, shape, or environment is used to shift the LSPR to a desired value.2,3 This imposes rather strict limits on the (active) tunability of such systems as well as hinders “static” at-fabrication scalability because to red-shift LSPRs beyond the visible, large metal particles need to be used. This approach makes the spectra rather complicated through, typically, the introduction of higher order modes and causes scattering to dominate over absorption4 due to their respective volume scaling. While it is possible to modify the scattering-toabsorption ratio and other characteristics of a given nanoparticle, for example, by coupling within an array,5 in general, the degree of manipulation is limited. To obtain a strong plasmon resonance peak in a nanometersized structure, a low carrier density is a necessity,6 a feat made possible by the use of semiconductors. In a pure state their conductivity is small, however, it can be varied at fabrication. Through careful doping of semiconductor nanocrystals,7,8 their LSPRs, originating from p-type9 and n-type10,11 carriers, can be identified in the infrared (IR) range of the spectrum.12,13 Furthermore, it was recently demonstrated that photodoped

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© 2017 American Chemical Society

ZnO nanocrystals with nonequilibrium charge populations are also able to support LSPR modes.14,15 One of the peculiar properties of these low carrier density nanocrystals (sizes in the few nanometer range, carrier densities on the order of 1020 cm−3) are their continuous IR absorption spectra having the shape of a regular (metallic-like) LSPR.9,14,15 This experimental finding is surprising because a sphere of radius R = 2 nm with the above value of the electron density has three to four conduction electrons; therefore a discrete absorption spectrum, similar to that of atoms, should be expected, yet recently Faucheaux and Jain reported a photoinduced LSPR formed by four conduction band electrons.14 It is thus important to understand mechanisms that govern the formation of a resonance with such few conduction electrons. The calculations of the absorption spectra of free-electrons in a box reported in ref 16 show that a broad band emerges as the envelope of a collection of singleelectron transitions for a sufficiently large number of electrons, but the spectrum remains discrete if only some tens of electrons occupy the dot. Calculations performed previously for Ag clusters yielded similar trends.17 In previous publications18,19 we showed that at ultralow electron densities diffuse scattering at the surface of nanospheres plays an unexpectedly important role in determining the maximum and line width of the LSPRs. Diffuse scattering Received: December 15, 2016 Accepted: January 6, 2017 Published: January 9, 2017 524

DOI: 10.1021/acs.jpclett.6b02953 J. Phys. Chem. Lett. 2017, 8, 524−530

Letter

The Journal of Physical Chemistry Letters

In eq 2 e and me are the electron charge and mass, respectively, c is the speed of light, |ϕi⟩ and |ϕf⟩ are the electronic initial and final states, respectively, p̂ = −iℏ∇⃗ is the momentum operator, and A⃗ is the vector potential of the electromagnetic field. In the c Coulomb gauge A⃗ = −i ω E ⃗ , where E⃗ is the electric field vector, so that Mi,f is rewritten as

originates from the existence of imperfections, inhomogeneities, or “roughness” at the surface and causes, on average, an electron arriving to the surface from the bulk to be scattered back in a random direction. It contributes to the blue shift and, more importantly, to the width of the LSPRs because coherence between scattering events is lost. On the basis of a pure electromagnetic theory, we calculated the position and width of the LSPRs as a function of size, finding good agreement with the experiments of IR absorption by photodoped ZnO colloidal nanocrystals in toluene reported by Schimpf et al.15 The purpose of this work is to look at IR absorption by rough spheres having such a small number of conduction electrons from a different perspective and realize that while ideally spherical particles have well-defined energy levels and consequently show an atomic-like spectrum, surface roughness causes the energy levels of these spheres to be randomly distributed and causes blurring of the atomic features. To investigate this effect, we calculate IR absorption spectra of ZnO rough spheres, having radii between 2 and 4 nm, starting from Fermi’s golden rule. We assume a Gaussian distribution of energy levels and use the standard deviation σ as a ruler to measure surface roughness. Although a small value of σ produces a spectrum with a few well-defined peaks, by increasing σ we demonstrate how a continuous LSPR resonance grows out from a few conduction electrons. We need σ to be ca. 15% of the Fermi energy of the nanosphere to obtain a good account of the experimental IR absorption spectra.15 We also investigate hot-electron production in these small nanosystems, finding broad features instead of narrow peaks of electrons at well-defined energies. We conclude that a detailed description of the energy levels of individual particles is not necessary, and a continuous model, such as is implicit in dielectric theories, is enough for obtaining the IR absorption and hot electron production, even of a four-conduction electron nanoparticle. Our system consists of nanospheres of radius R, with typical electron densities of ne = (1−1.8) × 1020 cm−3, such as in photodoped ZnO colloidal quantum dots,15 which are illuminated with light of frequency ω near their corresponding classical LSPR. The absorbed photons excite surface plasmons and electron−hole pairs (which are coupled excitations), yielding electrons above the Fermi energy with a distribution in energies that we want to calculate. To this end, we use Fermi’s golden rule of perturbation theory and, for simplicity, work at zero temperature so that only absorption processes are possible. This is not a drastic simplification because, as it can be seen below, the excitation energies are larger than the typical 0.1 eV and therefore much larger than the thermal energy at room temperature. Accordingly, the rate at which a photon of frequency ω is absorbed, bringing the electronic system from an initial state i of energy ei to a final state f of energy ef, is written as 1 2π = τ(ω) ℏ

∑ f (ei)(1 − f (ef ))|Mi ,f |2 δ(ef

Mi , f = −

eℏ ⟨ϕ |∇⃗ ·E ⃗ + 2E ⃗ ·∇⃗|ϕi⟩ 2meω f

(3)

In a self-consistent field (SCF) approximation, the electronic states appearing in eq 3 should be these of unperturbed electrons, that is, in the absence of the electromagnetic field, while the electric field E⃗ , which is the total field acting on the electrons, includes all of the effects caused by the perturbation via the dielectric response. For the small spheres we will deal with, the electronic energy levels are quantized. In the case of perfect spheres, the appropriate set of quantum numbers is (l, n, m, s), where (l, m) are the angular momentum quantum numbers, n quantizes the energy for each l (energies el,n) and s is the spin quantum number. Then, eq 1 reads 1 4π = τ (ω , R ) ℏ

∑ ∑

f (el , n)(1 − f (el ′ , n ′))|Mll,′n, n, m′ , m ′|2

l ,n,m l′n′m′

δ(el ′ , n ′ − el , n − ℏω)

(4)

where we have included a factor of 2 coming from the spin summation. Equation 4 holds for ideally spherical particles. For real spheres, however, because of the unavoidable existence of imperfections in their shape and morphology, it is nearly impossible to calculate a detailed level distribution. A common approach has been to assume the level distribution to be completely random.20 Following this approach and to take into account a statistically random distribution of energy levels in a simple way, we first make use of the identity δ(el ′ , n ′ − el , n − ℏω) =

∫ dei δ(ei − el ,n) ∫ def δ(ef − el′,n′)δ(ef − ei − ℏω) (5)

We then substitute the function δ(ei − el,n) by a Gaussian distribution of probability function as δ(ei − el , n) → G(ei − el , n) =

⎡ (e − e )2 ⎤ 1 i l ,n ⎥ exp⎢ − ⎢⎣ ⎥⎦ σ 2π 2σ 2 (6)

and likewise for δ(ef − el′,n′). The basic idea behind eq 6 is to consider the energy levels of perfectly spherical particles el,n as mean values for rough spheres and let the energies of the initial and final states deviate from these mean values with a standard deviation σ. We take σ as a parameter, which allows us to follow the evolution of the IR absorption and hot electron spectra as functions of “roughness.” A small value of σ describes perfectly spherical particles, but when σ is comparable to a characteristic value of el,n, as is the Fermi energy, it indicates that the particles cannot be considered as approximately spherical, which renders the present approach inappropriate. Moreover, σ should depend on size R and on electron density because in the case of a large sphere or a large electron density the relevant energy levels form a continuum, and surface roughness cannot randomize them further. The matrix elements of eq 3, however, are evaluated using wave functions for a perfect sphere, ϕl,n,m(r)⃗

− ei − ℏω)

i,f

(1)

where f(ex) is the Fermi factor giving the occupancy of state x of energy ex, and the δ function expresses energy conservation. The matrix elements Mi,f for the transition are e Mi , f = ⟨ϕ |p ̂ ·A⃗ + A⃗ ·p ̂|ϕi⟩ 2mec f (2) 525


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The Journal of Physical Chemistry Letters = Nlnψln(r)Yml (θ, φ), where the radial part ψln(r) is a spherical Bessel function, the angular part is the spherical harmonic Yml (θ, φ), and Nln is the normalization constant. Equation 4 thus reads 1 4π = τ(ω , R ) ℏ

∫ dei ∫ def ∑ ∑

and is obtained from the static polarizability of small spheres calculated by Gorkov and Eliashberg20 as18,19,21 Δ = ℏωp

f (el , n)(1 − f (el ′ , n ′))

where ωp is the plasma frequency and R0 can be estimated from a simple model as R 0 =

(7)

The electric field E⃗ entering the matrix elements in eq 3 is the total electric field acting on the unperturbed electrons. It is constructed within a dielectric response theory that includes diffuse scattering at the surface of a rough sphere in a phenomenological way, as explained in ref 18. Because of the small size of our spheres, only the dipolar component of E⃗ is non-negligible, and the quasi-static approximation is a good one, which allows us to express E⃗ as

⎧ ⎛ α(ω , R ) ⎞ ⎪ E 0 ⎜1 − ⎟ cos θ cos φ for r < R R3 ⎠ ⎪ ⎝ Eθ (r , θ , φ) = ⎨ ⎪ ⎛ α(ω , R ) ⎞ ⎟ cos θ cos φ for r > R ⎪ E0⎜⎝1 − r3 ⎠ ⎩

σabs(ω , R )

(9)

2

2πR

for the radial Er and angular Eθ components of the field, respectively. E0 is the amplitude of the incident electromagnetic field, assumed to be a plane wave propagating in the x̂ direction. Note that the dipolar approximation implies that only transitions with l′ = l ± 1 and m′ = m ± 1 are allowed. This selection rule is the consequence of assuming that the angular momentum quantum numbers (l,m) are still good numbers for rough spheres with diffuse surface scattering, which will not be the case, and some consequences of this will be commented in the presentation of the results. However, this assumption makes the problem less complex. The polarizability of the sphere α(ω, R) enters into the evaluation of E⃗ . It is given by18,19,22

=

2 ω εm Im[α(ω , R )] R2 c

electrons/photon =

(13)

ℏω τ(ω , R ) c | E0 |2 2πR2 8π

(14)

with the denominator of eq 14 expressing the incident power on the sphere surface. Equation 14 thus provides an alternative way of calculating absorption. Hence our main objective is the comparison of eqs 13 and 14. To have full consistency between eqs 13 and 14, the same set of electronic levels should be used in the evaluation of the matrix elements in eq 7 as in the calculation of E⃗ and the sphere polarizability α(ω, R), because the polarizability is the result of the response of such a system of electrons to an incident electromagnetic perturbation. Such consistency is not present in our model calculations because, as stated above, α(ω, R) is obtained from a nonlocal dielectric theory modeling diffuse surface scattering, in which quantization is included only through Δ (in a way similar to eq 11), while eq 7 specifically uses the energy levels of small perfect spheres and calculates matrix elements using these states but randomizes the energy distribution. However, we will compare the results of eqs 13 and 14 and see that they are compatible when an appropriate value of σ is used in the calculation of τ−1(ω, R). From eqs 14 and 7, we calculate the number of excited (hot) electrons having energies between e and e + de per incident photon of frequency ω as

ε(ω) − εm + 2

(10)

ωp2 Δ

ω 2 − ( ℏ )2 + iωγb

where Ne is the number of electrons

and the number of excited electrons per incident photon can be calculated as

where ε(ω) is the local permittivity of the conduction electrons ε(ω) = ε∞ −

, where m* is the effective

and a is the side of the box and, consequently, scales with Ne, m*, and particle size exactly as eq 12 does, so the same result is obtained from a different approach. Because Δ is the mean distance between levels and σ describes fluctuations in energy around these levels, it is convenient to scale σ with Δ. Then, σ/ Δ is the dimensionless parameter that we will change to follow the evolution of the spectra. We have to point out that within the SCF approximation eq 7 and our previous calculations of the absorption cross section, σabs(ω, R) based on the electromagnetic theory presented in refs 18 and 19 should give consistent results. This is because the total number of electrons excited at frequency ω should be exactly the same as the number of absorbed photons of that frequency. The number of absorbed photons per incident photon is

and

ε(ω) + 2εm +

3Ne 1/3 ℏ2 , π m * a2

( )

Δbox = π 3/2

(8)

dθ(ω , R ) R dθ(ω , R ) 2 R

3π a 0 4m * k F

electron mass (in units of the electron mass), a0 is the Bohr radius, and kF = (3π2ne)1/3.21 It is interesting to note that a local dielectric function for few electrons in a box has been calculated by Jain.16 It has the form of eq 11 with

⎧ ⎛ α(ω , R ) ⎞ ⎪ E 0 ⎜1 − ⎟ sin θ cos φ for r < R R3 ⎠ ⎪ ⎝ Er(r , θ , φ) = ⎨ ⎪ ⎛ α(ω , R ) ⎞ ⎟ sin θ cos φ for r > R ⎪ E0⎜⎝1 + 2 r3 ⎠ ⎩

α(ω , R ) = R

(12)

l ,n,m l′n′m′

× G(ei − el , n)G(ef − el ′ , n ′)|Mll,′n, n, m′ , m ′|2 δ(ef − ei − ℏω)

3

R0 R

(11)

εm is the permittivity of the medium surrounding the sphere (assumed to be frequency independent), and the complex length dθ(ω, R) takes into account the effects of diffuse surface scattering of the electrons.18,19 Quantization due to size affects the dielectric properties of nanosystems, especially at small frequencies, because a minimum energy is necessary to excite an electron from/to the Fermi energy. Thus the energy Δ in eq 11 is the average distance between the levels at the Fermi level 526


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The Journal of Physical Chemistry Letters dN (ω , e) 16πω = de c|E0|2 R2

∑ ∑

f (el , n)(1 − f (el ′ , n ′))

l ,n,m l′n′m′

× G(e − ℏω − el , n)G(e − el ′ , n ′)|Mll,′n, n, m′ , m ′|2 (15)

We perform calculations for the case of ultralow electrondensity photodoped ZnO nanocrystals in toluene, using the experimental values: ε∞ = 3.72, εm = 2.25, m* = 0.28, and γb = 0.1 eV.15 In the experiments the electron density was fixed within the experimental error, and as in previous work18 we use the lower limit, ne = 1 × 1020 cm−3 (ℏωp = 0.7 eV and R0 = 0.56 nm), to match the experimental surface plasmon energies and widths at the largest radii. The polarizability of the sphere is then calculated according to eq 10, which also gives the electric field E⃗ of eqs 8 and 9 entering in the matrix elements of eq 3. Electronic wave functions and energy levels for perfect spheres of radius R are obtained by solving the Schrödinger equation for electrons of mass m* confined by a step potential barrier at the sphere surface of height V0 = 4 eV.23 Here we denote the levels (n, l) according to the conventional notation as nS, nP, nD, nF..., where S for l = 0, P for l = 1, D for l = 2, F for l = 3, and so on. An even number N of electrons (N/2 of each spin) fills these levels according to their energy and degeneracy up to an energy εF defining the Fermi energy. The number N is 4 chosen as the even integer closest to ne × 3 πR3. In the present calculations, because of the fact that the relevant energies are on the order of a few tenths of an electronvolt and therefore much smaller than V0, the level energy increases with (n, l) in the same way as for an infinite potential barrier: 1S, 1P, 1D, 2S, 1F, 2P, 1G... Figure 1 shows the number of absorbed photons per incident photon, eq 13 (open circles), and the number of excited electrons per incident photon, eq 14, for spheres of increasing sizes. We begin our analysis, whose results are shown in Figure 1a, with a sphere of R = 2 nm having N = 4 conduction electrons, the Fermi energy εF = 0.57 eV, and for different values of the Gaussian standard deviation σ. To facilitate comparison among the different curves, each of them has been normalized to its maximum value. (For absolute values, we refer to Figure S1 in the Supporting Information, SI.) The red line is obtained for a small value of σ in eq 6, σ = Δ/15 = 0.017 eV, and displays an atomic-like spectrum with three peaks. The lowest energy peak corresponds to single-electron transitions of electrons in the lowest energy level 1S being excited to the Fermi level 1P. The most intense peak corresponds to singleelectron transitions from the Fermi level to the next highenergy state 1D. Notice that the energy of this transition is very close the energy of the LSPR. The highest energy peak corresponds to electronic transitions from the Fermi level 1P to the 2S state. When we increase σ, these three peaks broaden and move in energy in such a way that for σ = Δ/5 we already obtain a continuous curve showing a shoulder at high energy. For σ = Δ/2 = 0.1 eV a continuous resonance has grown and the calculation of the number of excited electrons nearly reproduces the LSPR of the IR photon absorption curve, except for the oscillation appearing in its large energy tail. We think this reflects the inaccuracy of assuming (lm) to be good quantum numbers for the rough sphere with diffuse surface scattering. Other single-electron transitions prohibited by the selection rule l′ = l ± 1 should become possible if this assumption is relaxed.

Figure 1. Normalized number of absorbed photons per incident photon (open circles) and number of excited electrons per incident photon versus the photon energy for spheres of (a) R = 2 nm having 4 conduction electrons, (b) R = 3 nm with 12 conduction electrons, and (c) R = 4 nm with 28 conduction electrons for increasing values of the standard deviation σ: σ = Δ/15 (continuous red line), σ = Δ/5 (green line), and σ = Δ/2 (blue line). As σ increases, the spectra evolve from an atomic-like one to a continuous resonance, obtaining the LSPR of the absorption spectrum for σ = Δ/2, which for (a) is 0.1 eV, for (b) is 65 meV, and for (c) is 50 meV. The insets in (b) and (c) are magnifications of the high-energy region of the electron spectra to better appreciate the growing-up of the high-energy oscillations.

In Figure 1b we increase the radius to 3 nm and consequently N = 12 conduction electrons. The Fermi energy is εF = 0.455 eV. As before, an atomic-like spectrum is obtained for σ = Δ/15. Two transitions contribute to the lowest energy peak, one from the Fermi level 1D to the 1F state and another from the occupied 1P level to the Fermi level. The next peak with increasing energy originates from single-electron transitions from the occupied 1P level to the empty 2S level. The third prominent peak is due to excitation from the Fermi level 1D to the 2P level. Other single-electron transitions show up as small high-energy peaks. Introducing the statistical distribution of energy levels with σ = Δ/5 already produces a spectrum with two broad peaks, and for σ = Δ/2 = 0.065 eV the LSPR is very well-reproduced in position and width. Because the most intense atomic-like peaks appear at energies not as close to the LSPR as for the case of Figure 1a, we could say that all three single-electron transitions contribute approximately equally in making the local surface plasmon this case. It is also worth noticing how the high-energy atomic-like peaks evolve as σ increases into a small oscillations on the high-energy tail of the 527


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same conclusion has been reached by Tserkezis et. al25 for a statistical distribution of metallic free-electron nanoparticles. In this respect we should mention that other experiments performed on assemblies of ZnO quantum dots report absorption spectra showing shoulders that are identified as originating from atomic-like electronic transitions instead of a continuous resonance.26 This points out that different systems and fabrication procedures produce samples with different degree of roughness so that electronic transitions between welldefined energy levels may or may not show up in the IR absorption spectra. We now analyze the evolution of the hot electron spectra eq 15 with the standard deviation σ. This is shown in Figure 2a,b

main resonance, similar to the one appearing in the absorption curve, as shown in the inset of Figure 1b. Equivalent calculations for a sphere of R = 4 nm and N = 28 conduction electrons and Fermi energy εF = 0.38 eV are shown in Figure 1c. As in Figure 1a,b, an atomic-like spectrum is obtained for σ = Δ/15 and a continuous resonance for σ = Δ/2 = 0.05 eV. Although the agreement between both kinds of calculations is not as good as for R = 3 nm, the main LSPR is reasonably obtained in position and width. Also, the oscillation on the high-energy tail of the main resonance is more prominent for this large size. Other single-electron transitions not strictly allowed by the selection rules for angular momentum may also contribute to produce these oscillations. We should comment that the fact that in Figure 1 there are energy regions where the number of excited electrons is larger than the number of absorbed photons is just an artifact of the selected normalization. In these calculations we used a potential step barrier of height V0 = 4 eV; however, it is known that the presence of absorbed molecules at the surface can increase the barrier height considerably.23 To this end we verified that our results do not depend essentially on this parameter. We repeated the calculations presented in Figure 1 with only one difference: The electronic states were obtained for a potential well of V0 = 6 eV.23 Comparison of both results (not shown here) confirms that, although the atomic-like spectra are different, the statistically random distribution of the energy levels washes out the differences. To summarize the results we have presented so far, the calculation of the number of excited electrons per incident photon, eq 14, reproduces the LSPRs of the IR absorption spectra (which are in good agreement with the experimental results of Schimpf et al.15) for a value of the standard deviation σ ≃ Δ/2. This is a reasonable result because Δ/2 is half the mean distance between levels near the Fermi level, so two consecutive levels cannot be clearly distinguished when the standard deviation σ reaches this value and the atomic structure is lost. When compared with the Fermi energies, we find σ/εF = 0.17 for R = 2 nm, σ/εF = 0.14 for R = 3 nm, and σ/εF = 0.13 for R = 4 nm. We deem these figures to indicate that the experimental nanoparticles, although approximately spherical, have a large and similar degree of surface roughness. Fluctuations in energy due to roughness can be related to fluctuations in size via el,n ∝ R−2. Because the Fermi energy is the relevant one, a relative change in the Fermi energy of ca. 15% can be related to deviations in radius of ca. 8%. Therefore, diffuse scattering at the surface of such inhomogeneous particles is a very probable surface scattering mechanism that plays an essential role in giving their optical properties, confirming the conclusions of refs 18 and 19. We note that our model for spheres of radius R with a random distribution of energy levels equivalently models a statistical ensemble of spheres with a random distribution of sizes (assuming the matrix elements to be weakly dependent on the size within the ensemble), such as is experimentally realized in colloidal quantum dots. In these systems, the particles are only approximately spherical in shape and have a dispersion in size with an uncertainty in the measured radii ranging between 524 and 10%15 depending on the system, which is similar to our estimate and justifies our approach. Consequently, a detailed description of the energy levels of individual particles is not necessary, and a continuous model of energy levels is enough for obtaining the IR absorption of these nanosystems. The

Figure 2. Number of excited electrons of energy e per incident photon versus the electron energy (with respect to the Fermi energy) calculated for (a) a sphere of R = 2 nm illuminated at ωR = 0.38 eV and (b) R = 3 nm, ωR = 0.305 eV for increasing values of the standard deviation σ: σ = Δ/15 (red line), σ = Δ/5 (green line), and σ = Δ/2 (blue line). (a) While for R = 2 nm the electron spectrum of small σ only shows nearly resonant excitation of electrons from the Fermi level, a broad structure near the Fermi level develops for increasing σ that is made possible by quasi-resonant excitation of electrons to the Fermi level from below. (b) Even though the atomic-like spectrum for R = 3 nm is quite different from that of the smaller sphere, the spectra for σ = Δ/2 are similar, showing a low-energy broad structure of excitations to the Fermi level and another one near ωR originating from electrons at the Fermi level being promoted to a higher energy.

for a sphere with R = 2 nm illuminated at frequency ωR = 0.38 eV and for R = 3 nm illuminated at ωR = 0.305 eV, respectively, which are their corresponding surface plasmon frequencies for which the coupling of light to nanoparticle is the strongest. Each spectrum has been rescaled so it integrates to the number of absorbed photons of frequency ωR obtained for the corresponding R, according to eq 13. The color of each line in Figure 2a,b is for the same value of σ as in Figure 1. For R = 2 nm and small σ only one intense peak is observed in Figure 528


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We have analyzed theoretically how surface roughness, always present in small particles, impacts the shape of the IR absorption and hot electron spectra in ultralow-carrier-density nanospheres through the random distribution of their energy levels. Specifically, we studied spheres of 2−4 nm in radius having 4−28 conduction electrons, respectively. We assumed a Gaussian distribution of energy levels with their mean values being those of ideally spherical particles and used the standard deviation σ as the parameter that allows us to follow the evolution of the spectra. Starting with very small values of σ, where we found atomic-like spectra with a few well-defined peaks, we increased σ and found that a continuous LSPR is fully developed for σ ≃ Δ/2, where Δ is the distance between levels at the Fermi energy, or σ ≃ 15% of the Fermi energy. This can be related to fluctuations in radius of ca. 8%, which is in accordance with the experimental uncertainty in size obtained in several systems of colloidal quantum dots. We also investigated hot-electron production in these nanospheres, finding broad structures instead of peaks of electrons at welldefined energies. We conclude that surface roughness influences the optical properties of low-density-electron nanosystems in two related ways: It randomizes the energy level distribution and induces diffuse surface scattering of the electrons in such a way that a continuous resonance can be observed even for a four-conduction electron nanocrystal. Consequently, a detailed description of the energy levels of individual particles is not necessary, and a continuous model, such as is implicit in dielectric theories, is enough for obtaining the IR absorption of these nanosystems.

2a, which corresponds to nearly resonant excitation of electrons from the Fermi level 1P to the 1D state. When σ increases, other quasi-resonant transitions are made possible by energy fluctuations, and another peak near the Fermi level develops coming from electrons in the lowest energy state 1S absorbing a photon of frequency ωR and being excited to the Fermi level. The results for R = 3 nm, displayed in Figure 2b, are somewhat different in that a small peak appears near the Fermi level already at very small values of σ, which comes from electrons in the 1P level being excited to the 2S level in a resonant transition. The most intense peak is again due to electrons at the Fermi level absorbing a photon of frequency ωR and being excited to a higher energy state. These two peaks broaden and move in energy as σ increases, producing a spectrum not unlike that of R = 2 nm presented in Figure 2a. Finally, Figure 3 compares the hot electron spectra for R = 2, 3, and 4 nm. The illuminating frequency ω equals the

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Figure 3. Number of excited electrons of energy e per incident photon of frequency ωR versus the electron energy (relative to the Fermi energy) calculated for spheres of R = 2, 3, and 4 nm, σ = Δ/2 in all cases. The black curves are for an electron density of ∼1020 cm−3 in all three cases: continuous line, R = 2 nm; dotted line, R = 3 nm; dashed line, R = 4 nm. Red dashed line, an R = 4 nm sphere with an electron density of ∼1.8 × 1020 cm−3.

ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.6b02953. Figure S1. Replotted Figure 1 with absolute values of the probability to excite electrons by an incident photon. (PDF)

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corresponding LSPR frequency ωR for each R: ωR = 0.38 eV for R = 2 nm, ωR = 0.305 eV for R = 3 nm, and ωR = 0.27 eV for R = 4 nm, and σ = Δ/2 in all cases. Each spectrum has been rescaled so it integrates to the number of absorbed photons of frequency ωR obtained for the corresponding R. The black lines are obtained for the same number of electrons in each sphere as in Figure 1 (ne ≃ 1020 cm−3). For these small spheres containing a small number of conduction electrons, the spectra show two resonances. As we said above, the high-energy resonance is always near ωR and basically comes from atomiclike transitions from the Fermi level to higher energy states, while the low-energy resonance comes from electrons below the Fermi level being excited to the Fermi level. These two resonances tend to merge, producing only one resonance near the Fermi level, as the size or the number of electrons increases. Actually, the red dashed curve in Figure 3, showing only one resonance, is obtained when we put the larger value of N = 38 electrons in the R = 4 nm sphere (ne ≃ 1.8 × 1020 cm−3). Also note that the intensity of the resonances increases with R as the IR absorption does. We should note that the maximum energy of the hot electrons we obtain is ca. 1 eV for R = 2 nm, which is well below the potential barrier height. Therefore, these electrons have very little probability for migrating away and will degrade their energy by heating of the nanoparticle.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +48 22 55 46 679. ORCID

Tomasz J. Antosiewicz: 0000-0003-2535-4174 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS R.C.M. acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the Mariá de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0377) and the project MAT2014-53432-C5-5-R. T.J.A. thanks the Polish Ministry of Science and Higher Education for support via the Iuventus Plus project IP2014 000473 and the Polish National Science Center via the project 2012/07/D/ST3/02152. T.J.A. and S.P.A. acknowledge financial support from the Swedish Foundation for Strategic Research via the Functional Electromagnetic Metamaterials for Optical Sensing project SSF RMA 11. 529


Letter


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(22) Apell, P.; Ljungbert, A. A General Non-Local Theory for the Electromagnetic Response of a Small Metal Particle. Phys. Scr. 1982, 26, 113−118. (23) Schlesinger, R.; Xu, Y.; Hofmann, O. T.; Winkler, S.; Frisch, J.; Niederhausen, J.; Vollmer, A.; Blumstengel, S.; Henneberger, F.; Rinke, P.; Scheffler, M.; Koch, N. Controlling the Work Function of ZnO and the Energy-Level Alignment at the Interface to Organic Semiconductors with a Molecular Electron Acceptor. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 87, 155311. (24) Shen, G.; Guyot-Sionnest, P. HgS and HgS/CdS Colloidal Quantum Dots with Infrared Intraband Transitions and Emergence of a Surface Plasmon. J. Phys. Chem. C 2016, 120, 11744−11753. (25) Tserkezis, C.; Maack, J. R.; Liu, Z.; Wubs, M.; Mortensen, N. A. Robustness of the Far-Field Response of Nonlocal Plasmonic Ensembles. Sci. Rep. 2016, 6, 28441. (26) Germeau, A.; Roest, A. L.; Vanmaekelbergh, D.; Allan, G.; Delerue, C.; Meulenkamp, E. A. Optical Transitions in Artificial FewElectron Atoms Strongly Confined inside ZnO Nanocrystals. Phys. Rev. Lett. 2003, 90, 097401.

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