1

An effective size window for impact ionization in PbS nanoparticles

and experimental evidence of dispersive screening of coulomb

interaction in hot excitons

Padmashri V. Patil and Shouvik Datta

Department of Physics, Indian Institute of Science Education and Research-Pune,

1st Floor, Central Tower, Sai Trinity Building, Pashan, Pune – 411021, Maharashtra, India.

Email : shouvik@iiserpune.ac.in, padmashriv.patil@gmail.com

ABSTRACT

Photo-induced carrier multiplications are predicted to overcome the limiting efficiency of single junction solar cells by

generating more than one electron-hole pair (exciton) per absorbed photon. Here, we have studied collisional

broadening of excitonic absorption spectra of PbS nano-crystallites at energies much above its fundamental band gap.

Observed spectral features necessitate the role of extended band structure of semiconductors to understand the physics

of carrier multiplication in nanoparticles. Our analysis also shows that quasi ballistic transport of hot excitons can

actually suppress exciton-exciton scattering events required for photo-induced carrier multiplication in very small

quantum dots. Measured variations of excitonic broadening clearly indicate that impact ionization of excitons may be

efficient only inside an intermediate „size window‟. This can explain the current debates on reported efficiencies of

carrier multiplication in semiconductor nanoparticles as compared to bulk materials. Moreover, we will discuss the

importance of „effective Bohr exciton radius‟ as a direct consequence of significant departure from the usual dielectric

screening limits of coulomb interactions of „hot‟ excitons in the region of strong dispersion at energies much above the

fundamental band gap.

Keywords: Exciton, Carrier Multiplication, Multiple Exciton Generation, Nanoparticles, Quantum Dots, Absorption

Spectroscopy, Dielectric Response, Auger Processes, Ballistic transport.

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I. INTRODUCTION

In solids, attractive coulomb interaction generates electron-hole pairs to form hydrogen atom like

bound states called excitons. In semiconductors, such exciton like quasi particles can form during optical

absorptions. Here we will study the optical absorption spectra of excitons generated much above the

fundamental bang gap in Lead Sulfide (PbS) nanoparticles. It is known that PbS is a narrow band gap (E0Bulk

~ 0.41eV at 300K) group IV-VI semiconductor having a seemingly large (~18nm) Bohr exciton radius1.

Optical absorption spectra of lowest excitonic transitions in PbS nanoparticles were extensively studied in

the past1-4

. It was also predicted earlier that carrier multiplication (CM) or multiple exciton generation5,6

can

be very efficient in these quantum confined structures of semiconductors. Consequently, one expects that

solar cells using these nanoparticles may be able to exploit CM to generate more than one electron-hole pairs

from single photon absorption events to enhance the efficiency beyond the maximum attainable

thermodynamic efficiency7 of a single junction solar cell. Efficient CM was subsequently reported by several

groups8-11

in many different semiconductor nanoparticles. However, some recent experimental results and

theoretical analyses generated further controversies12-15

about the precise nature of efficiency of such CM

processes in semiconductor nanoparticles as compared to their bulk counterparts.

Here we report, the broadening of an excitonic feature of PbS nano-crystallites in room temperature

optical absorption spectra at excitation energies (> 5.3 eV) much higher than its bulk band gap. This is in

contrast to most spectral studies where the focus were primarily on the lowest excitonic transitions in which

one can comfortably neglect the presence of strong dispersions in the dielectric response. In this report, we

will analyze – (i) the role of collisional broadening of „hot‟ excitons via „zero phonon coupling‟ to the

extended band structure of PbS and (ii) also the reduction of collisional broadening in the strong confinement

regimes where the nano-crystallites can be quasi ballistic for inverse Auger type of events. It will be

3

demonstrated that this collisional broadening decreases for both smaller and larger nano-crystallites.

Therefore, we will argue in favor of the existence of an effective intermediate size window for efficient CM

via photo-induced impact ionization in semiconductor nanoparticles. Moreover, we will also talk about - (iii)

the role of asymmetric effective masses of electron and holes at high energy critical points of PbS band

structure and (iv) the concept of ‘effective Bohr exciton radius’ (a*Ex) at photon energies much higher than

the fundamental band gap. There we will argue that within such energy regions of strong dielectric

dispersion, the usual high frequency limit of dielectric screening of coulomb interactions may no longer be

valid for such „hot‟ excitons. Furthermore, we deliberately allowed larger PbS nano-particles to age to get

more physical insights about the spectral broadening. We will discuss - (v) how the aging of excitonic

spectra due to spatial delocalization of excitons can affect such collisional broadening, and (vi) the „quasi-

reversible‟ evolution of room temperature optical absorption spectra of „hot‟ excitons of weakly confined

(particle radius > 3 times Bohr exciton radius ) PbS nano-crystallites to that of bulk like absorption spectra at

temperatures higher than room temperature. We will argue that all these above observations actually support

our analysis about the effect of exciton scattering on the collisional broadening of optical absorption spectra.

We hope that this work will be useful to optimize the size of semiconductor nanoparticles in nano-

photovoltaic cells designed to exploit the benefits of carrier multiplications to enhance the power conversion

efficiency.

II. EXPERIMENTAL METHODS

A. Synthesis of PbS Nanoparticles. Chemicals used for the synthesis of PbS nano-crystallites were

Lead Acetate Pb(CH3COO)2 and Sodium Sulfide (Na2S). Thio-glycerol (3-Mercapto-propane-1,2 diol) or TG

was used as capping agents. All chemicals were purchased from Sigma Aldrich and used without further

4

purifications. Synthesis was done in a two neck, round bottom flask with 18M de-ionized water as solvent.

10mL of 0.02 M Lead Acetate was heated to 80o C. Then 10mL of 0.02 M Sodium Sulfide and TG was

added drop wise. The quantity of TG was varied during the reaction (from none for the uncapped sample to

120 L in steps of 20 L) to get a wide range of nano-crystallite sizes. The temperature is maintained at 80o C

for half an hour for each reaction. After the reaction, particles were further washed several times with de-

ionized water using a 4000 rpm centrifuge for 10 minutes and finally re-dispersed in de-ionized water for

further characterizations. 20 L of this final aqueous dispersion of PbS nanoparticles is then mixed with 3mL

of de-ionized water inside a quartz cuvette for optical absorption studies. Similar characterization studies

were also performed on PbS nanoparticles dispersed in non-aqueous medium.

B. Characterizations of PbS Nanoparticles. Perkin Elmer lambda 950 was used for UV-VIS-NIR

optical absorption spectroscopy with scan steps of 0.2nm. Crystallite sizes were determined by X-ray powder

diffraction (XRD) spectra of drop casted thin films of these nanoparticles using a Bruker D8 Advanced X-

ray diffractometer with = 0.154 nm and increment of 0.01 per step. We use these crystallite sizes in our

analysis. Moreover, hydrodynamic sizes were also determined by Dynamic Light Scattering (DLS) studies

using Malvern Zeta-Sizer Nano-ZS90. DLS experiments were carried out at 25 C using 633nm, 3mW He-Ne

laser using 90 optics having 10 m apertures and autocorrelation time window of 20s. Transmission

Electron Microscopy (TEM) was done with FEI‟s Technai T-20 electron microscope using tungsten filament

at 200kV.

III. RESULTS AND DISCUSSION

A. Nano-Crystallite size vs hydrodynamic size. Figure 1a shows the XRD spectra for PbS

nanoparticles synthesized with varying Thio-glycerol (TG) concentrations. The 2 peak positions at 26.0 ,

5

30.1 , 43.1 , and 51.1 can be indexed respectively to planes (111), (200), (220), and (311) of cubic rocksalt

structure of PbS. Each of these peaks are used to calculate the final mean size of PbS nano-crystallites using

the Debye-Scherrer formula , where D is the average size(diameter) of the crystallite, K is the

shape factor ≈ 0.9, wavelength = 0.154 nm for Cu-K X-Ray, is the FWHM of the Bragg peaks and 2

is the Bragg angle. Estimated mean crystallite sizes (diameters) are also mentioned in figure 1a. Clearly the

width of the Bragg peaks increases as the crystallite size are reduced. Further analysis of the powder

diffraction spectra using William-Hall analysis showed very little strain (<0.05%) in these crystallites, which

substantiate the validity of our calculation based on Debye-Scherrer formula. A comparison of

hydrodynamic size of these particles as determined by DLS is also plotted along with the nano-crystallite

sizes in figure 1b. There we see that the hydrodynamic sizes and crystallite sizes are somewhat comparable

only below a certain crystallite diameter of 16nm. Below this 16nm of mean crystallite size, DLS data

possibly underestimate the hydrodynamic size. Above that critical size of 16nm, inherent agglomeration of

these nanoparticles is responsible for much larger hydrodynamic sizes. We will come back to this issue while

analyzing the ballistic limits of collisional broadening of excitonic line width and the aging of larger

nanoparticles.

B. Size Dependence of Excitonic Line Width. Figure 2a shows room

temperature optical absorption spectrum of freshly prepared uncapped PbS nano-crystallites dispersed in de-

ionized water. The term „freshly prepared‟ means the time (t = 0) starts 30mins after the end of reaction. We

have observed a strong excitonic feature (~5.9eV) above E3 16,17

critical point18

of bulk PbS (5.3eV).

Absorption edges are also observed near ~3.5 eV and ~2.1 eV which are close to E2 (3.14eV) and E1 (1.94

eV) critical points of bulk PbS respectively. Similar spectral features of excitonic transitions are also

6

reproduced in PbS nanoparticles prepared with different chemical reagents and capping processes as well as

particles dispersed in non-aqueous solution too. It is known, that E3 critical points of PbS are due16

to

electronic transitions ( 4 to 7) around M1 critical point and transitions ( 4 to 6) around M2 critical point.

Excitonic transitions around such saddle points in the electronic band structure are known as „hyperbolic

excitons‟18

as the constant energy surfaces around these critical points in the joint density of states are

hyperboloids. Presence of these critical point features clearly shows the need to consider the effect of

extended band structures for CM or MEG in semiconductor nanoparticles and this will be emphasized in the

next two sections IIIC and IIID.

In figure 2b, we compare the optical absorption spectra of uncapped and Thio-glycerol (TG)19

capped

nano-crystallites of various sizes having same molar concentrations of PbS in aqueous solution. We notice

(figure 2b) that absorbance of E3 exciton increases monotonically with decreasing nano-crystallite size. The

inset of figure 2b shows the absorption spectra at photon energies >5.0 eV only. We also notice that the E3

excitonic spectra of uncapped PbS nano-crystallite with mean diameter ~24nm is gradually broadened into a

shoulder like feature in 16nm PbS. Yet, below 16nm mean diameter, this gradual broadening with decreasing

crystallite size is ceased (figure 2b). Instead, thereafter we observe some kind of transition towards weak

sharpening of the excitonic spectrum as the crystallite sizes reduces. These E3 excitonic peaks as displayed in

figure 2b are then fitted with Voigt line shapes (convolution of Lorentzian and Gaussian shapes). From that,

we estimated both the Gaussian width (inhomogeneous component) and the Lorentzian width (homogeneous

component) of the E3 excitonic spectra. The Lorentzian line widths are nearly ~1.4 to 1.5 times the

respective Gaussian line widths for particles larger than 16nm. However, at this stage, we must note that

excitonic absorption spectra of samples with sizes smaller than 16nm are having asymmetric profiles and

these are not very well fitted by any „single’ peak like profile e.g a Gaussian or a Lorentzian or a Voigt line

shape.

7

In general, the magnitude of any excitonic transition (both Lorentzian and Gaussian shapes) is

inversely proportional to its broadening parameter. However, contrary to the usual expectations, the

inhomogeneous width of E3 exciton is not monotonically decreasing with size (see figure 3a). So it is very

clear that we do not witness any straightforward inverse type relationship between the peak absorbance and

the line width of spectral broadening for E3 exciton. Additionally, we also see monotonic blue shift (figure

3b) of the E3 excitonic energy and monotonic increase of peak absorbance with decreasing size. These results

indicate that oscillator strength for excitonic absorption is still size dependent in PbS nano-crystallites even

in the intermediate (16nm) to weak (24 nm) confinement regimes20

. Therefore, we conclude that the origin

of this large line width (~few hundreds of meV) may not be from any simple homogenous or inhomogeneous

type of broadening mechanisms. This is reinforced by the facts that spectral shapes are asymmetric and

unlike any single peak like feature for samples smaller than 16nm.

Moreover, the magnitude of the maximum broadening width Emax = [ E(RMin) - E(RMax)] is

calculated using , where Emax is the difference between the energy position

corresponding to the minimum (R=Rmin) and maximum (R=Rmax) nano-crystallite radius, me* and mh* are

effective mass of electron and holes, h is the Planck constant and R=D/2 is the radius of the crystallite as

estimated by XRD. These Emax values (figure 3a) are far less than the estimated inhomogeneous

(Gaussian) width and homogeneous (Lorentzian) width of the E3 excitonic peaks for all nano-crystallite

diameters. We also notice that the size dependence of both estimated homogeneous and inhomogeneous

widths are qualitatively different from that of the calculated Emax. Therefore, we can rule out any significant

contribution from variation of size distributions of these nano-crystallites into the energy broadening of E3

exciton. We will discuss the physical origin of this significant inhomogeneous component at the end of

section III.D.

8

Surprisingly, these excitonic transitions (figure 2b) are located much above the E3 critical point of

bulk PbS (> 0.6eV larger than 5.3eV). This difference is also much larger than the optical phonon energy of

PbS ~ 20-27meV, which may be preventing it from thermalizing to the band edge. Our estimates

also shows that this excess energy of E3 exciton cannot be explained by size

confinement induced enhancement of energy alone (see figure 3b), except

for the strongly confined PbS nano-crystallites of mean diameter ≤ 3nm. This unusually

large departure of the peak excitonic energy position above the bulk E3 critical point of PbS may be

attributed to (at least for smaller crystallites) – a) the failure of effective mass theory21-22

at small sizes, b)

non-parabolicity of the band structure at high energies and at small sizes. However, the measured size

dependence of excitonic energy is rather „much‟ slower than even the usual type of behavior 23

(figure

3b). So, it seems that, E3 exciton can qualify as a „hot exciton‟ with excess center-of-mass kinetic energy

above the E3 band edge. We will further elucidate this point in connection with the temperature dependence

of PbS nano-crystallites (figure 7) where we will observe how this „hot‟ E3 exciton loses its extra energy

above certain temperature.

C. Dispersive Screening of Coulomb Interaction and Effective Bohr Exciton Radius. Generally,

one uses the so called “zero frequency” dielectric constant or optical dielectric constant 18

of PbS as = e +

lattice = e(0) ~ 17 to estimate the exciton binding energy and Bohr radius etc, where the subscript „e‟

stands for the electronic contribution and the subscript „lattice‟ stands for lattice contribution to dielectric

response. In contrast, here in the presence of strong dispersion around E3 critical point, we have used the

empirical value ( ) = 7 around 5.9 eV of photon energy as determined by the spectroscopic ellipsometry

measurements17

on bulk PbS. In principle, one can use experimentally determined values of ( ) for the

9

same set of nanoparticles which is currently beyond the scope (for wavelength ≤ 210 nm). To continue, we

find that – (i) there is presence of significant dispersion16, 17

above the fundamental band gap of PbS and (ii)

this E3 exciton posses excess center-of-mass kinetic energy as discussed above. As a consequence, the

calculated exciton binding energy (EEx~1/ϵ, using ( ) = 7) of this E3 exciton can be substantially higher (as

reported in the past16

) than the usual limiting value of = 17 (See Table I). We also see in Table I that the

excitonic binding energy (for all sizes) in case of dispersive ( ) = 7 is larger than the reported1,6 optical

phonon energy of PbS. This fact along with size confinement can explain the

existence of this E3 excitonic transition in PbS nano-crystallites even at room temperature. Interestingly, the

mean nano-crystallite radius of 8nm at the transition point of excitonic line width broadening (figure 3a) is

comparable to the effective Bohr exciton radius (a*

Ex) = 7.6 nm [assuming17

( ) = 7 ( 17) at E > 5.9eV and

approximately using , where me* and mh* are effective mass of electron and

holes, me is the free electron mass and is the reduced mass of exciton at E3 critical point of PbS]. We also

want to state that approximating the dielectric function with an energy independent limiting value such as

e(0) ~ is only valid 18

for , where Eg is the band gap and T is the frequency of transverse

optical phonon. However, in case of this above band gap E3 transition, . Therefore, we

understand that this revised value for the excitonic Bohr radius (a*Ex) is a direct consequence of departure

from the usual high frequency limit of dielectric constant of coulomb interactions at high photon energies

due the presence of significant dispersion16,17

much above the fundamental band gap for this „hot‟ E3 exciton.

Later we will demonstrate further connection of this new length scale termed as „effective Bohr exciton

radius’ with the ballistic limit of impact ionization which can really affect the collisional broadening E3

exciton. We will also explore the physics of this unusual variation of the broadening of E3 exciton in more

detail with respect to aging and temperature dependence of optical absorption spectra.

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D. How Ballistic Limit of Impact Ionization Affects the Collisional Broadening? In order to

explore this connection, we first note that the electronic band structure near E3 critical point in PbS is not

symmetric16,24

at all (e.g. me*

mh* around E3). This is unlike the usual situation near its fundamental band

gap. So, we rather expect the broadening of E3 exciton25

to be dominated by Fröhlich type strong polar

exciton-phonon interactions. Usually this kind of strong coupling of excitons to optical phonons is neglected

in PbS because of its quasi symmetrical nature of electron and hole bands near its fundamental band gap

(E0). However, it is a different situation for the E3 transition in PbS. Despite the obvious presence of such

strong coupling with the optical phonons, the E3 excitonic feature still survives at room temperature and

hardly shows any temperature dependent shift in peak position (figure 7) even for weakly confined particles.

Therefore, it seems likely that quantum confinement induced non-phonon energy relaxation mechanisms26

like inverse Auger type of events (e.g impact ionization) can be dominating over phonon induced processes

for this highly energetic E3 exciton. We also know that extended band structures of a solid often play crucial

roles27-29

for inverse Auger processes. We note that this „hot‟ E3 exciton can easily avail a large number of

final density of states needed for efficient impact ionization via quantum confinement induced zero phonon

coupling to other parts of the PbS band structure. Currently, probing the CM or the impact ionization events

of this „hot‟ E3 exciton ( for wavelengths ≤ 210nm) with time domain transient absorption spectroscopy is

beyond the scope of this study. Instead, we concentrate more on the inverse Auger type of collisional

broadening30,31

of E3 excitonic peak as the mean diameter of PbS crystallite is changed from 24nm to 16nm

(figure 3a). We attribute this progressive enhancement of spectral broadening as collisional broadening due

to increased „zero phonon transitions‟ leading to efficient impact ionization and possibly CM of the E3

exciton with reducing size. Further evidences about the unusual nature of this size confinement induced

collisional broadening will be presented in connection with aging and temperature dependence.

11

As mentioned earlier in Section IIIB, fitting of Voigt lineshapes to the E3 excitonic spectra in figure

2b, also reveals significant presence of both homogeneous (Lorentzian) and inhomogeneous (Gaussian)

components. We have already shown in figure 3a, that size distribution of these PbS nano-crystallites cannot

account for such large inhomogeneous broadening. Usually, collisional broadening results in homogeneously

broadened line shapes for gas molecules. However, the presence of this significant inhomogeneous

broadening can be explained in terms of – a) size confinement induced momentum uncertainty which

connects E3 excitons to different symmetry points in the PbS band structure and b) a distribution in excitonic

K space for such hot excitons. These can cause anisotropic exciton-exciton collisions leading to

inhomogeneously broadened absorption peaks of E3 exciton. The asymmetric nature of E3 excitonic line

shape for smaller particles may also be coming from the fact that below 16nm of crystallite size, the ballistic

nature of exciton scattering can be influencing the spectral line shape. Therefore, we conclude that these

results also support the model of size confinement induced exciton scattering as the cause for spectral

broadening of E3 exciton.

Unlike the expected monotonic32-34

size variation of the rates of Auger type processes, spectral

broadening of E3 exciton is seemingly arrested (figure 2b and 3a) for sizes smaller than the „effective Bohr

exciton radius‟. Moreover, we even noticed weak sharpening of excitonic spectra in the intermediate to

strong confinement regime ( ≤ 16nm ). This observation can be explained if the impact ionization of E3

exciton and subsequent collisional broadening have a minimum size cut off. It is well known, that impact

ionization35-37

can be suppressed in very small structures due to the presence of quasi ballistic transport. This

usually happens when the nanoparticle size becomes smaller than the inverse Auger type mean free path

( )38

of exciton scattering events within the material. Our estimates based on the reported calculation38

indeed show that this for PbS nano-crystallites with

12

mean diameter of 16nm. Here we use M as the effective translation mass of exciton = me*+mh

* ≈ 0.21me, as

the reduced mass , a*Ex = 7.6nm is the effective Bohr exciton radius around E3, kB

is Boltzmann constant, T is temperature in degree Kelvin and EEx is the excitonic binding energy.

Interestingly, the estimated mean free path ( ) for exciton scattering for 16nm PbS nano-crystallite is very

much comparable - (i) to the size (diameter) at the transition point of exciton broadening (figure 2b and 3a)

and (ii) to the ‘effective Bohr exciton diameter of E3 Exciton and (iii) also to the size where the

hydrodynamic size and the crystallite size become comparable [figure 1b]! However, we must note that the

above formula was calculated based on symmetric S state wavefunctions only and may needs revision.

Moreover, we plot the estimated size dependent exciton mean free paths of PbS in figure 4 for both

( ) = 7 at 5.9eV and = 17. It is also very clear from the plot that use of the standard high frequency limit

of dielectric constant as = 17 will produce a mean free path for exciton scattering at a value > 30nm which

is much greater than the diameter of any of the PbS nano-crystallites used in this work. That way you hardly

expect any PbS nanoparticles having sizes < 30nm to undergo exciton-exciton collision processes and

subsequently produce any significant CM mediated by photo-induced impact ionization events. On the

contrary, we can invoke dispersive (dynamic) screening of coulomb interaction39-41

and an experimentally

measured value of ( ) = 7 can be used. Then it allows for the possibility of impact ionization of E3 exciton

and subsequent collisional broadening within a size window of operation in the weak to intermediate

confinement regimes. Although, our experimental results and subsequent explanations clearly resonate with

the theoretical statement42

that strong confinement may not be exclusively needed for efficient CM, but our

approach based on mean free path for inverse Auger type of exciton scattering, departure from standard static

limits of dielectric screening and the role of extended band structure is much different from those reported

earlier. Moreover, the above analysis clearly shows that we seriously need to revisit43, 44

the commonplace

13

excitonic terminologies to understand the physics of carrier multiplication of such „hot‟ excitons – a) having

very high center-of-mass velocity and b) created inside a region of strong dielectric dispersion. The dielectric

function of a material near such energetic quasi particle excitations can deviate considerably from its usual

constant (static) high frequency or optical frequency limit ( ) in such circumstances. Further support for the

above analysis will be discussed in connection with aging and temperature dependence of the excitonic

spectra.

E. Aging Dependence of Excitonic Line Width and the Role of Exciton Scattering. Aqueous

dispersion of PbS nanoparticles are deliberately allowed to age to get better physical insights into the spectral

broadening of the E3 excitonic spectra. We notice that at room temperature, E3 excitonic spectra definitely

sharpen up with aging (figure 5a and 5b) within the weak to intermediate size confinement regimes. In order

to understand this spectral property, we first focus on the structural aspects of these nanoparticles during the

whole aging process. We notice that hydrodynamic size of uncapped PbS was already very large (>2 micron)

compared to its mean nano-crystallite size of 24nm. However, hydrodynamic size of fresh 17nm particles is

comparable to its nano-crystallite diameter of 16nm. This hydrodynamic size of 17nm increases gradually to

over a micron due to aging (figure 6a). Surprisingly, XRD indicates that the size of these 16 nm nano-

crystallites do not increase at all subsequent to its aging in aqueous medium. Thus it clearly excludes– (a)

any Ostwald type ripening, (b) growth of PbS nano-crystallite by incorporation45

of sulfur atoms from the

thiol capping group. Therefore, we understand this observed increase of hydrodynamic size is a result of

agglomeration (figure 6b) of PbS nano-crystallites via Van der Walls type of attractive interactions only.

Such agglomeration can actually affect the dielectric confinement46

of excitons and we notice that

overall absorbance (in the spectral range <5eV) of nano particles also comes down with aging. We attribute

this to gradual increase of spatial extension of excitonic wavefunction into surrounding material induced by

14

increased agglomeration. Subsequently, it decreases size dependent oscillator strength21-22, 47

, which further

decreases absorbance in the low energy spectral regions. On the other hand, we notice in figure 5a that the E3

excitonic peak is rather being enhanced with time and is becoming sharper with aging! We understand that

enhancement of spatial delocalization of excitons with aging can decrease the confinement induced

momentum uncertainty. So, lesser numbers of density of states are available for exciton scattering (say

impact ionizations) via confinement induced zero phonon coupling of hot E3 exciton within the extended

bandstructure of PbS. Therefore, the probability of exciton scattering and possibility of impact ionization of

E3 exciton can also come down due to such spatial delocalization 48

. This subsequently decreases the amount

of collisional broadening of E3 exciton with aging. Overall, the observation of such aging behavior

substantiates our claim on the effect of exciton scattering and possibly impact ionizations towards

broadening of optical absorption spectra of E3 exciton. Similar phenomenon, although less pronounced, is

observed in the capped PbS nano-crystallite of mean diameter of 16nm (figure 5b), where we also see aging

induced sharpening of the E3 excitonic peak. The peak position of the corresponding E3 exciton in the 16nm

sample also show slight red shift (figure 5b) during the preliminary phases of aging. However, we do not

notice such red shifts in the peak position in larger uncapped samples due to its quasi-bulk nature.

On the contrary, absorbance of 3nm PbS is not changing much with aging (figure 5c) at all. Unlike,

16nm PbS nano-crystallites, hydrodynamic diameter (3nm) of these smaller PbS nano-crystallites are not

changing with aging either. This can only happen if these smaller nanoparticles are intrinsically charged as

compared to the bigger ones (≥16nm mean diameter, see figure 1b). Similarly charged nanoparticles tend to

repel each other and that can prevent agglomeration of this 3nm PbS. Subsequently, there can be no change

of dielectric confinement in the absence of agglomeration of PbS nano-crystallites and thereby the optical

absorption spectra of 3nm PbS remain unaffected due to the absence of spatial delocalization of exciton

wavefunction. Reader should note that any additional defect formation during aging or incorporation of

15

water49

into the nano-crystallites cannot explain the observed systematic but seemingly opposite variations

of – (a) the E3 excitonic peak and (b) the absorption spectra in lower energy ranges for crystallite sizes from

24nm to 16nm.

F. Temperature Variation of Excitonic Line Width and the Role of Exciton Scattering. In figure

7a, we notice that the E3 exciton broadens slowly with increasing temperature from 20 C to 100 C for the

uncapped quasi bulk sample of mean crystallite diameter of 24nm. As the original E3 excitonic peak is fading

out, another peak slowly emerges ~5.3 eV at the expense of this E3 exciton. This 5.3 eV is the bulk E3 critical

point of PbS! It must be clearly noted that this is not a gradual energy shift of the E3 peak position with

increasing temperature but rather a coexistence of E3 exciton and the bulk like spectral feature from 80 C to

95 C. Thereafter, the E3 exciton vanishes at higher temperatures. However, we again get back the original E3

excitonic peak at 5.9eV (without the additional bulk like spectral feature) once the temperature is reduced to

20 C, except for some aging induced reduction in absorbance and sharpening of the E3 peak. Most strikingly,

there is no temperature related shift of the peak position of E3 exciton. This indicates that events like lattice

dilation or electron-phonon coupling or temperature induced expansion of wavefunction are not responsible

for the evolution of E3 exciton with temperature. Usually, PbS has a positive coefficient of temperature for

band gap variation except in the strong confinement regime, where the reported50

peak position hardly varied

with temperature. In contrast, we cannot use the same reasoning (Ref 50) to explain these observations for

the quasi bulk and uncapped PbS of mean diameter of 24nm! Therefore, we attribute this behavior to the

decrease of impact ionization probability of E3 exciton with increasing temperature. With increasing phonon

scattering, E3 exciton can have much less access to other parts of PbS band structure (less number of final

density of states) for an efficient impact ionization at higher temperatures. Therefore, the emergence of the

bulk like spectral feature ~ 5.3eV with increasing temperature can be ascribed to the demise of the energetic

16

„hot‟ E3 excitons to its quasi bulk E3 band edge via Auger type cooling mechanisms51,52

by transferring its

excess energy mostly to heavier holes (note that me*

mh*at E3

16). Qualitatively similar changes were found

in the temperature dependence of 16nm PbS nano-crystallite where the corresponding E3 excitonic peak

broadens with increasing temperature (figure 7b). Moreover, the peak position of E3 exciton remains

temperature independent in both cases. The observed agglomeration of these uncapped nano-crystallites as

evidenced by DLS (figure 6a) and TEM (figure 6b) studies also point towards their non-charged nature

which may have prevented the reported suppression of Auger Cooling as observed51,53

in charged

nanoparticles. Nano-crystallite diameter of these particles as determined by XRD also remain same which

shows that significant irreversible changes like oxide formation or defect formation are not taking place in

this range of temperature. However, the emergence of bulk like E3 band edge is not so prominently visible in

16 nm samples because of much stronger size confinement and consequently lesser density of available

ground states as compared to uncapped PbS nano-crystallite with mean diameter of 24nm. At this stage, we

also wish to emphasize that line width broadening of E3 exciton with decreasing size and that with increasing

temperature are qualitatively different. We see the emergence of bulk E3 like spectral feature in the

temperature dependence studies of large PbS nano-crystallites only.

It has recently been argued that exciton-exciton scattering54

and impact ionization55,56

is the main

cause of carrier multiplication in semiconductor nanoparticles. This is in line with our above analysis about

the involvement of impact ionization towards collisional broadening of E3 excitonic spectra. We further note

that the temperature (T) coefficient (TCII) of impact ionization not only depends on the threshold energy for

impact ionization (Ei), but also on the mean free path or relaxation path length ( ) of exciton scattering

events as37

. It can be positive or negative depending on the relative balance of these

two terms. Temperature dependence of optical absorption spectra of 3nm PbS nanoparticles follows the

17

literature57

for annealing temperature >40 C, where it undergoes Ostwald ripening and we see that

hydrodynamic size of the particles increases to >275nm. However, there are hardly any irreversible changes

in the optical absorption spectra of E3 exciton till 30 C (figure 7c). This apparent temperature independence

of exciton line width broadening for strongly confined PbS can be ascribed to a set of reasons like – (a)

vanishing of TCII37

due to mutual cancellation of the above two terms, (b) reduction of impact ionization due

to the charged nature of these strongly confined particles, (c) reduction of collisional broadening in the

„quasi‟ ballistic regime, (d) strong surface effects (inelastic scattering at the hetero-interface) at very small

sizes etc. Further work is going on to understand these issues. The connection between the effects of

photocharging58

and/or photoinduced surface trapping59,

60

in smaller nanoparticles with our observation (see

figures 1b and 6a) of size dependence of hydrodynamic size and subsequent aging of this hydrodynamic size

in both capped and uncapped nanoparticles are currently being investigated in more detail.

IV. CONCLUSIONS

In summary, we have studied the broadening of very high energy excitonic feature in the

optical absorption spectra of PbS nano-crystallites. Presence of impact ionization and the effect of extended

band structure of PbS were sought to explain the observed collisional broadening of E3 excitonic absorption

spectra and its observed variation with the size of these nano-crystallites, aging and temperature. We have

also argued why quantum confinement induced collisional broadening can have significant inhomogeneous

component due to the anisotropic nature of exciton-exciton scattering events.

We find that inverse Auger type of mean free path for exciton scattering can be equally important

like the effective Bohr excitonic radius as one of the critical length scales for efficient CM in semiconductor

nanoparticles. Moreover, this concept of effective excitonic Bohr radius is a direct consequence of the

18

dispersive screening of coulomb interactions at high photon energies where the dielectric function of the

semiconductor can have a strong dependence on photon energy. We hope that our experimental analysis in

terms of dispersive (dynamical) screening of „hot‟ excitons in PbS nanoparticles will lead to better

conceptual understanding of the condensed matter physics of exciton generation and similar processes

involving dielectric contribution to coulomb interactions in semiconductors materials and devices. The

analysis of our experimental observations to some extent complements the recent theoretical prediction61

that

the effective coulomb interaction governing the carrier multiplication can be energy dependent.

To explain our experimental observations, we finally predict an intermediate size window for

semiconductor nanoparticles, where impact ionization can dominate over other kinds of exciton relaxation

pathways. Therefore, we suggest size optimization of these nanoparticles to achieve nano-photovoltaic cells

exploiting such carrier multiplications events to improve the power conversion efficiency. Finally, time

resolved studies of CM and quantum yield measurements on various semiconductor nanoparticles of

different sizes are necessary to complement these observations and additional details will be published in

future.

19

FIGURE 1

Figure 1: (a) This shows the XRD spectra of PbS nano-crystallites of different mean crystallite diameters.

Bragg peaks clearly broaden with the reduction in crystallite sizes. (b) Here we compare the mean crystallite

diameter as determined by XRD and the mean hydrodynamic diameter as determined by DLS against the

amount of TG (capping agent) added during the synthesis. The nano-crystallite diameter below which

crystallite sizes are somewhat comparable to hydrodynamic size is marked clearly.

20 30 40 50 60 70 80

(a)

16nm

8 nm

7 nm

6 nm

3 nm(311)(220)(200)

Inte

ns

ity

(a.u

.)

2 (degree)

24nm

18nm

(111)

0 30 60 90 1201

10

100

1000

Dia

me

ter

(nm

)

Amount of TG Added ( L)

DLS

XRD

(b)

16 nm

20

FIGURE 2

Figure 2: Room temperature optical absorption spectra of freshly prepared PbS nano-crystallites. (a) The

plot shows the position of various spectral features ascribed to three different critical point transitions in

uncapped PbS nano-crystallites of mean diameter of 24nm. This necessitates the role of extended band

structure of semiconductor nanoparticles to analyze CM or MEG. (b) This plot demonstrates the variation of

optical absorption spectra with PbS nanoparticles having different mean crystallite diameters as determined

by XRD. The arrow represents the direction of size variation of E3 excitonic peak. Molar concentrations of

PbS were kept same for all particle sizes during the measurements except for 3nm. There the molar

concentration was kept at 50% of the rest to avoid the saturation of optical absorption. The inset shows the

close up of the E3 excitonic peak (> 5.0 eV). It is clearly visible that the E3 excitonic spectra sharpen up for

nano-crystallites with mean diameter either smaller or bigger than 16nm. We also notice that the E3

excitonic spectral shapes for sizes smaller than 16nm are not at all symmetric unlike bigger particles.

Therefore any „single’ Gaussian or Lorentzian or even a Voigt type line shape is not a good fit to these

peaks.

2 3 4 5 6

0.15

0.30

0.45

0.60

(a)

E1

(2.1eV)

E2

(3.5eV)

Ab

so

rba

nc

e

Energy (eV)

E3Exciton

(5.9eV)

24nm

2 3 4 5 60.0

0.5

1.0

1.5

2.0

5.0 5.5 6.0 6.5

Ab

so

rba

nc

e

Energy (eV)

3nm

6nm

7nm

8nm

16nm

18nm

24nm

(b)

21

FIGURE 3

Figure 3: (a) De-convoluted Gaussian component of the E3 excitonic line width increases for crystallite sizes

from 24nm to 16nm. Anisotropic exciton-exciton scatterings are responsible for this large inhomogeneous

line width. However, this line width decreases below 16nm, showing that further broadening is ceased for

smaller particles. Maximum possible energy broadening ( Emax) due to the size distribution of these nano-

crystallites is much smaller than both the estimated homogeneous and inhomogeneous widths of the E3

exciton for all sizes. (b) This shows monotonic blue shifts of E3 excitonic spectra ( = E + 5.3eV) with

decreasing mean crystallite diameter of PbS nanoparticles. No single power law behavior can describe the

variation for all sizes. The size dependence of the energy position of peak absorbance is

also very much slower than 1/R2

behavior.

0 5 10 15 20 25

0.0

0.1

0.2

0.3

0.4 (a)

=7

=17

Gaussian

Bro

ad

en

ing

wid

th(e

V)

Diameter (nm)

5 10 15 20 25

6.0

6.1

6.2

6.3

Diameter(nm)En

erg

y @

Pe

ak

Ab

so

rba

nc

e (

eV

)

(b)

y=a x

= -0.053

= -0.076

22

FIGURE 4

Figure 4 : This plot demonstrates the comparison of exciton mean free path values calculated using ε = 17 and

ε = 7. In case of ε =17, the calculated mean free path for exciton-exciton scattering event is >30nm. In that

case, any impact ionization is not possible in any PbS nanoparticles under our investigation due to the ballistic

nature of the exciton scattering. The solid lines are just a guide to the eye only.

0 5 10 15 20 25

15

20

25

30

3540

Me

an

fre

e p

ath

(n

m)

Diameter (nm)

~ 17

~ 7

23

FIGURE 5

Figure 5: Aqueous dispersion of PbS nanoparticles were not sonicated and deliberately allowed to age to get

better physical insights of the spectral origin of E3 exciton. (a) Aging behavior of room temperature

absorption spectra of uncapped PbS nano-crystallites with mean diameter of 24nm. The variation of

absorbance of E3 exciton is characteristically opposite to the portion of the spectra at photon energies lower

than 5eV. The abrupt changes between 3eV to 4eV are due to instrumental artifacts for lamp changes etc at

small absorbance. This artifact is absent in similar samples having large absorbance due to higher

concentration (figure 7a) of the PbS nano-crystallites. (b) Aging of TG capped PbS nano-crystallites of mean

diameter 16nm. Evidently we see qualitatively similar kind of sharpening of E3 excitonic peak with aging. (c)

The absorption spectra for strongly confined PbS nano-particle with mean diameter 3 nm hardly changes

with aging as compared to that of figure 5a and 5b.

2 3 4 5 6

0.02

0.04

0.06

0.08 (a)

Ab

so

rba

nc

e

Energy (eV)

Fresh

50 mins

100 mins

200 mins

400 mins

12 hrs

24nm

2 3 4 5 60.0

0.5

1.0

1.5

2.0

16nm

(b)

Ab

so

rba

nc

e

Energy (eV)

Fresh

100 mins

150 mins

250 mins

450 mins

12 hrs

2 3 4 5 60

1

2

3

4(c)

Ab

so

rba

nc

e

Energy (eV)

Fresh

1 hr

2 hrs

4 hrs

5 hrs

24 hrs

3 nm

24

FIGURE 6

Figure 6: (a) This shows DLS results of 16nm PbS nano-crystallites during different stages of aging. These

nano-crystallites agglomerate and the hydrodynamic size increase to around a micron. (b) This shows

Transmission Electron Microscopy (TEM) image which reveals the presence of granular agglomerate of ≤

20nm PbS nano-crystallites. Samples are not sonicated deliberately to preserve the agglomerate structure.

This clearly says that aging did not increase the crystallite size for bigger nano-crystallites.

10 100 1000Particle size (nm)

fresh

Siz

e D

istr

ibu

tio

n %

(a) 16nm

1hr

2hrs

3hrs

5hrs

24 hrs

25

FIGURE 7

Figure 7: (a) Temperature variation of uncapped PbS nano-crystallite with mean diameter of 24nm. The E3

bulk edge (~5.3 eV) appears at the expense of E3 exciton and „coexist‟ for temperatures >80 C. E3 excitonic

feature nearly vanishes by 100 C but reversibly recovers around 20 C during cooling. (b) Almost similar but

less pronounced spectral changes with temperature are also observed for 16nm PbS during cooling. (c) 3 nm

PbS shows relative less temperature dependence upto 30 C. The arrows indicate the direction of temperature

variation in all three graphs.

2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

1.2 (a)

E3 Bulk edge

E3 Exciton

Ab

so

rba

nc

e

Energy (eV)

20 oC

60 oC

80 oC

90 oC

95 oC

100 oC

20 oC

cooling

24nm

2 3 4 5 60.0

0.3

0.6

0.9

1.2

1.5

1.8

Ab

so

rba

nc

e

Energy (eV)

20oC

60oC

80oC

100oC

20oC cooling

(b)

16 nm

2 3 4 5 60.0

0.5

1.0

1.5

2.0

2.5

3.0

Ab

so

rba

nc

e

Energy (eV)

20o C

25o C

30o C

20o C Cooling

(c)

3nm

26

TABLE I.

Samples ; Nano-Crystallite

Diameter (nm)

Exciton Binding Energy

EEx (eV)

assuming ε = 17

Exciton Binding Energy

EEx (eV)

assuming ε = 7

NO TG PbS ; 24 nm 0.013 0.030

20 TG PbS ; 18 nm 0.017 0.040

40 TG PbS ; 16 nm 0.020 0.045

60 TG PbS ; 8 nm 0.038 0.090

80 TG PbS ; 7 nm 0.044 0.103

100 TG PbS ; 6 nm 0.051 0.120

120 TG PbS ; 3 nm 0.102 0.240

27

Acknowledgements: Authors want to thank Dept. of Science and Technology, India for DST Nano

Unit grant SR/NM/NS-42/2009 and IISER-Pune for their support. We thank National Chemical Laboratory,

India for allowing us to get the TEM image. Authors also wish to thank Dr. Sandip Ghosh, Prof. K. L.

Narasimhan, Dr. G. V. Pavan Kumar, Dr. Harsh Chaturvedi, Dr. Pankaj Mondal, Dr. Shivaprasad Patil for

valuable discussions and Prof. K. N. Ganesh for encouragements and support.

28

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