sp2âsp3-Hybridized Atomic Domains Determine Optical Features of

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Sp–Sp-Hybridized Atomic Domains Determine Optical Features of Carbon Dots Nikita V. Tepliakov, Evgeny V. Kundelev, Pavel D Khavlyuk, Yuan Xiong, Mikhail Yu. Leonov, Weiren Zhu, Alexander V. Baranov, Anatoly V. Fedorov, Andrey L. Rogach, and Ivan D. Rukhlenko ACS Nano, Just Accepted Manuscript • DOI: 10.1021/acsnano.9b05444 • Publication Date (Web): 14 Aug 2019 Downloaded from pubs.acs.org on August 15, 2019

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Sp2Sp3-Hybridized Atomic Domains Determine Optical Features of Carbon Dots ∗,†

Nikita V. Tepliakov,

Evgeny V. Kundelev,

Mikhail Yu. Leonov,

Anatoly V. Fedorov,

†Information

†

†

Weiren Zhu,

†

†

Pavel D. Khavlyuk,

∗,¶

Andrey L. Rogach,

Yuan Xiong,

‡

†

Alexander V. Baranov,

‡,†

and Ivan D. Rukhlenko

∗,§,†

Optical Technologies Center, ITMO University, Saint Petersburg 197101, Russia

‡Department

of Materials Science and Engineering and Centre for Functional Photonics

(CFP), City University of Hong Kong, Kowloon, Hong Kong SAR, China

¶Department

of Electronic Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

§Institute

of Photonics and Optical Science (IPOS), School of Physics, The University of Sydney, Camperdown 2006, NSW, Australia

E-mail: [email protected]; [email protected]; [email protected]

Abstract Carbon dots (CDots) are a promising biocompatible nanoscale source of light, yet the origin of their emission remains under debate. Here, we show that all the distinctive optical properties of CDots, including the giant Stokes shift of photoluminescence and the strong dependence of emission color on excitation wavelength, can be explained by the linear optical response of the partially sp2 -hybridized carbon domains located on the surface of the CDots' sp3 -hybridized amorphous cores. Using a simple quantum 1

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chemical approach, we show that the domain hybridization factor determines the localization of electrons and the electronic bandgap inside the domains, and analyze how the distribution of this factor aects the emission properties of CDots. Our calculation data fully agree with the experimental optical properties of CDots, conrming the overall theoretical picture underlying the model. It is also demonstrated that fabrication of CDots with large hybridization factors of carbon domains shifts their emission to the red side of the visible spectrum, without a need to modify the size or shape of the CDots. Our theoretical model provides a useful tool for experimentalists and may lead to extending the applications of CDots in biophysics, optoelectronics, and photovoltaics.

Keywords carbon domains, hybridization, absorption, photoluminescence, quantum chemistry, extended Hückler method

Carbon nanostructures have always been an important topical subject of material science, owing to their exceptional optical and electronic properties and the availability of precursor materials.

1,2

Fullerenes,

3

nanotubes,

4

nanodiamonds,

5

nanographite,

6

and graphene

79

this famous list has been recently replenished with three nanoscale objects: graphene quantum dots (GQDs),

10

carbon dots (CDots),

11

and polymer carbon dots (PCDs).

12

Struc-

turally, CDots and PCDs are spherical or nearly spherical carbon nanoparticles whose surface is stabilized by some functional groups.

12,13

In contrast to traditional quantum dots (QDs)

based on inorganic semiconductors such as CdSe, CDots and PCDs possess low toxicity, high biocompatibility and low production costs, which make them a promising substitute for QDs in biological applications.

14,15

Furthermore, a broad range of useful optical fea-

tures renders CDots and PCDs equally appealing for photovoltaics, optoelectronics,

20,21

and chiral sensing.

22

2


16,17

photocatalysis,

18,19

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While CDots and PCDs have similar optical features and are often synthesized from the same precursors, the synthetic routes to them are dierent and so are the obtained nanoparticles. The synthesis of PCDs can be carried out at room temperature and typically takes a short time, precursors.

23

producing nanoparticles that predominantly consist of polymerized

As a consequence, the optical properties of PCDs are mainly determined by

non-conjugated polymeric chains.

24

CDots on the other hand are synthesized for several

◦ 25 hours at about 200 C, which leads to the carbonization of the initial polymeric chains and the formation of conjugated carbon structures.

26

Despite a great deal of recent research activities in the CDots eld, the physical mechanism behind their photoluminescence (PL) is still under debate. similarity of CDots to GQDs

28,29

24,27

Several studies assumed

while ignoring the fact that they do not exhibit the size-

quantization eect, which is a signature characteristic of QDs. The absence of size quantization clearly indicates that the light is emitted by localized optical centers inside CDots rather than by their entire volume. An alternative explanation assumed that the emission of CDots comes from the functional molecular groups attached to their surface.

11

This hypothesis is

supported by the dependence of the CDots' luminescence on surface oxidization.

30

Yet, it

remains unclear if the light is emitted by the surface groups themselves or they aect the luminescence by altering the internal structure of the CDots. After some TEM images revealed the presence of crystalline layers inside CDots, also been argued that the emission of CDots may originate from sp

2

31

it has

crystalline domains.

Although the exact form and appearance of these domains is still vague, they are often envisioned as individual uorescent aromatic molecules or their dimers.

32

For example, it

was demonstrated that a system of three types of polycyclic aromatic hydrocarbon (PAH) molecules embedded in a polymer matrix can mimic the Stokes shift, width, and excitation energy dependence of the CDots' PL.

31

However, because such molecules cannot exist in

the free form inside the solid cores of CDots, the true nature of the emission centers is still vague.

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Here we address this important issue by building a simple and intuitive, yet comprehensive and accurate semi-analytical model of optical centers in CDots.

We describe CDots

3 as polymer-like structures with sp -hybridized amorphous carbon cores that contain small 2 domains of partially sp -hybridized carbon atoms.

We show that the energy gap in the

2 domain's electronic spectrum narrows down linearly with the sp -hybridization factor, due to the delocalization of the valence electrons over the entire domain area.

By assuming

that the hybridization factor is distributed according to the Boltzmann law, we successfully describe the inhomogeneous broadening of the absorption and PL spectra of the CDots, as well as the excitation energy dependence of the emission. The comparison of our theoretical and experimental data shows that the developed model is accurate in predicting the giant Stokes shift of the PL, which is attributed to the strong charge separation in the rst excited state of the domains.

Most importantly, our model predicts that the emission of CDots

can be shifted to the long-wavelength side of the visible spectrum by increasing the number

2 of strongly sp -hybridized domains inside the CDots' cores.

Thus, by reproducing all the

characteristic experimental data on CDots, our model provides a strong evidence that the

2 optical properties of CDots are mainly determined by the partially sp -hybridized carbon 3 domains of their sp -hybridized amorphous cores.

Results and Discussion In order to consistently describe the optical response of CDots, we assume that it is fully

2 determined by the quantum states of the sp -hybridized atomic domains that are naturally formed on the amorphous cores of CDots during their synthesis. These domains are much more likely to form on the surface of CDots, as the formation energy is lower at the interface with vacuum. The schematic of the CDot's core is shown in Fig. 1a. The core (represented

3 by the grey area) is a polymer-like structure made of the sp -hybridized amorphous carbon 2 which contains small sp -hybridized atomic domains of various geometries. We do not model

4


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(a)

(c)

(b)

109°

120°

η= 0

η= 1 sp2 hybrydization

sp3 hybrydization

Figure 1: (a) Schematic representation of a core of CDot and its small domains made of fully 3 2 (b) sp - and (c) sp -hybridized carbon atoms. Blue and red spheres are C atoms (colored for better visibility) and white spheres are H atoms. The angle between the CC bonds is ◦ ◦ 109.5 in (b) and 120 in (c).

the internal structure of the amorphous core while assuming that it does not contribute to the optical properties of CDots in the visible spectral range. It is signicant that the domains (represented by pairs of coupled polycyclic aromatic molecules) are not like dimers of at aromatic systems, because of their non-planar geometry and the close proximity of the atomic layers. The spacing of the domain layers is much smaller than the characteristic distance between the layers of a molecular dimer, due to the inclusion of the domains into the rigid core of the CDot.

3 2 Figures 1b and 1c visualize small domains of carbon atoms with the sp - and sp hybridizations.

Each domain is represented by a pair of coupled pyrene molecules (com-

posed of four benzene rings) constrained by hydrogen atoms. The valence electrons of the

3 sp -hybridized carbon atoms are localized at the stable are approximately

109.5◦ .

σ -bonds,

the angles between which

Because of the wide gap in the energy spectrum of such electrons,

they can only interact with the high-energy UV irradiation and are insensitive to visible light.

2 In sharp contrast to this, the sp -hybridization implies that the valence electrons are highly delocalized over the entire domain area and have a strong response to visible irradiation.

2 The CC bonds within each atomic layer of the sp -hybridized domain form

120◦

angles.

2 The carbon domains are unlikely to exist in the pure sp -hybridized form as shown in

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2 Fig. 1c. Their constituting atoms are most probably hybridized into an intermediate sp 3 sp -conguration, while their valence electrons are partially delocalized over the domain area and partially localized on the chemical bonds. To account for this partial hybridization, we

2 introduce the sp -hybridization factor Apart from

η,

η

of carbon atoms inside the domains

(0 6 η 6 1).

the optical properties of CDots also depend on the lateral dimensions of

the partially hybridized domains. Domains with larger lateral dimensions (e.g., of the size of perylene molecules) absorb and luminesce at longer wavelengths, just like single PAH molecules.

33

We obtained the best agreement between our theory and experiment using the

pyrene-sized domains shown in Fig. 1, which indicates that these domains may be prevailing in CDots; this allows us to use a single tting parameter obtained

η

in our model. Note that CDots

via other synthetic routes may have domains of dierent lateral dimensions.

The energy spectra and wave functions of carbon domains are calculated using the extended Hückel formalism outlined in section Methods. Figure 2a shows the energies of the highest occupied molecular orbital (HOMO) and of the lowest unoccupied molecular orbital (LUMO) of the domains as functions of

η.

One can see that the energy of the LUMO weakly

depends on the hybridization factor whereas the energy of the HOMO steeply increases with it, due to the delocalization of electrons over the domain area.

Therefore, the delocaliza-

2 tion of electrons reduces the HOMOLUMO energy gap inside the sp -hybridized domain as compared to the surrounding matrix (see Fig. 2b), making the entire structure sensitive to visible light. The HOMOLUMO energy gap scales almost linearly with

Egap ≈ 3.1 − 1.8 η

(eV).

η

as

(1)

2 This linear dependency slightly diverges from the simulation data for weak sp -hybridizations (with

η . 0.1).

The divergence results from the replacement of the amorphous carbon matrix

surrounding the atomic domains by the bounding hydrogen atoms. The domain electrons should be strongly aected by the matrix and have a much larger energy gap, which is close

6


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-8

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

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(a) Egap

-9

LUMO

3

HOMO

2

-10 -11 0

(b)

HOMO-LUMO gap

1 20 40 60 80 100 2 sp hybridization degree (%)

0 0

20 40 60 80 100 sp hybridization degree (%) 2

Figure 2: (a) Energies of HOMO (red) and LUMO (blue) states and (b) HOMOLUMO 2 energy gap as functions of the sp -hybridization factor of carbon domains. Solid lines are the linear ts to the numerically calculated values shown by open circles.

to that of the core material. As a consequence, the domains with

η . 0.1

do not contribute

to the optical response of CDots and are ignored in the further analysis. One of the key distinctive features of CDots is their giant Stokes shift of PL, which can reach hundreds of meV. Our model elegantly explains this shift by the reconguration of the electron density occurring upon the excitation of the carbon domains (see Fig. 3). We rst note that the electron density of the HOMO state shown in Fig. 3a is predominantly localized at the domain center. As the LUMO state in Fig. 3b is excited, this density is transferred to the outer layers of the domain, leaving a positively charged vacancy in the space between the domain layers. The charge separation results in the strong Coulomb attraction of the layers and creates a compressing stress.

Relaxation of this stress reduces the distance between

2 the domain layers and increases the sp hybridization factor

η

of the domain. This in turn

narrows the HOMOLUMO energy gap of the domain (see eq 1) in the rst excited state, which is eventually observed as the giant wavelength dierence between the maxima of the absorption and emission spectra. The displacement

∆h of the domain layers can be estimated using a simple parallel-plate

capacitor model shown in Fig. 3c. The two layers of the excited domain are represented by the external capacitor plates of negative surface charge density

σ− = −e/(2S)

each (S is

the eective area of the domain) whereas the vacancy between the layers is modelled by a

7


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(a)

Page 8 of 25

(b)

(c)

P

σ–

S

σ+

h

σ–

HOMO

P

LUMO

2 Figure 3: (a) HOMO and (b) LUMO states of a fully sp -hybridized

(η = 1)

carbon domain

and (c) three-plate capacitor model used to estimate the displacement of the domain layers upon the redistribution of the electron density.

positively charged plate of surface density

σ+ = +e/S .

The domain is thus exposed to stress

P = −2π(σ+ + σ− )σ− =

πe2 , 2S 2

(2)

which causes strain

P πe2 ∆h = = , h E 2S 2 E where

(3)

h is the distance between the negatively charged layers and E

is the Young's modulus

of the domain in the direction perpendicular to the layers. Assuming for value between the Young's moduli of diamond (∼

100

GPa, we get

∆h/h ∼ 0.1

for

S = 0.4 × 0.4

E

an intermediate

1000 GPa) and graphite (∼ 10 GPa), E =

2 nm . Hence, the redistribution of the electric

charge accompanying the optical excitation of the domain compresses it by approximately 10%, resulting in a similar change of the hybridization factor.

The compression of this

magnitude corresponds to the experimentally observed giant red shift of PL of the order of hundreds of meV. Due to the high rigidity of the domain, the dynamic compression should occur at very short times it can possibly explain the relaxation channel with a time constant of 400 fs, revealed in CDots by an ultrafast time-resolved study. The optical spectra of CDots are often inhomogeneously broadened.

34

35

This fact is ex-

plained in our model by the variation of the hybridization of the atomic domains inside the CDots. If

ρ(η) is the probability distribution of the hybridization factor, then the absorption

8


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spectrum of CDots is given by

Z

1

A(λ) =

A(λ, η)ρ(η)dη,

(4)

0

where the absorption spectrum

A(λ, η)

of an individual domain of hybridization

η

is calcu-

lated according to eq 14 derived in section Methods. Note that the probability distribution

ρ(η)

includes variations of the domain geometries both between dierent dots and within

individual nanoparticles.

The PL spectrum of CDots excited at wavelength

λ0

is dened

similarly as

λ0 PL(λ, λ0 ) = λ where the spectral lineshape function

Z

1

Γ(λ, η)A(λ0 , η)ρ(η)dη,

(5)

0

Γ(λ, η)

takes into account the red shift estimated by

eq 3. Figure 4a compares the typical absorption and emission spectra of CDots (synthesized according to the standard procedure

25

outlined in section Methods) with the respective the-

oretical ttings obtained using eqs 4 and 5. The ttings assume that the hybridization factor obeys the Boltzmann distribution valence electron energies with

η

ρ(η) ∝ e−10η , which is justied by the linear growth of the

[see eq 1]. The absorption of CDots has a pronounced peak

A1 at 340 nm and a long-wavelength tail A2 centered around 450 nm. Peak A1 is produced

2 by the domains of low sp -hybridizations,

η . 40%, which prevail in the CDot ensemble.

Tail

A2 originates from the rest of the carbon domains whose absorption edges are signicantly red-shifted due to the higher hybridization factors. Although such domains are much less abundant, their contribution is still considerable, because the oscillator strengths of optical transitions grow with

η

as the electrons become more and more delocalized over the domain

area. This is clearly illustrated by measured and calculated PL lifetimes (Fig. 4b), which decrease with both the excitation and emission wavelengths, indicating that the oscillator strength of strongly hybridized domains are larger. Note that our simple quantum-chemical method cannot predict accurate values of PL lifetimes, but it reproduces well a general

9


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(a)

25

Abs (Th) PL (Th) Abs (Exp) PL (Exp)

E1

0.5

0 300

400 500 Wavelength (nm)

15

320 nm (Th) 405 nm (Th) 320 nm (Exp) 405 nm (Exp)

10 5

E2

A2

(b)

20 Lifetime (ns)

A1

1.0 Intensity (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

Page 10 of 25

600

0

400

500 600 Emission wavelength (nm)

Figure 4: (a) Absorption (Abs) and photoluminescence (PL) spectra of CDots with Boltzmann distribution of hybridization factors of carbon domains. Experimental data and their theoretical ts are shown by open circles and solid curves, respectively. The PL is excited at the center of the absorption peak A1, at

λ0 = 340

nm. (b) Measured and calculated PL

lifetimes vs. the emission wavelength upon excitation at 320 and 405 nm.

trend. The PL spectrum has a pronounced peak E1 at 450 nm and a much broader tail E2 at 550 nm, which are directly related to the respective features of the absorption spectrum. As expected, peak E1 is strongly red-shifted with respect to peak A1 by about 110 nm. The absorption and emission spectra in Fig. 4a have signicant inhomogeneous broadening.

While the broadening of the absorption peak is solely due to the variation of the

domain hybridization within the CDot ensemble, the broadening of the PL peak is also contributed by its inhomogeneous red shift due to the dependence of the domain's rigidity on the hybridization factor.

Another notable feature of the theoretical PL spectrum is that

it lies above the experimental one at longer wavelengths, which is due to the fact that the PL quenching is not taken into account in our model. Even though the radiative decay rate increases with

η , the PL quantum yield of the domains reduces with the hybridization factor,

as the nonradiative relaxation is also enhanced in highly-hybridized domains.

Our model

cannot predict the absolute value of quantum yield of CDots, because it does not take into account the eect of nonradiative decay(s) in CDots. These may result either from imperfections on the CDots surface, or from the internal structure of the CDot core. Schneider

10


et

1.0 (a)

Theory

0.8

340 nm 380 nm 400 nm 420 nm

0.4

0

Energy gap

e

0.6

(c)

0.2

(nm)

340 nm 380 nm 400 nm 420 nm

e

0.6

e

Experiment

0.4 0.2 0.0

400

500 600 Wavelength (nm)

700

Emission wavelength,

1.0 (b) 0.8

Bandgap Excited state

e 0 e

0.0

PL intensity (a.u.)

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PL intensity (a.u.)

Page 11 of 25

sp2 hybridization 600 (d)

Experiment Theory

550 500 450 300 350 400 450 500 Excitation wavelength, 0 (nm)

Figure 5: (a) Theoretical and (b) experimental PL spectra of CDots for dierent excitation wavelengths, (c) energy-level diagram illustrating the origin of the excitation wavelength dependence of the PL peak's position, and (d) PL emission wavelength of excitation wavelength

λ0 .

λe

as a function

The gradient in (c) represents the relative number of carbon

domains with dierent hybridization factors.

al measured the PL quantum yield in the citric-acid-based CDots modelled here, and found it to be 53% upon excitation at 320 nm.

36

It should be emphasized that despite the overlap of the absorption tail A2 with the emission peak E1, the strongly hybridized domains forming tail A2 do not contribute to peak E1, emitting at longer wavelengths (Figure 4a). Similarly, the domains whose emission forms peak E1 have lower hybridization factors and larger energy gaps, thus absorbing at peak A1. The described feature is responsible for arguably the most distinctive characteristic of CDots the dependence of their emission color on the excitation wavelength. Theoretical and experimental PL spectra demonstrating this eect are shown in Figs. 5a and 5b.

One can see that the increase of the excitation wavelength from 340 to 420 nm

notably weakens the PL peak and shifts it to longer wavelengths by nearly 50 nm.

11


The

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Page 12 of 25

origin of the excitation wavelength dependence of the peak's position is explained by the energy level diagram in Fig. 5c. The solid line in the diagram shows the absorption bandgap of carbon domains approximated by eq 1 while the dashed line corresponds to the energy of the excited domains with the redistributed electron density. One can see that the high energy light that can excite all the domains of the CDot is mainly absorbed by the prevailing domains with small

η s,

resulting in short emission wavelengths and high PL intensities. In

2 contrast, the low energy light exciting much less abundant and highly sp -hybridized atomic domains is emitted by them at longer wavelengths.

This results in the red-shifting and

weakening of the PL signal with the excitation wavelength. The central wavelength of the PL peak (emission wavelength,

λe )

is plotted as the func-

tion of the excitation wavelength in Fig. 5d. The position of the peak is seen to be almost independent of the excitation wavelength when the CDots are excited by the UV radiation shorter than 400 nm. This cuto wavelength corresponds to the energy gap of almost com-

3 pletely sp -hybridized domains (Egap

≈ 3.1

eV for

η 1,

see eq 1).

The irradiation of

shorter wavelengths excites the quasi-continuous spectrum of the carbon domains and does not aect the shape of the PL spectrum. theory and experiment for

λ0 & 440

quenching in the domains with large

Much like before, the discrepancy between the

nm is a consequence of not taking into account the PL

ηs

in our model.

One of the key eorts of the contemporary CDots' research has been the intensication of their emission on the red side of the visible spectrum.

2830,3739

According to the

present model, this can be achieved by producing CDots with a large number of strongly

2 sp -hybridized atomic domains. Figure 6 compares the absorption and PL spectra of CDots with two distributions of the hybridization factor: the previously used Boltzmann distribution (monotonously decaying with

η)

and the Gaussian distribution

2

ρ(η) ∝ e−(10η−6)

centered at a relatively high hybridization factor of 60%. One can see that the increase in

2 the number of strongly sp -hybridized domains shifts the fundamental absorption peak from 340 to 630 n,m and the main emission peak from 450 to 725 nm. The PL shift of this magni-

12


Page 13 of 25

(a)

0.5 Abs

0

(b) e

1.0 Intensity (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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350

e

PL

450

550 650 750 Wavelength (nm)

850

Figure 6: Absorption and PL spectra of CDots with (a) Boltzmann and (b) Gaussian distri2 butions of the sp -hybridization factors. The PL spectra are excited at (a) 340 nm and (b) 640 nm.

tude has been recently demonstrated by Li

et al. 37 It was realized through the modication

of the CDots' surface with oxygen/sulfur-rich molecules, which led to the distortion of the surface carbon layers and a decreased bandgap of the domains at the surface. Recently we have also predicted that CDots' PL can be shifted to red by surface functionalization with NH2 groups conjugating with the

π -network

of carbon domains.

40

As a concluding remark we would like to reiterate that our model of CDots agrees well with the experimental data (see Figs. 4 and 5), indicating that most of the optical features

2 of CDots are bound to the partially sp -hybridized domains of their cores.

The model

provides the basis for further research aimed at designing CDots with highly ecient PL in various regions of the visible and infrared spectra for applications in such elds as bioimaging, biomedicine, and optoelectronics. It can also be used for studying the dynamics of elementary excitations in CDots and the eect of doping on the optical spectra of CDots, which will be reported in our forthcoming publications.

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Conclusions We have developed a rst semi-analytical model of CDots, treating them as polymer-like

2 nanoparticles of amorphous carbon with embedded partially sp -hybridized atomic domains, which determine the optical response of the CDots. Electrons in such domains are partially delocalized over the domain area while maintaining strong coupling to the amorphous host matrix.

It was shown that the delocalization of the electrons increases with the domain

hybridization factor, leading to the linear narrowing of the electronic bandgap, and is accompanied by the growth of the optical transition rates. By comparing the predictions of the model with experimental data, we showed that the broad distribution of the domain hybridization factors is responsible for the inhomogeneous broadening of the absorption and PL spectra of CDots and for the strong dependence of their emission color on the excitation wavelength.

The rst excited state of the carbon domains is characterized by the spatial

separation of the electric charge density, which reduces the electronic bandgap and causes a large red shift of PL. Finally, it was shown that by altering the distribution of the domains' hybridizations, one can shift the emission of CDots to the red side of the optical spectrum, which is desired for a variety of applications.

20

Our simple analytical model thus describes

all the main optical features of CDots and oers a useful tool for both interpreting the experimental data and guiding further experiments on CDots.

Methods Materials.

Citric acid, ethylenediamine, butanol were purchased from Sigma-Aldrich. Iso-

propanol, carbon tetrachloride, xylene were purchased from ECOS-1 JSC, Russia. Methanol, chloroform, toluene were purchased from Vekton CJSC, Russia. Acetonitrile and hexane were purchased from Cryochrom LLC, Russia. Ethanol was purchased from Luks, Russia. All chemical reagents were used as received. Ultrapure water (Milli-Q water) was used throughout the experiments.

14


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ACS Nano

Synthesis of CDots.

The CDots used in this study were synthesized according to the

previously reported procedure.

25

First, citric acid (1.05 g) and ethanediamine (335

µL) were

dissolved in distilled water (10 mL) to form a transparent solution. Then, the solution was transferred into a poly(p -phenol)-lined stainless steel autoclave for solvothermal reaction. After heating treatment at 200

◦

C for 5 h and cooling down to room temperature, the

colloidal solution was used as obtained.

Energy levels and molecular orbitals of CDots.

We calculated the energy levels and

2 molecular orbitals of the partially sp -hybridized domains using the extended Hückel method (EHM). This method only takes into account the valence shells of atoms and ignores the closed shells, which do not form chemical bonds. The full wave functions of electrons in the carbon domains,

ϕ(r),

are assumed to be linear combinations of the valence atomic orbitals

χi (r), ϕ(r) =

N X

ci χi (r),

(6)

i=1 where the summation is taken over all Let

ˆ H

N

valence electrons of the constituting atoms.

be the Hamiltonian in the Hartree-Fock approximation, describing interaction of

each electron with all the atomic cores and other valence electrons in the domain. The full energy of an electron is then given by

E=

Replacing

ϕ

ˆ hϕ|H|ϕi . hϕ|ϕi

(7)

by the sum of atomic orbitals, we get

P i,j ci cj Hij E= P , i,j ci cj Sij where

ˆ ji Hij = hχi |H|χ

and

Sij = hχi |χj i.

combination of atomic orbitals. Equating

(8)

This energy must be minimal for a given linear

N

derivatives

15

∂E/∂ci


to zero, we get a system of

ACS Nano 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

N

Page 16 of 25

Fock equations

N X

(Hij − ESij )cj = 0,

(9)

j=1 which generally yields

N

dierent eigensolutions for the energies and wave functions of the

molecular orbitals. The diagonal matrix element

Hii

of the Hamiltonian is the ionization energy of the

orbital in the isolated atoms. For H and C atoms we take

Hii (H1s ) = −13.60

eV,

ith

Hii (C2s ) =

−24.38 eV, and Hii (C2p ) = −11.26 eV. The nondiagonal matrix elements (resonant integrals) are approximated as

Hij = K where 1.75.

K

Hii + Hjj Sij 2

(i 6= j),

(10)

is the phenomenological parameter whose value normally lies between 1.00 and

41

Although the EHM does not explicitly take into account the spins of electrons, it can be readily modied to allow for the spin in the rst approximation. The ground state of the carbon domain is a singlet state produced by the interaction of and

N/2

N/2 electrons with spins `up'

electrons with spins `down'. Strictly speaking, matrix elements

Hij

and

Sij

both

depend on whether the interacting orbitals have the same or dierent spin orientations. To keep the treatment simple, we assume that the resonant integrals the relative spins and are given by eq 10, where functions and constant

K

Sij

Hij

are independent of

is the overlapping of the spatial wave

is chosen such as to best describe the averaged interaction between

the valence electrons of arbitrary spins. We take

K = 1,

which yields the best match with

experimental data. The overlap integrals of the full wave functions vanish for opposite electron spins and equal to the overlap integral

Sij

of the spatial parts of the wave functions if the electrons

spins are the same. The lowest energy of a singlet state is achieved if the overlapping of the full atomic orbitals is minimal, in which case one can simplify the system of Fock equations to the form

P

j (Hij −Eδij )cj

= 0.

Energies

E

and coecients

16


cj

are then simply the eigenvalues

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ACS Nano

and eigenvectors of

Hij .

The overlap integrals

Sij

are directly calculated from the domain geometry and the form

of atomic orbitals. We approximate the latter using the Slater-type orbitals (STOs) of the form

χnlm (r) = (2ζ)n+1/2 [(2n)!]−1/2 rn−1 e−ζr Ylm (θ, φ), where and

ζ = Z/n, Z

Ylm (θ, φ)

is the eective nuclei charge,

(11)

n is the eective principal quantum number,

is the spherical harmonic. Note that even though the atomic orbitals chosen

are just an approximation to the carbon atoms, they behave correctly at the large distance from the nuclei.

And since only the tails of the wave functions contribute to the overlap

integrals, the STOs are well suited for our calculations. The exponential factors for H and C atoms are

ζ(H1s ) = 1.000

and

ζ(C2s ) = ζ(C2p ) = 1.625.

In order to consider an arbitrary sp

2

hybridization of C atoms, we transform the AOs in

eq 11 to their linear combinations as follows. First, we replace the spherical harmonics for

m = 0, ±1

with the standard

|Px i, |Py i,

and

|Pz i

orbitals directed along the respective

Cartesian axes. We then introduce four hybridized orbitals are the linear combinations of the three

|P i

Y1m

|F1 i, |F2 i, |F3 i,

orbitals and one

|Si

and

state with

|F4 i,

l = 0.

which These

hybridized orbitals are given by

|F1 i =

|F2 i = |F3 i =

where

p

p p p 1 − 2/3 csc2 α|Si + 2/3|Px i + 2/3 cot α|Pz i,

p 1 1 1 − 2/3 csc2 α|Si − √ |Px i + √ |Py i + 2/3 cot α|Pz i, 6 2

p p 1 1 1 − 2/3 csc2 α|Si − √ |Px i − √ |Py i + 2/3 cot α|Pz i, 6 2 p √ |F4 i = − 2 cot α|Si + 1 − 2 cot2 α|Pz i,

α = 109◦ − 19◦ η

is the angle between orbital

In the limiting case of complete sp

2

hybridization,

17

|F4 i

η = 1,

(12a)

(12b)

(12c)

(12d)

and the other hybridized orbitals. we have


α = 90◦

and orbitals

|F1 i,

ACS Nano 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

|F2 i,

and

|F3 i

lie in the

xy -plane,

while orbital

opposite limiting case of complete sp

3

Page 18 of 25

|F4 i = |Pz i

hybridization,

is perpendicular to them. In the

η = 0,

we get

α ≈ 109◦

hybridized orbitals form a tetrapod. Finally, we calculate the overlap matrix

and the four

Sij

using the

hybridized orbitals, substitute the overlap matrix into eq 10, and diagonalize Hamiltonian

Hij

2 to obtain energies and MOs of the partially sp -hybridized atomic domains.

Absorption and emission spectra of CDots.

Since carbon domains are much smaller

than the wavelength of light, their absorption and emission can be described in the dipole approximation. Electronic transition

fβα =

where

me

α→β

is characterized by the oscillator strength

X 2 me |hϕβ |ξ|ϕα i|2 , (E − E ) β α 3 ~2 ξ=x,y,z

(13)

is the free-electron mass. The absorption spectrum of the domains with hybridiza-

tion degree

η

is then given by

A(λ, η) ∝

X

fβα Γβα (λ),

(14)

β,α

where

Γβα (λ)

states

α

is the transition lineshape and the summation is taken over all the occupied

and free states

β.

The time-dependent PL spectrum is calculated as

λ0 I(λ, λ0 , t) = λ where

k(η)

Z

1

Γ(λ, η)A(λ0 , η)ρ(η)e−k(η)t dη,

is the radiative decay rate at the respective domains. The PL lifetime

calculating by tting

I(λ, λ0 , t)

(15)

0

with a single exponent

τ

is then

I0 e−t/τ .

Acknowledgements The authors thank the Ministry of Science and Higher Education of the Russian Federation for its Project 16.8981.2017/8.9 and Grant 14.Y26.31.0028, National Natural Science Foun-

18


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2019,

accepted on

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J. Chem. Phys.

1963, 39,

Page 25 of 25

Graphical TOC Entry LUMO Gap

Energy

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HOMO

sp2 hybridization

25


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