Phenomenological theory of optical broadening in zero-dimensional systems applied tosilicon nanocrystals
V. V. Nikolaev, N. S. Averkiev, and Minoru Fujii
Citation: Applied Physics Letters 108, 153107 (2016); doi: 10.1063/1.4946862
View online: http://dx.doi.org/10.1063/1.4946862
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APPLIED PHYSICS LETTERS 108, 153107 (2016)

Phenomenological theory of optical broadening in zero-dimensional systems
applied to silicon nanocrystals
V. V. Nikolaev,1,a) N. S. Averkiev,1 and Minoru Fujii2
1

Ioffe Institute, St. Petersburg 194021, Russia
Department of Electrical and Electronic Engineering, Graduate School of Engineering, Kobe University,
Kobe 657-8501, Japan
2(Received 10 March 2016; accepted 4 April 2016; published online 12 April 2016)
We develop a phenomenological theory of inhomogeneous broadening in zero-dimensional systems
and apply it to study photoluminescence (PL) spectra of silicon nanocrystals measured at helium
and room temperatures. The proposed approach allowed us to explain experimentally observed PL
peak asymmetry, linear dependence of the peak width on its maximum, and anomalous alteration ofspectral characteristics with temperature increase. Published by AIP Publishing.
[http://dx.doi.org/10.1063/1.4946862]

Silicon nanocrystals (Si-NCs) embedded in a silicon
dioxide matrix have been actively studied due to their potential for future-generation opto-electronics and photovoltaics.1
One of the advantageous features of Si-NCs is a significant
dependence of optical properties on the nanocrystal size,
which can provide tunability across a wide optical range. At
the same time, this leads to demand for Si-NC size control2–4
for device applications.
A typical PL spectrum of a Si-NC ensemble exhibits a
broad peak with the width between 100 and 400 meV even athelium temperatures,5–7 whereas single Si-NCs at low temperatures demonstrate a narrow few-meV-wide emission
lines.8 It may be expected that the PL peak broadening isrelated to the NC size dispersion7,9,10 through the size dependence of the optical-transition energies. However, finding
the exact relation between NC size and optical-transitionenergy is a difficult task due to complex structural propertiesof Si-NCs in silicon dioxide surrounding.11,12 Additionally,
the mechanism of the radiative recombination in Si-NCs is
not yet thoroughly understood.12–15
In this letter, we develop a general phenomenologicaltheory of inhomogeneous broadening in zero-dimensional
systems which does not rely on a particular microscopic
model of choice, and we apply it to interpret our experimental results on Si-NC ensembles.
Let us suppose that there is a one-to-one correspondence
between the energy E of a particular optical transition andNC size parameter d, which is represented by a smooth function E ¼ f(d). If the NC size dispersion in an ensemble isdescribed by a distribution function P(d), then the density of
optical transitions (DoOT) per energy interval q(E) is proportional to Ref. 9 1 df 1 ðEÞ;(1)qðEÞ / P f ðEÞ dE 
where d ¼ f 1 ðEÞ is the inverse of E ¼ f(d).
The function f(d) is defined by structural and material
properties of nano-objects and is generally unknown. Here,a)valia.nikolaev@gmail.com

0003-6951/2016/108(15)/153107/4/$30.00

similarly to our previous work,16 we presume that in a certain interval of the NC size variation, the microscopic size
dependence can be approximated by a power function13 (PF)E ¼ f ðd Þ ¼

Aþ E~g ;
dc

(2)

where A > 0, c > 0 and E~g are the phenomenological parameters. In this letter, we consider size dispersion described by
the asymmetric log-normal (LN) distribution9,17,18
   2!
1
1
d
:
(3)
PLN ðd Þ ¼ pﬃﬃﬃﬃﬃﬃ exp  2 ln 
2r
d
dr 2p
Here d is the median of the distribution,and r defines
thepﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ﬃ
relative standard deviation dd=dav ¼ exp ðr2 Þ  1 of thesize parameter d from its average value dav ¼ d exp ðr2 =2Þ.
Observation of LN distributions has been reported in manysystems of nano-objects,18 including ensembles of SiNCs.2–4,9,19–21
Substitution of Eqs. (2) and (3) into RHS of Eq. (1) produces a new normalized asymmetric distribution (E > E~g )0"!# 2 1~E  Eg1A
exp@ 2 ln
2v
E  E~g
PPFLN ðEÞ ¼;(4)
pﬃﬃﬃﬃﬃﬃ
v E  E~g 2p
where v ¼ cr; E is the median of the transition-energy distribution directly connected to the median size d by means of ¼ A=dc þ E~g .
PF Eq. (2): E ¼ f ðdÞ
Similarly to the size-dispersion parameter r in the LN
distribution in Eq. (3), v defines the asymmetry of the peak
and the relative standard deviation of the “effective quantized energy” e ¼ E  E~g from its average value eav ¼ Eav
 E~g through the expression de=eav ¼ dE=ðEav  E~g Þ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ¼ expðv2 Þ  1. Here, Eav is the average transition energy.
The full-width at half-maximum (FWHM) of the peaked
distribution function PPF–LN can be expressed in an analytical form

108, 153107-1

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153107-2

Nikolaev, Averkiev, and Fujii

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
DE ¼ 2ðEmax  E~g Þshðv 2 lnð2ÞÞ;Appl. Phys. Lett. 108, 153107 (2016)(5)

where Emax is the peak energy (mode) of the PF–LN distribution. As can be seen from this expression, the peak broadens with the increase of the parameter v. Since v is the
product of r (which characterizes the relative width of a size
distribution) and the exponent c, we naturally arrive at theconclusion that the width and the asymmetry of the DoOT
peaks depend on the size dispersion as well as on the character of the transition-energy dependence on size. We note that
PPF–LN(E) in Eq. (4) approaches the Gaussian distribution
when v ! 0.
Eq. (5) can be rewritten in the following form:
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ADE ¼ 2 exp v2 sh v 2 lnð2Þ c ;d(6)

to reveal that, at constant level of size dispersion set by r,
the FWHM of an inhomogeneously broadened optical transi
tion will increase with the decrease of the median NC size d,and the character of FWHM dependence on size will follow
the power law described by the same exponent c which
defines the transition energy dependence.
We argue that the DoOT in the form of Eq. (4) can beused to model the experimental PL spectra of silicon nanocrystals. Indeed, if the redistribution of electrons and holes
between NCs is prohibited by high potential barriers, variations of the pump-radiation absorption and internal quantum
efficiency22 across an ensemble are slow then the PL intensity at a certain photon energy will be defined by the numberof NCs with the related optical-transition energy, i.e., PL
spectra will replicate the shape of the DoOT. In this work,
we neglect the dependence of the internal quantum efficiency
on size justifying it by the results of experimental investigations which show that, despite the strong reduction of theradiative lifetime with the NC size decrease evident from thetime-resolved experiments,6,22 internal quantum efficiency
varies moderately, especially in the long wavelength range
for the samples with relatively large NCs.22
In Fig. 1, PL spectra measured at helium and room temperatures (RT) are displayed alongside with the diameter
distributions in five Si-NC/SiO2 samples under investigation.
The PL experiments were conducted at a low excitation
power density of 4 mW/cm2, in order to avoid NC groundlevel saturation and related effects. Experimental histograms,
obtained by high-resolution transmission electron microscopy (TEM), were fitted by scaled LN distributions (seeFig. 1(c)); the size-dispersion parameter r was found close
to be rTEM  0.2 in all five samples. Information on thesample preparation and experimental setup can be found in
Ref. 6.
The measured temperature dependence of PL decay
time for these samples6 was successfully described by the
two-level Calcott’s model.23 The model assumes that PL at
low temperatures originates from a ground “semi-dark” exciton level, which is predominantly spin-triplet with moderate
admixture of spin-singlet character facilitated by the spin–orbit interaction. Our measurements at 4 K (Fig. 1(a)) show anadditional peak associated with Pb centers at Si/SiO2 interface.24 At RT, PL is dominated by recombination of an

FIG. 1. Measured and calculated PL spectra at T ¼ 4 K (a) and T ¼ 295 K(b) from five samples with different NC size distributions (shown in (c)).
Spectra are normalised to their maximum intensities; dashed curves in (b)show modelled RT spectra without homogeneous broadening.

excited “semi-bright” exciton level which is split from“semi-dark” level by the electron-hole exchange interaction23 and has mainly spin-singlet character. The main PL
peaks at both temperatures demonstrate significant degree ofasymmetry alongside with the blueshift and linewidth broadening with the decrease of NC size.
Measurements on single Si-NCs show8 that at cryogenic
temperatures, the homogeneous broadening of an emission
line is less than 3 meV, and therefore at T ¼ 4 K, it can besafely neglected compared to peak widths exceeding
100 meV observed in our experiments. In this case, Eq. (4)and related equations can be used directly to model experimental data at T ¼ 4 K, provided the assumptions listed
above are valid.
Figure 2(a) shows the FWHM DE of the main PL peaks
from Figs. 1(a) and 1(b) as functions of peak energies Emax.
Experimental points lay close to straight lines. Analysis of Eq.
(5) reveals that linear dependence can take place when PF in
Eq. (2) describes transition energies in all five samples, andNC diameters are distributed log-normally with an approximately equal value of the size-dispersion parameter r. The
values of phenomenological parameters for T ¼ 4 K E~g4 K¼ 1:02660:013 eV and v4 K ¼ 0.31 6 0.02 are obtained
straightforwardly by fitting the linear dependence (Eq. (5)) to
the experimental data. These values were used to model the v4 K ; E~g4 K ; E
PL spectra with the scaled functions PPFLN ðE;¼ hxÞ (Eq. (4)), where median energies E (which are close to
the PL peak energies) are found by fitting. In Fig. 1(a), a good
agreement between calculated and experimental data can be

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153107-3

Nikolaev, Averkiev, and Fujii

FIG. 2. ((a) and (b)) PL FWHM measured at T ¼ 4 K (blue circles),T ¼ 295 K (red squares), and extracted inhomogeneous linewidths for
T ¼ 295 K (red triangles) as functions of PL peak energy Emax (a) and NC
median diameter d (b) (note log-log scale in (b)). (c) Energy shift of the PL
peak Emax with the temperature increase from 4 K to 295 K as a function of
Emax at 4 K. Lines in ((a)–(c)) represent modelling.

seen, with the modelled spectra perfectly replicating the asymmetry and the linewidth variation of the measured PL peaks.
In Fig. 2(b), FWHM values are plotted as functions of
the NC median diameter. According to our model (see Eq.
(6)), this dependence should be close to linear in log-log
scale, and the slope of the line gives the value of the exponent (c4 K ¼ 1.4 6 0.2 for T ¼ 4 K). The extracted parameters
allow us to estimate the level of size dispersion from PL
spectra alone r ¼ v4 K/c4 K  0.22, approximately matching
TEM results. Additionally, the dependence of the retrieved
median effective quantized energies e ¼ E  E~g on median
diameter, plotted in log-log scale (not shown), is approximately linear, which is in accordance with Eq. (2); theobtained absolute value of the slope 1.4 6 0.2 equals theprevious result for c4 K. These findings suggest that the presented theoretical approach is consistent with the experimental results at T ¼ 4 K.
The retrieved value of exponent c4 K agrees well withc ¼ 1.39 calculated by Delerue et al.13 The effective bandgap
E~g4 K  1:025 eV is smaller than the bandgap of crystalline
Si (1.17 eV), which is possibly partly due to an effectivehydrostatic pressure25 related to the NC surface,19 the energyof an emitted optical phonon,26 and the bulk-exciton binding
energy. In addition, E~g4 K may differ from the value of thematerial bandgap, because the formula (Eq. (2)) approximates the transition energy dependence only in a limited
diameter range.
Comparing the PL spectra measured at 4 K and 295 K,
one may find some unexpected effects. First, the FWHM
growth with the increase of PL peak energy (Fig. 2(a)) or
decrease of NC size (Fig. 2(b)) is less steep at RT. More surprisingly, the two samples with the smallest NCs (d ¼ 4:2

Appl. Phys. Lett. 108, 153107 (2016)

and 4.7 nm) demonstrate PL peak narrowing with theincrease of the temperature (see Fig. 2(b)). The shift of the
PL peak energy with the temperature increase is plotted inFig. 2(c) as a function of the peak energy at 4 K. One can seethat the two samples with smallest PL peak energies (andlargest median NC diameters d ¼ 6:8 and 9.1 nm) exhibit
blueshift with the temperature increase. This is a counterintuitive behavior, since the corresponding Si bandgap
reduction is approximately 50 meV (shown by a dashed line
in Fig. 2(c)). Saturation of NC ground levels at 4 K cannot
serve as an explanation for such a blueshift, since it has the
opposite effect.27
An attempt to directly apply the approach used for
T ¼ 4 K to the experimental results at T ¼ 295 K leads to thefollowing contradiction. Fitting the linear dependence (Eq. (5))
to the experimental FWHM in Fig. 2(a) gives vFW ¼ 0.18
6 0.01 and an unexpectedly low value for the effective
bandgap EgFW ¼ 0.75 6 0.03 eV; application of Eq. (6) to themeasured data (red squares in Fig. 2(b)) produces the value of
exponent cFW ¼ 0.6 6 0.08 leading to the estimate of the sizedispersion parameter r ¼ vFW/cFW  0.3, which exceeds theexperimentally obtained rTEM  0.2 by half.
This contradiction occurred because homogeneous
broadening of Si-NC optical transitions was not accounted
for by the analysis above. Experiments show8,28 that single
Si-NCs exhibit quite large PL linewidths (100–150 meV) at
RT. Manifestation of homogeneous broadening of more than
100 meV was also seen in ensemble investigations.26
To account for homogeneous broadening, we convolute
DoOT given by PPF–LN(E) (Eq. (4)) with the Lorentzianfunction LðEÞ ¼ 2Chom =½pð4E2 þ C2hom Þ, where Chom is the
FWHM of a single-NC transition. It was found that the best
fit to experimental data can be achieved if Chom ¼ 130 meV
is used, which is in accordance with the reported measurments.28 The parameters v295 K  0.195 and E~g295 K 0:98 eV were chosen to match the measured RT FWHM
in Fig. 2(a). The resulting spectra simulations are shown in
Fig. 1(b), demonstrating good agreement with the experiment. Slight overestimation of PL intensity in the lowphoton-energy region might be due to the deviation of theactual size dependence from PF (Eq. (2)) for larger NCs.
The value of the exponent c295 K ¼ 0.99 6 0.13 can bederived from the dependence of the obtained inhomogeneouspart of the linewidth on NC size (Fig. 2(b), red triangles).
The size dependence of the median effective quantized energies e ¼ E  E~g295 K (not shown) produces the same value of
exponent. The ratio v295 K/c295 K  0.2 coincides with rTEM,
which shows that our model is consistent.
Now, an explanation to the anomalous PL linewidth
narrowing and blueshift with temperature increase can beproposed. With the elevation of the temperature from 4 K to
295 K, the dominant luminescence channel switches fromrecombination of semi-dark lower exciton states to uppersemi-bright states, and the energy dependence on the size ofsemi-bright states at RT differs from that of semi-dark statesat 4 K. Both dependencies can be approximated by Eq. (2)but with remarkably different parameters. Effective bandgap
E~g shrinks from 1.025 eV to 0.98 eV, which is in line with
the Si bandgap reduction by 50 meV. The other two phenomenological parameters change as follows: c4 K  1.4 toReuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 133.30.106.15 On: Fri, 29 Apr 2016
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153107-4

Nikolaev, Averkiev, and Fujii

c295 K  1.0 and A4 K  3.3 eV nm1.4 to A295 K  1.8 eV nm(values of A are obtained by fitting Eq. (6) to experimental
data in Fig. 2(b)).
Decrease in the value of the exponent c means that thevariation of the optical-transition energy with the NC size
becomes slower, which leads to the reduction of the inhomogeneous broadening with temperature increase (compare
blue and dashed red lines in Figs. 2(a) and 2(b)). The thermal
broadening of a single-NC transition leads to the increase of
the total peak width by 70–80 meV, which partly compensates the decrease of inhomogeneous broadening. For samples with small NC sizes, the decrease of inhomogeneous
broadening is most pronounced, and the homogeneous
broadening is unable to compensate it, resulting in the total
linewidth narrowing with temperature elevation.
We calculated the difference between PL peak maxima
at 4 K and 295 K using phenomenological parameters listed
above (continuous red line in Fig. 2(c)); the size-dispersion
parameter was set to r ¼ 0.2. One can see that the modelling
reproduces the anomalous blueshift for structures with larger
NC diameters. The origins of this effect are as follows. First,
the semi-bright state at 295 K approaches bandgap with thesize increase more slowly than the semi-dark state at 4 K,
which is due to c reduction. At certain interval of (comparatively large) NC diameters, semi-bright transition energies at
T ¼ 295 slightly exceed semi-dark transition energies at 4 K
despite the reduced RT effective bandgap E~g295 K (45 meVsmaller than E~g4 K ). Second, enhancement of thermal broadening blueshifts the PL peak by 5–10 meV.
Significant reduction of c with temperature rise indicatesthat physical properties of Si-NCs in SiO2 matrix strongly
differ from those of homogeneous crystalline Si. The mechanism that leads to c alteration may be due to complex effects
which the temperature elevation imposes on stress profiles,29
the transition region between a Si core and SiO2 matrix11,12
or defect/surface states,14 still needs to be clarified. It should
be noted that, according to experimental observations, forsmaller Si-NCs in SiO2 matrix, the blueshift with the size
reduction is quenched and the PL peak energy normally does
not exceed 1.8–2.0 eV. From this, one may conclude that thedivergent PF (Eq. (2)) will not adequately describe NCs withsmall diameters, and therefore, our method is not applicableto such samples.
In conclusion, employing log-normal size distribution,17,18 we developed a concise phenomenological theoryof inhomogeneous broadening in zero-dimensional systems
which contains few fitting parameters. Obtained analyticalexpressions allow for relatively straightforward interpretation of experimental results. The proposed approach clarifies

Appl. Phys. Lett. 108, 153107 (2016)

the measured line shape and linewidth variations of the PL
spectra of silicon nanocrystals. We explain anomalous PLline narrowing and blueshift with temperature increase bycompetition between opposite variations of homogeneousand inhomogeneous broadening and by modification of thetransition-energy size dependence.
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