# Temperature induced giant shift of phonon energy in epitaxial boron nitride
layers

Jakub Iwański<EMAIL_ADDRESS>Piotr Tatarczak  Mateusz Tokarczyk
Grzegorz Kowalski
Aleksandra K. Dąbrowska  Johannes Binder  Roman Stępniewski  Andrzej Wysmołek
Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw,
Poland

###### Abstract

The recent progress in the growth of large-area boron nitride epilayers opens
up new possibilities for future applications. However, it remains largely
unclear how weakly attached two-dimensional BN layers interact with their
substrate and how their properties are influenced by defects. In this work, we
investigate hBN layers grown by Metal Organic Vapor Phase Epitaxy (MOVPE)
using Fourier-transform Infrared (FTIR) spectroscopy in the temperature range
of 160-540 K. Our measurements reveal strong differences in the character of
layer-substrate interaction for as-grown and delaminated epitaxial layers. A
much weaker interaction of as-grown layers is explained by wrinkles formation
that reduces strain at the layer-substrate interface, which for layers
transferred to other substrates occurs only in a limited temperature range.
The most striking result is the observation of a giant increase in the
$E_{1u}$ phonon energy of up to $\sim 6$ cm-1 in a narrow temperature range.
We show that the amplitude and temperature range of the anomaly is strongly
modified by UV light illumination. The observed giant effect is explained in
terms of strain generation resulting from charge redistribution between
shallow traps and different defects, which can be interpreted as a result of
strong electron-phonon coupling in hBN. The observed narrow temperature range
of the anomaly indicates that the effect may be further enhanced for example
by electrostrictive effects, expected for sp2 boron nitride.

epitaxial boron nitride, hBN, FTIR, MOVPE, $E_{1u}$ phonon, defect charged
states

††preprint: APS/123-QED

##  I INTRODUCTION

Hexagonal boron nitride (hBN) is a wide bandgap layered material [1] with
strong sp2 covalent bonds in-plane and weak van der Waals bonds between
adjacent layers [2]. An additional asset of hBN is the high thermal and
chemical stability [3, 4, 5, 6]. These properties make hBN an important
component in many novel van der Waals heterostructures [7, 8, 9, 10] or as an
encapsulating layer that increases the durability of other less durable
materials such as perovskites [11] or transition metal dichalcogenides [12,
13].

However, the effective usage of many systems and microdevices in everyday life
is dependent on the possibility to be scaled up. The solution for this problem
are epitaxial techniques which enable few-inch area layer growth. In case of
hBN the structural and optical quality of the epitaxial layers is still
inferior to those of better-studied flakes exfoliated from the bulk crystal.
The difference between epitaxial and bulk hBN is also observed in the
interaction with other materials [14, 15]. Hence, in order to combine
epitaxial hBN layers with other materials it is important to understand the
different properties of defects in epitaxial hBN and the mechanisms of their
creation as well as the rules that govern interactions between epitaxial hBN
and other materials.

In this work, we present $E_{1u}$ hBN phonon energy-temperature dependences
measured with Fourier-transform infrared (FTIR) spectroscopy. We measured
delaminated and as-grown epitaxial layers. For as-grown samples characteristic
wrinkles are created during cooling after growth, because of the relaxation of
the accumulated strain energy caused by the negative thermal expansion
coefficient of hBN [16]. We revealed a significant reduction in the layer-
substrate interaction, which is a consequence of wrinkles creation. In
addition, we observe an anomalous, giant upshift of phonon energy that occurs
in a narrow temperature range. This effect is explained in terms of charge
redistribution of hBN defects. The observed effects highlight that defects in
hBN play not only an important role for electrical conductivity, mid-gap
photoluminescence or single photon emission, but can also strongly affect the
phonon modes in a complex manner.

##  II METHODS

###  II.1 Samples

The boron nitride samples used in this work were obtained with Metal Organic
Vapor Phase Epitaxy (MOVPE) using an Aixtron CCS 3$\times$2” system. Two-inch
sapphire wafers were used as substrates, and ammonia and triethylboron (TEB)
were used as precursors of nitrogen and boron, respectively. In most
processes, hydrogen was used as a carrier gas, but a few samples were obtained
using a nitrogen atmosphere. Thin epitaxial boron nitride samples (from atomic
monolayer to 70 nm), oriented parallel to the sapphire surface, were obtained
by three growth regimes: Continuous Flow Growth [17, 18], Flow Modulation
Epitaxy [18, 19] and Two-stage Epitaxy [20]. Due to a pre-growth nitridation
step a thin AlN layer is formed at the sapphire/hBN interface. For
measurements, we prepared $\sim 8\times 8$ mm pieces of investigated hBN
layers. We are aware that our samples may consist of boron and nitrogen atoms
bonded with sp2 bonding in different phases such as hexagonal (hBN),
rhombohedral (rBN), turbostratic (tBN), Bernal stacking (bBN) [20, 21].
However, to simplify the notation in the present work, we will use hBN having
in mind all layered boron nitride phases.

For temperature-dependent FTIR measurements, we chose a 41 nm thick hBN layer
grown in Flow Modulation Epitaxy regime and a 20 nm thick hBN layer grown in
Two-stage Epitaxy regime. The thicker layer was also delaminated and deposited
onto a new sapphire substrate to obtain information about the layer-substrate
interaction. The delamination procedure is already described in the literature
[22, 18].

###  II.2 Experimental details

X-ray measurements were performed using a Panalytical X’pert diffractometer
equipped with a Cu sealed X-ray tube and a parallel beam Bragg X-ray mirror.
For all samples, X-ray reflectometry (XRR) measurements were taken
($\omega/2\theta$ scans). To obtain the thickness of the hBN layer from the
XRR measurements, the PANalytical X’Pert Reflectivity software package was
used. This program makes use of the recursive Parratt formalism for
reflectivity [23, 24, 25]. In the fitting procedure, the thickness, density,
and roughness of the hBN and AlN layers were taken into account, as well as
the roughness of the sapphire substrate. It should be pointed out that in the
calculations only a one-dimensional model is considered.

A Thermo Scientific Nicolet Continu$\mu$m Infrared Microscope with a 32x
Schwarzschild infinity corrected objective (NA 0.65) was used for FTIR
reflection measurements. All samples presented were measured at 3 points
approximately 3 mm apart with an illuminated area of 70$\times$70 $\mu$m. The
spectra were collected in the range of 650 to 4000 cm-1 with a resolution of
0.964 cm-1 for samples thinner than 4 nm and 0.482 cm-1 for the rest of the
samples. The light beam was perpendicular to the surface of the sample.

A Linkam THMS350EV cryostat was used for temperature dependence measurements.
The real temperature value was measured with a thermocouple soldered to the
sample surface with indium. The calibration allowed for a range of possible
temperatures between 157-538 K. Before each temperature-dependent measurement,
the sample was measured at room temperature on 6 different locations to be
sure that the sample is isotropic.

For UV sample irradiation, we used an EQ-99X LDLS lamp emitting light in the
range 170-2500 nm.

###  II.3 Characterization method

To characterize hBN layers, we simulate the whole FTIR spectrum. To do so, we
assume that the material is a medium composed of harmonic damped oscillators.
This allows us to introduce Dynamic Dielectric Function (DDF). In the function
there are parameters such as oscillator self-energy $\omega_{0}$ and its
damping parameter $\gamma$ which provide us with information about the phonon
properties in the material. In our analysis method, we consider a thin layer
on a thick substrate, also described by its DDF. This approach also allows to
describe heterostrucutres that consist of several layers or materials. It is
possible because of the phase difference analysis that comes from the
interference of light beams reflecting on the subsequent interfaces. This
method allows to obtain information about the thickness of the layer $d$ and
interface roughness $\sigma$ considered for $d>100$ nm. All the details about
the characterization method are available in the Supplementary Materials.

Figure 1:  FTIR spectra measured at room temperature for three different samples ($S_{1},S_{2},S_{3}$) of epitaxial hBN layers on a Al2O3 substrate (circles) with modeled curves (lines). The inset shows zoom of the hBN $E_{1u}$ vibrational mode peak. Table 1:  Best fit parameters of curves modeled for FTIR spectra in FIG. 1 Sample |  $\omega_{BN}$ (cm-1) |  $\gamma_{BN}$ (cm-1) |  $d_{BN}$ (nm) |  $\omega_{sapph}$ (cm-1) |  $\gamma_{sapph}$ (cm-1)
---|---|---|---|---|---
$S_{1}$ | 1367.69 | 22.60 | 8.47 | 917.45 | 37.18
$S_{2}$ | 1367.15 | 20.25 | 15.66 | 917.58 | 38.75
$S_{3}$ | 1368.66 | 22.40 | 60.28 | 918.69 | 33.72

In FIG. 1 we present an exemplary spectrum modelling for three different
samples. The samples were grown using three different growth conditions that
resulted in different material properties (i.e. strain, amount of defects,
thickness). The modelling works well for each of the samples. The values of
the modelled parameters are presented in TAB. 1. The peak position at about
1367 cm-1 corresponds to the $E_{1u}$ phonon mode of sp2 hybridized boron
nitride [26]. The high quality is manifested by the values of the
$\gamma_{BN}$ parameter for the $E_{1u}$ mode, which are relatively small
compared to the values reported for epitaxial layers [18, 27, 28, 29]. The
inset in FIG. 1 shows that the $E_{1u}$ peak intensity directly depends on the
thickness of hBN. The high reflectivity below 1000 cm-1 is connected to the
presence of sapphire substrate [30]. One can see that the modeled curves in
FIG. 1 describe experimental data almost entirely. All important features of
the spectrum are reproduced and the values in TAB. 1 are very precisely
determined with relative error values $\simeq 0,01\%$ for $\omega_{BN}$ and
$\simeq 1\%$ for $\gamma_{BN}$ and $d_{BN}$. This precision enables to detect
smallest deviations in strain, quality and thickness of epitaxial layers.
Furthermore, due to modeling parameters related to sapphire ($\omega_{sapph}$,
$\gamma_{sapph}$), one can monitor the quality of the substrate on which hBN
was grown or deposited.

The principal advantage of the described method over the very basic peak
Lorentzian fitting is the access to the physical properties of the sample,
even if the sample structure is complex. The absorption shape modified by the
damping parameter $\gamma$ which influences the peak position or layer
thickness is out of range for simple fitting. The thickness results coming
from our method are in very good agreement with the results obtained with XRR
for the samples in the range 2-70 nm (see Supplementary Materials FIG. S1).
The method can be applied for samples in the thickness range from a single
atomic layer to several $\mu$m (see Supplementary Materials FIG. S2).

##  III RESULTS AND DISCUSSION

Figure 2:  hBN phonon energy as a function of temperature. Grey spheres
represent data for bulk material obtained by Segura et al. [31]. a) thinner
lines represent contribution to the general trend that comes from lattice
thermal expansion ($\Delta\omega_{L}$) (yellow), anharmonic phonon processes
($\Delta\omega_{AN}$) (pink) and material-substrate interaction
($\Delta\omega_{S}$) (brown) considered for non-bulk samples. The thicker dark
red line is described by the introduced model and does not include material-
substrate interaction. b) blue and green spheres represent data for as-grown
and delaminated layer respectively. The dashed dark red line is created by
adding a constant offset to the modeled curve for bulk material. The dashed
brown line is created by adding offset and $\Delta\omega_{S}$ contribution to
the modeled curve for bulk material. The pictures in the insets schematically
show the appearance of as-grown (wrinkled) and delaminated (flat) layers.

Since the phonon energy is a good indicator for strain [28], we investigated
as-grown and delaminated samples. Due to the delamination process, both
samples should exhibit different layer-substrate interactions. The
interactions change with temperature because of the difference in the thermal
expansion coefficients of the layer and the substrate. As we provided a
reliable method for the identification of the phonon energy of hBN, it is
possible to analyze the temperature dependencies of the lattice vibrations of
the epitaxial layers $\omega_{BN}(T)$. As we present in FIG. 2, the phonon
energy for as-grown and delaminated layers follows different trends. To be
able to extract information about layer-substrate interactions, we need to
know what influences this $\omega_{BN}(T)$ dependency. It is governed by 4
dominant factors: the undisturbed phonon energy at 0 K $\omega_{0}$, the
lattice constant change (thermal expansion) $\Delta\omega_{L}$, the anharmonic
phonon decay $\Delta\omega_{AN}$, the layer-substrate interaction
$\Delta\omega_{S}$ caused by the difference in thermal expansion coefficients
of layer and substrate:

$\omega_{BN}(T)=\omega_{0}+\Delta\omega_{L}(T)+\Delta\omega_{AN}(T)+\Delta\omega_{S}(T)$
(1)

Taking the hBN thermal expansion coefficient provided by Yates et al. [16],
and using it for the Density Functional Theory (DFT) calculations as presented
by Cusco et al. [32], one can obtain the energy change due to the thermal
expansion coefficient:

$\Delta\omega_{L}(T)=-2.347+2.137\times 10^{-2}T-1.177\times 10^{-5}T^{2}$ (2)

According to DFT calculations, the $E_{1u}$ phonon has two dominant decay
paths: three-phonon process and four-phonon process [31]. Both give the
following contribution to $\Delta\omega_{AN}$ [33]:

$\displaystyle\Delta\omega_{AN}(T)$ $\displaystyle=$ $\displaystyle
A\left(1+\frac{1}{e^{\theta_{1}/T}-1}+\frac{1}{e^{(\theta_{0}-\theta_{1})/T}-1}\right)+$
(3) $\displaystyle+$ $\displaystyle
B\left(1+\frac{2}{e^{\theta_{1}/T}-1}+\frac{1}{e^{(\theta_{0}-2\theta_{1})/T}-1}+\frac{2}{\left(e^{\theta_{1}/T}-1\right)(e^{(\theta_{0}-2\theta_{1})/T)}-1)}+\frac{2}{(e^{\theta_{1}/T}-1)^{2}}\right)$

where $A,B$ \- constants are related to three- and four-phonon processes
respectively, $\theta_{i}=(hc/k)\omega_{i}$ ($h$ \- Planck constant, $c$ \-
speed of light, $k$ \- Boltzman constant, $\omega_{i}$ \- phonon energy). In
Eq. 3 we assumed that in both three- and four-phonon processes, there is the
same energy decay product. This assumption is justified since in both decays
the produced lower energy phonons are along the $M$-$K$ line in the dispersion
relation, which is flat. The change related to the layer interaction with the
substrate can be described with the following formula:

$\Delta\omega_{S}(T)=\beta_{BN}\int_{T_{r}}^{T}\alpha_{sapphire}(T^{\prime})-\alpha_{BN}(T^{\prime})dT^{\prime}$
(4)

where $\alpha_{sapphire,BN}$ is the thermal expansion coefficient for sapphire
and BN, $\beta_{BN}$ is the hBN Grüneisen parameter and $T_{r}$ is the
temperature at which the layer on the substrate is relaxed. For the as-grown
layer, it is the growth temperature, and for the delaminated layer it is room
temperature. All the described contributions of $\omega_{BN}(T)$ dependency
are plotted in FIG. 2a.

Using the model that does not include the layer-substrate interaction, we can
fit the line that follows experimental data for bulk material reported by
Segura et al. [31]. This gives the following values of parameters:
$\omega_{0}=1376$ cm-1, $A=-7.25$ cm-1, $B=-1.19$ cm-1 and $\theta_{1}=369.78$
K. It corresponds to the decay to two phonons of energies 257 cm-1 and 1119
cm-1 in the three-phonon process, two phonons of energy 257 cm-1 and one
phonon of energy 862 cm-1 in four-phonon process. These results are in good
agreement with the literature values [26, 32].

In FIG. 2b we present the dependence of $\omega_{BN}(T)$ for the epitaxial
layers. Both sets of results were collected for samples grown in the same
MOVPE process. However, one of them was delaminated and transferred to a new
sapphire substrate. The method used for delamination and transfer is as
described in the literature [22]. The blue spheres related to the as-grown
sample follow the same trend as the data for bulk material, which illustrates
a dark red dashed line. The line is drawn by adding a constant offset to the
modeled curve for bulk hBN. The shift towards higher energies is due to a
higher amount of defects in the epitaxial layer, which leads to additional
strain. This additional effect does not allow for reliable parameters modeling
because of an unknown temperature-dependent behaviour of defects. It is
suggested that the defect contribution should be linear [28]. Adding a further
linear or quadratic term to the model would strongly influence other
parameters which at this point are in good agreement with the literature
values. We later show that defects indeed play an important role and that the
assumption of a linear defect contribution is not always justified. For this
reason, we provide only qualitative results.

The green spheres related to the delaminated sample exhibit a more significant
decrease in phonon energy with increasing temperature. This decrease is most
likely caused by a stronger layer-substrate interaction, as suggested by the
brown dashed line which was created by adding $\Delta\omega_{S}$ term to the
dark red dashed line. To cover the experimental data in a as wide as possible
range, we assumed the hBN Grüneisen parameter $\beta_{BN}=-21$ cm-1/%. This
value is over two times lower than reported for hBN before [34]. The main
reason for the change of layer-substrate interaction $\Delta\omega_{S}$ before
and after delamination is the presence and lack of characteristic wrinkles.
They are created during the cooling of the material after the growth process
and can be removed in the delamination process [22]. The results presented in
FIG. 2b imply that in the as-grown layer, the wrinkles can compensate for the
layer-substrate strain. Consequently, the blue spheres run parallel to the
modeled curve for the bulk material, as the dark red dashed line suggests. The
strain in the case of the as-grown epitaxial layer is eliminated by excess
material stored in the wrinkles. The same behaviour is observed for the
delaminated layer below room temperature at which the delamination process
occurred. When cooling the relaxed delaminated layer, the wrinkles can be
created and compensate for the layer-substrate strain. Consequently, the
$\omega_{BN}(T)$ trend below room temperature is similar for both delaminated
and as-grown layers. Equally, the effect of the decrease in phonon energy
related to $\Delta\omega_{S}$ contribution for as-grown layers could be
observed at much higher temperatures, which are closer to the growth
temperature at which the wrinkles should disappear.

Figure 3:  $E_{1u}$ phonon energy as a function of temperature for the 20 nm
thick sample grown in the Two-stage Epitaxy regime. Arrows indicate the
anomalous behaviour. The initial four temperature-dependent measurements lead
to a softening of the anomalous phonon energy shift. After UV irradiation the
anomalous effect is enhanced. Lines are guide to the eye only.

The most striking result that emerges from our measurements is a giant,
anomalous shift of the $E_{1u}$ phonon energy which we observed for some
samples. In FIG. 3 we present the $\omega_{BN}(T)$ dependence for a 20 nm
thick layer grown in the Two-stage Epitaxy regime. One can notice an anomalous
increase of phonon energy at 425 K for the 1st measurement. This increase
cannot be explained by the presented model. When performing consecutive
temperature-dependent measurements, the anomaly softens and its maximum
slightly moves towards lower temperatures as presented for 2nd and 3rd
measurement. In the 4th temperature run, no anomalous phonon behaviour is
observed. To make sure that the effect is not an artifact of our data analysis
method, we performed basic peak Lorentzian fitting on the measured data set.
In this approach, the anomaly is still observed at the same temperatures (see
Supplementary Materials).

We also observed the described anomalous behavior of $\omega_{BN}(T)$
dependence for other samples grown under different regimes under NH3-rich
conditions before and after the delamination process (see Supplementary
Materials). The anomaly occurrence for different samples implies that the
phonon upshift is not an accidental effect related to one specific sample, but
can be reproduced when a special set of growth parameters is applied. The
observation of the effect occurrence for both as-grown and delaminated layers
proves that it does not come from the substrate or thin AlN layer on Al2O3-hBN
interface but it is attributed to the hBN layer itself.

In the following measurements presented in FIG. 3, there is a more steep
decrease in energy with increasing temperature. The same effect is observed
when we compare the results for as-grown and delaminated layers in FIG. 2b. In
this case, it is most likely related to changes in the layer wrinkle pattern.
As the result suggests, the wrinkle pattern can be changed for temperatures
much lower than the growth temperature. Wrinkles rearrangement leads to an
increase in the layer-substrate interaction contribution what is observed in
the energy downshift for higher temperatures, as suggested by the brown curve
in FIG. 2a.

Figure 4:  Schematic drawing of the most probable carriers transitions that
explain the anomalous phonon behaviour (the distances between defect levels
and bands are not in scale). The inset shows a simulation (see Supplementary
Materials) of occupation of the middle energy level $D_{2}$.

Once the anomaly vanishes after consecutive temperature runs, we can restore
it by UV irradiation. The anomaly measured after UV irradiation in the 5th run
is much stronger and appears at higher temperature as presented in FIG. 3. The
increase in phonon energy is $\sim 6$ cm-1. We propose the mechanism presented
in FIG. 4 to explain the observed phenomenon. When the sample is exposed to UV
light, electrons from the valence band are transferred to the conduction band
(CB) (\raisebox{-0.9pt}{1}⃝). Then they can be trapped on a shallow defect
$D_{1}$ just below the conduction band (\raisebox{-0.9pt}{2}⃝). Here, we
assume that $D_{1}$ defect energy level is less than 1 eV below the conduction
band, which has high degeneracy. The $D_{1}$ state can be filled during the
growth process as well. These kinds of traps can store carriers for a
significant time [35]. If the material has an additional defect state $D_{2}$
in between state $D_{1}$ and the CB, we can change the occupation probability
of all three states during the measurement by changing the temperature
(\raisebox{-0.9pt}{3}⃝). For a specific temperature, the occupation of the
$D_{2}$ defect state reaches the maximum value. The change in electrical state
of $D_{2}$ leads to the expansion of the defect. Consequently, an
inhomogeneous compressive strain is introduced. For such conditions, the
phonon energy increases. This dependence between the electric state of the
defect and the phonon energy suggests the presence of a strong electron-phonon
coupling in those epitaxial hBN, in particular as comparing to the bulk
material [35]. A further temperature increase transfers electrons to the
conduction band. Because the traps are situated close to the conduction band,
they can be refilled with electrons. Then the anomalous effect is repeated for
the next temperature-dependent measurement until we empty most of the shallow
traps, and electrons occupy deeper donor levels. Further evidence for the
presented mechanism is provided by the fact that the anomaly is observed in
both directions, for an increase as well as for a consecutive decrease in
temperature (see Supplementary Materials). We do not exclude that the
presented mechanism could be occurring symmetrically for the valence band and
acceptor states.

The defects properties in hBN are complicated and not fully understood. For
this reason, it is hard to pinpoint defects $D_{1}$ and $D_{2}$. However, we
have candidates that meet the described criteria. It has been reported that
the energy levels of $O_{N}$ are shallow traps placed below the conduction
band in hBN [36, 37, 38, 39, 40]. At very high temperature during the growth
process, oxygen atoms diffuse and desorp from the Al2O3 substrate [41].
Consequently, the substrate can be a source of oxygen atoms in the hBN layer
that replace nitrogen atoms. Such a defect has a low formation energy [36], so
it should be present in all high-temperature grown epitaxial layers. The
$O_{N}$ defect is therefore a good candidate for the $D_{1}$ defect.

The studied hBN layer was grown in the Two-stage Epitaxy regime in NH3 rich
conditions [20] which supports the creation of $V_{B}$ [36]. $V_{B}$ in 1- and
2- charge states increases the distortion between nitrogen atoms [36] which
leads to lattice compressive strain in vicinity of the defect. Moreover,
$V_{B}$ has an energy similar to that of the $O_{N}$ defect and the
$V_{B}-O_{N}$ complex [36]. Therefore, we speculate that the $D_{2}$ defect
can be a negatively charged boron vacancy.

The anomaly presented in FIG. 3 occurs in a narrower range of temperatures in
comparison to the simulation in the inset in FIG. 4. That suggests that there
are other effects that enhance the anomalous phonon energy behaviour. Such an
effect could be the piezoelectric response of boron nitride which is observed
for different phases of sp2-BN [42, 43, 44]. Such non-centrosymetric phases,
which exhibit piezoelectricity, are present in epitaxial boron nitride layers
[21]. Moreover, there is evidence of a strong electrostrictive effect in boron
nitride nanotubes [45, 46]. Electrostriction depends quadratically on the
electric field. Therefore, the reverse of the field does not reverse the
direction of strain. Consequently, the strain does not disappear when
considering diferent randomly arranged BN grains. The deformation related to
the electrostrictive effect is much greater compared to traditional
piezoceramics [46]. Consequently, the electric field induced by charge
redistribution on defect states would significantly influence the phonon
energy change as observed in the experiment.

##  IV CONCLUSIONS

In this work, we studied the thermal response of the $E_{1u}$ phonon mode in
boron nitride epitaxial layers using Fourier-transform infrared spectroscopy.
The developed FTIR spectra analysis provided reliable information about the
physical properties of epitaxial layers which are in good agreement with the
X-ray reflectometry results. Both methods are complementary. However, our
method allows to analyze a wide range of samples, from a single atomic
monolayer to a few tens of $\mu$m. The upper thickness limit depends on the
quality of the sample. We can also obtain information about surface roughness
or material porosity. The method can be easily extended to other materials and
structures.

In our study, we investigated as-grown as well as delaminated layers. The
results of temperature-dependent measurements emphasize the importance of
layer wrinkles in the layer-substrate interaction. The wrinkles significantly
reduce the strain on the interface. Consequently, the behaviour of the
epitaxial layer is similar to that of the bulk material.

Furthermore, we observed strong electron-phonon coupling, which can lead to a
phonon energy increase up to 6 cm-1. This anomalous behaviour vanishes after
sample heating and cooling cycles. Importantly, the process can be enhanced by
UV irradiation. This effect can be explained by the presence of defects
($D_{1}$, $D_{2}$) close to the conduction band. The exact nature of such
defects is still unknown. However, we speculate that these defects could be
$O_{N}$ and $V_{B}$ or their complexes. By increasing the occupation of the
upper $D_{2}$ energy state, we introduce an inhomogeneous compressive strain
which leads to an anomalous increase in phonon energy. Most likely, the
observed anomaly is enhanced by other effects such as piezoelectricity and
electrostriction. However, more research is needed to investigate the exact
properties of shallow impurity states and their influence on the properties of
hBN.

###### Acknowledgements.

This work was supported by the National Science Centre, Poland, under the
decisions 2019/33/B/ST5/02766 and 2020/39/D/ST7/02811.

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## Supplementary materials

## FTIR spectra analysis method

To simulate the whole FTIR spectrum, we assume that the medium is composed of
harmonic damped oscillators. Then Dynamic Dielectric Function (DDF) for hBN
$\varepsilon_{BN}$ is given by [26]:

$\varepsilon_{BN}(\omega)=\varepsilon_{\infty}+\frac{s_{1}^{2}}{\omega_{1}^{2}-\omega^{2}-i\gamma_{1}\omega}+\frac{s_{2}^{2}}{\omega_{2}^{2}-\omega^{2}-i\gamma_{2}\omega}$
(5)

where $\varepsilon_{\infty}=4.95$, $s_{1}^{2}=1.23\times 10^{5}$ cm-2,
$s_{2}^{2}=3.49\times 10^{5}$ cm-2, $\omega_{1}$ and $\omega_{2}$ are the
frequencies of the normal modes $A_{2u}$ and $E_{1u}$ respectively and
$\gamma_{1,2}$ are the corresponding damping parameters. For sapphire
dielectric function $\varepsilon_{sapph}$ is described as follows [30]:

$\varepsilon_{sapph}(\omega)=\varepsilon_{\infty,sapph}\prod_{k=1}^{4}\frac{\omega_{LO,k}^{2}-\omega^{2}-i\omega\gamma_{LO,k}}{\omega_{TO,k}^{2}-\omega^{2}-i\omega\gamma_{TO,k}}$
(6)

where $\omega_{LO,k}$ and $\omega_{TO,k}$ are the frequencies of longitudinal
and transverse optic normal modes in sapphire, $\gamma_{LO,k}$ and
$\gamma_{TO,k}$ are their damping parameters. $\varepsilon_{\infty,sapph}$ is
given by the formula [47]:

$\varepsilon_{\infty,sapph}(\omega)=1+\frac{A_{1}\omega_{1,sapph}^{2}}{\omega_{1,sapph}^{2}-\omega^{2}}+\frac{A_{2}\omega_{2,sapph}^{2}}{\omega_{2,sapph}^{2}-\omega^{2}}$
(7)

The values of the parameters connected to sapphire are taken from the papers
by Lee et al. [30] and Malitson [47]. In the next step, we assume that
incident light is reflected from air-hBN and hBN-Al2O3 interfaces and the
light transmitted into the sapphire is fully absorbed and does not interact
with the beam reflected on the interfaces. The light beams reflected from the
following interfaces interfere with each other. That leads to the formula for
the reflection coefficient $r$ for the structure of a thin layer grown on the
sapphire substrate [48].

$\displaystyle
r=\frac{r_{1}+r_{2}e^{i\Delta\varphi}}{1+r_{1}r_{2}e^{i\Delta\varphi}}$ (8)
$\displaystyle r_{1}=\frac{1-\widetilde{n}_{BN}}{1+\widetilde{n}_{BN}},\qquad
r_{2}=\frac{\widetilde{n}_{BN}-\widetilde{n}_{Al_{2}O_{3}}}{\widetilde{n}_{BN}+\widetilde{n}_{Al_{2}O_{3}}}$
(9)

where $\Delta\varphi=4\pi dn_{BN}$ is the optical path difference for light
that penetrates the hBN layer, $d$ \- thickness of the hBN layer and
$\widetilde{n}_{BN}$, $\widetilde{n}_{Al_{2}O_{3}}$ are complex refractive
indices of hBN and sapphire described by the square root of their dielectric
functions ($\widetilde{n_{i}}=\sqrt{\varepsilon_{i}(\omega)}$). Finally, by
calculating the square of the absolute value of the reflection coefficient, it
is possible to simulate the reflectivity curve of the analyzed structure
($R=\alpha|r|^{2}$, where $\alpha$ is a normalization factor).

To verify the propriety of the results coming from the described FTIR spectra
analysis method, we compared them with the results obtained with an
independent measurement technique, which is X-ray reflectometry. The values of
hBN thickness $d$ for the samples in range $2-70$ nm obtained from FTIR
($d_{FTIR}$) and XRR ($d_{XRR}$) measurements are plotted in FIG. S1. The
$d_{FTIR}$ values presented are the average of values coming from the 3
measured areas. The error bars are their standard deviation. The uncertainty
of the $d_{XRR}$ fitted values is about 1 nm. The points lie on the dashed
line described by the formula $d_{FTIR}=d_{XRR}$. This indicates a very good
agreement between both methods over a wide range of sample thickness.

Figure S1: hBN thickness values obtained with FTIR spectra analysis method as
a function of hBN thickness values obtained with X-ray reflectometry
measurement (black spheres) and $d_{FTIR}=d_{XRR}$ dashed red line.

The next step in the introduction of the method is to define its limitations.
In FIG. S2a we present the spectrum for the thinnest measured sample. The peak
related to the $E_{1u}$ mode in hBN is barely visible. Implementation of our
method revealed $d_{BN}=0.25$ nm. This value is slightly lower than the
thickness of the hBN monolayer [49]. This is because the measured area of
$70\times 70$ $\mu$m is not fully covered by the hBN layer. However, the data
presented in FIG. S2a proves that our method has no lower limitations in the
context of sp2-BN detection and enables atomic monolayer characterization.

The spectrum for the thickest measured layer is presented in FIG. S2b. For
this sample, we measured spectra in the range up to 6500 cm-1 for better
observation of characteristic oscillations related to interference within the
hBN layer. Due to the roughness and porosity of thick epitaxial layers, we
cannot use our model to describe the experimental data. However, we can
introduce some amendments to the model. To take into account the roughness
$\sigma$, we change the expression for the reflection coefficient as follows:

$r=\frac{r_{1}e^{-8(\pi\omega\sigma)^{2}}+r_{2}Ae^{i4\pi\omega
n_{BN}d}}{1+r_{1}r_{2}e^{-8(\pi\omega\sigma n_{BN})^{2}}e^{i4\pi\omega
n_{BN}d}},\quad
A=(1-r_{1}^{2})e^{-4(\pi\omega\sigma(n_{BN}-1))^{2}}+r_{1}^{2}e^{-8(n_{BN}^{2}+1)(\pi\omega\sigma)^{2}}$
(10)

To consider the porosity of the material, we use a simple model in which we
decrease the dielectric function with the parameter $a$.

$\varepsilon_{BN}^{\prime}=1+a(\varepsilon_{BN}-1),\quad a\in(0,1)$ (11)

When Eq. 10-11 are introduced it is possible to model a curve which describes
the experimental data much better as presented in FIG. S2b. The fitting
procedure revealed the thickness of the hBN layer $d_{BN}=1122$ nm, surface
roughness $\sigma=46$ nm, porosity $a=0.25$. The simulation presented in FIG.
S2b inset suggests that using our method it would be possible to analyze
crystals that are tens of micrometers thick. All this shows high universality
of our method, which allows analyzing layers in the range from single atomic
layers up to several $\mu$m.

Figure S2:  FTIR spectra of hBN a) monolayer (zoom on hBN $E_{1u}$ vibrational
mode peak), (b) 1122 nm thick rough and porous layer on Al2O3 substrate. Lines
represent DDF simulations with and without including roughness and porosity.
Inset shows simulation of 15 $\mu$m thick good quality hBN.

## Simulation of temperature dependence of $D_{2}$ defect occupation

We assume a three-level energy model
($\varepsilon_{0}<\varepsilon_{1}<\varepsilon_{2}$). The highest energy level
is the conduction band, which is represented by its high degeneracy $g_{2}\gg
1$. The middle energy level can be degenerated as well ($g_{1}\sim 1$).
Occupation is governed by the Boltzmann distribution. We allow only thermal
electron excitation. Then the occupation $n_{i}$ for the following energy
levels will be given by:

$\displaystyle n_{0}=N\frac{1}{Z},\quad
n_{1}=N\frac{g_{1}e^{-e_{01}/kT}}{Z},\quad N\frac{g_{2}e^{-e_{02}/kT}}{Z}$
$\displaystyle Z=1+g_{1}e^{-e_{01}/kT}+g_{2}e^{-e_{02}/kT}$

where $T$-temperature, $k$-Boltzmann constant, $e_{ij}$-energy distance
between states $i$ and $j$. For the case of analyzed defects in hBN:
$e_{02}<1$ eV, $e_{01}<e_{12}$ [36, 39]. FIG. S3 presents the results of
simulations with different sets of modeled parameters. The values of the
parameters for each simulation are presented in TAB. S1. One can observe the
simulated peak movement and its intensity change while changing parameters.
The curve and parameters related to Sim0 correspond to the simulation
presented in the main text in FIG. 4.

Figure S3:  Simulation of occupation of the middle energy level which we named $D_{2}$ defect. The values of modeled parameters are presented in TAB. S1. Table S1:  Values of parameters for the curves simulated in FIG. S3. Simulation |  $e_{01}$ (eV) |  $e_{12}$ (eV) | $g_{1}$ | $g_{2}$
---|---|---|---|---
Sim0 | 0.158 | 0.573 | 2 | $10^{8}$
Sim1 | 0.158 | 0.400 | 2 | $10^{8}$
Sim2 | 0.250 | 0.573 | 2 | $10^{8}$
Sim3 | 0.158 | 0.573 | 1 | $10^{8}$
Sim4 | 0.158 | 0.573 | 2 | $10^{7}$
Sim5 | 0.100 | 0.400 | 1 | $10^{7}$

## Anomaly observed for different samples

Figure S4:  $E_{1u}$ phonon energy (blue circles) and peak integral coming from Lorentzian fitting (orange squares) as a function of temperature for the 20 nm thick sample grown in the Two-stage Epitaxy regime. Arrows point out eh anomalous behaviour occurrence. Initial 4 temperature-dependent measurements lead to softening of the anomalous phonon energy increase. After UV irradiation the anomalous effect is enhanced. Lines are guide to the eye only. Due to the strong correlation between peak integral and other fitted parameters, there is no direct relation between peak integral intensity and the harmonic oscillator strength. At the temperatures at which the anomaly occurs, a peak integral dip is observed. For the measurement after UV irradiation, the minimal peak integral value is 5 times lower than the average for this temperature run. Figure S5:  Anomalous behaviour of $E_{1u}$ phonon energy as a function of temperature for the samples grown in different conditions. Growth parameter details for each sample are presented in TAB. S2. Arrows point out the anomalous behaviour occurrence. Table S2:  Growth parameters for the samples presented in FIG. S5. The subscripts refer to the corresponding panels in FIG. S5. Sample $S_{c}$ comes from the same growth process as the 20 nm thick Two-stage Epitaxy sample from the main text, but before measurement it was delaminated. The same for the pair of samples $S_{e}$ (as-grown) and $S_{f}$ (delaminated). The abbreviations used in the table for growth modes come from: CFG - Continous Flow Growth, FME - Flow Modulation Epitaxy, Two-stage - Two-stage Epitaxy. $T_{g}$ is the growth temperature. Sample | Growth mode |  $T_{g}$ (∘C) | NH3 flow (ccm) | TEB flow (ccm) |  $d_{BN}$ (nm)
---|---|---|---|---|---
$S_{a}$ | FME | 1295 | 4000 | 15 | 10.7
$S_{b}$ | CFG | 1315 | 4000 | 10 | 5.2
$S_{c}$ | Two-stage | 1309/1300 | 4000 | 10/15 | 15.4
$S_{d}$ | Two-stage | 1305/1300 | 4000 | 10/15 | 36.8
$S_{e}$ | Two-stage | 1295/1285 | 4000 | 10/15 | 8.5
$S_{f}$ | Two-stage | 1295/1285 | 4000 | 10/15 | 8.0
Figure S6:  $E_{1u}$ phonon energy as a function of temperature measured for
increasing (red circles) and decreasing temperature (blue squares). Results
refer to 1st measurement from the main text. Dashed lines are guide for eye
only.


# Probing Curvature Effects in The Fermi GRB 110920

A. Shenoy11affiliationmark:  , E. Sonbas22affiliationmark:  33affiliationmark:
, C. Dermer,44affiliationmark:  , L. C. Maximon11affiliationmark:  , K. S.
Dhuga11affiliationmark:  , P. N. Bhat55affiliationmark:  , J.
Hakkila66affiliationmark:  , W. C. Parke11affiliationmark:  , G. A.
Maclachlan11affiliationmark:  , T. N. Ukwatta77affiliationmark:  1Department
of Physics, The George Washington University, Washington, DC 20052, USA
2University of Adiyaman, Department of Physics, 02040 Adiyaman, Turkey  3NASA
Goddard Space Flight Center, Greenbelt, MD 20771, USA  4Space Science
Division, Code 7653, Naval Research Laboratory, Washington, D.C. 20375, USA
5CSPAR, University of Alabama in Huntsville, Huntsville, AL 35805, USA
6Department of Physics and Astronomy, College of Charleston, Charleston, S.C.
29424, USA  7Department of Physics and Astronomy, Michigan State University,
East Lansing, MI 48824, USA<EMAIL_ADDRESS>
###### Abstract

Curvature effects in Gamma-ray bursts (GRBs) have long been a source of
considerable interest. In a collimated relativistic GRB jet, photons that are
off-axis relative to the observer arrive at later times than on-axis photons
and are also expected to be spectrally softer. In this work, we invoke a
relatively simple kinematic two-shell collision model for a uniform jet
profile and compare its predictions to GRB prompt-emission data for
observations that have been attributed to curvature effects such as the peak-
flux–peak-frequency relation, i.e., the relation between the $\nu$Fν flux and
the spectral peak, Epk in the decay phase of a GRB pulse, and spectral lags.
In addition, we explore the behavior of pulse widths with energy. We present
the case of the single-pulse Fermi GRB 110920, as a test for the predictions
of the model against observations.

###### Subject headings:

Gamma-ray bursts: general

##  1\. Introduction

Pulses in GRB light curves are thought to be produced by collisions between
relativistic shells ejected from an active central engine (Rees and Meszaros
1994). The interception of a more slowly moving shell by a second shell that
is ejected at a later time, but with greater speed, produces a shock that
dissipates internal energy and accelerates the particles that emit the GRB
radiation. This scenario is widely adopted in order to model pulses in GRB
light curves (e.g., Daigne and Mochkovitch 1998; Zhang et al. 2009). Studies
of pulses are important to determine if GRB sources require engines that are
long-lasting or impulsive, and to determine the likely radiation mechanism(s),
with important implications for the nature of the central engine.

Spectral lags, where low energy photons reach the observer at later times than
high-energy photons, are seen in a significant fraction of GRBs. Cheng et al.
(1995) were the first to analyze the spectral lag of GRBs, which they
determined as the time delay between the peaks in the Burst and Transient
Source Experiment (BATSE) Large Area Detector (LAD) channel 1 (25 - 50 keV)
and channel 3 (100 - 300 keV) light curves. Since then, several authors have
analyzed spectral lags in GRBs, while also extending these observations to the
Swift and Fermi GRB samples (e.g., Norris et al. 1996; Norris, Marani &
Bonnell 2000, Wu & Fenimore 2000; Chen et al. 2005; Ukwatta et al. 2010;
Sonbas et. al. 2012) The leading model to explain the spectral lag is the
curvature effect, i.e. the kinematic effect due to the observer looking at an
increasingly off-axis annulus area relative to the line-of-sight (Fenimore et
al. 1996; Salmonson 2000; Kumar & Panaitescu 2000; Ioka and Nakumura 2001; Qin
2002; Qin et al. 2004; Dermer 2004; Shen et al. 2005; Lu et al. 2006). Softer
low-energy radiation comes from the off-axis annulus area due to smaller
Doppler factors. This radiation is also delayed at the observer end with
respect to on-axis observation due to the geometric curvature of the shell.

The existing models as well as observations suggest that a connection exists
between the observed hard-to-soft spectral evolution of GRB pulses and
spectral lags. It is therefore important to understand the mechanism that
produces this evolution. Tavani (1996) proposed that this hard-to-soft
spectral evolution is caused by the variation of the average Lorentz factor of
pre-accelerated particles and the strength of the local magnetic field at the
GRB site as the synchrotron emission evolves within the burst. Liang (1997)
proposed a physical model of hard-to-soft spectral evolution in which
impulsively accelerated non-thermal leptons cool by saturated Compton up-
scattering of soft photons. Kocevski & Liang (2003) have analyzed a sample of
19 GRBs and found a positive correlation between the decay rate of the peak
energy and the spectral lag. Ukwatta et al. (2010) have analyzed a sample of
31 Swift GRBs with known red-shifts and determined the spectral lags between
fixed source frame bands, 100 – 150 keV and 200 – 250 keV. They also
determined that the source-frame Epk lies beyond the higher-energy band, 100 –
250 keV for a majority of these bursts. Based on this result, they suggest
that spectral evolution may not be the dominant process causing the observed
spectral lag.

Borgonovo and Ryde (2001) studied the spectral evolution in the prompt-
emission in GRBs by performing a time-resolved spectral analysis of BATSE
single pulses and showed that in many cases the $\nu F\nu$ flux at the Epk for
each time segment was $\propto$ E${}_{pk}^{\eta}$ (hereafter referred to as
the peak-flux–peak-frequency relation), with $\eta$ ranging from $\approx 0.6$
to $3$. The exponent $\eta$ was found to stay roughly constant for pulses
within the same GRB. Dermer (2004) modeled and analyzed GRB pulses based on
curvature effects with a Broken-Power-Law (BPL) rest-frame spectrum and showed
that in the curvature limit, $\eta$ was equal to 3 for pulses with a wide
range of temporal properties.

While several studies (Qin 2002; Qin et al. 2004; Shen et al. 2005; Lu et al.
2006) have been performed to determine the role played by curvature effects,
both in the spectral evolution of GRB prompt-emission as well as in producing
spectral lags, a number of questions remain unanswered. Specifically, we note
that the dependence of the lag on the radius at which the emission takes place
is quite unclear at this stage with seemingly contradictory results being
reported in the literature (Shen et al. 2005; Lu et al. 2006). While that is
not the focus of our current study, it is a matter of considerable interest
and will be the central topic of a forthcoming work. In this work we focus on
the role played by the thickness of colliding shells on observables such
spectral lags, the $\nu F\nu$ flux vs. Epk relation, and the behavior of the
corresponding pulse widths as a function of energy. Other studies have also
attempted to determine the role played by the evolution of the rest-frame
spectrum on the observables such as the spectral lags and the evolution of the
pulse-widths with energy within the context of a curvature model(Qin et
al.2005; Lu et al. 2006; Qin et al. 2009; Peng et al. 2011). Again, such
considerations are very important but we do not specifically consider the
effects of the evolution of the rest-frame spectrum in this paper.

The paper is organized as follows: the basic features of the model are
presented in section 2, followed by a description of the sample selection
criteria, analysis methodology, and a case study in section 3. The discussion
of our main results is presented in section 4, followed by a summary of our
conclusions in section 5.

Table 1Selected parameters for the generation of the model light curves unless otherwise stated. $\eta_{r}$ | $\eta_{t}$ | $\eta_{\Delta}$ | $t_{var}(s)$ | $\Gamma$ | z | $\theta_{jet}$ | Epk,0 (keV) |  $d_{L}$ (cm) |  $u_{0}$ (ergs cm-3)
---|---|---|---|---|---|---|---|---|---
1.0 | 1.0 | 1.0 | 1.0 | 300 | 1.0 | 4.0/$\Gamma$ | 250.0 | $2.2\times 10^{28}$ | 1.0

##  2\. The Model

We have used a particular representation of the internal shock model for our
purposes (Dermer 2004). This model consists of a single two-shell collision
event occurring at a radius $r_{0}$ from the source, generating a uniform
spherical shell, with Lorentz factor $\Gamma$, that radiates for a co-moving
time between $t^{\prime}_{0}$ and $t^{\prime}_{0}+\Delta t^{\prime}$ with
$\Delta t^{\prime}=\eta_{t}\Gamma t_{var}/(1+z)$, where $t_{var}$ is the
observed variability time scale. The rest-frame–emission profile is assumed to
be rectangular with instantaneous rise and decay phases. The opening angle of
the jet is assumed to be $4/\Gamma$. Emission from angles greater than
$4/\Gamma$ are ignored. This is a suitable compromise between considering
emissions from an entire fireball surface ($0<\theta<\pi/2$) and a collimated
jet ($\theta\sim 1/\Gamma$). As noted by Qin et al. 2004, limiting the
radiation to $\theta<1/\Gamma$ leads to a cutoff-tail problem whereas the
contributions from areas at $\theta>1/\Gamma$ fall off very rapidly. We find
such a choice to be suitable for producing pulse profiles that may be directly
compared with observations. The co-moving width of the shell $\Delta
r^{\prime}$ is assumed to remain constant during the period of illumination
and given by $\Delta r^{\prime}=\eta_{\Delta}\Gamma~{}c~{}t_{var}/(1+z)$. It
is in this respect that the chosen model differs from previous studies.
Previous studies (see for instance Qin et al. 2004, Shen et al. 2005) have
studied the effects of curvature from a spherical surface in great detail. As
shown subsequently, the effect of a finite shell where $\Delta r^{\prime}$ is
comparable to $c\Delta t^{\prime}$, has a significant effect on the predicted
observables such as spectral lags and pulse widths as a function of energy
when compared to the models that study curvature effects from a surface. As
$\Delta r^{\prime}<<c\Delta t^{\prime}$, we approach the infinitesimal shell
or emission-surface situation and are able to recover many of the predictions
of the aforementioned models. The emission spectrum in the co-moving frame may
be described by any suitable spectral function such as a Broken Power-Law
(BPL), Band, or Comptonized–Epk, peaking at a co-moving photon energy
E${}^{\prime}_{pk,0}$ = (1 + z) Epk,0/2$\Gamma$ where Epk,0 is the observer
frame Epk at the start of the pulse. The spectral indices at E $<$ Epk,0 and
at E $>$ Epk,0 in counts space are denoted $\alpha$ and $\beta$, respectively.
The curvature constraint requires that $r\lesssim
2\Gamma^{2}c~{}t_{var}/(1+z)$ (Fenimore et al. 1996). The radius is thus
written using the expression $r_{0}=2\eta_{r}\Gamma^{2}c~{}t_{var}/(1+z)$,
with $0\lesssim\eta_{r}\lesssim 1$. The parameters $\eta_{t}$, $\eta_{\Delta}$
and $\eta_{r}$ thus control the blast-wave duration, shell-thickness and
radius of emission respectively.

Figure 1.— Normalized light curves obtained using selected parameters except:
Thin-shell pulse, $\eta_{\Delta}=0.1$; and the Curvature pulse,
$\eta_{\Delta}=\eta_{t}=0.1$. The case of $\eta_{t}=\eta_{\Delta}=\eta_{r}=1$
is referred to as a Causal pulse (see Dermer 2004 for a detailed discussion of
these three generic types of pulses). Figure 2.— Evolution of the spectral
energy distribution due to curvature effects for the case of the causal pulse
in Fig. 1. In the declining phase of the pulse, the value of
$f_{E_{pk}}\propto$ E${}_{pk}^{3}$ is shown by the red line. Figure 3.— Lag
vs. Energy for selected parameters except shell thickness $\eta_{\Delta}$.
Red: $\eta_{\Delta}=1.0$; Green: $\eta_{\Delta}=0.5$; Blue:
$\eta_{\Delta}=0.1$. Figure 4.— Lag vs. Energy due to different rest-frame
spectral functions with selected parameters except Epk,0 = 200 keV. Here, Red:
Broken-power-law with $\alpha=-1/3$, $\beta=-2.5$, Green: Band function with
$\alpha=-0.8$, $\beta=-2.25$ and Blue: Comptonized-Epk with $\alpha=-0.8$.
Figure 5.— Pulse FWHM vs. Energy for different shell-collision parameters
$\eta_{\Delta}$ and $\eta_{t}$ for selected parameters except E${}_{pk}=200$
keV, a Band spectrum with $\alpha=-1.0$ and $\beta=-2.25$ and $t_{var}=1.0$ s,
with: Pink: A pulse with Infinitesimal duration and shell thickness with
$\eta_{\Delta}=0.001$ and $\eta_{t}=0.001$ Red: Curvature pulse with
$\eta_{\Delta}=0.1$ and $\eta_{t}=0.1$; Green: Thin-shell pulse with
$\eta_{\Delta}=0.1$ and $\eta_{t}=1$; and Blue: Causal pulse with
$\eta_{\Delta}=1$ and $\eta_{t}=1$. Also shown is the exponent of the power-
law that best fits the model points for the thin-shell and Causal pulses.

Unless otherwise stated, the selected parameters used for the numerical
calculations of the light curves, spectra and spectral lags are shown in
Table. 1. Fig. 1 shows the normalized, generic pulse shapes obtained using the
selected parameters. As can be seen, the cases of the thin-shell pulse
($\eta_{\Delta}<<\eta_{t}$) and the curvature pulse
($\eta_{\Delta}=\eta_{t}<<\eta_{r}$) produce the sharp featured light curves
noted by Qin et. al. 2004 for emission from a rectangular pulse profile in the
co-moving frame, and which are attributed to the effect of a suddenly-dimming
emission profile. These two cases most closely correspond to a long duration
and a short duration pulse in the co-moving frame respectively, and where the
effects of a finite shell are suppressed. The case of the causal pulse
($\eta_{\Delta}=\eta_{t}=\eta_{r}=1$) however, shows that emission from a
finite shell can produce smooth light curves without the need for a slowly
dimming co-moving emission profile. Fig. 2 shows the evolution of the spectral
energy distribution for the case of the curvature pulse shown in Fig. 1. The
$\nu$Fν peak flux $f_{E_{pk}}$ $\propto$ E${}_{pk}^{3}$ equality line is shown
in the decay portion of the pulse. Dermer (2004) shows that this equality
holds in the declining phase for all pulses (a similar result has been derived
by Qin et al. 2009). We also note that the presence of a finite shell affects
the low energy and high energy fluxes equally and therefore does not effect
the shape of the spectrum as a function of time as first noted by Qin et. al.
2002 in the context of emission from a fireball surface. Fig. 3 shows the
spectral lag as a function of energy for a case with selected parameters but
with varying shell thickness parameter, $\eta_{\Delta}$. Fig. 4 shows the lag
as a function of energy for a case with selected parameters but with different
rest-frame spectral functions (BPL, Band, and Comptonized–Epk). Fig. 5 shows
the pulse Full-width at half-maximum (FWHM) as a function of energy for the
various profiles shown in Fig. 1. For the purpose of comparison we have used a
Band function with $\alpha=-1.0$ and $\beta=-2.25$, identical to Qin et. al.
2005. These authors have shown that the Doppler effect of a relativistically
expanding fireball could lead to a power-law trend for the pulse width as a
function of energy within a certain energy range. By taking a sizable sample
of BATSE GRBs, they demonstrated that the pulse widths exhibit a
plateau/power-law/plateau feature as a function of energy. They also note that
the power-law index depends strongly on Epk and the rest-frame radiation
spectrum. The plateau/power-law/plateau feature reported by Qin et al. 2005 is
well reproduced here. Furthermore, we note that the power-law exponent is
sensitive to the assumed thickness of the shell.

Figure 6.— KRL pulse-fit for the light curve for GRB 110920 with residuals.
Figure 7.— Light curve segments for 110920 with equal fluences. Figure 8.—
Comparable pulses generated using the BPL (E${}_{peak}=300$ keV,
$\alpha=-\frac{1}{3}$, $\beta=-2.5$), Band (E${}_{peak}=334$ keV,
$\alpha=-0.2$, $\beta=-2.65$)and Comptonized–Epk (E${}_{peak}=280.1$ keV,
$\alpha=-0.46$) spectral functions. The light curves have been offset by 20
seconds for better viewing. Figure 9.— $\nu F\nu$ Flux vs. Epk for the data
from GRB 110920. The data were fit with the best-fit Comptonized–Epk function
in the range 100-985 keV. Figure 10.—  $\nu F\nu$ Flux vs. Epk from the model
for the three rest-frame spectral functions described in the text with Blue:
BPL; slope = 3.11 +/- 0.04, Pink: Band; slope = 3.39 +/- 0.10 and Black:
Comptonized–Epk; slope = 2.57 +/- 0.01, for time segments identical to those
used for the data. The flux scale has been offset for better viewing. Figure
11.— Lag vs. Energy from the model for the pulses shown in Fig. 7 with Red:
Data, Solid blue squares: Comptonized–Epk, Hollow black squares: Band function
and Pink crosses: BPL. The Band 2 energies are the mid-point of the energy in
the second band. Figure 12.— Pulse FWHM vs. Energy for the data and the model
for GRB 110920 for identical energy bands. Also shown is the exponent of the
power-law that best fits the model points.

##  3\. Sample Selection and Methodology.

As a first step we analyze either single-pulse GRBs, or GRBs with relatively
simple light curves where the individual pulses within a multi-pulse structure
in the light curve can be distinguished. In addition, we require that the GRB
pulses be bright enough and of sufficient duration (the duration of the pulse
is particularly relevant to tests of the peak-flux – peak-frequency relation)
so that we may obtain reliable results from our analyses.

After identifying a potential candidate GRB, we pulse-fit the GRB light curves
using a suitable pulse function (such as the Kocevski-Ryde-Liang (KRL) pulse
function: see Kocevski Ryde & Liang 2003 or the Norris pulse function: see
Norris et al. 2005) in multiple energy bands. In order to support the
supposition that a given strong pulse (obtained from a suitable pulse-fit) is
not made from overlapping multiple pulses (within statistics), we perform a
wavelet based minimum-variability time-scale (MTS) extraction. In essence, the
MTS is a measure of the smallest temporal structure in a lightcurve. The full
details of its extraction, and the technique in general, are given in
MacLachlan et al. (2013). The correlation between MTS and pulse properties
such as rise times and widths is discussed in MacLachlan et al. (2012). The
best-fit pulse profile is then used as a representation of the light curve
from the data.

The time-integrated spectrum of the GRB is fit with a suitable function (Band,
Comptonized–Epk etc.). The best-fit spectral function is used as the rest-
frame emission spectrum in the model. The model parameters are then varied to
generate a light curve that best matches the best-fit pulse profile. The light
curve is subdivided into time segments with equal, and sufficiently high
background-subtracted fluence in order to minimize the effects of varying
signal-to-noise and an Epk is extracted via a spectral fit for each time
segment. The $\nu$Fν flux is extracted at Epk in a range spanned by the Epk-
error. The model light curve is treated in an identical fashion as the data
with regard to segments. Model fluxes and Epk’s are extracted and the peak-
flux – peak-frequency relation is tested. In addition, we extract spectral
lags in suitable, identical energy bands from the data and the model and
compare the predicted and the observed spectral-lag-energy evolution. The
spectral lags are extracted using the cross-correlation-function analysis
method as described in Ukwatta et. al (2010).

###  3.1. GRB 110920 - A Test Case

The Fermi GRB 110920 is a single-pulse burst with a relatively long fast-rise,
exponential-decay structure with a $T_{90}$ of 170 $\pm$ 17 seconds. The best
fit Band parameters (see McGlynn et al. 2012 for a detailed discussion on the
properties of this GRB) for the time interval [$T_{0}+0.003,T_{0}+52.737$]
(where $T_{0}$ is the trigger time) were $\alpha=-0.20\pm 0.02$,
$\beta=-2.65^{+0.07}_{-0.09}$ and $E_{peak}=334\pm 5$ keV with C-stat = 3206.5
(485 d.o.f.). However, when a blackbody component was included in the fit, the
C-stat was reduced to 2848.3 (483 d.o.f.). The peak energy of the Band
component was shifted up to $E_{peak}=978^{+154}_{-121}$ keV and the
temperature of the blackbody was found to be $kT=61.3^{+0.7}_{-0.6}$ keV. The
low energy index $\alpha$ became ($-1.05\pm 0.04$). McGlynn et al. (2012) have
attributed this blackbody component to the photospheric emission (see for
instance: Ryde, 2005; Rees & Meszaros, 2005). Since the redshift (z) is
undetermined for this GRB, they have assumed a value of z = 2 and then used
photospheric emission models to obtain a bulk Lorentz factor, $\Gamma$ of
$\sim$ 440\.

A careful analysis, based partly on the results presented above, allowed us to
separate the underlying pulse structure from the overall structure of the
light curve. Fig. 6 shows the full light curve for the GRB in the energy range
100 – 985 keV along with the best-fit KRL pulse function. The energy range was
so chosen to avoid contamination from the soft component (below 100 keV).

The best-fit KRL function was then used as the representation of the light
curve to be matched by the model. We assumed a Bulk Lorentz factor $\Gamma$ =
440 and z = 2 (as in McGlynn et al. 2012). Fig. 7 shows the segmentation of
the light curve into equal-fluence segments. The time-integrated energy-
spectrum of the pulse was fit with a Band function (as in McGlynn et al 2012,
in the energy range, 8 - 985 keV) and a Comptonized–Epk function (in the range
100-985 keV) and these, along with a theoretical BPL function (originally used
by Dermer 2004) were used as rest-frame spectra for the model. We varied the
shell-collision parameters and extracted best-fit pulses using these three
spectral functions (see Fig. 8 for the pulses as well as details of the
corresponding spectral parameters). The resulting best-fit shell-collision
parameters, together with a best-fit value of $t_{var}=44$ seconds and the
chosen values of $\Gamma$ and z, yielded a radius, $r_{0}$ of 1.5 $\times
10^{17}$ cm, a shell thickness, $\Delta r^{\prime}$ of 8.8 $\times 10^{12}$ cm
and a co-moving frame pulse-duration, $\Delta t^{\prime}$ of 2.0 $\times
10^{3}$ seconds for the Comptonized–Epk spectral function. The corresponding
values for the Band and BPL functions were very similar. Figs. 9 and 10 show
our results for the peak-flux–peak-frequency relation for the data with the
best-fit Comptonized-Epk function and the model using the three different
rest-frame spectra. Fig. 11 shows the spectral lags for the light curves of
Fig. 8.

In order to explore the evolution of the pulse width with energy, we extracted
the FWHM and plotted this as a function of energy. The plot is shown in Fig.
12. The energy bands chosen (in keV) were 8–25, 25–50, 50–100, 100–150,
150–200, 200–250, 250–350, 350–985. These energy bands were so chosen as to
ensure a sufficient number of counts in each energy band for a reliable
measurement of pulse FWHM while also providing a sufficient number and spacing
of bands to show the trend curve. As the KRL function does not fit the pulses
below 150 keV accurately, we employed a Monte-Carlo simulation where 1000
light curves were simulated in each energy band using the square-root of the
counts as their errors (assuming independent Poisson distributions for the
counts), the pulse FWHM was extracted for each light curve, and the mean and
standard deviation of the 1000 pulse FWHMs were used as the pulse FWHM and its
error respectively for each band. We fit a power-law of the form C$E^{a}$ to
the model points and extracted an exponent of -0.31 +/- 0.03.

##  4\. Discussion

Before we turn to the test-case GRB, we note that in the model calculation for
a broken-power-law rest-frame spectrum, the lag shows a relatively well-
defined trend with no lags at energies below $\sim$ 0.3Epk,0 and a constant
lag for all energies above Epk,0 (Fig. 4). Increasing or decreasing the value
of Epk,0, while keeping all other model parameters fixed, only shifts the
entire curve in the direction of increase or decrease. Shen et al. (2005)
studied the lags due to curvature effects using different rest-frame emission
profiles. They found that for an infinitesimal shell, a rectangular profile
produces no lags. We find the situation to be quite different for the case of
a finite shell thickness. In addition, as depicted in Fig 4, a change in the
rest-frame spectrum also has a significant effect in the evolution of the
spectral lag with energy. In the case of a Band or Comptonized–Epk function,
the lags are small for energies small compared to Epk,0, and the lags for a
Comptonized–Epk function do not show a saturation energy. This is primarily
because the Comptonized–Epk function varies monotonically at all energies and
does not have a well-defined break energy. The value of $r_{0}$ obtained for
the test case is consistent with estimates for an internal shock model (see
for e.g. Hascoet et al. 2012). While the value of $\Delta r^{\prime}$ ($\sim
10^{13}$ cm) seems reasonable, it is difficult to infer the significance of
its absolute magnitude in the context of the current analysis. The inclusion
of a finite-shell-thickness component in our curvature model also produces
relatively large lags ($\sim$ a few seconds) without a need for extreme
physical parameters such as $\Gamma<$ 50 (as concluded by Shen et al. 2005),
or a large local pulse width ($\sim 10^{7}$ seconds as concluded by Lu et al.
2006). We find that a rest-frame pulse duration, $\Delta t^{\prime}\sim
10^{3}$ seconds is sufficient to produce such lags.

Our results for the test case show that an internal shock model with a rest-
frame spectrum identical to that used to fit the data (the Comptonized–Epk and
Band functions), reproduces the observed pulse profile as well as the observed
spectral lags. It was difficult to determine if there was a saturation energy
present in the lag-energy-evolution (Fig. 11) as there were insufficient
counts to extract lags in higher energy bands. As noted above, a finite shell
thickness can account for observed lags even for the case of a rectangular
rest-frame emission profile. A second Band function component with a peak
shifted to 1 MeV (when a blackbody component is included in the fit) would
imply a 1 MeV break energy in the lag-energy plot, and would also predict no
lags (or small lags in the case of the Comptonized–Epk) below $\sim$300 keV.
This does not match the observations. It can be seen from Fig. 10 that the
exponent in the peak-flux–peak-frequency relation is close to 3 in all cases
for the model pulses shown in Fig. 8. This confirms the observations of Dermer
(2004) that the exponent at times after the peak of the light curve is close
to 3 even with different choices of rest-frame spectra. However, this does not
match the exponent obtained from the data (Fig. 9). While a connection may
exist between the observed spectral evolution and the spectral lags, it
appears that curvature effects alone cannot describe this connection and
additional emission mechanisms may be needed. We note in passing that Guirec
et al. (2012) have included a blackbody component to describe the temporal and
spectral properties of a number of GRBs.

We have also explored the behavior of the pulse width with energy. As shown in
Fig. 12, the pulse-width decreases with energy approximately as a power law.
Both the data and the model predictions exhibit similar trends although the
low-energy agreement is marginal. The exponent of the power law (-0.31 +/-
0.03) matches well the exponent extracted by Fenimore et. al. (1995), who
analyzed a large sample of bright BATSE bursts and obtained an average power-
law exponent of about -0.4. A similar result was also obtained by Peng et al.
2006 who analyzed a sizeable sample of bright single pulses in BATSE GRBs. In
addition, in a recent work based on an analysis of 51 long-duration FRED-like
single-pulses from the BATSE data, Peng et. al. 2012 showed that the curvature
effect combined with a Band rest-frame spectrum can explain the energy
dependence of the pulse widths. Cohen et al. (1997) have suggested that such
an exponent is consistent with a population of electrons losing energy via
synchrotron radiation, a process for which the exponent is predicted to be
-0.5.

##  5\. Conclusions

We have used a simple two-shell collision model to investigate curvature
effects in the prompt emission of GRBs. We have examined the effects of
emission spectra such as the Band, the Comptonized–Epk, and the BPL functions.
We have focused primarily on the peak-flux – peak-frequency relation and the
evolution of the spectral lags and the pulse widths with energy. We compare
our model results with the results of similar models in the literature and
also present a test case study of GRB 110920.

We summarize our main findings as follows:

  * •

We find that introduction of a finite shell thickness in the curvature
formulation can produce smooth light curves, i.e., without a rapid transition
from the rise to the decay portions even for the case of a rectangular rest-
frame emission profile. As we approach the infinitesimal shell (surface)
approximation, we recover the sharp featured light-curve profile of Qin et.
al. 2004 for a rapidly dimming intrinsic emission profile.

  * •

While we agree with Shen et al. (2005) that an infinitesimal shell produces no
discernible spectral lag using a rectangular emission-pulse-profile, we find
the situation to be different for a shell of finite thickness i.e., a finite
spectral lag can be produced even with a rectangular pulse profile;

  * •

The spectral lag evolution as a function of energy is quite sensitive to the
type of rest-frame spectrum. For example, the Comptonized–Epk model does not
appear to exhibit a saturation energy at which the spectral lags reach a
plateau phase as in the case of the Band and the broken-power-law functions.
We agree with Shen at al (2005) that the spectral lags seem to approach a
maximum when Epk is near the high-energy channel used in extracting the lag.
Most likely this simply reflects the break energy present in the assumed rest-
frame spectrum (i.e., Band and BPL);

  * •

All rest-frame spectral models tested exhibit the peak-flux – peak-frequency
relation although with exponents that differ from the predicted exponent of 3.
The significance of this discrepancy is not clear at this stage and warrants
further investigation;

  * •

The peak-flux – peak-frequency test for GRB 110920 yields an exponent of 1.64
+/- 0.012 compared to the theoretical one of 2.57 +/- 0.01 (with
Comptonized–Epk as the rest-frame spectrum). We consider this discrepancy to
be significant and the result to be in disagreement with the prediction based
purely on effects of curvature. Similar conclusions were reached by Dermer
(2004), Qin (2009) and Borgonovo and Ryde (2001);

  * •

Both the data (test GRB) and the model exhibit a very similar power-law trend
for the pulse width with energy. The plateau/power-law/plateau feature noted
by Qin et al. 2005 is well reproduced with a given choice of key model
parameters. In addition, we note that the power-law exponent is sensitive to
the assumed shell thickness. For the test-case GRB, the power-law exponent
matches well with exponents extracted from a larger sample of GRBs from
earlier studies (Fenimore et al. 1995; Peng et. al. 2006, 2012); and

  * •

Relatively good agreement is obtained with all rest-frame spectral models for
the spectral lag versus energy for the test GRB. This is somewhat surprising
given the result of the peak-flux – peak-frequency test noted above. It would
seem that some complex interplay is at work between various model parameters
such as shell thickness, variability time scale, the energy evolution of Epk
and the Lorentz factor. The investigation of the dependencies of these various
parameters is ongoing.

The role of the reported soft component of the light curve for GRB 110920 has
not been fully investigated in this study and is worth pursuing, particularly
with regard to the behavior of the peak-flux–peak-frequency relation. Finally,
we note that these studies are being extended to a larger sample of GRBs.

## Acknowledgements

The authors (AS and KSD) would like to acknowledge A. Eskandarian and O.
Kargaltsev (both from the George Washington University) for their valuable
contributions to the discussions as well as the financial support provided by
them to A. Shenoy at various stages of this work.

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