**Anomalous quantum confinement of the longitudinal optical phonon mode in PbSe quantum dots**

J. Habinshuti,1,2O. Kilian,1,3O. Cristini-Robbe,4A. Sashchiuk,5A. Addad,6S. Turrell,2E. Lifshitz,5
B. Grandidier,1_{and L. Wirtz}1,7,*

1_{Institut d’Electronique, de Micro´electronique et de Nanotechnologie (IEMN), CNRS, UMR 8520, D´epartement ISEN,}*41 bd Vauban, 59046 Lille C´edex, France*

2_{Laboratoire de Spectrochimie Infrarouge et Raman, LASIR, Universit´e des Sciences et Technologies de Lille,}*Bˆat C5, 59655 Villeneuve d’Ascq Cedex, France*

3_{Department of Experimental Physics, Comenius University, Mlynska dolina F1, 842 48 Bratislava 4, Slovakia}

4_{Laboratoire de Physique des Lasers, Atomes et Mol´ecules (CNRS, UMR 8523), Bˆat P-5, Universit´e des Sciences et Technologies de Lille,}*59655 Villeneuve d’Ascq Cedex, France*

5_{Schulich Faculty of Chemistry, Russell Berrie Nanotechnology Institute, Technion, Haifa 32000, Israel}

6* _{Laboratoire de Structure et Propri´et´es de l’Etat Solide, Universit´e des Sciences et Technologies de Lille, 59655 Villeneuve d’Ascq, France}*
7

_{Physics and Materials Science Research Unit, University of Luxembourg, 162a Avenue de la Fa¨ıencerie, L-1511 Luxembourg, Luxembourg}(Received 14 February 2013; published 24 September 2013)

We have investigated the diameter dependence of the Raman spectra of lead selenide nanocrystals. The
first-order Raman peak at about 136 cm−1 and its second-order overtone at twice this wavenumber move up
in energy with decreasing nanocrystal diameter. This anomalous behavior is interpreted in terms of quantum
*confinement of the longitudinal optical (LO) phonon whose frequency displays a minimum at in the dispersion*
*of bulk PbSe. We perform ab initio calculations of the phonons of PbSe slabs with up to 15 layers. The LO mode*
perpendicular to the slab shifts indeed upwards with decreasing layer thickness, thus validating the interpretation
of the anomalous radius dependence of the Raman spectra in terms of quantum confinement.

DOI:10.1103/PhysRevB.88.115313 *PACS number(s): 63.22.Kn, 78.67.Bf*

**I. INTRODUCTION**

*The lead chalcogenides PbX (X*= S, Se, or Te) are unusual
IV-VI semiconductors with a rock salt crystal structure and
with very small direct band gaps (410–280 meV at room
temperature1_{) at the high-symmetry point L. This band gap can}

be reduced to zero upon compressing the lattice,2,3 _{and in its}

calculation, the effects of both spin-orbit coupling and electron
correlation must be carefully taken into account.3,4 _{Closely}

related to the electronic band structure, the lattice dynamics
*of PbX (Ref.*5) also display some unusual features, e.g., the
transverse optical (TO) phonon mode displays a minimum at

*Gamma* and is very soft [60 cm−1 for PbS,6 _{47 cm}−1 _{for}

PbSe,7 30 cm−1 for PbTe (Ref. 8)], which means that the
materials are close to a ferroelectric instability.9_{Furthermore,}

the longitudinal optical (LO) mode displays an anomalous dip
*at which can be explained as due to a pseudo-Jahn-Teller*
effect10_{or as a near Kohn anomaly.}11

The phonon dispersion of PbS (Ref.6) and PbTe (Ref.8) have been measured by inelastic neutron diffraction. Only scarce neutron diffraction data is available for PbSe.12,13

*Raman characterization of PbX is delicate because both TO*
and LO optical modes are in principle not Raman active in
the rock salt crystal structure. Nevertheless, in some Raman
*experiments, the LO mode (which is infrared, but a priori,*
not Raman active) has been observed. In PbSe, the LO mode
was detected around 135 cm−1 for PbSe films on a glass
substrate14_{and on a (111) surface of BaF}

2.15In these cases, the observability of the LO mode may be explained by a reduction of symmetry in the thin film geometry. The results coincide well with the 133 cm−1measured by tunneling experiments16

and the 136 cm−1 from neutron diffraction.12 Similarly, the LO mode had been detected in the first-order Raman spectrum of a PbTe sample coated with a thin Pb layer, which induces

surface electric fields and energy-band bending.17 _{In recent}

Raman measurements on PbSe single crystals under resonant
conditions, a peak at 130 cm−1is found.18 _{The authors argue}

that the LO mode observation may be due to the resonant Fr¨ohlich mechanism.19

The small band gap of lead chalcogenides, together with
a large exciton diameter (e.g., 20 nm in PbS and 46 nm in
PbSe) and small effective electron and hole masses, make
them an ideal candidate for strong quantum confinement in
the nanocrystalline form.20 _{While many studies on optical}

absorption and luminescence spectroscopy as a function
*of the diameter of PbX quantum dots (nanocrystals) have*
been performed,21–23 _{relatively few works on the Raman}

spectroscopy of these materials exist, and the results are
contradictory.24–27 _{Some results suggest a redshift of the}

*LO-mode line in the Raman spectra of nanocrystalline PbX as*
*compared to the spectrum of bulk PbX, but other studies give*
evidence for a blueshift. Since the spectra may be strongly
dependent on the experimental conditions, only a systematic
Raman study of nanocrystals with different diameters under
constant experimental conditions can give a definite answer.
This is the purpose of our paper in which we provide such a
study for spherical PbSe nanocrystals of varying diameters.
We observe a blueshift of the LO-mode-related Raman line
with decreasing crystal diameter. Our findings are interpreted
in terms of a simple quantum-confinement model for the LO
*mode as well as by ab initio calculations of phonons in PbSe*
slabs of varying thickness.

**II. EXPERIMENTS**

PbSe nanocrystals (NCs) were synthesized according to
literature methods.28 _{Briefly, lead oleate precursors were}

FIG. 1. (Color online) (a) Normalized absorbance spectra of PbSe
nanocrystals for a range of mean diameter between 7.1 nm (bottom)
*and 4.5 nm (top). The optical transitions are labelled α, β, γ , and*

*E*1. Inset: TEM image of a monolayer of PbSe nanocrystals with

a mean diameter of 6.0 nm. (b) and (c) Raman spectra of PbSe nanocrystals with a mean diameter of 6.8 nm for two optical excitation wavelengths.

of oleic acid and 1-octadecene that was stirred at a tem-perature of 110◦C. Tri-n-octylphosphine selenide precursors were prepared from the dissolution of selenium in tri-n-octylphosphine. Then they were injected into the solution containing lead oleate precursors that was heated at 180◦C. The reaction was allowed to continue for 1 to 15 min at a temperature between 110 and 145◦C, depending on the nanocrystal size that had to be reached. After centrifugation and redispersing the precipitated nanocrystals, the suspension was thoroughly washed to remove unreacted species.

The nanocrystal size was determined from the absorbance
spectrum of the nanocrystal suspension and compared with
the size measured with high-resolution transmission electronic
microscopy (HR-TEM). Figure 1(a) shows representative
absorption spectra of six PbSe nanocrystal suspensions. The
nanocrystal mean diameter varies between 4.5 and 7.0 nm.
The narrow size distribution of the nanocrystals results in
sharp optical transitions that were assigned based on the
nomenclature used in Ref.21. As seen in Fig.1(a), the lowest
*energy transitions α, β, γ are blueshifted as the nanocrystal*

*size decreases. A higher transition, labelled E*1, is also visible.
It is not affected by quantum confinement and corresponds
to a transition that occurs at critical points of the PbSe
bulk band structure.29,30_{From HR-TEM measurements of the}

suspensions characterized in Fig.1(a), the standard deviation
for the size distribution was found to be 4%. For smaller and
larger nanocrystal sizes, the size distribution slightly increases.
Seven samples of PbSe nanocrystals with an average size
ranging from 2.5 to 8.5 nm were then studied by Raman
spectroscopy. For that experiment, solutions of nanocrystals
in toluene were drop-cast on a Si wafer or on glass. The
Raman spectra were excited by an Ar-ion laser (514.5 nm)
or by a He-Ne laser (632.8 nm) at several laser powers and
were analyzed by a LabRAM Dilor Jobin Yvon spectrometer
or a T64000 confocal micro-Raman spectrometer. The incident
*light was focused into a spot diameter of 1 μm. Spectral*
resolution of the Raman spectrometer was about 1 cm−1.
Each spectrum was measured with an accumulation time of
100–300 s. In order to perform micro-Raman measurements
between room temperature and 250◦C, a Linkam THMS 600
heating stage was used.

Starting with the Ar-ion laser, the analysis of PbSe
nanocrystals with a mean diameter of 6.8 nm reveals the
existence of two peaks at 140 and 281 cm−1 that are due
to the excitation of one LO phonon (1-LO peak) and two
LO phonons (2-LO peak), respectively.26,27_{Choosing another}

*laser wavelength of 632.8 nm that is resonant with the E*1
transition yields a similar spectrum but improves the quality of
the measurements. As a result, this excitation wavelength was
preferred in the following experiments. While this result favors
a resonant Fr¨ohlich mechanism to account for the excitation of
the LO modes in PbSe nanocrystals, the Raman activity could
also be caused by a reduction of the nanocrystal symmetry,
since PbSe nanocrystals are known to be faceted.31,32

As the laser power has been shown to induce structural
changes in semiconductor nanocrystals,33 _{we first turn our}

attention to the positions of the 1-LO and 2-LO peaks as a function of the laser power. As shown in Fig.2(a)for PbSe nanocrystals with a mean diameter of 5.5 nm, an increase of the laser power causes a significant shift of the peak position for the 1-LO mode. The peak decreases from 142.4 to 138.8 cm−1 as the laser power increases from 1 to 10 mW. A similar trend is observed for PbSe powder [Fig.2(c)] that can be considered as bulk material due to the micrometer size of the PbSe grains. Interestingly, working with the smallest laser power, as well as raising the sample temperature, both lead to a systematic redshift of the 1-LO mode [Figs.2(b)and2(d)] that is quite consistent with the one found when the laser power changes. Although heating the nanocrystals leads to a larger lattice and thus a reduction of the phonon wavenumber in agreement with our observation, it could also contribute to an increase of the oxidation of the nanocrystal surfaces.

yti
s
n
et
nI
523 K
373 K
473 K
300 K
(a)
138
139
140
141
142
143
2 4 6 8 10
0
Laser power (mW)
138
139
140
141
143
142
500
450
400
350
300
Temperature (K)
PbSe bulk
PbSe NC
= 5.5 nm
*D*
)
d
(
)
c
(
)
mc
(
r
e
b
m
u
n
ev
a
W
-1
140 150
130
Wavenumber (cm )-1
1 LO ( )Γ
PbSe NC
= 5.5 nm
*D*
PbSe NC,*D*= 5.5 nm
(b)
1 LO ( )Γ
PbSe NC,*D*= 5.5 nm
140 150
130
Wavenumber (cm )-1
120
1.0 mW
2.5 mW
5.0 mW
10.0 mW
160
(arb. units)

FIG. 2. (Color online) (a) Raman spectra of a 5.5-nm PbSe nanocrystal film for different laser powers. The experiment was performed at 300 K. (b) Raman spectra of a 5.5-nm PbSe nanocrystal film for different sample temperatures. The laser power was set to 1.0 mW. (c) Raman wavenumbers of the 1-LO peak as a function of the laser power for the 5.5-nm PbSe nanocrystal film and a bulk PbSe sample. (d) Raman wavenumbers of the 1-LO phonon mode as a function of the temperature for the 5.5-nm PbSe nanocrystal film. redshifts and broadens the peak related to the 2-LO mode of PbSe. Therefore, in order to precisely measure the energy of the phonon modes in PbSe nanocrystals with different sizes, it is necessary to avoid their exposure to air and use a low laser power to preserve the nanocrystals from any chemical or structural modifications. The subsequent experiments were thus performed at the lowest laser power of 1 mW, limiting the nanocrystal exposition to air as much as possible.

Finally, Raman spectra were measured on different places of the same sample. As shown in Fig. 3(b) for a sample consisting of a thin film of 5.5-nm PbSe nanocrystals, small but still significant shifts are observed in the peak position of the 1-LO mode. As an increase of the heat leads to a redshift

100 150 200 250 300 120 140 160 180
1 LO ( )Γ (b) 1 LO ( )Γ
)
a
(
Wavenumber (cm )-1
(arb. units)
yti
s
n
et
nI
Wavenumber (cm )-1
PbSe NC
= 5.5 nm
*D*
PbSe NC
= 5.2 nm
*D*
2 LO ( )Γ
air-free
air-exposed
Spot 1
Spot 2
Spot 3

FIG. 3. (Color online) (a) Comparison of Raman spectra for a 5.2-nm PbSe nanocrystal film measured in an air-free environment (red) and air exposed (black), both excited at 5 mW. (b) Spatial fluctuations in the position of the 1-LO peak of 5.5-nm PbSe nanocrystal films. The laser power was set to 1.0 mW in (b).

FIG. 4. (Color online) Raman shift of the 1-LO peak as a function of the PbSe nanostructure size. Open circles: calculated frequencies of the 1-LO peak for PbSe (100) slabs with a width varying from 6 layers up to 15 layers. Open squares: 1-LO frequencies for PbSe(100) slabs according to the quantum-confinement model (extracted from the dispersion in Fig.5). Filled squares: measured Raman frequencies for films of PbSe NCswith different sizes. Inset: Raman wavenumbers of the 2-LO peak as a function of the PbSe nanocrystal size. The right panel shows the vibrational pattern of the quantum confined LO mode (with polarization perpendicular to the slab) of a 12-layer slab. The double column of Pb (gray) and Se (orange) atoms represents the motif of the elementary unit cell of the slab.

of the 1-LO mode, and the heat dissipation might be more or less efficient depending on the separation between neighbors in the nanocrystal film, we assign these fluctuations to inhomogeneities in the compactness of the nanocrystal array and the degree of disorder. On average, we find a standard deviation of 0.85 cm−1for the peak position on a given sample. Reiterating the measurements for the other samples, we were able to plot the energy of the 1-LO and 2-LO peaks as a function of the nanocrystal size. Starting with the bulk value of the 1-LO peak at 136 cm−1, Fig.4reveals a blueshift of the 1-LO peak as the nanocrystal size decreases from 8.5 to 6 nm. Then the peak position of the 1-LO mode fluctuates around 142 cm−1for smaller nanocrystal sizes. A similar behavior is obtained for the 2-LO peak, indicating that the energy of the LO mode readily varies with the nanocrystal sizes.

**III. RESULTS**

For the modelling of Raman spectroscopy of nanocrystals,
three levels of approximation have been used in the past
(for an overview see, e.g., Ref.35): (i) In the “confinement
model” one assumes that the restriction of the phonon mode to
*a nanocrystal of diameter D leads to an uncertainty q* ≈

110
120
130
140
150
mc(
re
b
mu
ne
va
w
1- ) _{3 layers}
4 layers
5
6
7
8
10 9
X
14

FIG. 5. (Color online) Calculated dispersion of the LO phonon
*branch between and X (Ref.*2). The phonon dispersion has been
upscaled by 6% in order to match the experimental value for the LO
*phonon at . The red circles mark the wave vectors and frequencies*
*of the perpendicular LO mode of an n-layer slab according to the*
zone-folding model (see text). For comparison with Fig.4: the 6-layer
slab has a width of 3.45 nm, the 14-layer slab has a width of 8 nm.
character of the vibrations. This equation of motion is coupled
with the Poisson equation in order to take into account the
coupling of the vibrations to the electric field that may result
from the vibrations of a polar material. (iii) Phonons of some
nanocrystals [e.g., GaP (Ref.39)] have been calculated on the
atomistic level by diagonalizing the dynamical matrix, which
was calculated from force constants and effective charges that
were both fitted to the bulk phonon dispersion.

The quantum confinement model has been used to explain
the Raman spectra of microcrystalline hexagonal boron nitride
(hBN): with decreasing crystal size, the Raman peak (which
is due to the TO phonon in that case) shifts upwards in
wavenumber.40 The reason comes from the pronounced local
*minimum at Γ of the TO mode, as confirmed by calculations*41

and measurements42_{of the phonon dispersion. Thus, a similar}

analysis may hold for the LO mode of PbSe which also
*displays a minimum at Γ .*2,10This situation is demonstrated in
Fig. 5 *where we show the LO mode dispersion in the Γ X*
**direction (i.e., for the phonon wave vector q in the [001]**
direction). While it is not possible to associate the phonon
modes of spherical nanocrystals directly with a bulk mode,
this analysis can be done easily for the phonon modes of slabs
(2D-systems). The wave vector in the direction perpendicular
to the slab surface is quantized. This leads to the mapping
*of the node free LO mode of an n-layer (100) slab to the*
**bulk LO phonon with wave vector q***n***= q**max*/(n*− 1), where
**q**max denotes the wave vector at the high-symmetry point

*X* on the boundary of the first Brillouin zone. The wave
**vectors q***n* and the corresponding frequencies are shown by

red circles in Fig. 5. In this zone-folding scheme, the LO
mode of slabs does indeed blueshift with decreasing number
*of layers until the maximum is reached for n*= 5. For smaller
layers, the frequency should decrease again, but here the limit
of the validity of the zone-folding method is reached. The
wavenumber difference between bulk and the 5-layer slab is
almost 10 cm−1, in good agreement with the experimental data
of Fig.4.

*The continuum model was used by Krauss et al.*24 _{for}

the analysis of their Raman data of PbS nanocrystals with

diameter of 4 nm. Since it uses as an input the bulk phonon
*dispersion which displays a dip for the LO mode at ,*6
the model correctly predicts a blueshift of the Raman peak
with respect to the bulk LO model. Thus, both the quantum
confinement and the continuum model describe qualitatively
correctly the blueshift of the Raman peak with decreasing
nanocrystal diameter. Nevertheless, both models do not take
into account the changes of the electronic structure (band gap,
dielectric screening, effective charges) with decreasing crystal
diameter. It would thus be desirable to perform microscopic
lattice dynamical calculations (such as in Ref. 39 for GaP
quantum dots). Due to the very peculiar electron and phonon
dispersions, a good semiempirical model for the interatomic
force constants is not available (and would also not take
into account the changes in dielectric screening and effective
*charges). In principle, one should therefore perform ab initio*
calculations of the phonons. Since this is currently still
not feasible for nanocrystals containing several hundreds or
*thousands of atoms, we performed ab initio calculations of*
phonons in a slab geometry which allows studying the effect
of a one-dimensional quantum confinement.

We use density functional theory (DFT) in the local density
approximation (LDA) as implemented in the codeQUANTUM
ESPRESSO.43 The slabs consist of 6 to 15 layers in the (001)
orientation. The slabs are arranged in a periodic supercell with
a vacuum of a width of 14 a.u. (before geometry relaxation)
between adjacent slabs. Wave functions are expanded in
plane-waves with an energy cutoff of 30 Ry and the first
Brillouin zone is sampled by a 10× 10 × 2 k-point mesh. For
*Pb, we use a Vanderbilt ultrasoft pseudopotential with the 5d*
electrons in the valence, and for Se we use a norm-conserving
Bachelet-Hamann-Schl¨uter pseudopotential. For the in-plane
lattice constant, we choose 6.124 ˚A, which is the experimental
value at room temperature.1In the out-of-plane direction, the
geometry is optimized, which leads to a 1.3% decrease in the
*nearest neighbor distance and to a surface rumpling (z*Se*− z*Pb,
*where z denotes the coordinate perpendicular to the layer) of*
0.072 ˚A.

The phonon frequencies are obtained from the equation det√ 1

*MsMt*

˜

*C*_{st}*αβ***(q)***− ω*2**(q)*** = 0.* (1)

The dynamical matrix ˜*C*_{st}*αβ***(q) corresponds to the change of**
*the force acting on atom t in direction β with respect to a*
*displacement of atom s in direction α. In Raman spectroscopy,*
**only modes with a wave vector q**→ 0 are excited. The
elements of the dynamical matrix are calculated by density
functional perturbation theory (DFPT).44,45

the mode, one has to take into account the macroscopic electric
field that accompanies the collective atomic displacements
and adds an additional “nonanalytical” term to the dynamical
matrix44
˜
*C*_{st}*αβ***(q)**=an*C*˜_{st}*αβ***(q)**+na*C*˜_{st}*αβ (q),* (2)
with
na

_{C}_{˜}

*αβ*st

**(q)**=

*4π*

*e*2

**(q· Z**∗

*s*)

*α*

**(q· Z**∗

*s*)

*β*

**q**

*· ε*

_{∞}

**· q**

*.*(3)

*Here, is the unit-cell volume. The electronic dielectric*

*tensor ε*

_{∞}

*and the effective charge tensor Z*∗ are calculated within DFPT, just as the analytical part of the dynamical matrix an

_{s}

_{C}_{˜}

*αβ*

st **(q). The name “nonanalytical” stems from the fact that**
**in the limit q**→ 0, the value ofna_{C}_{˜}*αβ*

st **(q) in Eq.** (3)depends
on the direction in which the limit is taken. In our calculations,
the limit is taken in the direction perpendicular to the layers.

The results of our calculations are displayed in Fig. 4
together with the experimental data points and the results
from the quantum-confinement model (extracted from Fig.5).
The calculations confirm the experimentally observed trend
of increasing LO frequency with decreasing nanocrystal size.
*For slabs with more than 10 layers, the ab initio results are*
parallel to the predictions of the quantum-confinement model.
The difference of about 3 cm−1 can be attributed to surface
effects and to the fact that in the slab calculation we perform
a geometry optimization in the perpendicular direction while
the (in-plane) lattice constant is the experimental bulk lattice
constant.46

An important uncertainty in our comparison of calculations and experiment consists in the dielectric environment of the nanocrystals, which can play a role in the coupling of the induced macroscopic electric field with the LO mode. On one hand, in our calculations, we use vacuum (with a dielectric constant of 1) in between the layers. On the other hand, the periodic stacking of PbSe slabs leads to an average macroscopic dielectric constant (in the perpendicular direction) between 4.1 (for the 6-layer slab) and 7.2 (for the 15-layer slab)47and thus to an effective dielectric environment which is probably not very far from the situation of a stacking of nanocrystals separated by layers of organic materials. Increasing the vacuum width between the slabs reduces the average dielectric constant but also reduces the effective charges such that the net influence of the vacuum width in our calculations is rather weak.

Finally, we shall examine the line shape of the peaks. As seen in Fig.1(c), the peak of the 1-LO mode shows a small shoulder at 125 cm−1and a broad shoulder around 200 cm−1. Such features are better seen in Fig. 6 that highlights the experimental variation of the peak intensity as a function of the laser power for a sample consisting of nanocrystals and a bulk sample. The shoulder at low frequency, labelled SP, appears for the nanocrystal and bulk samples and has been discussed above as originating from the Pb-O-Pb stretching mode at the surface of the nanocrystals. In contrast to this low-frequency component, the broad shoulder at high frequency is not always visible. Only for the highest laser powers, the peak of the 1-LO mode becomes really asymmetric. As this asymmetry is also seen for the bulk sample [Fig.6(b)], we rule out any confinement effect as the cause of this line shape modification.

FIG. 6. (Color online) Raman spectra of (a) a 5.5-nm PbSe nanocrystal film and (b) a bulk PbSe sample measured for different laser powers.

Interestingly, we note that the change of the line shape takes place with a reduction of the peak intensity of the 2-LO mode. Measurement of the ratio between the peak intensity of the 2-LO mode yields a reproducible value, around 0.65, when the spectra at the lowest and highest laser powers are compared. Since this ratio reflects the electron-phonon coupling strength, a reproducible decrease of the peak intensity indicates a stronger screening of the electron-LO-phonon interaction, consistent with a higher density of free charge carriers in the samples as the excitation power increases. Therefore, we suspect the asymmetry to correspond to a Fano effect48

rather than a nonadiabatic effect.38 _{Indeed, the interaction of}

electronic Raman transitions with discrete optical phonons
results in Fano interference, and severe asymmetries have
been found for heavily doped semiconductor crystals48,49 _{and}

nanostructures.50 _{In the case of PbSe nanocrystals, due to the}

eightfold degeneracy of the lowest conduction and valence band levels that can be slightly lifted up under confinement51

*and the high polarizability of the p-like orbitals,*32we think that
the intraband electronic transitions between different valleys
could be resonant with the LO phonon mode52 _{and give rive}

rise to the asymmetric line shape.

**IV. CONCLUSION**

The measured6,12 _{and calculated}2 _{anomaly of the }

**ACKNOWLEDGMENTS**

We acknowledge support from the EU Seventh Frame-work Program (EU-FP7 ITN Herodot, Grant No. PITN-GA-2008-214954). Calculations were performed at the IDRIS supercomputing center, Orsay (Proj. No. 091827).

Technion laboratory was supported by the EU-FP7 SANS (Self-Assembled Nanostructure System) project, Israel Sci-ence Foundation Project No. 1009/07 and 1425/04, and USA-Israel Binational Science Foundation (Project No. 2006-225).

*_{Corresponding author: ludger.wirtz@uni.lu}

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45_{A relevant question for the calculation of phonon frequencies in}

lead chalcogenides is the inclusion of spin-orbit coupling (SOC).
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**78, 224302 (2008)]**have shown that SOC causes a slight softening

LDA calculations with SOC give accidentally a worse agreement
with experimental data. This is due to the fact that SOC reduces the
band gap and leads to an underestimation (or even an inversion) of
the band gap. Since the very small band gap of lead chalcogenides
*gives rise to the anomalous dip of the LO mode [Kilian et al.*
(Ref. 2)], an underestimation of the gap will lead to an exaggerated
dip of the LO mode. For a completely correct calculation of the
*LO mode at , one would have to correct the band gap of lead*
*chalcogenides with the GW method on top of LDA* + SOC or
by using a hybrid functional together with SOC [as has been done
*in Hummer et al. (Ref. 4)]. Calculating phonons in that context*
has not been achieved yet. Thus, the best compromise to date is
the use of LDA without SOC, which (by cancellation of errors)
yields a reasonable band gap of all three lead chalcogenides and
thus reproduces quantitatively the anomalous phonon dip of the LO
*mode at .*

46_{The fact that the calculated frequencies are higher than the measured}

ones is due to the fact that the local density approximation tends to overbind, i.e., it underestimates the bond length (overbinding)

and thus overestimates the phonon frequencies when a geometry optimization is performed.

47_{For details of the dependence of the dielectric constant on the}

slab width and on the relation between the average macroscopic
dielectric constant and the dielectric constant in the interior of the
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