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Journal ArticleDOI

Size dependence of confined acoustic phonons in CuCl nanocrystals

15 Aug 1999-Physical Review B (American Physical Society)-Vol. 60, Iss: 7, pp 4481-4484

Abstract: Persistent spectral hole burning spectroscopy was used to study the size dependence of the confined acoustic phonons in CuCl nanocrystals embedded in silicate glass, NaCl, and KCl. It is found that the energies of the confined acoustic phonons in the nanocrystals in glass and KCl are almost the same, but twice larger than those in NaCl, which were obtained from the Stokes-side acoustic phonon holes with the same excitation energies. The confined acoustic phonons in the nanocrystals in glass and KCl can be well explained in terms of the lowest-frequency vibrational modes calculated on a sphere model with a free boundary condition. However, the energies of the confined acoustic phonons in the nanocrystals in NaCl are lower than the frequencies of the lowest-frequency vibrational modes predicted by a cube model with a free boundary condition. This observation shows that the energies of the confined acoustic phonons in CuCl nanocrystals depend on the size, the shape, and the boundary condition of the nanocrystals.
Topics: Phonon (51%)

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Summary

  • Persistent spectral hole burning spectroscopy was used to study the size dependence of the confined acoustic phonons in CuCl nanocrystals embedded in silicate glass, NaCl, and KCl.
  • It is found that the energies of the confined acoustic phonons in the nanocrystals in glass and KCl are almost the same, but twice larger than those in NaCl, which were obtained from the Stokes-side acoustic phonon holes with the same excitation energies.
  • The confined acoustic phonons in the nanocrystals in glass and KCl can be well explained in terms of the lowest-frequency vibrational modes calculated on a sphere model with a free boundary condition.
  • The energies of the confined acoustic phonons in the nanocrystals in NaCl are lower than the frequencies of the lowest-frequency vibrational modes predicted by a cube model with a free boundary condition.
  • This observation shows that the energies of the confined acoustic phonons in CuCl nanocrystals depend on the size, the shape, and the boundary condition of the nanocrystals.

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Size dependence of confined acoustic phonons in CuCl nanocrystals
Jialong Zhao and Yasuaki Masumoto
Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan
Received 19 March 1999
Persistent spectral hole burning spectroscopy was used to study the size dependence of the confined acoustic
phonons in CuCl nanocrystals embedded in silicate glass, NaCl, and KCl. It is found that the energies of the
confined acoustic phonons in the nanocrystals in glass and KCl are almost the same, but twice larger than those
in NaCl, which were obtained from the Stokes-side acoustic phonon holes with the same excitation energies.
The confined acoustic phonons in the nanocrystals in glass and KCl can be well explained in terms of the
lowest-frequency vibrational modes calculated on a sphere model with a free boundary condition. However,
the energies of the confined acoustic phonons in the nanocrystals in NaCl are lower than the frequencies of the
lowest-frequency vibrational modes predicted by a cube model with a free boundary condition. This observa-
tion shows that the energies of the confined acoustic phonons in CuCl nanocrystals depend on the size, the
shape, and the boundary condition of the nanocrystals. S0163-18299913431-3
Recently, semiconductor nanocrystals have attracted con-
siderable attention from the viewpoint of the fundamental
physics and the application possibility to functional devices.
1
In the nanocrystals, not only the electronic energy levels but
also the lattice vibrational modes become discrete due to the
three-dimensional confinement. The size dependence of elec-
tronic and vibrational spectra and the electron-phonon inter-
action in the nanocrystals is very important for basic under-
standing of these materials. However, the knowledge on the
size dependence of vibrational spectra in the nanocrystals is
still in the elementary stage.
Raman scattering is one of the most popular methods to
obtain information about the lattice vibrational modes in sol-
ids. The low-frequency Raman scattering in the region of
several milielectron volts was first observed in
MgCr
2
O
4
-MgAlO
4
microcrystals by Duval, Boukenter, and
Champagnon.
2
After that, similar low-frequency Raman
scattering spectra for various nanocrystals such as Ag, CdS,
and Si quantum dots were reported by several researchers.
3–7
The frequencies of the low-frequency Raman scattering
peaks were found to be inversely proportional to the diam-
eter of the nanocrystal and consistent with those of the low-
frequency vibrations of spherical particles predicted by
Lamb’s theory.
8
Therefore, they were attributed to the con-
fined acoustic phonons in the nanocrystals.
2–7
The persistent spectral hole burning PSHB phenomena
have been widely observed in II-VI and I-VII quantum dots
and applied to site-selective spectroscopy of quantum
dots,
9–12
and, therefore, the PSHB method as a simple
method can be used to study the confined acoustic phonons
in nanocrystals. CuCl nanocrystals provide a typical example
of quantum dots in the weak confinement regime. The nar-
row homogeneous linewidths of excitons in CuCl nanocrys-
tals in glass and NaCl were observed by using PSHB, tran-
sient four-wave mixing, and accumulated photon-echo
methods.
13–15
In order to understand their origin and the
exciton-phonon interaction with confined acoustic phonons,
it is necessary to study systematically the confined acoustic
phonons in CuCl nanocrystals. In the previous work,
16
the
observation of the confined acoustic phonons in CuCl nanoc-
rystals was reported briefly, but the acoustic phonon side-
band holes in CuCl nanocrystals in NaCl were not well dis-
tinguished due to the limitation of the spectral resolution.
In this paper, we study in detail the size dependence of the
confined acoustic phonons in CuCl nanocrystals in glass,
NaCl, and KCl by the PSHB method with improved spectral
resolution. Further, we explain the confined acoustic
phonons of CuCl nanocrystals in glass, KCl, and NaCl in
terms of the lowest-frequency vibrations of the nanocrystals
with spherical and cubic shapes, respectively.
CuCl nanocrystals used in the experiment were embedded
in sodium aluminoborosilicate glass and NaCl and KCl crys-
tals. The size of the nanocrystals was controlled by heat
treatment with different temperature and time. The average
size of the nanocrystals was estimated by small-angle x-ray
scattering. The samples were directly immersed in super-
fluid helium at2Kinanoptical cryostat. A narrow-band dye
laser pumped by the third harmonics of the output of a
Q-switched Nd
3
: YAG laser 355 nm was used as a pump
source. The pulse duration and repetition rate were about 5
ns and 30 Hz, respectively. The spectral linewidth was about
0.014 meV. A halogen lamp was used as a probe source. The
PSHB spectrum was measured as follows: First, the absorp-
tion spectrum was obtained and then the sample was exposed
to 9000 shots of dye laser pulses to burn a persistent spectral
hole at an excitation energy. The absorption spectral change
d is defined as the difference between the spectra be-
fore and after the laser exposure. The subsequent measure-
ments were performed at the new position of the samples and
were not carried out at the position burnt previously. The
transmitted light of the samples was detected by a liquid-
nitrogen-cooled charge-coupled device in conjunction with a
75-cm spectrometer involving a 1800 grooves/mm grating
operated in the mode of the second order of diffraction. The
spectral resolution of the experiment was about 0.13 meV.
Figure 1 shows the absorption a and the PSHB spectra
b of CuCl nanocrystals with an average radius of 1.4 nm in
glass at 2 K. The Z
3
exciton absorption band is inhomoge-
neously broadened due to the size distribution of the nanoc-
rystals. A large blue shift of 64.9 meV is observed in Fig.
PHYSICAL REVIEW B 15 AUGUST 1999-IVOLUME 60, NUMBER 7
PRB 60
0163-1829/99/607/44814/$15.00 4481 ©1999 The American Physical Society

1a. The vibrational structures in the absorption spectrum of
the nanocrystals cannot be distinguished since the Z
3
exciton
absorption band is not from the single-sized nanocrystals, but
from an ensemble of various sized nanocrystals. As seen in
Fig. 1b, on the other hand, the narrow zero-phonon holes
that coincide with the energies of pump beams, and both
broad and asymmetric acoustic phonon sideband holes
around the zero-phonon line as well as antiholes, are clearly
observed. The low-energy side hole is referred to as a
pseudophonon wing or a Stokes-side acoustic phonon
hole,
10,17
corresponding to the acoustic phonon-assisted ab-
sorption of the nanocrystals. The high-energy side one is
called a real phonon hole or an anti-Stokes-side acoustic
phonon hole, coming from the decrease in the acoustic
phonon-assisted absorption of the resonant hole. The interval
between the zero-phonon line and the Stokes-side acoustic
phonon hole in the PSHB spectra is considered to be the
energy of the confined acoustic phonon in the
nanocrystal.
16,17
Thus, the energy of the confined acoustic
phonon from the Stokes-side acoustic phonon hole in spec-
trum C in Fig. 1b was estimated to be 2.2 meV. By de-
creasing the nanocrystal size, the confined acoustic phonon
energy increases gradually. In addition, a sharp peak with a
Stokes shift of about 23.5 meV observed in the PSHB spec-
tra was related to the softened LO phonon.
12
Figure 2 shows the absorption a and the PSHB spectra
b of CuCl nanocrystals in NaCl at 2 K. Only in NaCl,
oscillatory fine structures are observed between 3.22 and
3.28 eV in the inhomogeneously broadened Z
3
exciton ab-
sorption band and can be explained by the size-quantized
energy states of the Z
3
excitons confined in CuCl quantum
cubes whose side changes stepwise in a unit of a/2, where a
is the lattice constant of the CuCl crystal.
11,18
The vertical
solid lines in Fig. 2 indicate the calculated energies of the Z
3
exciton states in the nanocrystals assumed as quantum cubes.
There is a well-resolved peak in both Stokes- and anti-
Stokes-side acoustic phonon holes when the excitation en-
ergy is in the Z
3
exciton absorption band. The energies of the
confined acoustic phonons from both holes were estimated to
be 0.7 meV at excitation energy of 3.2621 eV and much
smaller than those in CuCl nanocrystals in glass and KCl. In
addition, the lower-energy satellite holes in the PSHB spec-
tra were supposed to originate from the hole burning of the
exciton energy states of CuCl quantum cubes.
11
Here, we
must point out that the peaks around the zero-phonon holes
in CuCl nanocrystals in NaCl reported in the previous work
16
were from the hole burning of the lowest exciton energy
states relaxed from the photoexcited exciton state
11
while the
confined acoustic phonon energies were too small to be ob-
served due to the limitation of the spectral resolution.
The PSHB spectra of CuCl nanocrystals with different
sizes in glass, NaCl, and KCl were measured systematically
to determine the relations between the confined acoustic pho-
non energy and the size of the nanocrystal as shown in Fig.
3. Both of them were estimated from the Stokes-side acous-
tic phonon hole at 2 K. The radius and the side length of
spherical and cubic CuCl particles in the weak confinement
regime
1,11,19,20
were calculated by the confinement energies
E
Sphere
E
B
2
2
/2MR
2
and E
Cube
E
B
3
2
2
/2ML
2
assuming that the Stokes-shifted phonon sideband energy co-
incides with the lowest exciton state energies for spherical
and cubic CuCl particles, where E
B
is bulk Z
3
exciton en-
ergy (E
B
3.2022 eV at 2 K, M 2.3m
0
is the translational
mass of the bulk exciton, and R and L are the radius and the
side length of the spherical and cubic nanocrystals, respec-
tively.
The free vibrations of a homogeneous elastic sphere under
stress-free boundary conditions were first studied theoreti-
cally by Lamb.
8
Two kinds of eigenmodes, the spheroidal
and torsional modes, were derived. The eigenmodes are char-
acterized by the angular momentum quantum number l and
FIG. 1. Linear absorption a and the PSHB spectra b of CuCl
nanocrystals with an average radius of 1.4 nm in glass. The excita-
tion intensity is about 100 nJ/cm
2
and excitation energies for spectra
A, B, C, D, E, F, G, and H are 3.2583, 3.2626, 3.2669, 3.2712,
3.2755, 3.2798, 3.2841, and 3.2882 eV, respectively.
FIG. 2. Linear absorption a and the PSHB spectra b of CuCl
nanocrystals in NaCl. The vertical solid lines in a indicate the
calculated energies of the Z
3
exciton states under the assumption of
a quantum cube. The excitation intensity is about 50 nJ/cm
2
and
excitation energies for spectra A, B, C, D, E, F, G, and H are
3.2336, 3.2368, 3.2410, 3.2452, 3.2494, 3.2535, 3.2578, and 3.2621
eV, respectively.
4482 PRB 60
BRIEF REPORTS

branch number n. The longitudinal and transverse sound ve-
locities of bulk CuCl are
l
4172 m/s and
t
2021 m/s de-
rived from the elastic constants
21
c
11
0.72 and c
44
0.169
10
12
dyn/cm
2
at 4.5 K and the mass density
4.136 g/cm
3
. The frequencies of the lowest eigenmodes of
a spherical particle are
10
1.17
t
/2Rc
n 0, l1, spheroidal mode
, 1
20
0.82
t
/2Rc
n 0, l2, torsional mode
, 2
20
0.85
t
/2Rc
n 0, l2, spheroidal mode
, 3
where c is the speed of the light in the vacuum. In compari-
son with the experimental data, the calculated energies of the
confined acoustic phonons as a function of the nanocrystal
diameter are shown in Fig. 3 with the dashed lines. Figure 3
clearly indicates that the energies of the confined acoustic
phonons are inversely proportional to the nanocrystal diam-
eter, which are in good agreement with the frequencies of the
lowest-frequency vibrational modes of a spherical particle
predicted with Lamb’s theory with a free boundary condi-
tion. Therefore, we may consider that the shape of CuCl
nanocrystals in glass and KCl is spherical and the surface of
the nanocrystals is stress-free. In other words, the contact
between the nanocrystals and glass and KCl is considered to
be weak.
The energies of the confined acoustic phonons of CuCl
nanocrystals in NaCl was found to be much smaller than
those in glass and KCl obtained with the same excitation
energies and can hardly be explained by the elastic sphere
model. The previous works have shown that the oscillatory
fine structures observed only in the Z
3
exciton absorption
band of the nanocrystals in NaCl, the resonantly burned hole,
and the lower-energy satellite holes were related to quantum
cubes.
11,18
However, the problem of determining the vibra-
tion frequencies of a free cube cannot be solved exactly like
a free sphere. The free vibrations of an isotropic cube were
calculated numerically by Demarest, Jr. using the Rayleigh-
Ritz technique.
22
The frequencies of the lowest vibrational
modes T
d1
, F
a1
, and S
s1
of a cubic particle are expressed as
d1
0.45
t
/Lc
torsional mode
, 4
a1
0.59
t
/Lc
flexural mode
, 5
s1
0.61
t
/Lc
shear mode
, 6
and are shown in Fig. 3 as the solid lines. Figure 3 shows
that the energies of the confined acoustic phonons in CuCl
nanocrystals in NaCl are nearer to the frequencies of the
vibration mode T
d1
of a free cube than a free sphere. Thus,
the confined acoustic phonons in the nanocrystals in NaCl
are rather considered as the lowest-frequency vibrations of
cubic nanocrystals, which is in accordance with the previous
results.
11,18,23
This shows that the confined acoustic phonon
energy is associated with the shape of the nanocrystals in
matrices. One can still see the significant discrepancy be-
tween the measured and the calculated results. The most pos-
sible reason is that an ideal cube with a free boundary con-
dition was simply assumed for the calculation of the
frequencies of the confined acoustic phonons of CuCl nanoc-
rystals in NaCl. It is noted that only in NaCl, the orientation
of CuCl nanocrystals was demonstrated to be aligned parallel
to that of the NaCl crystal by two-photon absorption.
23
The
elastic interaction between the nanocrystals and the sur-
rounding lattice was studied and the existence of deforma-
tion fields around the nanocrystals was observed by means of
a scanning near-field microscope.
24,25
These results show
that the surface of CuCl nanocrystals in NaCl is highly
strained and the atomic arrangement at the interface between
the nanocrystals and the matrix NaCl is more regular than
those at the interfaces between the nanocrystals and the sur-
rounding glass and KCl, perhaps resulting in the formation
of cubic nanocrystals in NaCl. As a result, CuCl nanocrystals
interact with the surrounding NaCl and cannot vibrate freely,
similarly to damped oscillators, leading to the softening of
the frequencies of the low-frequency vibrations.
The influence of the surrounding matrix on the confined
acoustic phonons in nanocrystals have recently been studied
by an elastic continuum theory.
26,27
Contradictory conclu-
sions were made. Ovsyuk and Novikov
26
claimed that the
matrix effects are important, while Montagna and Dusi
27
re-
ported that the influence of matrices are rather small and can
be negligible. In this experiment, the effect of the surround-
ing matrix on the confined acoustic phonons cannot be ob-
served in the nanocrystals in glass and KCl, but exists in the
nanocrystals in NaCl. Therefore, the vibrational frequencies
FIG. 3. Confined acoustic phonon energies of CuCl nanocrystals
in glass solid circles,KClempty circles, and NaCl solid tri-
angles as a function of the size of the nanocrystal. The horizontal
axes are drawn in proportion to the inverse of the diameter the
scales below the axis and the side length the scales above the axis
of the spherical and cubic nanocrystals, respectively. The upper
horizontal axis is drawn in proportion to the square root of the
confinement energy. The dashed lines represent the size dependence
of the calculated frequencies of the lowest spheroidal (n0, l 1
and n 0, l 2) and torsional modes (n 0, l 2) of a spherical
particle with a free boundary condition, respectively. The solid lines
drawn from top to bottom are the results of the theoretical calcula-
tion for the lowest torsional, flexural, and shear modes of a cubic
particle with a free boundary condition.
PRB 60
4483BRIEF REPORTS

of the confined acoustic phonons in CuCl nanocrystals de-
pend on the boundary conditions between nanocrystals and
matrices.
In summary, we have studied systematically the size de-
pendence of the confined acoustic phonons in CuCl nanoc-
rystals embedded in glass, NaCl, and KCl by the PSHB spec-
troscopy. The confined acoustic phonons in CuCl
nanocrystals in glass and KCl are explained as the lowest-
frequency vibrations of the spherical nanocrystals with the
free boundary condition. Those in CuCl nanocrystals in NaCl
are explained as the softened lowest-frequency vibrations of
the cubic nanocrystals with the strained boundary condition.
We would like to thank Dr. T. Mishina, Dr. T. Okuno, Dr.
J. Qi, and Dr. M. Ikezawa for valuable discussions. This
work was partially supported by the Scientific Research
Grant-in-Aid No. 10554011 from the Ministry of Education,
Science, Sports, and Culture of Japan. Small angle x-ray
scattering experiments were done in the Photon Factory of
the National Laboratory for High-Energy Physics 98G071.
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On the nanoscale, the combination of discrete mode properties with significant phonon coupling strengths has the potential for the coherent coupling of the NV center to selected distinct modes. We analyze the nanodiamond size dependence of the coupling to low-energy acoustical phonon modes, thereby focusing on its (strong) impact revealed in energy shifts on the orbitally distinct NV ground-excited state transition. In this way, the potential for coherent coupling for crystal sizes . 30 nm is shown. Moreover, for the specific example of an elastic sphere, the breathing mode is identified as promising, pairing close to homogeneous interaction with stable mode properties. Schemes for exploiting this phonon mechanism for the conditional manipulation of NV-centers are constructed. This is achieved by introducing the phonon coupling to the ground state by Raman transitions and a direct conditional coupling amongst NV centers is obtained in a dispersive regime. Moreover, several concepts for improving the speed and robustness of such conditional gates are addressed. Scaling NV-centers in a controllable coherently-interacting way remains a crucial milestone for a wide range of quantum applications. In contrast to the implantation approach in bulk crystals, we propose an inverse bottom-up approach by assembling nanodiamonds on the nanoscale in designed biological (protein) lattices. This allows for controlled distances of around ten nanometers, thus overcoming fundamental resolution limitations of implantation techniques imposed by scattering. A complete framework for the implementation of controlled quantum operations in such assembled nanodiamond networks is presented. In particular, we show that the challenging random distribution of the NV symmetry axis, combined with a magnetic field, allows for both beneficial individual addressing and a uniform Ising-type coupling. A detailed study is carried out to analyze the decoherence properties based on a diamond surface noise model; this then leads to the construction of fully decoupled gate interactions by time-addition or second order driving methods. Applications, such as cluster state computation or the simulation of Heisenberg chains are proposed and their viability in this framework is supported by numerical simulations. Interferometry with massive particles has potential in studying decoherence mechanisms, but also for testing fundamental limitations of quantum mechanics. We propose and analyze the interferometry with nanodiamonds in a Ramsey-Bordé setup in view of identifying mass enhanced quantum gravity (QG) modifications of the energy dispersion. A phase suppression mechanism associated with the thermal motion and gravitation turns out to render QG signatures inaccessible in such systems; as a remedy a revised setup based on gravitational momentum inversion is constructed. A careful analysis of the interference pattern for different particles sizes, temperatures and decoherence influences suggests typical nanodiamonds capable to shed light on the controversial discussions in the field of QG. Moreover, simple widely applicable formulas for calculating the interference phase and path contributions are derived. Last, a framework for the interpretation and customization of coherent hyperfine-couplings between a ‘pulsed decoupled’ NV-center and single nuclear spins is introduced. This is based on the filter formalism widely known from decoherence descriptions, with the filter describing the external control on the NV-center. Based on an analysis in the perturbative limit, thereby identifying the relation to semiclassical frameworks and resonance conditions for the pulse spacing, this concept is extended beyond that limitation by relying on ‘sliced evolutions’ of non-equidistant decoupling pulses. This then allows for an intuitive interpretation and design of gate interactions both in the weak and strong coupling limit, thus applicable for single spin sensing and the intended use of nuclear spins as a qubit. List of Publications Parts of this thesis are based on or have been taken from material first published in the following peer-reviewed journals: [A1] A. Albrecht, A. Retzker, F. Jelezko and M. B. Plenio, Coupling of nitrogen vacancy centres in nanodiamonds by means of phonons, New. J. Phys. 15, 083014 (2013), arXiv: 1304.2192. Copyright (2013) by IOP Publishing and Deutsche Physikalische Gesellschaft (Creative Commons Attribution Unported (CC BY 3.0)). Chapter 4. [A2] A. Albrecht, G. Koplovitz, A. Retzker, F. Jelezko, S. Yochelis, D. Porath, Y. Nevo, O. Shoseyov, Y. Paltiel and M. B. Plenio, Self-assembling hybrid diamond-biological quantum devices, New. J. Phys. 16 093002 (2014), arXiv: 1301.1871. Copyright (2014) by IOP Publishing and Deutsche Physikalische Gesellschaft (Creative Commons Attribution Unported (CC BY 3.0)). Chapter 5 and parts of Chapter 3. [A3] A. Albrecht, A. Retzker and M. B. Plenio, Testing quantum gravity by nanodiamond interferometry with nitrogen-vacancy centers, Phys. Rev. A 90 033834 (2014), arXiv: 1403.6038 (2014). Copyright (2014) by the American Physical Society (APS Journals). Chapter 6. Publications not covered in this thesis: [A4] A. Albrecht, A. Retzker, Ch. Wunderlich and M. B. Plenio, Enhancement of laser cooling by the use of magnetic gradients, New. J. Phys. 13, 033009 (2011), arXiv: 1009.2441.

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Cites background from "Size dependence of confined acousti..."

  • ...Such modified phonon properties in nanoparticles have been observed in numerous experiments, as in Raman scattering [263] and spectral hole burning spectroscopy [302], modified specific heat properties [262] or by the characteristic scaling of the exitonic dephasing rate in semiconductor nanocrystals [257]....

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