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

共Received 19 March 1999兲

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. 关S0163-1829共99兲13431-3兴

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 共PSHB兲 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 共355 nm兲 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 共a兲 and the PSHB spectra

共b兲 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

0163-1829/99/60共7兲/4481共4兲/$15.00 4481 ©1999 The American Physical Society

1共a兲. 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. 1共b兲, 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. 1共b兲 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 共a兲 and the PSHB spectra

共b兲 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

⫽ ប

2

2

/2MR

2

and E

Cube

⫺ E

B

⫽ 3ប

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兲, 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 共a兲 and the PSHB spectra 共b兲 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 共a兲 and the PSHB spectra 共b兲 of CuCl

nanocrystals in NaCl. The vertical solid lines in 共a兲 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

共

n⫽ 0, l⫽1, spheroidal mode

兲

, 共1兲

20

⫽ 0.82

t

/2Rc

共

n⫽ 0, l⫽2, torsional mode

兲

, 共2兲

20

⫽ 0.85

t

/2Rc

共

n⫽ 0, l⫽2, spheroidal mode

兲

, 共3兲

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

共

torsional mode

兲

, 共4兲

a1

⫽ 0.59

t

/Lc

共

ﬂexural mode

兲

, 共5兲

s1

⫽ 0.61

t

/Lc

共

shear mode

兲

, 共6兲

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 共solid circles兲,KCl共empty circles兲, and NaCl 共solid tri-

angles兲 as a function of the size of the nanocrystal. The horizontal

axes are drawn in proportion to the inverse of the diameter 共the

scales below the axis兲 and the side length 共the scales above the axis兲

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 (n⫽0, 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 共98G071兲.

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