научная статья по теме APPLYING AN IMPROVED PHONON CONFINEMENT MODEL TO THE ANALYSIS OF RAMAN SPECTRA OF GERMANIUM NANOCRYSTALS Физика

Текст научной статьи на тему «APPLYING AN IMPROVED PHONON CONFINEMENT MODEL TO THE ANALYSIS OF RAMAN SPECTRA OF GERMANIUM NANOCRYSTALS»

APPLYING AN IMPROVED PHONON CONFINEMENT MODEL TO THE ANALYSIS OF RAMAN SPECTRA OF GERMANIUM NANOCRYSTALS

V. A. Volodina'h* D. V. Marina'h, V. A. Sachkovc, E. B. Gorokhova, H. Rinnertd, M. Vergnatd

a Rzhanov Institute of Semiconductor Physics, Siberian Branch. Russian Academy of Sciences

630090, Novosibirsk, Russia

b Novosibirsk State University 630090, Novosibirsk, Russia

€ Omsk Scientific Center, Siberian Branch, Russian Academy of Sciences 644040> Omsk, Russia

dInstitut Jean Lamour UMR CNRS 719S Universite de Lorraine, B.P. 70239 54506, Vandrmivre-les-Nancy Cedex, France

Received April 20, 2013

The improved phonon confinement model developed previously [11] is applied for definition of germanium nanocrystal sizes from the analysis of its Raman scattering spectra. The calculations based on the model allow determining the sizes of germanium nanocrystals more precisely from the analysis of their Raman spectra. In some cases, the comparative analysis of Raman data and electron microscopy data is carried out, and good agreement is observed.

DOI: 10.7868/S0044451014010076

1. INTRODUCTION

Semiconductor nanocrystals (NCs) in dielectric films have attracted the interest of researchers because of their electrical and optical properties tunable by altering the size and also because of their potential in new optoelectronic and nonvolatile memory devices. Due to electron confinement, the optical gap of semiconductor NCs is size-dependent [1,2], which is usually-called quantum size effect. A semiconductor NC embedded in a wide-gap insulating matrix lias a discrete electron spectrum due to localization of the electron wave function in three directions [3]. In germanium NCs, the quantum size effect should be stronger due to a larger exciton Bohr radius in Ge compared with Si. The vibrational spectra of quantum dots also differ from the vibrational spectra of bulk materials, and they are determined by the composition, size, shape,

* E-mail: volodin'flisp.nsc.ru

and mechanical stresses of the quantum dots. For this reason, analysis of the vibrational spectrum of nanos-tructures based 011 Raman spectroscopy is widely used for studying the quantum dots and can give information about their structure (size and shape).

Si NCs are studied more intensively than Ge NCs. For example, more than thirty years ago, it was experimentally observed that Raman spectra of polycrys-talline Si and bulk Si are different [4]. The polycrys-talline Si studied in [4] contains Si NCs, and the authors explained the shift and broadening of optical phonon Raman peaks by softening of the conservation law of the quasimomentum of phonons due to phonon confinement in NCs. The quasimomentum conservation law is softened according to the Heisenberg uncertainty principle. The phonon confinement model (PCM) proposed in [4] was developed and generalized in [5], where it was also applied to one dimensional objects (quantum well wires). In [5], the dependence of the calculated Raman spectra 011 phonon weight functions was first discussed. Because of its clear physical approach, the

PCM is being developed up to now fC 11]. The model was considerably improved for Si NCs by taking the angle dispersion of transverse and longitudinal phonons into account [11]. The phonon dispersion was calculated with the Keating model instead of being approximated by empirical expressions, as was done in earlier approaches. The PCM was also applied to the analysis of sizes of Ge NCs [12 18], but the above-mentioned factors were not taken into account. The present work is devoted to applying the improved PCM to Ge NCs and to comparative analysis of calculated and experimental Raman spectra.

2. EXPERIMENTAL

In this study, the correctness of the model is verified by comparative analysis of the calculated and experimental Raman spectra. The triple Raman spectrometer T64000 (Horiba Jobin Yvon) with micro-Raman setup was used. All experimental spectra were recorded in the backscattering geometry at room temperature using the 514.5-nm line of an Ar+ laser. The incident light was polarized linearly, the polarization of scattered light was not analyzed. The spectral resolution was not worse than 1.5 cm"1. The detector was a silicon-based CCD matrix, cooled with liquid nitrogen. The power of the laser beam reaching the sample was 2 niW. To avoid heating by the laser beam, the sample was placed slightly farther than the focus, and the spot size was about 30 //in.

Films containing Ge NCs were obtained using two methods. The first is deposition on a cooled substrate from supersaturated germanium monoxide (GeO) vapor. The GeO is metastable in the solid phase and dissociates at the temperature about 300 °C (solid-state chemical react ion):

2GeO(solid) —¥ Go(solid)—Go02(solid). (1)

Depending on the substrate temperature, undis-sociated GeO films or Ge:GeO-2 films with different sizes of Ge clusters can be deposited. Using postdeposition thermal annealing treatments at different temperatures, Ge clusters can be crystallized to obtain Ge NCs of different sizes. The higher the annealing temperature is, the bigger the Ge NCs are. The first method is described in more detail elsewhere [15,19]. If the deposition of GeO occurs at higher temperatures, the solid-state chemical reaction also takes place, but in this case the Ge NCs are formed, and post-growth annealing is not needed. According to the electron microscopy data, the sizes of Ge NCs can range

from 2.5 mil (growth at 450 °C) to 7 8 mil (growth at 580 °C).

The second method is successive evaporation of Ge02 and Si02 by an electron beam in high vacuum (about 10-8 Torr) and deposition onto substrates maintained at 100 °C. The pressure during the evaporation increases to 3 x 10-6 Torr due to the partial decomposition of G0O2. The deposition rate of 0.1 iim/'s was controlled by a quartz microbalancc. I11 fact, under electron bombardment of G0O2, its partial decomposition into Ge, O2 and GeO occurs. The last two components are more volatile, but, unlike O2, GeO is easily deposited onto a cool substrate. Hence, the deposited germanium oxide is substoichio-metric, namely Ge03., where x is close to 1 [20]. I11 the same paper [20], it was also shown that silicon oxide layers have a composition very close to Si02-Three multilayer structures containing 10 periods of Ge03.(4 11111) Si(),( I 11111), GeO., (2 am) SH),( I 11111), and Ge03.(l nm)/Si02(4 11111) were covered by a Si02 cap layer with a thickness equal to 100 11111. The samples were annealed in high-vacuum quartz tube with a tubular oven. The heating rate was 10°C/niin. When the annealing temperature (600 °C) was reached, the samples were held in the oven for 30 mill, then the oven was removed and the films cooled naturally. The second method is described in more detail elsewhere [21,22].

3. RESULTS AND DISCUSSION

3.1. Improved PCM and Raman spectra for Ge NCs

As mentioned above, the PCM allows calculating the Raman spectra for NCs of various sizes [9 11]. The PCM was considerably improved by taking dispersion of phonons into account not only in the magnitude of the quasinionient 11111 but also in its direction [11]. A significant refinement of the model was reached by using the widely approved Keating model to calculate the phonon dispersion instead of approximating it by empirical expressions as was done in earlier approaches [11]. I11 general, for crystals with a diamond-type lattice (like Si and Ge), there are six phonon branches with dispersions u;■(</).

The phonon dispersions of Ge and Si are similar, and hence the first-order Raman spectrum for the phonon weight function W(r,L) = exp(^4r2/L2) is [11]

g Qm a /

i= 1 n

q L

exp I---— ) x

4(g)

n(wi(?)) + l }q2dq

(c-^(g))2 + (r/2)2

, (2)

where L is the diameter of NCs,

'hco'(q)

n(u>'(q)) =

exp

kT

is the Bose-Einstein factor, and T is the full width at half maximum of the Raman peak of a single phonon. Wavenumbers q range from 0 to qmax (an edge of the Brillouin zone). We note that for directions with high symmetry ((100) and (111)), some phonon branches are degenerate. The density of states for phonons is proportional to q2dq.

In some approaches, the phonon frequencies are determined using quantum mechanical calculations from first principles [23,24], but this method requires large computational resources, while NCs with diameters more than 5 nm contain more than one thousand atoms. To calculate the phonon dispersion, the Keating model of valence forces [25] was therefore used, as in [11]. In this simple, but sufficiently adequate model, the elastic energy of the crystal depends on the bond length and on the deviation of the bond angle from ideal tetrahedral angles.

We consider interaction only between nearest-neighbor atoms. For a crystal with a diamond-type lattice, the elastic energy of the unit cell is

E

+

_3

16 „2

-

3o_2i 16

+

EE

k,j>k

i) ■ Oi -rk) +

a 16

(3)

where ki and k^ are elastic constants (Hooke's coefficients) and a is the lattice constant. TO and LO phonons at the Brillouin zone center are degenerate for crystals with the diamond type lattice. The frequency is given by

00 r =

8 (ki + 3

3 m

(4)

where m is the mass of Ge atoms. For germanium, the frequencies of TO and LO phonons at the Brillouin zone center are equal to 301 cm-1. Hence, the elastic constants k^ and ki are not independent (see formula (4)). The elastic constant ki was determined

co, cm 300

200

100

X W X

-1- i ^ i ^ i J* /+ /+

+t+ 1 1 1 1 t

1r + ++k + i + -+ j +4 1 1 1 1 1 1 V A. +\ \ + \ \ \ +-+++\\\ 1 y+++"H

A

Fig. 1. Phonon dispersion for bulk germanium: experimental [26] (figures) and calculated (lines) using the Keating model

from the approximation of calculated dispersions in directions (100), (110), and (111) to experimental dispersions obtained from neutron scattering data [26].

It is important to consider phonons of different directions, because in experiment, Raman signal comes from a l

Для дальнейшего прочтения статьи необходимо приобрести полный текст. Статьи высылаются в формате PDF на указанную при оплате почту. Время доставки составляет менее 10 минут. Стоимость одной статьи — 150 рублей.

Показать целиком