A model for predicting the thermal conductivity of SiO2-Ge nanoparticle composites.

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Cite this: Phys. Chem. Chem. Phys., 2015, 17, 13429

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A model for predicting the thermal conductivity of SiO2–Ge nanoparticle composites† Vasyl Kuryliuk,*a Andriy Nadtochiy,a Oleg Korotchenkov,a Chin-Chi Wangb and Pei-Wen Libc We present a simple theoretical model that predicts the thermal conductivity of SiO2 layers with embedded Ge quantum dots (QDs). Overall, the resulting nanoscale architecture comprising the structural relaxation in the SiO2 matrix, deviation in mass density of the QDs compared to the surrounding matrix and local strains associated with the dots are all likely to enhance phonon scattering and thus reduce the thermal conductivity in these systems. We have found that the conductivity reduction can be predicted by the dotinduced local elastic perturbations in SiO2. Our model is able to explain not only this large reduction but

Received 9th January 2015, Accepted 20th April 2015

also the magnitude and temperature variation of the thermal conductivity with size and density of the dots.

DOI: 10.1039/c5cp00129c

different samples are in close agreement with the experimental measurements. Including the details of the strain fields in oxidized Si nanostructured layers is therefore essential for a better prediction of the heat

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pathways in on-chip thermoelectric devices and circuits.

Within the error range, the theoretical calculations of the temperature-dependent thermal conductivity in

1 Introduction Germanium and silicon quantum dots (QDs), which exhibit unique physical properties, have received increasing interest for thermoelectric, photonic, and electronic applications1–7 because of their potential for integration into high density microelectronic circuits. The performance of thermoelectric energy conversion devices depends on the thermoelectric figure of merit (ZT) of a material defined as ZT = S2sT/k, where S, s, k, and T are the Seebeck coefficient, electrical conductivity, thermal conductivity, and absolute temperature, respectively.8 It is now widely appreciated that low-dimensional systems can have ZT much larger than that of bulk materials due to the enhanced electron transport and reduced thermal conductivity.9–13 Nanostructuring, particularly via a nanocomposite approach in thermoelectric materials, has been demonstrated as an effective strategy for improving ZT by significantly reducing the thermal conductivity while leaving the electronic properties either unaffected or improved in the process.14 The thermal conductivity of nanostructured materials itself is of great technological importance for micro- and nanoelectronics a

Department of Physics, Taras Shevchenko Kyiv National University, Kyiv 01601, Ukraine. E-mail: [email protected]; Tel: +380 44 526 2326 b Department of Electrical Engineering and the Center for Nano Science and Technology, National Central University, ChungLi 320, Taiwan c Department of Electronic Engineering, National Chiao-Tung University, Hsin-Chu 30010, Taiwan † Electronic supplementary information (ESI) available: A computer animation which graphically visualizes the effect of embedded Ge QDs. See DOI: 10.1039/ c5cp00129c

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heat management, and for micro/nanoscale energy conversion on a chip.10,15–17 Particular attention has been given to nanoparticle composites with improved performance issues related to the thermal conductivity.18,19 In some devices, such as computer processors or semiconductor lasers, the heat would escape away as efficiently as possible. These devices require materials with a high thermal conductivity. In other systems, such as thermal barriers or thermoelectric materials used for solid-state refrigeration, the thermal conductivity would be as small as possible. Nanostructuring can significantly reduce k of a single-crystalline material due to phonon barriers generated by multiply interfaces, where phonons are scattered. Thus, multilayered SiGe structures with a set of individual phonon-scattering nanodot barriers can significantly alter the thermal conduction properties of spatially defined nanosized regions, which results in decreased k that can be explained by the diffuse mismatch model for the barriers.20 The fundamental phonon mechanisms affecting the interfacial thermal conductance across a single layer of quantum dots on a planar substrate have also been examined.21 Particular attention has been given to the role of interfaces and associated elastic strain in relation to their influence on phonon scattering in crystalline semiconductors.22 Recent studies have demonstrated the growth of layer-cake Ge QDs in SiO2 and Si3N4 matrices through thermal oxidation of SiGe-on-SiO223 and SiGe-on-Si3N424,25 structures. This technique produces innovative materials with high-quality Ge QDs in close proximity to conductive Si layers through the alteration of different types of matrices and Si layers, which can offer new far-reaching implications for energy storage and conversion.26

Phys. Chem. Chem. Phys., 2015, 17, 13429--13441 | 13429

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The temperature-dependent thermal conductivity of these structures has been systematically investigated, showing much lower k than bulk Ge and SiO2, and the reduction of the thermal conductivity increases with increasing surface-to-volume ratio of the QD in SiO2.27 It has been tentatively suggested that the reduced thermal conductivity can be related to the QD-induced strains, although there have been attempts to explain the effect using the phonon confinement model and boundary scattering events. Obviously, the observed reduction in the thermal conductivity scales with the interface region between the Ge QD and SiO2 matrix. There have been a number of systematic theoretical treatments taking into account interface scattering of phonons,28,29 as well as including the influence of strains on lattice dynamics.30 The calculations are, however, quite complicated and the agreement between experiment and theory is not always perfect, which indicates the need for further study. The aim of this work is therefore to give a new and very simple theoretical description of the thermal conductivity of SiO2–Ge nanoparticle composites based on the effects of quantum dot strains. We find that the measured reduction of the conductivity can be predicted by properly taking into account the strains in the SiO2 matrix imposed by the Ge dots of different diameters and densities. The experimentally measured reduction is explained fairly well using this theory without any fitted parameters.

2 Thermal conductivity of amorphous silicon dioxide The thermal conductivity of noncrystalline dielectric solids differs significantly from that of crystalline materials.31 Fig. 1 is plotted to exemplify the temperature dependence of the thermal conductivity of amorphous silicon dioxide. In crystalline semiconductors and insulators, lattice vibrations are the carriers of heat currents. The expression for the lattice thermal conductivity in this case can be written as33 ð 1 oD hnCðo; TÞLðo; TÞ do; (1) kph ¼ 3 0 kB T where Cðo; TÞ ¼ 9Na kB

3 T xo 4 eðxo Þ y ðeðxo Þ 1Þ2

(2)

is the Debye specific heat function, Na and kB are the Avogadro and Boltzmann constants, respectively, T is the temperature, u is the the acoustic phonon velocity (or sound velocity), o is the phonon frequency, L is the phonon mean free path, y is the Debye temperature, oD = kBy/h is the Debye frequency, xo= ho/kBT is the dimensionless phonon frequency. The temperature dependence of the thermal conductivity can be described by calculating the total mean free path L, which is expressed as a combination of various scattering mechanisms through the Matthiessen rule as34 1 1 1 1 1 ¼ þ þ þ ; L Lph Lb Ld Li

13430 | Phys. Chem. Chem. Phys., 2015, 17, 13429--13441

(3)

Fig. 1 Thermal conductivity of amorphous SiO2 (solid curve), calculated by using the data of ref. 32. Dashed curve schematically illustrates a decrease in k at high T due to temperature-induced structural changes.

where Lph, Lb, Ld, and Li are the phonon mean free paths due to the phonon–phonon, phonon-boundary, phonon-dislocation, and phonon-point-defect scattering, respectively. At low temperatures, the phonon mean free path is long, i.e., L is of the order of the sample size, so that the boundary scattering dominates phonon relaxation. In this regime, the temperature dependence of the thermal conductivity follows the specific heat behavior, and k varies as T3. At greater temperatures, the anharmonic umklapp and normal three-phonon scattering processes become more frequent, and hence the phonon mean free path decreases. As a consequence, the thermal conductivity decreases with increasing temperature. In many cases of fundamental interest and clear applied relevance, interface scattering, which is insensitive to temperature, dominates the transport. The above discussion is relevant to conventional semiconductors but cannot be applied to amorphous oxides. Thermal conduction in bulk amorphous solids has been extensively discussed in the literature.35,36 In these materials, heat conduction is adequately treated by a simple model, in which heat conduction is described by a random walk of vibrational energy on the time and length scales of atomic vibrations and interatomic spacings. Therefore, amorphous solids have thermal conductivities that could be modeled as phonon liquids with very short phonon mean free paths, L E a, where a is comparable to interatomic spacing.37 The conductivity is several orders of magnitude smaller than in crystals, it decreases monotonically with decreasing temperature.31 The thermal conductivity at temperatures smaller than E1 K varies as T 2, which is interpreted as a mean free path for phonons going as ol instead of o4, expected for disordered systems, and which is seen at higher temperatures.38 At temperatures greater than E30 K the conductivity gradually increases with T, while the plateau is usually observed in the intermediate range of the k(T) dependence (the solid curve in Fig. 1). The two-level-system model successfully accounts for the thermal properties of amorphous materials at the lowest temperatures (T o 1 K),39 and it has also


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been extended to higher temperatures in the framework of the soft-potential model.40 Various additional associated models have been suggested to address the origin of the plateau in the thermal conductivity, and the mechanism responsible for the thermal conduction above the plateau.41 For example, experimental behaviour of the thermal conductivity in thin-film glasses indicates a saturation of k(T) above 100–150 K, which can be followed by a decrease in k.42 The essential physics of the experimentally observed saturation is that the phonons are so strongly scattered at high T that their mean free path approaches its limiting lower value, L E a, yielding a saturation value of k B (1/3)CVua,37 where CV is the heat capacity at constant volume. It should be emphasized that, unlike in the case of amorphous solids, the mean free path in crystals is much larger and decays as 1/T at high T due to increasing phonon–phonon scattering. Meanwhile, in glasses, k can also show a decrease at high T, which was observed experimentally42 and explained theoretically.43 This decrease in k is explicitly introduced by dashed curve in Fig. 1. As proposed by Lee and co-workers,43 a viable explanation for this decrease involves the temperature-induced structural changes. In this context, several key mechanisms were considered. First, the temperature-induced coordination defects act as new scattering centers for phonons. As the number of coordination defects increases with temperature, the value of k exhibits a small decrease. Next, one has to take into account (i) scattering from a static structural disorder, which is the analogue of impurity scattering in crystals, and (ii) phonon–phonon scattering, which is also temperature dependent. In what follows, we discuss the role of Ge quantum dots embedded into an amorphous SiO2 matrix in the thermal conductivity of this nanoparticle composite material.

3 Experimental details We compare a series of Ge QDs in SiO2 matrices grown in ChungLi by thermal oxidation of Si1xGex-on-insulator structures deposited on top of Si3N4/p-type (100) Si substrates with a resistivity of about 1 O cm. A 17 nm base, which catalyzed the hydrolysis of the tetraethylorthosilicate (TEOS) layer, was placed between Si1xGex and Si3N4. The oxidation was performed at 1120–1170 K, producing high-quality Ge QDs in amorphous SiO2 layers, as described in detail elsewhere.25,44 The resulting structures were capped with a 26 nm TEOS layer. The thickness of the resulting SiO2 layer with embedded Ge QDs was from 300 to 600 nm, whereas the dot size and density varied from about 3 to 15 nm and from 6.8 1011 to 1.4 1012 cm2, respectively, in different samples. Therefore, we analyze heat currents in the structure, as depicted pictorially in Fig. 2(a). Cross-sectional transmission electron microscopy (TEM) observations were performed using a JOEL JEM-2100 LB6 Transmission Electron Microscope and a FEI Tecnai Osiris Transmission Electron Microscope under conditions of minimal radiation damage. The typical operating voltage of the TEM is 200 kV and beam current is 100 mA.


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Fig. 2 Schematics of SiO2–Ge nanoparticle composites (a) and the size distribution of Ge QDs for one of our samples (b) obtained from crosssectional TEM. Averaging over more than 100 QDs, the Ge dot diameter is 4.92 0.75 nm, the density and volume fraction of Ge QDs are about 1.3 1012 cm3 and 9.1%, respectively. The thickness of the SiO2 layer varied from about 300 to 600 nm in different samples.

The thermal conductivity of our samples was measured by resistance thermometry. By noting the power dissipated by the lithographically patterned aluminum or platinum thin film heater, the conductance of the SiO2 layer with Ge QDs was calculated. The measurements were performed in a temperature controlled Lakeshore TTP-6 probe station and a CS204 (Advanced Research Systems) closed cycle cryostat. The temperature of the heater was obtained by monitoring its resistance and pre-calibrating the temperature coefficient of resistance using the four-point-probe technique. Measurements were also taken on several test structures by a differential 3o method.45 Their results exhibited very similar features to the resistance thermometry data given below. In our experiment, when the Ge QD sample is placed in ambient gas, the thermal conductivity is several times greater than that in a vacuum. This may be thought of as a consequence of atmospheric pressure, since high-pressure environments enable tuning of the thermal conductance.46 However, the measurements under pressure require identifying and controlling relevant boundary conditions for solving a heat-transport equation as the pressure is changed. Indeed, heat is lost from the top surface of the nanocomposite sample to the surroundings by radiation and, if the sample is not in a high vacuum, by convection. This heat loss through the surrounding air may therefore be interpreted as the seemingly increased thermal conductivity of our sample.


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Furthermore, small sample scales require the convective coefficient to be precisely determined. This requirement is drastically important for nanoscale structures directly exposed to air. Thus, the convective coefficient is B100 W m2 K1 for air when the scale is less than 100 mm, which is 5 to 10 times larger than that at macroscales.47 This enhanced heat transfer coefficient is explained in terms of better convection heat transfer due to compressed boundary layers at very small scales. Whether or not the increase in k obtained in ambient gas is in fact due to atmospheric pressure is, therefore, somewhat controversial. A more accurate account for the heat losses deserves a special emphasis. In this work, these complications were not taken into account and the thermal conductivity measurements were performed in a vacuum with a pressure of B106 Torr to prevent convective heat loss to the surrounding gaseous media. Details of the measuring procedure are given elsewhere.48 Infrared transmission measurements were carried out using a Nicolet model 6700 FTIR Fourier-transform infrared spectrometer. Transmission mode data in normal incidence were taken in ambient air using the air as a reference. The FTIR spectra given below are related to this reference spectrum and show the spectral changes relevant to the layer-cake fabrication process. The transmission spectra were measured in a frequency range from 400 to 1700 cm1, which includes the traces of Si–Si, Si–O and Ge–O bonds. Raman spectra were collected with a 514.5 nm line of an Ar+ laser using a Horiba Jobin-Yvon T64000 spectrometer. The laser light was focused onto the sample surface to a spot of 0.7 mm in diameter. The spectral resolution was 0.15 cm1. Spectroscopy investigations were carried out at room temperature.

4 Computational details We have attempted to calculate the temperature-dependent thermal conductivity of the SiO2–Ge nanocomposites taking into account all basic phonon scattering mechanisms, such as structural relaxation, deviation in mass density, which is obtained by adding Ge QDs into SiO2, and strains near the SiO2–Ge boundaries. In writing the mean free path of phonons as in eqn (3), we therefore equate 1 1 1 1 ¼ þ þ ; L Lo Lr Le

where t is the phonon relaxation time, NQD is the density of Ge QDs participating in the scattering processes with phonons, s is the phonon scattering cross-section, q is the phonon wavenumber, a is the radius of the quantum dot and d is the dot diameter. Therefore, for a near-surface region of Ge QDs with a deviation in mass density Dr compared to the surrounding SiO2 matrix with density r, the phonon mean free path is modeled as " # Dr 2 2Dn m 2 4 4 144n m þ 2ðodÞ þ3 r nm " Lr ¼ (7) # ; 2 Dr 2Dn m 2 pd 6 o4 NQD þ3 r nm where nm is the sound velocity in the matrix elastic medium, and Dnm is the difference between the sound velocity in the Ge QD material and that in the SiO2 matrix. The strains near the SiO2/Ge interface were taken into account by the introduction of Le in the form Le ¼

nm 2 ; 9NQD g2 e2 d 4 o2

(8)

¨neisen parameter and e is the strain imposed where g is the Gru by the dots. For linear elasticity, the components of the strain tensor ekl and stress tensor sij are given in terms of the displacement vector components uk by 1 @uk @ul ; (9) þ ekl ¼ 2 @xl @xk sij = cijkl(ekl e0kl),

(10)

where cijkl is the elastic module tensor, e0kl are the components of the initial strain tensor due to QDs embedded in SiO2, xl components are referred to a fixed Cartesian coordinate system shown in Fig. 3, all subscript indices range from 1 to 3, and summation over repeated indices is assumed. In mechanical equilibrium the forces inside any volume vanish, so that @sij ¼ 0: @xj

(11)

(4)

where Lo is the frequency dependent mean free path limited by structural relaxation, as defined by eqn (8) in the work of Goodson et al.,32 Lr is related to the deviation in mass density,49 and Le describes the strain effect on the phonon free path.22 In what follows, equations for Lr and Le from these studies are rewritten by taking into account that L ¼ nt ¼

1 ; NQD s

(5)

and od qa ¼ ; 2n


(6)

Fig. 3 Geometry of the problem including three unit cells of Ge QDs (diameter d) embedded in the SiO2 layer with the thickness of 2h, which is much greater than d.


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We employ three dimensional (3D) numerical simulations in modeling an infinite array of quantum dots, composed of a periodically repeated unit cell (dashed-line rectangle in Fig. 3). We apply periodic boundary conditions on the boundary planes of the cell in the x and y directions, which are given by ux|x=L/2 = ux|x=L/2 = 0,

(12)

uy|y=L/2 = uy|y=L/2 = 0.

(13)

Finally, the traction-free condition is imposed on the upper and bottom surfaces of the cell (at z = h) shown in Fig. 3. This is due to the fact that the force on the boundary equals the external pressure, and in the absence of external traction, which in the case of free space is zero, we get sijnj = 0

(14)

with nj the j-component of the unit vector normal to the surfaces. The e0kl strain can be obtained from the fact that the Ge dot placed into the SiO2 matrix is hydrostatically compressed out of equilibrium, which is originated from a liquid–solid phase transition in Ge.50 In view of the fact that the annealing is processed at high temperatures, where parts of the sample are molten, it is not obvious how much strain will build up in the sample while cooling it down to room temperature. Moreover, the growth and migration behavior of Ge QDs within SiO2 and Si3N4 matrices during post-growth thermal annealing in an oxidizing ambience appears to have been rather complex due to the cooperative influence of silicon and oxygen interstitials facilitated by the Ge QD acting as a catalyst.51 Meanwhile, the nearly spherical shape of Ge quantum dots obtained with the layer-cake technique allows us to simplify calculations by implicitly assuming that there is a melting of liquid Ge droplets embedded in the viscous SiO2 matrix. The volume Vl of a liquid phase of the Ge inclusion is about 6% smaller than that in a solid phase Vs. Assuming a completely stiff SiO2 matrix during cooling yields a lattice constant of Ge nanoparticles E2% less than that of unconfined Ge. Due to some strain relaxation during cooling, the remaining deviation of the lattice constant is reported to be about 1%.52 Since the volume fraction of the dots in the matrix material is significantly small, the SiO2 matrix is considered here as perfectly rigid and the inter-atomic spacings in the solid (as) and liquid (al) nanoparticle phases are related through their known volume pffiffiffiffiffiffiffiffiffiffiffiffiffi ratio as as =al ¼ 3 Vs =Vl . Therefore, the initial compressive strain in the quantum dot is rffiffiffiffiffiffi al as 3 Vs 0:02: (15) ¼1 e0 ¼ al Vl A difference in the thermal expansion coefficients between the dot and the matrix should also be taken into account in the expression for the initial strain. The thermal expansion coefficients are aGe = 5.7 106 K1 and aSiO2 = 0.55 106 K1 in Ge and SiO2, respectively.53 Taking the temperature difference DT varying from 500 to 1000 K gives the strain in Ge dots,


Table 1

Material parameters used in the calculations (ref. 58)

Elastic constants (GPa) C11 C12 C44 Density (kg m3) Sound velocity nm (m s1) ¨neisen parameter Gru

Ge

SiO2

128.9 48.3 67.1 5323 5300

78.5 16.1 31.2 2200 4100 1.5

(aGe aSiO2)DT, ranging from 2.6 102 to 5.2 102. Obviously, this is much smaller than e0 estimated above. As a final remark in our discussion of the strains, we point out that the strain values used in our modeling are also reported in a number of Raman studies.54–56 Since amorphous SiO2 layers are isotropic, e0xx = e0yy = e0zz. Therefore, we have implicitly assumed that the shear strain components are presumed to be zero and e0ij = e0dij,

(16)

where dij is the Kronecker delta. Then, the strain distribution around the dot is calculated from eqn (9)–(11) using the finite-element method (FEM). Further details of the calculations are given elsewhere.57 In Table 1, we give an overview of the numerical parameters, which are used in the calculations. The calculated strain values have then been used to evaluate Le of eqn (8). In what follows, we will only be interested in the average hydrostatic strain component, e¼

1 V

ð

exx þ eyy þ ezz dV;

(17)

V

where integration is carried out over the volume V of the SiO2 matrix influenced by strain effects. Now we can evaluate all of the terms in eqn (4), and then substitute the mean free path L into eqn (1). Then the temperature dependence of the thermal conductivity can be calculated for different densities and diameters of Ge QDs.

5 Results and discussion The bonding configuration of the nanocomposite samples is deduced from infrared absorption. Fig. 4 shows FTIR spectra of as-deposited SiO2-on-Si films (spectrum 1) and SiO2–Ge nanoparticle composites (spectrum 2). Three well defined sharp peaks are observed at around 460, 820 and 1090 cm1 in both samples, which are related to the rocking, bending and stretching vibration modes of the Si–O bonding, respectively.59 A subsidiary absorption band shifted to the higher wavenumber side of the main Si–O–Si stretching mode and ranging from 1150 to 1300 cm1 can be attributed to the Si–O network that consists of SiO2 and SiO3 groups.60,61 The decreased transmittance in this band seen in spectrum 2 of Fig. 4 is then considered due to amorphous SiO2 grown by using the oxidation process described in Section 3.


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Fig. 4 FTIR transmission spectra of samples with a 351 nm SiO2 layer grown on Si substrates (spectrum 1) and with 8-period Ge QDs in the SiO2 matrix shown in Fig. 2(a) having a thickness of the resulting SiO2–Ge nanocomposite structure of 555 nm (spectrum 2). Arrows a to c mark the Ge-related peak positions (see the text for details). The average diameter of Ge QDs in the composite is about 13 nm. Spectra are arbitrarily shifted on the vertical axis for a better separation.

The peak position of the Si–O stretching mode can be used to estimate the degree of oxygen content x in SiOx since the vibration frequency n(x) = (987 + 48.8x) cm1.62 Taking x = 2 yields n = 1085 cm1, which is very close to the transmittance minima observed at E1090 cm1 in spectra 1 and 2 of Fig. 4 and thus gives fairly clear-cut evidence that the chemical composition of the silicon oxide is stoichiometric SiO2. The stretching mode of Si–Si63 is observed at around 612 cm1 in spectrum 1 for as-deposited SiO2-on-Si and it remains nearly unchanged upon the sample annealing (spectrum 2). Meanwhile, the Ge–Ge mode at about 740 cm163 (arrow a) is not clearly distinguished in spectrum 2 of Fig. 4. In contrast, the formation of Ge QDs leads to a marked decrease in transmittance in the range from about 800 to 1000 cm1, which is evident in spectrum 2. One might be tempted to conclude that this result is a clear proof of the formation of chemical bonds between Ge and O atoms. Indeed, the structure of amorphous GeO2 is described as Ge–GeyO4y tetrahedral connected by bridging oxygen atoms. The asymmetric stretching vibration mode of Ge–O–Ge in stoichiometric GeO2 is at 885 cm1 (arrow b in Fig. 4). The vibration frequency of oxygen deficient structures, for y values varying from 0 to 4, changes linearly from 885 to 740 cm1 (between arrows b and a in Fig. 4) according to the empirical equation, n(x) = (72.4x + 743) cm1, where x is the oxygen content which ranges from two to zero.64 Another supporting evidence that the 800 to 1000 cm1 region in spectrum 2 of Fig. 4 has the Ge–O-related origin follows from the fact that values of the absorption peak between 995 and 1030 cm1 (arrow c in Fig. 4) are reported for the Si–O–Ge vibrational mode.65,66 However, it appears from our estimates that the experimental points in spectrum 2 of Fig. 4 show a pronounced discrepancy with the simulated curve that assumes the contribution of the Ge–O–Ge and Si–O–Ge bonds.


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As previously reported by Chen et al.,51 it would be dictated by chemical thermodynamics that oxygen predominantly bridges silicon interstitial atoms, so that the density of the Ge–O atomic bonds is believed to be rather low considering that the formation enthalpy for SiO2 (910.9 kJ mol1) differs considerably from that for GeO2 (477 kJ mol1).67 As also pointed out by Chen et al.,51 there is no evidence for forming GeO since the heat of formation for Ge + GeO2 - 2GeO is 226 kJ mol168 whereas the one for Si + SiO2 - 2SiO is 426.1 kJ mol1.69 This gives an argument to suggest that the possibility of forming GeO2 is extremely low. However, Ge QDs are inherently bonded to SiO2 since they are formed by thermally oxidizing SiGe via the preferential oxidation of Si from SiGe. Consequently, the segregated Ge is incorporated into the as-yet unoxidized SiGe regions. Thus, Ge QDs are ultimately formed by a progressive ‘‘concentration’’ of the Ge content within the remaining, unoxidized, SiGe grains until Si is entirely used up leaving Ge QDs embedded within the newly-formed SiO2 layer. The oxidation of the SiGe layer proceeds initially in a lattice-coherent manner, so that the Ge nanocrystallites become coherently bonded to the growing oxide. Therefore, it is thought that the enhanced density of Ge–O bonds can be considered as only partly reliable for explaining the FTIR spectroscopy data in the spectral range, which extends from about 800 to 1000 cm1. An alternative to the Ge–O-related origin of the above changes in spectrum 2 of Fig. 4 is the Si–N bonding for Si3N4 layers with strains imposed by Ge QDs. It is worth remarking at this point that Ge dots can migrate through the Si3N4 thin layer and may even become embedded into the Si substrate.44 The migration behavior of the Ge nanocrystallites is quite complex and has already been described in detail.51 In turn, it has been noticed that the peak position and absorption strength of the Si–N vibration mode are affected by the degree of strain in the nitride spacer films. The peak position shifts in the range from about 830 to 855 cm1 with stresses varying from high compressive (1.3 GPa) to high tensile (1 GPa).70 As the Si3N4 layer gains tensile stresses, the Si–N bond length increases making its strength weaker. Since the transmittance decreases in the region of interest in spectrum 2 of Fig. 4, the Si–N bond strengthens, which amounts to a compressive stress in the Si3N4 film and thus produces a larger stress on the close-lying Si substrate and the SiO2 layer. This is thought to originate from a thermal annealing and the burrowing of Ge QDs into the Si3N4 layer. The bonding configuration of the Si3N4 layers with embedded Ge dots can therefore be rather complicated and the layer stoichiometry and intrinsic strain would affect the absorption peak positions. For these reasons we can expect a significant smearing out and overlapping of the locally varying Si–N absorption bands, as indeed observed experimentally in spectrum 2 of Fig. 4. Fig. 5 shows the Raman spectra of two samples. The Raman spectrum of the Ge QD structure is composed of three dominating features, identified as the Ge–Ge (B300 cm1), Si–Ge (B410 cm1) and Si–Si (B520 cm1) vibrations (see Fig. 5(a)). Due to the small film thickness, the strong peak at B520 cm1 is due to the Si–Si vibrational mode from the substrate. As seen in Fig. 5(b), the substrate spectrum also includes three peaks at


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Fig. 5 (a) Raman spectra from four random points on the surface of an 8-period Ge QDs in the SiO2 matrix shown in Fig. 2(a) having a thickness of the resulting SiO2–Ge nanocomposite structure of 555 nm, (b) Raman spectrum for a sample with a 351 nm SiO2 layer grown on Si substrates, (c) detailed Ge–Ge bands for the four spectra given in (a).

about 230, 305 and 430 cm1, which originate from the twophonon scattering of 2TA(L), 2TA(X), and 2TA(S), respectively.71 It is most likely that appropriate weak spectral features in the Raman spectra of the SiO2–Ge nanoparticle composite shown in Fig. 5(a) are related to the Si substrate. The majority of the laser-excited points across the surface of the Ge QD structure exhibit essentially weak B410 cm1 bands of the Si–Ge vibrations, as exemplified by three line spectra in Fig. 5(a). This is obviously due to the fact that the samples demonstrate a low intermixing between silicon and Ge QDs due to the rapid diffusion of Si in Ge during the prolonged thermal oxidation at B1150 K, as has already been pointed out.48 The appearance of a stronger Si–Ge band is accompanied by the emergence of a subsidiary Raman peak at 509 cm1, as evidenced by a dotted spectrum in Fig. 5(a). The latter peak is characteristic of emergence of nanocrystalline silicon,72,73 and therefore the appropriate volume of the nanocomposite consists of small Si aggregates embedded in the Ge QDs. The Si–Si band is shifted to lower wavenumbers with respect to crystalline Si (B520 cm1) due to the finite size of Si. It may implicitly be assumed that this can occur near inhomogeneusly distributed small defect clusters, where the rapid diffusion of Si starts to deteriorate. It is also seen in Fig. 5(a) that the Raman spectra do not include any amorphous fractions of Ge and Si in the form of a shoulder at the low-frequency side of the Ge–Ge and Si–Si bands, thus proving that pure crystalline Ge QDs grow in the nanocomposite material. Amorphous SiO2 tissues are also absent in the Raman spectra, due to the small film thickness and small Raman scattering cross-section. The Ge–Ge phonon mode at about 300 cm1 exhibits a noticeable broadening to the high-frequency side of the band, as shown


in Fig. 5(c). We can decompose the measured band into its two components shown by I1 and I2 in Fig. 6. The I1 component in spectra 2 and 3 peaks in the range from 302 to 304 cm1 in different points on the sample surface and can be related to the Ge–Ge mode in SiO2. The peak intensity of I2 is smaller than that of I1 and it appears at 309 to 311 cm1 in different fixed points. This shift to higher wavenumbers for the second spectral component and the varying I2/I1 intensity ratio can be attributed to locally varying stresses mentioned above. In a cubic Ge crystal, the Raman active optical phonon is triply degenerate, including one longitudinal (LO) and two transverse (TO) optical phonons. It is well known that the degeneracy can be lifted in mechanical stress (or strain), either present in a crystal lattice or externally introduced, breaking the cubic symmetry. Clearly, since the stress tensor shifts the vibration frequency of each one of these three phonon modes, a full determination of the stress state from the Raman data can only be done if the stress dependence of all the three phonon modes can be carefully quantified. However, the Raman spectra shown in Fig. 5(a) are taken in scattering geometry z(x,x)%z + z(x,y)%z, where the labels x, y and z refer to directions parallel to the [1,0,0], [0,1,0] and [0,0,1] crystal axes, respectively, and thus would be expected to exhibit only the allowed LO mode, while the TO mode is forbidden in this configuration and is not Raman active. On the other hand, the above I1 and I2 components deduced from Fig. 6 may be thought of as arising from the collective response of all the three phonon modes, provided that the appearance of the TO mode is attributed to the presence of a short-range perturbation in the nanocomposite, which relaxes the Raman selection rules. In the most general case, the lift of degeneracy results in both frequency splitting and shift in the position of the


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Consequently, in the absence of frequency splitting, the above I1 and I2 components can indeed be used for the hydrostatic stress estimates. It is therefore possible to simply include the stress effects into the Ge–Ge phonon mode frequency shift seen in Fig. 6 via varying hydrostatic stress s as follows:77


Do = 3.85s 0.039s2,

Fig. 6 Deconvolution of the two detail Raman spectra in the region associated with the atomic vibrations of the Ge–Ge bond (points), which are given in Fig. 5(c), into two Lorentzian spectral shapes I1 and I2. The solid-line curves are sums of the two fitting curves. Spectra 1 and 2 are arbitrarily shifted in the vertical direction for a better separation of curves.

Ge–Ge peak. While the shift is due to biaxial and hydrostatic components of the stress, the splitting of the three modes into a doublet and a singlet is solely determined by the biaxial component.74–76 In our description of spherical QDs contained within a matrix of SiO2, the biaxial stress component is presumed to be nearly zero, so that the stress field is almost purely hydrostatic.

(18)

where Do = o o0, o and o0 are the peak frequency in the locally stressed and unstressed regions, respectively. Taking the literature value of o0 = 300.6 cm1 and experimentally observed o ranging from 309 to 311 cm1 yields s = 2.4 to 2.9 GPa. This turns out to give the strain values in the range from 1.1 to 1.4%. Based on the experimental results, stress has been attributed to several key mechanisms involving (i) liquid–solid phase transition in Ge during a high-temperature heat treatment, (ii) lattice mismatch between the Ge and SiO2 matrix, (iii) composition of the surrounding oxide matrix such as silicon suboxide (SiOx) or Ge suboxides (GeOx) and (iv) imperfect surface reconstruction during growth of nanoparticle composites.54,77,78 These indicate a compressive stress and a lattice contraction of less than E1%. Our estimates based on Fig. 6 lead to a slightly greater overall strain values. This result is consistent with the above mentioned observation that Ge QDs penetrate the underlying Si3N4 buffer layer. Indeed, a lattice contraction of the QD placed in the Si3N4 matrix amounts to E4%, which is greater than that in SiO2 and thus results in an overall increase of the strain. Fig. 7 shows the calculated hydrostatic strains for two densities of the QDs. It is seen that the compressive interfacial strain (negative e values) is observed in the horizontal direction of the QD chain, which results in spreading of the strain over the distance of E10 nm in Fig. 7(a) with the amount of compressive strain as high as E1%. The tensile strain with e 4 0 is imposed in the vertical direction, where the dots are absent,

Fig. 7 Hydrostatic strain distributions in the SiO2 matrix material due to embedded Ge QDs (white circle areas centered at x = 0, z = 0) for NQD = 1011 cm2 (a) and 1012 cm2 (b), which correspond to inter-dot distances of E32 nm and 10 nm, respectively. Diameter of the dot d = 5 nm.



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exhibiting the strain accumulation of about 1% near the Ge/SiO2 interface. The compressively strained region is expanded outward at the expense of the expanded region with the strain increase beyond 1.8% in Fig. 7(b) for an inter-dot distance of 10 nm. We therefore expect that strains will have a notable influence on phonon scattering at and across such SiO2/Ge interfaces. It is also seen in Fig. 7 that the amount of the hydrostatic strain decreases rather sharply in the vicinity of the dot. This allowed us to restrict our analysis of e in eqn (17) to a distance of 2d, when the integration extends over the volume " 3 # 4 d 3 V¼ p d : (19) 3 2 On the other hand, it is well known that elastic anisotropy strongly affects the growth and patterning of self-assembled nanostructures. Usually, a thin film layer has different material properties as compared to the growing surface material. In many cases, the size distribution of self-organized islands is fairly broad, and due to the statistical nature of the island nucleation process, lateral ordering is typically absent. However, the interaction among islands via their elastic strain fields may lead to a

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gradual improvement in size homogeneity, as well as to a more uniform lateral island spacing. Holy´ et al.79 showed that the vertical and lateral correlations in self-organized QD superlattices can be explained by taking into account the elastic anisotropy of the materials. Furthermore, without strain energy, randomly ordered islands are usually formed and often associated with Ostwald ripening. However, with the strain energy involved, isolated and ordered islands (for both size and distribution) can be observed.80 Our cross-sectional TEM micrographs of several layers of Ge QDs in the nanocomposite shown in Fig. 8 are in broad general agreement with this picture. Both the randomly distributed QDs and more or less ordered nanoparticles are clearly visible in all four images displayed in Fig. 8. The principal calculation results are summarized in Fig. 9, which shows the modeled temperature dependence of the thermal conductivity of SiO2/Ge nanoparticle composites depending on the density (a) and diameter (b) of Ge QDs. As expected from Section 2, the thermal conductivity decreases steadily with decreasing temperature. A specific feature is the reduction of the thermal conductivity of the nanocomposite (curves 2) as compared with the value of the bulk SiO2 material (curves 1). Increasing the density and diameter of Ge QDs, its overall value

Fig. 8 Cross-sectional TEM micrographs of several layers of Ge QDs in the SiO2–Ge nanocomposite material with d = 4.9 nm and NQD = 1.4 1012 cm2.



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Fig. 9 Thermal conductivity as a function of temperature in a SiO2 layer without Ge QDs (curves 1) and in Ge QDs–SiO2 for: (a) varying QD density NQD = 2.2 1011 (2), 6.3 1011 (3), 1012 (4) cm2, which correspond to inter-dot distances of E21 nm, 13 nm, 10 nm, respectively, at a given dot diameter d = 5 nm, and (b) for varying d = 3 (2), 5 (3), 7 (4) nm at a given NQD = 1012 cm2 and a corresponding inter-dot distance of 10 nm.

is reduced even further, as seen in curves 2 to 4 in panels (a) and (b) of Fig. 9, respectively. This trend is fairly obvious just looking at pictures of Fig. 7, because phonons of wavelength comparable to the SiO2/Ge boundaries and associated strain lengthscales are scattered strongly by the disorder. The denser and the larger Ge QDs therefore produce higher disorder, which tends to increase the scattering of phonons. To allow for comparison of theory and experiment, we have fitted the experimental points shown in Fig. 10 for a number of samples with a combination of the scattering mechanisms given above. The measured data provide evidence of a large reduction of the thermal conductivity of the nanocomposite (filled points) as compared with the value of a SiO2 layer (open circles). This result is in quantitative agreement with the predictions of the theory of Section 4 (curves labeled 1 to 6 in Fig. 10). The fit shown as the dotted curve 1 in Fig. 10 consistently underestimates the measured thermal conductivity of a SiO2 layer (open circles). This is basically due to the fact that the boundary scattering in thin SiO2 layers is not properly taken into account by the theory emphasized in Section 4. However, this is not so important in the quantum dot layers with a considerable number of the QD scatterers. Indeed, within the


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Fig. 10 Comparison of the experimental (points) and calculated (curves) thermal conductivity as a function of temperature for a SiO2 layer without Ge QDs (open circles, curve 1) and for SiO2–Ge nanoparticle composites with d = 3.7 0.61 nm and NQD = 1.4 1012 cm2 (filled circles, curves 2 and 3), d = 4.9 0.75 nm and NQD = 1.4 1012 cm2 (triangles, curve 4), d = 6.3 0.85 nm and NQD = 7.2 1011 cm2 (diamonds, curve 5), d = 7.6 0.78 nm and NQD = 6.8 1011 cm2 (squares, curve 6). Curve 1 represents the contribution of Lo in eqn (4) for pure SiO2, curve 2 indicates a subsidiary contribution of Lr, and curves 3 to 6 are obtained by taking into account all the Lo, Lr, and Le contributions in eqn (4). Error bars in the measured data mainly result from instrumental errors.

margin of error, the agreement between experimental (filled circles, triangles, diamonds and squares in Fig. 10) and appropriate calculated (curves 3 to 6) data for the four samples may be considered to be fairly good. Our calculations provide compelling evidence in support of the argument that including the QD-induced strain is clearly a prerequisite for a proper analysis of the observed reduction of the thermal conductivity. The fit shown as the dashed curve 2 in Fig. 10 summarizes the contributions of the two scattering mechanisms including frequency dependent phonon scattering in bulk SiO2 [Lo in eqn (4)] and phonon scattering due to the difference in mass density of Ge QDs and the SiO2 matrix (Lr). It is seen that the prediction (curve 2) significantly overestimates the measured data (filled circles). In contrast, taking into account an additional contribution due to the strains near the SiO2/Ge interface [Le in eqn (4)] yields the fit shown as curve 3 in Fig. 10, which describes the experimental behavior of the thermal conductivity (filled circles in Fig. 10) quite well. A computer


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6 Conclusions

Fig. 11 Calculated frequency dependence of the phonon mean free path Le in the SiO2–Ge nanoparticle composite (curve 1). Dotted lines mark the dot diameter of the four samples used in our experiments.

animation which graphically visualizes our model is available as ESI† (see two video clips illustrating the time-resolved heat transfer from the top to the bottom of the SiO2 layer and the one with embedded Ge QDs). An aspect that needs discussion is the deviation of some experimental results (diamonds and squares in Fig. 10) from the shape of the calculated curves (curves 5 and 6). The experimental results show a clear maximum at around 250– 300 K. As noted above (Section 2), the temperature-induced structural changes may constitute a viable approach to account for this discrepancy. As shown in Fig. 10, to the experimental accuracy we are now discussing, the main discrepancy appears for larger dot diameters d of about 6 nm. Therefore, it may happen that inclusion of sufficiently large Ge dots may generate a local structural distortortion in SiO2 so as to ensure that phonon scattering occurs at high temperatures. It is then important to understand how the temperature increase affects the number of dangling and floating bonds in the QD/surrounding matrix environment. Clearly much more work and detailed simulations including more physical effects are needed to establish the importance of the various contributions that are not negligible. Based on the physical concepts of Section 4, we have calculated the frequency dependencies of a phonon mean free path for the phonon scattering mechanism related to the SiO2/ Ge interface and associated strain, which is shown in Fig. 11. It is seen that, including the QD-induced strained region, which varies from a few nanometers up to some tens of nanometers in size (see Fig. 7), the phonon scattering is intensive within a broad frequency range (Fig. 11). This can be understood by noting that within the approximation applied in Section 4, strains cause strong phonon scattering with frequencies above E0.2oD. Though somewhat intuitively obvious, the strain contribution provides a coherent account reflecting the consistency between experimental observations and theoretical calculations of the QD-induced phonon scattering in SiO2 layers given in Fig. 10, and thus explains a greatly reduced thermal conductivity of the SiO2–Ge nanoparticle composite material.


In summary, we present a simplified theoretical explanation for the experimentally measured temperature-dependent thermal conductivity of SiO2 layers with embedded layer-cake Ge quantum dots. The key contribution of this work is the derivation of a very time consuming model for predicting the performance of the thermal conductivity of SiO2–Ge nanoparticle composites. Our model gives us a reasonable account of both the large reduction of the thermal conductivity and its variation with dot diameter and density. The detailed analysis presented above may be readily extended to cover more general cases of nanoparticle composites exhibiting strain distribution at a well defined length scale. The results achieved here can be used with benefit for modeling functional devices with thermal pathways, including micro- and nanoscale energy converters.

Acknowledgements V. K. acknowledges financial support by Taras Shevchenko Kyiv National University (Project No. 15BF051-03). The work at NCU was supported by the National Science Council of ROC (NSC 101-3113-P-008-008 and NSC-99-2221-E-008-095-MY3). Work performed at NCTU was supported by the National Science Council of ROC through Grant No. MOST 102-2221-E-009-195-MY3.

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