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Theoretical and experimental study of heat conduction in as-prepared and oxidized meso-porous silicon
Microelectronics Journal Microelectronics Journal 30 (1999) 1141–1147 www.elsevier.com/locate/mejo
Theoretical and experimental study of heat conduction in as-prepared and oxidized meso-porous silicon V. Lysenko a,*, L. Boarino b, M. Bertola b, B. Remaki a, A. Dittmar a, G. Amato b, D. Barbier a a
Laboratoire de Physique de la Matie`re, CNRS UMR 5511, INSA de Lyon, Baˆt. 401, 20, av. Albert Einstein, 69621, Villeurbanne Cedex, France b Istituto Elettrotecnico Nazionale Galileo Ferraris, Strada della Cacce 91, 10135 Turin, Italy
Abstract Recently measured low thermal conductivity of as-prepared and slightly oxidized meso-porous silicon (meso-PS) offers new possibility to apply this promising material for thermal isolation in microsensors and microsystems. We report here a theoretical model describing specific mechanisms of heat transport in as-prepared and oxidized meso-PS. The model is in good agreement with experimental data obtained earlier. In order to compare the thermal conductivity values of meso-PS layers oxidized at different temperatures with each other and with thermal conductivity of monocrystalline Si, a series of photoacoustic measurements was carried out. Evolution of the thermal conductivity along with oxidized fraction of Si in meso-PS has the same dynamics as that described by the theoretical model. 䉷 1999 Elsevier Science Ltd. All rights reserved. Keywords: Meso-porous silicon; Heat conduction; Photoacoustic measurements
1. Introduction Till now, the physical properties of porous silicon (PS) have been studied mainly from the structural, electrical and optical points of view [1,2]. This very promising material was also an object of numerous applications in silicon-oninsulator [3], microsystem [4] and microsensor [5] technologies. Recently, a new interesting phenomenon related to the thermal properties of PS has been discovered [6–10]. The authors report about low values of thermal conductivity (TC) of PS layers formed on p-type Si wafers. These values are two or three orders lower than that of monocrystalline silicon and close to the TC of SiO2 which is known as a good thermal insulator. Due to the low TC values some PS structures offer new possibilities to be applied as thermal insulating substrates for microsensor design [6,7]. An application of thick (about 100 mm) oxidized meso-PS layers for the thermal isolation of thermopile-based microsensors has been proposed and studied [11,12] as alternative approach to usually used thin Si membranes and cantilever beams [13]. At a fixed thickness, PS layers obtained on heavily doped p-type Si (meso-PS) are mechanically more stable than those formed on lightly doped p-type Si (nano-PS). TC of * Corresponding author. Tel.: ⫹ 33-4724-38859; fax: ⫹ 33-472438987.
as-prepared meso-porous silicon (meso-PS) with moderate porosity (45–64%) [7–9] as well as of oxidized meso-PS [7] has been recently measured at room temperature. Results of the measurements are given in Table 1. The main aim of this paper is to propose possible mechanisms of heat transport in as-prepared and oxidized meso-PS layers. In the theoretical part of this work we propose physical interpretations of the measured TC values of meso-PS [7–9] and study the influence of thermal oxidation on the TC values. In the experimental part of this paper we describe a series of photoacoustic measurements to study qualitatively the dynamics of TC evolution along with the oxidized fraction of meso-PS.
2. Theoretical 2.1. Thermal conductivity of as-prepared meso-PS In order to explain the measured TC values of meso-PS structures (see Table 1), some morphological peculiarities of this material have to be reconsidered. As it has been already reported [14,15], meso-PS layers formed on highly doped p-type Si wafers (electrical resistivity of about 0.01 V cm) exhibit a clear column structure. The layers consist of many long main pores running perpendicularly to the wafer surface and having small side branches. Characteristic structure size of interpore spacing (mean
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Table 1 Data (at room temperature) of thermal conductivity (TC) of meso-PS structures Reference
Year
Original Si wafer
Porosity (%)
TC of asprepared PS (W m ⫺1 K ⫺1)
TC after preoxidation (W m ⫺1 K ⫺1)
Drost et al. [7] Gesele et al. [8] Benedetto et al. [9]
1995 1997 1997
p ⫹-type, (100) p ⫹-type, (100) p ⫹-type, (111)
45 64 50 60
80 0.8 3.9 2.5
2.7 – – –
column width value) for this PS type is typically from 10 to 100 nm [7]. We consider here that a meso-PS layer with small porosity (⬍50%) is a relatively regular arrangement of quasiparallel continuous waved-side silicon columns (or wires) which are schematically shown in Fig. 1(a). In the case of enhanced porosity (ⱖ50%), the columns can be represented as stacking of quasi-spherical crystallites with a mean size, rcr, of about 10 nm. According to cross-sectional transmission electron micrographs of meso-PS presented in [14,15], our model is not far from reality. In the case of small porosity (Fig. 1(a)), by assuming that heat transfer in as-prepared meso-PS occurs only through the Si columns and not through the pores, the total TC of the meso-PS layer can be defined as: kmeso-PS kcol 1 ⫺ P
1
where kcol is the TC of columns (or wires) along the column height and P is the porosity of the as-prepared meso-PS layer. Since the height of each continuous silicon column is significantly longer than phonon mean free path in monocrystalline silicon, the TC of the columns, kcol, approaches to the TC value of monocrystalline silicon (kSi): kcol ⬇ kSi.
In the case of enhanced porosity (Fig. 1(b)), total TC of the meso-PS layer can be defined as: kmeso-PS kcol 1 ⫺ Pg0
2
where g0 is a so-called percolation strength which can be interpreted as the fraction of the solid phase which is interconnected and thus contributing to the heat conduction. Parameter g0 is introduced here in the same way as proposed by Gesele et al. [8] where a fixed relation between g0 and the porosity was given: g0 1 ⫺ P2
3
In this case, kcol is determined as mean TC of the quasicr . Comparspherical crystallites constituting each column keff ing characteristic mean size of the crystallites (rcr ⬇ 10 nm) with the phonon mean free path in monocrystalline Si at room temperature (LSi 43 nm [16]), we can remark that rcr ⬍ LSi. It means that heat transport in a single nanocrystallite cannot be explained by classic Fourier heat conduction theory for which rcr should be much larger than L. In the case when rcr ⬍ L, there is no phonon scattering inside the crystallites and consequently, neither temperature gradient nor the notions of temperature and thermal conductivity can be defined. Phonons scatter only at the crystallite boundaries which restore local thermodynamic equilibrium. Therefore, only at the boundaries, which act as artificial scatter sites, the notion of temperature can be introduced. Between two opposite boundaries the phonon transport has a ballistic nature. In spite of this, a notion of effective thercr ; can be introduced, mal conductivity of the crystallites, keff considering that the phonons scatter at an effective length, Leff [16,17]: Leff
LSi 4 LSi 1⫹ 3 rcr
4
cr can be determined as: Therefore, keff cr 1=3cvLeff keff
Fig. 1. Schematic presentation of the columnar structure of meso-PS: (a) small porosity, non-oxidized columns; (b) high porosity, non-oxidized columns; (c) small porosity, partially oxidized columns; (d) high porosity, partially oxidized columns.
kSi 4 LSi 1⫹ 3 rcr
5
where c is the specific heat per unit volume, v is the average speed of sound and kSi is the TC of monocrystalline Si. cr and g0 from Eqs. (5) and (3) into Eq. (2), we Putting keff
V. Lysenko et al. / Microelectronics Journal 30 (1999) 1141–1147
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90 25
80
20 70 15 60 10 50
5
40
0 0,01 0,05 0,09
30
0,13 0,17 0,21
Oxidized fraction
20 10 0 0
0,2
0,4
0,6
0,8
1
Oxidized fraction Fig. 2. Dependence of the thermal conductivity of meso-PS layer with small porosity on silicon oxidized fraction.
obtain for the case of enhanced porosity: kmeso-PS
kSi 1 ⫺ P3 4 LSi 1⫹ 3 rcr
6
of the column volume after the oxidation and j is the oxidized fraction of the origin silicon column and defined as:
j
For the case of small porosity (Eq. (1)), considering that kSi 150 (W m ⫺1 K ⫺1) and P 45%, we obtain kmeso-PS 82.5 (W m ⫺1 K ⫺1). In the case of enhanced porosity (Eq. (6)), considering that LSi 43 nm and rcr 10 nm, we obtain kmeso-PS 2.8 (W m ⫺1 K ⫺1) for P 50%, kmeso-PS 1.4 (W m ⫺1 K ⫺1) for P 60% and kmeso-PS 1.0 (W m ⫺1 K ⫺1) for P 64%. All these results are in good agreement with the experimental measurements performed earlier and summarized in Table 1.
V0ⴱ ⫺ VSiⴱ 2:27V0
8
where V0 is the initial volume of the silicon column, V0ⴱ is the total volume of the column after the partial oxidation ⴱ is the total volume of remained silicon cores of the and VSi oxidized column. ⴱ ; is The total TC of the oxidized meso-PS layer, kmeso-PS defined analogically to Eq. (1): ⴱ ⴱ kcol 1 ⫺ Pⴱ kmeso-PS
9
ⴱ
2.2. Dependence of the TC of meso-PS on oxidized fraction 2.2.1. Small porosity After partial oxidation, each Si column of the meso-PS layer is covered with a SiO2 sheet (Fig. 1(c)). First of all, the thinnest parts of the columns are completely oxidized and it occurs quickly even at very low temperatures. This fact and another one specifying that kSi q kSiO2 allow us to consider each column as a sequence of Si and SiO2 parts (Fig. 1(c)). ⴱ ; According to this, TC of a partially oxidized column, kcol can be determined as: ⴱ kcol
1 ⫹ 1:27j 1 ⫺ j 2:27j ⫹ kSi kSiO2
7
where kSi is the TC of monocrystalline silicon, kSiO2 is the TC of silicon dioxide, coefficient 2.27 considers expansion
where P is the porosity of the oxidized layer. By assuming external layer volume to be unchanged after the oxidation, P ⴱ depends on the porosity, P, of initial as-prepared mesoPS: Pⴱ 1 ⫺ 1 ⫺ P 1 ⫹ 1:27j
10
Putting Eqs. (7) and (10) in Eq. (9), we obtain: ⴱ kmeso-PS
1 ⫺ P 1 ⫹ 1:27j2 : 1 ⫺ j 2:27j ⫹ kSiO2 kSi
11
ⴱ f j is shown in Fig. 2 for the values The function kmeso-PS of kSi 150 W m ⫺1 K ⫺1, kSiO2 1:4 W m ⫺1 K ⫺1, and P 45%. As it can be seen from the figure, a slight oxidation of meso-PS layers leads to considerable decreasing of its TC value from 82.5 W m ⫺1 K ⫺1 for the as-prepared material down to 2.7 W m ⫺1 K ⫺1 for partially (19%) oxidized
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Porosity...........................................56 % Phonon mean free path in Si.........43 nm Thermal conductivity of Si.............150 W/m K Thermal conductivity of SiO2.........1.4 W/m K
Thermal conductivity, (W/mK
3,5 3
r=5 nm r=10 nm r=20 nm
2,5 2 1,5 1 0,5
0
25 50 75 Si oxidized fraction, (%)
100
Fig. 3. Dependence of the thermal conductivity of meso-PS layer with high porosity on silicon oxidized fraction for three different sizes of silicon particles containing the columns.
material. TC of the layer is almost unchanged for oxidized fractions above 25%. Chemical composition analysis (made by means of energy dispersive X-ray fluorescence spectroscopy (EDXRFS)) of partially oxidized meso-PS layers with about 50% porosity has shown that the oxidized fraction of 15–20% can be reached after oxidation process in dry O2 atmosphere at 300⬚C for 1 h (so-called “pre-oxidation step”). Almost the same value of oxidized fraction (about 12%) in a meso-PS layer with 51% porosity after the “pre-oxidation step” has been obtained by means of another technique [18] based on the measurements of dissolution rates of thermally grown oxides in buffered hydrofluoric acid solutions. The theoretical calculations made just above are in excellent agreement with experimental results described in [7] where the TC of as-prepared meso-PS equal 80 W m ⫺1 K ⫺1 decreased drastically to 2.7 W m ⫺1 K ⫺1 after the “pre-oxidation step”. 2.2.2. Enhanced porosity After partial oxidation, silicon crystallites of each column are covered with a SiO2 sheet and their initial sizes are reduced (Fig. 1(d)). In this case, TC of the remained Si ⴱ ; can cores of the partially oxidized Si/SiO2 crystallite, kSi be expressed by analogy with Eq. (5) as following: ⴱ kSi
kSi p 4 1 ⫹ LSi rcr 3 1 ⫺ j 3
12
where j is the oxidized fraction of the original Si crystallite and defined by analogy with Eq. (8).
It can be easily proved (by analogy with Eq. (7)) that the ⴱ TC of a partially oxidized column, kcol ; can be written as: ⴱ kcol
1 ⫹ 1:27j 4 LSi p3 1 ⫺ j 1 ⫹ 2:27j 3 rcr 1 ⫺ j ⫹ kSiO2 kSi
13
where kSiO2 is the TC of silicon dioxide. The total TC of the oxidized meso-PS layer with ⴱ ; is defined by analogy with Eq. enhanced porosity, kmeso-PS (2) as: ⴱ ⴱ kcol 1 ⫺ Pⴱ gⴱ0 kmeso-PS ⴱ
14 gⴱ0
where P is the porosity and is the percolation strength of the oxidized meso-PS layer. Considering that P ⴱ is determined by Eq. (10) and gⴱ0 1 ⫺ Pⴱ 2 ; we obtain: ⴱ kmeso-PS
1 ⫺ P3 1 ⫹ 1:27j4 4 LSi p3 1 ⫺ j 1 ⫹ 2:27j 3 rcr 1 ⫺ j ⫹ kSiO2 kSi
15
Using kSi 150 W m ⫺1 K ⫺1, kSiO2 1:4 W m ⫺1 K ⫺1, ⴱ LSi 43 nm and P 56%, the function kmeso-PS f j is plotted in Fig. 3 for three different sizes of the original nonoxidized Si nanocrystallites. The layer porosity of 56% is chosen to test the function, because for this porosity value, there is no change of the layer volume when the layer is completely oxidized [19], i.e. the pores are completely filled by the grown silicon oxide at j 100% and initial
V. Lysenko et al. / Microelectronics Journal 30 (1999) 1141–1147
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Fig. 4. Schematic view of experimental set-up for photoacoustics measurements.
as-prepared PS layer is completely transformed into a single ⴱ should be equal to block of SiO2. Therefore, kmeso-PS ⫺1 ⫺1 1.4 W m K at j 100%. As it can be seen in the Fig. 3, at a fixed porosity value (⬎50%), the initial TC value of meso-PS depends on the characteristic size of the non-oxidized crystallites. That is why the TC value depends on distribution of the crystallites upon their sizes and, according to our calculations, for P 56%, will be in the range of 1.0–3.3 W m ⫺1 K ⫺1. For a fixed ⴱ value of rcr, the function kmeso-PS f j decreases in the range of small j values and has a minimum value at j ⬇ 20%. It corresponds to heat transport preferably through the ⴱ increases along the decreased Si cores. Then, kmeso-PS increasing of oxidized fraction of silicon, because of the heat transport occurring through the growing oxide sheet, and comes up to the TC value of silicon dioxide (1.4 W m ⫺1 K ⫺1) when the PS layer is completely oxidized and transformed into a SiO2 layer.
3. Experimental 3.1. Photoacoustic measurements In order to estimate TC values of as-prepared and oxidized meso-PS layers and to compare them with TC of monocrystalline Si, a series of photoacoustic (PA)
measurements has been performed. This kind of measurements was already used to evaluate TC of different PS types [9]. The experimental setup for PA measurements is schematically described in Fig. 4. A photoacoustic cell of small dimensions (530 mm 3), which consists of a cylindrical aluminum body, delimited by two optical quartz windows was used [20]. The samples were glued to one window in some points by a small amount of adhesive paste so as to leave a thin air gap between the window and the substrate. The front surface of the sample was illuminated with the light beam at 514 nm from an Ar ⫹ ion laser, modulated with an acousto-optical modulator in the frequency range 10 Hz–1 kHz. The experimental results were normalized to the transfer function of the measurement system (cell volume, microphone and preamplifier), which is determined on a carbon black sample [21]. Thermal diffusion length in investigated material is decreased with increasing modulation frequency. The surface of PS layers becomes to be important in the thermal transport for the laser modulation frequency value of 100 Hz. Temperature variations at the sample surface determining the amplitude of air pressure variations detected by the microphone increase with decreasing TC of the samples. Therefore, PA signals corresponding to PS samples are larger than those of Si samples because of the lower TC of the PS layers with respect to that of monocrystalline Si substrates.
V. Lysenko et al. / Microelectronics Journal 30 (1999) 1141–1147
Photoacoustic signal, (mV)
1146
1
oxidized meso-PS
0,1 Si PS150C PS300C
0,01
PS450C PS600C
mono-Si
PS700C
10
100
1000
Modulation Frequency, (Hz) Fig. 5. Amplitude of photoacoustics signal versus modulation frequency of laser beam: monocrystalline Si (B); Si/meso-PS structure oxidized at 150⬚C (X); at 300⬚C (K); at 450⬚C (P); at 600⬚C (V); at 700⬚C (✠).
The absorption is assumed to be localized to the sample surface (typically the light absorption coefficient of the investigated samples is in the range 10 3 –10 4 cm ⫺1). The pressure variation as a function of the TC of meso-PS is calculated assuming that the experimental situation can be represented by a four-homogeneous-layer structure “air– meso-PS–monocrystalline Si–air”. This function was described in details in [9]. Formation of ‘Si/meso-PS’ structures was carried out by means of well-known anodic dissolution process [1] that is usually achieved in HF-based electrolytes. In this work, monocrystalline (100)-oriented highly doped p ⫹-type Si wafers with resulting electrical resistivity of about 0.02 V cm and a standard electrochemical cell with metallic back-side electrode were used. Due to the enhanced wafer doping concentration, ohmic contacts to the back-side of the wafers were not necessary. PS circular samples with the area of 1 cm 2 were fabricated at constant anodization current density value of 100 mA cm ⫺2 in a solution of HF (40%) and ethanol, the ratio of which was HF (40%):ethanol 2:1. After fabrication, the samples were rinsed in deionized
Photoacoustic signal, (mV)
0,36
water. The 60 mm thick layers were oxidized in dry O2 atmosphere for 1 h at different temperatures. PA signals from the oxidized Si/PS structures illuminated by the modulated laser light beam as a function of the modulation frequency are shown in the Fig. 5. As one can see in this figure, the PA signals corresponding to the Si/PS structures are almost the same and quite different from the PA signal of monocrystalline Si. It means that the TC values of the as-prepared and oxidized PS layers are close to one another and quite different (estimated qualitatively to be about two order lower) from the TC of monocrystalline Si. Having such small TC values, oxidized meso-PS can be successfully applied as thermal insulating substrate. The PA signals from the oxidized PS layers corresponding to the modulation frequency of 100 Hz are plotted in Fig. 6 as a function of the oxidation temperature. As can be seen in the figure, the PA signals reflecting the TC evolution of the PS layers when increasing their oxidation temperature and consequently the PS oxidized fraction, have a maximum in the range of low temperatures (150–200⬚C) and then decrease. It means that at these low temperatures producing small oxidation fraction values there is a slight minimum of the TC of oxidized meso-PS layers and then the TC slightly increases when the oxidation temperature increases too. This evolution of TC of the meso-PS samples oxidized at different temperatures along the oxidized fraction values (corresponding to the oxidation temperatures) has the same dynamics as that described by the theoretical curves shown in Fig. 3. The TC increase for the temperatures above 200⬚C can be explained by a volume expansion of the oxidized Si particles of PS which fills partially the pores decreasing the layer porosity, and consequently, increasing the TC. The experiment does not give the absolute TC values of the oxidized meso-PS layers. However, it qualitatively brings to the fore two important points described in Section 2 of the paper: (i) quite smaller TC values of the as-prepared and oxidized meso-PS compared with TC of monocrystalline silicon and (ii) existence of minimum of the TC of oxidized meso-PS samples as function of oxidized fraction of the samples.
modulation frequency - 100 Hz
0,34
4. Conclusions
0,32 0,30 0,28 0,26 0,24 0
200
400
600
Oxidation temperature, (˚C) Fig. 6. Amplitude of photoacoustics signal versus oxidation temperature at 100 Hz of laser beam modulation frequency.
A theoretical model describing specific mechanisms of the heat transport in as-prepared and oxidized meso-PS was proposed. All theoretical estimations presented above are in good agreement with experimental results reported earlier and with photoacoustic measurements described in this paper. A new approach which uses oxidized meso-PS with low TC values (two order lower than TC of monocrystalline silicon) for thermal isolation of numerous Si-based microsensors and microsystems can be employed in future.
V. Lysenko et al. / Microelectronics Journal 30 (1999) 1141–1147
Acknowledgement This work was partially supported by the French Government through the a Ph.D. fellowship for V. Lysenko.
[11]
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Theoretical and experimental study of heat conduction in as-prepared and oxidized meso-porous silicon V. Lysenko a,*, L. Boarino b, M. Bertola b, B. Remaki a, A. Dittmar a, G. Amato b, D. Barbier a a
Laboratoire de Physique de la Matie`re, CNRS UMR 5511, INSA de Lyon, Baˆt. 401, 20, av. Albert Einstein, 69621, Villeurbanne Cedex, France b Istituto Elettrotecnico Nazionale Galileo Ferraris, Strada della Cacce 91, 10135 Turin, Italy
Abstract Recently measured low thermal conductivity of as-prepared and slightly oxidized meso-porous silicon (meso-PS) offers new possibility to apply this promising material for thermal isolation in microsensors and microsystems. We report here a theoretical model describing specific mechanisms of heat transport in as-prepared and oxidized meso-PS. The model is in good agreement with experimental data obtained earlier. In order to compare the thermal conductivity values of meso-PS layers oxidized at different temperatures with each other and with thermal conductivity of monocrystalline Si, a series of photoacoustic measurements was carried out. Evolution of the thermal conductivity along with oxidized fraction of Si in meso-PS has the same dynamics as that described by the theoretical model. 䉷 1999 Elsevier Science Ltd. All rights reserved. Keywords: Meso-porous silicon; Heat conduction; Photoacoustic measurements
1. Introduction Till now, the physical properties of porous silicon (PS) have been studied mainly from the structural, electrical and optical points of view [1,2]. This very promising material was also an object of numerous applications in silicon-oninsulator [3], microsystem [4] and microsensor [5] technologies. Recently, a new interesting phenomenon related to the thermal properties of PS has been discovered [6–10]. The authors report about low values of thermal conductivity (TC) of PS layers formed on p-type Si wafers. These values are two or three orders lower than that of monocrystalline silicon and close to the TC of SiO2 which is known as a good thermal insulator. Due to the low TC values some PS structures offer new possibilities to be applied as thermal insulating substrates for microsensor design [6,7]. An application of thick (about 100 mm) oxidized meso-PS layers for the thermal isolation of thermopile-based microsensors has been proposed and studied [11,12] as alternative approach to usually used thin Si membranes and cantilever beams [13]. At a fixed thickness, PS layers obtained on heavily doped p-type Si (meso-PS) are mechanically more stable than those formed on lightly doped p-type Si (nano-PS). TC of * Corresponding author. Tel.: ⫹ 33-4724-38859; fax: ⫹ 33-472438987.
as-prepared meso-porous silicon (meso-PS) with moderate porosity (45–64%) [7–9] as well as of oxidized meso-PS [7] has been recently measured at room temperature. Results of the measurements are given in Table 1. The main aim of this paper is to propose possible mechanisms of heat transport in as-prepared and oxidized meso-PS layers. In the theoretical part of this work we propose physical interpretations of the measured TC values of meso-PS [7–9] and study the influence of thermal oxidation on the TC values. In the experimental part of this paper we describe a series of photoacoustic measurements to study qualitatively the dynamics of TC evolution along with the oxidized fraction of meso-PS.
2. Theoretical 2.1. Thermal conductivity of as-prepared meso-PS In order to explain the measured TC values of meso-PS structures (see Table 1), some morphological peculiarities of this material have to be reconsidered. As it has been already reported [14,15], meso-PS layers formed on highly doped p-type Si wafers (electrical resistivity of about 0.01 V cm) exhibit a clear column structure. The layers consist of many long main pores running perpendicularly to the wafer surface and having small side branches. Characteristic structure size of interpore spacing (mean
0026-2692/99/$ - see front matter 䉷 1999 Elsevier Science Ltd. All rights reserved. PII: S0026-269 2(99)00077-4
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V. Lysenko et al. / Microelectronics Journal 30 (1999) 1141–1147
Table 1 Data (at room temperature) of thermal conductivity (TC) of meso-PS structures Reference
Year
Original Si wafer
Porosity (%)
TC of asprepared PS (W m ⫺1 K ⫺1)
TC after preoxidation (W m ⫺1 K ⫺1)
Drost et al. [7] Gesele et al. [8] Benedetto et al. [9]
1995 1997 1997
p ⫹-type, (100) p ⫹-type, (100) p ⫹-type, (111)
45 64 50 60
80 0.8 3.9 2.5
2.7 – – –
column width value) for this PS type is typically from 10 to 100 nm [7]. We consider here that a meso-PS layer with small porosity (⬍50%) is a relatively regular arrangement of quasiparallel continuous waved-side silicon columns (or wires) which are schematically shown in Fig. 1(a). In the case of enhanced porosity (ⱖ50%), the columns can be represented as stacking of quasi-spherical crystallites with a mean size, rcr, of about 10 nm. According to cross-sectional transmission electron micrographs of meso-PS presented in [14,15], our model is not far from reality. In the case of small porosity (Fig. 1(a)), by assuming that heat transfer in as-prepared meso-PS occurs only through the Si columns and not through the pores, the total TC of the meso-PS layer can be defined as: kmeso-PS kcol 1 ⫺ P
1
where kcol is the TC of columns (or wires) along the column height and P is the porosity of the as-prepared meso-PS layer. Since the height of each continuous silicon column is significantly longer than phonon mean free path in monocrystalline silicon, the TC of the columns, kcol, approaches to the TC value of monocrystalline silicon (kSi): kcol ⬇ kSi.
In the case of enhanced porosity (Fig. 1(b)), total TC of the meso-PS layer can be defined as: kmeso-PS kcol 1 ⫺ Pg0
2
where g0 is a so-called percolation strength which can be interpreted as the fraction of the solid phase which is interconnected and thus contributing to the heat conduction. Parameter g0 is introduced here in the same way as proposed by Gesele et al. [8] where a fixed relation between g0 and the porosity was given: g0 1 ⫺ P2
3
In this case, kcol is determined as mean TC of the quasicr . Comparspherical crystallites constituting each column keff ing characteristic mean size of the crystallites (rcr ⬇ 10 nm) with the phonon mean free path in monocrystalline Si at room temperature (LSi 43 nm [16]), we can remark that rcr ⬍ LSi. It means that heat transport in a single nanocrystallite cannot be explained by classic Fourier heat conduction theory for which rcr should be much larger than L. In the case when rcr ⬍ L, there is no phonon scattering inside the crystallites and consequently, neither temperature gradient nor the notions of temperature and thermal conductivity can be defined. Phonons scatter only at the crystallite boundaries which restore local thermodynamic equilibrium. Therefore, only at the boundaries, which act as artificial scatter sites, the notion of temperature can be introduced. Between two opposite boundaries the phonon transport has a ballistic nature. In spite of this, a notion of effective thercr ; can be introduced, mal conductivity of the crystallites, keff considering that the phonons scatter at an effective length, Leff [16,17]: Leff
LSi 4 LSi 1⫹ 3 rcr
4
cr can be determined as: Therefore, keff cr 1=3cvLeff keff
Fig. 1. Schematic presentation of the columnar structure of meso-PS: (a) small porosity, non-oxidized columns; (b) high porosity, non-oxidized columns; (c) small porosity, partially oxidized columns; (d) high porosity, partially oxidized columns.
kSi 4 LSi 1⫹ 3 rcr
5
where c is the specific heat per unit volume, v is the average speed of sound and kSi is the TC of monocrystalline Si. cr and g0 from Eqs. (5) and (3) into Eq. (2), we Putting keff
V. Lysenko et al. / Microelectronics Journal 30 (1999) 1141–1147
1143
90 25
80
20 70 15 60 10 50
5
40
0 0,01 0,05 0,09
30
0,13 0,17 0,21
Oxidized fraction
20 10 0 0
0,2
0,4
0,6
0,8
1
Oxidized fraction Fig. 2. Dependence of the thermal conductivity of meso-PS layer with small porosity on silicon oxidized fraction.
obtain for the case of enhanced porosity: kmeso-PS
kSi 1 ⫺ P3 4 LSi 1⫹ 3 rcr
6
of the column volume after the oxidation and j is the oxidized fraction of the origin silicon column and defined as:
j
For the case of small porosity (Eq. (1)), considering that kSi 150 (W m ⫺1 K ⫺1) and P 45%, we obtain kmeso-PS 82.5 (W m ⫺1 K ⫺1). In the case of enhanced porosity (Eq. (6)), considering that LSi 43 nm and rcr 10 nm, we obtain kmeso-PS 2.8 (W m ⫺1 K ⫺1) for P 50%, kmeso-PS 1.4 (W m ⫺1 K ⫺1) for P 60% and kmeso-PS 1.0 (W m ⫺1 K ⫺1) for P 64%. All these results are in good agreement with the experimental measurements performed earlier and summarized in Table 1.
V0ⴱ ⫺ VSiⴱ 2:27V0
8
where V0 is the initial volume of the silicon column, V0ⴱ is the total volume of the column after the partial oxidation ⴱ is the total volume of remained silicon cores of the and VSi oxidized column. ⴱ ; is The total TC of the oxidized meso-PS layer, kmeso-PS defined analogically to Eq. (1): ⴱ ⴱ kcol 1 ⫺ Pⴱ kmeso-PS
9
ⴱ
2.2. Dependence of the TC of meso-PS on oxidized fraction 2.2.1. Small porosity After partial oxidation, each Si column of the meso-PS layer is covered with a SiO2 sheet (Fig. 1(c)). First of all, the thinnest parts of the columns are completely oxidized and it occurs quickly even at very low temperatures. This fact and another one specifying that kSi q kSiO2 allow us to consider each column as a sequence of Si and SiO2 parts (Fig. 1(c)). ⴱ ; According to this, TC of a partially oxidized column, kcol can be determined as: ⴱ kcol
1 ⫹ 1:27j 1 ⫺ j 2:27j ⫹ kSi kSiO2
7
where kSi is the TC of monocrystalline silicon, kSiO2 is the TC of silicon dioxide, coefficient 2.27 considers expansion
where P is the porosity of the oxidized layer. By assuming external layer volume to be unchanged after the oxidation, P ⴱ depends on the porosity, P, of initial as-prepared mesoPS: Pⴱ 1 ⫺ 1 ⫺ P 1 ⫹ 1:27j
10
Putting Eqs. (7) and (10) in Eq. (9), we obtain: ⴱ kmeso-PS
1 ⫺ P 1 ⫹ 1:27j2 : 1 ⫺ j 2:27j ⫹ kSiO2 kSi
11
ⴱ f j is shown in Fig. 2 for the values The function kmeso-PS of kSi 150 W m ⫺1 K ⫺1, kSiO2 1:4 W m ⫺1 K ⫺1, and P 45%. As it can be seen from the figure, a slight oxidation of meso-PS layers leads to considerable decreasing of its TC value from 82.5 W m ⫺1 K ⫺1 for the as-prepared material down to 2.7 W m ⫺1 K ⫺1 for partially (19%) oxidized
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V. Lysenko et al. / Microelectronics Journal 30 (1999) 1141–1147
Porosity...........................................56 % Phonon mean free path in Si.........43 nm Thermal conductivity of Si.............150 W/m K Thermal conductivity of SiO2.........1.4 W/m K
Thermal conductivity, (W/mK
3,5 3
r=5 nm r=10 nm r=20 nm
2,5 2 1,5 1 0,5
0
25 50 75 Si oxidized fraction, (%)
100
Fig. 3. Dependence of the thermal conductivity of meso-PS layer with high porosity on silicon oxidized fraction for three different sizes of silicon particles containing the columns.
material. TC of the layer is almost unchanged for oxidized fractions above 25%. Chemical composition analysis (made by means of energy dispersive X-ray fluorescence spectroscopy (EDXRFS)) of partially oxidized meso-PS layers with about 50% porosity has shown that the oxidized fraction of 15–20% can be reached after oxidation process in dry O2 atmosphere at 300⬚C for 1 h (so-called “pre-oxidation step”). Almost the same value of oxidized fraction (about 12%) in a meso-PS layer with 51% porosity after the “pre-oxidation step” has been obtained by means of another technique [18] based on the measurements of dissolution rates of thermally grown oxides in buffered hydrofluoric acid solutions. The theoretical calculations made just above are in excellent agreement with experimental results described in [7] where the TC of as-prepared meso-PS equal 80 W m ⫺1 K ⫺1 decreased drastically to 2.7 W m ⫺1 K ⫺1 after the “pre-oxidation step”. 2.2.2. Enhanced porosity After partial oxidation, silicon crystallites of each column are covered with a SiO2 sheet and their initial sizes are reduced (Fig. 1(d)). In this case, TC of the remained Si ⴱ ; can cores of the partially oxidized Si/SiO2 crystallite, kSi be expressed by analogy with Eq. (5) as following: ⴱ kSi
kSi p 4 1 ⫹ LSi rcr 3 1 ⫺ j 3
12
where j is the oxidized fraction of the original Si crystallite and defined by analogy with Eq. (8).
It can be easily proved (by analogy with Eq. (7)) that the ⴱ TC of a partially oxidized column, kcol ; can be written as: ⴱ kcol
1 ⫹ 1:27j 4 LSi p3 1 ⫺ j 1 ⫹ 2:27j 3 rcr 1 ⫺ j ⫹ kSiO2 kSi
13
where kSiO2 is the TC of silicon dioxide. The total TC of the oxidized meso-PS layer with ⴱ ; is defined by analogy with Eq. enhanced porosity, kmeso-PS (2) as: ⴱ ⴱ kcol 1 ⫺ Pⴱ gⴱ0 kmeso-PS ⴱ
14 gⴱ0
where P is the porosity and is the percolation strength of the oxidized meso-PS layer. Considering that P ⴱ is determined by Eq. (10) and gⴱ0 1 ⫺ Pⴱ 2 ; we obtain: ⴱ kmeso-PS
1 ⫺ P3 1 ⫹ 1:27j4 4 LSi p3 1 ⫺ j 1 ⫹ 2:27j 3 rcr 1 ⫺ j ⫹ kSiO2 kSi
15
Using kSi 150 W m ⫺1 K ⫺1, kSiO2 1:4 W m ⫺1 K ⫺1, ⴱ LSi 43 nm and P 56%, the function kmeso-PS f j is plotted in Fig. 3 for three different sizes of the original nonoxidized Si nanocrystallites. The layer porosity of 56% is chosen to test the function, because for this porosity value, there is no change of the layer volume when the layer is completely oxidized [19], i.e. the pores are completely filled by the grown silicon oxide at j 100% and initial
V. Lysenko et al. / Microelectronics Journal 30 (1999) 1141–1147
1145
Fig. 4. Schematic view of experimental set-up for photoacoustics measurements.
as-prepared PS layer is completely transformed into a single ⴱ should be equal to block of SiO2. Therefore, kmeso-PS ⫺1 ⫺1 1.4 W m K at j 100%. As it can be seen in the Fig. 3, at a fixed porosity value (⬎50%), the initial TC value of meso-PS depends on the characteristic size of the non-oxidized crystallites. That is why the TC value depends on distribution of the crystallites upon their sizes and, according to our calculations, for P 56%, will be in the range of 1.0–3.3 W m ⫺1 K ⫺1. For a fixed ⴱ value of rcr, the function kmeso-PS f j decreases in the range of small j values and has a minimum value at j ⬇ 20%. It corresponds to heat transport preferably through the ⴱ increases along the decreased Si cores. Then, kmeso-PS increasing of oxidized fraction of silicon, because of the heat transport occurring through the growing oxide sheet, and comes up to the TC value of silicon dioxide (1.4 W m ⫺1 K ⫺1) when the PS layer is completely oxidized and transformed into a SiO2 layer.
3. Experimental 3.1. Photoacoustic measurements In order to estimate TC values of as-prepared and oxidized meso-PS layers and to compare them with TC of monocrystalline Si, a series of photoacoustic (PA)
measurements has been performed. This kind of measurements was already used to evaluate TC of different PS types [9]. The experimental setup for PA measurements is schematically described in Fig. 4. A photoacoustic cell of small dimensions (530 mm 3), which consists of a cylindrical aluminum body, delimited by two optical quartz windows was used [20]. The samples were glued to one window in some points by a small amount of adhesive paste so as to leave a thin air gap between the window and the substrate. The front surface of the sample was illuminated with the light beam at 514 nm from an Ar ⫹ ion laser, modulated with an acousto-optical modulator in the frequency range 10 Hz–1 kHz. The experimental results were normalized to the transfer function of the measurement system (cell volume, microphone and preamplifier), which is determined on a carbon black sample [21]. Thermal diffusion length in investigated material is decreased with increasing modulation frequency. The surface of PS layers becomes to be important in the thermal transport for the laser modulation frequency value of 100 Hz. Temperature variations at the sample surface determining the amplitude of air pressure variations detected by the microphone increase with decreasing TC of the samples. Therefore, PA signals corresponding to PS samples are larger than those of Si samples because of the lower TC of the PS layers with respect to that of monocrystalline Si substrates.
V. Lysenko et al. / Microelectronics Journal 30 (1999) 1141–1147
Photoacoustic signal, (mV)
1146
1
oxidized meso-PS
0,1 Si PS150C PS300C
0,01
PS450C PS600C
mono-Si
PS700C
10
100
1000
Modulation Frequency, (Hz) Fig. 5. Amplitude of photoacoustics signal versus modulation frequency of laser beam: monocrystalline Si (B); Si/meso-PS structure oxidized at 150⬚C (X); at 300⬚C (K); at 450⬚C (P); at 600⬚C (V); at 700⬚C (✠).
The absorption is assumed to be localized to the sample surface (typically the light absorption coefficient of the investigated samples is in the range 10 3 –10 4 cm ⫺1). The pressure variation as a function of the TC of meso-PS is calculated assuming that the experimental situation can be represented by a four-homogeneous-layer structure “air– meso-PS–monocrystalline Si–air”. This function was described in details in [9]. Formation of ‘Si/meso-PS’ structures was carried out by means of well-known anodic dissolution process [1] that is usually achieved in HF-based electrolytes. In this work, monocrystalline (100)-oriented highly doped p ⫹-type Si wafers with resulting electrical resistivity of about 0.02 V cm and a standard electrochemical cell with metallic back-side electrode were used. Due to the enhanced wafer doping concentration, ohmic contacts to the back-side of the wafers were not necessary. PS circular samples with the area of 1 cm 2 were fabricated at constant anodization current density value of 100 mA cm ⫺2 in a solution of HF (40%) and ethanol, the ratio of which was HF (40%):ethanol 2:1. After fabrication, the samples were rinsed in deionized
Photoacoustic signal, (mV)
0,36
water. The 60 mm thick layers were oxidized in dry O2 atmosphere for 1 h at different temperatures. PA signals from the oxidized Si/PS structures illuminated by the modulated laser light beam as a function of the modulation frequency are shown in the Fig. 5. As one can see in this figure, the PA signals corresponding to the Si/PS structures are almost the same and quite different from the PA signal of monocrystalline Si. It means that the TC values of the as-prepared and oxidized PS layers are close to one another and quite different (estimated qualitatively to be about two order lower) from the TC of monocrystalline Si. Having such small TC values, oxidized meso-PS can be successfully applied as thermal insulating substrate. The PA signals from the oxidized PS layers corresponding to the modulation frequency of 100 Hz are plotted in Fig. 6 as a function of the oxidation temperature. As can be seen in the figure, the PA signals reflecting the TC evolution of the PS layers when increasing their oxidation temperature and consequently the PS oxidized fraction, have a maximum in the range of low temperatures (150–200⬚C) and then decrease. It means that at these low temperatures producing small oxidation fraction values there is a slight minimum of the TC of oxidized meso-PS layers and then the TC slightly increases when the oxidation temperature increases too. This evolution of TC of the meso-PS samples oxidized at different temperatures along the oxidized fraction values (corresponding to the oxidation temperatures) has the same dynamics as that described by the theoretical curves shown in Fig. 3. The TC increase for the temperatures above 200⬚C can be explained by a volume expansion of the oxidized Si particles of PS which fills partially the pores decreasing the layer porosity, and consequently, increasing the TC. The experiment does not give the absolute TC values of the oxidized meso-PS layers. However, it qualitatively brings to the fore two important points described in Section 2 of the paper: (i) quite smaller TC values of the as-prepared and oxidized meso-PS compared with TC of monocrystalline silicon and (ii) existence of minimum of the TC of oxidized meso-PS samples as function of oxidized fraction of the samples.
modulation frequency - 100 Hz
0,34
4. Conclusions
0,32 0,30 0,28 0,26 0,24 0
200
400
600
Oxidation temperature, (˚C) Fig. 6. Amplitude of photoacoustics signal versus oxidation temperature at 100 Hz of laser beam modulation frequency.
A theoretical model describing specific mechanisms of the heat transport in as-prepared and oxidized meso-PS was proposed. All theoretical estimations presented above are in good agreement with experimental results reported earlier and with photoacoustic measurements described in this paper. A new approach which uses oxidized meso-PS with low TC values (two order lower than TC of monocrystalline silicon) for thermal isolation of numerous Si-based microsensors and microsystems can be employed in future.
V. Lysenko et al. / Microelectronics Journal 30 (1999) 1141–1147
Acknowledgement This work was partially supported by the French Government through the a Ph.D. fellowship for V. Lysenko.
[11]
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