The Pennsylvania State University. The Graduate School. Department of Mechanical and Nuclear Engineering MEASUREMENT OF ONE-DIMENSIONAL NANOSTRUCTURES

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1 The Pennsylvania State University The Graduate School Department of Mechanical and Nuclear Engineering A MODIFIED 3! METHOD FOR THERMAL CONDUCTIVITY MEASUREMENT OF ONE-DIMENSIONAL NANOSTRUCTURES A Thesis in Mechanical Engineering by Hsiao-Fang Lee " 2009 Hsiao-Fang Lee Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science May 2009

2 ii The thesis of Hsiao-Fang Lee was reviewed and approved* by the following: M. Amanul Haque Associate Professor of Mechanical Engineering Department of Mechanical and Nuclear Engineering Thesis Advisor Anil K. Kulkarni Professor of Mechanical Engineering Department of Mechanical and Nuclear Engineering Karen A. Thole Professor of Mechanical Engineering Head of the Department of Mechanical and Nuclear Engineering *Signatures are on file in the Graduate School

3 ABSTRACT iii We present a new analytical model for thermal conductivity measurement of onedimensional nanostructures on substrates. The model expands the capability of the conventional 3! technique, to make it versatile with both in and out of plane thermal conductivity measurement on specimens either freestanding or attached to substrates. We demonstrate this new technique on both conducting (aluminum) and semi-conducting (focused ion beam deposited platinum) specimens. The agreement with the established values in the literature suggests the superiority of this technique in terms of convenience and robustness of measurement.

4 TABLE OF CONTENTS iv LIST OF FIGURES...vi ACKNOWLEDGEMENTS...viii Chapter 1 Introduction Thermal Characterization of Nanostructures Motivation of This Study Reference...4 Chapter 2 Literature Review Thermal Conductivity Measurement by 3! method Line Heater on Substrate Thin Film on Substrate Suspended Wire Strain Effects on Thermal Conductivity Reference...21 Chapter 3 Analytic Modeling of 3! Method Introduction of 3! Method Continuity Equation D Heat Conduction Equation with Heat Source D Heat Conduction Equation with Heat Source and Heat Sink Solution of 1-D Heat Conduction Equation with Heat Source and Heat Sink Reference...32 Chapter 4 Experimental Method Sample Preparation Experimental Set-up Main Apparatus Cryogenic Station Calibration Reference...38 Chapter 5 Results and Discussion Aluminum Film Temperature Dependence of Resistance of Aluminum Film Measurement of Thermal Conductivity...41

5 5.2 Focused Ion Beam Deposited Platinum Nanowire Conclusion Reference...50 v

6 LIST OF FIGURES vi Figure 2-1: Measurement structure schematic proposed by McConnel Figure 2-2: Side view of geometry of line heater on substrate Figure 2-3: Four pads patterned metal film on amorphous substrate Figure 2-4: Geometry of the thin file thermal conductivity measurement Figure 2-5: The real part of temperature oscillation as function of frequency Figure 2-6: A two-layer structure of 3! applications Figure 2-7: (a) a thin line (b) a wide line Figure 2-8: Illustration of the four-probe configuration of a filament-like specimen Figure 2-9: Circuits of voltage approximated current source Figure 2-10: Schematic of traditional and DC offset added 3! method Figure 2-11: Relationship between electrical and thermal functions Figure 2-12: The image of suspended silicon nanowire and the schematic of this device Figure 2-13: Relationship between thermal conductance and temperature Figure 2-14: Thermal conductivity as a function of temperature and strain Figure 2-15: Thermal conductivity and mechanical strains Figure 3-1: Principal of 3! method Figure 3-2: 3! method Figure 3-3: Geometry of wire-on-substrate Figure 4-1: SEM image of aluminum specimen Figure 4-2: Schematic of 3! technique applied on FIB-deposited Platinum

7 Figure 4-3: Experimental set-up of 3! technique Figure 4-4: 3! voltage measurement by using silver line heater on glass substrate sample geometry Figure 5-1: Temperature dependence of resistance of aluminum film Figure 5-2: The measured V 3" and expected V 3" of aluminum thin film Figure 5-3: Thermal conductivity of aluminum thin film Figure 5-4: The proportional relationship between V of aluminum thin film and I 3 3" Figure 5-5: The length dependence on thermal conductivity of aluminum thin film Figure 5-6: Resistance of FIB platinum Figure 5-7: Components of FIB platinum Figure 5-8: Temperature dependence of resistance of FIB platinum Figure 5-9: The measured V 3" for FIB-deposited Pt nanowire Figure 5-10: The average temperature change of FIB Pt nanowire Figure 5-11: The conventional 3! setup vii

8 ACKNOWLEDGEMENTS viii I would like to express my sincere gratitude to my adviser for bringing me into the research field of nanotechnology. I can t approach to the present achievement without his advice and encouragement, especially during my frustrated time. Besides, I want to thank my dear friends, Benedict and ChengIng, for helping me to solve problems and giving me practical suggestions. They are not only wise teachers but also nice friends. Moreover, I appreciate my lab mates, Mohan and Sandeep for preparing devices of my experiments, and Jiezhu for discussing 3! method with me. I am very grateful to my parents for supporting me to continue my study. I wish that they would be proud of my work and myself. Finally, I would like to thank Buddhas and Buddhadharma for bringing me the mental sustenance and wisdom.

9 1 Chapter 1 Introduction 1.1 Thermal Characterization of Nanostructures The rapid progress of fabrication technology of materials and devices with nanoscale dimensions has brought not only research opportunities of nanostructures but also industrial applications of nanostructures. There are few kinds of active nanostructures, for example, nanotubes, thin films, and nanowires. The most widely used nanostructure, carbon nanotube, has semi-conducting property and can play the role as electric wires. The novel mechanical, electric, photoelectric, chemical, and thermal conductive properties of carbon nanotube make it a highly potential material in biomedical application, energy storage, modern compound materials, and nano-electronic application. Nanoscale thin films can be made of metal particles, semi-conducting particles, insulating particles, and polymers. The structures of thin films are divided into single layer and multi layers (such as superlattice). Applications of nanoscale thin films are resistors, capacitors, magnetic tape, and materials in semi-conducting integrated circuits. As for nanowires, they have been used for sensor technology, solar cells, and also in electronics. Electrical and thermal transport properties exhibit significant size effects at the nanoscale because of the pronounced surface-to-volume ratio influence on the surface and grain boundary scattering processes. Characterization of such size effects and

10 2 understanding of the fundamental mechanism is especially important for the microelectronics industry where the critical dimension is rapidly approaching a few nanometers. The effect of miniaturization is profound on the energy (in the form of heat dissipation) density in this class of devices, and thermal management in nanoscale application has been increasingly noticed and studied [1]. Another class of devices, namely energy conversion, is also deeply impacted by thermal conductivity of materials. Therefore, recent efforts focus on the use of nano-structured materials for improved efficiency [2]. While classical concepts on thermal conductivity render it length-scale independent, there is growing evidence in the literature for the opposite to be true [3,4]. Therefore, thermal characterization of nanoscale materials has been a very active area of research in the last decade [5,6,7]. 1.2 Motivation of This Study To measure thermal conductivity, one needs to determine the heat flux (typically set up by an energy source such as electrical heater) across the specimen to acquire the cross-plane thermal conductivity, and another requires to measure the temperature drop between two separate points in the same plane to obtain the in-plane thermal conductivity. In this paper, we focus on the electrical heating and sensing based techniques, where the specimens are usually equipped with micro-fabricated heater and sensors [8]. Well-established techniques for measuring cross-plane thermal conductivity of thin films and nanostructures are the 3! and the steady-state method [9]. Moreover, the membrane and bridge methods, which require the substrate to be removed and make

11 3 the specimen freestanding, are used for in-plane thermal conductivity measurement [10,11]. The conventional 3! technique is versatile in terms of specimen configuration, such as: line-heater-on-substrate, thin-film-on-substrate, and multilayered-thin-film-onsubstrate. There is a metallic strip deposited on the sample to act as the heating source and sensing element [12,13]. The strip metal film acts both as a thermometer and a line heater driven by a periodically oscillating (1!) heat source, i.e. a.c. current source. The heat generated due to the a.c. current source and the resistance of the line heater (joule heating) causes the temperature fluctuation of the line heater itself at the frequency 2!, and it also heats up the underlying sample. Then, the temperature oscillations make the resistance of line heater varying at 2!. Therefore, voltage oscillations at 3! can be derived. By measuring the 3! voltage of the line heater, the thermal conductivity of the underlying sample can be extracted from the amplitude of 3! voltage. When the specimen is capable of self heating (metals and semiconductors), it is usually released from the substrate so that the entire heating energy is exploited to set up the temperature gradient in the specimen and the measured thermal conductivity is unaffected by that of the substrate [14]. However, the specimen on substrate configuration is closer than freestanding configuration in representing real life applications as in microelectronic devices. Also, releasing the specimen requires additional processing steps and careful considerations on not to affect the specimen itself, which may be difficult to execute on nanoscale specimens. Therefore, the motivation of the present study is to relax the requirement for freestanding specimens, which can increase the versatility of the 3!

12 4 technique. We achieve this by developing an analytical model that accounts for the heat loss to the substrate so that the thermal conductivity of both the specimen and the substrate could be measured simultaneously. We demonstrate the new technique on both metallic (lithographically patterned aluminum) specimen on glass substrate and semiconducting focused ion beam (FIB) deposited platinum specimen on silicon substrate with an insulating thermal oxide layer. It is important to note that the FIB deposited materials have received increasing attention in nano-heaters, nano-connectors for the applications of circuit repair, protection layers, and Transmission Electron Microscope specimen preparation. The electrical properties of FIB deposited platinum are well studied [15,16], but the thermal properties are not yet, which motivates us to report the value of thermal conductivity. 1.3 Reference [1] D. G. Cahill, W. K. Ford, K. E. Goodson, G. D. Mahan, A. Majumdar, H. J. Maris,R. Merlin, and S. R. Phillpot, Nanoscale thermal transport, J. Appl. Phys. 93(2), 2003, p.793~818 [2] Balaya, P., "Size effects and nonastuctured materials for energy applications." Journal of Energy & Environmental Science 1, 2008, p.645~654 [3] Stewart, D. and P. M. Norris, "Size effects on the thermal conductivity of thin metallic wires: microscale implications." Microscale Thermophysical Engineering 4, 2000, p.89~101

13 5 [4] Liang, L. H., and Li B., "Size-dependent thermal conductivity of nanoscale semiconducting systems " Journal of Physics Review B 73(15), 2006, p [5] Hone, J., M. Whiteney, C. Piskoti, and A. Zettl, "Thermal conductivity of singlewalled carbon nanotubes." Journal of Physics Review B 59(4), 1999, p.2514~2516 [6] Li, D., Y. Wu, P. Yang, and A. Majumdar, "Thermal conductivity of Si/SiGe superlattice nanowires." Journal of Applied Physics Letters 83(15), 2003, p.3186 ~3188 [7] Gang Chen, Thermal conductivity and heat conduction in nanostructures: modeling, experiments, and applications, AIAA 2004, p [8] Borca-Tasciuc, T. and G. Chen, Experimental techniques for thin-film thermal conductivity characterization Thermal Conductivity: Theory, Properties, and Applications, 2004, p.205~237 [9] Borca-Tasciuc, T., A. R. Kumar, and G. Chen, "Data reduction in 3! method for thin-flim conductivity determination." Review of Scientific Instruments 72(4), 2001, p.2139~2147 [10] Jansen, E., and E. Obermeier, "Thermal conductivity measurements on thin films based on micromechanical devices." J. Micromech. Microeng. 6, 1996, p.118~121 [11] Shi, L., and et. al., "Measuring thermal and thermoelectric properties of onedimensional nanostructures using a microfabricated device." Journal of Heat Transfer 125, 2003, p.881~888 [12] Cahill, D. G., "Thermal conductivity measurement from 30 to 750K: the 3! method " Review of Scientific Instruments 61(2), 1990, p.802~808 [13] Cahill, D. G., M. Katiyar, and J. R. Abelson, "Thermal conductivity of a-si:h thin films." Physical Review B 50(9), 1994, p.6077~6081

14 6 [14] Lu, L., Y. Wang, and D.L. Zhang, "3! method for specific heat and thermal conductivity measurement." Review of Scientific Instruments 72(7), 2001, p.2996~3003 [15] Marzi, G. D., D. Iacopino, A.J. Quinn, and G. Redmond, "Probing intrinsic transport properties of single metal nanowires: Direct-wire contact formation using a focused ion beam." Journal of Applied Physics 96(6), 2004, p.3458~3462 [16] Langford, R. M., T.-X. Wang, and D. Ozkaya, "Reducing the resistivity of electron and ion beam assisted deposited Pt." Microelectrical Engineering 84, 2007, p.784~788

15 7 Chapter 2 Literature Review The mechanism of heat transfer can be divided into three ways, heat conduction, heat convection, and heat radiation. In micro and nano eletromechanics field, the main mechanism of heat transfer is conduction, especially for semiconductor production process. Some methods have been developed these decades to measure thermal conductivity of micro and nano materials. For measuring thermal conductivity, there are two primary methods. One is thermal conductance method, and the other is thermal diffusivity method. The former is to extract the thermal conductivity directly from experiments according to Fourier s Law showing as equation (2-1). q x = "#A dt dx (2-1) Where q x is the heat flux rate in x direction, " is the thermal conductivity, A is the cross section of heat transfer, and dt dx is the temperature gradient in x direction. Different from direct measurement of thermal conductivity, the diffusivity method gets thermal diffusivity first and calculates the thermal conductivity through the relationship, " (thermal diffusivity) is equal to " (thermal conductivity) divided by the product of " (mass density) and C p (specific heat). In thermal conductance method, the most common application is to use the suspended bridge structure or membrane structure. Above the suspended thin film, a

16 8 heater and a thermal sensor are deposited to measure the in-plane thermal conductivity. McConnell et al. [1] used this method to measure the thermal conductivity of poly-silicon film, and Figure 2-1 shows the suspended device. When DC current is passing through the heater in the middle of the structure, heat flow is produced because of Joule heating. The resistance of thermometer varies with change of temperature. According to Fourier s Law, the thermal conductivity can be calculated by equation (2-2). Figure 2-1: Measurement structure schematic proposed by McConnel [1] " = (Q/2)(X thermometer # X heater ) A c (T heater # T thermometer ) (2-2) 2.1 Thermal Conductivity Measurement by 3! method An important technology, 3! method, is one kind of thermal conductance methods with the easy-applied and high-accurate characteristics. It can measure thermal

17 9 properties of several different structures of micro or nano materials. The solid sample geometry in 3! method can be divided into three types, Line Heater on Substrate, Thin Film on Substrate, and Suspended Wire. The measurement applications of 3! method are introduced by the following paragraphs according to different geometry catalogs Line Heater on Substrate To measure thermal conductivity of bulk materials in 3! technique, the bulk sample is prepared adjacent to a patterned strip metal film which is shown as Figure 2-2. The strip metal film acts both as a thermometer and a line heater driven by a periodically oscillating (1!) heat source. The heat causes temperature oscillations of line heater at 2!, and the temperature oscillations make the resistance of line heater varying at 2!. Therefore, voltage oscillations at 3! can be derived. The thermal conductivity information of the underlying sample can be extracted from the amplitude of voltage oscillations of the line heater. Figure 2-2: Side view of geometry of line heater on substrate Cahill [2] utilized 3! method to measuring the thermal conductivity of amorphous solids in the temperature range between 30K to 300K. A four-probe

18 10 configuration of metal line heater on top of sample substrate was proposed as shown in Figure 2-3. The outside two electrode pads were used to feed in the periodically oscillating heat source (a.c. constant current source), and the inside two pads are for measuring the voltage drop across the line heater. The temperature oscillation of line heater and the thermal conductivity of sample can be calculated from 1! voltage and 3! voltage of the line heater. Figure 2-3: Four pads patterned metal film on amorphous substrate [2] In 1990, Cahill [3] published thermal conductivity measurement with temperature range from 30K to 750K. Equation (2-3) presents calculation of temperature oscillation, and equation (2-4) shows the governing equation of thermal conductivity. "T = 4 dt dr R V 3# V 1# " s = V 3 1# ln( f 2 ) f 1 4$LR 2 (V 3#,1 %V 3#,2 ) & dr dt (2-3) (2-4) The wave penetration depth (WPD) means the wavelength of diffusive thermal wave and is defined as Equation (2-5). D presents the thermal diffusivity of line heater,

19 11 and! is the frequency of heat source (a.c. current source). If WPD is much smaller than sample thickness, the semi-infinite substrate condition can be assumed. The proper operating frequency of heat source also can be decided by the assumption: WPD is much larger than line heater width. WPD = D 2" (2-5) Thin Film on Substrate Cahill [4] proposed thermal conductivity measurements of a-si:h thin film on substrate. The sample was deposited on a substrate as a thin layer on the order of 100µm. Then, a strip of metal was added on the top of sample film, and it acted as a line heater and also a thermal sensor. Four electrode pads were using to feed in the a.c. constant current and measure the voltage drop as shown in Figure 2-4. The rule here is that the width of the metal strip must be at least five times larger than the thickness of sample film, then the heat transfer can be viewed as one dimension (cross-plane) problem. Figure 2-4: Geometry of the thin file thermal conductivity measurement [4]

20 12 If the thermal conductivity of thin film is small compared to the thermal conductivity of substrate, and WPD is much larger than the thickness of thin film at low frequency. Then, the temperature oscillation of the thin film can be shown as equation (2-6), which is independent of frequency. P l is the unit power density of the line heater, and t is the thickness of thin film. 2b stands for the width of line heater. The temperature at substrate is shown as equation (2-7). "T f = P lt # f 2b "T s = P % l sin 2 (kb) #$ s (kb) 2 (k 2 + q 2 ) dk, 1 & 1/ 2 q2 = WPD = 2' 2 D 0 (2-6) (2-7) Wang et al. [5] measured not only thermal conductivity of the SiO 2 thin film but also the thermal conductivity of Si substrate by extending 3! method to wide-frequency range. Figure 2-5 shows temperature oscillation results. Figure 2-5: The real part of temperature oscillation as function of frequency [5]

21 13 A practical extended 3! method has been proposed to measure thermal conductivity and thermal diffusivity of multilayer structures by B. W. Olson et al. [6]. The two-layer structure is shown as Figure 2-6. Part (a) in Figure 2-6 is the 3! line element, and part (b) is the plane view of it. Figure 2-7 presents how the width of 3! line element can affect the heat flow path. Figure 2-6: A two-layer structure of 3! applications [6] Figure 2-7: (a) a thin line (b) a wide line [6] There are two samples measured in this research. The first two-layer sample is composted of a borosilicate glass substrate, a consolidated zeolite film, and a copper line heater. The other one is made of a bulk silicon substrate, a SiO 2 thermal oxide film, and a aluminum line heater. Thermal conductivities and thermal diffusivities of substrates and films are all measured by 3! method.

22 2.1.3 Suspended Wire 14 Lu et al. [7] measured thermal conductivity of a filament-like specimen with fourprobe configuration by applying 3! method. The measurement schematic is shown in Figure 2-8. The specimen of this method restricts to be electrically conductive, and the resistance of the specimen should vary with temperature. The a.c. constant current (1!) is fed in from the outside two probes, and the three omega voltage (V 3! ) is obtained from the inside two probes. V 3! can be derived from the heat generation and diffusion equation (2-8) and boundary conditions (2-9). The value of V 3! is approximately proportional to the dimensions of the specimen, shown as equation (2-10). Once V 3! is obtained from experiments, the thermal conductivity can be back calculated. This research measured thermal conductivity and specific heat of the platinum wire and multiwall carbon nanotube bundles. In addition, the amplitude of current is restricted in suspended wire configuration shown as equation (2-11). Figure 2-8: Illustration of the four-probe configuration of a filament-like specimen [7] 2 # # "C p T(x,t) $% #t #x T(x,t) = I sin 2 &t LS [R + R '(T(x,t) $ T 0 )] (2-8)

23 15 where C p, ", R, and # are the specific heat, thermal conductivity, electrical resistance, and mass density of the specimen at the substrate temperature T 0. Boundary conditions, $ T(0,t) = T & 0 % T(L,t) = T 0 & ' T(x,"#) = T 0 4I 3 $ V 3" # e $ e % ) L, +. & 4 ' 1+ (2"() 2 * S - 3 (2-9) (2-10) where # e is the electrical resistivity of the specimen, and S is the cross section of the specimen. 2 I 0 R " L <<1 (2-11) n 2 # 2 $S Dames and Chen [8] used the voltage source to approximate a.c. current source as shown in Figure 2-9. They added DC offset to the heat source, therefore V 0! and V 2! can be obtained besides V 1! and V 3!. Figure 2-10 compares the traditional and DC-offset added 3! method. The voltage from measurement can be divided into in-phase (real) part X and out-of-phase (imaginary) part Y shown as equation (2-12). Figure 2-9: Circuits of voltage approximated current source [8]

24 16 Figure 2-10: Schematic of traditional and DC offset added 3! method [8] V n",rms = X 2#R 2 3 n (" 1,$) + jy n (" 1,$) (2-12) e0 I 1,rms The important result of this paper is the summary of relationship between (inphase and out-of-phase) electrical transfer functions and thermal transfer function Z shown as Figure Figure 2-11: Relationship between electrical and thermal functions [8]

25 17 Thermal conductivity measurement of a non-electrical conducting suspended wire can also be implemented by 3! method. O. Bourgeois et al. [9] patterned a NbN thermometer on top of Si nanowire and utilized suspended wire configuration 3! method. The individual single-crystalline silicon nanowires are fabricated by e-beam lithography and suspended between two separated SiO 2 pads as shown in Figure These experiments are executed at low temperature around 0K$ 2K and at low frequency around 0Hz $ 50Hz. The results of experiments showed that the conductance behaves as T 3 as shown in Figure Figure 2-12: The image of suspended silicon nanowire and the schematic of this device [9]

26 18 Figure 2-13: Relationship between thermal conductance and temperature [9] 2.2 Strain Effects on Thermal Conductivity Generally, thermal conductivity of a solid can be contributed from transports of phonons and electrons. At microscale and nanoscale limits, the classic model of heat conduction is inadequate for predicting thermal conductivity. There are two primary approaches applied on the new models used for predicting the thermal conductivity of the microscale and nanoscale devices. One is Boltzmann transport equation (BTE) which describes the statistical distribution of particles in a fluid. It is used to study how a fluid transports physical quantities (such as heat and charge), then some transport properties (such as electrical conductivity, hall conductivity, viscosity, and thermal conductivity) can be derived. The other approach is Molecular dynamics (MD). It is a form of computer simulation wherein atoms and molecules are allowed to interact for a period of time under known laws of physics, which gives a view of the motion of the atoms. There are two distinct types of parameters control effective thermal conductivity. The first type is thermodynamic parameters, such as temperature and pressure. The other

27 19 one is extrinsic parameters, such as impurities, defects, and bounding surfaces. Nanoscale devices, particularly thin film, usually contain residual strain after fabrication, which may considerably affect the thermal transport properties of the material and the device concerned. The effect of strain (tri-axial strain equivalent to the uniform pressure) on thermal conductivity is still an unexplored research field. Bhowmick and Shenoy [10] studied effect of strain on thermal conductivity of insulating solids. In this study, the thermal current was carried solely by phonons due to insulating solids. They performed classical molecular dynamics simulations and obtained the strain and temperature dependence on thermal conductivity. In addition, they changed the velocity of sound and relaxation time of phonons to observe how strain affects thermal conductivity. Figure 2-14 shows the relationship among thermal conductivity, temperature, and strain. Figure 2-14: Thermal conductivity as a function of temperature and strain [10] Tamma et al [11] discussed mechanical strain effect on thermal conductivity of single-wall carbon nanotubes. Compression, tension, and torsion were applied to SWNT in this study. They found that thermal conductivity didn t change much for small strained

28 20 tube. However, the bulking collapse reduced the thermal conductivity in a compressed and torsionally twisted tube. In tensile stretched case, no significant result was found. The result in detail is shown in Figure Figure 2-15: Thermal conductivity and mechanical strains [11]

29 2.3 Reference 21 [1] A. D. McConnell, S. Uma, and K. E. Goodson, Thermal Conductivity of Doped Polysilicon Layers, J. Microelectromechanical Sys. 10(3), September 2001, p.360~369 [2] D. G. Cahill and R. O. Pohl, Thermal Conductivity of amorphous solids above the plateau, Phys. Rev. B 35(8), March 1987, p.4067~4073 [3] D. G. Cahill, Thermal Conductivity measurement from 30 to 750K: the 3! method, Rev. Sci. Instrum. 61(2), February 1990, p.802~808 [4] D. G. Cahill, M. Katiyar, and J. R. Abelson, Thermal conductivity of a-si:h thin films, Phys. Rev. 50(9), September 1994, p.6077~6081 [5] Zhao Liang Wang, Da Wei Tang, and Xing Hua Zheng, Simultaneous determination of thermal conductivities of thin film and substrate by extending 3!- method to wide-frequency range, Appl. Surface Sci. 253, May 200, p.9024~9029 [6] B. W. Olson, S. Graham, and K. Chen, A pratical extension of 3! method to multilayer structures, Rev. Sci. Instrum. 76(5), April 2005 [7] L. Lu, W. Yi, and D. L. Zhang, 3! method for specific heat and thermal conductivity measurements, Rev. Sci. Instrum. 72(7), July 2001, p.2996~3003 [8] C. Dames and G. Chen, 1!, 2!, and 3! methods for measurements of thermal properties, Rev. Sci. Instrum. 76(12), December 2005 [9] O. Bourgeois, T. Fournier, and J. Chaussy, Measurement of the thermal conductance of silicon nanowires at a low temperature, J. Appl. Phys. 101(1), January 2007

30 22 [10] S. Bhowmick and V. B. Shenoy, Effect of strain on the thermal conductivity of solids, J. Chem. Phys. 125, October 2006 [11] C. Anderson, K. K. Tamma, and D. Srivastava, Prediction of thermal conductivity of single-wall carbon nanotubes with mechanical strains, AIAA

31 23 Chapter 3 Analytic Modeling of 3! Method 3.1 Introduction of 3! Method 3! method is a commonly used and easily applied technique to measure thermal properties of bulk materials, thin films, wires, and liquid as well. Generally, the sample preparation is limited to a patterned conducting line film deposited on top of the sample. The conducting film serves as both a heater and a sensor, and it is called line heater. The main principal of 3! method shows as Figure 3-1. Figure 3-1: Principal of 3! method 3! method utilizes periodically oscillating current source with 1! frequency to drive the line heater to generate joule heating at 0! and 2! frequency. Then, heating leads to temperature fluctuation at 0! and 2! frequency. The temperature difference causes resistance variation of heater at 0! and 2! frequency. Eventually, the voltage of heater is equal to current multiplied by resistance, and it contains 1! and 3! components.

32 The thermal information that we are interested can be extracted from 3! voltage. Figure 3-2 explains the concept of 3! method. 24 Figure 3-2: 3! method There are several kinds of geometry of sample preparation for 3! method. For line-heater-on-substrate, thin-film-on-substrate, and multilayered-thin-film-on-substrate, there is a metal line deposited on the sample to act as the heating source and sensing element. Thermal conductivities of these thin films and substrate can be revealed by 3! method. Another geometry called suspended-wire, and this freestanding wire is not only the heater but also the sensor. 3! method is utilized to measure the thermal properties of the freestanding wire instead of thermal property of substrate. However, the fabrication technique to make freestanding films at nanoscale is still a challenge. Therefore, this study deposits the nano-/micro-wire on top of the substrate instead of suspended wire. In this wire-on-substrate geometry, the heat diffused to the

33 25 substrate needs to be considered. The main object of this study is to verify the 1-D heat conduction model by considering heat loss into substrate and to get thermal conductivity of the nano-/micro-wire from the third harmonic voltage information. The following sections start from the continuity equation to heat conduction equation and reveal the modeling of wire-on-substrate. 3.2 Continuity Equation The conservative transport of energy can be described by the continuity equation. The divergence of flux density is equal to the negative rate change of energy density with no generation or removal rate of energy, as shown in equation (3-1). " J = # $% energy $t (3-1) For one-dimensional heat conducting, J means heat flux density (W/m 2 ) and presents as equation (3-2). # energy stands for heat density (J/m 3 ), and %(x,t) is temperature difference. J = "#$% = "# &%(x,t) &x (3-2) " J = " (#$ %&(x,t) ) = #$ % 2 &(x,t) (3-3) %x %x 2 "# energy "t = "(#C p$(x,t)) "t = #C p "$(x,t) "t (3-4) Therefore, the one-dimensional heat conduction equation without any heat source or sink can be derived as equation (3-5).

34 "C p #$(x,t) #t = % # 2 $(x,t) #x 2 (3-5) $ " : mass density ( & & % C p : specific heat ) & '# : thermal conductivity & * D Heat Conduction Equation with Heat Source In 3! method, a periodically oscillating heat source (I=I 0 sin!t) is applied to a metal wire and leads to joule heating which is equal to I 2 (R 0 +&R). &R means the resistance fluctuation of the metal wire and is proportional to temperature difference. The one-dimensional heat conduction equation of freestanding wire with heat source is showing as equation (3-6). L and S are the length and the cross-section of metal wire, and R " is resistance derivative of temperature. "C p #$(x,t) #t = % # 2 $(x,t) + I 2 0 sin 2 &t (R #x R ' $(x,t)) (3-6) LS D Heat Conduction Equation with Heat Source and Heat Sink Before calculating the heat loss to substrate, we need to define the temperature profile along y direction that points into the substrate as shown in Figure 3-3. We assume temperature of the surface of substrate is the same as temperature of the nanostructure and ignore the interface resistance. In addition, the temperature change of our sample along the longitudinal direction can be viewed as. Then, difference of temperature decays exponentially along y direction and the temperature profile can be viewed as

35 "(x,t)e # y $. The parameter of exponential term relates to thermal wave penetration depth ' 27 which is equal to " s 2#. ( s is thermal diffusivity of substrate, and! is frequency of heat source. L Sample X Y Substrate Figure 3-3: Geometry of wire-on-substrate The heat flux through the interface (y=0) from wire to substrate is shown as equation (3-7). " s is thermal conductivity of substrate, and A is contacting area between wire and substrate.

36 28 q = " A #($(x,t)e% A y= 0 s #y y & ) y= 0 = " s A[% 1 & $(x,t)e% y & ] y= 0 = % " s A$(x,t) & (3-7) Therefore, the 1-D heat conduction equation of wire-on-substrate model can be derived as equation (3-8). "C p #$(x,t) #t = % # 2 $(x,t) + I 2 0 sin 2 &t (R #x R ' $(x,t)) ( % sa $(x,t) (3-8) LS LS) 3.5 Solution of 1-D Heat Conduction Equation with Heat Source and Heat Sink "#(x,t) "t " #$(x,t) #t & ( A = " #C ( p B = I 2 ( 0R $ ( LS#C ' p ( " C = s A ( LS%#C p ( D = I 2 0R ( 0 ) LS#C p From equation (3-8), we can get $ % " 2 #(x,t) $ ( I 2 0 sin 2 't R ( $ &C p "x 2 LS&C p % s A )#(x,t) = I 2 0R 0 sin 2 't (3-9) LS)&C p LS&C p % A # 2 $(x,t) #x 2 % (Bsin 2 &t % C)$(x,t) = Dsin 2 &t (3-10) Boundary conditions and initial condition are shown as following. % "(0,t) = 0 ' & "(L,t) = 0 ' ("(x,#$) = 0 (3-11)

37 Equation (3-10) shows a non-homogenous PDE equation. We use impulse theorem to modify it to be a homogenous PDE equation. Let z(x," + ) = Dsin 2 #" and "(x,t) = t & $% z(x,t;#)d#. We can get "z "t # A " 2 z "x # 2 (Bsin2 $t # C)z = 0 (3-12) 29 $ z(0,t) = 0 & % z(l,t) = 0 & ' z(x," + ) = Dsin 2 #" (3-13) From boundary condition, we can assume z(x,t;") = $ U n (t;")sin n#x L (3-14) n Substitute equation (3-14) into equation (3-12) + % du n + (A n 2 " 2 ( ' # Bsin 2 $t + C)U dt L 2 n * sin n"x & ) L = 0 (3-15) n For non-trivial solution of equation (3-15) du n dt + (A n 2 " 2 L 2 # Bsin 2 $t + C)U n = 0 (3-16) When B << A n 2 " 2 L 2 + C # 2 I 0 R $ L% <<1, equation (3-16) becomes n 2 " 2 &%S + & s LA du n dt + (A n 2 " 2 L 2 + C)U n = 0 (3-17) " U n (t;#) = E n e $(A n 2 % 2 +C )(t$# ) L 2 " Z n (x,t;#) = E n e $(A n 2 % 2 & n L 2 +C )(t$# ) sin n%x L (3-18) (3-19)

38 From initial condition, we know 30 Z n (x," + ) = # E n sin n$x L = Dsin2 %" (3-20) n " E n = D# 2 L L sin2 $% sin n&x 2D ( dx = L n& [1' ('1)n ]sin 2 $% (3-21) 0 n 2 $ 2 2D " z(x,t;#) = ' n$ [1% (%1)n ]sin 2 &#e %(A +C )(t%# ) n$x L 2 sin L (3-22) Now, let s find out "(x,t) "(x,t) = t n & z(x,t;#)d# = $% Let G = A n 2 " 2 6 n / 12D 0 n' [1$ ($1)n ]sin n'x 21 L 2 + C, and solve equation (3-23) e$(a n 2 ' 2 L L 2 +C )t ) + * + t & $% e (A n 2 ' 2 +C )# L 2, 3 1 sin 2 (#d# (3-23) "(x,t) = =, n, n 2D n# [1$ ($1)n ]sin n#x 1 & G 2 + 4% 2 $ Gcos(2%t) + 2%Gsin(2%t) ) ( L G ' 2(G 2 + 4% 2 + ) * D n# [1$ ($1)n ]sin n#x 1 & L G 1$ G2 cos(2%t) + 2%Gsin(2%t) ) ( ' G 2 + 4% 2 + * (3-24) Let $ sin" = & % cos" = '& G G 2 + 4# 2 2# G 2 + 4# 2 (3-25) D "(x,t) = n# [1$ ($1)n ]sin n#x 1 ' L G 1$ G * - ) sin& cos(2%t) + cos& sin(2%t), (3-26) G 2 2 ( + 4% + n D " #(x,t) = n$ [1% (%1)n ]sin n$x 1 ( 1% sin& sin(2't + &) L G [ ] (3-27) n sin" = 1 1+ cot 2 "

39 31 D " #(x,t) =. n$ [1% (%1)n ]sin n$x 1 ( sin(2&t + ') + * 1% - L G n )* 1+ cot 2 ',- (3-28) Take n=1, the temperature difference becomes as equation (3-29) "(x,t) = 2D # sin #x L 1 ' sin(2%t + &) * ) 1$, G () 1+ cot 2 & +, (3-29) Then, resistance variation can be obtained like equation (3-30) "R = R # L L $(x,t)dx = 4D R # ) sin(2't + (), / + 1&. (3-30) % 2 0 G * + 1+ cot 2 ( -. The voltage of wire is showing as equation (3-31) -/ V (t) = I 0 sin"t R + 4D R # ' sin(2"t + &) * 1 /. ) 1%, 2 0 / $ 2 G () 1+ cot 2 & +, 3 / (3-31) Because of sin(2"t + #)sin"t = $ 1 [cos(3"t + #) $ cos("t + #)] 2 and 1 1+ cot 2 " = G G 2 + 4# = # 2 G 2 Therefore, $ V (t) = I 0 R + 4I 0 R " D' & ) sin*t + 2I 0 R " D % # 2 G ( # 2 G + 2I 0 R " D # 2 G 1 cos(3*t + +) 1+ 4* 2 2 G 1 cos(*t + +) 1+ 4* 2 2 G (3-32) ( $ & 2 If 4" 2 << G 2 # 2" << %C p L + 1 $ s A + * ) 2 %C p LS' -, the magnitude of 3! voltage becomes, V 3",rms # 4I 3 rms R 0 R $ L % 4 &S + L% 2 & s A ' (3-33)

40 G!sin" = G 2 + 4# $ " = & G ) 2 sin%1 ( + ' G 2 + 4# 2 * &G 2 >> 4# 2 32," - sin %1 (1) =. 2 Therefore, the magnitude of 1! voltage is V 1",rms # 2I rms R I 3 rmsr 0 R $ L % 4 &S + L% 2 & s A ' (3-34) The average temperature change of the wire can be calculated from equation (3-29) " # 8I 2 rms R 0 L $ 4 %S + L$ 2 % s A & (3-35) Therefore, the thermal conductivity of wire can be derived from equation (3-33) ) + * " # 4I 3 rms R 0 R $ L V 3% ' 4 S & L' 2 " s A (,. - (3-36) 3.6 Reference [1] L. Lu, W. Yi, and D. L. Zhang, 3! method for specific heat and thermal conductivity measurements, Rev. Sci. Instrum. 72(7), July 2001, p.2996~3003 [2] Z. L. Wang et al, Length-dependent thermal conductivity of an individual singlewall carbon nanotube, Appl. Phys. Letters 91, 2007, p

41 33 Chapter 4 Experimental Method method. This chapter introduces the sample preparation and experimental set-up of 3! 4.1 Sample Preparation In this study, we utilize 3! technique to measure thermal conductivity of metal thin films and nanoscale semiconductors. The metal used is evaporated aluminum, which is 99.99% pure evaporated aluminum of glass substrate and subsequently patterned using photo-lithography and wet etching. The mean grain size of the nm thick specimens is about 50 nm as obtained by transmission electron microscopy. Figure 4-1 shows scanning electron microscope (SEM) image of our aluminum specimen with the dimensions of 100"m long, 10" wide, and 130nm thick. In addition, we also have a focused-ion-beam deposited platinum nanowire across lithographically patterned gold electrodes on substrate. The substrate is a commercially available silicon wafer with about 500 nm thick thermally grown oxide layer at the top. The FIB Pt specimen is 7µm long, 600nm wide, and 800nm thick showing as Figure 4-2.

42 Electrodes 34 Aluminum thin film Electrodes Figure 4-1: SEM image of aluminum specimen 10µm KEITHLEY 6221 SR830 Lock-in Amp A B Phase marker Reference in FIB Pt nanowire sample Figure 4-2: Schematic of 3! technique applied on FIB-deposited Platinum

43 4.2 Experimental Set-up 35 Figure 4-2 shows the experimental setup schematic along with a scanning electron micrograph of a focused-ion-beam deposited platinum nanowire across lithographically patterned gold electrodes. For conducting nanowires, the parasitic/contact resistances usually are of the same magnitude as the measured DUT (device-under-test) resistance. Therefore, the electrodes with four-probe configuration are used to eliminate contact resistance and minimize experimental error of measurement. The experiments are performed in vacuum environment to prevent heat loss through convection. The main components of the experimental setup are the Keithley 6221 constant current source and the Stanford SR830 lock-in amplifier [1] Main Apparatus The power supply, Keithley 6221 constant current source, has periodical wave output function. It s utilized to provide a constant alternating current I 0 sin!t through the specimen. The phase marker function of Keithley 6221 allows user to set a pulse marker that defines a specific point of a waveform over a range of 0 to 360. The phase marker signal is a 1"s pulse that appears on the selected line of the external trigger connector. Then, we use this external trigger signal inputs into the lock-in amplifier as a reference of frequency detection. The Stanford SR830 lock-in amplifier is proficient for measuring small a.c. signals that can reach nano-volt levels with a very low level of noise floor. It has the reference in function that can recognize the specific frequency which users are interested

44 36 in, and it can measure not only voltage but also current signal at this specific frequency and it s harmonics. There are single and differential input models in SR830 lock-in amplifier. The differential input method is utilized in this study to eliminate common mode noise signal. The key features of a 3! experiment are the choice of amplitude of the a.c. current, the low frequency operating settings of the lock-in amplifier, and use of a high vacuum environment. From the analytical model we recognize a current limitation: 2 I 0 R " L# n 2 $ 2 %#S + % s LA <<1 implicit in the derivation of "(x,t). Therefore, the amplitude of the applied current can t be too large. However, from a measurement stand-point it has to be large enough so that the signal to noise ratio of the 3! signal is satisfactory. The most important parameter in using the lock-in amplifier is the time constant which decides the bandwidth of low pass filter. For example, we use 30~300 seconds here for the frequency range of a.c. current from 30Hz to 1Hz to fulfill the frequency ' # % 2 limitation: 2" << $C p L + 1 # s A * ) ( 2 $C p LS&, Cryogenic Station The 3! experiments are performed in a cryogenic station at a high vacuum environment to prevent heat convection losses. This cryogenic station as showing in Figure 4-3 is a digital Deep Level Transient Spectroscopy (DLTS) system that can provide the environment temperature between 50F ~ 300F.

45 37 Figure 4-3: Experimental set-up of 3! technique 4.3 Calibration One of these traditional 3! geometries, line heater on substrate, is utilized to calibrate our experimental setting. The metallic material of line heater is silver thin film patterned to be 100µm long, 20µm wide, and 200nm thick, and the sample is glass substrate. The thermal conductivity value of glass from literature is 1.1mW/K. Cahill s model showing as equation (4-1) is used to obtain thermal conductivity of glass substrate. Figure 4-4 shows the measured 3! voltage, then we can derive the thermal conductivity value of glass substrate from these data. Thermal conductivity value is equal to 1.07mW/K, and the error is about 2.7%.

46 38 " s = V 3 1# ln f 2 f 1 dr 4$lR 2 (V 3#,1 %V 3#,2 ) dt (4-1) Where, V 1" is the voltage across the metal strip at frequency!, V 3",1 and V 3",2 are the voltages at the third harmonic for frequency f 1 and f 2 respectively, R and l are the metal line resistance and length respectively, and R) is the slope of resistance as a function of temperature at the temperature of measurement. Figure 4-4: 3! voltage measurement by using silver line heater on glass substrate sample geometry 4.4 Reference [1] T. Y. Choi et al, Measurement of thermal conductivity of individual multiwalled carbon nanotubes by the 3-! method, Appl. Phys. Letters 87, 2005, p

47 39 Chapter 5 Results and Discussion The new 3! model developed in this study is first experimentally characterized and calibrated on a metal specimen of known thermal conductivity. The metal used is evaporated aluminum that is lithographically patterned with four electrodes to present four-configuration measurement. The following section shows our results of thermal conductivity of aluminum agreed with the literature value nicely. Afterward, our 3! model is utilized to measure thermal conductivity of focused ion beam deposited platinum nanowire without releasing it from substrate. Then, following by the conclusion. 5.1 Aluminum Film According to our 3! analytic model, the thermal conductivity of sample can be extracted from the 3! voltage by using equation (5-1). For eliminate the measuring error, we first use measured data of 3! voltage to fit into the traditional 3! model, Cahill s line-heater-on-substrate model showing like equation (5-2), to obtain the " s (thermal conductivity of substrate). Then, substitute the derived value of " s and measured data of 3! voltage into our 3! model to obtain thermal conductivity of our aluminum thin film sample. V 3",rms # 4I 3 rms R 0 R $ L % 4 &S + L% 2 & s A ' (5-1)

48 40 " s = V 3 1# ln f 2 f 1 dr 4$lR 2 (V 3#,1 %V 3#,2 ) dt (5-2) Temperature Dependence of Resistance of Aluminum Film The parameter, R), needs to be known before executing this model. R) is the slope of resistance of the sample as a function of temperature at the temperature of measurement, and we can measure it by performing four-point resistance measurement and using the cryogenic station. Figure 5-1 shows that dr dt is equal to #/K. Figure 5-1: Temperature dependence of resistance of aluminum film

49 5.1.2 Measurement of Thermal Conductivity 41 The aluminum thin film sample reported here is lithographically patterned to be 150µm long, 10µm wide, and 130nm thick. Figure 5-2, Series 1 presents the measured rms value of 3! voltages in the frequency range 1 Hz ~ 30Hz which matches well with the expected data, based on the analytical model, showing as Series 2. The difference between measured data and expected data is showing as Series 3 in Figure 5-2. We observe that the 3! voltages slightly increase with a decrease in the frequency, which is a reasonable result because the 3! voltages are related to thermal penetration depth ' according to equation (5-1). We fit this data to the Cahill s model, i.e. equation (5-2), to find out the thermal conductivity of glass substrate, resulting in a value of 1.07 W/mK at 297K. This result matches the value gotten from our calibration experiment. By substituting this value of thermal conductivity of the glass substrate into our 3! model, i.e. equation (5-1), we obtain the thermal conductivity of the aluminum thin film as shown in Figure 5-3. Additionally, according to our model equation (5-1) implies a linear relationship between 3! voltage (V 3" ) and cube of the current amplitude ( I 3 0 ), and this is experimentally verified and graphically shown in Figure 5-4. Here we chose the value of amp as the amplitude of a.c current in order to fulfill the current limitation criteria, 2 I 0 R " L# n 2 $ 2 %#S + % s LA <<1.

50 42 Figure 5-5 shows the length dependence of thermal conductivity of aluminum thin film with the same width. The result presents there is no obvious variance at thermal conductivity of aluminum films with the micro-length scale. Figure 5-2: The measured V 3" and expected V 3" of aluminum thin film

51 43 Figure 5-3: Thermal conductivity of aluminum thin film Figure 5-4: The proportional relationship between V 3" of aluminum thin film and I 0 3

52 44 Figure 5-5: The length dependence on thermal conductivity of aluminum thin film 5.2 Focused Ion Beam Deposited Platinum Nanowire After verifying the analytical model and calibrating the experimental setup, the experiments on FIB-deposited platinum nanowires are performed. The specimen is 7µm long, 600nm wide, and 800nm thick. The resistance of FIB platinum is measured by I-V measurement and showing as Figure 5-6. The chemical composition of FIB platinum shows like Figure 5-7. Carbon (45%~55%) and platinum (40%~50%) are the major components [1]. Therefore, we can expect the behavior of temperature dependence of resistance of FIB platinum acts like semiconductors showing as Figure 5-8.

53 45 Figure 5-6: Resistance of FIB platinum Figure 5-7: Components of FIB platinum

54 46 Figure 5-8: Temperature dependence of resistance of FIB platinum Figure 5-9 and Figure 5-10 individually show the V 3" value and average temperature change measured experimentally. The amplitude of a.c. current and the operating frequency used here are 7.07 µa and 1000Hz~400Hz, which satisfies the restrictions built into the analytical 3! model. The value of thermal conductivity of FIB platinum nanowire obtained from measurement is 40.97W/mK at room temperature. For pure platinum, the thermal conductivity is 71.6W/mK at room temperature. The variation seen is because the FIB-deposited Pt is not just purely Pt metal, but is a conducting metal-organic polymer (C 9 H 16 Pt) which in its as-deposited form creates a nanostructure made of Pt and amorphous carbon mixture (The Ga + ion in the ion beam causes amorphization of the carbon). The thermal conductivity of amorphous carbon varies from 0.3 to 10W/mK at room temperature. If we use a simple rule of mixtures like equation