Thermal and Electrical Transport Characteristics of ...

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,+7& IHTC14-22328 THERMAL AND ELECTRICAL TRANSPORT CHARACTERISTICS OF POLYCRYSTALLINE GOLD NANOFILMS Weigang Ma, Tingting Miao, Xing Zhang* Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, People’s Republic of China *Corresponding Author: [email protected]

ABSTRACT

NOMENCLATURE

The in-plane thermal and electrical conductivities of several suspended polycrystalline gold nanofilms with thickness of 20.0-54.0 nm have been measured simultaneously at 100-310 K. Both the thermal and electrical conductivities drop greatly compared to the corresponding bulk value, and the electrical conductivity reduction is larger. Fits to the temperature-dependent electrical conductivity confirm that the scattering of electrons by softened phonons is significant and cannot be reconciled with the classical size-effect model considering only surface and grain boundary. Taking into account the enhanced electron-phonon scattering, the electrical conductivity is well predicted over the whole temperature range and the obtained Debye temperature agrees well with the calculated value from the elastic continuum model. Furthermore, a new model on the thermal transport of metallic nanofilm is proposed based on the Energy Conservation Law, in which the electron-phonon scattering induced electron energy decrease is supposed to be counteracted by the phonon energy increase. The present model greatly improves the prediction of thermal conductivity in thin films compared to the corresponding result directly from electrical thermal analogy applied to bulk metals.

t w l

Keywords: thermal conductivity, electrical conductivity, phonon softening, gold nanofilm, Debye temperature

δ

d T ħ e ω na σ

λ

l0 ρR β p R K α κ u V kB L ΘD q

time (s) width (nm) length (nm) thickness (nm) grain size (nm) Temperature (K) Plank constant (J s) electrical charge (C) phonon frequency (Hz) atom number density (m-3) electrical conductivity (Ω-1 m-1) thermal conductivity (W m-1 K-1) bulk electron mean free path (m) residual electrical resistivity (Ω m) temperature coefficient of resistance (K-1) surface scattering parameter, dimensionless grain-boundary scattering parameter, dimensionless prefactor of the Bloch-Grüneisen theory (Ω m) grain diameter parameter, dimensionless thickness parameter, dimensionless lattice displacement vector (m) phonon group velocity (m s-1) Boltzmann constant (J K-1) Lorenz number (W Ω K-2) Debye temperature (K) wave vector (m-1)

Subscripts and superscripts 0 bulk material f film l longitudinal wave t transverse wave

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1 INTRODUCTION Transistors operate faster as their dimensions are scaled down. The metallic nanofilms that connect these transistors to form a circuit, however, do not exhibit the same benefit of scaling. It is a confirmed fact that the electrical and thermal conductivities of metallic nanofilms dramatically decrease with diminishing film thickness [1, 2]. The dramatic decrease in electrical conductivity can induce significant signal delay, and the thermal conductivity decrease will lead to integrated circuit (IC) failure due to higher temperature increase. Therefore, it is crucial to take a deep insight into the electrical and thermal transport characteristics of metallic nanofilms. For the electrical conductivity decrease, it is usually explained by two classical size-effect mechanisms: one is due to increased electron scattering at the surface proposed by Fuchs [3] and Sondheimer [4] (FS), and the other is enhanced scattering at the grain boundary proposed by Mayadas and Shatzkes (MS) [1]. However, recent studies [5-9] have found that the temperature-dependent electrical conductivity cannot be perfectly matched by only the FS and MS theory. Further, they found that the electron scattering by softened phonons in thin film plays a significant role due to the phonon spatial confinement. Marzi [6] even suggests that electron-phonon scattering is the dominant mechanism at all temperature even though the surface and grain boundaries cannot be neglected in 70-nm-diameter platinum nanowires. The studies on the thermal conductivity of metallic thin films are rather scarce compared to those on the electrical conductivity for the difficulties in measurement. By applying the electrical thermal analogy introduced by the WiedemannFranz (WF) law, which is widely used to relate the electronic thermal conductivity and the electrical conductivity in bulk metals [10], the researchers [11, 12] studied the surface and grain boundary scattering on thermal conductivity of polycrystalline metallic films. The underlying assumption is that the WF law is still valid in the polycrystalline metallic thin films. Recent theoretical studies on the disordered metals [13] and experimental studies on the copper-oxide superconductor [14] showed violation of the WF law. Zhang et al. [15, 16] found that the relationship between the thermal and electrical conductivities of Pt and Au nanofilms does not follow the WF law and the Lorenz numbers are even several times over the bulk value. Choi et al. [17] also found that the thermal conductivity drop in the Mo film was much smaller than the electrical conductivity drop and hence led to a violation of the WF law. In the present paper, the thermal and electrical transport characteristics of four suspended polycrystalline gold nanofilms with thickness of 20.0-54.0 nm have been studied simultaneously at 100-310 K using the direct current heating method. Both the thermal and electrical conductivities drop greatly compared to the corresponding bulk value, while the electrical conductivity reduction is larger. The theoretical predicted temperature-dependent electrical conductivity by

only considering electron scattering of sample surface and grain boundary deviates from the measured value, which reveals that the scattering by softened phonons is significant. Taking into account the enhanced electron-phonon scattering, the electrical conductivity is well predicted over the entire temperature range and the obtained Debye temperatures agree well with the calculated value from the elastic continuum model. Based on the Energy Conservation Law, a new model on the thermal transport in thin films is proposed, in which the electron-phonon induced electron energy decrease is supposed to equal the phonon energy increase. The present model can greatly improve the prediction of thermal conductivity compared to the corresponding result directly from Wiedemann-Franz law. 2 EXPERIMENT To eliminate the substrate influence on measurement of the intrinsic in-plane thermal and electrical conductivities and to get rid of contact resistance between the thin film and probe electrodes, suspended thin gold film integral with the probing electrodes was fabricated with electron beam-physical vapour deposition under vacuum conditions, assisted by the electron beam lithography and the isotropic/anisotropic etching techniques. The width and length of the thin film are measured with a scanning electron microscope (SEM), and the film thickness is measured with a calibrated quartz crystal thin film thickness monitor (CRTM-7000 with the resolution of 0.01 nm). Figure 1 presents the SEM image of the fabricated nanofilm and schematic diagram of the corresponding fourprobe measurements. From Fig.1, it can be found there is no contact resistance between the tested film and the probing electrodes. A pair of electrodes is used to pass current through the suspended film and the other is to measure the voltage. The average grain diameter of the nanofilms is investigated by x-ray diffraction (XRD); Figure 2 displays the diffraction spectra of gold nanofilms. The dimensions and the average grain size are listed in Table1.

Fig. 1 SEM IMAGE OF THE SUSPENDED THIN FILM AND SCHEMATIC DIAGRAM OF THE CORRESPONDING FOUR-PROBE MEASUREMENTS.

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⎡ 3 ⎤ ⎛ 1⎞ 6 2 3 ⎢1 − 2 α + 3α − 3α ln ⎜1 + α ⎟ − πκ (1 − p ) × ⎥ ⎝ ⎠ ⎥, σ = σ0 ⎢ 2 ⎢ π2 ∞ cos φ ⎛ 1 1 ⎞ 1 − exp ( −κ tH ( t , φ ) ) ⎥ ⎢ ∫ dφ ∫ dt 2 ⎥ ⎜ − ⎟ 1 H ( t , φ ) ⎝ t 3 t 5 ⎠ 1 − p exp ( −κ tH ( t , φ ) ) ⎥⎦ ⎢⎣ 0

where H (t,φ ) = 1 +

Fig. 2 X-RAY DIFFRACTION SPECTRA OF GOLD NANOFILMS. Table 1 SIZE PARAMETERS OF FABRICATED GOLD NANOFILMS.

Nanofilm sample Thickness, δ (nm) Width, w (nm) Length, l (μm) Grain diameter, d (nm)

(1) 20.0 371.0 10.03 18.2

(2) 23.0 455.2 5.62 19.9

(3) 36.0 329.5 10.23 29.3

(4) 54.0 378.0 10.26 38.3

From the thermal analyses, when all the measurements are carried out in high vacuum (-10-4 Pa), the residual gas heat conduction and thermal radiation loss can be ignored [18]. The heat transfer process in the nanofilm is simplified into onedimensional steady-state heat conduction. Both the thermal and electrical conductivities of the film can be obtained from current heating induced resistance, which is measured by the standard four-probe method using two high-precision digital multimeters (Keitheley 2002, 8.5 digits) and a standard resistance (Yokogawa 2792). In measurement, the silicon chip with the suspended nanofilm is mounted on a sample holder of a liquid nitrogen cryostat (Oxford Instruments, Optistat DNV). The sample chamber is maintained a high vacuum level of ~10−4 Pa by using a two-stage vacuum-pumping system composed of a roughing pump and a high-vacuum pump (turbo molecular pump). The temperature of the sample holder can be adjusted to any point between 77 and 500 K using the intelligent temperature controller (ITC 503, Oxford Instruments DNV). 3 RESULTS AND DISCUSSION 3.1 Temperature-dependent electrical conductivity It is well-known that the electrons of the nanofilm are scattered by sample surfaces and grain boundaries. Fuchs [3] and Sondheimer [4] (FS) explained the size effect on the resistivity of thin film based on the surface scattering. Mayadas and Shatzkes (MS) [1] further took into account the scattering by grain boundaries as well as by film surfaces, and obtained the total electrical conductivity of a polycrystalline metallic film as,

α

cos φ (1 − t −2 )

12

.

(1)

(2)

It should be noted that FS-MS theory just takes the background conductivity of thin films as bulk value and does not consider the effect of enhanced electron-phonon scattering on the background conductivity. The temperature-dependent electrical conductivities of the gold nanofilms are plotted in Fig. 3. It indicates that the electrical conductivity and the corresponding temperature dependence (slope of σ versus T) of the films decrease compared to the corresponding bulk value. A reduced temperature coefficient of resistance (TCR) at 220 K in response to the weakened temperature dependence is plotted in Fig. 4.

Fig. 3 TEMPERATURE-DEPENDENT ELECTRICAL CONDUCTIVITY OF THE GOLD NANOFILMS.

Fig. 4 TEMPERATURE COEFFICIENT OF RESISTANCE OF THE GOLD NANOFILMS (220 K).

The decreased conductivity and TCR reveal enhanced scattering of electrons in gold nanofilms. The dash curve in Fig. 3 and open triangle in Fig. 4 show the best fit of electrical conductivity and TCR from FS-MS theory taking into account the enhanced scattering from film surfaces and grain

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boundaries. We can find pronounced deviation between theoretical and experimental results on the temperature dependence of resistivity, since the classical model does not consider the enhanced electron-phonon scattering in thin films, and attributes all the increased resistivity to surface and grain boundary scattering. The deviation illustrates softened phonons scattering plays an irreplaceable role. In thin films, due to the phonon spatial confinement, the phonons are softened. At a higher temperature, there will be more excited phonons and hence the number of scattering events between electrons and phonons is increased. The temperature-dependent electrical conductivity related to the Debye temperature, which is the effect of electron-phonon coupling, has been formulated analytically by the BlochGrüneisen (BG) theory [19], ⎡ 5 ΘD σ 0 (T ) = ⎢ ρ R + K ( T Θ D ) ∫ 0 ⎢⎣

−1

T

⎤ x5 dx ⎥ , x −x ( e − 1)(1 − e ) ⎥⎦

(3)

where the prefactor and residual resistivity are both taken as bulk gold value K=5.2525 μΩ cm [7] and ρR=0.022 μΩ cm [20], respectively. Eq. (3) suggests that the temperaturedependent electrical conductivity of the films is related to the vibration of the phonons through the Debye temperature. To take into account the effect of electron-phonon softening on the electrical conductivity, the background conductivity of the film can be obtained through the Bloch-Grüneisen theory for a given Debye temperature, then substitute the background conductivity into the FS-MS theory instead of the bulk value. The theoretical predictions of conductivity and TCR from BG and FS-MS theories, taking into account enhanced electron scattering from softened phonons along with film surface and grain boundary, are presented in Fig. 3 (solid curves) and Fig.4 (open circles). The theoretical predictions of both electrical conductivity and TCR are in good agreement with the experimental results over the entire temperature range. Table 2 THE VALUES OF ΘD AND R OBTAINED FROM ELECTRICAL CONDUITIVITY DATA.

δ (nm) ΘD (K) R

20.0 98.7 0.31

23.0 84.4 0.32

36.0 121.9 0.33

54.0 103 0.28

Table 2 shows the values of ΘD and R obtained from electrical conductivity data. The Debye temperature of bulk gold is 165 K [10]. A sharp reduction of the Debye temperature can be found, which reveals the phonons of the gold films are softened. This is consistent with previous experiments on Au crystals, in which the surface Debye temperature was found to be reduced to about 83 K [21] due to the phonon spatial confinement. In addition, the measured reflection coefficient of electrons striking the grain boundaries is very close to the value in earlier studies [22-24]. To inquire into the underlying mechanism of phonon softening, the low-dimensional phonon system in the nanofilm is characterized by the elastic continuum model. This treatment has been applied with successes in nanofilm [25, 26] and nanowire [27]. According to the Debye model, the lattice vibrates as if it is an elastic continuum, but the vibration wave number can only lie within the first Brillouin zone. The corresponding Debye temperature derived from the maximum wave number can be expressed as, hV ( 6π 2 na )

13

ΘD =

kB

.

(4)

where V is the average phonon group velocity in the freestanding gold nanofilm instead of bulk value. Considering the film as an elastic continuum, the lattice displacement vector is governed by the elasticity equation [25], ∂ 2u = Vt 2∇ 2u + (Vl 2 − Vt 2 ) ∇ ( ∇ ⋅ u ) . ∂t 2

(5)

The boundary conditions at the free surface are derived from the requirement of vanishing normal stress component, instead of periodic boundary conditions of the bulk gold. Solving numerically the elasticity equation, three different types of confined acoustic modes characterized by their distinctive symmetries have been identified [25]: shear, dilatational, and flexural waves. Then, the phonon group velocities in the nth branch defined as Vn=∂ωn/∂q are determined by numerical differentiation. The calculated phonon dispersion relations of these three modes for the 20.0-nm-thick gold film are shown in Fig. 5.

Fig. 5 PHONO DISPERSIONS OF 20.0-nm-THICK GOLD NANOFILMSGOLD NAFILMS.

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There appear multiple dispersion branches for each polarization type and some of the dispersion curves are within the forbidden energy regions for bulk phonons. This is the result of quantization of phonon modes due to phonon spatial confinement and is similar to anomalous surface phonon dispersion relations for Ag(111) measured by inelastic scattering of He atoms [28]. Moreover, all the calculated dispersion curves show reductions in phonon group velocities, which is embodied in the dispersion curves flatter than that of bulk. Quantitatively, the average group velocities are approximately 653.7, 690.8, 824.5, and 892.7 m s−1 for the gold nanofilms with 20.0, 23.0, 36.0 and 54.0 nm in thickness, respectively, dropping below 1406.7 m s−1 for bulk gold. The calculated Debye temperatures derived from the average phonon group velocity are compared to the experiment results in Fig. 6. The two values match well with each other and show a sharp decrease compared to that of bulk gold. The Debye temperature tends to increase as the film thickness is increased, since the phonon spatial confinement becomes weaker. The reduced and thickness-dependent Debye temperature reflects the softening of phonons in gold films.

mechanisms also reduce the electronic thermal conductivity and we believe that the electronic thermal conductivity and the electrical conductivity still obey the Wiedemann-Franz law. However, the enhanced electron-phonon scattering increases the heat exchange between the electrons and phonons while decreased the electron thermal conduction. Based on the Energy Conservation Law, the electron energy decrease due to the electron-phonon scattering equals to the phonon energy increase, which leads to the result that the total thermal and electrical conductivities do not obey the Wiedemann-Franz law any more. In the present model, the electron thermal conduction decrease due to enhanced electron-phonon interaction can be counteracted by the increased heat exchange between the electrons and phonons. And hence, the thermal conductivity of the polycrystalline nanofilms can be predicted by applying the FS-MS theory, ⎡ 3 ⎤ ⎛ 1⎞ 6 2 3 ⎢1 − 2 α + 3α − 3α ln ⎜1 + α ⎟ − πκ (1 − p ) × ⎥ ⎝ ⎠ ⎥. λ = λ0 ⎢ ⎢ π2 ∞ cos 2 φ ⎛ 1 1 ⎞ 1 − exp ( −κ tH ( t , φ ) ) ⎥ ⎢ ∫ dφ ∫ dt 2 ⎥ ⎜ − ⎟ 1 H ( t , φ ) ⎝ t 3 t 5 ⎠ 1 − p exp ( −κ tH ( t , φ ) ) ⎥⎦ ⎢⎣ 0

(7)

It is important to note that the background thermal conductivity is taken as bulk value instead of the modified value by phonon softening in the electrical conductivity calculation, and all the other parameters are the same as Eq. (1). The specularity parameter and the reflection coefficient are taken the values previously obtained from the fit of the electrical conductivity.

Fig. 6 VATIATIONS OF THE DEBYE TEMPERATURE ALONG THE FILMS THICKNESS.

3.2 Thermal conductivity In bulk metals, the well-known Wiedemann-Franz law is widely applied, which considers the electron is a mutual carrier of charge and heat. The Wiedemann-Franz law [10] states that for metals at not too low temperature the ratio of the thermal conductivity to the electrical conductivity is directly to the temperature, 2

λ π 2 ⎛ kB ⎞ = L0 ≡ ⎜ ⎟ . σT 3 ⎝ e ⎠

(6)

According to the WF law, the value of Lorenz number is 2.45 × 10-8 W Ω K-2, independent of the particular metal. However, it is still an open question whether the WF law is valid in the polycrystalline metallic thin films. Fits to the reduced temperature-dependent electrical conductivity indicate that in the gold nanofilms, there are three mechanisms enhanced electron scattering including surface, grain boundary and softened phonon. All these three

Fig. 7 TEMPERATURE-DEPENDENT THERMAL CONDUCTIVITY OF THE GOLD NANOFILMS.

In Fig.7, The scatters are the measured thermal conductivities of the nanofilm and the dash-dot curve is the corresponding bulk value. It indicates that the thermal conductivity dramatically decreases and the temperaturedependent tendency even entirely differs from that of bulk gold. The dash curves are the thermal conductivities derived directly from the electrical conductivities by using the Wiedemann-Franz law. Large deviation between the theoretical and experimental data can be found, which reveals that the

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Wiedemann-Franz law is inapplicable in nanofilms and cannot be directly used any more. The solid curves are the prediction using Eq. (7), considering the electron thermal conduction decrease due to enhanced electron-phonon interaction can be counteracted by the increased heat exchange between the electrons and phonons. We can find that the present model have greatly improved the prediction compared to the corresponding result directly from Wiedemann-Franz law, although the predictions cannot match the experimental data perfectly. The possible reason for the deviation between the theoretical predictions and the experimental value may come from that the present model have not considered the effect of phonon softening on the thermal conduction of itself. 4 CONCLUSION The thermal and electrical conductivities of several suspended polycrystalline gold nanofilms with different thickness and grain size have been measured in a temperature range from 100 to 310K. Taking into account the enhanced electron scattering of softened phonons due to spatial confinement along with surface and grain boundary, the temperature-dependent electrical conductivity of the film is well predicted over the whole temperature range. The characteristic Debye temperature obtained from temperaturedependent electrical conductivity agrees well with the value derived from the calculation of phonon dispersion relations using the elastic continuum model. The reduced and thicknessdependent Debye temperature provides convictive evidence for the softening of phonons in gold nanofilms. By taking into account the softening of phonons in nanofilms, a new model on the thermal transport of metallic nanofilm is proposed based on the Energy Conservation Law, in which the electron-phonon scattering induced electron energy decrease is supposed to be counteracted by the phonon energy increase. The prediction of thermal conductivity has been greatly improved by using the present model compared to the corresponding result directly from Wiedemann-Franz law. ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Grant Nos 50730006 and 50976053).

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