Influence of Magnetic Domain Walls and Magnetic Field on the

Letter pubs.acs.org/NanoLett

Influence of Magnetic Domain Walls and Magnetic Field on the Thermal Conductivity of Magnetic Nanowires Hao-Ting Huang,† Mei-Feng Lai,† Yun-Fang Hou,‡ and Zung-Hang Wei*,‡ †

Institute of NanoEngineering and MicroSystems and ‡Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu 30013, Taiwan ABSTRACT: We investigated the influence of magnetic domain walls and magnetic fields on the thermal conductivity of suspended magnetic nanowires. The thermal conductivity of the nanowires was obtained using steady-state Joule heating to measure the change in resistance caused by spontaneous heating. The results showed that the thermal conductivity coefficients of straight and wavy magnetic nanowires decreased with an increase in the magnetic domain wall number, implying that the scattering between magnons and domain walls hindered the heat transport process. In addition, we proved that the magnetic field considerably reduced the thermal conductivity of a magnetic nanowire. The influence of magnetic domain walls and magnetic fields on the thermal conductivity of polycrystalline magnetic nanowires can be attributed to the scattering of long-wavelength spin waves mediated by intergrain exchange coupling. KEYWORDS: Thermal conductivity, magnetic domain wall, magnon, magnetic nanowire

H

eat in a solid crystal is manifested by atomic or lattice vibration from a microscopic perspective. A phonon is the quantized vibration of a crystal lattice. Among various types of quantized vibrations, ferromagnetic materials exhibit magnetic ordering between neighboring magnetic moments because of an exchange interaction that is a quantum mechanical effect, and a magnon is a quantized spin wave. Heat transport in a crystal is mediated by quantized vibration waves such as phonons or magnons; in addition, for most metallic materials electrons significantly contribute to heat transport. Phonon transport is affected by the scattering and collision of phonons, which can be caused by temperature, lattice defects, grain boundaries, and structural boundaries. Similarly, magnon transport is influenced by the scattering and collision of magnons that can be induced by specific arrangements of magnetic moments, domain walls (DWs), and applied magnetic fields. Therefore, studying the mechanism of transport behaviors of various heat carriers is crucial for understanding the heat transport of materials. For micro- and nanostructured materials, electron or phonon behaviors are easily affected by boundary scattering due to structure miniaturization.1 When the characteristic dimension of a nanostructure is comparable with the mean free path of the heat carriers, the thermal conductivity of the nanostructure might significantly differ from that of bulk materials. For fundamental physics research and to meet the increasing demand for nanodevice applications, understanding the underlying mechanism of heat transport in nanostructures such as nanotubes and nanowires is crucial. Some studies have investigated the heat transport properties of carbon nanotubes,2 silicon nanowires,3−5 ZnO nanowires,6 and Ni nanowires.7 According to a previous study, the Wiedemann−Franz law is invalid for nanostructured materials.8 © XXXX American Chemical Society

This is believed to be due to the boundary scattering effect in nanostructures, which causes the Lorenz number to deviate from the constant. Therefore, the thermal conductivity of nanostructures cannot be derived directly by using the Lorenz number and measured electrical conductivity. Most studies on heat transport in nanostructures have focused on semiconductor nanowires and carbon nanotubes. Some studies have investigated heat transport in magnetic microstructures and bulk materials.9−13 However, few studies have examined heat transport in nanostructured magnetic materials. Kimling et al. reported that the Lorenz number and thermal conductivity are anisotropic and sensitive to the saturated magnetic field direction along easy and hard axes in Ni nanowires.14 However, literature on the difference in the thermal conductivity of various magnetization configurations at zero magnetic field and the continuous variation in the magnetic field at various temperatures is scarce. Ferromagnets exhibit various types of magnetic domains depending on the geometry, applied magnetic field, and magnetic history. The number of magnetic DWs influences not only the hysteresis loop but also electron transport properties such as magnetoresistance. We expect the transport of magnons and the thermal conductivity of magnetic materials to be affected by the distribution of magnetic domains and DWs; this effect has not yet been studied. In this study, we report on the thermal conductivity of straight and wavy ferromagnetic nanowires with various magnetic DW densities to understand the role of magnon scattering in the heat transport of magnetic materials. In addition, because magnetic Received: July 8, 2014 Revised: March 18, 2015

A

DOI: 10.1021/nl502577y Nano Lett. XXXX, XXX, XXX−XXX

Letter

Nano Letters fields suppress the spin waves or magnons caused by thermal excitation, we measured the thermal conductivity of magnetic nanowires at various magnetic field intensities. Twenty nanometer oxide layers were grown on the top of silicon substrates using dry oxidation in a horizontal quartz tube furnace. Poly(methyl methacrylate) (PMMA) electron beam resist was spin coated onto the substrates and then electronbeam (e-beam) lithography was used to define the rectangle windows on the substrate for further oxide etching by buffered oxide etchant (BOE). The remaining oxide layers served as the mask for subsequent silicon etching. E-beam lithography was used again to define wavy nanowires with 480 nm width and 62.83 μm curve length, and straight nanowires with 860 nm width and 37 μm length. After defining the wires, patterned magnetic thin films were derived using e-beam evaporation and lift-off technique with thickness of 100 nm. The purity of the nickel target (Admat Midas Inc.) we used to deposit nickel nanowire is 99.99%. The atomic proportion of the Permalloy target (Admat Midas Inc.) we used to deposit Permalloy nanowire is Ni/Fe = 81:19, and the purity is above 99.95%. Subsequently, the same process was repeated to fabricate the Au electrodes for four-probe measurement. In order to avoid the thermal contact between the nanowires and the substrates, we suspended the wires by etching the underlying silicon substrates with tetramethylammonium hydroxide (TMAH). Figure 1a,b shows the SEM images of the fabricated straight nickel nanowires and wavy permalloy nanowires, respectively. Evaluating the thermal conductivity κ is a crucial parameter for investigating the heat transport properties of nano- and microstructures. Some well-established methods, such as steady-state Joule heating,15 the 3ω method,16 optical reflectance,17 and micromachined thermal methods,18 have been used to measure thermal conductivity. In the present study, we first obtained the temperature coefficient of electrical resistance (α) and then used the steady-state Joule heating method to analyze the electrical resistance variation in the structure due to spontaneous heating. Finally, we derived the thermal conductivity coefficient by using the procedure described as follows. To obtain the temperature coefficient of electrical resistance, the devices were placed in a vacuum chamber (at 5 × 10−4 Torr) equipped with a temperature controller. Each sample was maintained at a specific ambient temperature for 1 h to ensure that the sample had reached thermal equilibrium. We then measured the resistance by using a four-probe method and obtained the resistance−temperature characteristics. The measured resistance increased with an increase in temperature, as expressed in the following equation R f = R i[1 + α(Tf − Ti)]

Figure 1. Scanning electron microscopy (SEM) image of the suspended straight Ni nanowires. The nanowires are 37 μm, 860 nm, and 100 nm in length, width, and thickness, respectively. (b) SEM image of the suspended permalloy wavy nanowires. The nanowires are 62.83 μm, 480 nm, and 60 nm in arc length, wire width, and thickness, respectively. The silicon substrate was covered with a 20 nm SiO2 layer. In addition, Au electrodes at both ends of each nanowire were used for four-point probe measurements.

The three-dimensional (3D) heat conduction equation can be expressed as qgen ̇ ρc ∂T ∇2 T + = (2) κ κ ∂t where q̇gen is the rate of energy generation per unit volume due to Ohmic heating, ρ and c represent the density and specific heat of the material, respectively, and ∂T/∂t represents the time variation of the temperature. In this study, because the length of the nanowire was considerably greater than its thickness and width, the thermal conductivity was restricted to the 1D direction. Equation 2 can be modified to obtain the following 1D steady-state heat conduction equation (eq 3)

(1)

where Rf, Ri, Tf, and Ti are the final resistance, initial resistance, final temperature, and initial temperature, respectively. Thus, the temperature coefficient of resistance α can be derived. The steady-state Joule heating method was used subsequently. A weak current of 2 μA flowed through the nanowires, resulting in an initial resistance R0. A strong current of 10 μA was then applied to the nanowires for spontaneous heating. The heat was transported from both ends of the nanowires to the substrate, and a steady-state heat distribution was formed in the nanowires. The steady-state resistance R was measured during this time. We referred to the previously obtained resistance− temperature curve and determined temperatures T0 and T of the nanowires in the initial and steady states, respectively.

I 2R 0[1 + α(T − T0)] d2T + =0 (3) wbl d2x where w, l, and b represent the width, length, and thickness of the nanowire, respectively, κ(T) is the thermal conductivity coefficient that is a function of temperature, I is the applied current, and x is the position along the axis of the nanowire. The time-dependent term ∂T/∂t in eq 2 cancels out here when the system is in a steady state, and the energy generation term q̇gen in eq 2 is replaced by the spontaneous heating rate of the nanowire. By solving eq 3, the relationship between the resistances before and after the Joule heat was generated on the κ (T )

B

DOI: 10.1021/nl502577y Nano Lett. XXXX, XXX, XXX−XXX

Letter

Nano Letters nanowire was derived15,19−21 under steady-state conditions (eq 4) ⎡⎛ 2 ⎞ ⎛ β ( T ) l ⎞⎤ ⎟⎥ , R ( T ) = R 0 ⎢⎜ ⎟tan⎜ ⎣⎝ β ( T ) l ⎠ ⎝ 2 ⎠⎦

[β(T )]2 =

curvature patterns tend to induce magnetic vortices such that a single-domain state can transform into other stable states. Therefore, there were periodic DWs in our magnetic nanowires. Systematic theoretical and simulation studies on the magnetic nanowires of various scales are limited, and most available data are experimentally observed magnetization patterns.24,28−30 Similarly, various DWs were generated in the wavy nanowires. After we applied an initial magnetic field perpendicular to the wavy nanowires and then turning off the field, magnetic DWs appeared on the peaks and troughs of the wavy structure, which corresponded to head-to-head (H−H) DWs and tail-to-tail (T−T) DWs, respectively (Figure 2c). This was because when the initial strong magnetic field perpendicular to the nanowire was turned off, the magnetic dipole moment was aligned with the boundary of the nanowire because of shape anisotropy. When the initial field was applied parallel to the wavy nanowires, the magnetic moments within the nanowires were arranged in a successive head-to-tail configuration; therefore, no magnetic DWs were created (Figure 2d). The spin directions are indicated in Figure 2a−d by arrows. Figure 3 shows the resistance−temperature relationship of the magnetic nanowires; Figure 3a illustrates that for the straight Ni nanowires with sparse and dense DWs; Figure 3b

I 2R 0α wblκ(T ) (4)

When a current I flows through a nanowire, the resistance of the nanowire increases because of spontaneous heating. By measuring the resistances R0 and R before and after applying the current, respectively, and substituting them into eq 4, β can be derived. We can then obtain the thermal conductivity κ of the nanowire at an environmental temperature T by using β2 = (I2R0α)/(wblκ). All measurements were performed in a vacuum; therefore, the heat loss due to heat convection can be ignored. In addition, according to the Stefan−Boltzmann law,22,23 the thermal radiation can be neglected because the temperature difference between the environment and the nanowire that was subjected to spontaneous Joule heating was small (approximately 1 K); furthermore, the overall surface area of the nanowire was negligible. We placed the straight Ni nanowires between a pair of electromagnets that provided a large initial magnetic field (2600 G) along the long axis of the nanowires. We then turned off the field to obtain the normalized magnetization state. By using a process similar to that used in ref 24, two remanent states with different magnetic DW densities were generated by applying magnetic fields perpendicular and parallel to the straight nanowires.24 Magnetic force microscopy (MFM) was used to measure the magnetic force gradient that originated from the magnetic pole density between the tip and the sample. Figure 2a,b depicts MFM images of the magnetic nanowires with

Figure 2. MFM images of (a,b) the straight Ni nanowire and (c,d) the permalloy wavy nanowire after we applied a 2600 G magnetic field along its (a,c) short and (b,d) long axis, and the magnetic field was then turned off. The scale bars in (a,b) and (c,d) represent 1 and 5 μm, respectively.

various DW densities. The magnetization of the magnetic domains formed highly periodic patterns in continuous upward and downward directions. The distances between the adjacent magnetic DWs were 800 and 380 nm under perpendicular and parallel magnetic field conditions, respectively. According to previous studies,25−27 infinitely long magnetic wires and highaspect-ratio ellipses exhibit a highly periodic buckling mode when nucleation begins, and these highly periodic and high

Figure 3. Resistance−temperature curves of (a) the straight Ni nanowire with various magnetic DW densities, (b) the permalloy wavy nanowire with and without magnetic DWs, and (c) the straight Ni nanowire under various magnetic fields applied along its long axis. C

DOI: 10.1021/nl502577y Nano Lett. XXXX, XXX, XXX−XXX

Letter

Nano Letters illustrates that for the wavy permalloy nanowires with and without DWs; and Figure 3c illustrates that for the straight Ni nanowires under various magnetic fields (330, 670, and 1010 G). According to the figures, the electrical resistance increased with an increase in the ambient temperature in all cases. The resistance leveled out and reached a constant for each nanowire at a low temperature. This was because of the defects and impurity scattering within the nanowires. According to the overlapped curves in Figure 3a−c, for a specific nanowire the DW numbers and the magnitude of the applied magnetic field had little influence on the resistance−temperature curves. According to the Wiedemann−Franz law, the electronic contribution to thermal conductivity is proportional to the electrical conductivity, implying that the DW numbers or the magnitude of the applied magnetic field had little effect on the electronic contribution to thermal conductivity. Figure 4a shows the relationship between the temperature and the thermal conductivity of the straight Ni nanowires with various magnetic DW densities. In our experiments, the thermal conductivity measurements for the cases of sparse and dense DWs were performed using the same straight nanowires; therefore, the phonon scattering effect due to boundaries and defects was identical and thus could be neglected. In addition, the results in Figure 3 suggest that the DWs had little influence on the electronic contribution to thermal conductivity. Therefore, the difference between the two curves in Figure 4a can be attributed to the distinct magnon scattering effects in the magnetic nanowire due to differences in DW distributions. For single crystals, which are continuous media for lattice vibration, the thermal conductivity exhibits a Gaussian feature31−35 (pink curve in Figure 4b). In the high-temperature range (right-hand side of the peak), the number of phonons increases with an increase in the temperature. In addition, the probability of phonon−phonon scattering increases such that the mean free path λ of the phonons (λ ∝ 1/T) and the thermal conductivity32 κ = Cvλν/3 ̅ decrease, where Cv is the heat capacity of the material, which is a constant at high temperatures32 and ν̅ is the mean velocity of the heat carriers. In the low-temperature range (left-hand side of the peak), the mean free path theoretically increases with a decrease in temperature. However, in most cases the mean free path is limited by the boundary effect and the imperfection and impurity of the crystal. Therefore, the thermal conductivity κ is proportional to T3, inherited from the behavior of the heat capacity Cv at low temperatures.32 In this study, the nanowires were polycrystalline structures and prepared using e-beam evaporation.36−41 The grain boundaries, dislocation, and vacancies led to the discontinuity of crystals in the nanowires, thus increasing the probability of phonon scattering. Therefore, the temperature-dependent thermal conductivity of a polycrystalline system became a monotonic decreasing function instead of a Gaussian curve7,42−44 (green curve in Figure 4b). In addition, the thermal conductivity of the Ni nanowires was lower than that of bulk materials7,10,33 partly because of boundary scattering caused by a size effect and partly because 3D bulk materials might have many channels for thermal conduction42 compared with 1D polycrystalline nanowires.7 For ferromagnetic materials, neighboring magnetic moments are attached through an exchange force mediated mainly by 3d orbital electrons (Figure 4c; where the small springs with a large spring constant represent the exchange force between atomic magnetic moments). Because strong springs are

Figure 4. (a) Thermal conductivity versus temperature curves of the straight Ni nanowire with various magnetic DW densities. More than 15 nanowires were considered in this study, and each data point shown in the figure was derived by averaging the data of more than 10 measurements. (b) Schematic relationship between temperature and thermal conductivity in crystalline (solid pink curve) and polycrystalline or amorphous (solid green curve) structures. The dashed orange curve is the summation of the two solid curves. (c) Schematic model of exchange springs in the magnetic polycrystalline thin film. The blue stripes represent the direction of the crystal orientation or magnetic easy axis in a grain. The blue spring represents the atomic scale exchange force between magnetic moments. The green spring represents the intergrain exchange force between the effective magnetic moments of adjacent grains.

localized in small-scale atomic spins, only spin waves with short wavelengths can exist therein. In general, such spin waves do not dominate in the thermal conduction of polycrystalline nanowires because of the scattering and localization of grain boundaries. The nanowires used in this study were polycrystalline, and the effective magnetic moment of each grain was attached to its neighboring effective magnetic moment through a large spring with a smaller spring constant that represents the exchange force between the effective magnetic moments of adjacent D

DOI: 10.1021/nl502577y Nano Lett. XXXX, XXX, XXX−XXX

Letter

Nano Letters grains (Figure 4c). In contrast to the exchange force between neighboring atomic magnetic moments, the exchange force between the effective magnetic moments of adjacent grains is mediated mainly by intergrain exchange coupling.45−47 The spring attachment between the effective magnetic moments of grains is weak, and the spring size is equivalent to the grain size such that only spin waves with long wavelengths, which span over several grains, are allowed to exist. Such spin waves are not limited by the grain boundaries of polycrystalline nanowires and cannot easily be scattered by lattice imperfection because of their long wavelengths. This polycrystalline structure can be regarded as a continuous medium for spin waves with long wavelengths. At low temperatures, spin waves with long wavelengths dominate in the thermal conduction process, and the corresponding temperature-dependent thermal conductivity resembles the low-temperature part of the Gaussian curve (pink curve). Therefore, phonons and short-wavelength spin waves are significantly suppressed in the thermal conduction process for polycrystalline magnetic thin films, which can provide a suitable physical system for investigating the thermal conduction caused by long-wavelength spin waves and various types of magnon scattering. The mentioned model can be used to explain the experimental results shown in Figure 4a. For the curve obtained for sparse DWs, when the environmental temperature decreased from room temperature to a lower temperature, the thermal conductivity increased slightly at 130 K and decreased with a decrease in the temperature. This curve (dashed orange curve) can be regarded as the summation of a typical Gaussian curve (solid pink curve) dominated by long-wavelength spin waves and the monotonically decreasing curve (solid green curve) dominated by phonons in polycrystalline systems (Figure 4b). By comparing the curves of nanowires with sparse and dense DWs (Figure 4a), we observed that the thermal conductivity of the nanowires with dense DWs was lower than that of the nanowires with sparse DWs. A possible reason is the interaction between the magnons and DWs. The mean free path of magnons was approximately 200 nm at room temperature,10 and this mean free path increased with a decrease in temperature. When the mean free path of magnons was comparable with the magnetic domain widths, which were 380 (dense) and 800 nm (sparse), the scattering between magnons and DWs started to dominate the heat transport process. The nanowires with denser DWs experienced magnon scattering at high temperatures in the cooling process; therefore, the thermal conductivity decreased at high temperatures compared with the nanowires with sparser DWs. This phenomenon is more remarkable for the nanowires with dense DWs, implying that the DWs inhibited magnon contribution to heat transport. The largest difference of 21.69 W/mK between the two thermal conductivity curves was observed at an ambient temperature of 130 K. Figure 5a shows the relationship between the temperature and thermal conductivity of the wavy permalloy nanowires. The presence and absence of DWs caused a significant difference in the thermal conductivity, which is consistent with the result shown in Figure 4a. The wavy nanowires with magnetic DWs had lower thermal conductivity than that of the nanowires without DWs. The largest difference of 15.75 W/mK between the two thermal conductivity curves was obtained at an ambient temperature of 60 K. As the temperature was reduced, the amount of excited magnons and phonons decreased, and the boundary and defect scattering effect started to dominate over

Figure 5. Thermal conductivity versus temperature curves of the wavy permalloy nanowire without and with magnetic DWs. The length of one period is 5 μm in (a) and 10 μm in (b).

the magnon−DW scattering effect on the heat transport. The thermal conductivity difference between the wavy nanowires with and without DWs gradually became smaller, and the two curves eventually merged at approximately 0.4 W/mK when the temperature was reduced to 8.5 K. Figure 5b shows the thermal conductivity versus temperature curve of a wavy nanowire with approximately half the number of waves of the wavy nanowire in Figure 5a. Similarly, we observed that the thermal conductivity of the wavy nanowire with DWs was lower than that without DWs, validating our previous results. When the environmental temperature was reduced from room temperature, the mean free path of magnons was considerably smaller than that of DW separation; therefore, the DWs did not dominate the thermal conduction process until 90 K, and when the environmental temperature was reduced to 45 K, the thermal conductivity difference between the wavy nanowires with and without DWs was the maximum. Figure 6 shows the relationship between the applied magnetic field and the thermal conductivity of the straight Ni nanowires at various ambient temperatures. To ensure the same

Figure 6. Thermal conductivity versus magnetic field curves of the straight Ni nanowire at various ambient temperatures. The magnetic field was applied along the long axis of the nanowire. E

DOI: 10.1021/nl502577y Nano Lett. XXXX, XXX, XXX−XXX

Letter

Nano Letters initial magnetic state in the nanowire, an initial magnetic field of 2600 G was applied along the long axis of the nanowires and was then removed before each thermal conductivity measurement. As the magnetic field H was increased, the magnetic moment was gradually restricted, inhibiting the generation of magnons and reducing the thermal conductivity. For example, for the curve at room temperature (297 K), the total thermal conductivity can be expressed as κtotal = κe + κph + κm_short + κm_long, where κe, κph, κm_short, and κm_long represent the thermal conductivity of electrons, phonons, magnons (with short wavelength and high frequency), and magnons (with long wavelength and low frequency), respectively. According to aforementioned discussion, we infer that the thermal conductivity of electrons, phonons, and magnons (with short wavelength and high frequency) can be suppressed in a polycrystalline nanowire. However, magnons (with long wavelength and low frequency) play a crucial role in determining the thermal conductivity. After we applied a magnetic field of 1010 G along the long axis of the nanowire (Figure 6), magnons (with long wavelength and low frequency) were significantly suppressed and few magnons were generated to transport the heat. Therefore, under a magnetic field of 1010 G, the contribution of magnons (with long wavelength and low frequency) to the total heat transport can be neglected; that is, κ2 ≈ κe + κph + κm_short. The thermal conductivity difference κ1 between 0 and 1010 G was actually the contribution of magnons (with long wavelength and low frequency) to the thermal conductivity at 0 G at which the magnons (with long wavelength and low frequency) were not suppressed; that is, κ1 ≈ κm_long. In general, the contribution of electrons and phonons to heat transport is believed to be far more significant than that of magnons;10,11 that is, κ2 ≫ κ1. However, our results showed that κ2 < κ1 at room temperature (Figure 6). The decrease in the measured thermal conductivity coefficient due to the suppression of magnons under a high magnetic field was up to 62.77%, which was considerably higher than those reported in literature. As previously mentioned, one possible reason is that the nanowires used in this study were polycrystalline. In general, phonons and short-wavelength spin waves do not dominate the thermal conduction of polycrystalline nanowires because of the scattering and localization of grain boundaries. The polycrystalline structure can be regarded as a continuous medium for spin waves with long wavelengths. Long-wavelength spin waves dominate the total thermal conductivity, thus making κ1 > κ2. According to ref 48, spin waves in crystals can be suppressed when a weak magnetic field is applied at a low temperature, and in this situation the dominant spin waves are of a long wavelength and low frequency. Because long-wavelength spin waves dominated in the thermal conduction process of the polycrystalline thin films in this study, a magnetic field of only 1010 G was sufficient to have a considerable influence on the thermal conductivity. In addition, according to Figure 6, as the temperature was reduced the thermal conductivity decreased because of a decrease in the amount of thermally excited magnons and phonons. In summary, we investigated the influence of magnetic DWs, magnons, and magnetic field on the thermal conductivity of suspended magnetic nanowires. The thermal conductivity of straight and wavy nanowires was obtained using steady-state Joule heating to measure the resistance change caused by spontaneous heating. Our results showed that for straight and

wavy nanowires, the thermal conductivity coefficient decreased with an increase in the magnetic DW number at low temperatures, implying that the scattering between magnons and DWs hindered the heat transport process. In addition, we proved that the magnetic field can significantly reduce the thermal conductivity of a magnetic polycrystalline nanowire. The influence of magnetic DWs and magnetic fields on the thermal conductivity of magnetic polycrystalline nanowires can be attributed to the scattering of long-wavelength spin waves mediated by intergrain exchange coupling.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: (Z.H.W.) [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors would like to thank ROC National Science Council (Grants NSC 102-2112-M-007-012-MY3 and NSC 103-2221-E-007-017-MY2).

■

REFERENCES

(1) Liu, W.; Asheghi, M. Appl. Phys. Lett. 2004, 84, 3819−3821. (2) Kim, P.; Shi, L.; Majumdar, A.; McEuen, P. L. Phys. Rev. Lett. 2001, 87, 215502. (3) Li, D.; Wu, Y.; Kim, P.; Shi, L.; Yang, P.; Majumdar, A. Appl. Phys. Lett. 2003, 83, 2934−2936. (4) Liu, W.; Asheghi, M. J. Appl. Phys. 2005, 98, 123523. (5) Sultan, R.; Avery, A. D.; Stiehl, G.; Zink, B. L. J. Appl. Phys. 2009, 105, 043501. (6) Bui, C. T.; Xie, R.; Zheng, M.; Zhang, Q.; Sow, C. H.; Li, B.; Thong, J. T. L. Small 2012, 8, 738−745. (7) Ou, M. N.; Yang, T. J.; Harutyunyan, S. R.; Chen, Y. Y.; Chen, C. D.; Lai, S. J. Appl. Phys. Lett. 2008, 92, 063101. (8) Zink, B. L.; Avery, A. D.; Sultan, R.; Bassett, D.; Pufall, M. R. Solid State Commun. 2010, 150, 514−518. (9) Yang, Y.; White, R. M.; Asheghi, M. J. Heat Transfer-Trans. ASME 2006, 28, 113−120. (10) Yelon, W. B.; Berger, L. Phys. Rev. Lett. 1970, 25, 1207−1210. (11) Yelon, W. B.; Berger, L. Phys. Rev. B 1972, 6, 1974−1985. (12) Lue, C. S.; Kuo, C. N.; Tasi, D. S.; Kuo, Y. K.; He, Z.; Itoh, M. Phys. Rev. B 2008, 78, 012406. (13) Hess, C.; Baumann, C.; Ammerahl, U.; Büchner, B.; HeidrichMeisner, F.; Brenig, W.; Revcolevschi, A. Phys. Rev. B 2001, 64, 184305. (14) Kimling, J.; Gooth, J.; Nielsch, K. Phys Rev. B 2013, 87, 094409. (15) Tai, Y. C.; Mastrangelo, C. H.; Muller, R. S. J. Appl. Phys. 1988, 63, 1442−1447. (16) Cahill, D. G. Rev. Sci. Instrum. 1990, 61, 802−808. (17) Rosencwaig, A.; Opsal, J.; Smith, W. L.; Willenborg, D. L. Appl. Phys. Lett. 1985, 46, 1013−1015. (18) Zawilski, B. M.; Littleton, R. T., IV; Tritt, T. M. Rev. Sci. Instrum. 2001, 72, 1770−1774. (19) Sayer, R. A.; Zeng, J.; Hsu, H. H.; Peroulis, D. J. Microelectromech. Syst. 2012, 21, 850−858. (20) Pop, E.; Mann, D.; Wang, Q.; Goodson, K.; Dai, H. Nano Lett. 2006, 6, 96−100. (21) Völklein, F.; Reith, H.; Cornelius, T. W.; Rauber, M.; Neumann, R. Nanotechnology 2009, 20, 325706. (22) Stefan, J. Wien. Ber. II 1879, 79, 391−428. (23) Boltzmann, L. Wiedemann’s Ann. 1884, 22, 291−294. (24) Rudiger, U.; Yu, J.; Kent, A. D.; Parkin, S. S. P. Appl. Phys. Lett. 1998, 73, 1298−1300. (25) Usov, N. A.; Chang, C. R.; Wei, Z. H. Phys. Rev. B 2002, 66, 184431. F

DOI: 10.1021/nl502577y Nano Lett. XXXX, XXX, XXX−XXX

Letter

Nano Letters (26) Wei, Z. H.; Lai, M. F. J. Appl. Phys. 2007, 101, 09F515. (27) Lai, J. Y.; Wei, Z. H.; Lai, M. F.; Chang, C. R.; Wu, J. C.; Lo, I. C. J. Appl. Phys. 2006, 99, 08G313. (28) Chung, T. K.; Keller, S.; Carman, G. P. Appl. Phys. Lett. 2009, 94, 132501. (29) Tetienne, J. P.; Hingant, T.; Kim, J. V.; Diez, L. H.; Adam, J. P.; Garcia, K.; Roch, J. F.; Rohart, S.; Thiaville, A.; Ravelosona, D.; Jacques, V. Science 2014, 344, 1366. (30) Franken, J. H.; Hoeijmakers, M.; Swagten, H. J. M.; Koopmans, B. Phys. Rev. Lett. 2012, 108, 037205. (31) Makinson, R. E. B. Math. Proc. Cambridge Philos. Soc. 1938, 34, 474−497. (32) Uher, C. Thermal Conductivity of Metals; Tritt, T. M., Eds.; Kluwer Academic: New York, 2004; pp 22−91. (33) White, G. K.; Woods, S. B. Philos. Trans. R. Soc. London, Ser. A 1959, 251, 273. (34) Hust, J. G.; Clark, A. F. Mater. Res. Stand. 1971, 11, 22−24. (35) Powell, W. Int. J. Heat Mass Transfer 1965, 8, 1033−1045. (36) Delamoreanu, A.; Rabot, C.; Vallee, C.; Zenasni, A. Carbon 2014, 66, 48−56. (37) Atuchin, V. V.; Grigorieva, T. I.; Pokrovsky, L. D.; Kruchinin, V. N.; Lychagin, D. V.; Ramana, C. V. Mod. Phys. Lett. B 2012, 26, 1150029. (38) Umair, A.; Raza, H. Nanoscale Res. Lett. 2012, 7, 437. (39) Müller, G. A.; Carpene, E.; Gupta, R.; Schaaf, P.; Zhang, K.; Lieb, K. P. Eur. Phys. J. B 2005, 48, 449−462. (40) Durkan, C.; Welland, M. E. Phys. Rev. B 2000, 61, 14215− 14218. (41) Prutton, M.; Bradley, E. M. Proc. Phys. Soc. 1960, 75, 557−574. (42) Alvarez-Quintana, J.; Martínez, E.; Pérez-Tijerina, E.; PérezGarcía, S. A.; Rodríguez-Viejo, J. J. Appl. Phys. 2010, 107, 063713. (43) Uma, S.; McConnell, A. D.; Asheghi, M.; Kurabayashi, K.; Goodson, K. E. Int. J. Thermophys. 2001, 22, 605−616. (44) Cahill, D. G.; Katiyar, M.; Abelson, J. R. Phys. Rev. B 1994, 50, 6077−6081. (45) Yonemura, S.; Saito, S.; Hashimoto, A.; Takahashi, M. J. Magn. Magn. Mater. 2008, 320, 3053−3056. (46) Takahashi, M.; Kikuchi, A.; Kawakita, S. IEEE Trans. Magn. 1997, 33, 2938−2943. (47) Kikuchi, A.; Kawakita, S.; Nakai, J.; Shimatsu, T.; Takahashi, M. J. Appl. Phys. 1996, 79, 5339. (48) Raquet, B.; Viret, M.; Sondergard, E.; Cespedes, O.; Mamy, R. Phys. Rev. B 2002, 66, 024433.

G

DOI: 10.1021/nl502577y Nano Lett. XXXX, XXX, XXX−XXX