Phonon-Assisted Field Emission in Silicon Nanomembranes for Time

Apr 29, 2013 - Time-of-flight (TOF) mass spectrometry has been considered as the method of ... at high masses, the detectors require an expensive cryo...
3 downloads 3 Views 2MB Size
Download PDF
Letter pubs.acs.org/NanoLett

Phonon-Assisted Field Emission in Silicon Nanomembranes for Timeof-Flight Mass Spectrometry of Proteins Jonghoo Park,†,‡ Zlatan Aksamija,‡ Hyun-Cheol Shin,§ Hyunseok Kim,‡ and Robert H. Blick*,‡,§,∥ †

Department of Electrical Engineering, Kyungpook National University, Daegu, Korea Department of Electrical and Computer Engineering and §Department of Material Science and Engineering, University of Wisconsin-Madison, Madison, Wisconsin 53706, United States ∥ Angewandte Physik, Universitat Hamburg, Jungiussstrasse 11, 20355 Hamburg, Germany ‡

S Supporting Information *

ABSTRACT: Time-of-flight (TOF) mass spectrometry has been considered as the method of choice for mass analysis of large intact biomolecules, which are ionized in low charge states by matrixassisted-laser-desorption/ionization (MALDI). However, it remains predominantly restricted to the mass analysis of biomolecules with a mass below about 50 000 Da. This limitation mainly stems from the fact that the sensitivity of the standard detectors decreases with increasing ion mass. We describe here a new principle for ion detection in TOF mass spectrometry, which is based upon suspended silicon nanomembranes. Impinging ion packets on one side of the suspended silicon nanomembrane generate nonequilibrium phonons, which propagate quasi-diffusively and deliver thermal energy to electrons within the silicon nanomembrane. This enhances electron emission from the nanomembrane surface with an electric field applied to it. The nonequilibrium phonon-assisted field emission in the suspended nanomembrane connected to an effective cooling of the nanomembrane via field emission allows mass analysis of megadalton ions with high mass resolution at room temperature. The high resolution of the detector will give better insight into high mass proteins and their functions. KEYWORDS: NEMS, nanomembrane, phonons, mass spectrometry 1a. the silicon nanomembrane is placed at the end of the flight tube of a commercial MALDI-TOF mass spectrometer (Perseptive Biosystems Voyager-DE STR). The detector consists of four parts, a silicon nanomembrane, an extraction gate, microchannel plates (MCPs), and an anode, as shown in Figure 1b. The silicon nanomembrane was fabricated from silicon-on-insulator (SOI) material by wafer thinning and wet etching to form an array of two suspended silicon nanomembranes with an area of (2 × 2) mm2 each and a thickness of 180 nm. The operating principle of the detector is illustrated in a band diagram in Figure 1c. Applying an electric field at the surface of the silicon nanomembrane via the extraction gate lowers and “thins” the potential barrier, resulting in electron emission from the surface of the nanomembrane. The applied electric field we estimate to be about 1.9 × 107 V/m, which alone is not sufficient to place the nanomembrane in the strong Fowler− Nordheim tunneling regime.13 Instead, the nanomembrane is in the so-called Schottky emission regime,14,15 where both thermionic and tunneling components contribute, and the field emission current is well approximated by the expression16

I

n time-of-flight (TOF) mass spectrometry,1 the low charge state ions, such as those generated by the matrix-assisted laser desorption/ionization (MALDI) process,2,3 are accelerated in an electric field and drift to a detector with different velocities. Although TOF mass spectrometry has been known to operate with an unlimited mass range, in practice, however, its mass range is limited by the sensitivity of the detector. The sensitivity of microchannel plate (MCP) detectors, which are used in most TOF mass spectrometers, decreases as v4.4, where v is the velocity of the incident ion.4 This leads to a remarkable decrease in sensitivity of MCP detectors for heavier ions, which drift more slowly down to the detector than lighter ones. Phonon-mediated particle detectors such as cryogenic microcalorimeters and superconducting tunnel junctions have been demonstrated to show mass (i.e., velocity) independent sensitivity by measuring the thermal energy deposited by ion bombardment at temperatures lower than hundred millikelvin.5−12 Although these cryogenic particle detectors deliver exceptional mass sensitivity at high masses, the detectors require an expensive cryogenic cooling unit. Here we describe a nonconventional phonon-mediated particle detector for the detection of ultra large ions in TOF mass spectrometry operating at room temperature. Our approach is based upon nonequilibrium phonon-assisted field emission (PAFE) in silicon nanomembranes. The nanomembrane detector we describe here is illustrated in Figure © 2013 American Chemical Society

Received: March 8, 2013 Revised: April 17, 2013 Published: April 29, 2013 2698

dx.doi.org/10.1021/nl400873m | Nano Lett. 2013, 13, 2698−2703

Nano Letters

Letter

away from the location of impact. These nonequilibrium phonons deliver sufficient thermal energy to the electrons within the nanomembrane and excite them to higher energy states, thereby allowing them to overcome the vacuum barrier (∼4.6 eV for undoped silicon) and escape from the nanomembrane surface via field emission. This results in the increased field emission current and can be referred to as phonon-assisted field emission. When electrons escape from the nanomembrane by PAFE, they carry an energy amount equal to the average energy difference between the emitted and replaced electrons, resulting in cooling of the cathode-side (front-side) of the nanomembrane. Then, the field emission current is amplified by the microchannel plates (MCPs), which are placed behind the nanomembrane, and collected by the anode. Finally, the current is recorded using an oscilloscope in real time followed by a 1−2 MHz low pass filter. This is removing the high-frequency noise induced by the mechanical vibrations of the nanomembrane excited through ion bombardment.17 Figure 2a shows a MALDI mass spectrum obtained using the silicon nanomembrane detector for human immunoglobulin G

Figure 1. Operating principle of the silicon nanomembrane detector. (a) Schematic of the detector coupled to a MALDI-TOF mass spectrometer. Proteins are desorbed and ionized by MALDI and separated by their mass to charge ratio in the time-of-flight tube before they reach the detector. (b) Schematic representation of the detector, consisting of a 180 nm thin silicon nanomembrane, an extraction gate, MCPs, and an anode. The electric field is applied via the extraction gate, allowing field emission of electrons and enhancing thermionic emission. MCPs amplify the number of electrons before they are collected by the anode. The emission current is displayed by the oscilloscope in real-time. (c) Energy band diagram of the silicon nanomembrane. After absorbing thermal energy from the phonons, generated through the ion bombardment, electrons are excited to higher energy states. This allows electrons to overcome and tunnel through the barrier more easily, which results in the large field emission current. When electrons escape from the nanomembrane by PAFE, they carry an energy amount in the range of Eph, resulting in cooling of the nanomembrane.

J (T , F ) =

mekB 2π ℏ3

T 2e−(W −ΔW )/ kBT

Figure 2. MALDI TOF mass spectrum for IgG obtained using the silicon nanomembrane detector. (a) Mass spectrum for IgG with sinapinic acid, showing multiple peaks at the time-of-flight of the singly and doubly charged IgG. Inset: the magnified view of the peaks enclosed by dashed lines in semilog scale. This clearly represents the sharp onset due to the quasi-diffusive transport of phonons, and exponential decay due to the slow lateral heat diffusion. (b) Histogram of IgG, showing a relative abundance of IgG ions.

(1)

(IgG, ∼150 000 Da, 1 pmol/μL, Sigma Aldrich, U.S.A.) in a sinapinic acid matrix (see Supporting Information for sample preparation and MALDI-TOF operating conditions). In addition to the major peaks for the intact IgG, labeled by singly and doubly charged IgG, small peaks, corresponding to fragment ions or neutrals are also observed. The dominant feature of the spectrum is formed by the peaks found around the time-of-flight of the singly charged intact IgG: the m/z of the peaks shown are 139.2, 143.4, 145.6, and 148.1 kDa and are well-resolved owing to the high resolution (m/Δm = 73) of our silicon nanomembrane detector. The averaged full width at

where W is the work function of the material, and ΔW = (eF/ 4πε0)1/2 is the field-dependent correction to the work function due to tunneling through the barrier with F being the electric field strength, T is the temperature, and e and m are the electron’s charge and mass, respectively. When accelerated ions propagate through the flight tube and bombard the backside of the silicon nanomembrane, almost half of the kinetic energy is transformed into thermal energy.7 This raises the temperature in the vicinity of the impact site and produces nonequilibrium phonons, which carry thermal energy 2699

dx.doi.org/10.1021/nl400873m | Nano Lett. 2013, 13, 2698−2703

Nano Letters

Letter

half-maximum (fwhm) resolution of the detector for 18 IgG spectra is 129 with a standard deviation of 44 (see Supporting Information Figure S1 for the spectrum with higher resolution). It is also important to note that due to the high sensitivity of the detector it was only necessary to average 10 laser shots in comparison to other ion detectors often requiring several hundred shots. The detector provides a signal-to-noise ratio (SNR) higher than 10 for IgG with a molar concentration of 100 fmol/μL. Another interesting feature of the peaks in the spectrum is the sharp onset at the time-of-flight of the ions followed by an exponential decay, which is described well by nonequilibrium phonons traveling through the 180 nm thick (100)-oriented silicon nanomembrane. The process can be described in detail as follows: nonequilibrium phonons produced by ion bombardment with a kinetic energy of 23.5 keV are rich in high-frequency states near the Debye frequency ωD. At frequencies comparable to ωD, phonons have a very short mean free path (mpf) and their relaxation is dominated by anharmonic decay (τA−1 ∝ ω5) and isotope scattering (τI−1 ∝ ω4).18,19 Anharmonic decay converts the high-energy phonons into two lower-energy phonons, which have a longer mfp than their parent phonons. After several anharmonic decay steps, high-frequency phonons are finally down-converted to phonons with frequencies well below ωD. These low frequency phonons travel ballistically in the out-of-plane direction (perpendicular to the surface of the nanomembrane), if their mfp is comparable to the thickness of the nanomembrane. Therefore, phonons travel much faster than via normal diffusion but slower than in pure ballistic transport. This leads to a sharp onset in the MALDI spectrum. Such phonon transport is typically referred to as quasi-diffusive.20−23 The exponential decay followed by the sharp onset mainly stems from the much slower process of the in-plane heat diffusion. Exactly, this behavior is shown in the inset of Figure 2a, where a magnified view of the peaks is given (see Supporting Information Figure S2 for the full spectrum in time scale). As seen, the sharp onset with a rise time of 350 ns is followed by an exponential decay with a decay constant of 3 μs. The histogram of the obtained MALDI-TOF spectrum for IgG is plotted in Figure 2b and shows a relative abundance of the IgG ions in the sample under test with the bin size of 1 kDa. The size of the bin is determined from the average mass resolution of 129, which yields Δm of ∼1.1 kDa. The m/z of the two most abundant IgG ions are found to be (146.5 ± 0.5) and (149.5 ± 0.5) kDa. Fifty spectra were gathered for the histogram and each spectrum was obtained by averaging 10 laser shots. To demonstrate high mass detection capabilities of the silicon nanomembrane detector, human immunoglobulin M (IgM, ∼1 MDa, 1.44 pmol/μL, Calbiochem, U.S.A.) in a sinapinic acid was analyzed. The resulting spectrum is shown in Figure 3a. In addition to the IgM peak at the m/z of 987.5 kDa, peaks for fragment ions or neutrals are also observed at the m/z of 621, 623.4, and 696.5 kDa. We underline that our detector has similar detection efficiency for neutrals as it has for charged molecules. The shape of the peaks is consistent with that shown and described in Figure 2. The average resolution of the IgM was found to be 250 with a standard deviation of 48. However, the microheterogeneity of IgM essentially causes a much broader peak width than that shown in Figure 3a. We speculate that this narrow peak width is mainly due because the number of 100 laser shots for the spectrum is not sufficient to accumulate all microheterogeneity of IgM. However, the m/z of

Figure 3. MALDI TOF mass spectrum for IgM obtained using the silicon nanomembrane detector. (a) Mass spectrum for IgM with sinapinic acid. Inset: histogram of IgM, showing the microheterogeity of IgM. (b) The fwhm mass resolution of the detector showing increasing mass resolution as mass increases.

the IgM peak in each spectrum varies from 849 kDa to 1.213 MDa, which exhibit the microheterogeity of the IgM and is shown in inset of Figure 3a as a histogram. Eighteen spectra were gathered for the histogram and each spectrum was obtained by averaging 100 laser shots and the size of the bin is 40 kDa. Figure 3b shows the average resolutions and their standard deviations of the silicon nanomembrane detector in the mass range from bovine serum albumin (BSA, ∼66 kDa) to IgM (∼1 MDa). In addition to intact BSA, IgG, and IgM, we also plot the average resolution of the fragment ions. It was found that the rise time and decay constant are highly independent of both mass and kinetic energy of the ion. Therefore, with the time resolution being fixed for all masses, the mass resolution (m/ Δm) increases with increasing molecular mass. The time resolution is highly dependent on the effective cooling of the silicon nanomembrane via field emission. It is crucial to cool the silicon nanomembrane after ion bombardment before successive ion packets deposit additional thermal energy onto the silicon nanomembrane. Only then the nanomembrane provides the required resolution between ion packets. Cooling through field emission is provided by energy exchange between the emitted and replaced electrons. When the average energy of the emitted electrons is higher than that of the replaced electrons, it cools the cathode-side of the nanomembrane. Such a phenomenon in general can also be referred to as the Nottingham effect.24−26 It has been known that the thermal conductivity of the silicon nanomembrane drops dramatically due to phonon-boundary scattering and the presence of defects.27,28 Therefore, high resolution will not be achieved by only cooling the nanomembrane through conventional inplane heat conduction. For modeling, the cooling via PAFE, we perform numerical calculations of the coupled heat diffusion and field emission processes occurring within the nanomembrane after the high2700

dx.doi.org/10.1021/nl400873m | Nano Lett. 2013, 13, 2698−2703

Nano Letters

Letter

Figure 4. Heat flux and termparature profile of the nanomembrane. (a) Heat flux due to the kinetic energy of the impacting proteins as a function of time (red line) and the resulting field emission current caused by heating of the nanomembrane (black line). (b) The temperature profile of the nanomembrane shows a rising peak due to the heating, corresponding to the peak in the input heat flux at time t = 0.75 μs. (c) The temperature reaches a maximum and then flattens due to the enhanced cooling from the field emission of electrons at time t = 1 μs. (d) Once the temperature falls below the threshold for strong fields through the Schottky process, the heat “slowly” diffuses radially outward from the center and the peak is reduced at t = 5 μs.

tensor as a function of lattice temperature and both the in-plane (along the nanomembrane) and out-of-plane (normal to the surfaces of the nanomembrane) components of the tensor and their respective contributions to thermal transfer within the nanomembrane. Having obtained the thermal conductivity tensor for the nanomembrane, we then proceed to solve the HDE at all points throughout the nanomembrane

energy ion packet bombardment deposits its thermal energy on the backside of the nanomembrane. When electrons escape from the nanomembrane by PAFE, they do carry an energy amount equal to or exceeding that of the distance between the vacuum level and the bottom of the conduction band. Therefore the energy ΔE removed by each electron is given by the difference between the work function and the conduction band level ΔE = 3.5 eV, which allows for significant cooling of the nanomembrane through field emission.29 The total energy removed by field emission is Qout (T) = ΔEJ(T,F)/e, where e is the electron charge and J(T,F) is the temperature and field-dependent field emission current described in eq 1. Because of the field emission current, and consequently the heat loss due to cooling by the field emitted electrons, being a strong function of temperature, we must also solve the heat diffusion equation (HDE) of the nanomembrane together with the combined effects of heating from the ion impact and the cooling due to field emission. First, we obtain the lattice thermal conductivity of the nanomembrane by solving the phonon Boltzmann transport equation in the steady-state relaxation time approximation. We consider the full phonon dispersion and all the relevant phonon scattering mechanisms, including boundary, phonon−phonon, isotope, and defect scattering.30 We obtain the thermal conductivity

ρCp

T = ∇(k(T ) ·∇T ) + Q in(t ) − Q out(T ) t

(2)

with the heating term Qin(t) representing the heating effect of the kinetic energy deposited by the ions onto the backside of the nanomembrane and the heat loss and the cooling term Qout(T) representing the energy loss due to the electrons escaping the front-side of the nanomembrane through field emission. The prefactor ρ is the density of the material and Cp is the heat capacity (at constant pressure) of crystalline silicon. We use the standard finite-difference method to discretize the Laplacian operator and the variable order adaptive method to advance the solution in time. The input term Qin(t) is a function of position on the surface of the nanomembrane as well as time. We assume that the heating effect of the ion 2701

dx.doi.org/10.1021/nl400873m | Nano Lett. 2013, 13, 2698−2703

Nano Letters

Letter

stress that the total area of the nanomembrane detector used here is about 200 times smaller than that of a typical MCP detector with a 2 in. diameter. Because the size of the silicon nanomembrane is scalable using conventional CMOS-compatible fabrication processes, the overall sensitivity can be improved further by increasing the total area of the silicon nanomembrane. The resolution of the detector can also be improved further by enhancing the in-plane heat diffusion via applying materials such as diamond with a larger thermal conductivity. On the other hand, the cooling by field emission can be increased with an adjustement of the work function. This is also attainable using diamond as the nanomembrane material or even a monolayer of graphene.

packet is represented by a Gaussian function in both time and space with a standard deviation (spread) in space of 1 mm around the center of the membrane and a standard deviation in time of 100 ns. This narrow time pulse (shown in Figure 4a, red line) delivers a large quantity of energy to the nanomembrane in a short duration of time, causing a rapid spike in temperature, which quickly reaches the opposite side of the nanomembrane. Phonon velocities in silicon are 5 × 103 m/s for transverse and 9 × 103 m/s for longitudinal phonon modes, which means that the lattice waves excited by ion bombardment carry the thermal (vibrational) energy across the thin (L = 180 nm) membrane in some 10 ps, several orders of magnitude faster than even the duration of the short pulse of impact. Therefore, the heat transfer process across the nanomembrane in the outof-plane direction is very rapid because the phonon motion is indeed quasi-diffusive until the phonons reach the opposite surface of the nanomembrane where they scatter diffusively from the atomic roughness, thereby dispersing their energy and causing local heating and elevating the lattice temperature in the nanomembrane as shown in Figure 4a (red line). Subsequently, the rising temperature first causes an increase in field emission (indicated in Figure 4a, black line), which results in increased cooling in the center of the nanomembrane, thereby dulling the sharp peak, as shown comparing Figure 4b (t = 0.75 μs) and 4c (t = 1 μs). Since the cooling term Qout(T) due to field emission is a very strong function of the local temperature at any point, the temperature peak quickly saturates. Next the diffusive heat conduction in the plane of the nanomembrane begins to take over and spreads the heat away from the area of impact, thereby lowering the lattice temperature and slowly shutting off the field emission process, as shown in Figure 4d. The net effect of this chain of events is that the measured field emission rises quickly (trise ≈ 350 ns) as heat is transferred across the nanomembrane and electrons are emitted, followed by a slower decay (tdecay ≈ 3 μs) due to heat conduction laterally through the nanomembrane. The rising slope is limited by the spread in time of the incoming heating term Qin(t) and the time it takes to accumulate all the deposited energy to the point where field emission begins. On the other hand, the trailing slope is limited by the much slower process of lateral heat diffusion in-plane of the nanomembrane. The inplane thermal diffusivity, given by the ratio of thermal conductivity and volumetric heat capacity, is also smaller in the nanomembrane than in bulk silicon due to frequent boundary scattering of phonons from the rough walls of the nanomembrane. This scattering process limits the phonon mfp to equal the thickness of the nanomembrane (Λp ≈ L), which in turn means that the thermal diffusivity is limited to about 8 × 10−5 m2/s at room temperature. This value leads to a radial heat diffusion velocity of 445 m/s in the L = 180 nm nanomembrane. Consequently, heat takes 2.25 μs to reach from the center to the edges of the (2 × 2) mm 2 nanomembrane, which gives a qualitative explanation of the slower decay time constant in both our measurements and simulations, which agree very well each other. In summary, we have shown how a silicon nanomembrane can be employed as a mass spectrometer with exceptional mass range and sensitivity for ultrahigh protein masses. The nanomembrane detector also operates at acceleration voltages of only 9 kV for the protein masses up to 39 212 Da (aldolase), enabling application of the detector in other types of mass spectrometers such as ESI-TOF. Finally, it is important to

■

ASSOCIATED CONTENT

S Supporting Information *

Fabrication of the silicon nanomembranes, configurations of the detector, sample preparations, and high mass resolution of IgG, mass spectra in time scale. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare the following competing financial interest(s): RHB has a stake in Prospero Biosciences LLC, a Madison based venture, pursuing the development of detectors for mass spectrometry applications.

■

ACKNOWLEDGMENTS This work was supported in part by the Air Force (AFOSR) through MURI′08: FA9559-08-1-0337, the Wisconsin Alumni Research Foundation (WARF) through an accelerator program grant, the Kyungpook National University Research funding in 2012, and National Science Foundation (NSF) CI TraCS fellowship.

■

REFERENCES

(1) Wiley, W. C.; Mclaren, I. H. Rev. Sci. Instrum. 1955, 26, 1150− 1157. (2) Tanaka, K.; Waki, H.; Ido, Y.; Akita, S.; Yoshida, Y.; Yoshida, T.; Matsuo, T. Rapid Commun. Mass Spectrom. 1988, 2, 151−153. (3) Karas, M.; Hillenkamp, F. Anal. Chem. 1988, 60, 2299−2301. (4) Chen, X.; Westphall, M. S.; Smith, L. M. Anal. Chem. 2003, 75, 5944−5952. (5) Twerenbold, D.; Vuilleumier, J.-L.; Gerber, D.; Tadsen, A.; van den Brandt, B.; Gillevet, P. M. Appl. Phys. Lett. 1996, 68, 3503−3505. (6) Frank, M.; Mears, C. A.; Labov, S. E.; Benner, W. H. Rapid Commun. Mass Spectrom. 1996, 10, 1946−1950. (7) Hilton, G. C.; Martinis, J. M.; Wollman, D. A.; Irwin, K. D.; Dulcie, L. L.; Gerber, D.; Gillevet, P. M.; Twerenbold, D. Nature 1998, 391, 672−675. (8) Wenzel, R. J.; Matter, U.; Schultheis, L.; Zenobi, R. Anal. Chem. 2005, 77, 4329−4337. (9) Frank, M.; Labov, S. E.; Westmacott, G.; Benner, W. H. Mass Spectrom. Rev. 1999, 18, 155−186. (10) Twerenbold, D. Rep. Prog. Phys. 1996, 59, 349−426. (11) Booth, N. E.; Goldie, D. J. Supercond. Sci. Technol. 1996, 9, 493− 516. (12) Booth, N. E.; Cabrera, B.; Fiorini, E. Annu. Rev. Nucl. Part. Sci. 1996, 46, 471−532. (13) Fowler, R. H.; Nordheim, L. Proc. R. Soc. London, Ser. A 1928, 119, 173−181.

2702

dx.doi.org/10.1021/nl400873m | Nano Lett. 2013, 13, 2698−2703

Nano Letters

Letter

(14) Seitz, F. Modern Theory of Solids; McGraw-Hill Book Company: New York, 1940; pp 161−168. (15) Mead, C. A. J. Appl. Phys. 1961, 32, 646−652. (16) Murphy, E. L.; Good, H. R. Phys. Rev. 1956, 102, 1464−1473. (17) Park, J.; Qin, H.; Scalf, M.; Hilger, R. T.; Westphall, M. S.; Smith, L. M.; Blick, R. H. Nano Lett. 2011, 11, 3681−3684. (18) Tamura, S. Phys. Rev. B 1983, 27, 858−866. (19) Tamura, S. Phys. Rev. B 1985, 31, 2574−2577. (20) Siemens, M. E.; Li, Q.; Yang, R.; Nelson, K. A.; Anderson, E. H.; Murnane, M. M.; Kapteyn, H. C. Nat. Mater. 2010, 9, 26−30. (21) Bron, W. E.; Levinson, Y. B.; O’Conno, J. M. Phys. Rev. Lett. 1982, 49, 209−211. (22) Kazakovtsev, D. V.; Levinson, Y. B. Phys. Status Solidi B 1979, 96, 117−127. (23) Msall, M. E.; Tamura, S.; Esipov, S. E.; Wolfe, J. P. Phys. Rev. Lett. 1993, 70, 3463−3466. (24) Richardson, O. A. Philos. Trans. R. Soc. London, Ser. A 1903, 201, 497−549. (25) Nottingham, W. B. Phys. Rev. 1941, 59, 906−907. (26) Charbonnier, F. M.; Strayer, R. W.; Swanson, L. W.; Martin, E. E. Phys. Rev. Lett. 1964, 13, 397−401. (27) Asheghi, M.; Touzelbaev, M. N.; Goodson, K. E.; Leung, Y. K.; Wong, S. S. J. Heat Transfer 1998, 120, 30−36. (28) Liu, W.; Sheghi, M. J. Appl. Phys. 2005, 98, 123523. (29) Chung, M. S.; Hong, S. C.; Mayer, A.; Cutler, P. H.; Weiss, B. L.; Miskovsky, N. M. Appl. Phys. Lett. 2008, 92, 083505. (30) Aksamija, Z.; Knezevic, I. Phys. Rev. B 2009, 82, 045319.

2703

dx.doi.org/10.1021/nl400873m | Nano Lett. 2013, 13, 2698−2703