Extreme Near-Field Heat Transfer Between Gold Surfaces
Takuro Tokunaga, Amun Jarzembski, Takuma Shiga, Keunhan Park, Mathieu Francoeur
EExtreme Near-Field Heat Transfer Between Gold Surfaces
Takuro Tokunaga, Amun Jarzembski, TakumaShiga, Keunhan Park, ∗ and Mathieu Francoeur † Department of Mechanical Engineering, University of Utah,Salt Lake City, Utah 84112, United States. Sandia National Laboratories, Albuquerque, NM 87185, United States. Department of Mechanical Engineering,The University of Tokyo, Bunkyo, Tokyo 113-8656, Japan.
Extreme near-field heat transfer between metallic surfaces is a subject of debateas the state-of-the-art theory and experiments are in disagreement on the energycarriers driving heat transport. In an effort to elucidate the physics of extreme near-field heat transfer between metallic surfaces, this Letter presents a comprehensivemodel combining radiation, acoustic phonon and electron transport across sub-10-nm vacuum gaps. The results obtained for gold surfaces show that in the absenceof bias voltage, acoustic phonon transport is dominant for vacuum gaps smallerthan ∼ a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b contributions of acoustic phonon and electron as a function of the bias voltage. Radiative heat transfer between two surfaces separated by a sub-wavelength vacuum gapcan exceed the far-field blackbody limit by a few orders of magnitude owing to tunneling ofevanescent electromagnetic (EM) waves [1]. Theoretical predictions of near-field radiativeheat transfer based on fluctuational electrodynamics [2] are well-established, and have beenexperimentally validated in various configurations for nanosized vacuum gaps [3–17]. How-ever, fluctuational electrodynamics may not be able to accurately describe heat transfer inthe extreme near-field regime, defined here as sub-10-nm vacuum gap distances, due to itsmacroscopic nature involving local field averaging (i.e., local dielectric function) [18]. In ad-dition, fluctuational electrodynamics solely accounts for EM waves (i.e., radiation), whereasacoustic phonons and electrons may also contribute to heat transport between surfaces sep-arated by single-digit nanometer vacuum gaps prior to contact. A few theoretical workshave investigated acoustic phonon transport across vacuum gaps [18–26], and some haveexplicitly shown the inadequacy of fluctuational electrodynamics for modeling heat transferin the extreme near field [18, 20, 22–26].Experimental measurements of extreme near-field heat transfer are scarce [27–32].Jarzembski et al . [32] measured a thermal conductance exceeding fluctuational electro-dynamics predictions by three orders of magnitude between a silicon tip and a platinumnanoheater separated by sub-10-nm vacuum gaps down to contact. By considering all energycarriers, it was shown quantitatively that heat transfer across vacuum gaps was dominatedby acoustic phonon transport mediated by van der Waals and Coulomb forces. Extremenear-field heat transfer between metallic surfaces have also been measured, but drasticallydifferent results have been reported. Kim et al. [29] measured heat transfer between athermocouple-embedded tip fabricated at the free-end of a stiff cantilever and a suspendedresistive microheater at 305 K. Their measurements between a gold (Au)-coated tip and anAu surface are in good agreement with fluctuational electrodynamics down to a vacuum gapof ∼ et al. [30] reported the measurement of heattransfer between an Au thermocouple-embedded scanning tunneling microscope tip and acryogenically-cooled Au surface, observing heat transfer largely exceeding fluctuational elec-trodynamics for vacuum gaps from 7 nm down to 0.2 nm. However, no comprehensivemodeling was performed to quantitatively support their observation.Subsequently, Cui et al. [31] measured heat transfer between an Au-coated tip and anAu surface, both subjected to various surface-cleaning procedures, for vacuum gaps from 5nm down to a few ˚A. They hypothesized that the large heat transfer reported by Kloppstech et al . [30] may be due to a low apparent potential barrier for electron tunneling mediatedby surface contaminants bridging the tip and the surface prior to bulk contact.A few theoretical works have analyzed extreme near-field heat transfer between Au sur-faces [22, 24–26]. Using a surface perturbation approach, Pendry et al. [22] predicted thatthe heat transfer coefficient due to acoustic phonon exceeds that obtained with fluctuationalelectrodynamics for vacuum gaps smaller than 0.4 nm. Alkurdi et al. [26] compared thecontributions of radiation, phonons and electrons to heat transfer between Au surfaces sepa-rated by vacuum gaps varying from 1.5 nm down to 0.2 nm. Using a three-dimensional (3D)lattice dynamics framework, it was found that acoustic phonon transport exceeds radiationfor all vacuum gaps considered, whereas electron tunneling slightly surpasses the phononcontribution for vacuum gaps smaller than 0.4 nm. In stark contrast, Messina et al. [25]predicted that acoustic phonon transport does not play any role in extreme near-field heattransfer between Au surfaces. Their results suggested that for vacuum gaps smaller than ∼ ∼ G betweena tip and a surface separated by a vacuum gap of thickness d is calculated from local heattransfer coefficients h between two parallel surfaces, modeled as semi-infinite layers, usingthe Derjaguin approximation [3, 33]: G = Z r tip drh ( ˜ d )2 πr (1)where r is the radial direction, r tip is the tip radius, and ˜ d = d + r tip − p r tip2 − r is thevacuum gap distance between the surfaces (see Fig. 1). The heat transfer coefficient in Eq.(1) is the sum of contributions from radiation, phonon, and electron, as detailed hereafter.The heat transfer coefficient due to radiation between two parallel surfaces (media L andR) separated by a vacuum gap of thickness ˜ d (medium 0) is calculated using fluctuational FIG. 1:
The thermal conductance between a tip and a surface separated by a vacuum gap d is calculatedfrom the heat transfer coefficients between two parallel surfaces, modeled as semi-infinite layers, separatedby a vacuum gap ˜ d using the Derjaguin approximation. Top-most atoms are located at the Au-vacuuminterfaces. The interatomic vacuum distance d used for predicting acoustic phonon transport is the sameas the vacuum gap thickness ˜ d used for radiation and electron tunneling calculations. The left (L) and right(R) semi-infinite layers are assumed to be at constant and uniform temperatures T L and T R . electrodynamics to account for the near-field effects [1, 2]: h rad = 1 π ( T L − T R ) Z ∞ dω [Θ ( ω, T L ) − Θ ( ω, T R )] Z ∞ dk ρ k ρ X γ =TE , TM T γ rad ( ω, k ρ ) (2)where ω is the angular frequency, k ρ is the component of the wavevector parallel to aninterface, and Θ( ω, T ) is the mean energy of an EM state calculated as ~ ω/ [exp( ~ ω/k b T ) − γ for propagating ( k ρ ≤ k ) and evanescent( k ρ > k ) EM waves in vacuum are respectively given by: T γ rad , prop ( ω, k ρ ) = (cid:16) − | r γ | (cid:17) (cid:16) − | r γ | (cid:17) (cid:12)(cid:12) − r γ r γ e i Re( k z0 ) ˜ d (cid:12)(cid:12) (3) T γ rad , evan ( ω, k ρ ) = e − k z0 ) ˜ d Im ( r γ ) Im (r γ ) (cid:12)(cid:12) − r γ r γ e − k z0 ) ˜ d (cid:12)(cid:12) (4)where k z0 is the component of the vacuum wavevector perpendicular to an interface, and r γ j ( j = L , R) is the Fresnel reflection coefficient [34]. The frequency-dependent dielectricfunction of Au is calculated using the Drude model provided in Ref. [35].Acoustic phonons can tunnel across vacuum gaps via force interactions. Acoustic phonontransport is modeled using the one-dimensional (1D) atomistic Green’s function (AGF)method [36, 37]. The heat transfer coefficient due to acoustic phonon transport across aninteratomic vacuum distance d for a 1D atomic chain is written as: h ph = 1 A ( T L − T R ) Z ∞ dω ~ ω π T ph ( ω )[ N ( ω, T L ) − N ( ω, T R )] (5)where N = 1 / [exp( ~ ω/k b T ) −
1] is the Bose-Einstein distribution function and A is thecross-sectional area of an atom. The atomic radius of Au is taken as 1.740 ˚A [38]. The top-most atoms are located at the Au-vacuum interfaces [39], such that the interatomic vacuumdistance d is the same as the vacuum gap thickness ˜ d used for radiation and electrontunneling calculations (see Fig. 1). Note that the lattice constant of Au (4.065 ˚A) is used asthe criterion for bulk contact. The minimum distance at which the heat transfer coefficientis calculated is therefore 4.065 ˚A. The phonon transmission function is given by the Caroliformula [40]: T ph ( ω ) = Trace[Γ L G d Γ R G † d ] (6)where the superscript † denotes conjugate transpose, Γ L , R is the escape rate of phonons fromthe device region to the semi-infinite layers, and G d is the Green’s function of the deviceregion. In the present study, the device region encompasses the vacuum gap and five atomsin each of the semi-infinite layers. Increasing the number of atoms in the device region doesnot affect the phonon transmission function. Details about the 1D AGF method have beenprovided in Refs. [32, 36, 37].For the case of two Au surfaces, the short- and long-range interactions across the vacuumgap are respectively described by the Lennard-Jones and Coulomb force models [18, 26]. TheLennard-Jones model accounts for the van der Waals force and overlapping electron cloudrepulsive force, whereas the Coulomb force is effective only when there is a bias voltage[41–43]. The rationale behind the selection of these forces and the force models are providedin Sec. S1 of the Supplemental Material [44].Electrons can contribute to heat transfer across vacuum gaps via tunneling and thermionicemission. Owing to the low temperatures considered ( ∼
300 K), the contribution ofthermionic emission is negligible. The heat transfer coefficient due to electron tunnelingacross a vacuum gap can be written as [45–47]: h el = 1( T L − T R ) Z W max −∞ dE z [ ( E z + k b T L ) N L ( E z , T L ) − ( E z + k b T R ) N R ( E z − eV bias , T R )] T el ( E z ) (7)where e = 1 . × − C is the electron charge, V bias is the bias voltage, E z is the electronenergy perpendicular to the surfaces, and W max is the maximum potential barrier. The term N j ( E z , T j ), denoting the number of electrons at energy level E z per unit area and per unittime, is calculated as [45]: N j ( E z , T j ) = m e k b T j π ~ ln (cid:20) (cid:18) − E z − E F,j k b T j (cid:19)(cid:21) (8)where m e = 9 . × − kg is the electron mass, and E F,j is the Fermi level used as areference for computing the electron energy [45]. The electron transmission function iscalculated using the Wentzel-Kramers-Brillouin (WKB) approximation [48]: T el ( E z ) = exp (cid:20) − √ m e ~ Z z z dz p W ( z ) − E z (cid:21) (9)where W ( z ) is the potential barrier profile in the vacuum gap, while z and z are the rootsof W ( z ) − E z = 0 delimiting the width of the electron tunneling barrier E z . The potentialbarrier profile can be expressed as [45]: W ( z ) = W id ( z ) + W ic ( z ) (10)Note that the space-charge effect is assumed to be fully suppressed for sub-10-nm vacuumgaps [47]. The ideal barrier profile [47] and image-charge perturbation [49] are respectivelycalculated as: W id ( z ) = Φ L − (Φ L − Φ R − eV bias ) (cid:18) z ˜ d (cid:19) (11) W ic ( z ) = e π(cid:15) d (cid:20) −
2Ψ (1) + Ψ (cid:18) z ˜ d (cid:19) + Ψ (cid:18) − z ˜ d (cid:19)(cid:21) (12)where Φ L , R = 5 .
10 eV is the work function of Au [50] and Ψ is the digamma function.The WKB approximation may lead to an overestimation of the electron heat trans-fer coefficient below vacuum gaps of ∼ T el ( E z ) = Trace[Γ L G R Γ R (cid:0) G R (cid:1) † ] , (13)where Γ L , R is the energy broadening matrix, whereas G R is the retarded Green’s function.The heat transfer coefficients between two Au surfaces due to radiation, acoustic phonontransport, and electron tunneling are presented in Fig. 2(a) for the experimental conditionstaken from Kim et al. [29], where T L = 305 K, T R = 300 K, and V bias = 0 V. The Au-vacuum interfaces support surface plasmon polaritons at a high frequency ( ∼ . × rad/s) that cannot be thermally excited at room temperature. As such, the radiation heattransfer coefficient saturates as the vacuum gap decreases [35]. It should be noted thatwhile fluctuational electrodynamics is not expected to be valid for sub-2-nm vacuum gapsowing to non-local effects [35], radiation predictions below 2 nm vacuum gaps are shownthroughout the paper for reference. The accuracy of the 1D AGF approach for predictingacoustic phonon transport between Au surfaces is verified in Fig. 2(a) by comparison againstthe results obtained from a 3D lattice dynamics framework [26]. In the absence of biasvoltage, only the van der Waals and overlapping electron cloud forces, represented by theLennard-Jones model, contribute to acoustic phonon transport. The phonon and electronheat transfer coefficients exceed that of radiation for vacuum gaps smaller than ∼ ∼ ∼ FIG. 2: (a) Heat transfer coefficients due to radiation, phonon, and electron between two Au surfacesseparated by a vacuum gap ˜ d ( T L = 305 K, T R = 300 K, V bias = 0 V). The electron heat transfer coefficientis calculated via the WKB approximation and the NEGF. The phonon heat transfer coefficient predicted viathe 1D AGF method is compared against the 3D lattice dynamics results of Alkurdi et al. [26]. (b) Thermalconductance between a 450-nm-radius tip (300 K) and a surface (305 K), both made of Au, separated bya vacuum gap d . The total conductance is calculated using the electron heat transfer coefficient from theWKB approximation. The predicted conductance is compared against the experimental results of Kim etal. [29]. ∼ ∼ et al. [29]. Vacuum gaps approximately equal to or smaller than 1 nm would have beenrequired to observe a significant enhancement of the conductance due to acoustic phonontransport.Figure 3(a) presents the heat transfer coefficients between two Au surfaces due to ra-diation, acoustic phonon transport, and electron tunneling for the experimental conditionstaken from Kloppstech et al . [30], where T L = 280 K, T R = 120 K, and V bias = 0.6 V. Itshould be noted that an electric field generated by a bias voltage not only enhances electrontunneling but also induces surface charges [42, 43], which are at the origin of the Coulombforce [41]. Since the long-range Coulomb force can drive acoustic phonon transport acrossvacuum gaps [18], the surface charge density under a bias voltage should be determined andtaken into account for the calculations. Jarzembski et al . [32] estimated the surface chargedensity of a biased platinum surface based on their experimental conditions. A surfacecharge density of 8 × − C / m was estimated for a bias voltage of 0.8 V. In the presentwork, a value of 6 × − C / m is used under the assumption that the surface charge densityvaries linearly with the bias voltage [52, 53]. Note that the phonon heat transfer coefficientis calculated with and without bias voltage, and the shaded area in Fig. 3(a) shows itspossible values.Owing to the long-range Coulomb force, the phonon heat transfer coefficient exceeds thatof radiation for vacuum gaps smaller than ∼ FIG. 3: (a) Heat transfer coefficients due to radiation, phonon, and electron between two Au surfacesseparated by a vacuum gap ˜ d ( T L = 280 K, T R = 120 K, V bias = 0.6 V). The electron heat transfer coefficientis calculated via the WKB approximation and the NEGF. The phonon heat transfer coefficient is calculatedboth with and without bias voltage. (b) Thermal conductance between a 30-nm-radius tip (280 K) and asurface (120 K), both made of Au, separated by a vacuum gap d . The total conductance is calculated usingthe electron heat transfer coefficient from the WKB approximation. The predicted conductance is comparedagainst the experimental results of Kloppstech et al. [30] and Messina et al. [25]. V bias = 0.6 V). The results are compared against the experimental dataof Kloppstech et al. [30], in addition to a more recent set of data from the same group(Messina et al. [25]). Note that the most recent experimental data were obtained for surfaceand tip temperatures of respectively 195 K and 295 K. These slight temperature differenceshave no noticeable impact on the heat transfer coefficients (see Sec. S2 of the SupplementalMaterial [44]). The total conductance, which include the radiation, phonon, and electroncontributions, display three distinct regimes. For vacuum gaps larger than 5 nm, radiationtransport via evanescent EM waves drives heat transfer. For vacuum gaps from 5 nm down to ∼ d − . power law. Within that range, the data of Messina et al. [25] vary as d − . , which isclose to the theory. This regime arises due to the Coulomb force induced by the bias voltage.For vacuum gaps smaller than 0.9 nm where electron tunneling dominates heat transport,the total conductance follows a d − . power law. Interestingly, a similar trend is observedin the experimental data of Kloppstech et al. [30] but at much larger vacuum gaps ( d & et al. [25] estimated an apparent potential barrier of 1 eV,which induces an enhancement of the electron heat transfer coefficient. Fig. 4 shows thethermal conductance predicted by considering a simple potential barrier of 1 eV for electrontunneling. As expected, electron tunneling for that case becomes dominant at a largervacuum gap ( ∼ et al. [25], but the trends are different. Electron-mediated heat transportfollows a d − . power law, which is not experimentally observed. In the gap range between1.5 nm and 2 nm where all energy carriers contribute to the conductance, a d − . powerlaw is predicted. This is again close to the d − . power law of Messina et al. [25] whichhowever arises in a slightly different vacuum gap range (1 nm to 1.5 nm). It is concludedthat the measured conductance reported by Messina et al. [25] is not solely due to electrontunneling. Heat transport for that case is likely to be a combination of acoustic phonontransport, enhanced via the bias voltage, and electron tunneling possibly enhanced by thelow apparent potential barrier mediated by contaminants. It has been pointed out that thedata of Kloppstech et al. [30] were obtained at different apparent potential barrier heights[25], which make their interpretation challenging. Yet, the variation of the conductance with5 FIG. 4:
Thermal conductance between a 30-nm-radius tip (280 K) and a surface (120 K), both madeof Au, separated by a vacuum gap d . The electron heat transfer coefficient is calculated with the WKBapproximation using an apparent potential barrier of 1 eV. The phonon heat transfer coefficient is calculatedwith a bias voltage of 0.6 V. The predicted conductance is compared against the experimental results ofMessina et al. [25]. ∼ ∼ ∗ Electronic address: [email protected] † Electronic address: [email protected][1] D. Polder and M. Van Hove, Phys. Rev. B , 3303 (1971).[2] S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 3 Elements of Random Fields (Springer, New York, 1989).[3] E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J. J. Greffet, Nat.Photonics , 514 (2009).[4] S. Shen, A. Narayanaswamy, and G. Chen, Nano Lett. , 2909 (2009).[5] B. Song, Y. Ganjeh, S. Sadat, D. Thompson, A. Fiorino, V. Fern´andez-Hurtado, J. Feist, F. 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Department of Mechanical Engineering,The University of Tokyo, Bunkyo, Tokyo 113-8656, Japan.
S1. LENNARD-JONES AND COULOMB FORCE MODELS
Short- and long-range force interactions are considered for modeling acoustic phonontransport across vacuum gaps. The short-range interactions include the repulsive force dueto overlapping electron clouds and the van der Waals force [1]. The possible long-rangeinteractions are the Maxwell stress of the electric field (capacitance force) [2], the Maxwellstress of the magnetic field [2], and the Coulomb force [3].The short-range force interactions are modeled via the first order derivative of theLennard-Jones potential, U L − J , with respect to the interatomic distance, d [4]: F L − J = ∂U L − J ∂d = − (cid:16) εd (cid:17) (cid:20) (cid:16) σd (cid:17) − (cid:16) σd (cid:17) (cid:21) (1)where ε = 8 . × − J and σ = 2 . × − m [5]. In Eq. (1), the term varying as d − describes the overlapping electron cloud repulsive force [6, 7], whereas the d − term describesthe van der Waals force interactions [6, 8]. a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b For the long-range force interactions, the Maxwell stress of the magnetic field does notaffect low-frequency phonon heat transfer and can thus be ignored [2]. The application of abias voltage induces surface charges that in turn generate capacitance and Coulomb forces[9–11]. The capacitance force induced by a bias voltage between a tip and a surface is givenby [12]: F Capacitance = (cid:15) V bias A tip d (2)where (cid:15) is the permittivity of free space, V bias is the bias voltage, A tip is the tip surface, and d is the vacuum gap thickness. Eq. (2) describes the capacitance force over the entire tipsurface. For acoustic phonon transport calculations based on the atomistic Green’s functionmethod, the force constants between gold (Au) atoms must be calculated. The capacitanceforce between Au atoms is obtained by normalizing Eq. (2) by the tip surface, A tip , and bythen multiplying the resulting expression by the cross-sectional area of an Au atom, A : F Capacitance = (cid:15) V bias Ad (3)where the vacuum gap d has been replaced by the interatomic distance d .The Coulomb force due to surface charges between a tip and a surface is calculated asfollows [12]: F Coulomb = Q s Q t π(cid:15) d (4)where Q s is the surface charge, and Q t is the tip charge defined as Q t = − ( Q s + Q e ). Theinduced capacitive charge is given by Q e = CV bias , where C (= (cid:15) A tip /d ) is the parallel platecapacitance [13]. Following the same procedure as for the capacitance force, the Coulombforce between Au atoms is given by: F Coulomb = Q s Q t π(cid:15) d AA tip (5)where the vacuum gap d has been replaced by the interatomic distance d .The capacitance and Coulomb forces between Au atoms, calculated via Eqs. (3) and (5),as a function of the interatomic vacuum distance, d , are compared in Fig. S1 for a biasvoltage, V bias , of 0.6 V and a surface charge density, Q s /A tip , of 6 × − C / m . Note thatthe tip radius of 30 nm reported in Refs. [14, 15] is considered in the simulations. FIG. S1:
Comparison of the capacitance and Coulomb forces between Au atoms ( V bias = 0.6 V, Q s /A tip =6 × − C / m , 30-nm-radius tip). The Coulomb force exceeds the capacitance force for all interatomic vacuum distances. Assuch, the Coulomb force is the only long-range interactions considered for acoustic phonontransport calculations.The total force driving acoustic phonon transport across vacuum gaps is written as: F total = F L − J + F Coulomb (6)The inputs for atomistic Green’s function calculations are the force constants which areobtained by taking the absolute value of the first order derivative of Eq. (6) with respect to d [16].At the onset of contact, the Coulomb force vanishes due to charge neutralization [17].The reduction of the surface charge, Q s , with respect to the vacuum gap between a tip anda surface has been previously predicted in Refs. [18, 19]. In addition, an experimental efforthas demonstrated a vanishing capacitance between a tip and a substrate with the reductionof the vacuum gap distance [20]. As such, the surface charge should ideally be treated asa function of the interatomic distance [19]. However, since it is challenging to develop agap-dependent surface charge model, Q s is treated here as a constant value that vanishesbelow a cut-off vacuum distance [21]. The cut-off value is approximated as the interatomicvacuum distance for which electron tunneling becomes significant, corresponding to d ≈ S2. IMPACT OF TEMPERATURE ON THE HEAT TRANSFER COEFFICIENT
The thermal conductance between a 30-nm-radius tip and a surface, calculated via theheat transfer coefficients of Fig. 3(a) ( T L = 280 K, T H = 120 K, V bias = 0.6 V) and theDerjaguin approximation, is reported in Fig. 3(b). The predicted conductance is comparedagainst the experimental data of Kloppstech et al. ( T L = 280 K, T H = 120 K, V bias = 0.6V) [14] in addition to those of Messina et al. ( T L = 295 K, T H = 195 K, V bias = 0.6 V)[15]. Fig. S2 shows the heat transfer coefficients between two Au surfaces due to radiation,acoustic phonon transport, and electron tunneling for the experimental conditions of Messina et al. [15]. It is clear by comparing Fig. S2 against Fig. 3(a) that the slight temperaturedifferences have no noticeable impact on the heat transfer coefficients, thus justifying directcomparison of the data of Refs. [14, 15] in Fig. 3(b). FIG. S2:
Heat transfer coefficients due to radiation, phonon, and electron between two Au surfaces separatedby a vacuum gap ˜ d ( T L = 295 K, T R = 195 K , V bias = 0.6 V). The electron heat transfer coefficientis calculated via the Wentzel-Kramers-Brillouin (WKB) approximation and the non-equilibrium Green’sfunction (NEGF) approach. ∗ Electronic address: [email protected] † Electronic address: [email protected][1] S. Sadewasser and M. C. Lux-Steiner, Phys. Rev. Lett. , 266101 (2003).[2] J. B. Pendry, K. Sasihithlu, and R. V. Craster, Phys. Rev. B , 075414 (2016).[3] V. Chiloyan, J. Garg, K. Esfarjani, and G. Chen, Nat. Commun. , 6755 (2015).[4] L.-J. John, Proc. Math. Phys. Eng. Sci. , 463 (1924).[5] N. Yu and A. A. Polycarpou, J. Colloid Interface Sci. , 428 (2004).[6] J. E. Jones, Proc. Math. Phys. Eng. Sci. , 463 (1924).[7] M. Mohebifar, E. R. Johnson, and C. N. Rowley, J. Chem. Theory Comput. , 6146 (2017).[8] C. Li and T. W. Chou, Compos Sci Technol , 1517 (2003).[9] R. G. Horn and D. T. Smith, Science , 362 (1992).[10] S. Heinze, X. Nie, S. Bl¨ugel, and M. Weinert, Chem. Phys. Lett. , 167 (1999).[11] N. D. Lang and W. Kohn, Phys. Rev. B , 3541 (1973).[12] B. D. Terris, J. E. Stern, D. Rugar, and H. J. Mamin, Phys. Rev. Lett. , 2669 (1989).[13] D. El Khoury, R. Arinero, J. C. Laurentie, and J. Castellon, AIP Adv. , (2016).[14] K. Kloppstech, N. K¨onne, S. A. Biehs, A. W. Rodriguez, L. Worbes, D. Hellmann, andA. Kittel, Nat. Commun. , 14475 (2017).[15] R. Messina, S.-A. Biehs, T. Ziehm, A. Kittel, and P. Ben-Abdallah, arXiv:1810.02628v1.[16] Y. Ezzahri and K. Joulain, Phys. Rev. B , 115433 (2014).[17] S. Srisonphan, M. Kim, and H. K. Kim, Sci. Rep. , 3764 (2014).[18] J. Hong, K. Noh, and S. il Park, Phys. Rev. B , 5078 (1998). [19] S. H. Behrens and D. G. Grier, J. Chem. Phys. , 6716 (2001).[20] M. J. Yoo, T. A. Fulton, H. F. Hess, R. L. Willett, L. N. Dunkleberger, R. J. Chichester, L. N.Pfeiffer, and K. W. West, Science , 579 (1997).[21] H. J. Butt, Biophys. J. , 777 (1991).[22] D. Anselmetti, T. Richmond, G. Borer, M. Dreier, M. Bernasconi, and H.-J. Guntherodt, EPL , 297 (1994).[23] L. S. McCarty and G. M. Whitesides, Angew. Chem. Int. Ed.47