Optimizing the interfacial thermal conductance at gold-alkane junctions from 'First Principles'
AAPS/123-QED
Optimizing the interfacial thermal conductance at gold-alkanejunctions from ‘First Principles’
Jingjie Zhang, ∗ Carlos A. Polanco, and Avik W. Ghosh
1, 2, † Department of Electrical and Computer Engineering,University of Virginia, Charlottesville,VA-22904. Department of Physics, University of Virginia, Charlottesville,VA-22904. (Dated: November 21, 2018)
Abstract
We theoretically explore the influence of end-group chemistry (bond stiffness and mass) on theinterfacial thermal conductance at a gold-alkane interface. We accomplish this using the Non-Equilibrium Green’s Function (NEGF) coupled with first principle parameters in Density Func-tional Theory (DFT) within the harmonic approximation. Our results indicate that the interfacialthermal conductance is not a monotonic function of either chemical parameters, but instead max-imizes at an optimal set of mass and bonding strength. This maximum is a result of the interplaybetween the overlap in local density of states of the device and that in the contacts, as well asthe phonon group velocity. We also demonstrate the intrinsic relationship between the DiffusiveMismatch Model (DMM) and the properties from NEGF, and provide an approach to get DMMfrom first principles NEGF. By comparing the NEGF based DMM conductance and range of con-ductance while altering the mass and bonding strength, we show that DMM provides an upperbound for elastic transport in this dimensionally mismatched system. We thus have a prescriptionto enhance the thermal conductance of systems at low temperatures or at low dimensions whereinelastic scattering is considerably suppressed. ∗ [email protected] † [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec . INTRODUCTION Hybrid junctions between inorganic components and organic molecules have been of greatinterest due to their potential applications in electronic devices [1–3], thermoelectrics [4, 5],solar cells [6, 7] and optoelectronics [8]. In recent years, the hybrid junctions, particularlythe self-assembled monolayers (SAMs) have been intensively explored for thermal propertiesbecause of the discovery that the interfacial thermal conductance can be enhanced by morethan a factor of four at room temperature simply by embedding the commonly assumedthermally insulating polymers (alkane chains) at the metal/dielectric interface [9, 10]. Thisenhancement can even be tuned over one order of magnitude by carefully varying the inter-facial properties (bond stiffness and mass) with end-group chemistry [9].The observations of the influence of the interfacial properties on the interfacial thermalconductance diverge in the existing literature. One set of articles [11–13] reports that theconductance increases monotonically with the bonding strength. This observation is com-monly predicted by molecular dynamics simulations of interfaces between solids. A contraryviewpoint [14–18] is arrived at by lattice dynamics and Green’s Function on one dimensionalharmonic chains, showing that a maximum conductance exists when the interface propertieshelp match the impedance from two sides. This maximum conductance can be obtainedwith an optimal interfacial bonding (the harmonic mean of the bonding from two sides)and the favored connecting mass (the geometric mean of the masses from two sides) [19]like scattering treatment would be consistent with this viewpoint, since the transmissiondepends quadratically on the mass deviation, producing a sweet spot along the transmissiongraph.The discrepancy can arise from two sources. The first is the presence of additional in-elastic channels created by phonon-phonon interactions across interfaces. Note that thetheoretical observations showing a monotonically increasing conductance with interfacialbonding strength involve molecular dynamic simulations, where anharmonicity plays a sig-nificant role [11, 12, 20]. On the contrary, the maximum conductance arises in the absenceof phonon-phonon interaction, relevant at low temperatures and low dimensions [14–17]. Aseparate origin for the discrepancy is a dimensionality mismatch across the interface. Amonotonic increase is found in 3D-3D interfaces (e.g. thin films) where the system can bedecoupled into separate 1D chains for each transverse momentum. In contrast, 3D-1D in-2erfaces such as molecular wires on substrates end up coupling the 1D modes strongly. Theinterfacial bonding strength needed to achieve maximum conductance will be unrealisticallylarge in the former, 3D-3D case.Recent research shows that anharmonicity has limited influence on the metal-SAM sys-tems, which are dominated by elastic transport of phonons across the metal-alkane junctions[21–25]. Phonon interference in the metal-SAM systems was predicted both theoretically[21, 23] and experimentally [24], demonstrating that phonons across the chains encounteronly weak phonon-phonon interaction. S. Majumdar [25] recently reported experimentallythat the interfacial thermal conductance of this system decreases with the increased mis-match in frequency spectra between the two metal contacts, in fact behaving opposite tomolecular dynamics simulations. This observation further validates elastic phonon transportalong the chains.Motivated by the tuning capability of surface chemistry on the metal-alkane interfacialthermal conductance and the fact that phonon-phonon interaction in this system is trivial,we use ab-initio Non-Equilibrium Green’s Function(NEGF) method under harmonic approx-imation to explore the influence of the bond stiffness and end-group mass on the interfacialthermal conductance of the Au-alkane junctions. We show that the maximum conductancecan be obtained at a set of optimal interfacial properties. This result clarifies that in a di-mension mismatched heterojunction, even in the case that bulk contact serves as a reservoirof channels, matched impedance still improves the conductance. We generalize our resultswith a simple one degree of freedom 3D-1D model and show that the conductance can betracked by the density of states(DOS), but is also influenced by the phonon group velocity.The rest of the paper is organized as follows: First, we briefly discuss the structures ofAu-alkane surfaces with both thiol and amine as end groups. Then we describe the waythe bond stiffness is varied in this study. After that we introduce the first principle NEGFmethod with the ”scooping” technique for the dimensionality transition followed by a shortdecription of NEGF based DMM method. Finally we discuss and analyze our first principleinterfacial thermal conductance results and generalize them by using a simple 3D-1D model.3
I. MODEL AND SIMULATION METHODA. Au/alkane surface structure
We model the Au-alkane surface using the √ × √ R o [26, 27] unit cell. This structurecorresponds to the highest density possible of alkane chains assembled on Au surface, whichalso leads to the highly ordered chains [27]. Fig. 1 shows the top view of the absorptionposition of alkanes on Au surface and the stacked view of the junctions. We use the hcpposition as the adsorption position in this simulation (Fig. 1(b)). The alkane chains beforegeometry optimization are set to tilt along the hcp-fcc dash lines in Fig. 1(b), with tiltangles of 10 o , 20 o and 30 o . This geometry is very close to the optimized ones reported inthe existing literature (20 o ∼ o between the tilted alkanes and the normal to the goldsurface). During the first principle simulations, the alkane length is set to be 6 C atoms inthe chain with the end group as thiol or amine .We also considered the reconstruction ofthe gold surface[28–31]. In this case, a gold adatom is initially centered on top of the hcpposition with the alkane end-group connected to it. The whole system is then relaxed duringthe optimization process. Both the clean surface and the adatom surface are simulated asshown in Fig. 1(c).The geometry optimization and the force constants are calculated using the first principlespackage Quantum Espresso [32]. Ultrasoft pseudopotentials with Local Density Approxima-tion exchange-correlation function are used in the simulation. The cutoff energy for wavefunctions is calibrated for convergence with a minimum of 35 Ry, while the cutoff energyfor charge density is set to be 300 Ry. The threshold for the convergence of ground stateenergy is set to be 1 . × − Ry.Before the geometry relaxation of the alkane chains assembled on the gold surface, weperformed the structure optimization of bulk gold and infinite polyethylene. An 8 × × × × × ×
12 grid is used in both the groundstate calculation and DFPT in which the chain orientation is along the z direction. Thedistance between chains is set to be fairly large (10˚A) to avoid interactions between them.4 igure 1. (a) Stacking order of Au slab in (111) direction, the color difference represents the goldatoms in difference layers; (b) top view of the gold (111) surface, the black lines highlight thesimulation supercell; (c) schematic representation of the systems being considered.
The optimized lattice constant for gold is 4.03 ˚A and the optimized length for one C H unit is 2.5215 ˚A. The dispersions using calculated force constants are shown in Fig. 2(a) andFig. 2(b), along with the existing experimental data in the literature [34–37].The distance between the radicals and the gold surface before relaxation is set to be 2˚A.A supercell contains a √ × √ R o unit cell and four (111) Au layers is used as the model.20˚A vacuum is set between the adsorbed alkane chains and the neighboring gold slab. A3 × × . × − Ry/˚A.After the calculation of the lattice dynamic matrix by DFPT, a Fourier transformation ofthe dynamic matrix is performed to obtain the real space force constant parameters. The qgrid in DFPT calculations corresponds to the atomic interaction range. Up to 2nd neighbor-ing atomic interactions are calculated for gold; up to 7th neighboring unit cell interactions5 [rad/s] × N u m b e r o f m od e s × Au(CH ) n DMM
T[K] h k [ M W m - K - ] G X K G L ω [r a d / s ] × qa ω [r a d / s ] × G [ M W m - K - ] (a) (b) (c) (d) Figure 2. (a)
Dispersion of Au compared with experimental data; (b)
Dispersion of Polyethylenecompared with experimental data; (c)
The number of modes of Au and polyethylene. The insetfigure shows the MT of DMM taking the harmonic mean of modes from each side; (d)
Theconductance of Au-SAMs junction with NEGF-based DMM. are calculated for polyethylene. Although the 3 × × B. NEGF and Landauer Formula
In the Landauer description of phonon transport, the interfacial thermal conductance canbe written as: G = (cid:90) ∞ (cid:126) ω π M ¯ T ∂N∂T dw (1)where M is the number of modes and ¯ T is the mode-averaged transmission coefficient. M ¯ T can be calculated by the NEGF approach with the simulated system divided into leftcontact, right contact and a central device, as shown in Fig. 3: M ¯ T = T race [Γ l G d Γ r G † d ] (2)where Γ l and Γ r are the broadening matrices for the left and right contact, and G d is theretarded Green’s Function for the device: G d = ( M d w − K d − Σ l − Σ r ) − (3)6n the above equation, M d and K d are the mass matrix and the force constant matrixfor the device respectively, while Σ l and Σ r are the self-energies for the contacts whoseanti-Hermitian parts give us the broadening matrices Γ l,r . Figure 3. Schematic diagram of left, device and right regions of the alkane-gold junction in theNEGF simulation
In this dimension mismatched system, we invoke an additional step that we call ”scoop-ing” [38]. As shown in Fig. 3, the device contains 3 gold atoms around the hcp position(in the case of reconstructed gold surface with adatom, the scooped area contains 4 goldatoms with the adatom gold atom sitting on top of the triangle) in the gold surface layer,the end-group and the end C H unit of the alkane. The left contact is the semi-infinitebulk gold with the adsorption surface scooped out of it, while the right contact continuesas the semi-infinite alkane chain. Based on the optimized tilt angle, we rotate the forceconstants of alkanes to minimize the mismatch at the right contact. The self-energy of theright contact is then evaluated as: Σ r = K dr g r K rd (4)where g r is the surface green’s function of the right contact. For the left contact with thescooped area, the self-energy can be calculated by:Σ l = M sp w − K sp − (cid:101) g − (5)with M sp being the mass matrix for the scooped area, and K sp the renormalized force7onstants following the acoustic sum rule(ASR) for the bare gold surface, with (cid:101) g the baresurface green’s function of the scooped area. C. DMM from ab-initio NEGF
The Diffusive mismatch model (DMM) is commonly used to predict the thermal boundaryconductance. In order to set a reference conductance for Au-SAM junctions, we calculatethe DMM conductance from the first principle parameters. Instead of using the frequencydependent density of states and velocity from the full non-equilibrium lattice dynamics, wedevelope a quick estimate using DMM-NEGF, using variables calculated from a Green’sfunction with ab-initio parameters.We start with the description of heat current across the interface in Landauer formula asin Eq. 1: q → = (cid:90) ∞ ( (cid:126) ω π M ¯ T → )∆ N dw (6) q → = (cid:90) ∞ ( (cid:126) ω π M ¯ T → )∆ N dw (7)The DMM equations arise under the assumption of detailed balance[39, 40], the heat sothat the current from material 1 to material 2 is equal to the current from material 2 tomaterial 1: q → = q → ⇒ M ¯ T → = M ¯ T → (8)Diffusive process of phonons across the interface assumes that the phonons random-ize their phases at one-shot at the interface, consequently the transmission probability ofphonons from material 1 to material 2 should equal to the reflection probability of phononsfrom material 2 to material 1: ¯ T → = 1 − ¯ T → (9)Combining the Eq. 8 and Eq. 9, we get: M ¯ T DMM = M ¯ T → = M ¯ T → = M M M + M (10) M and M are the number of modes of the bulk materials at each side of the interfaces.We can get M and M similarly in Green’s function by defining the contacts and junction8s the same materials, using M ¯ T = T race [Γ l G Γ r G † ]. In the bulk materials, for each mode,transmission probability is 1, so M ¯ T is just the number of modes.In a Green’s function description, the number of modes is proportional to AD ( ω ) v ( w ), where A is the transverse area of the simulated junction unit cell, D ( w ) and v ( w ) arethe frequency dependent density of states and velocity respectively. If we write the parallelcombination of modes in terms of density of states and velocity, we can recover the generalexpression often used to describe the DMM from lattice dynamics. For the Au-SAM system,we need to consider the modes of Au and modes of SAMs in the same area. The DMMconductance can be obtained from the combined mode density in parallel: G DMM = (cid:90) ∞ (cid:126) ω π ( M /A )( M /A )( M /A ) + ( M /A ) ∂N∂T dw (11)In our simulations, we use a compact Au-SAM density with the surface geometry as √ × √ R o . Each alkane chain connects with 3 Au atoms in an area of 2 . × − m .The simulated number of modes for Au and alkane chain in that area are shown in Fig. 2(c),with the inset graph showing the M ¯ T DMM . The number of modes for the Au with thatspecific density is 3 times as the number of modes of Au in primitive unit cell in (111)direction. The M ¯ T DMM equals the number of modes of materials at two sides in parallel,so it should set the upper limit reference if the number of modes at two sides are vastlydifferent. In the case where the number of modes at two sides are similar, DMM sets up thelower limit conductance reference only if the interface is a well matched interface withoutany roughness and disorder.In the Au-SAM system, the dimensionality and property differ at the the interface whenonly gold atoms are connected to the sulfur atom. Even though the number of modes of Auand alkane chain in the same area are similar to each other, the DMM conductance still setsup the upper bound to the interfacial conductance, as shown in Fig. 4.
III. INFLUENCE OF BONDING AND END GROUP MASSA. Comparison of different end groups
Fig. 5 shows the force constants for different end groups obtained from first principlescalculations, and the corresponding interfacial thermal conductance extracted from NEGF.9 emperature [K] G [ M W m - K - ] Available range of tuningDMMOptimized structure
Figure 4. Tunable range of interfacial thermal conductance by varying the interface parameterscompares to DMM conductance and the first principle optimized one.
The calculated trend of interfacial thermal conductances of different end-groups is consis-tent with the experimental data from Losego et al. [10]. The Au-thiol-alkane junctionshave a larger interfacial bonding strength compared to Au-amine-alkane junctions, and acorrespondingly larger interfacial thermal conductance. The interfacial thermal conductancefollows the bonding strength in this case. However this is not conclusive because the bondingstrengths are not large enough to attain maximum conductance.The Au-thiol-alkane junction and Au-amine-alkane junction show different trends relatingto the roughness of the gold surface. In the Au-amine-alkane junction, the reconstructedsurface (adatom case) shows a larger conductance than that of the clean surface junction. Inthe Au-thiol-alkane junction, the interfacial thermal conductance is larger for the clean goldsurface. This discrepancy can be explained by analyzing the bond strength. The strengthof all the bonds in all directions for Au-amine-alkane junction with adatom is larger thanthat of Au-amine-junction without adatom, hence the conductance is larger in the formercase. In contrast, the reconstructed Au-alkanethiol junction has a bond only along thez direction is much larger than the clean surface Au case, which limits the coupling oftransverse vibrational modes in gold to those within the alkane chain.We note that the experimentally reported interfacial thermal conductance of a single Au-thiol-alkane junction is around 115 MWm − K − [25] and 220 MWm − K − [41], while ourresults are around 110 MWm − K − to 210 MWm − K − , depending on details of the gold10 [ N / m ] K xxKyyKzz
SHfcc SHadatom NH2fcc NH2adatom G [ M W m - K - ] T=300KT=100K
NH2adatomNH2fccSHadatomSHfcc
Figure 5. The first principle NEGF result of the interfacial thermal conductance for the Au-thiol-alkane junctions and the Au-amine-alkane junctions. (a)
The bonding strength of different endgroups on different adhesion sites. (b)
The corresponding interfacial thermal conductance. Theerror bars show the range of conductance when the initial tilting angle is varied between 10 o and30 o . surface reconstruction. The calculated results are close to the previous experimental work,especially that of the clean gold surface compared to Wang et al. for the same system [41]. B. Role of interfacial parameters
We use two different mechanisms to vary the interfacial bonding strength: varying theadsorption distance, or directly varying the bonding energy λ relative to the Au-alkane thioljunction. In the case of varying adsorption distance, we assume that the alkane chainsare normal to the gold surface, with the S atom centered on top the equilateral triangleformed by 3 gold atoms. In this step, the structure is not optimized during the groundstate or during the lattice dynamic matrix calculations. The force constants are calculatedby the small displacement method [42–44] by only moving the atoms around the interfaces.Alternately, we directly vary the interfacial bonding strength between the gold and the end-group relative to its calculated bond strength at an adsorption distance of 2˚A by a ratio of λ . For adjusting the end-group mass, we assume once again that the bond stiffness is fixedat its 2˚A value, and then change the corresponding mass matrix of the end-group.In order to focus on the influence of bonding strength and mass of the end-group attachedto gold, we assume there is no impedance mismatch between the alkane end C H unit andthe thiol end-group, and set their bonding strength equal to the one between two C H units. This would be satisfied by enforcing the acoustic sum rule for the Sulfur atom.11 istance [Å] G [ M W m - K - ] K A u - S [ N / m ] GK Au-S, xx K Au-S, yy K Au-S, zz
Figure 6. The stiffness of the bonding varied with adhesion distance, and the corresponding thermalconductance of Au-SAMs junction. G [ M W m - K - ] End group mass [a.m.u.] G [ M W m - K - ] Figure 7. Variation of the thermal conductance with directly changing (a) the stiffness of thebonding and (b) the mass of the adhesion species.
Fig. 6 shows the variation of bond strength K and thermal conductance G with varyingadsorption distance between 1.6˚ A and 2.2˚ A . The interfacial bond strength shows an inverseproportionality to the adsorption distance. Curiously, the corresponding interfacial thermalconductance reaches a maximum at a distance 1.9˚A, which is close to the relaxed distance foralkanethiols self-assembled on gold. The existence of a maximum is independently validatedwhen we directly vary the bond strength, as shown in Fig. 7(a). A maximum conductanceis seen to arise when the bond strength is around 1.3 times the initial 2˚ A distance value forthiol. When the bond strength continues to increase, the interfacial thermal conductancereduces and ultimately seems to saturate.A maximum is also observed around 10 a.m.u. when we plot the thermal conductanceagainst the mass of the end group. This mass is lower than the mass of the C atom, indicating12hat the vibrational dynamics of the hydrogen atoms can not be ignored. For an impedancematched to maximize thermal transport across the interfaces, we would normally expect thepreferred bonding mass to lie in between the masses of the two contacts. The maximizingmass also suggests that the prevailing Hautman-Klein model [45] in molecular dynamicsmethod used to simulate the alkane chains, where we neglect hydrogen atoms and add theirmasses instead to the carbon atoms, might not be suitable in the harmonic limit.We can use the NEGF extracted mode count in the DMM formula to directly estimatethe thermal conductance (Fig. 4), as discussed in the section III. The shaded area in thisplot represents the tunable range of the interfacial thermal conductance reached by varyingthe interfacial strength and the end-group atomic mass. The range is predicted by the twoapproaches we described earlier in this section, which as we cautioned ignores the impedancebetween the alkane end C H unit and the thiol anchoring group. Ignoring this contributionimplies that the upper-limit of the tunable range could well be smaller than the shaded areain the harmonic assumption. The solid black line represents the contribution from a cleangold surface optimally bonded to an alkanethiol. Comparing with the tunable range, we cansee that, in the harmonic limit the interfacial thermal conductance can at most be enhancedby less than 30%. This maximum is still less than the conductance that DMM predictsfor this system. Since DMM arises in the limit of complete dephasing, it seems that suchdephasing events must break restrictive symmetry selection rules that would otherwise limitthe bandwidth of the transmitting phonon modes across the two dissimilar materials. Inreality, we expect the answer to be slightly lower than DMM, because true dephasing eventsretain partial memory, and also tends to reduce the average transmission per mode. C. 3D-1D generalized results and discussions
To explore whether our conclusions generalize to dimensional mismatch, we also studieda simplified one degree of freedom 3D-1D interface with only one degree of mechanicalfreedom (Fig. 8(a)). In this model, the force constants exist in three dimensions to accountfor the transverse momenta in the 3D system, but the atoms are only allowed to move in thetransport direction. The mass of the atoms on the 3D side is set to that of gold, and the bondstrength is set to match its cutoff frequency. Similarly, the atoms in the 1D chain have themass of carbon while their bond strength matches the cutoff frequency of the alkane chain13coustic branches. The alkane channel is represented by a 1D channel between a 1D contactrepresenting its extension on one side, and the 3D contact representing the Au substrate.
Figure 8. (a)A structural sketch of the simple 3D-1D model; (b)The thermal conductance (in unit[W/K]) map with respect to end group mass and stiffness of the interfacial bonding in the simple3D-1D model.
Fig. 8(b) shows a map of the thermal conductance to visualize its dependence on interfa-cial parameters.The thermal conductance is scaled by the conductance in the classical limit.The figure confirms the existence of a maximum in the interfacial thermal conductance fora set of optimal interfacial parameters. We also see that the conductance is most sensi-tive to mass variations when the bond energy hits its maximizing value. In contrast, bondvariations are critical when the endgroup masses are relatively high.The existence of a maximum conductance with interfacial parameters can be explainedby considering the dynamics of the density of states (DOS). Fig. 9(a), Fig. 9(c), Fig. 9(d)correspond to varying the bond strength with the end group mass set to 50 a.m.u. (uptriangle symbols in Fig. 8(b)), while Fig. 9(b), Fig. 9(d), and Fig. 9(e) correspond to varyingthe end group mass, with the bonding strength set to 35N/m (square symbols in Fig. 8(b)).14 [Trad/s] DO S [ A r b . U n it s ] × -13
3D DOS 1D DOS ω [Trad/s] DO S [ A r b . U n it s ] × -13 ω [Trad/s] T r a n s m i ss i on ω [Trad/s] DO S [ A r b . U n it s ] × -13
25 a.m.u.50 a.m.u.75 a.m.u.100 a.m.u.125 a.m.u.
3D DOS 1D DOS ω [Trad/s] DO S [ A r b . U n it s ] × -13
25 a.m.u.50 a.m.u.75 a.m.u.100 a.m.u.125 a.m.u. ω [Trad/s] T r a n s m i ss i on
25 a.m.u.50 a.m.u.75 a.m.u.100 a.m.u.125 a.m.u. (a) (b) (c) (d) (e) (f)
Figure 9. Density of states of the 3D contact, 1D contact and the device with different bondingstrength (a) and end group mass (b) as shown in Fig. 8(b) with square symbols and up trianglesymbols respectively. The shadowed areas in (c) and (d) shows the averaged overlap densityof states for the 3D and 1D contacts, dictating an density of states window for the DOS of thejunctions. (e) and (f ) shows the corresponding transmission for each set of interfacial parameters.
Let us first look at varying bonding strength. Fig. 9(a) shows the DOS of the 3D and 1Dcontacts and the average local density of states (LDOS) of the device with various interfacialparameters. As the bonding strength between gold and end-group atom increases, the LDOSshifts towards the higher frequency states spanned by the 1D contact. This can be explainedby the acoustic sum rule that once the bond strength increases, the on-site energy on theend-group atom will also increase, resulting in a up-shifted spectrum. These high frequencystates cannot be used as transport states within a harmonic assumption, so that elastictransport cannot arise above the 3D cut-off frequency, in this case 30 Trad/s.15he shaded area in Fig. 9(c) shows the overlap of DOS (ODOS) between the contacts :
ODOS ( ω ) = D D ( ω ) D D ( ω ) (cid:82) ∞ D D ( ω ) D D ( ω ) dω (12)where D D ( ω ) and D D ( ω ) are the DOS of 3D and 1D contact respectively. Within aharmonic assumption, the ODOS of contacts should serve as a weighted window for thechannel states. As the carbon-carbon bonding strength increases, one of the LDOS peaksgets pushed out of the 3D band, resulting in an eventual decrease in transmission andits corresponding non-monotonicity with a preferred sweet spot. Comparing the DOS inFig. 9(c) and transmission in Fig. 9(e), we see that the transmission approximately followsthe LDOS of the devices within the weighted window. As described earlier, the differencebetween LDOS and transmission is the phonon group velocity. High frequency phonons havelower velocity, which explains the suppressed transmission of the LDOS peaks around thecutoff frequency. Weaker bonding strength also create lower phonon velocity, so that thelowest transmission arises for the weakest bonding strengths.The transmission dependence on end group mass has an origin similar to above, seen bycomparing the transmission in Fig. 9(f) and the DOS in Fig. 9(d). Within the weightedwindow of the ODOS, the DOS for the lighter end group atom is much smaller, makings itstransmission correspondingly weak. On the other hand, heavier end group mass reduces thephonon group velocity especially at higher frequency, reducing their corresponding trans-missions as well. The end result is once again a maximum in the conductance when varyingthe mass of the end group atom. IV. CONCLUSION
In summary, using First principles phonon bands coupled with NEGF, we studied theinfluence of interfacial chemistry (bond strength and mass of the connector atoms) on the in-terfacial thermal conductance of gold-alkane junctions. Within a harmoonic approximation,we predict the existence of a maximum conductance that is mirrored by a simpler 3D-1Dmodel. We attribute the maximum conductance to the interplay between local density ofstates and phonon group velocity in determining the phonon transmission. To estimate an16pper limit to the conductance, we developed a DMM model based on the number of modesof the materials at each side of the interface extracted from the DFT-NEGF calculations.
ACKNOWLEDGMENTS
J.Z., C.A.P. and A.W.G. are grateful for the support from NSF-CAREER (QMHP1028883)and from NSF-IDR (CBET 1134311). They also acknowledge the School of Engineering andApplied Sciences at University of Virginia that covered the registration fees for a QuantumEspresso workshop. This work used the Extreme Science and Engineering Discovery Envi-ronment (XSEDE)(DMR130123) [46], which is supported by National Science Foundationgrant number ACI-1053575. [1] A. Nitzan and M. A. Ratner, Science , 1384 (2003).[2] C. Joachim, J. K. Gimzewski, and A. Aviram, Nature , 541 (2000).[3] B. Xu, Science , 1221 (2003).[4] P. Reddy, S.-Y. Jang, R. A. Segalman, and A. Majumdar, Science , 1568 (2007).[5] K. Baheti, J. A. Malen, P. Doak, P. Reddy, S.-Y. Jang, T. D. Tilley, A. Majumdar, and R. A.Segalman, Nano Letters , 715 (2008).[6] J. Tang, H. Liu, D. Zhitomirsky, S. Hoogland, X. Wang, M. Furukawa, L. Levina, and E. H.Sargent, Nano Letters , 4889 (2012).[7] A. H. Ip, S. M. Thon, S. Hoogland, O. Voznyy, D. Zhitomirsky, R. Debnath, L. Levina, L. R.Rollny, G. H. Carey, A. Fischer, K. W. Kemp, I. J. Kramer, Z. Ning, A. J. Labelle, K. W.Chou, A. Amassian, and E. H. Sargent, Nat Nano , 577 (2012).[8] M. C. Gather, A. K¨ohnen, and K. Meerholz, Advanced Materials , 233 (2011).[9] P. J. O’Brien, S. Shenogin, J. Liu, P. K. Chow, D. Laurencin, P. H. Mutin, M. Yamaguchi,P. Keblinski, and G. Ramanath, Nat Mater , 118 (2013).[10] M. D. Losego, M. E. Grady, N. R. Sottos, D. G. Cahill, and P. V. Braun, Nat Mater , 502(2012).[11] M. Shen, W. J. Evans, D. Cahill, and P. Keblinski, Phys. Rev. B , 195432 (2011).
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