Atomistic simulation of phonon heat transport across metallic vacuum nanogaps

The understanding and modeling of heat transport across nanometer and subnanometer gaps, where the distinction between thermal radiation and conduction becomes blurred, remains an open question. In this work, we present a three-dimensional atomistic simulation framework by combining the molecular dynamics (MD) and phonon nonequilibrium Green's function (NEGF) methods. The relaxed atomic configuration and interaction force constants of metallic vacuum nanogaps are generated from MD as inputs into harmonic phonon NEGF. Phonon tunneling across gold-gold and copper-copper nanogaps is quantified, and is shown to be a significant heat transport channel below a gap size of 1 nm. We demonstrate that lattice anharmonicity contributes to within 20%–30% of phonon tunneling depending on gap size, whereas electrostatic interactions turn out to have a weak effect for the small bias voltage typically used in experimental measurements. This work provides detailed information of the heat current spectrum and interprets the recent experimental determination of thermal conductance across gold-gold nanogaps. Our study contributes to deeper insight into heat transport in the extremely near-field regime, as well as hints for future experimental investigation.

Figure 1

Atomistic simulation of phonon heat transport across a metallic nanogap: (a) schematic of the Au-Au nanogap with a gap size $d$ and periodic cross section; (b) Lennard-Jones potential for interaction between Au atoms; (c) energy accumulations at the hot and cold thermostats in MD (molecular dynamics) simulation of the Au-Au gap with $d=3.94\AA$ at 300 K; the discrete points represent the MD data while the solid lines are linear fitting.

Figure 2

Thermal conductance versus gap size at room temperature (300 K): (a) Au-Au gap, (b) Cu-Cu gap. The blue squares denote the direct NEMD (nonequilibrium molecular dynamics) result; the magenta diamonds and the black circles represent the phonon NEGF (nonequilibrium Green's function) results with and without tethering of the surface atomic layers adjacent to the nanogap. The dashed line represents a power-law scaling of $d^{-9}$. The conduction limit (Cond. Limit) corresponds to the contact situation (with a gap size of half the lattice constant).

Figure 3

Thermal conductance versus gap size around room temperature: (a) plate-plate configuration, (b) tip-plate configuration corresponding to experimental setup. The magenta diamonds denote the present 3D phonon NEGF (nonequilibrium Green's function) result with tethering of the surface atomic layers adjacent to the nanogap, the black dash-dotted line represents the 1D phonon NEGF result from Ref. [17], the black squares denote the lattice dynamic (LD) model result of phonons from Ref. [24], the blue solid line denotes the elastic continuum model (ECM) result of phonons from Ref. [25], the red solid line denotes the NFRHT (near-field radiative heat transfer) result predicted by the fluctuational electrodynamic theory, and the green solid line denotes the contribution of electron tunneling to heat transport, whereas the blue circles represent the experimental data [12].

Figure 4

Spectral thermal conductance of Au-Au nanogaps at room temperature (300 K): NEMD result versus NEGF results (a) without and (b) with tethering of the surface atomic layers adjacent to the nanogap. The solid lines represent the NEMD (nonequilibrium molecular dynamics) results, whereas the dashed and dash-dotted lines represent the phonon NEGF (nonequilibrium Green's function) results without and with tethering, respectively.

Figure 5

Spectral thermal conductance of Cu-Cu nanogaps at room temperature (300 K): NEMD result versus NEGF results (a) without and (b) with tethering of the surface atomic layers adjacent to the nanogap. The solid lines represent the NEMD (nonequilibrium molecular dynamics) results, whereas the dashed and dash-dotted lines represent the phonon NEGF (nonequilibrium Green's function) results without and with tethering, respectively.

Figure 6

Spectral thermal conductance of Au-Au nanogap with $d=3.94\AA$ at different temperatures: (a) 1 K, (b) 10 K, (c) 300 K, and (d) 500 K. The black solid line represents the NEMD (nonequilibrium molecular dynamics) result, whereas the blue solid line and magenta dashed line represent the phonon NEGF (nonequilibrium Green's function) results without and with tethering of the surface atomic layers adjacent to the nanogap, respectively.

Figure 7

Spectral thermal conductance of Cu-Cu nanogap with $d=3.45\AA$ at different temperatures: (a) 1 K, (b) 10 K, and (c) 300 K. The black solid line represents the NEMD (nonequilibrium molecular dynamics) result, whereas the blue solid line and magenta dashed line represent the phonon NEGF (nonequilibrium Green's function) results without and with tethering of the surface atomic layers adjacent to the nanogap, respectively.

Figure 8

Temperature-dependent thermal conductance of nanogaps: (a) Au-Au gap with $d=3.94\AA$, (b) Cu-Cu gap with $d=3.45\AA$. The squares and diamonds with lines denote the direct NEMD result and NEGF result with tethering, respectively, whereas the circles with line represent the ratio of the thermal conductance by NEGF over that by direct NEMD. The cross symbols represent the thermal conductance by integrating the SHC from NEMD.

Figure 9

Electrostatic effect on thermal conductance of Au-Au gap via direct NEMD at room temperature (300 K): (a) thermal conductance; (b) ratio of thermal conductance over bare case. The squares denote the result of bare nanogaps; the diamonds denote the result of nanogaps with uniformly charged surfaces under a constant voltage of 600 mV, corresponding to the experimental setup of Kloppstech et al. [12]; and the circles denote the result of nanogaps with electrical double layer (EDL) at the surfaces; the dash-dotted line represents the result of electrostatic phonon heat transfer across nanogaps with EDL based on the elastic continuum model (ECM) in Ref. [49].

Figure 10

Electrostatic effect on spectral thermal conductance of Au-Au gap via molecular dynamics at room temperature (300 K): (a) $d=3.94\AA$, (b) $d=4.43\AA$, and (c) $d=6.28\AA$. The blue solid lines denote the results of bare nanogaps; the red solid lines denote the results of nanogaps with uniformly charged surfaces under a constant voltage of 600 mV, corresponding to the experimental setup of Kloppstech et al. [12]; and the black lines denote the results of nanogaps with electrical double layer (EDL) at the surfaces. The relaxed gap sizes of the charged nanogaps are 3.97, 4.42, and 6.25 Å, respectively; the relaxed gap sizes of the nanogaps with EDL are 3.96, 4.46, and 6.40 Å, respectively.

Figure 11

Comparison of Lennard-Jones potential and Grimme's DFT-D3 dispersion correction for gold.