# The Effect of Anchor Group on the Phonon Thermal Conductance of Single Molecule Junctions
## 2 Results and Discussion
Using \(T_{p}\), the phonon thermal conductance \(\kappa_{p}\) at temperature \(T\) was calculated from \(\kappa_{p}(T)=(2\pi)^{-1}\int_{0}^{\infty}\hbar\omega T_{p}(\omega)(\partial f _{BE}(\omega,T)/\partial T)d\omega\) where \(f_{BE}(\omega,T)=\left(e^{\hbar\omega/k_{B}T}-1\right)^{-1}\) is the Bose-Einstein distribution function, \(\hbar\) is the reduced [PERSON]'s constant and \(k_{B}\) is the Boltzmann's constant [12]. Figure 2 shows thermal conductance for OPE3 with the PY, S, SMe and CS anchor groups. \(\kappa_{p}\) increases with temperature \(T\) and saturates for temperatures higher than 150 K. This saturation of \(\kappa_{p}\) is mainly because of the small Debye frequency of gold electrodes [24]. The order of thermal conductances for different anchors is as follows: OPE3-CS (34 pW/K) > OPE3-PY (30 pW/K) > OPE3-S (25 pW/K) > OPE3-SMe (19 pW/K). Clearly, thermal conductance due to phonons is influenced strongly by the choice of anchor groups and the lowest thermal conductance is obtained for the molecule with the SMe anchor group. The thermal conductance of OPE3 with the SMe anchor was about two times lower than that with the CS anchor. To understand the DFT result further, we construct a simple tight binding (TB) model of ball and springs with one degree of freedom per site and the spring constant \(\gamma=61.3\times 10^{-3}eV\) (Figure 3a) connected to two one dimensional leads through a week coupling. Figure 3b shows the phonon transmission coefficient using the TB model for junctions with different anchor groups. Note that for simplicity, we have considered all spring constants \(\gamma\) the same. The phonon thermal conductance showed a similar trend to the DFT result \(\kappa_{p}^{CS}>\kappa_{p}^{PY}>\kappa_{p}^{S}>\kappa_{p}^{SMe}\). The only difference between the junction with S and SMe anchors is the additional pendent side groups in SMe (Figure 3a). These pendent side groups attached to S clearly leads to the suppression of \(T_{p}\) resonances for the high frequency phonons and consequently to the decrease of the thermal conductance in OPE3 with the SMe anchor groups. The width of the \(T_{p}\) resonances with the CS anchor group was larger. CS anchors were connected to the electrodes from two points (inset of Figure 1d); thus the overall coupling strength to electrodes is higher. This leads to the larger broadening of \(T_{p}\) resonances in OPE3 with the CS anchor groups, leading to a high  
Figure 2: Phonon thermal conductance versus temperature T for OPE3 with different anchor groups. thermal conductance. Note that thermal conductance is proportional to the area under \(T_{p}\) curve that increases when width of a resonance increases. The thermal conductance of OPE3 with the PY anchor is larger than that of with the S anchor. The only difference between these two junctions in the simple TB model is the additional sites at the two ends of the molecule (Figure 3a). There are two competing effects associated with this. First, the level spacing between resonances decreases when the size of the system (the number of atoms) increases. As a result, the thermal conductance is expected to increase because more resonances moved into the energy window defined by the Bose-Einstein distribution function at room temperature. Secondly, the resonance width decreases. This is because the length of the junction increases and, consequently, the density of phonon state at the two end points decreases, leading to the smaller broadening of the resonances. Clearly, the first effect is dominant here and thermal conductance increased from the S to PY anchor. It is worth mentioning that thermal conductance is dominated by phonons in molecular junctions. For example, the room-temperature thermal conductance due to electrons and phonons in 4,4\({}^{\prime}\)-bipyridy connected to gold electrodes are 0.17 pW/K and 34.8 pW/K, respectively [17].