Heat conduction in molecular transport junctions

Heating and heat conduction in molecular junctions are considered within a general nonequilibrium Green’s function formalism. We obtain a unified description of heating in current-carrying molecular junctions as well as the electron and phonon contributions to the thermal flux, including their mutual influence. Ways to calculate these contributions, their relative importance, and ambiguities in their definitions are discussed. A general expression for the phonon thermal flux is derived and used in a different “measuring technique” to define and quantify “local temperature” in nonequilibrium systems. Superiority of this measuring technique over the usual approach that defines effective temperature using the equilibrium phonon distribution is demonstrated. Simple bridge models are used to illustrate the general approach, with numerical examples.

Figure 1

(Color online) Electron fluxes through the junction interfaces. $L$ and $R$ represent the left and right leads, respectively.

Figure 2

A model for examining the definitions of electron and phonon energy currents where a two-electronic-site/one-phonon bridge connect between two electronic reservoirs as described by the Hamiltonian of Eq. (28).

Figure 3

(Color online) Vibrational population $n$ (solid line, blue) and deviation $\sigma$ (dashed line, green) vs the applied bias. The inset shows the low-voltage region. See text for parameters. In this calculation, the GFs and SEs are evaluated within the strong electron-phonon coupling scheme of Ref. 49.

Figure 4

(Color online) The “local temperature” (left vertical axis) defined by the equilibrium distribution assumption [Eq. (41)] (dashed line, blue) and by the measurement process explained in the text (solid line, red) plotted against the applied bias for the same model and parameters as in Fig. 3. The inset shows the low bias region. The dotted line (green) shows the junction current (right vertical axis). In this calculation, the GFs and SEs are evaluated within the strong electron-phonon coupling scheme of Ref. 49.

Figure 5

(Color online) Heat generation in a current-carrying junction characterized by one electronic level and one oscillator mode (taken to be at thermal equilibrium), plotted for a zero potential bias (dashed line, blue) and for bias $\Phi =0.1\mathrm{V}$ (solid line, red) as a function of the mode temperature. Parameters of the calculation are ${\epsilon }_0=2\mathrm{eV}$, ${\Gamma }_L={\Gamma }_R=0.5\mathrm{eV}$, $E_F=0$, ${\omega }_0=0.2\mathrm{eV}$, and $T_L=T_R=300\mathrm{K}$.

Figure 6

(Color online) The heat generation rate [Eq. (16)] in a current-carrying junction characterized by one electronic level and one primary phonon coupled to a thermal bath (solid line, red) plotted against the potential bias. The inset shows its second derivative (dashed line, blue) in the low bias region. Parameters of the calculation are ${\epsilon }_0=0.2\mathrm{eV}$, ${\Gamma }_L={\Gamma }_R=0.02\mathrm{eV}$, $E_F=0$, ${\omega }_0=0.2\mathrm{eV}$, ${\Omega }_L={\Omega }_R=0.005\mathrm{eV}$, $M=0.2\mathrm{eV}$, and $T_L=T_R=100\mathrm{K}$. The calculation of the needed Green’s functions was done utilizing the strong coupling procedure of Ref. 49.

Figure 7

(Color online) Contour plot of junction temperature $T_{\mathit{th}}$ vs the strength of the electronic molecule-lead coupling $\Gamma$ and the vibration–thermal-bath coupling $\Omega$, in a junction characterized by symmetric electronic coupling to leads $({\Gamma }_L={\Gamma }_R)$ and a junction bias $\Phi =0.1\mathrm{V}$. Other junction parameters are ${\epsilon }_0=1\mathrm{eV}$, $E_F=0$, ${\omega }_0=0.1\mathrm{eV}$, $M=0.2\mathrm{eV}$, and $T_L=T_R=300\mathrm{K}$.

Figure 8

(Color online) Thermal (energy) currents carried by electrons (solid line, red) and phonons (dashed line, blue) through a one-state junction (see text) connecting metal leads under an imposed temperature difference between the two sides. In the calculations shown, $T_R=300\mathrm{K}$ is kept fixed, while $T_L$ is varied. The junction parameters used in this calculation are $E_F=0$, ${\omega }_0=0.1\mathrm{eV}$, and ${\Omega }_0^L={\Omega }_0^R=0.005\mathrm{eV}$. In (a)–(c), ${\epsilon }_0=1\mathrm{eV}$ and the molecule-lead electronic coupling is varied: (a) ${\Gamma }_L={\Gamma }_R=0.1\mathrm{eV}$, (b) $0.25\mathrm{eV}$, and (c) $0.5\mathrm{eV}$. In (d)–(f), ${\Gamma }_L={\Gamma }_R=0.1\mathrm{eV}$ and ${\epsilon }_0$ is varied according to (d) ${\epsilon }_0=1\mathrm{eV}$, (e) $0.5\mathrm{eV}$, and (f) $0.2\mathrm{eV}$.

Figure 9

(Color online) The thermal electron flux $J_Q^{\mathit{el}}=J_{Q,L}^{\mathit{el}}=-J_{Q,R}^{\mathit{el}}$ [Eq. (15)] (dotted line, black; right axis) and the electric current (left axis) through a one-level/one-vibration junction, plotted for an unbiased junction against the left-side temperature where the temperature on the right is kept fixed, $T_R=300\mathrm{K}$. The molecular energy level is positioned at ${\epsilon }_0-E_F=0.5\mathrm{eV}$ (solid line, red) and $-0.5\mathrm{eV}$ (dashed line, blue). ${\Gamma }_L={\Gamma }_R=0.1\mathrm{eV}$ and the other junction parameters are those of Fig. 8. In particular, electron-phonon interaction is taken to be zero in this calculation.

Figure 10

(Color online) Thermal flux through a single level as a function of the temperature difference between the leads. ($T_R=300\mathrm{K}$ is kept fixed, while $T_L$ is varied.) The bridge parameters are ${\epsilon }_0=1\mathrm{eV}$, $E_F=0$, ${\Gamma }_L=0.05\mathrm{eV}$, ${\Gamma }_R=0.5\mathrm{eV}$, ${\omega }_0=0.1\mathrm{eV}$, and ${\Omega }^L={\Omega }^R=0.001\mathrm{eV}$. The electron-phonon coupling is taken at $M=0.5\mathrm{eV}$ when present. The total flux in the presence of electron-phonon coupling is represented by the solid line (red). The sum of electron and phonon contributions to the thermal flux in the $M=0$ case is given by the dash-dotted (black) line. The inset shows the difference between these results. The other two lines focus on the phonon flux (as defined in Sec. 3) calculated from Eq. (24), with (dotted line, green) and without (dashed line, blue) including the electron-phonon interaction $(M=0.5\mathrm{eV})$ in calculating the phonon Green’s function.

Figure 11

(Color online) Thermal flux through a junction as a function of bridge length under a given temperature difference between leads (see text for parameters). Shown are results for weak and strong intersite electronic interactions, $t=0.1\mathrm{eV}$ (dashed line, triangles, blue) and $t=0.5\mathrm{eV}$ (solid line, circles, red) [see Eq. (27)].

Figure 12

(Color online) The local temperature along a biased five-site bridge, displayed as function of the site number. See text for parameters.