Luttinger-field approach to thermoelectric transport in nanoscale conductors

Thermoelectric transport in nanoscale conductors is analyzed in terms of the response of the system to a thermomechanical field, first introduced by Luttinger, which couples to the electronic energy density. While in this approach, the temperature remains spatially uniform, we show that a spatially varying thermomechanical field effectively simulates a temperature gradient across the system and allows us to calculate the electric and thermal currents that flow due to the thermomechanical field. In particular, we show that in the long-time limit, the currents thus calculated reduce to those that one obtains from the Landauer-Büttiker formula, suitably generalized to allow for different temperatures in the reservoirs, if the thermomechanical field is applied to prepare the system, and subsequently turned off at $t=0$. Alternatively, we can drive the system out of equilibrium by switching the thermomechanical field after the initial preparation. We compare these two scenarios, employing a model noninteracting Hamiltonian, in the linear regime, in which they coincide, and in the nonlinear regime, in which they show marked differences. We also show how an operationally defined local effective temperature can be computed within this formalism.

Figure 1

This sketch shows a typical transport setup where a nanoscale junction (central region) is connected via leads to reservoirs. If the leads are held at different potentials ${\mu }_1,{\mu }_2,...,{\mu }_N$ and/or different temperatures $T_1,T_2,...,T_N$, a charge current $I$ and a heat current $Q$ will flow through the junction.

Figure 2

Sketch of a typical transport scenario between two leads. The upper panel depicts the situation in the LB approach and the lower panel in the TM approach. A potential bias $U$ is applied to the left lead and the temperature in the left lead is raised. Note that in the TM approach the effective potential, determining the position of the occupation function $f_{\mathrm{L}}$, is rescaled by $(1+\psi )=\frac{T_{\mathrm{L}}}{T}$. This is indicated by the dashed extension of the range (green) representing the applied bias $U$. Furthermore, the band of the left lead is rescaled by $(1+\psi )$ as shown by the dashed extension of the original bandwidth (blue).

Figure 3

Sketch of the Hamiltonian employed in the numerical examples. The left and right leads are modeled by infinite tight-binding chains, which are characterized by the nearest-neighbor hopping amplitude $t$. The hopping to the impurity site is described by the amplitude $V=0.1t$. The energy of the impurity site is at ${\epsilon }_{\mathrm{imp}}=\mu =0$.

Figure 4

Comparison of the steady-state particle currents $I$ in the TM and the LB approach. The currents are plotted against the potential bias $U$. The upper panel depicts the currents when the temperature in the left lead is raised to twice its initial value, i.e., $\psi =\delta T/T=1$. The lower panels shows the currents when the temperature in the left lead is reduced to half its initial value ($\psi =\delta T/T=-0.5$). The dashed, black curve shows the particle current at zero temperature difference for comparison. It is identical in the LB and the TM approach. The circles (red curve, labeled “$\mathrm{TM}\pm$”) correspond to the current in the TM approach and the squares (green curve, labeled “$\mathrm{LB}\pm$”) to the LB approach.

Figure 5

Comparison of the steady-state heat currents $Q$ in the TM and the LB approach. The heat currents are plotted against the potential bias $U$. The upper panel depicts the heat currents when the temperature in the left lead is raised to twice its initial value ($\psi =1$) and the lower panel shows the heat currents when the temperature in the left lead is reduced to half its initial value ($\psi =-0.5$). The dashed, black curve is the heat current at a uniform temperature throughout the system. The circles (red curve, labeled “$\mathrm{TM}\pm$”) correspond to the heat current in the TM approach and the squares (green curve, labeled “$\mathrm{LB}\pm$”) to the LB approach.

Figure 6

Comparison of the steady-state heat current $Q$ in the TM and the LB approach at zero potential bias as a function of the relative temperature difference. The circles (red curve, labeled “TM”) and squares (green curve, labeled “LB”) show the heat current in the TM and LB approaches, respectively. While in the main plot the temperature is only changed in the left lead, the inset shows the heat current when the TM field (or relative temperature difference) is applied antisymmetrically in the left and right leads, i.e., ${\psi }_{\mathrm{L}}=\delta T_{\mathrm{L}}/T=-{\psi }_{\mathrm{R}}=-\delta T_{\mathrm{R}}/T$. In the inset we have defined $\psi ={\psi }_{\mathrm{L}}-{\psi }_{\mathrm{R}}$. $Q$ at the left boundary ($\psi =-0.5$) of the main plot corresponds to $Q$ at $U=0$ in the lower panel of Fig. 5 and $Q$ at the right boundary ($\psi =1$) of the main plot to $Q$ at $U=0$ in the upper panel.

Figure 7

Sketch of the Hamiltonian employed in the numerical computation of an effective local temperature. In addition to the Hamiltonian sketched in Fig. 3, a probe lead is connected to the impurity. The probe lead is modeled by a half-filled tight-binding chain characterized by a hopping amplitude $t_{\mathrm{probe}}\gg t$ and the coupling to the impurity site is described by the amplitude $V_{\mathrm{probe}}\ll V$. This implies that the probe lead can be treated in the wide-band limit.

Figure 8

Comparison of the local temperature in the TM and the LB approach. The relative local temperature differences ${\psi }_{\mathrm{probe}}$ are plotted against the relative temperature bias $\psi$. In the upper panel, the TM field or temperature bias is only applied to the left lead. The dashed, black line depicts the approximation ${\psi }_{\mathrm{probe}}\approx 0.5\psi$, which corresponds to taking the average between the two leads as estimate for the temperature at the impurity. In the lower panel, the temperature bias is applied symmetrically to the left and right leads, i.e., the temperature in the right lead is change by the same amount as in the left lead but in the opposite direction. Accordingly, a simple estimate of the relative temperature difference at the impurity ${\psi }_{\mathrm{probe}}=0$, which is shown by the horizontal dashed, black line. The circles (red curve, labeled “TM”) show ${\psi }_{\mathrm{probe}}$ in the TM approach and the squares (green curve, labeled “LB”) in the LB approach.

Figure 9

Plot showing the real and imaginary part of the embedding self-energy for a lead modeled by a tight-binding chain in the limit of infinite sites.