Simulating time-dependent thermoelectric transport in quantum systems

Adel Kara Slimane, Phillipp Reck, and Geneviève Fleury

Phys. Rev. B 101, 235413 – Published 8 June 2020

Abstract

We put forward a gauge-invariant theoretical framework for studying time-resolved thermoelectric transport in an arbitrary multiterminal electronic quantum system described by a noninteracting tight-binding model. The system is driven out of equilibrium by an external time-dependent electromagnetic field (switched on at time $t_{0}$ ) and possibly by static temperature or electrochemical potential biases applied (from the remote past) between the electronic reservoirs. Numerical simulations are conducted by extending to energy transport the wave-function approach developed by Gaury et al. and implemented in the t-Kwant library. We provide a module that allows us to compute the time-resolved heat currents and powers in addition to the (already implemented) charge currents, and thus to simulate dynamical thermoelectric transport through realistic devices, when electron-electron and electron-phonon interactions can be neglected. We apply our method to the noninteracting resonant level model and verify that we recover the results reported in the literature for the time-resolved heat currents in the expected limits. Finally, we showcase the versatility of the library by simulating dynamical thermal transport in a quantum point contact subjected to voltage pulses.

Received 13 January 2020
Revised 4 May 2020
Accepted 7 May 2020

DOI:https://doi.org/10.1103/PhysRevB.101.235413

Physics Subject Headings (PhySH)

Mesoscopics Quantum thermodynamics Quantum transport Thermoelectric effects

Anderson impurity model Landauer formula Tight-binding model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Adel Kara Slimane , Phillipp Reck , and Geneviève Fleury ^*

Université Paris-Saclay, CEA, CNRS, SPEC, 91191, Gif-sur-Yvette, France

^*genevieve.fleury@cea.fr

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 101, Iss. 23 — 15 June 2020

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Left particle current $I_{L}^{N}$ (a) and left heat currents $I_{L}^{H}$ (b) and ${\tilde{I}}_{L}^{H}$ (c) as a function of time $t$ , for the 1D RLM defined by Eqs. (66, 67, 68, 69), when the dot energy level is modified as $ε_{0} (t) = ε_{0} + e V_{0} Θ (t)$ [inset of panel (a)]. Units of $x$ and $y$ axes are indicated in brackets. In all panels, data are computed numerically with t-Kwant for different values of $λ γ / Γ$ [1 (red lines), 6.25 (green lines), and 100 (black lines)]. The horizontal dashed lines plotted for $λ γ / Γ = 1$ (in red) and 100 (in black) correspond to the static limits at large times $Γ t / ℏ ≫ 1$ given by the Laudauer-Büttiker formulas (see Sec. 3c4). When $λ γ / Γ ≫ 1$ , the t-Kwant results converge to the NEGF results (circles) derived in the wideband limit (Appendix pp4). Inset of panel (c): Comparison of $I_{L}^{H} (t)$ (red dashed line) and ${\tilde{I}}_{L}^{H} (t)$ (black line) in the wideband limit. In all panels, $ε_{0} = 0.5 Γ$ , $e V_{0} = 2.5 Γ$ , $ε_{L} (t) = ε_{R} (t) = 0$ , $T_{L} = Γ / k_{B}$ , $T_{R} = 0$ , $μ_{L} = 0.5 Γ$ , and $μ_{R} = - 0.5 Γ$ . The NEGF curves are independent of $Γ$ . The t-Kwant curves are functions of $λ γ / Γ$ and not of the three parameters $λ$ , $γ$ , and $Γ$ taken separately.
Reuse & Permissions
Figure 2
Left heat current $I_{L}^{H}$ (in red) and right heat current $I_{R}^{H}$ (in blue) as a function of time $t$ , for the 1D RLM defined by Eqs. (66, 67, 68, 69), when a voltage step $ε_{L} (t) = e V_{L} Θ (t)$ is applied in the left lead (sketch in inset). Units are indicated in brackets. The data issued from Ref. [69] (solid lines) and those calculated with t-Kwant (circles) are superimposed. The horizontal dashed lines show the static limits $I_{L / R}^{H, \bar{s t}}$ at large times given by the Landauer-Büttiker formula (see Sec. 3c4). Parameters are fixed to $ε_{0} = 0.2 γ_{c}$ , $e V_{L} = 2 γ_{c}$ , $ε_{R} (t) = 0$ , $γ = 5 γ_{c}$ , $T_{L} = T_{R} = 0.01 γ_{c} / k_{B}$ , and $μ_{L} = μ_{R} = 0$ .
Reuse & Permissions
Figure 3
(a) QPC discretized model. The site color in the central region encodes the value of the on-site potential $U_{i}$ given by Eq. (75) [from 0 (white) to larger values (shades of blue)]. A few layers of the left and right semi-infinite leads are shown in red. A voltage pulse $V_{L} (t)$ is applied in the left lead. Currents are evaluated at the (green dashed) interface indicated by the green arrow. (b) Transmission function $T (E)$ of the QPC defined by ${\hat{H}}^{0}$ [see Eq. (74)]. (c) $\int_{0}^{\infty} d t [I_{L}^{N} (t) - I_{L}^{N} (t = 0)]$ (in blue, in units of $1 / 2 π$ ) and $\int_{0}^{\infty} d t I_{L}^{H} (t)$ (in red, in units of $γ / 2 π$ ) as a function of the pulse width $τ_{p}$ at fixed $n_{p} = 0.2$ . Squares with full lines are t-Kwant results; circles with dashed lines are Landauer-Büttiker adiabatic results. Lines are guides to the eye. (d) to (g) Left particle currents $I_{L}^{N}$ (in blue, in units of $100 γ / h$ ) and left heat currents $I_{L}^{H}$ (in red, in units of $γ^{2} / h$ ) as a function of time $t$ (in units of $ℏ / γ$ ), for different widths of the voltage pulse [ $τ_{p} = 20 ℏ / γ$ (d), $100 ℏ / γ$ (e), $200 ℏ / γ$ (f), and $800 ℏ / γ$ (g)] at fixed $n_{p} = 0.2$ . Full lines are t-Kwant results; dashed lines are Landauer-Büttiker adiabatic results. In all panels, parameters are fixed to $W = 18$ , $L = 48$ , $l_{x} = 50$ , $l_{y} = 5$ , $μ_{L} = 0.20607 γ$ , $μ_{R} = 0.2 γ$ , $T_{L} = 0.018 γ / k_{B}$ , and $T_{R} = 0.02 γ / k_{B}$ .
Reuse & Permissions
Figure 4
Comparison of the computation times needed for the evolution of a single wave function by a time step and for the evaluation of its contribution to the particle and energy operators. Data (bullets) are shown for the square system made of $N = L^{2}$ sites defined in the text. Its Hamiltonian includes hopping terms up to the $z$ th nearest neighbors. Dashed lines are linear fits. Left: CPU times for evaluating operators and evolving the wave function, as a function of the size of their input site/hopping tuples (varied by increasing $L$ , for fixed $z = 1$ ). The input size equals $N = L^{2}$ for the wave function and the density/source operators, while it equals the number of hoppings $M_{[z = 1]} / 2 = L (L - 1)$ for the current operators. Right: CPU times divided by the input size $N$ or $M_{z} / 2$ , as a function of the average number of neighbors per site $M_{z} / N$ , varied by taking $z = 1, 2, 3,$ and 4 at fixed $N = 10^{4}$ sites.
Reuse & Permissions

Physical Review B

covering condensed matter and materials physics