Thermal balance and quantum heat transport in nanostructures thermalized by local Langevin heat baths

Modeling of thermal transport in practical nanostructures requires making tradeoffs between the size of the system and the completeness of the model. We study quantum heat transfer in a self-consistent thermal bath setup consisting of two lead regions connected by a center region. Atoms both in the leads and in the center region are coupled to quantum Langevin heat baths that mimic the damping and dephasing of phonon waves by anharmonic scattering. This approach treats the leads and the center region on the same footing and thereby allows for a simple and physically transparent thermalization of the system, enabling also perfect acoustic matching between the leads and the center region. Increasing the strength of the coupling reduces the mean-free path of phonons and gradually shifts phonon transport from ballistic regime to diffusive regime. In the center region, the bath temperatures are determined self-consistently from the requirement of zero net energy exchange between the local heat bath and each atom. By solving the stochastic equations of motion in frequency space and averaging over noise using the general fluctuation-dissipation relation derived by Dhar and Roy [J. Stat. Phys. 125, 801 (2006)], we derive the formula for thermal current, which contains the Caroli formula for phonon transmission function and reduces to the Landauer-Büttiker formula in the limit of vanishing coupling to local heat baths. We prove that the bath temperatures measure local kinetic energy and can, therefore, be interpreted as true atomic temperatures. In a setup where phonon reflections are eliminated, the Boltzmann transport equation under gray approximation with full phonon dispersion is shown to be equivalent to the self-consistent heat bath model. We also study thermal transport through two-dimensional constrictions in square lattice and graphene and discuss the differences between the exact solution and linear approximations.

Figure 1

(a) A schematic illustration of the system under study. The structure is divided into the left lead, the center region, and the right lead. All atoms are coupled to spatially uncorrelated quantum Langevin heat baths, which are shown explicitly for one cross section in (b). In the left and right lead, the temperatures of the local heat baths have prescribed values $T_L$ and $T_R$, respectively. In the center region, on the other hand, temperature varies between $T_L$ and $T_R$ and the bath temperatures are determined self-consistently using the requirement that the average thermal current to each bath vanishes. The leads can contain an infinite number of atoms, but the center region is finite. Two-dimensional square lattice with nearest-neighbor interactions is shown for illustrative purposes, but the basic principle can be applied to any geometry.

Figure 2

Illustration of the systems studied in Secs. 4a and 4b: (a) a chain of length $N$ connected to two semi-infinite chains, (b) a constriction of width $w$ and length $l$ between two leads of width $W$. $L$ layers of atoms in the leads are included in the self-consistent calculation to account for the gradual temperature drop near the constriction.

Figure 3

Bath temperature profiles in a chain of length $N=5$ sandwiched between two semi-infinite leads at temperatures $T_L=0.2$ and $T_R=0.1$. Friction parameters are (a) $\gamma ={10}^{-3}$, i.e., the system is nearly ballistic and (b) $\gamma =0.1$. Symmetry requires that $T_3=0.15$ for the linearized models, but the quantum exact temperature is higher due to the nonlinearity of the Bose-Einstein function.

Figure 4

Comparison of temperature profiles in a chain of length $N=5$ for two kinds of external reservoirs: Rubin-Greer leads [Fig. 2a] and Ohmic reservoirs studied in Ref. [27]. In contrast to Ref. [27], we also couple the particles at the ends of the chain to self-consistent baths. The reservoirs are at temperatures $T_L=0.2$ and $T_R=0.1$ and the coupling constant to self-consistent baths inside the chain is $\gamma =0.1$. In the Rubin-Greer setup, the coupling constant in the leads is ${\gamma }_L={\gamma }_R=0.1$. In the case of Ohmic external reservoirs, coupling constant at the ends of the chain is denoted by ${\gamma }_o$. Choosing ${\gamma }_o=1$ reproduces the low-frequency self-energy of the Rubin-Greer chain, leading to similar temperature profiles at low temperatures. The geometries are shown in insets.

Figure 5

Thermal current $Q$ as a function of lead temperature $T_L$ for fixed $T_R=0.2$. Chain length is $N=10$ and the friction parameters are (a) $\gamma ={10}^{-3}$ and (b) $\gamma =0.1$. With both weak and strong friction, the linear response approximation reproduces the exact current up to very high values of bias. In (a), current $Q=G_Q(T_L-T_R)$ corresponding to the quantum of thermal conductance $G_Q=\pi T/6$ with $T=0.2$ is also shown (black dashed).

Figure 6

Exact thermal current as a function of chain length $N$. The friction parameter in the center region is $\gamma =0.1$. The BTE current (40) matches the self-consistent current if the leads are nearly ballistic, ${\gamma }_L={\gamma }_R=0.001$. The bath temperatures in the left and right semi-infinite chains are $T_L=0.2$ and $T_R=0.1$, but the currents also agree at other temperatures for ballistic leads.

Figure 7

Bath temperature profiles in a $w=5$, $l=9$ constriction coupled to leads of width $W=71$ and $L=35$ [see Fig. 2b]. Lead temperatures are $T_L=0.2$ and $T_R=0.1$. Figures show (a) quantum exact and (b) classical self-consistent bath temperature profiles. Friction parameter is $\gamma =0.01$. The separation of isolines is $0.05$ and four contour lines are labeled for convenience.

Figure 8

(a) Graphene nanoconstriction. The leads extend infinitely to the left and right, but the temperatures are determined self-consistently only for the gray atoms in the shown center region. (b) and (c) Self-consistent bath temperature profiles (K), in (b) quantum exact case and (c) classical approximation. The semi-infinite leads are at temperatures $T_L=300$ K and $T_R=280$ K. The relaxation time $\tau =1/\gamma$ is set to 1 ps.

Figure 9

The search for the self-consistent bath temperatures $T_1$ and $T_2$ that satisfy $Q_1(T_1,T_2)=Q_2(T_1,T_2)=0$ for an $N=2$ chain. The lead temperatures are $T_L=0.2$ and $T_R=0.1$ and the friction parameters are (a) $\gamma ={10}^{-2}$ and (b) $\gamma =1$. The two methods used for the search are Runge-Kutta-Fehlberg integration of Eq. (42) (dots connected by dashed lines) and Newton-Raphson iteration (crosses connected by dash-dotted lines). The contours belong to the target function $f(T_1,T_2)$ that is defined to be the squared sum of the currents flowing to the self-consistent reservoirs, $f(T_1,T_2)=Q_1{(T_1,T_2)}^2+Q_2{(T_1,T_2)}^2$. The self-consistent temperature configuration $({\tilde{T}}_1,{\tilde{T}}_2)$ satisfies $f({\tilde{T}}_1,{\tilde{T}}_2)=0$ and is also the global minimum of $f$. The initial guess is $T_1=0.2$, $T_2=0.1$.