**1. Introduction**

The accelerated pace of technological advances, which has taken place over the last half-century, has driven the continuous search for higher speed and cheaper computing with the concomitant developments of larger integration densities and miniaturization trends based on novel materials and processes [1]. The negative counterpart of this amazing technological developments consists in the increasing problems with the thermal management, which is ultimately leading to limiting the efficiency of many of these technological advances, mostly in the domain of nanoelectronics. As a response to this challenge, a novel field, nanophononics, has emerged, providing a large variety of interesting physical effects and potential applications [2–5]. The major goal of nanophononics is to develop efficient strategies for controlling the *heat flux* in organic and inorganic nanostructured materials, and it was originally aiming at realizing thermal devices such as diodes, logic gates, and thermal transistors [2,6]. However, more recent efforts in the field have triggered radically new applications in nanoelectronics [7,8], renewable energy harvesting [9,10], nano- and optomechanical devices [11], quantum technologies [12,13], and therapies, diagnostics, and medical imaging [14].

An important milestone in the field was the theoretically predicted [15] and subsequently measured *quantization* of the phononic thermal conductance in mesoscopic structures [16] at low temperatures, this result building the counterpart of the well-known quantization of the electrical conductance in quantum point contacts and other nanostructures (with conductance quantum given by *<sup>e</sup>*2/*h*, *e* being the electron charge and *h* Planck constant). Thermal conductance quantization has also been found in smaller nanostructures, such as gold wires, where quantized thermal conductance at room temperature was shown down to single-atom junctions [17,18]. The issue is, however, still a subject of debate (see, e.g., [19] for a recent discussion). In contrast to the electrical conductance quantization, the quantum of thermal conductance *κ*0 depends, however, on the absolute temperature *T* through: *κ*0 = *π*2*k*<sup>2</sup> B *T*/3*h*, with *k*B being the Boltzmann constant. This highlights a first important difference between charge and phonon transport. The second one is related to the different energy windows determining the corresponding transport properties: for electrons, the important window lies around the Fermi level, while, for phonons, the conductance results from an integral involving the full vibrational spectrum. Working with a broad spectrum of excitations poses major challenges when it comes to designing thermal devices such as cloaks and rectifiers [2,4], or for information processing in phonon-based computing [6].

From the experimental perspective, it is obviously more difficult to tune heat flow than electrical currents. Unlike electrons, phonons are quasi-particles with zero mass and zero charge, thus they cannot be directly controlled through electromagnetic fields in a straightforward way. Moreover, while considerable progress has been achieved in nanoelectronics in the implementation of local electrodes and gates over very short length scales, establishing temperature gradients over nanoscopic length scales remains a considerable challenge. In this respect, for characterizing thermal devices, novel sophisticated experimental techniques have been developed, such as the 3 *ω* method [20] and the frequency domain thermoeflectance [21], pioneered by Cahill et al. [22]. This has led, in turn, to the modification of atomic force microscopes for thermometry [17,23] and to the development of Scanning Thermal Microscopy [18,24].

Turning nanophononics into a practical field with specific applications in energy managemen<sup>t</sup> requires nanoscale engineering of the thermal transport properties. This approach has been successfully implemented in nanostructured thermoelectric materials. Even though the fundamental tool to understand nanophononics—non-equilibrium thermodynamics—is well established at the macroscale, many open issues remain at the nanoscale, having deep consequences for the development of relevant strategies to control heat transport in low dimensions. Low-dimensional materials have finite cross sections along one or more spatial dimensions and a large surface-to-volume ratio; as a result, their vibrational spectrum and, consequently, the heat transport mechanisms can be dramatically modified (see the recent reviews in [2–4]). Different simulation tools have been used to study heat transport in nanomaterials [4,25]. These approaches can be grouped into three categories, although overlaps between them are clearly possible. The first group includes methodologies based on molecular dynamics (MD) simulations, the equilibrium versions (EMD) based on the Green–Kubo formula [26], and the non-equilibrium versions (NEMD) exploiting Fourier's law [27,28]. The second category includes approaches based on the Boltzmann transport equation (BTE) [29,30] and lattice dynamics (LD) [31]. Finally, the last category covers methodologies relying on the Landauer approach, or more generally on non-equilibrium Green's functions (NEGF) [32–35]. All these methodologies have found extensive application in the prediction of the thermal transport properties of various low-dimensional materials, yielding correct trends and results in good agreemen<sup>t</sup> with experimental studies [36–38]. As it turns out, the thermal response is sensitively determined by different parameters such as surface boundaries, overall device geometry, spatial confinement, doping, and structural defects [37,39–43].

Moreover, in non-stationary situations with time-dependent external parameters able to affect the transport characteristics, phonon dynamics becomes crucial. For instance, time-varying temperature fields [44,45] or local heating mediated by laser fields [46,47] can be exploited to exert additional control over thermal transport. Thus, novel non-equilibrium effects such as heat pumping [48,49], cooling [50], and rectification [51,52] have been theoretically proposed. The description of such phenomena requires in many instances to work in the time domain, which is very challenging from a numerical point of view. Although noticeable progress has been achieved in dealing with time-dependent spin [53,54] and electron [55–60] transport, much less attention has been paid to vibrational degrees of freedom [61–63].

Despite the previously delineated methodological advances to model and understand nanoscale thermal transport, there are many basic questions about thermal managemen<sup>t</sup> of thermoelectric materials, phononic devices, and integrated circuits that must be addressed. In the current paper, we review our recently implemented atomistic models based on the NEGF technique, allowing to address transient and steady quantum phonon transport in low-dimensional systems. We have successfully used our methodology to propose different routes for improved thermal management, eventually leading to realizing novel nanoscale applications.

The paper is organized as follows. In Section 2, the basics of the NEGF approach to compute quantum ballistic transport are introduced. We proceed then to review few selected applications by using the NEGF in combination with a Density-Functional based Tight-Binding approach (DFTB), which allows addressing nanostructures at the atomistic level with considerable accuracy and large computational efficiency. The reviewed applications include 2D materials, BNC heteronanotubes, and molecular junctions. In Section 3, the NEGF formalism previously introduced is expanded to deal with time-dependent thermal transport by exploiting an auxiliary mode approach. This methodology is illustrated for a one-dimensional chain and simple nanoscale junctions based on polyethylene and polyacetylene dimers.
