## Periodic Report Summary 1 - CANTOR (Carbon-nanotube-based terahertz-to-optics rectenna)

The main goal of the project is to combine the efforts of scientific teams from Britain, Israel and Belarus in developing a theoretical basis for the use of carbon nanotube antennas for rectification and detection of electromagnetic radiation in terahertz and optical ranges. To reach the main project goal, we will combine the efforts of the international consortium on the following objectives:

- to extend the Landauer-Büttiker theory of electron transport to the case of photon-dressed electrons;

- to analyze theoretically the rectenna performance with photon-dressed electrons as working substance;

- to estimate thermodynamically the ultimate rectenna efficiency stipulated by the entropy transfer in photo-assisted tunneling.

The Project Consortium joins representatives of different scientific communities under the common roof of interdisciplinary research in theoretical modeling of nanorectenna which contributes to the problem of the solar energy conversion to electricity. The combined effort of teams with the different background and experience provides the complementarity/synergy between the partners stimulating the generation of new knowledge directly related to quantum optics of carbon nanostructures and aimed towards the progress in sunlight conversion into electric power.

The main scientific results obtained within the first two-year period of the project in accordance with the project tasks and deliverables are as follows:

We carried out physical analysis of the rectification mechanism and established its relation to the process of generation of terahertz radiation by Rabi oscillations in a two-level system with broken inversion symmetry. Localized modes in the form of single-peaked fundamental and vortex-like stationary Rabi solitons and self-trapped breathers have been found. The results for the stability, mobility and rectification properties of the Rabi modes suggest a concept of a self-assembled 2D soliton-based nanorectenna, which should be stable against imperfections. Our novel results illustrate possibilities to control its operation by means of optical tools. Soliton-based nanorectennas are shown to be promising for applications in solar cells and detectors of terahertz and optical radiation.

We extended the Landauer- Büttiker theory of electron transport to the case of ballistic electron transport over the 1D insulator layer sandwiched between two metals. The shape of the tunneling barrier is determined by the work function difference between these two metals, the applied bias voltage, and the electron affinities and thicknesses of the insulators. The described system provides rectification of the driving electromagnetic radiation. Nanorectenna is modeled by two segments of a transmission line with frequency dispersion (frequency dependence both inductance and capacitance per unit length) and identical impedance. The segments of transmission lines have been conceptually considered as two reservoirs, which dynamics is governed by asymptotic Hamiltonians. The total Hamiltonian of the system is presented as a sum of the electrode Hamiltonian components and the term, which describes the interaction between the electrodes and MIM- (metal-insulator-metal) diode. The rotating-wave approximation have been used. It was assumed that the rectification process occurs in the adiabatic regime.

As a major step towards the nanorectenna theory, we investigated two typical electromagnetic interference problems, namely, coupling and matching in nanoscale circuits composed of nano-interconnects and quantum devices in entangled state. Nano-interconnects under consideration are implemented by using carbon nanotubes or metallic nanowires (NWs), while quantum devices are represented by semiconductor quantum dots. The equivalent circuits of such nanocircuits contain additional elements arising at nanoscale due to quantum effects. As a result, the notions of coupling and impedance matching are reconsidered. Two examples are studied: in the first one, electromagnetically coupled NWs are connected to classical lumped devices; in the second one, electromagnetically uncoupled transmission lines are terminated on quantum devices in entangled states. In both circuits, the electromagnetic interference features differ both qualitatively and quantitatively from their classical analogs. We also demonstrated the existence of quantum coupling due to the entanglement, which exists in spite of the absence of classical electromagnetic coupling. The entanglement also modifies the matching condition introducing a dependence of the optimal value of load impedance on the line length.

We also carried out theoretical and experimental investigations of the effective permittivity of single-walled carbon nanotube (SWCNT)-based suspensions in the frequency range from 100 MHz to 20 GHz. Using the difference in absorption mechanisms realized in metallic and semiconducting SWCNTs we have theoretically demonstrated the possibility of separating the contributions of metallic and semiconducting SWCNTs to the electromagnetic response of SWCNT suspensions. Experimental data provide the evidence of the unique properties of SWCNTs - to enhance electromagnetic fields in a volume as much as 100 times larger than the volume occupied by a nanotube itself. Such field enhancement opens the way of increasing efficiency of nanorectennas and solar cells.

We studied the fundamental thermodynamic limitation of the rectenna efficiency. It was shown that the fundamental thermodynamic limitation of the rectenna efficiency is governed by the generation of the heat flux and coherence flux in nano-gaps. We have modeled a nanorectenna as a 1D tight-binding chain of two-level systems coupled through common dissipative Markovian reservoirs. We show that even for a few systems in a chain, a density matrix of this chain evolves according to the classical heat transfer equation or, alternatively, can be described by the classical random walk equation. However, the dynamics of the chain induces quantum correlations between the rectenna elements. The dynamics is non-trivial and can be tailored by choosing the initial state of this one-dimensional chain to model much more complex systems. The anomalous thermodynamic behaviour contradicting Fourier law was demonstrated. Besides the energy transport occurring in one dimension (along the rectenna), there is an additional motion. The Fourier law always breaks down in the presence of an additional motion (the simplest example is convection in liquids). In our case this additional motion is a flow of quantum coherence along the rectenna.

During the first two-year period of the project the Consortium organized a set of knowledge transfer events in accordance with project milestones. These events included four special sessions at major international conferences and a focused advance research workshop.

Over the reported period CANTOR consortium members published two book chapters and over 25 papers in peer-reviewed journals and conference proceedings, delivered 4 invited and about 25 oral and poster presentations at international conferences and workshops.

- to extend the Landauer-Büttiker theory of electron transport to the case of photon-dressed electrons;

- to analyze theoretically the rectenna performance with photon-dressed electrons as working substance;

- to estimate thermodynamically the ultimate rectenna efficiency stipulated by the entropy transfer in photo-assisted tunneling.

The Project Consortium joins representatives of different scientific communities under the common roof of interdisciplinary research in theoretical modeling of nanorectenna which contributes to the problem of the solar energy conversion to electricity. The combined effort of teams with the different background and experience provides the complementarity/synergy between the partners stimulating the generation of new knowledge directly related to quantum optics of carbon nanostructures and aimed towards the progress in sunlight conversion into electric power.

The main scientific results obtained within the first two-year period of the project in accordance with the project tasks and deliverables are as follows:

We carried out physical analysis of the rectification mechanism and established its relation to the process of generation of terahertz radiation by Rabi oscillations in a two-level system with broken inversion symmetry. Localized modes in the form of single-peaked fundamental and vortex-like stationary Rabi solitons and self-trapped breathers have been found. The results for the stability, mobility and rectification properties of the Rabi modes suggest a concept of a self-assembled 2D soliton-based nanorectenna, which should be stable against imperfections. Our novel results illustrate possibilities to control its operation by means of optical tools. Soliton-based nanorectennas are shown to be promising for applications in solar cells and detectors of terahertz and optical radiation.

We extended the Landauer- Büttiker theory of electron transport to the case of ballistic electron transport over the 1D insulator layer sandwiched between two metals. The shape of the tunneling barrier is determined by the work function difference between these two metals, the applied bias voltage, and the electron affinities and thicknesses of the insulators. The described system provides rectification of the driving electromagnetic radiation. Nanorectenna is modeled by two segments of a transmission line with frequency dispersion (frequency dependence both inductance and capacitance per unit length) and identical impedance. The segments of transmission lines have been conceptually considered as two reservoirs, which dynamics is governed by asymptotic Hamiltonians. The total Hamiltonian of the system is presented as a sum of the electrode Hamiltonian components and the term, which describes the interaction between the electrodes and MIM- (metal-insulator-metal) diode. The rotating-wave approximation have been used. It was assumed that the rectification process occurs in the adiabatic regime.

As a major step towards the nanorectenna theory, we investigated two typical electromagnetic interference problems, namely, coupling and matching in nanoscale circuits composed of nano-interconnects and quantum devices in entangled state. Nano-interconnects under consideration are implemented by using carbon nanotubes or metallic nanowires (NWs), while quantum devices are represented by semiconductor quantum dots. The equivalent circuits of such nanocircuits contain additional elements arising at nanoscale due to quantum effects. As a result, the notions of coupling and impedance matching are reconsidered. Two examples are studied: in the first one, electromagnetically coupled NWs are connected to classical lumped devices; in the second one, electromagnetically uncoupled transmission lines are terminated on quantum devices in entangled states. In both circuits, the electromagnetic interference features differ both qualitatively and quantitatively from their classical analogs. We also demonstrated the existence of quantum coupling due to the entanglement, which exists in spite of the absence of classical electromagnetic coupling. The entanglement also modifies the matching condition introducing a dependence of the optimal value of load impedance on the line length.

We also carried out theoretical and experimental investigations of the effective permittivity of single-walled carbon nanotube (SWCNT)-based suspensions in the frequency range from 100 MHz to 20 GHz. Using the difference in absorption mechanisms realized in metallic and semiconducting SWCNTs we have theoretically demonstrated the possibility of separating the contributions of metallic and semiconducting SWCNTs to the electromagnetic response of SWCNT suspensions. Experimental data provide the evidence of the unique properties of SWCNTs - to enhance electromagnetic fields in a volume as much as 100 times larger than the volume occupied by a nanotube itself. Such field enhancement opens the way of increasing efficiency of nanorectennas and solar cells.

We studied the fundamental thermodynamic limitation of the rectenna efficiency. It was shown that the fundamental thermodynamic limitation of the rectenna efficiency is governed by the generation of the heat flux and coherence flux in nano-gaps. We have modeled a nanorectenna as a 1D tight-binding chain of two-level systems coupled through common dissipative Markovian reservoirs. We show that even for a few systems in a chain, a density matrix of this chain evolves according to the classical heat transfer equation or, alternatively, can be described by the classical random walk equation. However, the dynamics of the chain induces quantum correlations between the rectenna elements. The dynamics is non-trivial and can be tailored by choosing the initial state of this one-dimensional chain to model much more complex systems. The anomalous thermodynamic behaviour contradicting Fourier law was demonstrated. Besides the energy transport occurring in one dimension (along the rectenna), there is an additional motion. The Fourier law always breaks down in the presence of an additional motion (the simplest example is convection in liquids). In our case this additional motion is a flow of quantum coherence along the rectenna.

During the first two-year period of the project the Consortium organized a set of knowledge transfer events in accordance with project milestones. These events included four special sessions at major international conferences and a focused advance research workshop.

Over the reported period CANTOR consortium members published two book chapters and over 25 papers in peer-reviewed journals and conference proceedings, delivered 4 invited and about 25 oral and poster presentations at international conferences and workshops.