Extremely efficient terahertz high harmonic generation in graphene by repetitive heating and cooling its free electrons
Graphene is an ultimate two-dimensional material, consisting of only one layer of carbon atoms arranged into a hexagonal lattice. The electrons in graphene in a wide range of energies and momenta behave as ultra-relativistic charge particles called Dirac fermions. Due to such a unique electronic structure, graphene has long been predicted to possess extremely strong nonlinear electronic response, permitting very efficient mixing or multiplication of very high frequency electronic signals.
H. A. Hafez et al., “Extremely efficient terahertz high-harmonic generation in graphene by hot Dirac fermions”, Nature (2018) DOI: 10.1038/s41586-018-0508-1
In particular, it was expected that graphene, in spite of being only 1 atomic layer thin, can support very efficient mixing of electronic frequencies in the terahertz (THz, 10^12 Hz) range, which would finally enable very fast graphene-based devices for data transmission and processing. In other electronic materials such as semiconductors, handling such high frequencies is technologically rather challenging, yet it is badly needed for the next generation of electronics operating at faster and faster rates. Many theoretical predictions have been made about the ability of graphene to efficiently mix or up-convert THz-frequency electronic signals, yet until recently all experimental demonstrations have failed.
In my group we focus on the studies of ultrafast dynamics of charge, lattice and spins in various electronic materials, and as a main experimental tool we use ultrafast terahertz (THz) spectroscopy implemented in many different modalities. The physics of graphene is a significant part of our current work, and for several years we were busy trying to understand what exactly mechanisms govern the electronic response of graphene on the (sub-)picosecond timescales, corresponding to the THz frequencies.
Our first breakthrough came around 2014, when my group was part of the Department of Molecular Spectroscopy headed by Mischa Bonn at Max Planck Institute for Polymer Research in Mainz. Then we discovered, that the free electrons in graphene respond to external excitation, as if they were a liquid, which can evaporate and become hot gas when exposed to ultrafast electric field.
This is how it happens. Initially the free electrons in graphene form a so-called Fermi liquid. If one applies an electric field to graphene, the electrons occupying the states at the surface of this Fermi liquid will start to move, and the electrical current will flow. The electrons occupying the states with lower energies, i.e. residing below the Fermi surface, cannot move due to the quantum-mechanical blocking mechanism, called Pauli blocking, and are thus not participating in the electrical current. As the Fermi surface electrons are moving in applied electric field, they increase their kinetic energy and momentum. And now the interesting thing will happen: these energetic, current-carrying electrons at the Fermi surface will start sharing their kinetic energy and momentum with all other electrons occupying the lower-lying energetic states. And this electron energy and momentum sharing, also known as electron-electron scattering, in graphene happens extremely fast, on the timescale faster than only 100 femtoseconds. As a result, the energy of electrical currents, initially concentrated to the electronic states at the Fermi surface, is very quickly converted into the collective kinetic energy of all other free electrons in graphene. In other words, this energy becomes the electronic heat.
What happens to the normal liquid, such as e.g. water, when the heat is supplied to it? It starts evaporating. And this is exactly what happens to the free electrons in graphene: the electrons initially forming a cold Fermi liquid will heat up under the influence of driven current, and will form a hot Fermi vapor. This very fast increase in the electronic temperature in turn will decrease the electrical conductivity of graphene, which is a consequence of certain basic conservation laws in solid state physics. Now, what happens to the hot water vapor, when it meets a cold surface? The water vapor molecules will transfer their thermal energy to the cold surface, will cool down, and will condense back into the cold liquid. The electrons forming the hot Fermi vapor in graphene will do the same: they will transfer their thermal energy to the crystal lattice of graphene in a process called phonon emission, and will condense back to form the cold Fermi liquid. This electron cooling, or “condensation”, as a result of interaction with the crystal lattice, takes several picoseconds, which is considerably longer than the initial heating time of < 100 femtoseconds. This is because the carbon atoms, forming the crystal lattice of graphene, are much heavier than the electrons: so it takes some time to transfer the energy of hot electrons to them, i.e. to make the atoms move. Cooling electrons down will also restore the initially high electrical conductivity of graphene.
So now we have established the connection between the electronic temperature in graphene and its electrical conductivity, and figured out the whole energy conversion chain in the process of launching the electrical currents in graphene: the energy is first transferred from the applied electric field to the electrons at the Fermi surface, making them move. Then these electrons very quickly share their energy and momentum with the rest of the electrons in the Fermi liquid, turning it into hot Fermi vapor. This is when the conductivity of graphene drastically decreases, and it becomes quite resistive. Then, the hot electrons will start transferring their energy to the crystal lattice of graphene, which will make the crystal lattice atoms oscillate with larger amplitude, but will also cool the electrons down: the Fermi vapor will condense back into the Fermi liquid, and graphene will become highly electrically conductive again. This is now known as the thermodynamic model of electronic conduction in graphene, which we have published in the following paper a few years ago: Z. Mics et al.,”Thermodynamic picture of ultrafast charge transport in graphene,” Nature Commun. 6, 7655 (2015) .
This heating and cooling dynamics of electrons in graphene, accompanied by the very quick reduction, and somewhat slower recovery, of graphene electrical conductivity in ultrafast electric fields looked very promising for highly efficient mixing of very high frequency electronics signals, or higher harmonics generation. This was the effect long sought after both in academia and in industry, as it would allow to drastically increase the operation speed of electronic components and devices. It was clear to us that the thermodynamic response of graphene electrons was the solution, but we could not prove this in my lab: we did not have a proper source of electromagnetic radiation, which would provide the THz frequency signals in a quasi-continuous manner, yet with a considerably high electric field strength, driving electronic temperature in graphene up and down repetitively many times around. In my lab we use optical lasers to generate THz electric fields, and we can only generate THz signals containing one single oscillation of the THz field. And we needed many.
And this is when I met my colleague Michael Gensch from Helmholtz-Zentrum Dresden-Rossendorf (HZDR). We both attended the spin physics conference in Greifswald at the German Baltic sea coast in September of 2015, and Michael was showing very interesting results achieved in first pilot experiments at a new type of THz source based on superradiant emission from relativistic electron bunches, called TELBE (B. Green et al., “High-Field High-Repetition-Rate Sources for the Coherent THz Control of Matter,” Sci. Rep. 6, 22256 (2016)). And this TELBE source could provide us with exactly what we needed: electromagnetic pulses containing many oscillations of the THz electric field with a well-defined, accurately tunable frequency, and a sufficiently large field strength. Most importantly, Michael and his group developed a sophisticated and very sensitive detection scheme for THz field signals emitted from photo-excited matter and employed it at that time to investigate magnetic nanofilms. As we discovered in the discussions after his talk, his very sensitive set-up could also be suitable to detect the THz higher harmonics emitted from graphene. So Michael and I went for lunch together on the picturesque Greifswald market square and it all went from there. Our two groups started a collaboration, which continues to this date.
2.5 years and 4 very intense TELBE beam times after our first meeting, we have achieved what we wanted: we could directly demonstrate the generation of high odd-order THz harmonics in graphene up to the 7th order. This means, that we could multiply the frequency of the incoming THz signal in graphene by as much as 7 times! And this works at quite modest THz driving electric fields of only 10s of kV/cm. Such fields are about an order of magnitude lower than what one finds in e.g. modern high-speed field effect transistors, and about three orders of magnitude lower (i.e. one million times less peak power!) than what is required for successful harmonics generation in other electronic materials.
Part of our trick was to make sure that our graphene sample possessed a significant concentration of free electrons, efficiently swaying between the conductive Fermi liquid and resistive Fermi vapor phases, as the high frequency electrical current passes through graphene. Such a free electron concentration can be created naturally by e.g. adsorption of oxygen molecules to the surface of graphene from ambient air, or by choosing the optimal substrate to put graphene on. It can also be generated electronically using a technique called gating. Another part of our trick was to use TELBE - the ideally suited source of multi-cycle strong THz field, equipped with a very sensitive THz field detector. It is the combination of these two, that let us succeed at the end.
In our studies we have also found, that graphene possesses an extremely large nonlinear coefficients in the THz frequency range, exceeding by very many orders of magnitude that of all other optical and electronic materials known to us. So, probably, graphene is the most nonlinear electronic material discovered to-date.
Our new results were just published in Nature: H. A. Hafez et al., “Extremely efficient terahertz high harmonics generation in graphene by hot Dirac fermions”,
which can be found here.
Image (c) Junik/HZDR.