Nanomagnetism

Nanomagnetism is a very broad field where one can find exciting physics ranging from the correlations induced in the Kondo effect, to the dynamics of spin excitation and de-excitation on the atomic scale, passing by exotic state such as Majorana bound states in superconductors. 

We study nanomagnetism from the point of view of atomic spins. Their dynamics and correlations entail the processes we just described. But quantum coherence and entanglement naturally appear when atomic spins interact with each other. 

Recently, a new development of the STM has produced the exciting possibility of fully characterizing the properties of atomic spins one by one. This is the study of Electron Spin Resonance using the STM (ESR-STM). Manassen and collaborators and Komeda and Manassen used GHz excitation frequencies to study single magnetic moments in Si. More recently, other experiments also using STM showed data that molecular spins could also be addressed in a GHz pulsed STM. 

 New highly-reproducible data with quantitative measurements and explanation recently came available in the low-temperature set-up of IBM (USA) and QNS (Republic of Korea). They showed compelling evidence that Rabi oscillations were established between to Zeeman-split spin states of an Fe atom on an MgO surface when a GHz modulation signal was injected on the electron current. Evidence was given that time-dependent electrons can drive the magnetic oscillations. 

 

Figure.- Electron Spin Resonance principle in a scanning probe junction. (a) At low electron-tunnelling rates, a radio frequency (RF generator) signal is added on the applied bias (VDC). The frequency is swept to match the Larmor frequency of the precessing spin under an external magnetic field B. For an atom of spin S, the Larmor frequency is the Zeeman splitting. The tip of the STM is itself polarized in order to inject electrons with a well-determined spin. In a very simplified picture, the tunnelling current increases when the atom spin is parallel to the tip's magnetisation and decreases when it is antiparallel. As a result, the current will experience a change when the radio frequency bias matches the precession frequency. (b) ESR spectrum of a Ti atom on two layers of MgO/Ag(100) with an external magnetic field of 0.7 Tesla parallel to the surface plane. The behavior of the current as a function of the frequency is a peak/dip-like function whose width is roughly the inverse of the coherence time T₂.
 

Coherent oscillations.- Let us assume that we have two clear spin states that can be connected via an oscillating driving term (the electric field between tip and sample in the STM setup). Then there are three possibilities: each state is populated (two states) and a cross term between the two states. This cross term measures how coherent the two spin states are. When this term goes to zero, the two states are totally disconnected they are incoherent. 

Incoherence means that their respective phases change in time randomly. If the phase is constant (or almost constant) in time between the two states, they will interfere. The lifetime of each state, the coherence time and the Rabi frequency and the resonance frequency. Here, we have one excited states then there is only one lifetime, T₁. 

The other obvious lifetime is the pure dephasing time, or the time when the relative phase between the two spin states is constant, T₂^*. Please be aware that in NV centers a different notation is followed. 

The third time is the one corresponding to the period of the Rabi oscillations. The Rabi oscillation frequency corresponds to population oscillations when quantum tunneling between two states controls the population dynamics. This frequency gives the strength of the coupling between the external driving force and the reaction of the system to that external field. 

The fourth time is given by the resonance frequency or the energy of the excited spin state, f₀ such that the excitation energy is E=h f₀ where h is Planck's constant. 

We have worked with our collaborators at QNS and achieved several exciting results, both studying the conitnuous-wave (CW) regime or the pulsed one. From the theory side, we have developed a quantum-master equation code that takes into account the DC bias, the driving and the conductance properties of the STM junction to reproduce both the CW and pulsed experiments.

 

Inelastic electron tunneling spectroscopy (IETS).-

Besides scanning, the STM can also be used to measure the differential conductance over selected points on a surface. This can be done at different vacuum gaps. Indeed, the tip can be approach as much as to press the surface and produce conductance measurements in contact. The STM is a very versatile measuring instrument that can give a lot of information on the atomic arrangement of unknown structures, their electronic properties and their conduction ones. The differential conductance, dI/dV (where I is the electronic current between the tip and the substrate, and V the applied bias), contains a lot of information.

When there is a vacuum gap between tip and substrate, the electrons form a tunneling current. In this conditions, the conductance is proportional to the spectral function. See the work by Tersoff and Hamann among others for some indications of how this can be obtained. The spectral function is also called the density of states, because it gives the number of states per unit energy. The idea is that the bias is related to energy through the charge of the electron, and the differential conductance is some sort of density of current per unit energy. In the Tersoff-Hamann picture, the changes in current are in part due to the changes of states in the sample.

The differential conductance is then proportional to the density of states. However the STM has the ability to focus the current into an area well below atomic dimensions. The density of states takes into account this by becoming the local density of states (LDOS). This is a density of states weighed by the spatial distribution of the wavefunctions corresponding to the states at the energy matching the bias. The spectral function is a generalization of the density of states. For the STM it is the local density of states.

And it contains all possible types of electronic states. Particularly, excitations of the system. The spectral function is mathematically defined by the function that describe how one electron propagates into a system of electrons in interaction with all possible degrees of freedom: vibrational, magnetic, etc. The conductance of the STM gives direct information of the system states that can be excited by an injected electron. The STM can then measure excitations. This gives rise to the Inelastic Electron Tunneling Spectroscopy (IETS) using the STM.

IETS was developed in the 60's for planar tunneling junctions using oxides or other insulators, in a thin film, between two metals (or superconductors). Dramatic signals of vibrational modes of impurities in the insulator were detected.

Theory of IETS.-

With the above physical picture in mind, we can then generalized the Tersoff and Hamann approximation to its many-body variant where all possible excitations of the system are included. To do this we just need to use the generalization of the LDOS. For that, the easiest way to go is to use the one-electron propagator in the system. The imaginary part of the propagator is proportional to the LDOS when it is evaluated in real space.

The propagator or one-electron Green's function evaluates the probability of creating a particle in a point r at a given time t and finds it in another point r' at a later time t'. This is a retarded propagator because it follows the time evolution. We can mathematically express it by:

This is a very complex equation that contains all possible physics of the transport problem of an electron into a complex system. But if you have some way of calculating it, then you can have a perfect IETS simulator.
The equation tells you that you inject an electron by creating a particle with the field operator:

Here the annihilation operator c, destroys a particle in state lambda. When you add up over all possible states, each with its spatial distribution given by the wavefunction of the state and its time-dependent phase, you have all information you need on the local and time properties of the electron. The above Green's function contains an anti-commutator and a theta function that take care of the correct time evolution.
There are a number of treaties and web documents where correlators, propagators and Green's function in general are reall well explain. Check the one  by Odashima, Prado and Vernek.
Despite their complexity, the advantage of the above formalism is that it includes everything and then you can choose your controlled approximation to evaluate it as realistically as possible.

In order to evaluate the LDOS with IETS information, you need to Fourier transform the above propagator (assume invariance under time translation, or a t-t' dependence) and in the spirit of the  Tersoff and Hamann approximation evaluate it in real space at the position of the STM tip.

Let us assume you would like to compute the IETS of a graphene sheet due to the excitation of phonons. There are awesome experimental data on this (i.e. the paper by the Crommie group, or the paper by the Stroscio group). And it is very interesting to have quantitative simulations that permit us to clearlyu ascribed the different physical processes at play. One way to go is to evaluate the electronic structure of graphene with DFT, also the phonons, and then evaluate the electron-phonon couplings. Nowadays this can be done using the software developed by the quantum espresso research effort. A particularly interesting tool is the one developed by the Giustino group. The EPW code allows you to compute the electron-phonon coupling with very large precision. You can easily use this to evaluate your Green's functions. Its imaginary part will be proportional to the IETS conductance.