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Modeling and simulation of electrically driven quantum light emitters

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and thanks to the organizers for giving me the opportunity to present my research here in the framework of the magnets and this day so this
is a joint work with my colleagues from biased as Institute Marcos McMahon spike to must porky will bundle and intervention it it's
on modeling and simulation of electrically driven quantum light sources and we speak about devices that are based on semiconductor quantum dots so I maybe should explain briefly what a quantum but is so it's briefly denoted as an artificial atoms because it has several properties that we are familiar with from atoms and mainly it's the fact that we have these discrete energy levels here that come from the confinement of carriers in all 3 dimensions such that they are trapped now and but this is now in a solid and not in a single atom and this is very interesting because these discrete energy levels you haven't is quantum dots now give rise to this create optical
transitions which is interesting for many applications in opto-electronics so it's
it's not only a trip for electrons it's also a trip for whole it's which are a quasiparticles that we have in semiconductors which is missing electron and you can see that they have some orbit which would look like atomic physics but in fact it's not it's not really in it's not an atom is more likely in monocrystalline structure that consists of 10 thousand to 100 thousand of atoms that creates this effective potential trap for the carriers and the quantum dots are very interesting for technological proposes because they have these tunable electro-optical properties that you can somehow controlled by the gross process and you can directly integrate them into your a semiconductor device
by standard manufacturing techniques and also integrate them into photonic resonators and they're accessible to electrical control and all these
features make them very promising candidates for a novel opt to electronic and photonic devices and
here I'd like to address maybe mainly the the the the quantum dot based light sources and most leader the sources of nonclassical lights which are single and so
single photon emitters and sources of entangled photon pairs which have emission characteristics that are fundamentally different from light bulbs and lasers and all the so light sources but we know from our daily life so because we have here a light source that emits quantum light which has a certain feature that is called aunty bunching and to understand this we need to measure the 2nd order coherence function of of emitted photons from the device and then you can see that there is some something like a dip here which corresponds to the anti bunching OK what is this most easily understood if we take a look on how this is actually measure it then you have this and very branch was experiment we have a light source of war year that's is emitting photons and you have 2 detectors and what do not want to see is that you have a decrease probability of the 2 attacked in a photo in here when you have measured at the same time a 41 year and so then you have a single photon source and this gives you this dip which is on the bunching isn't so they don't come in bunches and for a light bulbs so for example you have the completes a different situation that they they they show some punching behavior of and this is this is really a non-classical feature of light besides the single-photon sources there are also quantum but none lasers which have very interesting features I'm not going to go into details here and applications for these devices are mainly in the feud off from quantum information processing where you make use of the photo and as an carrier of quantum information and you make use of quantum mechanical observer effect and then you can use it for applications in quantum cryptography or for a linear optical quantum computing and there as applications in metrology and so on so there is apparently a need for a single photon emitters let's come to modeling so as I said
we have some electronic devices and of course we need to do with electronic transport so we need to somehow describe how electrons and holes move in our semiconductor device and this but the standard model for these problems is the so-called funnels book system that you all may know from the talk of demonstrable be read this is the standard model for these model pathway diagrams so what this model does is it's describing their the motion of L introns and holds it within their self consistently generated electrostatic and it's a system of nonlinearly coupled partial differential equations so the first one is apostles equation that gives you the electric field generated by the holes and electrons and then you have continuity equations for the motion of the electrons and the holes with a flux here and they call and also recombine so they get lost and radiator or some the crystal lattice and so on and you need some auxilliary relations here like for example you need an expression for the current which is consisting of a drift term in the electric field and then you have carrier diffusion and you need to will link carrier densities with their chemical potentials and electric field and this is what we are here according to offend duck statistics and to drive the system you need to apply of course a certain bias that gives you the current flow in this is done by a boundary conditions OK so
to give you some examples of this verbs this this year electrically driven single-photon source manufactured at Technical University Berlin so there you have some contacts at the top and the bottom and you have these layers here which are essentially and we have you have some some kind of an insulating layer here but in aperature in the middle and this approach you have you have this you have a quantum but that is some outside controlled Glier nucleated over there and the idea is now that you have current flow through the device which somehow gives you carriers exciting this quantum dot and then you have recombination in the quantum dot and that's giving you a singer for that you want to use for your technology so that's the hope that that's the idea but what you see in experiment is that apparently it does work as intended because you can see in their electroluminescence map you can see that there's a recombination not really from the central quantum dot there is there's a lot of detrimental and emission from quantum dots that are far away from the departure so apparently doesn't work like like intended and the question is what's what's going on here and when you use the finals book system and model the current flow in that device you can see that it's it's not looking like these these red lines it's more like a like this so you have a rapid lateral current spreading of office oxides right here because your devices operating under really extreme conditions so this is very low temperatures and also very low current where you have some counterintuitive phenomena and this electric current spreading leads leads to the excitation of quantum dots far away from the from the center and this explains the 3rd experiment so apparently simulation and modeling of current flow in such devices is a very important task and if you are interested in
that you can also look up the tip AV to see some nice videos of the carrier injection into a that device and the problem here is Of
course I told you about these non-classical properties of the light that we have no heat in these devices so I just say that that that the quantum optical properties and this is apparently not contained in the in the final spot modern so this is describing the current flow but not not the quantum optical properties so we need to we need to do some more and if
you want to describe quantum optics you need to go to quantum electrodynamics and he you starts a for example with the description of the transitions between discrete electronic states so you have a two-level system and imagine this is quantum dot where you have in ground state and you have an excited state for the carriers and they can now have some transitions and moreover this tool level system is now included into a photonic resonator birds interacting with the photonic so they are exciting this transitions and you can also have a mission again and so on and this is coupled and you described is in by using the the the quantum-mechanical density matrix and every physicist knows this you if my mind equation which is the evolution equation for the density matrix that is equivalent to Schroedinger's equation and in the Hamiltonian you'd know write down all the things that matter to you so it's describing the energies of the electrons and the holes there cool of interaction you describe the photo energy and you have to light matter interactions but it then you can describe as the problem is of course this this this is for a closed system so this is given you Hamiltonian system with unitary time evolution and so on this is for quantum system that some does not see its environment and what we what we have is we have a quantum system that is now an embedded into that semiconductor device and it's it's interacting with its environment and then it's not a closed system anymore so we have dissipated interactions with the environment for example we can we can pump our two-level system from the from the outside we have decay processes we have a mission of the photon and so on and to describe that we need to use a quantum last equation which is which is the link that mass equation year where you add on a dissipation superoperator onto your from an equation that somehow can deal with all these processes and and it gives you energy dissipation decoherence and all these things so and yeah equipped with that we have a 2 will bed and enables us to compute all this quantum optical properties in our system the question is
now can be somehow combine these 2 very different levels of description and so we have to find all spoke system which describes the carrier transport we have this quantum last equation that's giving us the microscopic physics and can be some combined as in a reasonable way and it's also a mathematical a hybrid model because we sum the 2 coupled partial differential equations with operator evolution equations so it's not really clear how to do that but
let me add that this may be up so as I said we want to describe their continuum carriers that do the transport coupled to the quantum dots have a problem we have to find all support model we have to you my mind equation and the 1st thing we need to worry about this of course carrier capturing escape so the electron needs to make some transition from the continuum states to the confined state in the quantum dot and this can be done by including certain dissipators here and to balance this these carriers some only to get lost for the for the continuum problem so we have an additional sing term on the right-hand side of continuity equations and on the other hand our quantum dot can of course be the the the charge so when it has an electron it contributes to the generation of electric field so somehow we need to include something on the right hand side of Paulson's equation OK and I'm going to specify this on the next slide this is what we call our hybrid quantum classical model for the description of these electrically driven quantum the devices and so some of this gives us know the full story so we apply some controlled apply some bias at the device we have the current flow from the contact to the active region the carriers are captured into the quantum dot voice according to some quantum-mechanical evolution equations and finally ended a photo on and we can do with the quantum optical characterization of the output beam so we need to specify a
this a little more so or Hamiltonian is describing the single-particle energies and their interactions as I said before the essential point here is that it's we have this constraint that our Hamiltonian evolution of the quantum system must conserve there the charge of the quantum system that means it must commute with the charge number operator which is going to be an important quantity on the next slides so it's simply the difference of the electron number operator and the whole when number operator and this is this quite clear that the Hamiltonian evolution must conserve the charge because this is not an interaction that goes on in the quantum system itself it's an effect that some are mediated by the interaction with the environment so it's contained in this dissipation superoperator which is describing our a bunch of processes like for example a carrier capture and escape and spontaneous emission the phasing relaxation and so on and so very lead these operators fall into 2 classes so we have we have these carrier capture and escape which change the charge of the quantum systems so they are not charge conserving and then the we have all these other processes which lead the charge invariant and structurally the dissipated as certain form looks a bit complicated on 1st sight so we have we have a
transition rate that is somehow depending on the transition energy on gun and we have these link blood soup operator which involves a certain jump operator for the process this discovery briquets here are the is this and commentator and
I Pfizer's labeling a certain process and we have many many processes you like these here and these rays always appear in pairs as you always have a forward and a backward processor yet capture you have escaped you have generation you have recombination and so on and this is what we need to include you in our evolution equation in our case the situation is a bit versus more complicated estes transition rates not only depend on the transition energy they also depend on the state of the of the macroscopic environments or of our PDE problems so we have a coupling already at that level now so that they did the electron capture rate is apparently bigger if you have many electrons in the vicinity of the quantum of and no we need to model the the terms that kappa back to the finals book system so we need to find an expression for the for the contribution on the pulse also equations right hand side so we what we do here is we take the expectation value of the charge number operator which the trace of the operator with the density matrix and to give this some spatial information because we know we need to include that into a spatially resolved transport model we invented this spatial profile function that somehow makes the wave functions square of our confined carriers and in a similar fashion we can write down the the the contributions to the right hand side of the continuity equations where we have now not the expectation value of the charge it's the expectation value of a certain charge flux namely the electron flux from the from the continuum levels to the to the quantum dots and they'll again include these spatial profile functions because that somehow localizes rather process happens and that if you do it like that you have a very nice property it didn't take the time derivative of possums equation and just in there these equations and exploit some some some other relations and then you find that the total current as divergent which means we have low could charge conservation OK that's a very nice property and so what
I want to show you on the next slides is that the system has several other very nice properties so let's discuss the Tamil wooden and properties and they want to show you that we
just did not just some merge to 2 different equations baby be we did it in a in a certain fashion that as he received a day's give something reasonable in the thermodynamic equilibrium they give
us some detailed balance relation and I'm going to show you is that they obey the 2nd law of thermodynamics so and this is very important as we learned and many talks in and in in this conference it's it's it's important to to to be consistent with thermodynamics and this is also true in semiconductor device simulation so in the
equilibrium we have to this grand potential that someone needs to be minimized which is included in the free energy of the problem and then there's a constrained for the charge conservation and what we need to do is to write down the free energy functional which has this contribution from the from the classic motion of electrons and holes left the quantum system and then there's some electrostatic interaction and minimization with respect to the density matrix gives us an expression that looks very similar to the ones from from the textbook so it's a it's a deep state and it has this in addition a contribution here that arises from the interaction with environment so the quantum system has some spatially averaged it it's it's fused a spatially averaged electrostatic potential you have which it gives you this off of set and yeah so apparently it interacts with its spatially averaged environment and this this is now a transferred to the to the non-equilibrium case where we now ordered this transition rates appearing in the dissipation superoperator as functions of the spatially averaged microscopic potentials and then we just now do some fits to microscopic calculations or experiments to model these transition rates and then we find that very deep inside that there is a relation hidden which is the core Wal-Mart injuring condition that gives you a certain relation between the forward and the backward rates they're related in such a fashion that you come up with the quantum detailed balance relation which essentially means that your dissipation superoperator of every poses I vanishes under equilibrium conditions so this is the quantum detailed balance relation that is encoded here so and so this this this is included and finally we are able to write down the entropy production rate after couplet problem which has several terms it's not important what they are the 1st line
is arising from the entropy production of the spoke system itself this is somewhat entropy production of the quantum system itself and these 2 arise from the coupling and what we can do
is that we can show that each contribution year itself is non-negative and therefore the entropy production rate is non-negative which means that our system is consistent with the 2nd law of thermodynamics which is I guess a very strong theoretical results
and OK I have something on the numerical method but I'm going to skip that it's essentially
finite volumes method under very
nasty a very nasty parameter regime that like to show you something on the
application so what we want to do now is to describe the physics of such an electrically driven singer single-photon stores where you have contacts on the top and the bottom and you have a quantum dots here in the center and we need to specify the Hamiltonian here so Hamiltonian is describing a set of microscopic states so we have a quantum dot that this and here you have no carriers captured the single-particle states would have single electron or hole you have accidents ones and the highest status of B exit on
and for example lexicon as an electron hole pairs you can see here and these estates on our linked by certain dissipate of interactions with the environment so you go from the empty set to the single-electron state by capturing an electron and in a similar
fashion and you can if you have already a whole would go to 1 of these accidents states by capturing an additional electron in the same fashion you can capture holes and you have a bunch of allow its optical transitions where for example the the decays gives your photo on it and you you lose Biden exit on which decays then under the quantum dot is empty and when we know do some calculations we find for example we can give to the single photon generation rate of of certain optical transitions like for example generation from exit honorably exit on and you can you also have injection current from the finals spoke model and you can bring this not together and it looks as if we are qualitatively similar like in experiments so the order of magnitude fits I don't know the parameters of this problem so looking very similar so the this is nice I would say we can also calculated that the 2nd order correlation function which gives us this characteristic dip that says we have and the bunching and it's really a quantum light source and it's also which is an experiment
and we can also do a transient calculations for we now have a look on the pulsed operation value we apply a biased pies and then we have a carrier injection into that intrinsic zone year the you have if from spread out of the of the current flow and to military Lexus back and here in the center of the quantum dot is placed and you can see that we can now do with that current
calculations have consistently with the quantum dynamics and we see that the the exit on his occupied and then decays giving us the decay of the exit on with some delay let me summarize I have presented
a hybrid quantum classical modeling approach for electrically driven quantum light sources I showed that our model whose system is consistent with fundamental laws of non-equilibrium thermodynamics and I showed a brief that briefly an application to electrical driven singer fought on images and the claim is now that we can have carrier transport and quantum optics out of 1 box yeah this is published and pure being thanks for funding to the as a southern at 7 semiconductor none of photonics and my colleagues from last as Institute and thanks for your attention
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Titel Modeling and simulation of electrically driven quantum light emitters
Serientitel The Leibniz "Mathematical Modeling and Simulation" (MMS) Days 2018
Autor Kantner, Markus
Lizenz CC-Namensnennung 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/35428
Herausgeber Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS), Technische Informationsbibliothek (TIB)
Erscheinungsjahr 2018
Sprache Englisch

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Fachgebiet Informatik, Mathematik
Abstract The design of electrically driven quantum light sources based on semiconductor quantum dots, such as single-photon emitters and nanolasers, asks for modeling approaches combining classical device physics with cavity quantum electrodynamics. In particular, one has to connect the well-established fields of semi-classical semiconductor transport theory and the theory of open quantum systems. We present a first step in this direction by coupling the van Roosbroeck system with a quantum master equation in Lindblad form. The resulting hybrid quantum-classical system obeys the fundamental laws of non-equilibrium thermodynamics and provides a comprehensive description of quantum dot devices on multiple scales: It enables the calculation of quantum optical figures of merit (e.g. the second order intensity correlation function) together with the spatially resolved simulation of the current flow in realistic semiconductor device geometries in a unified way.

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