Electronic Transport Calculation of a Selective Gas Sensor Based on an InAs/InP Triple-Barrier Heterostructure
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Slides of the oral presentation at the 9th Iberoamericano Congress on Sensors. Bogotá, October 2014
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Electronic Transport Calculation of a Selective Gas Sensor Based on an InAs/InP Triple-Barrier Heterostructure
- 1. Sebastián Caicedo Dávila José Ferney Rivera Miranda Jaime Velasco Medina
- 2. Taken from: www.vernier.com James, D., Scott S. M., Zulfiqur Ali, O’Hare, W. Chemical Sensors for Electronic Nose Systems. Microchimica Acta, 2005 Yi Cui, Qingqiao Wei Hongkun Park, Charles M. Lieber. Nanowire Nanosensors for Highly Sensitive and Selective Detection of Biological and Chemical Species. Science 2001 Larry Senesac and Thomas G. Thundat. Nanosensors for trace explosives Detection. Mat. Today 2008
- 3. Taken from: Ilia A. Solov'yov,. “Vibrationally assisted electron transfer mechanism of olfaction: myth or reality?” Phys. Chem. Chem. Phys., 2012,14 Vibration of protein alpha helix, taken from Wikipedia
- 4. Taken from 1995 BBC Horizon documentary “A Code in the Nose” about Luca Turin's vibration theory of olfaction.
- 5. Very sharp energy levels can be used as allowed states, lessing the effect of thermionic excitation’s current. Such levels (QB-states) can be created by confinement in 1D. Heterostructures are suitable candidates for the job. Semiconductor sensors are very effective and easily integrated with conventional electronics. A. P. Horsfield, L. Tong, Y.-A. Soh, and P. A. Warburton, “How to use a nanowire to measure vibrational frequencies: Device simulator results,” Journal of Applied Physics, vol. 108, no. 1, 2010.
- 6. Bias applied accross the TBH shifts the energy profile (and the QB states). Vibrational modes of an adsorbate at the middle PB can be excited by electrons that will lose energy and tunnel through the device, just as Turin’s theory proposes.
- 7. InP Barriers Cu Electrode Cu Electrode InAs Wells InAs Wire InAs Wire Right Well Left Well Source Drain
- 8. En E4 E3 E2 E1 H ... {Ψ} Stationary QM system: closed system represented WF doesn’t change in time. En E4 E3 E2 E1 H ... {Ψ} μ HR {ΦR} [τ] Coupling to the contacts broadens the discrete energy states.
- 9. e +i g 2 æ è ç ö ø ÷F = EF E -e -i g 2 æ è ç ö ø ÷F= 0 E -e -i g 2 æ è ç ö ø ÷F= S Homogeneous equation Non-homogeneous equation G = E -e -i g 2 æ è ç -1 ö ø ÷ Green’s Function {F} = [EI -H - S]-1 {S} G = i S- Sé * ë ù û
- 10. Spectral function A(E) = i G(E) -G(E)é + ë In the Eigenstates Basis ù û Non-equilibrium density of States
- 11. Current as change in Electron Density Remember {F} = [EI -H - S]-1 {S} i ¶ ¶t {F} = E{F} Equivalent to the Landauer Formula T (E) = Trace G1G G2G+ { }
- 13. Potential profile (dashed blue), LDOS in the wells regions (solid red and magenta) and transmission (solid black), showing the whole energy picture of the system at thermal equilibrium.
- 14. Transmission coefficient vs. Energy for the system at thermal equilibrium. The transmission peaks (double due to bonding-antibonding combinations) arise due to the alignment of quasi-bound states of the wells. The red dashed line shows the bottom of the conduction band.
- 15. I-V characteristic curve showing peaks and valleys (resonant device). The latter are the regions most suitable for gas sensing.
- 21. • A gas nanosensor, based on an InAs/InP triple-barrier heterostructure, devised to achieve high selectivity and detect vibrational modes of molecules was simulated at room temperature. • We proposed a simplified 1D model and used NEGF, and a single-band effective-mass Hamiltonian, to calculate the I-V characteristic curve of the semiconductor triple-barrier heterostructure at low computational expense. • We were able to demonstrate the plausibility of the device, by analyzing the LDOS in the wells regions, which showed that our model device would be able to sense different vibrational modes of SO2. • NEGF lets us include all kind of interactions between the system and external stimuli. We can include electron-phonon interactions and the interaction with vibrating molecules adsorbed at the device region, only by adding extra self-energy matrices.
- 22. • Consider a self-consistent potential in order to include space-charge effects. • Build and simulate a 3D model with finite cross-section, attached to metallic contacts. • Analyze the phonons of the simplified model, and build a self-energy matrix describing the electron-phonon interaction. • Calculate the Hamiltonian and self-energy matrices using self-consistent DFT. • Build additional self-energy matrices that describe the interaction of vibrating molecules with the device. • Determine the selectivity of the device by performing calculations of the interaction with several molecules.
- 24. Questions?
Editor's Notes
- Good morning to everyone! Today I want to share the story of this project: it actually started as an accident. As an undergrad student I discovered nanotechnology and the idea of being able to manipulate atoms to build something useful dazzled me. So, I started to look into it. I had a very exciting project with a friend of mine (she’s a chemist): we wanted to design a sensor for a biomarker of breast cancer. Unfortunately, due to university’s policies we weren’t able to do that, and that’s when Professor José Ferney came up with this very exciting idea. Of modeling a gas nanosensor.
- I’ve always been excited by the idea of transducers and I’m specially interested in chemical sensors, since everything around us is chemistry. From the sugar-coffee mix you drink every morning, to the enzymes that are driving you excited at this talk, right? Then, I decided to focus my research efforts in novel chemical sensors, which are very attractive to me and to multi-million dollar markets (that’s also kind of motivating) such as border and industrial security, food and pharmaceutical industries among others. They are so attractive because they are portable low cost devices, they consume low power, and they’re simple to use (no trained personnel required: I don’t see every single guard at airport security being an expert analytical chemist). However, sensors lack the sensitivity, selectivity, robustness and accuracy that analytical methods provide and hey are usually tuned to sense a specific chemical species (or more like a range of them). Here we see in the slide some examples of such conventional chemical sensors. See that the sensing areas, where the actual action happens, are covered by molecules with certain shapes (this is usually called functionalization), that will allow target analytes to bind to the device.
- Well, nanoengineered structures are key devices to overcome some of the problems I mentioned before. That is because they are VERY sensitive, due to the surface-area-to-volume ratio. Now, when we go down to the nanoscale, quantum mechanical effects start playing an important role, so, you have to actually learn quantum mechanics in order to be able to design, simulate, build and work with these nanodevices. I’m an engineer not a physicist, but I had to go through intensive quantum mechanics learning the past two years, only to understand what I was going to do and I stumbled upon this phrase of one of the finest and most inspiring physicist for us nantechnologist, Richard Feynman. It turns out he’s right. As engineering is driven to the nanoscale, we will have to learn how to use quantum mechanics.
- Ok, let’s just finish the foreplay and go to what’s interesting. The idea of this sensor was inspired by the theory of smell proposed by Luca Turin in the early 90’s. The idea is that smell vibrations, instead of the classical shaped-based theory we get from high-school biology. But how? Well, let’s say we have a device with two contacts at different energy levels, here in the picture called donor and acceptor levels (say the donor is the one that puts the electrons into the system). Electrons are fuzzy particles that can jump through this gap, but only at equal energy, that means, the electron here, cannot just go to this other side, because what would happen with the excess of energy? Well there is a simple solution to that: excite vibrational modes. That will absorb the energy excess. Let’s see a short cute video I extracted from a documentary that explains Turin’s theory.
- The battery drives the energy of the contacts to different levels, if a molecule with the right vibrational energy is adsorbed in the gap, electrons can tunnel through the device and produce current. That is actually the principle of inelastic electron tunneling spectroscopy.
- Using this ide, Prof. Horsfield at UCL devised a nanostructure that could serve the purpose of vibration recognition His idea was to use very sharp levels of energy, that electrons can populate, and by moving those up and down in energy, he could create different “energy gaps” and therefore differente vibrational energies (so molecules) can be detected. He used for this device a triple-barrier heterostructure, a structure with different kinds of materials, with different energy profiles, to form potential barriers and quantum as we see here in this cartoon schematic. The 1D confinement inside this QWs creates this sharp levels known as Quasi-bound levels.
- When we start applying bias to this thing, the QB-states start shifting, and the difference in energy between the states in one well and the states in the adjacent well can be used as the gap that the molecular vibrational energy should equal., as we see in this slide.
- Stationary states, separable WF, Potential independent of time Wavefunction (WF): Fully describes the state of a system Experimentally measurable quantities can be calculated by applying a hamiltonian operator to eh WF The time-spatial evolution of the WF is governed by the Time Dependent Schrödinger Equation (TDSE) For Stationary states, we can build the Time Independent Schrödinger Equation (TISE), which we can relate to the TDSE using:
- TISE can only be solved analytically for the Hydrogen atom. For bigger systems we represent the WF as a linear combination of known functions, usually atomic orbitals, The approximation known as Linear Combination of Atomic Orbitals (LCAO) LCAO allows us to represent the TDSE in form of a matrix eigenvalue problem, with operators as nxn Matrices and WF as n-dimensional column vectors.
- Well, this was a very interesting idea, and Horsfield modeled this using a very interesting semi-classical approach, but he left something open for discussion: How well can be this model improved, and how can we actually model this device to see if it would actually work as a sensor. This is where we come in. We wanted to try to solve this questions, and as a first step, we built a 1D model that we can easily solve in a personal computer. We used the finite differences method to discretize the whole device, using the lattice constant of each materials respectively. Here in the slide we point out some of the interest points we used to analyze the device. But let’s start with the boring part of this story, the theory. And please try to stay awake while I try to quickly go through this, ok?
- Well, to start, we describe electronic transport in a quantum system as teo Using QM is mandatory to accurately describe the properties of atomic-sized devices. Electrodes with a scatterer in between. Contacts can be described using Bloch’s theorem. Metallic contacts preserve local charge neutrality, energy level is shifted rigidly, different from the SR, calculated self-consistently. Continuity of EP necessary to avoid spurious scattering. The separation lets us calculate an otherwise impossible system. We want to model two probe molecular electronic devices. Electrodes are modeled as semi-infinite, perfectly crystalline structures. To ensure the continuity of the electrostatic potential at the interface between contacts and the molecule, the extended molecule is built. We can treat each region separately and the coupling can be described by a self-energy matrix, which depends on the potential and charge distribution in the electrodes.
- In an isolated quantum mechanical system we have a system described by a wavefunction Psi and a Hamiltonian operator (a matrix for calculation purposes) that describes the energy of the system. In such a system no electrons are exchanged, and if we assume the system is stationary, the WF does not vary in time (this of course is not true for real systems). But when we couple this isolated system to a contact, the once discrete sharp levels broaden and the system starts exchanging electrons with the cintact, here described by a new function Phi and a Hamiltonian Hr. The coupling is described by the matrix Tau.
- Let’s check a one level example: For an open system, we have to add a complex quantity to the energy, so the system is no longer hermitian (the non-hermicity describes the open system). We write an expression similar to TISE: This is however a system in thermal equilibrium (no bias voltage applied). When we apply voltage, the contacts drive current according to a source term S turning this homogeneous equation into a non-homogeneous one, like this. From here, we can define the Green’s function as this, and if you see, this is kind of the impulse response of the wavefunction, if we see this as transfer functions. We see here that this is extended to a more general treatment in matrix form for several energy levels. Sigmas, are called the self-energy matrix and describe the coupling to the contacts. They are non-hermitian, and its anti-hermitian part, Gamma (the capital Gamma) is the broadening matrix (this one here in the one level example), and as the name says, it describes how the levels of the isolated system broaden with the coupling.
- Now, the density of states at a given basis is defined by this equation. If we use a real space representation, as is our case (we discretized in real space), we can then obtain LDOS at each point of the real-space lattice. Now the spectral function is defined by this equation. We see a certain similarity in this two equations. What we see is that the DOS is kind of contained in the Spectral function when alpha equals beta. Now, if we use the matrix form of the spectral function (here in the eigenvalue representation) we see that the diagonal elements are the DOS. Now from this definition here, and with some algebra that for time reasons we are not able to develop, we can define the spectral function in terms of the Green’s function. And this we will use to calculate LDOS and to demonstrate how this device works
- DM seminal concept of non-equilibrium statistical mechanics. IN MATRIX FORM AND EIGENSTATES REP. Using the definition of Dirac’s delta as the limit of the Lorentzian function. The first is the retarded and the second the advanced Green’s function. The DOS is then the anti-hermitian part of GF So the DM is deduced from NEGF and therefore everything else. Approximations only to calculate the Hamiltonian.
- Now to finish, and I promise this is the last slide filled with equations (so now you can wake up), let’s see how we can calculate the current from the Green’s Function. We can see the current as the change in time of the electron density (or the density matrix, defined as the norm squared of the WF). Now we use the definitions of Green’s function and the time-dependent Schrödinger equation, the chai rule and some love for math, to obtain this equation here. And if we inspect this in detail is nothing more than the Landauer equation, where we can define the Transmission function as the trace of this sandwich here. Now that we have all the math we need, let’s go to the fun part: THE RESULTS!!!
- Well for all these calculations, we had to make some assumptions to make them possible. The first ones is that we assumed a potential profile with three barriers, obtained from the difference in bandgaps of the semiconductors. In more accurate calculations, this has to be calculated self-consistently. The other assumption is that we assumed incident and reflected waves at the contacts interface. This is what we used to build the self-energy matrices. The metallic nature of the contacts, was included by controlling the position of the Electrochemical potential in the Landauer formula.
- Here we see the LDOS in each QW (sum of LDOS of each point within the QW region). We can see here the highly localized quasi-bound states that will serve as the energy levels for electrons. See this splitting here? That is due to the formation of bonding-and anti-bonding states. Let’s say the QB states are described by a WF, well this fuzzy WF overlap over the middle barrier anf form this bonding and anti-bonding states combinations. The states over the barriers are known as shape resonances, which are nothing more than states that feel the potential beneath and localize a little We see that aligned states generate resonance peaks, which is logical, because the probability of electrons tunneling between states at the same energy level, is high.
- Here we see a clearer picture of the transmission at thermal equlibrium. This transmission peaks are due to QB states. Note that the splitting is lower for the more localized states. These peaks here, are due to shape resonances as we mentioned before.
- Now, after we calculated the I_V curve, we noted the typical curve with negative differential resistance that we expect from a resonant tunneling device. We can distinguish two regions for active sensing, which are the regions where the states are unaligned. The fuzzy peak here is the consequence of the alignment of states in the adjacent wells (the lower QB state of the left well, and the second QB state in the right had side well). The peak at the beginning is the result of the initially aligned states, that start to getting unaligned as we apply bias, thus the current reduces.
- Now, what can we do with a device like this? Well here we have a set of molecules and the energies f it’s vibrational modes. Some of these can be used as a toy example, but we see that the vibrational energies of organic molecules are high, and we wouldn’t be able to sense those with this device. For gases like the ones in the table below, we do can use the device proposed in this work, so we chose Sulfur dioxide as an example.
- Now, see these three little cutes dancing around. Well this explain the vibrational mode of sulfur dioxide: Symmetrical stretching, bending, and anti-symmetrical stretching. The table here below, summarizes the vibrational modes energies, and the voltages at which they could be detected with our proposed device. See that for bending, the middle dancer, we have only one detection voltage, since its vibrational energy is low, but for higher energies, we found two detection voltages, an we sill explain why.
- Here we see that at 0.34V approximately, we obtain an QB states separation equal to the vibrational energy of the bending mode. The lowest QB states of each wells are implied. Note as well that the splitting completely disappeared, sinche there is no more WF overlapping.
- Now, for the stretching modes, we see that we get two detection voltages: one, due to the separation of the lowest QB states in each well, and other due to the separation of the second, or upper QB states in each well, like here. The second voltage starts to play a role in detection only when it lowers below the electrochemical potential of the left contact, like this. That is why the second QB states are not useful for detection of the bending vibrational mode.
- Here we see a picture very similar to the prior one, for the anti-symmetrical stretching mode.
- To summarize, we have developed a simple 1D model of a Triple barrier semiconductor heterostructure, that can be used as a gas sensor. We simulated that simple model at room tempreature, which is one of the biggest drawbacks of current Inelastic electron tunneling spectroscopy that has to be carried out at low temperatures. We demonstrated the plausibility of such device by using a toy example of a Sulfur dioxide molecule. And we used a simple method, the non equilibium Green’s function, that allows to include any perturbation to the system and in future studies may help study this device in more detail.
- However, there’s still too much work o do on this. We would have to consider a self consistent potential and hamiltonian to include space-charge effects. We have to build a full atomistic 3D model, and solve it using techniques like DFT, to anlyze things as important as phonons in the structure (that would affect its performance) And finally build self- energy matrices to include the adsrobate in the calculation, obtaining spectra for different molecules.
- I want to express my gratitude to all the memebers of the bionanolelctronics research group of universidad del Valle for their support to this project, as well as the German academic exchange service, and its academic literature program.