High performance models for charge transport in large scale materials systems
- Thomas Frauenheim (University of Bremen, Germany)
- Tim O. Wehling (University of Bremen, Germany)
In the last decades the amount of materials employed in the fabrication of devices for optoelectronics, energy harvesting or energy storage has largely increased, including in the family relatively novel materials like organic semiconductors, nanostructured organic and inorganic compounds and two-dimensional layered materials. A reliable modeling of charge transport properties in these systems need to meet several requirements: physical accuracy at a quantum-chemistry level is needed due to the breakdown of semi-classical physics at nano-scale or chemistry scale; a broad range of applications, as new materials and novel design strategies are constantly advancing; scalability, as device and materials properties are strongly influenced by geometrical or morphological details on length scales up to micrometers. These requirements can be fulfilled only by combining hierarchical and multi-scale numerical and analytical approaches at atomistic scale, which have to be founded on ab-initio techniques.
Combining efficient atomistic methods based on ab-initio theory, powerful analytical techniques and multi-scale hierarchical approaches up to continuum models is most promising to fulfill those requirements. Most of these methods are based on efficient tight binding representation or, at a more sophisticated level, ab-initio methods with localized basis such as Wannier functions or Gaussian basis. As very different communities are involved in development of methods and related software based on similar grounding, the aim of this workshop is to bring together these communities to foster interdisciplinary collaborations within the area of computational/theoretical charge transport simulations. We are aiming to attract and invite the internationally leading scientists in the field to share their state of the art developments for initializing the next generation release of high performance models for charge transport in large scale materials systems. Furthermore we will stimulate young researchers to broaden their methodological basis for addressing novel interesting applications.
The rise of novel materials for energy harvesting, energy storage and optoelectronic applications and the rush for device miniaturization, driven by the needs of computer industry, pose new challenges in modeling the physics of charge transport. Industry is moving massively towards novel materials: polymer and organic materials nowadays are routinely employed. Organic and inorganic 1D and 2D materials are gathering large attention, single molecule devices could play a role in the emerging field of spintronics [1 - Sanvito2010]. The landscape of novel materials is that rich that high-troughput Computer Aided Design (CAD) techniques are foreseen for the near future [2 - Curtarolo2013]. Even in “traditional” electronic devices, the design of silicon-based transistor has been moved to multi-gate structures to keep up with Moore’s scaling law [3 - Zaima2013], and the boundary between device and materials design is becoming more and more evanescent. Despite the deep differences between the various types of materials, some common challenges connect different communities: atomistic details are of fundamental importance, an efficient transfer between ab-initio and approximate charge transport methods and a deeper understanding of non-equilibrium charge dynamics are indispensable.
A fundamental starting point is the choice of appropriate quantum mechanical methods for calculating the electronic structure at atomistic level. Depending on the system size, empirical and semi-empirical tight binding methods and ab-initio Density Functional Theory (DFT) techniques based on Gaussian and Wannier orbitals are commonly employed [4 - DiCarlo2003, 5 - Pecchia2004]. The choice of a particular methods is dictated by a tradeoff between range of applicability, physical grounding and basis size. DFT is nowadays the reference method, as it ensures a parameter free approach vital in the investigation of novel materials. Impressive improvement have been made in the field of O(N) parallel DFT, and codes capable to scale up to several thousands of atoms have been developed [6 - Bowler2006]. In bulk systems, localized basis sets can be obtained by determining the optimally localized Wannier functions associated with the Bloch states, as pioneered by Marzari [7 - Marzari1997]. Semi-empirical methods as Density Functional Tight Binding (DFTB) [8 - Elstner1998] have been successfully applied to inorganic and organic systems and combine a complexity close to ab initio methods with a physical grounding which ensures, among the others, a high transferability compared to empirical methods. Empirical methods can scale up to tens of millions of atoms [9 - Usman2011] and are routinely employed in the study of inorganic nanostructures; in order to tackle the cumbersome process of deriving a good parameterization, automatic tools have been proposed. In organic bulk systems, where geometrical fluctuations in the length scale of micrometers can determine electrical properties, fragment orbital coarse-grained methods are commonly employed: hierarchical coupling with ab-initio calculations at a different level of approximation ensures a certain degree of generality [10 - Nelson2009, 11 - Troisi2009, 12 - Kubar2010].
Despite the huge variety of electronic structure methods, the localized basis ansatz allows for similar techniques to investigate charge transport properties in the limit of linear response, non-equilibrium steady state (NESS) or full time-dependent. NESS has been for long investigated by means of Keldysh Green’s function formalism [5 - Pecchia2004]. The advantage of this technique is that it is very general and has been indeed applied in combination with all the previously listed electronic structure methods.
Microscopically, the transport properties of most materials are governed by a Hamiltonian H of some dimension D. Even for nm-size samples, the number of involved orbitals and lattice sites put D generally beyond a value, where response functions like the DC or AC conductivity or the time evolution of quantum states could be constructed from the full diagonalization of H. The problem is that the computational cost for full diagonalization of H scales as D^3. Fortunately, there are very efficient linear scaling alternatives called kernel polynomial methods (KPM) [13 - Weiße2006]. These iterative approaches require only multiplications of H with a small set of vectors to calculate dynamical response functions or also to simulate the explicit time evolution of quantum states. In this way, KPMs facilitate the treatment of systems containing up to D~10^9 degrees of freedom and they have been applied to various materials from bulk semiconductors to low dimensional materials like graphene. In low dimensional structures, system sizes on micron scale come into reach and systems of up experimental sample sizes can be treated [14 - Yuan2010]. In carbon materials like disordered graphene, KPM simulations played an important role to explain AC and DC electronic transport or electronic screening. Various physical regimes have been addressed, from ballistic to diffusive electron dynamics and finally Anderson localization [15 - Lherbier2008, 16 - Wehling2010, 17 - Lherbier2011, 18 - Leconte2011, 19 - Cresti2013]. The explicit time evolution of trial wave functions can be tracked by using KPM / Chebyshev polynomial methods to extract localization lengths in disordered graphene samples and nanoribbons [20 - Schubert2009, 21 - Schubert2012]. In graphene the low energy band structure is very simple (Dirac states derived from carbon pz orbitals) and realistic models for impurities like adatoms and adsorbed organic groups could be derived from DFT calculations (see e.g. [16 - Wehling2010, 18 - Leconte2011]). It is generally however quite man power consuming to bring real material aspects of host, disorder and impurities reliably into tight binding models for the treatment with KPM methods and therefore linking to the field of DFTB simulations appears very promising.
On the other hand, in order to model charge transport at a device level non-equilibrium and geometrical features must be taken into account. Full devices up to millions of atoms have been modeled by means of Non Equilibrium Green’s Function (NEGF) formalism [5 - Pecchia2004]. The advantage of this technique is that it is very general and has been indeed applied in combination with several localized basis electronic structure methods. As a “trivial” implementation scales as O(N^3) and includes very fine energy integration grids, several schemes to develop iterative parallelizable algorithms and to optimize the calculation of lead self-energies have been proposed [22 - Rungger2008][23 - Culey2011][24 - Pecchia2008]. Recently, multi-scale approaches coupling NEGF atomistic simulations with Finite Element continuum calculations have been proposed [25 - Yam2011, 26 - AufderMaur2011] in order to avoid full quantum mechanics description. NEGF techniques have been also extended to include time-dependent response [27 - Chen2013], and more recently a large interest has being devoted to electron-phonon interaction in non-equilibrium. This issue is particularly relevant in the community of organic materials: in such systems polaronic effects, solvent reorganization and off-diagonal disorder go beyond the perturbation level; excited states and non-adiabatic phenomena can play a crucial role. Different techniques have been proposed to calculate charge mobility, including multi-scale approach combining Classical Molecular Dynamics (CMD), ab-initio Density Functional Theory (DFT) and Langevin Dynamics [28 - Troisi2006], combining CMD and semi-empirical DFT with Born-Oppenheimer [12 - Kubar2010] or Ehrenfest [29 - Kubar2009] wave function propagation. Due to the exceptional computational cost of explicit time integration, all these methods rely on some kind of hierarchical multi-scale approach, where only a relevant subset of states, built from ab-initio calculations, is propagated in time. In this way, material on length scales up to micrometers can be modeled.
[1 - Sanvito2010] Sanvito S. Organic Spintronics: Filtering spin with molecules, Nature Materials 10, 484 (2010)
[2 - Curtarolo2013] Curtarolo, S.; Hart, G. L. W.; Nardelli, M. B.; Mingo, N.; Sanvito, S. and Levy O. The high-throughput highway to computational materials design, Nature Materials 12, 191 (2013).
[3 - Zaima2013] Zaima S.: Technology evolution for silicon nanoelectronics: Postscaling technology, Japanese Journal of Applied Physics 52 (2013).
[4 - DiCarlo2003] Di Carlo A. Microscopic theory of nanostructured semiconductor devices: beyond the envelope-function approximation, Semiconductor Science and Technology 18, 1 (2003).
[5 - Pecchia2004] Pecchia A., Di Carlo A.: Atomistic theory of transport in organic and inorganic nanostructures, Reports on Progress in Physics 67, 1497 (2004)
[6 - Bowler2006] Bowler, D. R.; Choudhury, R.; Gillan, M. J. and Miyazaki, T. Recent progress with large-scaleab initio calculations: the CONQUEST code. Phys. Status Solidi B, 243, 989 (2006).
[7 - Marzari1997] Marzari N. and Vanderbilt D. Maximally localized generalized Wannier functions for composite energy bands, Phys. Rev. B 56, 12847 (1997).
[8 - Elstner1998] Elstner, M.; Porezag, D.; Jungnickel, G.; Elsner, J.; Haugk, M.; Frauenheim, T.; Suhai, S. and Seifert, G. Self consistent density-functional tight-binding method for simulations of complex materials properties Physical Review B, 58, 7260 (1998).
[9 - Usman2011] Usman, M.; Tan, Y.-H. M.; Ryu, H.; Ahmed, S. S.; Krenner, H. J.; Boykin T. B. and Klimeck G.: Quantitative excited state spectroscopy of a single InGaAs quantum dot molecule through multi-million-atom electronic structure calculations, Nanotechnology 22, 315709 (2011).
[10 - Nelson2009] Nelson, J.; Kwiatkowski, J, J.; Kirkpatrick, J. & Frost, J. M. Modeling charge transport in Organic Photovoltaic Materials Accounts of Chemical Research, 42, 1768 (2009).
[11 - Troisi2009] Troisi, A.; Cheung, D. L. & Andrienko, D. Charge Transport in Semiconductors with Multiscale Conformational Dynamics Physical Review Letters, 102, 116602 (2009).
[12 - Kubar2010] Kubar, T. & Elstner, M. Coarse-Grained Time-Dependent Density Functional Simulation of Charge Transfer in Complex Systems: Application to Hole Transfer in DNA The Journal of Physical Chemistry A, 114, 11221-11240 (2010).
[13 - Weiße2006] Weiße, A.; Wellein, G. and Fehske, H. The kernel polynomial method, Rev. Mod. Phys. 78, 275 (2006).
[14 - Yuan2010] Yuan, S.; De Raedt, H. and Katsnelson, M. I. Modeling electronic structure and transport properties of graphene with resonant scattering centers, Phys. Rev. B 82, 115448 (2010).
[15 - Lherbier2008] Lherbier, A.; Biel, B.; Niquet Y.M. and Roche, S. Transport Length Scales in Disordered Graphene-based Materials: Strong Localization Regimes and Dimensionality Effects, Phys. Rev. Lett. 100, 036803 (2008)
[16 - Wehling2010] Wehling, T. O.;Yuan, S.; Lichtenstein, A. I.; Geim, A. K. and M. Katsnelson, M. I. Resonant scattering by realistic impurities in graphene, Phys. Rev. Lett. 105, 056802 (2010).
[17 - Lherbier2011] A. Lherbier et al., Two-Dimensional Graphene with Structural Defects: Elastic Mean Free Path, Minimum Conductivity, and Anderson Transition, Phys. Rev. Lett. 106, 046803 (2011).
[18 - Leconte2011] N. Leconte, A. Lherbier, F. Varchon, P. Ordejon, S. Roche, and J.-C. Charlier, Quantum transport in chemically modified two-dimensional graphene: From minimal conductivity to Anderson localization, Phys. Rev. B 84, 235420 (2011)
[19 - Cresti2013] Cresti, A. Ortmann, F. Louvet, T. Van Tuan, D. and Roche, S. Broken Symmetries, Zero-Energy Modes, and Quantum Transport in Disordered Graphene: From Supermetallic to Insulating Regimes, Phys. Rev. Lett. 110, 196601 (2013)
[20 - Schubert2009] Gerald Schubert Jens Schleede, and Holger Fehske, Anderson disorder in graphene nanoribbons: A local distribution approach, Phys. Rev. B 79, 235116 (2009)
[21 - Schubert2012] Gerald Schubert and Holger Fehske, Metal-to-Insulator Transition and Electron-Hole Puddle Formation in Disordered Graphene Nanoribbons, Phys. Rev. Lett. 108, 066402 (2012).
[22 - Rungger2008] Rungger, I and Sanvito, S. Algorithm for the construction of self-energies for electronic transport calculations based on singularity elimination and singular value decompositionI. Phys. Rev. B, 78, 035407, (2008)
[23 - Culey2011] Cauley, S.; Luisier, M.; Balakrishnan, V.; Klimeck, G. and Koh, C.-K. Distributed non-equilibrium Green's function algorithms for the simulation of nanoelectronic devices with scattering. Journal of Applied Physics , 110 , 043713 (2011)
[24 - Pecchia2008] Pecchia, A.; Penazzi, G.; Salvucci, L. & Di Carlo, A. Non-equilibrium Green's functions in density functional tight-binding: method and applications New Journal of Physics, 2008, 10, 065022
[25 - Yam2011] Yam, C.; Meng, L.; Chen, G.; Chen, Q. and Wong, N. Multiscale quantum mechanics/electromagnetics simulation for electronic devices, Phys. Chem. Chem. Phys. 13, 14365 (2011).
[26 - AufderMaur2011] M. Auf der Maur, G. Penazzi, G. Romano, F. Sacconi, A. Pecchia, A. Di Carlo: The multiscale paradigm in electronic device simulation, IEEE Transactions on Electron Devices 58(5), 1425 (2011).
[27 - Chen2013] Zhang, Y.; Chen, S. G. and Chen, G. H. First-principles time-dependent quantum transport theory Chen, Phys. Rev. B 87, 085110 (2013)
[28 - Troisi2006] Troisi, A. & Orlandi, G. Charge-Transport Regime of Crystalline Organic Semiconductors Diffusion Limited by Thermal Off-Diagonal Disorder Physical Review Letters, 2006, 96, 086601
[29 - Kubar2009] Kubar, T.; Kleinekathöfer, U. & Elstner, M. Solvent Fluctuations Drive the Hole Transfer in DNA: A Mixed Quantum-Classical Study Journal of Physical Chemistry B, 2009, 113, 13107-13117