From quantum interference to Coulomb blockade: the role of realistic modeling

1. From quantum interference to Coulomb blockade: the role of realistic modeling Massimo Macucci Dipartimento di Ingegneria dell’Informazione, Università di Pisa Via Caruso 16, I-56122, Pisa, Italy. macucci@mercurio.iet.unipi.it

2. Summary • The meaning of modeling • The case of the Tstub transistor • A view of the capacitance of quantum dots • Shell-filling effects in quantum dots • Modeling applied to the interpretation of experiments: shot noise suppression in “chaotic” cavities • Modeling applied to the development of new devices and architectures: limits in the performance of QCA circuits

3. The meaning of modeling • As a tool to support the design of new device concepts: at this level modeling must not loose contact with the intuitive understanding of the underlying physics • As a tool to facilitate the understanding of some physical concepts and to provide a realistic description of device operation: here modeling must include a detailed description of the relevant phenomena • As a tool to support the design of an industrial product: in this case modeling must have capabilities for quantitatively reliable predictions and span multiple dimensional scales, with the introduction of appropriate approximations to make the problem tractable

4. TSTUB transistor (I) • In the late 80’s conductance quantization and ballistic transport in quasi 1-D nanostructures were experimentally demonstrated • Karl had the idea of using quantum interference to control conductance in a T-shaped device • Adjusting the length of the stub, one could switch the interference from constructive to destructive F. Sols, M. Macucci, U. Ravaioli, Karl Hess, J. Appl. Phys. 66, 3892 (1989)

5. TSTUB transistor (II) In 1996 J. Appenzeller and coworkers gave the first clear experimental demonstration of a QMT effect J. Appenzeller, Ch. Schroer, Th. Schäpers, A. v. d. Hart, A. Förster, B. Lengeler, and H. Lüth, Phys. Rev. B 53, 9959 (1996)

6. TSUB transistor (III) The effect is rather small, mainly due to the presence of elastic scattering associated with impurities and donors and inelastic scattering due to phonons, but there are other reasons for deviations from ideality, for example the actual shape of the potential

7. TSTUB transistor (IV) TSTUB Transistor, F. Sols, M. Macucci, U. Ravaioli, K. Hess, APL 54, 350 (1989)

8. TSTUB transistor (V) • It may be worth performing new simulations with the inclusion of the effect of random potential fluctuations due to impurities and donors, and evaluate whether stronger interference effects are to be expected • The experiment by the group in Jülich were done at 300 mK, now we can go lower in temperature and reduce the importance of inelastic phonon scattering • Therefore, if a realistic simulation showed the possibility of observing stronger interference effects, it would certainly be interesting to do the experiment again

9. Coulomb Blockade Karl, when discussing Coulomb Blockade, had a simple, yet profound, explanation of why charge on a capacitor can be non integer, based on the continuous displacement of electron orbitals with respect to positive nuclei

10. The concept of capacitance for quantum dots (I) Looking for an intuitive description of the chemical potential of quantum dots M. Macucci, Karl Hess, G. J. Iafrate, Phys. Rev. B 48, 17354 (1993).

11. The concept of capacitance for quantum dots (II)

12. The concept of capacitance for quantum dots (III)

13. Circular quantum dots (I) Charging energy for the addition of each electron G. J. Iafrate, Karl Hess, J. B. Krieger, M. Macucci, Phys. Rev. B 52, 10737 (1995).

14. Circular quantum dots (II) A model based on a simple parabolic confinement potential, but with a self-consistent treatment of electron-electron interaction within the LDA framework was able to provide a good agreement with experiments S. Tarucha, D. G. Austing, T. Honda, R. J. van der Hage, L. P. Kouwenhoven, Phys. Rev. Lett. 77, 3613 (1996) M. Macucci, Karl Hess, G. J. Iafrate, Phys. Rev. B 55, R4879 (1997)

15. Shot noise is the result of the granularity of charge In the case of independent electrons, we obtain Schottky’s result: S=2qI In a perfect quantum wire S=0 In general, shot noise in quantum devices is linked to the transmission eigenvalues (M. Buettiker, Phys. Rev. Lett. 65 (1990)) The ratio of the actual noise power spectral density to that of full shot noise is defined “Fano factor” Shot noise in mesoscopic devices Poly metal Source Drain L Shallow trench isolation

16. Noise for B=0 P. Marconcini, M. Macucci, G. Iannaccone, B. Pellegrini, G. Marola, Europhys. Lett. 73, 574 (2006) S. Oberholzer, E. V. Sukhorukov, C. Schönenberger, Nature 415, 765 (2002)

17. Cavities with different shapes Poly metal

18. Chaotic dynamics in a classically regular cavity Quantum diffraction occurs at the constrictions, thus leading to multiple trajectories originating from each impinging one The shape of the cavity is not important!

19. Thanks, Karl!

20. Role of modeling (I) • Numerical modeling has had a fundamental role in the development of mesoscopic physics and of nanoelectronics • It has helped interpreting the experiments and formulating new device proposals • While at an early stage it was sufficient to capture the main physical properties, we now need to be able to include nonideality effects and fabrication tolerances • If low-dimensional devices have to move from physics to engineering, we need simulation tools with good capabilities for quantitative prediction, as those which have been developed for traditional microelectronics and for optoelectronics

21. Role of modeling (II) • Future modeling must be able to deal with complete systems • Therefore hierarchical approaches are needed, allowing increasing abstraction as we move from single devices or single molecules, to circuits, to systems • It is the intermediate step that is still largely missing • A strongly interdisciplinary approach is needed, involving physics, chemistry, engineering, if we want this effort to be successful • Modeling must at the same time support the design of new devices and receive inputs from experimentalists to validate the theoretical results

22. Noise as a function of magnetic field S. Oberholzer, E. V. Sukhorukov, C. Schönenberger, Nature 415, 765 (2002) P. Marconcini, M. Macucci, G. Iannaccone, B. Pellegrini, G. Marola, Europhys. Lett. 73, 574 (2006)

23. Stacked quantum dots (I) • Inspiration came listening to a talk by Pierre Petroff on capacitance spectroscopy performed on layers of self-assembled quantum dots • We aimed at the formulation of a rigorous theory quantitatively explaining these experiments

24. Stacked quantum dots (II) For two separated dots, solving two distinct Schroedinger equations: For three strongly coupled dots, solving one single Schroedinger equation:

25. In 1991 Bakshi and coworkers proposed a novel computing architecture based on “Quantum dashes” (JAP 70, 5150 (1991)) This architecture was affected by a fundamental fault that was recognized by the Notre Dame group: it did not have “signal” gain, but, rather, attenuation The way around it consisted in introducing a barrier, which localizes the electron on either side of the dash Quantum dashes Poly metal Source Drain L

26. To create something more complex than a simple “binary wire,” lateral branching is needed: this can be accomplished placing an orthogonal chain of dashes, interacting with two dashes of the main row From dashes to cells Poly The basic four-dot cell is thus obtained: tunneling between the two halves of the cell is in general not needed, and preventing it may improve performance metal Drain L

27. Ground state computation • The original QCA proposal involved relaxation of the system to the ground state once appropriate logic inputs have been provided The main problem with this approach is in the speed with which convergence to the ground state is actually achieved Furthermore, pipelining of data is not possible

28. Effect of fabrication tolerances Fabrication tolerances for QCA cells, M. Governale, M. Macucci, G. Iannaccone, C. Ungarelli, J. Martorell, JAP 85, 2962 (1999) 0.1 angstrom error on one dot

29. Clocked QCA • The basic clocked QCA cell is a variation of the Parametron concept (Korotkov APL 67, 2412, Likharev and Korotkov Science 273, 763) proposed by Lent (Proc. IEEE 85, 541 (1997)) A periodic clock signal has to be distributed across the whole QCA array, pipelining of data is now allowed

30. Realistic modeling of clocked QCA circuits L. Bonci, M. Gattobigio, G. Iannaccone, M. Macucci, JAP 92, 3169 (2002) RC=10-12 s !