Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 1 di 28 Giornata di lavoro Mathematical modeling and numerical analysis of quantum systems with applications to nanosciences Firenze, 16 dicembre 2005 Giovanni Frosali Dipartimento di Matematica Applicata “G.Sansone” MULTIBAND TRANSPORT MODELS FOR SEMICONDUCTOR DEVICES

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 2 di 28 Research group on semicoductor modeling at University of Florence Dipartimento di Matematica Applicata “G.Sansone”  Giovanni Frosali  Chiara Manzini (Munster)  Michele Modugno (Lens-INFN) Dipartimento di Matematica “U.Dini”  Luigi Barletti Dipartimento di Elettronica e Telecomunicazioni  Stefano Biondini  Giovanni Borgioli  Omar Morandi Università di Ancona  Lucio Demeio Others: G.Alì (Napoli), C.DeFalco (Milano), A.Majorana(Catania), C.Jacoboni, P.Bordone et. al. (Modena)

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 3 di 28 The spectrum of the Hamiltonian of a quantum particle in a periodic potential is continuous and characterized by (allowed) "energy bands“ separated by (forbidden) “band gaps". In the presence of additional potentials, the projections of the wave function on the energy eigenspaces (Floquet subspaces) are coupled by the Schrödinger equation, which allows interband transitions to occur. RITD Band Diagram Negibible coupling: single-band approximation This is no longer possible when the architect- ure of the device is such that other bands are accessible to the carriers. In some nanometric semiconductor device like Interband Resonant Tunneling Diode, transport due to valence electrons becomes important. TWO-BAND APPROXIMATION

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 4 di 28 Multiband models are needed: the charge carriers can be found in a super-position of quantum states belonging to different bands. Different methods are currently employed for characterizing the band structures and the optical properties of heterostructures, such as envelope functions methods (effective mass theory), tight-binding, pseudopotential methods,…  Schrödinger-like models (Barletti, Borgioli, Modugno, Morandi, etc.)  Wigner function approach (Bertoni, Jacoboni, Borgioli, Frosali, Zweifel, Barletti, Manzini, etc.)  Hydrodynamics multiband formalisms (Alì, Barletti, Borgioli, Frosali, Manzini, etc) OUR APPROACH TO THE PROBLEM

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 5 di 28 MULTIBAND TRANSPORT General Multiband Models General Multiband Models KANE model MeF model WIGNER APPROACH WIGNER APPROACH SCHRÖDINGER APPROACH SCHRÖDINGER APPROACH HYDRODYNAMIC MODELS HYDRODYNAMIC MODELS QUANTUM DRIFT-DIFFUSION MODELS QUANTUM DRIFT-DIFFUSION MODELS Isothermal QDD Isothermal QDD Chapman- Enskog expansion Chapman- Enskog expansion

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 6 di 28 We filter the solution Envelope function models Multiband “KP” system S chrödinger equation Hamiltonian. The envelope functions and are the projections of on the Wannier basis, and therefore the corresponding multi-band densities represent the (cell-averaged) probability amplitude of finding an electron on the conduction or valence bands, respectively.

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 7 di 28 The quantity represents the mean probability density to find the electron into the n-th band, in a lattice cell. MEF model: first order Physical meaning of the envelope function:

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 8 di 28 intraband dynamic Zero external electric field: exact electron dynamic MEF model: first order Effective mass dynamics:

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 9 di 28 intraband dynamic interband dynamic first order contribution of transition rate of Fermi Golden rule Coupling terms: MEF model: first order

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 10 di 28 Wigner function: Phase plane representation: pseudo probability function Wigner equation Liouville equation CLASSICAL LIMIT Moments of Wigner function: Wigner picture:

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 11 di 28 Density matrix Multiband Wigner function Evolution equation WIGNER APPROACH Wigner picture for Schrödinger-like models

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 12 di 28 Wigner picture: Two-band Wigner model pseudo-differential operators:

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 13 di 28  intraband dynamic: zero coupling if the external potential is null Wigner picture: Two-band Wigner model

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 14 di 28 intraband dynamic: zero coupling if the external potential is null interband dynamic: coupling like G-R via Wigner picture: Two-band Wigner model

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 15 di 28 Mathematical setting Hilbert space: Weighted spaces: 1 D problem: If the external potential the two-band Wigner system admits a unique solution

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 16 di 28 unbounded operator unitary group on Stone theorem Mathematical setting

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 17 di 28 Symmetric bounded operators Mathematical setting If the external potential a unique solution the two band Wigner system admits The operator generates semigroup The unique solution is given by Simulation

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 18 di 28

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 19 di 28 Hydrodynamic version of the MEF MODEL We can derive the hydrodynamic version of the MEF model using the WKB method (quantum system at zero temperature). Look for solutions in the form we introduce the particle densities Then is the electron density in conduction and valence bands. We write the coupling terms in a more manageable way, introducing the complex quantity with Vai alla 21

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 20 di 28 We introduce the rescaled Planck constant parameter MEF model reads in the rescaled form: with with the dimensional where are typical dimensional quantities and the effective mass is assumed to be equal in the two bands

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 21 di 28 Quantum hydrodynamic quantities Quantum electron current densities when i=j, we recover the classical current densities Complex velocities given by osmotic and current velocities can be expressed in terms of plus the phase difference Osmotic and current velocities

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 22 di 28 The quantum counterpart of the classical continuity equation Taking account of the wave form, the MEF system gives rise to Summing the previous equations, we obtain the balance law where, compared to the Kane model, the “interband density” Is missing. The previous balance law is just the quantum counterpart of the classical continuity equation.

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 23 di 28 Next, we derive a system of coupled equations for phases, obtaining a system equivalent to the coupled Schrödinger equations. Then we obtain a system for the currents and The equations can be put in a more familiar form with the quantum Bohm potentials It is important to notice that, differently from the uncoupled model, equations for densities and currents are not equivalent to the original equations, due to the presence of.

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 24 di 28 Recalling that and are given by the hydrodynamic quantities and, we have the HYDRODYNAMIC SYSTEM for the MEF model

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 25 di 28 The DRIFT-DIFFUSION scaling We rewrite the current equations, introducing a relaxation time, in order to simulate all the mechanisms which force the system towards the statistical mechanical equilibrium. In analogy with the classical diffusive limit for a one-band system, we introduce the scaling Finally, after having expressed the osmotic and current velocities, in terms of the other hydrodynamic quantities, as tends to zero, we formally obtain the ZER0-TEMPERATURE QUANTUM DRIFT-DIFFUSION MODEL for the MEF system.

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 26 di 28 Hydrodynamic version of the MEF MODEL

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 27 di 28 NON ZERO TEMPERATURE hydrodynamic model We consider an electron ensemble which is represented by a mixed quantum mechanical state, to obtain a nonzero temperature model for a Kane system. We rewrite the MEF system for the k-th state, with occupation probability We use the Madelung-type transform We define the densities and the currents corresponding to the two mixed states We define Performing the analogous procedure and with an appropriate closure, we get

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 28 di 28 Isothermal QUANTUM DRIFT-DIFFUSION for the MEF MODEL

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 29 di 28 Thanks for your attention !!!!!

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 30 di 28 REMARKS We derived a set of quantum hydrodynamic equations from the two-band MEF model. This system, which is closed, can be considered as a zero- temperature quantum fluid model. In addition to other quantities, we have the tensors and Starting from a mixed-states condition, we derived the corresponding non zero-temperature quantum fluid model, which is not closed. Closure of the quantum hydrodynamic system Numerical treatment Heterogeneous materials Generalized MEF model NEXT STEPS similar to the temperature tensor of kinetic theory.

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 31 di 28 Problems in the practical use of the Kane model: Kane model Strong coupling between envelope function related to different band index, even if the external field is null Critical choice in the cut off for the band index Poor physical interpretation

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 32 di 28 Electromagnetic and spin effects are disregarded, just like the field generated by the charge carriers themselves. Dissipative phenomena like electron- phonon collisions are not taken into account. The dynamics of charge carriers is considered as confined in the two highest energy bands of the semiconductor, i.e. the conduction and the (non- degenerate) valence band, around the point is the "crystal" wave vector. The point is assumed to be a minimum for the conduction band and a maximum for the valence band. The physical environment where The Hamiltonian introduced in the Schrödinger equation is where is the periodic potential of the crystal and V an external potential.

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 33 di 28 Interband Tunneling: PHYSICAL PICTURE Interband transition in the 3-d dispersion diagram. The transition is from the bottom of the conduction band to the top of the val-ence band, with the wave number becoming imaginary. Then the electron continues propagating into the valence band. Kane model Vai alla 9

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 34 di 28 KANE MODEL The Kane model consists into a couple of Schrödinger-like equations for the conduction and the valence band envelope functions. be the valence band envelope function. Letbe the conduction band electron envelope function and m is the bare mass of the carriers, is the minimum (maximum) of the conduction (valence) band energy P is the coupling coefficient between the two bands (the matrix element of the gradient operator between the Bloch functions)

Dipartimento di Matematica Applicata Università di Firenze Multiband transport models for semiconductor devices Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 n. 35 di 28 Remarks on the Kane model The external potential V affects the band energy terms, but it does not appear in the coupling coefficient P. There is an interband coupling even in absence of an external potential. The interband coefficient P increases when the energy gap between the two bands increases (the opposite of physical evidence). The envelope functions are obtained expanding the wave function on the basis of the periodic part of the Bloch functions evaluated at k=0, where.