Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Dipartimento di Matematica Applicata Università di Firenze Giornata di lavoro Mathematical modeling and numerical analysis of quantum systems with applications to nanosciences Firenze, 16 dicembre 2005 MULTIBAND TRANSPORT MODELS FOR SEMICONDUCTOR DEVICES Giovanni Frosali Dipartimento di Matematica Applicata “G.Sansone” [email protected] Multiband transport models for semiconductor devices n. 1 di 28 Dipartimento di Matematica Applicata Università di Firenze Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 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) Multiband transport models for semiconductor devices n. 2 di 28 Dipartimento di Matematica Applicata Università di Firenze Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 TWO-BAND APPROXIMATION • 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. • Negibible coupling: single-band approximation • In some nanometric semiconductor device like Interband Resonant Tunneling Diode, transport due to valence electrons becomes important. Multiband transport models for semiconductor devices 2 1 Energy (ev) • This is no longer possible when the architecture of the device is such that other bands are accessible to the carriers. RITD Band Diagram 0 -1 -2 0 10 20 30 40 Position (nm) 50 60 n. 3 di 28 Dipartimento di Matematica Applicata Università di Firenze Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 • 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,… OUR APPROACH TO THE PROBLEM 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) Multiband transport models for semiconductor devices n. 4 di 28 Dipartimento di Matematica Applicata Università di Firenze Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 MULTIBAND TRANSPORT Isothermal QDD General Multiband Models WIGNER APPROACH HYDRODYNAMIC MODELS SCHRÖDINGER APPROACH QUANTUM DRIFT-DIFFUSION MODELS KANE model MeF model Multiband transport models for semiconductor devices ChapmanEnskog expansion n. 5 di 28 Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Dipartimento di Matematica Applicata Università di Firenze Envelope function models Schrödinger equation H E H 2 2m0 We filter the solution ( x) Hamiltonian Multiband “KP” system 2 V per ( x ) U ext ( x ) 1 ( x ) 2 ( x ) n ( x ) The envelope functions c and v 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. Multiband transport models for semiconductor devices n. 6 di 28 Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Dipartimento di Matematica Applicata Università di Firenze MEF model: first order 2 1 i t 2m* c 2 2 i * t 2mv 2 2 1 P U Ec U 1 2 2 x m0 E g x 2 2 2 P U Ev U 2 1 2 x m0 E g x Physical meaning of the envelope function: | n dx n ( Ri ) Ri cell The quantity i x 2 2 represents the mean probability density to find the electron into the n-th band, in a lattice cell. Multiband transport models for semiconductor devices n. 7 di 28 Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Dipartimento di Matematica Applicata Università di Firenze MEF model: first order 2 1 i t 2m* c 2 2 i * t 2mv 2 2 1 P U Ec U 1 2 2 x m0 E g x 2 2 2 P U Ev U 2 1 2 x m0 E g x Effective mass dynamics: • intraband dynamic Zero external electric field: exact electron dynamic Multiband transport models for semiconductor devices n. 8 di 28 Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Dipartimento di Matematica Applicata Università di Firenze MEF model: first order 2 1 i t 2m* c 2 2 i * t 2mv 2 2 1 P U Ec U 1 2 2 x m0 E g x 2 2 2 P U Ev U 2 1 2 x m0 E g x Coupling terms: • intraband dynamic • interband dynamic T ( n n, k k ) first order contribution of transition rate of Fermi Golden rule Multiband transport models for semiconductor devices n. 9 di 28 Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Dipartimento di Matematica Applicata Università di Firenze Wigner picture: Wigner function: f x, p 1 ip x / 2 m x / 2 m e d 2 Phase plane representation: f x, p pseudo probability function CLASSICAL LIMIT 0 Wigner equation Liouville equation Moments of Wigner function: x n x f x, p dp 2 m J x p f x, p dp Multiband transport models for semiconductor devices n. 10 di 28 Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Dipartimento di Matematica Applicata Università di Firenze WIGNER APPROACH Wigner picture for Schrödinger-like models Density matrix 1 x 1 y x, y n 1 1 n n n d i H, H x H y dt Multiband Wigner function f ij x, p W 1 ip x / 2 m , x / 2 m e d ij 2 Evolution equation Multiband transport models for semiconductor devices df i W H x H y W -1 f dt n. 11 di 28 Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Dipartimento di Matematica Applicata Università di Firenze Wigner picture: Two-band Wigner model f cc p f cc icc f cc * m t f p vv f vv ivv f vv * m t f i cv i * p 2 f cv icv f cv 4m t ij 2 P f cv m0 E g 2 P f cv m0 E g P f cc f vv i m0 E g pseudo-differential operators: Fp ij f ij Vi x / 2m V j x / 2m Fp1 f ij Fp f ij V x / 2m Fp -1 f ij Multiband transport models for semiconductor devices n. 12 di 28 Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Dipartimento di Matematica Applicata Università di Firenze Wigner picture: Two-band Wigner model f cc p f cc icc f cc * m t f p vv f vv ivv f vv * m t f i cv i * p 2 f cv icv f cv 4m t 2 P f cv m0 E g 2 P f cv m0 E g P f cc f vv i m0 E g fii x, v W ii intraband dynamic: zero coupling if the external potential is null Multiband transport models for semiconductor devices n. 13 di 28 Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Dipartimento di Matematica Applicata Università di Firenze Wigner picture: Two-band Wigner model f cc p f cc icc f cc * m t f p vv f vv ivv f vv * m t f i cv i * p 2 f cv icv f cv 4m t 2 P f cv m0 E g 2 P f cv m0 E g P f cc f vv i m0 E g fii x, v W ii • intraband dynamic: zero coupling if the external potential is null • interband dynamic: coupling like G-R via f cv x, p Multiband transport models for semiconductor devices n. 14 di 28 Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Dipartimento di Matematica Applicata Università di Firenze Mathematical setting 1 D problem: Hilbert space: H Weighted spaces: X 1 If the external potential U ext W 2, ( x X1 X1 X1 f : f L2 Multiband transport models for semiconductor devices ; 1 p 2 dx dp ) the two-band Wigner system admits a unique solution df Af B C f i dt f (0) f0 2 f H f f cc , f vv , f cv T f0 D A H n. 15 di 28 Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Dipartimento di Matematica Applicata Università di Firenze Mathematical setting Stone theorem p p 2 1 2 A diag i * , i * , * 2 p m x m x 4m x unbounded operator e unitary group on f1 f 2 2 f 3 D( A) f H : , , 2 X1 x x x B diag cc ,vv ,cv 0 P C m E 0 0 g i iAt 2 2 0 0 i H 0 F p ij f ij Vi x / 2m V j x / 2m F p 1 f ij F p f ij U x / 2m F p -1 f ij Multiband transport models for semiconductor devices n. 16 di 28 Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Dipartimento di Matematica Applicata Università di Firenze Mathematical setting Symmetric bounded operators B, C B H If the external potential U ext W 2, ( a unique solution f H ij fij X c Uext W 1, ( fij X c Uext W 2, ( x A B C fij fij x) X X ) the two band Wigner system admits df Af B C f i dt f (0) f0 The operator x) f f cc , f vv , f cv T generates semigroup The unique solution is given by Multiband transport models for semiconductor devices i A B C t f e f0 Simulation n. 17 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. 18 di 28 Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Dipartimento di Matematica Applicata Università di Firenze 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 iS ( x, t ) c ( x, t ) nc ( x, t ) exp c v ( x, t ) nv ( x, t ) exp iSv ( x, t ) we introduce the particle densities Then n nij ( x, t ) i ( x, t ) j ( x, t ). c c v v is the electron density in conduction and valence bands. We write the coupling terms in a more manageable way, introducing the complex quantity ncv : c v nc nv e Multiband transport models for semiconductor devices i with : Sc Sc Vai alla 21 n. 19 di 28 Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Dipartimento di Matematica Applicata Università di Firenze We introduce the rescaled Planck constant mlR2 parameter tR and the effective mass where m lR , t R with the dimensional are typical dimensional quantities is assumed to be equal in the two bands MEF model reads in the rescaled form: m P V with K mE g Multiband transport models for semiconductor devices n. 20 di 28 Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Dipartimento di Matematica Applicata Università di Firenze Quantum hydrodynamic quantities • Quantum electron current densities J ij Im( i j ) when i=j , we recover the classical current densities J c ncSc J v nvSv • Osmotic and current velocities uc uos ,c iuel ,c uos ,i ni , ni uel ,i uv uos ,v iuel ,v Ji Si , i c, v ni • Complex velocities given by osmotic and current velocities can be expressed in terms of nc , nv , J c , J v Multiband transport models for semiconductor devices plus the phase difference n. 21 di 28 Dipartimento di Matematica Applicata Università di Firenze Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 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. c v The previous balance law is just the quantum counterpart of the classical continuity equation. Multiband transport models for semiconductor devices n. 22 di 28 Dipartimento di Matematica Applicata Università di Firenze Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Next, we derive a system of coupled equations for phases S c , S v , obtaining a system equivalent to the coupled Schrödinger equations. Then we obtain a system for the currents J c and J v 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 . Multiband transport models for semiconductor devices n. 23 di 28 Dipartimento di Matematica Applicata Università di Firenze Recalling that and Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 ncv , uc , and uv are given by the hydrodynamic quantities nc , nv , J c , J v , we have the HYDRODYNAMIC SYSTEM for the MEF model Multiband transport models for semiconductor devices n. 24 di 28 Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Dipartimento di Matematica Applicata Università di Firenze 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 t t , J c J c , J v J v , , 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. Multiband transport models for semiconductor devices n. 25 di 28 Dipartimento di Matematica Applicata Università di Firenze Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Hydrodynamic version of the MEF MODEL Multiband transport models for semiconductor devices n. 26 di 28 Dipartimento di Matematica Applicata Università di Firenze Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 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 k We use the Madelung-type transform We define ik nik exp iSik / , i c, v J ck , J vk , k , ncvk , uck , uvk . We define the densities and the currents corresponding to the two mixed states Performing the analogous procedure and with an appropriate closure, we get Multiband transport models for semiconductor devices n. 27 di 28 Dipartimento di Matematica Applicata Università di Firenze Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Isothermal QUANTUM DRIFT-DIFFUSION for the MEF MODEL Multiband transport models for semiconductor devices n. 28 di 28 Dipartimento di Matematica Applicata Università di Firenze Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Thanks for your attention !!!!! Multiband transport models for semiconductor devices n. 29 di 28 Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Dipartimento di Matematica Applicata Università di Firenze 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 zerotemperature quantum fluid model. Starting from a mixed-states condition, we derived the corresponding non zero-temperature quantum fluid model, which is not closed. In addition to other quantities, we have the tensors vc and c , v , cv similar to the temperature tensor of kinetic theory. NEXT STEPS • Closure of the quantum hydrodynamic system • Numerical treatment • Heterogeneous materials • Generalized MEF model Multiband transport models for semiconductor devices n. 30 di 28 Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Dipartimento di Matematica Applicata Università di Firenze Kane model 2 2 1Kane 2 1Kane 2Kane Kane Vc 1 P i 2 t 2m0 x m0 x Kane 2 2 Kane 2 Kane Kane 2 2 1 i V P v 2 t 2m0 x 2 m0 x Problems in the practical use of the Kane model: • Strong coupling between envelope function related to different band index, even if the external field is null • Poor physical interpretation n( x ) Kane i x 2 i • Critical choice in the cut off for the band index Multiband transport models for semiconductor devices n. 31 di 28 Dipartimento di Matematica Applicata Università di Firenze Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 The physical environment Electromagnetic and spin effects are disregarded, just like the field generated by the charge carriers themselves. Dissipative phenomena like electronphonon 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 (nondegenerate) valence band, around the point k 0 where kis the "crystal" wave vector. The point k 0 is assumed to be a minimum for the conduction band and a maximum for the valence band. The Hamiltonian introduced in the Schrödinger equation is H Ho V , where h2 H o Vper 2m V per is the periodic potential of the crystal and V an external potential. Multiband transport models for semiconductor devices n. 32 di 28 Dipartimento di Matematica Applicata Università di Firenze Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 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 Multiband transport models for semiconductor devices Vai alla 9 n. 33 di 28 Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Dipartimento di Matematica Applicata Università di Firenze KANE MODEL The Kane model consists into a couple of Schrödinger-like equations for the conduction and the valence band envelope functions. Let c ( x , t ) be the conduction band electron envelope function and be the valence band envelope function. v ( x, t ) • m is the bare mass of the carriers, Vi Ei V , i c, v • Ec ( Ev ) 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) Multiband transport models for semiconductor devices n. 34 di 28 Giornata di lavoro sulle Nanoscienze Firenze 16 dicembre 2005 Dipartimento di Matematica Applicata Università di Firenze Remarks on the Kane model • The envelope functions c ,v are obtained expanding the wave function on the ikx basis of the periodic part of the Bloch functions bn ( x, t ) e un (k , x ), evaluated at k=0, 0 0 ( x) c ( x)uc v ( x)uv where uc0,v ( x) uc,v (0, x) . • The external potential V affects the band energy terms Vc (Vv ), 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 E g increases (the opposite of physical evidence). Multiband transport models for semiconductor devices n. 35 di 28