Presentazione di PowerPoint - UniFI

Presentazione di PowerPoint - UniFI

Facolt di Ingegneria ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 ICTT19 19th International Conference on Transport Theory Budapest, July 24-30, 2005 Wigner Wigner approach approach to to aa new new two-band two-band envelope envelope function function model model for for quantum quantum transport transport Omar Morandi Dipartimento di Elettronica e Telecomunicazioni [email protected] Giovanni Frosali Dipartimento di Matematica Applicata G.Sansone [email protected] Wigner approach to a new two-band envelope function model for quantum transport n. 1 di 22 ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Facolt di Ingegneria Plain of the talk Description of of the the model model Description Multiband (MEF) model Multiband-Wigner picture Mathematical problem problem Mathematical Mathematical setting Well posedness of the Multiband-Wigner system Numerical applications

applications Numerical Description of the numerical algorithm Application to IRTD Wigner approach to a new two-band envelope function model for quantum transport n. 2 di 22 ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Facolt di Ingegneria Problem setting: unperturbed system Homogeneous periodic crystal lattice: d i ( x) H 0 x dt 2 2 H 0 V per ( x) 2m0 Time-dependent evolution semigroup: ( x, t0 ) n (k , x) ( x, t ) e i H0 t ( x, t0 ) ( x, t ) e i En ( k ) t n (k , x) No interband transition are possible if ( x, t0 ) is a Bloch function H 0 n (k , x) Ek n (k , x) Wigner approach to a new two-band envelope function model for quantum transport n. 3 di 22 Facolt di Ingegneria

ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Multiband models: models: derivation derivation Multiband Wannier envelope function 1 2 Bloch envelope function wn x Ri n Ri n Ri Wannier function 1 ik x Ri wn x Ri u k , x e dk n 2 Non Homogeneous lattices 1 ikx k u k , x e dk n n 2 Bloch function 2 n Ri is the (cell. averaged) probability to find the electron

in the site R and into n-th band i Wigner approach to a new two-band envelope function model for quantum transport n. 4 di 22 ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Facolt di Ingegneria Multiband models: models: derivation derivation Multiband Wannier envelope function In literature are proposed different approximations of un k , x We loose the simple interpretation of the envelope function 1 2 w x R R n n i n i Ri kp methods Wannier function 1 ik x Ri Bloch Kane wn x Ri u k , x e

k un , k und n ,0 Kane 2 un k , x High oscillating behaviour Bloch u n ,k unL,k K unBloch L-K ,0 The direct use of Wannier basis is a difficult task!! k k k 0 Wigner approach to a new two-band envelope function model for quantum transport n. 5 di 22 Facolt di Ingegneria ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 MEF model: model: derivation derivation MEF Wannier function 1 2 Bloch function wn x Ri n Ri n Ri 1 n Ri 2

1 ikx k u k , x e dk n n 2 ikRi k e dk n FBZ Our approach To get our multiband model in Wannier basis: We recover un approximate set of equation for n k in the Bloch basis (momentum space) We Fourier transform the equations obtained (coordinate space) Wigner approach to a new two-band envelope function model for quantum transport n. 6 di 22 Facolt di Ingegneria ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 MEF model model characteristics: characteristics: MEF Hierarchy of kp multiband effective mass models, where the asimptotic parmeter is the quasi-momentum of the electron Direct physical meaning of the envelope function Easy approximation (cut off on the index band) Highlight the action of the electric field in the interband transition phenomena Easy implementation: Wigner and quantum-hydrodynamic formalism

Wigner approach to a new two-band envelope function model for quantum transport n. 7 di 22 ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Facolt di Ingegneria MEF model: model: derivation derivation MEF H 0 U ext | E | ikx | n , k u ( k , x ) e First approximation: n , k U ext k k ' un ,k | un ',k ' n , k | U | n ext ( E En )n (k ) dk n, k | U ext | n , k n ' (k ') 0 n un,k | un ',k ' un (k , x)un ' (k ', x)dx | un ,k un ( k , x) cell By exploiting the periodicity of potentials un (k , x) and for slow varying external Wigner approach to a new two-band envelope function model for quantum transport

n. 8 di 22 ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Facolt di Ingegneria MEF model: model: formal formal derivation derivation MEF Evaluation of matrix elements un,k | un ',k ' un (k , x)un ' (k ', x)dx cell un ,k | un ',k ' k k ' Pn ,n ' (k,k') m0 En ,n ' (k,k') Pn,n k , k ' 2 u (k , x) u n x n' (k ', x )dx Kane momentum matrix cell 2 En ,n ' ( k , k ) En (k ) En ' (k ') k 2 k 2 2m0 Wigner approach to a new two-band envelope function model for quantum transport n. 9 di 22 Facolt di Ingegneria ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 MEF model:

model: derivation derivation MEF ( E En )n (k ) dk n, k | U ext | n, k n ' (k ') 0 n En (k ) En (k )n (k ) U ( k k ')n (k ')dk 2 i m0 k k ' U (k k ')Pn ,n ' (k,k')n ' (k ')dk nn En , n ' Wigner approach to a new two-band envelope function model for quantum transport n. 10 di 22 Facolt di Ingegneria ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 MEF model: model: derivation derivation MEF Our aim: simplify the above equation. Interband Interbandterm: term: 2 i m0 2 k k ' P (k,k') dk i U n ,n ' n E k , k ' m0

nn n,n ' 2 i m0 * 2 Pn ,n ' (0,0) Second approximation: M n ,n ' dk We retain only the first k ' k k ' U d k ik k k ' U We retain only the first n ' n'

E (0,0) m E ( 0 , 0 ) nn n,n ' 0 nn n,n ' k k ' U Pn ,n ' (0,0)n 'dk nn En , n ' order orderterm term Wigner approach to a new two-band envelope function model for quantum transport n. 11 di 22 ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Facolt di Ingegneria MEF model: model: derivation derivation MEF ( E En )n (k ) dk n, k | U ext | n, k n ' (k ') 0 n Approximate Approximatesystem system En (k ) En (k )n (k ) U (k k ')n ' (k ')dk Fk 2 i m0 k k ' U Pn ,n ' (0,0)n 'dk

nn En , n ' We Wewrite writeititin incoordinate coordinatespace space n x = F k n Wigner approach to a new two-band envelope function model for quantum transport n. 12 di 22 ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Facolt di Ingegneria MEF model: model: first first order order MEF 1 2 i t 2m* c 2 i 2 * t 2 m v 2 1 2 P U Ec U 1 2 2 x m0 Eg x 2 2 2 P U Ev U 2 1

2 x m0 Eg x Physical meaning meaning of of the the envelope envelope function: function: Physical | n dx n ( Ri ) 2 |n Ri cell 2 The quantity i x represents the mean probability density to find the electron into n-th band, in a lattice cell. Wigner approach to a new two-band envelope function model for quantum transport n. 13 di 22 ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Facolt di Ingegneria MEF model: model: first first order order MEF 1 2 i t 2m* c 2 i 2 * t

2 m v 2 1 2 P U Ec U 1 2 2 x m0 Eg x 2 2 2 P U Ev U 2 1 2 x m0 Eg x Effective Effectivemass massdynamics: dynamics: intraband dynamic Zero external electric field: exact electron dynamic Wigner approach to a new two-band envelope function model for quantum transport n. 14 di 22 ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Facolt di Ingegneria MEF model: model: first first order order MEF 1 2 i t 2m* c 2 i 2 *

t 2 m v 2 1 2 P U Ec U 1 2 2 x m0 Eg x 2 2 2 P U Ev U 2 1 2 x m0 Eg x Coupling Couplingterms: terms: intraband dynamic interband dynamic first order contribution of T (n n, k k ) transition rate of Fermi Golden rule Wigner approach to a new two-band envelope function model for quantum transport n. 15 di 22 Facolt di Ingegneria ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Kane model model Kane 1Kane 2 2 1Kane 2 2Kane Kane i t 2m x 2 Vc 1 m P x 0 0 Kane 2 Kane Kane

2 2 Kane i 2 2 1 V P v 2 t 2m0 x 2 m0 x Problems Problemsin inthe thepractical practicaluse useof ofthe theKane Kanemodel: 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 Wigner approach to a new two-band envelope function model for quantum transport n. 16 di 22 ICTT19 19thInternational Conference on Transport Theory

Budapest, July 24-30, 2005 Facolt di Ingegneria Wigner picture: picture: Wigner 1 f x, p x / 2m x / 2m eip d 2 Wigner function: Phase plane representation: f x, p pseudo probability function Classical limit Wigner equation 0 Liouville equation Moments of Wigner function: 2 x n x f x, p dp J x p f x, p dp m Wigner approach to a new two-band envelope function model for quantum transport n. 17 di 22 ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Facolt di Ingegneria Wigner picture: picture: Wigner 1 ,.., n t n n-th band component General Schrdinger-like model matrix of operator d i

H dt Density matrix 1 x 1 y 1 n x, y n 1 n n d i H, H x H y dt Wigner approach to a new two-band envelope function model for quantum transport n. 18 di 22 Facolt di Ingegneria ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Wigner picture: picture: Wigner Multiband Wigner function f ij x, p W 1 ip x / 2 m

, x / 2 m e d ij 2 Evolution equation df i W H x H y W -1 f dt Introduced by Borgioli, Frosali, Zweifel [1] Well-posedness of the two band Kane-Wigner System [1] G. Borgioli, G. Frosali and P. Zweifel, Wigner approach to the two-band Kane model for a tunneling diode, Transp. Teor.Stat. Phys. 32 3, 347-366 (2003). Wigner approach to a new two-band envelope function model for quantum transport n. 19 di 22 ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Facolt di Ingegneria Wigner picture: picture: Wigner Multiband Wigner function f ij x, p W 1 ip x / 2 m , x

/ 2 m e d ij 2 Evolution equation Two band MEF model df i W H x H y W -1 f dt 2 2 Ec U * 2 2mc x H= 2 P U m0 Eg x 2 P U m0 Eg x 2 2 Ev U * 2 2mv x Two band Wigner model

Wigner approach to a new two-band envelope function model for quantum transport n. 20 di 22 ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Facolt di Ingegneria Wigner picture: picture: Wigner Two band Wigner model f cc p f cc i cc f cc * m t f p vv f vv i vv f vv * m t f cv i * i p 2 f cv i cv f cv t 4m i 2 P f cv m0 E g 2 P f cv m0 E g P

f cc f vv m0 Eg Moments of the multiband Wigner function: fii x, p dp i x f x, p dp ii 2 represents the mean probability density to find the electron into n-th band, in a lattice cell. Wigner approach to a new two-band envelope function model for quantum transport n. 21 di 22 ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Facolt di Ingegneria Wigner picture: picture: Wigner Two band Wigner model f cc p f cc i cc f cc * m t f p vv f vv i vv f vv * m t f cv i * i p 2 f cv i cv f cv t 4m i

2 P f cv m0 E g 2 P f cv m0 E g P f cc f vv m0 Eg F p ij f ij Vi x / 2m V j x / 2m F p 1 f ij F p f ij V x / 2m F p -1 f ij Wigner approach to a new two-band envelope function model for quantum transport n. 22 di 22 ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Facolt di Ingegneria Wigner picture: picture: Wigner Two band Wigner model f cc p f cc i cc f cc * m t f p vv f vv i vv f vv * m t f

cv i * i p 2 f cv i cv f cv t 4m i 2 P f cv m0 Eg 2 P f cv m0 Eg P f cc f vv m0 Eg f ii x, v W ii intraband dynamic: zero coupling if the external potential is null Wigner approach to a new two-band envelope function model for quantum transport n. 23 di 22 ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Facolt di Ingegneria Wigner picture: picture: Wigner Two band Wigner model f cc p f cc i cc f cc * m t f p vv

f vv i vv f vv * m t f cv i * i p 2 f cv i cv f cv t 4m i 2 P f cv m0 Eg 2 P f cv m0 Eg P f cc f vv m0 Eg f ii 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 Wigner approach to a new two-band envelope function model for quantum transport n. 24 di 22 ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Facolt di Ingegneria Mathematical setting setting Mathematical x, p 2 1 D problem: Hilbert space: H X 1 X 1 X 1 Weighted spaces:

X 1 f : f L2 2 ; 1 p 2 dx dp f , g H fi , gi X i f,g H 1 Wigner approach to a new two-band envelope function model for quantum transport n. 25 di 22 Facolt di Ingegneria ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Mathematical setting setting Mathematical 2, If the external potential U ext W ( x ) the two band Wigner system admits a unique solution f df Af B C f i dt f (0) f0 H f f cc , f vv , f cv T f0 D A H Wigner approach to a new two-band envelope function model for quantum transport n. 26 di 22 Facolt di Ingegneria ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Mathematical setting

setting Mathematical 2, If the external potential U ext W ( x ) the two band Wigner system admits a unique solution f df Af B C f i dt f (0) f0 f f cc , f vv , f cv p p 2 1 2 A diag i * , i * , * 2 p m x m x 4m x Unbounded operator H T Stone theorem e iAt unitary semigroup on H f1 f 2 2 f3 D( A) f H : , , 2 X1 x x x Wigner approach to a new two-band envelope function model for quantum transport n. 27 di 22 Facolt di Ingegneria ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Mathematical setting setting Mathematical

2, If the external potential U ext W ( x ) the two band Wigner system admits a unique solution f df Af B C f i dt f (0) f0 B diag cc , vv , cv H f f cc , f vv , f cv 0 P C 0 m0 Eg i 0 0 i T 2 2 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 Wigner approach to a new two-band envelope function model for quantum transport n. 28 di 22 Facolt di Ingegneria ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Mathematical setting setting Mathematical 2, If the external potential U ext W ( x ) the two band Wigner system admits a unique solution f

df Af B C f i dt f (0) f0 B diag cc , vv , cv H f f cc , f vv , f cv 0 P C 0 m0 Eg i Symmetric bounded operators ij fij X c U ext B, C B H fij X c U ext 2 2 0 0 0 i W 1, ( x ) T fij

W 2, ( x ) Wigner approach to a new two-band envelope function model for quantum transport X f ij X n. 29 di 22 Facolt di Ingegneria ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Mathematical setting setting Mathematical 2, If the external potential U ext W ( x ) the two band Wigner system admits a unique solution f df Af B C f i dt f (0) f0 The operator A B C The unique solution of (1) is H f f cc , f vv , f cv T generate semigroup f ei A B C t f0 Wigner approach to a new two-band envelope function model for quantum transport n. 30 di 22 ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Facolt di Ingegneria Numerical implementation:

implementation: splitting splitting scheme scheme Numerical i A+ B+C t f e f0 Uniform mesh xi i x Linear evolution semigroup p j j p fi , j fi , j = f ( xi , p j ) e i A+ B+C t f ( xi , p j ) e is a three element vector i A+ B +C t n f ij t n t Discrete operator e i A+ B +C t e iA t 2 e iBt iA t iC t 2 e

e Wigner approach to a new two-band envelope function model for quantum transport n. 31 di 22 ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Facolt di Ingegneria Numerical implementation: implementation: splitting splitting scheme scheme Numerical e i A+ B +C t e iA t 2 e iA t iBt 2 e e iA t iA -1 2 f ij F x e f ij t iC t 2 e

f ij F x f ij f.f.t. pj pj 1 2 A diag i * 2 , i * 2 , 2 p * m 4m m Approximate solution of df Af i dt f (0) f 0 p p 2 1 2 A diag i * , i * , p * 2 m x m x 4m x Wigner approach to a new two-band envelope function model for quantum transport n. 32 di 22 ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Facolt di Ingegneria Numerical implementation: implementation: splitting splitting scheme scheme Numerical e i A+ B +C t

e iB t 2 e iA t iBt 2 e e iA t iC t 2 e t iB -1 2 f ij F p e f ij f.f.t. B diag cc , vv , cv ij Approximate solution of df Bf i dt f (0) f 0 f ij F p f ij 1 Vi x / 2m V j x / 2m 2

B diag cc , vv , cv ij f ij 1 2 Vi x / 2m V j x / 2m v' Wigner approach to a new two-band envelope function model for quantum transport i v v ' e dv ' d n. 33 di 22 ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Facolt di Ingegneria f cc x, p Conduction band x Spa c ec p oor din a te o M nt me um co

or a din te Wigner approach to a new two-band envelope function model for quantum transport n. 34 di 22 ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Facolt di Ingegneria f vv x, p Valence band f cc x, p Conduction band x x Spa c ec p oor din a te o M u nt e m m co or a din p

te Sp ace coo rd ina te inat d r o co tum n e Mom Wigner approach to a new two-band envelope function model for quantum transport e n. 35 di 22 Facolt di Ingegneria ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Wigner approach to a new two-band envelope function model for quantum transport n. 36 di 22 Facolt di Ingegneria ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Stationary state: Thermal distribution f cc x, p Conduction band f cv x, p f vv x, p Valence band f cc x, p Wigner approach to a new two-band envelope function model for quantum transport n. 37 di 22

Facolt di Ingegneria ICTT19 19thInternational Conference on Transport Theory Budapest, July 24-30, 2005 Conclusion Multiband-Wigner model Well posedness of the Multiband-Wigner system Application to IRTD Next steps Extention of MEF model to more general semiconductor Well posedness of Multiband-Wigner model coupled with Poisson eq. Calculation of I-V IRDT characteristic for self-consistent model Wigner approach to a new two-band envelope function model for quantum transport n. 38 di 22

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