Beyond the effective-mass approximation: multi-band models of semiconductor devices

Transcription

1 Beyond the eective-mass approximation: multi-band models o semiconductor devices Luigi Barletti Università di Firenze - Dipartimento di Matematica Ulisse Dini Symposium: Mathematical modeling o electron transport in semiconductors SIMI 2004 Conerence Venezia, Isola di San Servolo, Settembre 2004 Beyond the eective-mass approximation p1/28

2 Contributors G lì IC-CNR, Napoli G Borgioli Dip di Elettronica e Telecomunicazioni, Firenze M Camprini Post-Doc Dip di Elettronica e Tel, Firenze L Demeio Dip di Matematica, ncona G Frosali Dip di Matematica pplicata, Firenze C Manzini Perezionamento MTI-SNS, Pisa M Modugno Post-Doc, Dip di Fisica, Firenze O Morandi Dottorato in Elettronica dello Stato Solido, Firenze Beyond the eective-mass approximation p2/28

3 Support GNFM Research Project: Mathematical Models or Microelectronics, 2004 MIUR-COFIN: Mathematical problems o Kinetic Theories, 2002 Beyond the eective-mass approximation p3/28

4 The eective-mass approximation Most o mathematical models o quantum transport in semiconductor devices makes use o the so-called eective-mass approximation This amounts to substituting the true Hamiltonian with the ollowing: Beyond the eective-mass approximation p4/28

5 The eective-mass approximation (continued) The ective-mass tensor arises rom a parabolic approximation o the conduction band: Energy E c (p)! " p 0 Momentum In such approximation the electron[hole] belongs exclusively to the conduction[valence] band Beyond the eective-mass approximation p5/28

6 Interband devices The eective-mass approximation is unable to describe interband tunneling, a quantum eect which plays an important role in modern devices Scheme o the interband diode developed by P Berger s team (Ohio State University, US) Beyond the eective-mass approximation p6/28

7 Multi-band models We need thereore to go beyond the eective-mass approximation and consider more suitable models in which the electron[hole] eels the presence o at least two-bands Beyond the eective-mass approximation p7/28

8 ) ( ) (, * ' ) ( ) ( * Multi-band models We need thereore to go beyond the eective-mass approximation and consider more suitable models in which the electron[hole] eels the presence o at least two-bands 1 - two-band Kane model: -/ +* $#&% -/ +* E Kane, J Phys Chem Solids, 15 Beyond the eective-mass approximation p7/28

9 ) ( ) (, * 8 0 ) ( ) ( * Multi-band models We need thereore to go beyond the eective-mass approximation and consider more suitable models in which the electron[hole] eels the presence o at least two bands 2 - two-band order-1 M-M model: -/ * M-M -/ 17) +* 4/5 O Morandi & M Modugno, 2004 (to appear) Beyond the eective-mass approximation p8/28

10 Multi-band models (continued) ll these models urnish an approximation o the real multi-band dispersion relation: Energy E c (p) E v (p) p 0 Momentum Beyond the eective-mass approximation p/28

11 Multi-band models (continued) s an example, here is the dispersion relation computed with the Kane Hamiltonian or Gas: Energy (ev) E c 10 0 E v Momentum (10 24 Kg m/s) Beyond the eective-mass approximation p10/28

12 Research program Beyond the eective-mass approximation p11/28

13 Research program Quantum kinetic theory Beyond the eective-mass approximation p11/28

14 Research program Quantum kinetic theory Nonlinear/dissipative Wigner equations MB Wigner equations MB thermal equilibrium Beyond the eective-mass approximation p11/28

15 Research program Quantum kinetic theory Nonlinear/dissipative Wigner equations MB Wigner equations MB thermal equilibrium Quantum hydrodynamics Beyond the eective-mass approximation p11/28

16 Research program Quantum kinetic theory Nonlinear/dissipative Wigner equations MB Wigner equations MB thermal equilibrium Quantum hydrodynamics MB-QDD MB-QET MB-QHD Beyond the eective-mass approximation p11/28

17 Research program Quantum kinetic theory Nonlinear/dissipative Wigner equations MB Wigner equations MB thermal equilibrium Quantum hydrodynamics MB-QDD MB-QET MB-QHD Spintronics Beyond the eective-mass approximation p11/28

18 Research program Quantum kinetic theory Nonlinear/dissipative Wigner equations MB Wigner equations MB thermal equilibrium Quantum hydrodynamics MB-QDD MB-QET MB-QHD Spintronics pplications to electronic devices and BEC Beyond the eective-mass approximation p11/28

19 E E > F : D : CB Q N M The Wigner transorm The Wigner transorm E (LK J ;:3 is a unitary mapping o PO N into itsel It allows a quasi-kinetic ormulation o statistical QM E Wigner, Phys Rev, 132 Beyond the eective-mass approximation p12/28

20 W D ) ( V R The Wigner equation The quantum Liouville equation SUT D +* Beyond the eective-mass approximation p13/28

21 W D ) ( V F V * [ W ( Beyond the eective-mass approximation p13/28 The Wigner equation The quantum Liouville equation D SUT R +* is equivalent to the Wigner equation Z : [ ( F Z : YX S T ( F

22 ^ ` _ c ^ ^ How MB Transport Eqs should look like? Single band case (no discrete degrees o reedom): ;:3 Wigner transorm ^ ;b ` ;a \] mixed state Wigner unction Beyond the eective-mass approximation p14/28

23 ^ ` _ c ^ ^ ` _ c ^ ^ Fd How MB Transport Eqs should look like? Single band case (no discrete degrees o reedom): ;:3 Wigner transorm ^ ;b ` ;a \] mixed state Wigner unction Multi-band/spin case (one discrete degree o reedom): ;:3 F d Wigner transorm ^ ;b ` d ^ ;a F \] mixed state Wigner matrix where ;:3 F d ;:3 Beyond the eective-mass approximation p14/28

24 How MBTE should look like? (contd) Now assume >e R e and recall that the Pauli matrices gih j k k j g k j j k g k l m m k g j k k l j are a orthonormal basis o O hermitian matrices over N Beyond the eective-mass approximation p15/28

25 e R >e k j m k j k k j l g g g k k j j k j k m l N O n ^ How MBTE should look like? (contd) Now assume and recall that the Pauli matrices gih are a orthonormal basis o hermitian matrices over Thus, we can decompose the Wigner matrix F d as: r u oqu r t oqt r s oqs r p oqp where the unctions are real Beyond the eective-mass approximation p15/28

26 ~ } { z " ~ How MBTE should look like? (contd) z) ) z) 2 z) 2 ƒ { z2 2 { z2 ) z2 ) 2 ) +ƒl { { z2 z) v wiw wiwyx Explicitly: z) ) z2 2 l 2 ) { zc Beyond the eective-mass approximation p16/28

27 ~ } { z " ~ How MBTE should look like? (contd) z) ) z) 2 z) 2 ƒ { z2 2 { z2 ) z2 ) 2 ) +ƒl { { z2 z) v wiw wiwyx Explicitly: z) ) z2 2 l 2 ) { zc, we have K :3 ;: Putting or a pure state, h or a mixed state, 3 h in analogy with Stokes parameters describing a polarized light beam Beyond the eective-mass approximation p16/28

28 ˆ > g ^ K : Œ Interpretation we have R For Š K gf ;:3 ^ gf h ^L, implies d ^ gf g which, since Beyond the eective-mass approximation p17/28

29 ˆ > g ^ K : Œ > K : Interpretation we have R For Š K gf ;:3 ^ gf h ^L, implies d ^ gf g which, since K ;:3 h Beyond the eective-mass approximation p17/28

30 ˆ > g ^ K : Œ > K : ˆ > > R K : O Interpretation we have R For Š K gf ;:3 ^ gf h ^L, implies d ^ gf g which, since K ;:3 h, R and, or spin expectation in direction K ;:3 F Beyond the eective-mass approximation p17/28

31 > g Interpretation (continued) In the case o Kane or M-M model: >is the observable band index Beyond the eective-mass approximation p18/28

32 > g Interpretation (continued) In the case o Kane or M-M model: >is the observable band index The Wigner unction can thus be given a local meaning: h local expectation o band-index Beyond the eective-mass approximation p18/28

33 [ Z g, g R > - Dynamics The Kane Hamiltonian can be decomposed as ollows: - g h where we put ssume or simplicity that we are describing electrons in a bulk crystal (constant ), and Thus the dynamics o Wigner unctions is given by the ollowing set o equations Beyond the eective-mass approximation p1/28

34 Z Z [, [ Z, [ Z F Dynamics (continued) X - h [ Ž R X SUT -/ Ž7 R X S T h X - Ž7 R X SUT v wiwywiwiwyw wiwiwywiwywix -/ Ž7 R X SUT [ ( Z : [ ( F Z : Ž Beyond the eective-mass approximation p20/28 where

35 [ Z [ Z, [ Z, [ Z Plane-wave dynamics statistical superimposition o plane waves corresponds in the Wigner picture to a space-homogeneous Wigner unction, the : ;: ssuming space-homogeneity and previous equations reduce to h S T - S T S T v wiwywiwywiwiw wywiwiwywiwywix -/ S T Beyond the eective-mass approximation p21/28

36 , [ Z Plane-wave dynamics (continued) Putting and - the spinorial part o the previous system can be presented in the ollowing simple orm: š SUT momentum drit band transitions Beyond the eective-mass approximation p22/28

37 Plane-wave dynamics (continued) Path o a plane wave on the Poincaré sphere: 1 08 w 1 w 2 w Beyond the eective-mass approximation p23/28

38 ˆ > Beyond the eective-mass approximation p24/28 Moment equations R For, deine the local averages: K ;:3 F ;: F K ;:3 F ;: F K ;:3 F ;: œf

39 -, -, Order-0 moment equations h -/ h S T S T S T S T v wywiwywiw wiwiwywiwyx h - Beyond the eective-mass approximation p25/28

40 -, -, - Order-0 moment equations h -/ h S T S T S T S T v wywiwywiw wiwiwywiwyx h - Continuity equation or the total density: h h SUT Beyond the eective-mass approximation p25/28

41 -, -, - - ž Order-0 moment equations h -/ h S T S T S T S T v wywiwywiw wiwiwywiwyx h - Continuity equation or the total density: h h SUT interband current Beyond the eective-mass approximation p25/28

42 , - Ÿ Ÿ Ÿ ( Order-1 moment equations - h œh œ -/ œ, h œ h S T S T S T S T v wywiwywiw wiwiwywiwyx œ -/ œ Where: F F, F d Ÿ d Ÿ œf Ÿ Bohm term F ) F temperature term F Beyond the eective-mass approximation p26/28

43 > Two-band Madelung equations Theorem I are the Wigner unctions o a pure state, then the temperature terms vanish: h ˆ ª R F Beyond the eective-mass approximation p27/28

44 > Two-band Madelung equations Theorem I are the Wigner unctions o a pure state, then the temperature terms vanish: h ˆ ª R F Thereore, the order-0 and order-1 moment equations are a closed system yelding Madelung-like equations or the Kane model, equivalent to the Schrödinger equation E Madelung, Zeitschr Phys, 126 Beyond the eective-mass approximation p27/28

45 Conclusions Beyond the eective-mass approximation p28/28

46 Conclusions We have discussed the meaning o the multi-band approach to quantum transport in semiconductor devices Beyond the eective-mass approximation p28/28

47 Conclusions We have discussed the meaning o the multi-band approach to quantum transport in semiconductor devices We have ocused our attention on the description o a 2-B system by means o Wigner unctions Beyond the eective-mass approximation p28/28

48 Conclusions We have discussed the meaning o the multi-band approach to quantum transport in semiconductor devices We have ocused our attention on the description o a 2-B system by means o Wigner unctions We have seen the orm o 2-B transport equations or the Kane model Beyond the eective-mass approximation p28/28

49 Conclusions We have discussed the meaning o the multi-band approach to quantum transport in semiconductor devices We have ocused our attention on the description o a 2-B system by means o Wigner unctions We have seen the orm o 2-B transport equations or the Kane model We have seen the orm o 2-B Madelung-like QHD equation or the Kane model Beyond the eective-mass approximation p28/28