Strained Silicon, Electronic Band Structure and Related Issues.

Transcription

1 Strained Silicon, Electronic Band Structure and Related Issues. D. Rideau, F. Gilibert, M. Minondo, C. Tavernier and H. Jaouen STMicroelectronics,, Device Modeling 850 rue Jean Monnet, BP 16, F Crolles CEDEX, France

2 OUTLOOK (1/4) INPUT STRAIN Strain Matrix matrix Ansys... What to do?? DESIRED VALUES Current & Capacitances 2/35

3 Electronic Structure upon Strain OUTLOOK (2/4) Dispersion relation and Gaps Abinitio: LDA RPA GW Empirical: TB KP EPM 3/35

4 Electronic Structure upon Strain OUTLOOK (3/4) Dispersion relation and Gaps Abinitio: LDA RPA GW Empirical: TB KP EPM Density of States GR Algorithm G. Gilat and J. Raubenheimer, PR 144, 390 (1966) Scattering Times: Fermi Golden rule modified GR Algorithm Integration over Brillouin Zone Carrier Density Mean Carrier Energy Fermi Dirac Statistics Mobility Linear Response Theory Kubo-Greenwood formula 4/35

5 Electronic Structure upon Strain OUTLOOK (4/4) Dispersion relation and Gaps Abinitio: LDA RPA GW Empirical: TB KP EPM Density of States GR Algorithm G. Gilat and J. Raubenheimer, PR 144, 390 (1966) Scattering Times: Fermi Golden rule modified GR Algorithm Integration over Brillouin Zone Carrier Density Mean Carrier Energy Fermi Dirac Statistics Mobility Linear Response Theory Kubo-Greenwood formula Semiconductor Equation Current & Capacitances Poisson Schrodinger Compact Models Current & Capacitances MC Mobility µαi 5/35

6 1 Review for Electronic Band Structure Method Review for Electronic Band Structure Ab initio vs Empirical methods Lower Dimension Aproximation 6X6 KP and Effective mass Hamiltonian A strain example: Si on SiGe buffer 6/35

7 Review for Electronic Band Structure Method Schrodinger Equation Hψ nk = E nk ψ nk Wave function Basis Gaussian-orbital Plane Waves Block Functions TB EPM KP Matrix Elements Evaluation Ab initio DFT + LDA Troullier-Martins psp from fhi98pp Hartwingsen psp including SO C.Hartwingsen et al, PRB, (1998) Fitting parameters (includes SO) EPM Best in Damocles TB Best in Y.M. Niquet et al, PRB (2000) and KP (UTOX) C. Tserbak et al, PRB (1993) Self-consistent evaluation Kohn-Sham Scheme GW correction H KS = Ec + V ion + V HF + Σ ψ nk dr 3 eigenvalues Simple eigenvalue problem Electronic Band Structure, Overlap integral... 7/35

8 Ab Initio: LDA KSS vs GW ABINIT V4.4.3 ENERGY (ev) L G X W K,U L W X K G WAVE VECTOR PSP: Hartwingsen psp (C.Hartwingsen et al, Phys. Rev. B, (1998)) 8/35

9 EPM (local) UTOX AFTER J.R. CHELIKOWSKY AND M.L. COHEN, PHYS. REV. B 14, 556 (1976) ENERGY (ev) L G X W K,U L W X K G WAVE VECTOR NON LOCAL EFFECT 9/35

10 KP UTOX ENERGY (ev) L G X W K,U L W X K G WAVE VECTOR KP GW 10/35

11 Effective mass approximation for Electronic Band Structure Valence Bands: KP and Effective Mass Hamiltonian KP MASS Ek (mev) (111) k(10 8 m 1 ) (100) k(10 8 m 1 ) (110) 11/35

12 Hole curvature mass for Effective Mass Hamiltonian Hole Conduction mass (m0 units) hh lh sh wafer in plane orientation (deg.) 12/35

13 Conduction Bands: KP and Effective Mass Hamiltonian MASS KP Ek (mev) (100) k(10 8 m 1 ) (001) 13/35

14 STRAIN (STUDIED CASE) Si SiGe lattice larger than Si lattice ε = Strain 0.01 e xx x 14/35

15 RELAXED SILICON ENERGY (ev) EQUIVALENT VALLEY 2 3 L G X,Z G Y,X G 15/35

16 2% TENSILE ENERGY (ev) L G X,Z G Y,X G 16/35

17 ENERGY SHIFT VS. STRAIN CONDUCTION BANDS ENERGY (ev) LH SH HH VALENCE BANDS a /a 0 17/35

18 CONDUCTION BANDS (RELAXED) 18/35

19 CONDUCTION BANDS (1% TENSILE) 19/35

20 CONDUCTION BANDS (1% COMPRESSIVE) 20/35

21 VALENCE BANDS (RELAXED) 21/35

22 VALENCE BANDS (1% TENSILE) 22/35

23 VALENCE BANDS (1% COMPRESSIVE) 23/35

24 VALENCE BANDS 1% Tensile Relaxed 1% Compressive Ek (mev) 400 Ek (mev) 400 Ek (mev) (111) k(10 8 m 1 ) (100) k(10 8 m 1 ) (110) (111) k(10 8 m 1 ) (100) k(10 8 m 1 ) (110) (111) k(10 8 m 1 ) (100) k(10 8 m 1 ) (110) 24/35

25 INTEGRATION OVER BRILLOUIN ZONE Electronic Structure Dispersion relation and Gaps Abinitio: LDA RPA GW Empirical: TB KP EPM 1 Density Of States and DOS masses 2 Carrier Density 3 Scattering Rates 25/35

26 DENSITY OF STATES Electronic Structure Dispersion relation and Gaps Abinitio: LDA RPA GW Empirical: TB KP EPM DENSITY OF STATES INTEGRATION ρ E ( E) = δ k [ E E n ( k) ] n BZ G. Gilat and J. Raubenheimer, PR 144, 390 (1966) L U SYMMETRIES W K X 1/48 1/8 26/35

27 DENSITY OF STATES 5 EPM (LINES) GW (DASHED LINES) 5 KP (LINES) GW (DASHED LINES) g E (10 22 cm 3 ev 1 ) g E (10 22 cm 3 ev 1 ) E (ev) E (ev) 27/35

28 DENSITY OF STATES (FB VS. EFF MASSES) HOLES ELECTRONS 28/35

29 MASSES Table 1 Experimental and theoretical band gap, conduction band curvature masses and valence band Luttinger parameters for Silicon. Exp. a k.p GW c EPM d E g (ev) mt (m o ) m l (m o ) γ b γ b b γ 3 a Ref. [11]; b Fit for the 6-level k.p; c with ABINIT V4.3.3 [3]; d Ref. [17]. 29/35

30 DOS MASSES IN SI/SIGE (VALENCE BANDS) m(dos) x=0 x= x=0.2 x= T (K) m lh (DOS) x=0 x= x=0.2 x= T (K) m sh (DOS) x= x=0.1 x=0.2 x= T (K) KP (UTOX) 30/35

31 CARRIER DENSITY VS. STRAIN: FB 1.5% tensile OB 1.5% tensile FB relaxed Si OB relaxed Si FB 1.5% tensile OB 1.5% tensile FB relaxed Si OB relaxed Si n (/cm3) p (/cm3) E F E C (ev) E V E F (ev) UTOX AFTER M. V. FISCHETTI ET AL. IN DAMOCLES 31/35

32 CAPACITANCE (MEASUREMENTS) NMOS PMOS x=0.2 C (µf/mm2) C (µf/mm2) relaxed VG (V) VG (V) 32/35

33 CAPACITANCE (SIMULATED CURVES) x= C (µf/mm2) relaxed C (µf/mm2) VG (V) VG (V) Charge Sheet Model Density Gradient 33/35

34 CURRENT (LOW FIELDS) ID(A/µm2) UTOX relaxed Si UTOX 1.5% tensile VG VFB(V) 34/35

35 CONCLUSIONS Methods for Band Structure STRAINED SILICON Band Structure DOS and Scattering times Capacitances 35/35