EE-612: Lecture 11: Effective Mobility Mark Lundstrom Electrical - PowerPoint PPT Presentation

EE-612: Lecture 11: Effective Mobility Mark Lundstrom Electrical and Computer Engineering Purdue University West Lafayette, IN USA Fall 2008 NCN www.nanohub.org Lundstrom EE-612 F08 1

outline 1) Review of mobility 2) “Effective” mobility 3) Physics of the effective mobility 4) Measuring effective mobility 5) Discussion 6) Summary Lundstrom EE-612 F08 2

mobility μ = q τ m * (1/ τ ) Δ t = probability per second of scattering [(1/ τ ) is ‘scattering rate’] τ = average time between scattering events μ 1 = q τ 1 μ 2 = q τ 2 μ 12 = ? μ 12 ≠ μ 1 + μ 2 m * m * − 1 ⎛ ⎞ 1 + 1 μ 12 = ⎜ ⎟ μ 1 μ 2 ⎝ ⎠ Lundstrom EE-612 F08 3 Mathiessen’s Rule

diffusion coefficient D = υ T λ = υ R λ 2 υ T unidirectional thermal velocity υ R Richardson velocity λ mean-free-path for scattering μ = k B T D D ↔ μ q Lundstrom EE-612 F08 4

mobility vs. doping density (silicon) μ L : T − 3/2 μ ( N I ) 1360(480 holes) μ II : T 3/2 N I − 1 ⎛ ⎞ 1 + 1 μ = ⎜ ⎟ μ L μ II ⎝ ⎠ 100(80) ( ) ≈ μ L μ N I << N CR ( ) N I = N D N A ( ) ≈ μ I μ N I >> N CR N CR ≈ 10 17 cm -3 Lundstrom EE-612 F08 5

μ II temperature dependence (silicon) μ ( T ) + μ II : T 3/2 2 m * υ 2 = 3 1 2 k B T T Lundstrom EE-612 F08 6

lattice scattering (silicon) N P ~ k B T (acoustic phonons) h ω − � ω k T μ ( T ) N ~ e (optical phonons) B P μ L : T − 3/2 μ II : T 3/2 μ TOT ( T ) − 1 ⎛ ⎞ 1 + 1 μ TOT = ⎜ ⎟ μ L μ II ⎝ ⎠ T Mathiessen’s Rule Lundstrom EE-612 F08 7

comment (added after lecture) phonons electrons ω ( k ) E ( k ) ω 0 OP AP υ S = ω k E C β − π − π π k π a a a a Phonons in a periodic structure are Electrons in a periodic structure are characterized by “dispersion curve” characterized by “dispersion curve” ω ( β ). Two types of phonons, E ( k ). acoustic (AP) and optical (OP). Lundstrom EE-612 F08 8

high-field transport (silicon) − μ o E y electric field along the υ y = ( ) 2 − υ y direction of current flow 1 + E y E cr υ sat 10 7 E y >> E cr → υ y = − μ o E y υ y → μ o E cr ≡ υ sat E cr = υ sat E y μ 10 4 Lundstrom EE-612 F08 9

field-dependent mobility ( ) E y υ y = μ E y μ ( E y ) μ ( E y ) ≡ − υ y μ o E y μ o = ( ) 2 1 + E y E cr E y 10 4 Lundstrom EE-612 F08 10

outline 1) Review of mobility 2) Effective mobility 3) Physics of the effective mobility 4) Measuring effective mobility 5) Discussion 6) Summary Lundstrom EE-612 F08 11

mobility in an inversion layer V G V S V D drain source ( ) μ n N A , T , E y , E x ? y x V B Lundstrom EE-612 F08 12

effective mobility [ ] μ n ( x , y ) dV ( ) υ y = WC G V G − V T − mV ( y ) I D = − WQ i y dy [ ] dV ∫ L ∫ V DS = WC G μ eff V G − V T − mV ( y ) I D dy 0 0 ∞ ∫ n ( x ) μ n ( x ) dx (If we only consider the x- μ eff = 0 ∞ dependence, i.e. normal to the ∫ n ( x ) dx surface) 0 Lundstrom EE-612 F08 13

effective normal field μ eff ( E eff ) ∞ ∫ n ( x ) E x ( x ) dx E eff ≡ 0 ∞ ∫ n ( x ) dx 0 E eff Lundstrom EE-612 F08 14

15 effective normal field x below threshold Lundstrom EE-612 F08 W − N A 0 ) = qN A W 2 ε Si E S = qN A W ε Si E W / 2 (

effective normal field (ii) above threshold E S = Q S ε Si − N A electron x 0 inversion Q I W DM layer Lundstrom EE-612 F08 16

effective normal field (iii) E x E S = Q S ε Si ( ) E S = E 0 ( ) E S = 1 Q DM + Q I ( ) E eff = E x t inv / 2 ε Si ( ) E ( t inv ) = Q DM E t inv ε Si ⎛ ⎞ x t inv W DM Q DM + Q I E eff = 1 ⎜ ⎟ ε Si ⎝ ⎠ 2 Lundstrom EE-612 F08 17

effective normal field (iv) ⎛ ⎞ Q DM + Q I E eff = 1 (1) ⎜ ⎟ ε Si ⎝ ⎠ 2 V T = V FB + 2 ψ B + Q DM ( ) Q DM = C ox V T − V FB − 2 ψ B C ox ( ) Q I ≈ C ox V GS − V T ( ) Q I = C G V GS − V T ⎛ ⎞ E eff = C ox V T − V FB − 2 ψ B + V G 2 − V T ⎜ ⎟ ⎝ ⎠ ε Si 2 Lundstrom EE-612 F08 18

effective normal field (v) ⎛ ⎞ E eff = C OX V T − V FB − 2 ψ B + V G 2 − V T ⎜ ⎟ ⎝ ⎠ ε Si 2 V FB = − E G 2 q − ψ B ε OX ⎡ V G + V T ⎤ ⎛ ⎞ ⎟ + E G E eff = 2 q − ψ B ⎜ ⎢ ⎥ ⎝ ⎠ ε Si t OX ⎣ ⎦ 2 Lundstrom EE-612 F08 19

outline 1) Review of mobility 2) Effective mobility 3) Physics of the effective mobility 4) Measuring effective mobility 5) Discussion 6) Summary Lundstrom EE-612 F08 20

transport in bulk Si z under low (and modest) fields: Si conduction -6 equivalent ellipsoids band y - n /6 electrons in each one − 1 ⎛ ⎞ * = 2 4 * + = 0.26 m 0 m C ⎜ ⎟ ⎝ ⎠ * 6 m l 6 m t x dominant scattering processes: * = 0.9 m 0 m l (low-field, room temperature) * = 0.19 m 0 m t -acoustic phonons (ADP) -ionized impurities (II) -intervalley phonons (IV) Lundstrom EE-612 F08 21

transport in Si inversion layers is different from transport in bulk Si….. ∞ ∞ ε 2 ′ z ε 3 Si conduction energy --> band y ε 2 ε 1 ′ E F x * = 0.9 m 0 m l ε 1 * = 0.19 m 0 m t 0 x W Lundstrom EE-612 F08 22

transport in Si inversion layers z expectations: Si conduction band y most carriers in unprimed subbands -lighter conductivity m * * ≈ 0.19 m 0 m C x * = 0.9 m 0 m l X * = 0.19 m 0 -suppressed intersubband scattering m t -enhanced intra subband phonon scattering Lundstrom EE-612 F08 23

surface roughness V G V S V D drain source Si : SiO 2 interface is rough ! − 1 ⎛ ⎞ (possibly present too) 1 + 1 + 1 μ eff = ⎜ ⎟ μ L μ II μ SR ⎝ ⎠ V B Lundstrom EE-612 F08 24

effective mobility vs. effective field μ II 1) screening μ eff μ L 2) electron confinement μ SR 3) proximity to the surface E x or V G Lundstrom EE-612 F08 25

26 δ t OX ⇒ δψ S surface roughness scattering SiO 2 Si Lundstrom EE-612 F08 δ t OX V G = V FB + ψ S − Q S C ox ( ) Δ r r

surface roughness scattering (ii) δ t OX ⇒ δψ S ε i E C E F V G E V from Jing Wang, et al., Appl. Phys. Lett ., Aug. 2005 Lundstrom EE-612 F08 27

‘universal’ mobility electrons: ⎛ ⎞ Q DM + Q I E eff = 1 ⎜ ⎟ ε Si increasing N A ⎝ ⎠ 2 μ eff holes: ⎛ ⎞ Q DM + Q I E eff = 1 ⎜ ⎟ N A 1 ε Si ⎝ ⎠ 3 universal N A 4 behavior E eff Lundstrom EE-612 F08 28

outline 1) Review of mobility 2) Effective mobility 3) Physics of the effective mobility 4) Measuring effective mobility 5) Discussion 6) Summary Lundstrom EE-612 F08 29

measuring μ eff I D = W L μ eff Q i V DS R CH = V DS L I D = W μ eff Q I D i V GS L μ eff ( V G ) = ( ) W R CH V G Q i ( V G ) ( ) Q i ( V G ) ≈ C G V GS − V T V DS R CH >> R SD Lundstrom EE-612 F08 30

measuring μ eff (ii) ( ) = C ox V GS − V T ( ) I D V DS = 10mV Q I Q I V G V G V T V G V G ( ) = ( ) dV G ∫ Q I V G C gs V G 0 Lundstrom EE-612 F08 31

universal mobility for electrons Effective mobility (cm 2 /V -s) Effective field (MV/cm) S. Takagi, A. Toriumi, M. Iwase, and H. Tango, IEEE Trans. Electron Dev ., 41 , pp. 2357-2362, 1994 Lundstrom EE-612 F08 32

universal mobility for holes Effective mobility (cm 2 /V -s) Effective field (MV/cm) S. Takagi, A. Toriumi, M. Iwase, and H. Tango, IEEE Trans. Electron Dev ., 41 , pp. 2357-2362, 1994 Lundstrom EE-612 F08 33

outline 1) Review of mobility 2) Effective mobility 3) Physics of the effective mobility 4) Measuring effective mobility 5) Discussion 6) Summary Lundstrom EE-612 F08 34

“field-effect” mobility ( ) ( ) I D = W V DS ≈ W L μ eff Q I V GS L μ eff C G V GS − V T V DS g m = ∂ I D = W L μ C G V DS ( assumes μ is independent of V G ) ∂ V GS V DS L μ FE ≡ g m “field-effect mobility” WC G V DS For a discussion of mobility measurement techniques, see: Narain Arora, MOSFET Models for VLSI Circuit Simulation, Theory and Practice , Springer-Verlag, New York, 1993. Lundstrom EE-612 F08 35

mobility and on-current Linear region current is proportional to the effective mobility: ( ) I D = W L μ eff C G V GS − V T V DS What about the on-current? ( ) I D = WC G υ sat V GS − V T Lundstrom EE-612 F08 36

mobility and on-current (ii) T ( ) I D = WC G υ T V GS − V % 2 − T T ( ) E T = λ λ + l ( ) μ n D n = υ T λ 2 = k B T q � ( ) ε 1 x also: x μ n = q τ m * υ T = 2 k B T π m * smaller m* implies larger ballistic velocity: Lundstrom EE-612 F08 37

effective mobility for the 45 nm technology node ε OX [ ] E eff = ( V G + V T ) 2 + E G 2 q − ψ B ε Si t OX t OX = 1.1 nm ( ) ≈ 2 MV/cm T ≈ 0.25 V E eff V G = 1V V G = V DD = 1.0 V V N A ≈ 2.7 × 10 18 cm -3 → ψ B = 0.48 Lundstrom EE-612 F08 38

universal mobility for electrons Effective mobility (cm 2 /V -s) Effective field (MV/cm) S. Takagi, A. Toriumi, M. Iwase, and H. Tango, IEEE Trans. Electron Dev ., 41 , pp. 2357-2362, 1994 Lundstrom EE-612 F08 39