Compact model for nanoscale MOSFETs in an intermediate regime - PowerPoint PPT Presentation

Compact model for nanoscale MOSFETs in an intermediate regime between ballistic and drift-diffusion transport G. Mugnaini, G. Iannaccone Dipartimento di Ingegneria dell’Informazione Università di Pisa, Italy Giuseppe Iannaccone Università di Pisa

Outline ! Motivation ! Ballistic DGMOSFET in a closed form ! Vertical electrostatics and local equilibrium ! Ballistic Segmentation of a drift-diffusion channel ! Effects of non-linear ballistic transport on the mobility ! Compact macromodel for the ballistic chain ! Bulk MOSFET under the effects of Fermi-Dirac statistics ! Conclusion Giuseppe Iannaccone Università di Pisa

Motivation Existing compact models do not provide a description for the transition from drift-diffusion to ballistic transport regime. Far from equilibrium transport is commonly described by the introduction of electron heating, which is not valid for totally ballistic MOSFETs. We expect that the partially ballistic transport regime will be important in nanoscale devices and it is already important for high mobility FETs. The Fermi-Dirac statistics is not typically considered in MOSFET models, but can be important for very thin devices or low temperatures. Giuseppe Iannaccone Università di Pisa

A remark EKV-like models = linearized charge models and opposite current fluxes ! EKV ! ACM ! USIM ! UCCM ! Maher-Mead model ! and so on!... Giuseppe Iannaccone Università di Pisa

Ballistic DG MOSFETs Giuseppe Iannaccone Università di Pisa

Ballistic DG MOSFET in a closed form Starting from the Natori theory [Natori,1994] of ballistic FETs, two hemi-maxwellian carrier populations are present on the peak of the barrier:   φ − φ − V V ( ) c s c d qN   φ φ − φ + χ − φ − = + c 2 C V Q e e t t   g g m c b 2    φ − φ −  V V c s c d qN   φ φ = − = − c I I I v e e t t   ds f r th 2   Giuseppe Iannaccone Università di Pisa

Effect of the anisotropy of the effective mass tensor If the anisotropy of the effective mass tensor is considered, the unidirectional thermal velocity is found to vary slowly with the silicon thickness:   ε ε −  −  2 kT 2 kT 2 kT ∑ ∑       + + n , l n , t N exp N exp       π φ π π φ n , l n , t     m  m m  = t t t l t v th N c − ε  − ε  ∑ ∑     = + n , l n , t N N exp 2 N exp     φ φ c n , l n , t     t t N c is the effective density of electron states in all conduction subbands. Remarkably v th does not depend on bias in the case of rectangular confinement. Giuseppe Iannaccone Università di Pisa

Charge based vertical electrostatics(I) Interestingly the vertical electrostatics can be written in a form that is similar to the EKV-like electrostatics:       2 Q Q   − − φ = + φ m m V V log   log   g 0 t t V V     − − 2 C qN s d φ φ +   g c e e t t $ ! ! ! # ! ! ! " V m   where:   2 = φ   V log m t V V   − − s d φ φ +   e e t t plays the role of an equivalent Fermi potential, mean of V s and V d Giuseppe Iannaccone Università di Pisa

Charge based vertical electrostatics(II) Analytic solution through the Lambert W-function [Corless,1993]:     V V − − s d −     V V  φ φ  + g T Q e e t t   φ φ = − φ − χ + − φ   b V W  e  t   c g m t   2 C 2     g     Where the normalisation mobile charge: = φ Q 2 C n g t It is useful to define a threshold voltage   Q qN   ≡ φ − χ + + φ b c V log   T m t   2 C Q g n Giuseppe Iannaccone Università di Pisa

Charge based vertical electrostatics(III) ! Mobile charge density upon the peak     V V − − s d −   V V φ φ   + g T e e t t φ =     Q Q W e t m n   2       ! Total Current     V V − − s d −   V V φ φ     + g T e e V t t φ   =   ds I Q v W  e  tanh t   φ ds n th     2 2   t     Giuseppe Iannaccone Università di Pisa

Gummel symmetry test The proposed MOSFET model is similar to Natori-Rahman- Lundstrom model. Some benefits: ! Explicit equations ! fully symmetrical form. The Natori-Rahman-Lundstrom model does not pass the Gummel simmetry test:   2 d I − = − =   I ( V ) I ( V ) and 0 x x 2   dV = x V 0 x Giuseppe Iannaccone Università di Pisa

Vertical electrostatics and local equilibrium The vertical ballistic electrostatics is fully consistent with local equilibrium transport (i.e. drift-diffusion transport with uniform mobility). If V s =V d =V Fn (V Fn the local quasi Fermi potential):   Q Q   − − = + φ m m V V V log   g T Fn t   2 C qN g c ! Same electrostatics of EKV-like models Giuseppe Iannaccone Università di Pisa

Dissipative MOSFETs Giuseppe Iannaccone Università di Pisa

Ballistic Segmentation of a drift- diffusion channel(I) Consistently with the • approach of Buttiker probes [Buttiker,1986], a drift-diffusion MOSFET can be interpreted as along enough chain of ballistic transistors. Fermi potentials are defined only at the -th contact. If N is the number of MOSFETs in the chain, current continuity imposes N equations for the N unknowns: Giuseppe Iannaccone Università di Pisa

Ballistic Segmentation of a drift- diffusion channel(II) ! For the k-th ballistic MOSFET:     V V + − − k k 1 −   V V φ φ    +  − g T e e V V t t   φ = + =   k 1 k I Q v W  e  tanh with k 0..N - 1 t   φ ds n th     2 2   t     = V V 0 s = V V N d − ! If N is large enough and is small enough: V V + 1 k k V → ! (discrete) (continuous) V k Fn x → ! (nonlinear) (linear) tanh( ) x ! (discrete) (continuous) − → λ ∇ V V V + k 1 k Fn λ Is the length of a ballistic transistor λ λ λ Giuseppe Iannaccone Università di Pisa

Ballistic Segmentation of a drift- diffusion channel(III) With aboveseen simplifications, it is:   − λ V V v   → ∇ + k 1 k th Q v tanh Q ( V ) V   φ φ m , k th m Fn Fn   2 2 t t It is natural to define a low field mobility: λ v µ = th φ n 2 t ! Mobility in terms of mean free path λ λ λ λ ! The length of a ballistic transistor is the mean free path Giuseppe Iannaccone Università di Pisa

Drift-diffusion limit(I)   − V V V   g T − Fn Q v dV  φ φ  =   n th Fn I W e e t t φ ds     2 dx   t ! By means of current continuity along the channel, for a long channel drift-diffusion MOSFET:  −  V V V   L g T − Fn   v dV ∫ = φ φ =   th Fn I L Q W e e dx t t   φ ds n   2 dx   t 0 $ ! ! ! # ! ! ! " Q ( V ) m Fn   =  −  V Vd Fn V V V g T − Fn     φ φ 2 W e e t t      −  V V V g T −   Fn v Q   φ φ =  +  th n W e e t t   φ   2 2   t       = V V Fn s Giuseppe Iannaccone Università di Pisa

Drift-diffusion limit (II) Finally it is:   Q Q   − − = + φ ms ms V V V log   g T s t   2 C qN g c   Q Q   − − = + φ md md V V V log   g T d t   2 C qN g c   µ φ − 2 2 Q Q   = + − n t ms md I Q Q   ds ms md L  2 Q  n ! EKV-like model for non-degenerate long DGMOSFETs subject to rectangular quantum confinement! Giuseppe Iannaccone Università di Pisa

Ballistic Segmentation of the channel(I) N=L/ λ λ λ λ ! A long enough ballistic chain ( N>>1 ) describes a drift- diffusion MOSFET rigorously. ! If N=1 , the ballistic chain reduces to a ballistic transistor ! For intermediate N the ballistic chain describes the transition from drift-diffusion to ballistic transport ! N depends on the low-field mobility and on the length L Giuseppe Iannaccone Università di Pisa

Ballistic Segmentation of the channel(II) 3000 2800 Drift diffusion 2600 2400 2200 2000 1800 I ds [A/m] 1600 1400 1200 1000 800 600 N=30 400 200 0 0.0 0.1 0.2 0.3 0.4 0.5 V ds [V] Giuseppe Iannaccone Università di Pisa