Towards a Scalable EKV Compact Model Including Ballistic and Quasi-Ballistic Transport

Transcription

1 2011 Workshop on Compact Modeling Towards a Scalable EKV Compact Model Including Ballistic and Quasi-Ballistic Transport Christian Enz 1,2, A. Mangla 2 and J.-M. Sallese 2 1) Swiss Center for Electronics and Microtechnology (CSEM), Neuchâtel, Switzerland 2) Swiss Federal Institute of Technology, Lausanne (EPFL), Switzerland

2 Outline Introduction State of the Art Behavior of the Continuous Model Conclusions Slide 2

3 Introduction Carrier Transport Mechanisms in MOSFETs L >> λ (carrier mean-free-path) Diffusive transport (drift and diffusion) Collision dominated Well described by conventional mobility theory L ~ λ Quasi-Ballistic transport where carriers encounter limited amount of scattering from source and drain Mobility theory no longer describes transport L < λ Ballistic transport Collision free Controlled by carrier injection from source into the channel source source source L drain drain drain K. Natori, Ballistic/quasi-ballistic transport in nanoscale transistor, Applied Surface Science, Jul Slide 3

4 Coefficients Introduction The Ballistic MOSFET r b r 0.6 b Rb r Te Channel Length [nm] The ballistic transistor remains an ideal device that represents the limit of scalability of a real device. Below 10nm, quasi-ballistic transport needs to be accounted for K. Natori, Ballistic/quasi-ballistic transport in nanoscale transistor, Applied Surface Science, Jul r: Back-scattering coefficient b: Ballisticity Te: Fraction of injected carriers reaching the drain without energy relaxation Rb: Portion of injected flux that rebounds from the drain to the channel due to scattering inside the drain Slide 4

5 Outline Introduction State of the Art Behavior of the Continuous Model Conclusions Slide 5

6 State-of-the-Art Lundstrom s Flux Scattering Theory Transmission view of the device where carriers injected into the channel from the source, across an energy barrier whose height is controlled by the gateto-source voltage The positive directed flux is injected over the source-channel barrier The negative flux is composed of two components: Backscattered flux, and Fraction of flux injected from drain The drain injected flux is suppressed at high drain bias Source Virtual Source Drain 0 L Position Along the Channel Source injected flux Drain injected flux Backscattered flux K. Natori, Ballistic metal-oxide-semiconductor field effect transistor, Journal of Applied Physics, J. McKelvey, R. Longini et al., Alternative Approach to the Solution of Added Carrier Transport Problems in Semiconductors, Physical Review, M. Lundstrom, Elementary scattering theory of the Si MOSFET, IEEE Electron Device Letters, Jul Slide 6 Electron Potential

7 Introduction Lundstrom s Flux Scattering Theory Assuming current continuity, evaluated at the virtual source ID Q 0 0 for V U W v 1 r vx(0) vt b vt 1 r i x DS T 0: inversion charge density at the virtual source 0: average carrier velocity at the virtual source : thermal velocity of electrons 01: back-scattering coefficient function of the gate and drain bias voltages Electron Potential Source Virtual Source Drain 0 L Position Along the Channel Source injected flux Drain injected flux Backscattered flux M. Lundstrom and Z. Ren, Essential Physics of Carrier Transport in Nanoscale MOSFETs, IEEE Transactions on Electron Devices, Slide 7

8 Introduction Lundstrom s Model The drain current including the flux injected from the drain and reaching the source is given by ID 0 1 r 1 e Qi vt W 1 r 1 r 1 1 r V where non-degenerate conditions have been assumed. In a fully ballistic device 0, and the current simplifies to e DS V U DS T U T I W D V U 1 e DS T Q 0v Q 0v for V U V 1 DS U e T i T i T DS T A. Rahman and M.S. Lundstrom, A compact scattering model for the nanoscale double-gate MOSFET, IEEE TED, Mar Slide 8

9 State-of-the-Art Lundstrom s Model: The Backscattering Coefficient The backscattering coefficient,, defined as the ratio of the backscattered flux to the injected flux, captures the scattering occurring in the channel r where l is the length across which the potential drops by a few U T s, which depends on the potential profile in the channel and is the carrier mean-free-path, which depends on the field-dependent mobility In Lundstrom s theory, the source-to-drain and drain-to-source backscattering coefficients are assumed to be equal Even though attempts have been made to evaluate it analytically, it remains largely empirical Slide 9

10 State-of-the-Art Scalable Models from Ballistic to Drift-Diffusion The classical approach to Ballistic modeling is quite simple However, it is not seamlessly scalable It cannot be directly scaled up to result in the conventional drift-diffusion model Designers would need to use different models for short and long channel devices at the same technology node It is apparent then that we need a compact model that either Extends the ballistic model to include the drift-diffusion model, or Extends the drift-diffusion model to include the ballistic case. Slide 10

11 State-of-the-Art Khakifirooz s Semi-empirical Model Follows Lundstrom s transport over the barrier approach. The drain current in the virtual source model is given by ID Q 0 0 F W v i x s Avoids expressing 0, which becomes a model parameter The model is very simple since it uses only 6 measured and 4 fitting parameters 0 is log-exp function including a suitable inversion transition function for continuity between weak and strong inversion V (0) ln 1 exp GS V T U T F Q f i Ceff nut nu T F f 1 V V U 1 exp U GS T T T 2 A. Khakifirooz, O. M. Nayfeh, and D. Antoniadis, A Simple Semiempirical Short-Channel MOSFET Current Voltage Model Continuous Across All Regions of Operation and Employing Only Physical Parameters, IEEE Transactions on Electron Devices, Aug Slide 11

12 State-of-the-Art Khakifirooz s Semi-empirical Model Comparison of electrostatics with the EKV model Normalized source charge density q i (0) Khakifirooz EKV Normalized overdrive voltage (v g -v t ) Slide 12

13 State-of-the-Art Khakifirooz s Semi-empirical Model An empirical saturation function modifies the velocity term to model the transition between saturation and linear regions according to ID Q 0 0 F W v i x s F s V DS V DSAT 1 V 1 DS VDSAT L c =35 nm L c =32 nm A. Khakifirooz, O. M. Nayfeh, and D. Antoniadis, A Simple Semiempirical Short-Channel MOSFET Current Voltage Model Continuous Across All Regions of Operation and Employing Only Physical Parameters, IEEE Transactions on Electron Devices, Aug Slide 13

14 State-of-the-Art Chain of Ballistic Transistors VG VD VS B B B Virtual reservoirs B It has been proposed that a Drift-Diffusion transistor can be represented as a chain of ballistic transistors At each node, the electrons are thermalized in accordance with the Buttiker virtual probe approach There are N-1 Fermi levels but the drain current continuity is ensured G. Mugnaini and G. Iannaccone, Physics-Based Compact Model of Nanoscale MOSFETs Part I: Transition From Drift-Diffusion to Ballistic Transport, IEEE Transactions on Electron Devices, Aug Slide 14

15 State-of-the-Art Chain of Ballistic Transistors The electrostatics formulation of the chain turns out to be similar to the EKV electrostatics Also, the drain current formulation is similar in structure to EKV A nonlinear term accounts for velocity saturation, similar to the field-dependent mobility term G. Mugnaini and G. Iannaccone, Physics-Based Compact Model of Nanoscale MOSFETs Part I: Transition From Drift-Diffusion to Ballistic Transport, IEEE Transactions on Electron Devices, Aug Slide 15

16 State-of-the-Art The Unified Model Whatever be the transport mechanism, there is always a limiting velocity: vlim min vsat, v The drain current can be expressed as a Mathiessen s rule : inj I I min I, I D dd bal sat : drift-diffusion component : ballistic component : velocity saturated component D. Fleury, G. Bidal, A. Cros, F. Boeuf, T. Skotnicki, and G. Ghibaudo, New Experimental Insight into Ballisticity of Transport in Strained Bulk MOSFETs, Symposium on VLSI Technology, Slide 16

17 State-of-the-Art The Unified Model Results of thermal measurements on 22nm strained-si devices Velocity limiting the current is the saturation velocity (due to collisions) and not the thermal injection velocity Even in 22nm effective length devices, the transport is scattering dominated D. Fleury, G. Bidal, A. Cros, F. Boeuf, T. Skotnicki, and G. Ghibaudo, New Experimental Insight into Ballisticity of Transport in Strained Bulk MOSFETs, Symposium on VLSI Technology, Slide 17

18 Outline Introduction State of the Art Behavior of the Continuous Model Conclusions Slide 18

19 Behavior of the Continuous Model EKV Continuous Model The classical DD EKV model has been extended to include ballistic transport The electrostatic is identical to the classical EKV model The ballistic behavior is added following a similar approach to Mugnaini, but with a simpler implementation Still early work and requires additional improvement and validation The characteristics of our continuous model at various channel lengths is shown, and the results are compared with the Ballistic (B) model and Drift-Diffusion (DD) model (without velocity saturation) G. Mugnaini and G. Iannaccone, Physics-Based Compact Model of Nanoscale MOSFETs Part I: Transition From Drift-Diffusion to Ballistic Transport, IEEE Transactions on Electron Devices, Aug Slide 19

20 Behavior of the Continuous Model Behavior of a Continuous Model Very Short Channel For very short devices (L=2nm), the model mostly follows the ballistic model Output Characteristics Transfer Characteristics L=2nm, V G -V T =1V, V S =0 10 L=2nm, V D =1V, V S =0, V T =0 I D.L/W [ma] This model Ballistic model EKV drift-diffusion model I D.L/W [ma] This model Ballistic model EKV drift-diffusion model V DS [V] V GS [V] Slide 20

21 Behavior of the Continuous Model Behavior of a Continuous Model Short Channel For short-channel devices (L=20nm), the model operates as quasi-ballistic, it follows the DD model at low bias and switches to B at higher bias Output Characteristics Transfer Characteristics I D.L/W [ma] L=20nm, V G -V T =1V, V S = This model Ballistic model EKV drift-diffusion model I D.L/W [ma] L=20nm, V D =1V, V S =0, V T = This model Ballistic model EKV drift-diffusion model V DS [V] V GS [V] Slide 21

22 Behavior of the Continuous Model Behavior of a Continuous Model Long Channel For long-channel devices (L=200nm), the model follows the DD model Output Characteristics Transfer Characteristics L=200nm, V G -V T =1V, V S =0 1 L=200nm, V D =1V, V S =0, V T =0 I D.L/W [ma] This model Ballistic model EKV drift-diffusion model I D.L/W [ma] This model Ballistic model EKV drift-diffusion model V DS [V] V GS [V] Slide 22

23 Behavior of the Continuous Model Saturation in Ballistic Transistors Ballistic: ID C W v V V ox T GS T DD in strong inversion: ID C V V W v ox sat GS T Saturation current expression of B is similar to that of DD velocity saturation in SI Saturation occurs because of velocity saturation at the source Velocity overshoot appears in the high field region close to the drain Similar to ballistic transistor, saturation of DD transistor current in weak inversion occurs for Do we also have velocity saturation in DD transistor biased in weak inversion? M. Lundstrom and Z. Ren, Essential physics of carrier transport in nanoscale MOSFETs, IEEE Transactions on Electron Devices, Slide 23

24 Behavior of the Continuous Model Effect of Velocity Saturation on the Current in WI Velocity saturation also affects the current in weak inversion (WI) in saturation i d 20 UT 2U T v L E L The normalized source transconductance is then given by c sat Gms q Ispec g s i with G 2n U G 1 2 U ms d spec T spec c T The G ms /I D ratio is not affected by velocity saturation and remains equal to unity as for the long channel case 0 I 2 D qs Qi Ispec nu where qs with I 1 2 Q Q 2nC U spec c spec spec ox T Velocity saturation parameter c 2 T Slide 24

25 Behavior of the Continuous Model G m / I D Characteristic Including VS c 20 UT 2U T v L E L sat c 1 g ms /i d 0.1 IC I I D spec (saturation) 1 c c 2 1 c Inversion Coefficient, IC A. Mangla, J.-M. Sallese and C. Enz, MIXDES 2011 Slide 25

26 Behavior of the Continuous Model Figure-of-Merit for Low Power RF Design Moderate inversion is a good trade-off for having at the same time high current efficiency and maximum gain at RF for a given current 1 FoM g ms i d f t id 1 i d WI SI FoM 0.1 moderate inversion c Inversion Coefficient, IC 20 UT 2U T c v L E L sat c A. Shameli and P. Heydari, ISLPED 2006 A. Mangla, J.-M. Sallese and C. Enz, MIXDES 2011 Slide 26

27 Outline Introduction State of the Art Behavior of the Continuous Model Conclusions Slide 27

28 Conclusion With the scaling of CMOS, the devices would soon start approaching the ballistic limit Significant advances have been made to model the ballistic and quasi-ballistic behavior in devices. However, a design oriented compact model, scalable from extremely short to long channel lengths and continuous at all inversion levels, is still elusive We are working towards a scalable EKV compact model which includes ballistic transport. The preliminary results look promising Our analysis shows that velocity saturation not only strongly affects the current in strong inversion but has also an impact in weak inversion For short channel devices operating at high frequencies, the optimum operating point lies in moderate inversion and would eventually shift towards weak inversion for nanoscale devices Slide 28