Simple Theory of the Ballistic Nanotransistor

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1 Simple Theory of the Ballistic Nanotransistor Mark Lundstrom Purdue University Network for Computational Nanoechnology

2 outline I) Traditional MOS theory II) A bottom-up approach III) The ballistic nanotransistor IV) Discussion V) Summary

3 nanoscale MOSFETs 130 nm technology (L G = 60 nm) Low V T I DS (ma/µm) Intel Technical J., Vol. 6, May 16, 00.

4 MOSFET IV: low V DS 0 V G V D I D V GS Q i ( x) = "C ox V GS "V T "V(x) ( ) I D = W Q i ( x)" x (x) = W Q i ( 0)" x (0) V DS I D = W C ox ( V GS "V T )µ eff E x E x = V DS L I D = W L µ effc ox ( V GS "V T )V DS

5 MOSFET IV: high V DS 0 V G V D I D V GS V ( x) = ( V GS "V T ) I D = W Q i ( x)" x (x) = W Q i ( 0)" x (0) V DS I D = W C ox ( V GS "V T )µ eff E x E x " V GS! V L T I D = W L µ eff C ox ( V GS "V T )

6 velocity saturation V DS 1.5V V/cm L = 60 nm! " velocity cm/s ---> 10 7 " = µe " = " sat 10 4 electric field V/cm --->

7 MOSFET IV: velocity saturation 0 V G V D E >>10 4 x I D = W Q i ( x)" x (x) = W Q i ( 0)" x (0) I D = W C ox V GS "V T ( )# sat I = W C! ( V " V ) D ox sat GS T

8 MOSFET IV: velocity overshoot µ Position along Channel (µm) Position along Channel (mm) Frank, Laux, and Fischetti, IEDM Tech. Dig., p. 553, 199

9 the MOSFET as a BJT electron energy vs. position S G D V D 0V V GS I D V DS V D = V DD E.O. Johnson, RCA Review, 34, 80, 1973

10 the MOSFET as a BJT BJT Theory: J C = q D MOSFET n Theory: e qv BE /k BT! e qv BC /k BT W B n i N A ( ) = q D n W B n i e qv BE /k BT N A ( 1! e qv CE /k BT ) V BE! " " W # " k S W B! L I DS! W t inv t inv k bt # q( V ) BT q ( ) ( ) GS! VT / mkbt qvds / kbt I D == µ eff Cox $ % m! 1 $ % e 1! e & L ' & q ' E S C V CE! V qn SD! E S = A S = ox V G = V G C ox + C D m (m! 1)C ox! n D n! k T B i $ q µ eff " # % & = e 'q( B /k T B = e 'qv T /m k B T Yuan Taur and Tak Ning, Cambridge Univ. Press, eqn. (3.36) on p. 18 of Fundamentals of Modern VLSI Devices, N A

11 the MOSFET as a BJT log 10 I DS ( V! V ) ~ GS T above threshold: ( ) Q = C V! V i ox GS T Q e! i q S / kbt ~ q( V! V )/ mk T GS T B ~ e! ~ ln( V " V ) S GS T ( " ) I e V V! S / B ~ k T D ~ GS T V GS E.O. Johnson, RCA Review, 34, 80, 1973

12 Outline I) Traditional MOS theory II) A bottom-up approach III) The ballistic nanotransistor IV) Discussion V) Summary

13 a general view of nano-devices Gate E F1 D( E! U SCF ) E F1 -qv D! 1 = h " = h " 1 #

14 top of the barrier model U = E " q! FB SCF C S energy E F1 LDOS E F1 -qv D E C (0) device L ε(x) contact 1 contact h h! 1 =! = = " L/ # x position

15 filling states from the left contact Gate E F1 D( E! U SCF )! 1 = h " 1 N ( E) = D( E! U ) f ( E) 0 1 SCF 1 0 d N( E) N1 ( E) " N = dt! 1

16 filling states from the right contact Gate D( E! U SCF ) Eµ F1 -qv D N ( E) = D( E! U ) f ( E) 0 SCF 0 d N( E) N ( E) " N = dt! " = h #

17 steady-state 0 0 d N( E) N1 " N N " N = + = dt!! 1 0 (ballistic) [ ( ) ( ) ( ) ( )] N =! D1 E f1 E + D E f E de D1 E! D E U " D E U!! " " ( ) 1 # ( $ ) = ( $ ) SCF SCF

18 steady-state current I D 1 ( 0 N ) N 0 N1 " N " " = =!! q I ( ) ( ) D = " M E f1( E)! f( E) de h hd( E) 1 M ( E) $ = " D( E )!! # + #! +! ( ) 1 1

19 NEGF theory Gate E F1 [" 1 ] [H] device [" S ] [" ] E F Non-equilibrium Green s Function Approach (NEGF) S. Datta, IEDM Tech. Dig., 00

20 outline I) Traditional MOS theory II) A Bottom-up approach III) The ballistic nanotransistor IV) Discussion V) Summary

21 assumptions 1) D, planar MOSFET ) 1 subband occupied energy E F1 LDOS E F1 -qv D U " E = E # q! FB SCF C C S L contact 1 contact! 1 =! = L " x ε(x) position 3) parabolic E(k) * D( E) = WL m " E # E C! h 1 ( ) D ( E) = D ( E) = D( E)! 1 M ( E) = L = = L ( ) " * x E m /# W! h * m E

22 procedure 1) assume a ψ S (sets top of the barrier energy) ) fill states energy E F1 LDOS L E F1 -qv D ε(x) +! N = N + N 3) self-consistent electrostatics contact 1 contact! 1 =! = L " x position 4) evaluate current q I D = M ( E) ( f1( E)! f( E) ) de h "

23 filling states (ballistic) N =![ D 1 (E) f 1 (E) + D (E) f (E)] de * m N = WL [ f1( E) f( E) ] de! h " + N [ ( ) ( )] D N = W L F0! F1 + F0! F N at the top of the barrier depends on: V G (through ψ S ) V D (through E F ) E(k) h k m * N * m kbt D =! h ( ) k b T! F1 = E F1 " E C FB + q# S E F1 FB E " q! S C k E F ( ) k b T! F = E F1 " qv D " E C FB + q# S

24 filled states in equilibrium 1 f ( E) = # e = e $ e ( E" EF )/ kbt 1+ e * ( k + k )/ m k T x y B f ( k, k ) e x y! h * F " B ( EF " EC )/ kbt! B ( E E)/ k T m / k T f 0 f (k x, k y )

25 filling states under bias ε 1 vs. x for V GS = 0.5V f (k x, k y ) ε 1 (ev) ---> Increasing V DS X (nm) --->

26 the ballistic current q I D = M ( E) * [ f1( E)! f( E) ] de m " W M ( E) =! h * m E I = qn W! " # " [ F ( ) F ( )] D D T 1/ F1 1/ F alternatively: I " W Q! D Q n n N = W L! "% T T = k B T * " m (! ) ( ) kbt $ F % 1/ F1 = * # m & 0! ' ( F F1 ) " "% # 1 % F (! ) / F (! ) $ 1/ F 1/ F1 = T & ' 1 + F0(! F ) / F0(! F1) ( )

27 carrier velocity in a ballistic MOSFET ε 1 vs. x for V GS = 0.5V E C (ev) ---> Increasing V DS Increasing V DS X (nm) --->

28 velocity saturation in a ballistic MOSFET ε 1 vs. x for V GS = 0.5V "(0) # " T ε 1 (ev) ---> Increasing V DS υ inj (10 7 cm/s) ---> υ(0) X (nm) ---> V DS ---> injection velocity "% T (! ) ( ) kbt $ F % 1/ F1 = * # m & 0! ' ( F F1 )

29 IV of a ballistic MOSFET N Key equations [ ( ) ( )] D N = W L F0! F1 + F0! F I = qn W! " # " [ F ( ) F ( )] D D T 1/ F1 1/ F FB ( E E q ) k T FB ( )! = # + " F1 F1 C S b! = E # qv # E + q" k T F F1 D C S b We must express ψ S in terms of or V G (1D electrostatics) V G and V D (D electrostatics)

30 1D MOS electrostatics (above threshold) Key equations N [ ( ) ( )] D N = W L F0! F1 + F0! F I = qn W! " # " [ F ( ) F ( )] D D T 1/ F1 1/ F (1) () ( ) C V! V = ox GS T qn WL (3) equations (1), (), and (3) give

31 1D MOS electrostatics (above threshold) I W C V V % $ F1/ (! F1 # qvds / kbt ) % 1# & F (! ) & 1/ F1 DS = ox ( GS # T )" T ' ( F0(! F1 # qvds / kbt ) ) & 1+ & F (! ) & & * 0 F1 + N Cox VG VT 0 F1 0 F1 qvds kbt D ( " ) = [ F (! ) + F (! " / )] for non-degenerate statistics: " qvds / kbt # $ 1" e IDS = W Cox ( VGS " VT )! T % " qvds / kbt & ' 1+ e (

32 the ballistic MOSFET I W C V V % $ F1/ (! F # U DS ) % 1# & F (! ) & 1/ F DS = ox ( GS # T )" T ' F0(! F # U DS ) ( & 1+ & F (! ) & & ) 0 F * ideal electrostatics ( ) I DS (on) =W C ox " T V GS #V T quantum conductance " M q h I DS ( V G! V T ) " K. Natori, JAP, 76, 4879, V DS

33 electrostatics (subthreshold and D) V G C G C S C D V S Q = "qn V D! S! = V # C $ + V # C $ + V # C $ % ( ) ( ) ( ) qn G D S S G & ' D & ' S & ' C" C" C" C" (! ) S

34 procedure for a given V G, V D: 1) guess ψ S ) fill states 3) compute improved ψ S! = V # C $ + V # C $ + V # C $ qn % ( ) ( ) ( ) G D S S G & ' D & ' S & ' C" C" C" C" 4) iterate between () and (3) 5) compute current (! ) S see FETToy at I = qn W! " # " [ F ( ) F ( )] D D T 1/ F1 1/ F 6) select new V G, V D, and go to 1

35 outline I) Traditional MOS theory II) A Bottom-up approach III) The ballistic nanotransistor IV) Discussion V) Summary

36 the quantum capacitance ( ) Q = C V! V C Gate Gate G T = C C C inc ins Q + C ( qn ) Q " # C q D E m ( ) S * Q = = D F ~ "! S if C >> C, C! C Q ins Gate ins C ins C S V G = C Q! S I D Q! i nj

37 bandstructure effects in nano-mosfets (001) (111) -tight binding model (sp 3 d 5 s*) (Boyken, Klimeck, et al.) (100) a (110) (010) -Si, Ge, SiGe, GaAs, InAs, (strained or unstrained) (heterostructure channels) -bulk, UTB, nanowire MOSFETs Top-of-the-barrier model k E( k) = h E( k) : tabulated * m analytical numerical

38 scattering in nano-mosfets measured ballistic Intel 30nm bulk MOSFET Chau et al, IEDM Technical Digest, 000, pp MOSFETs operate at 50% of their ballistic limit

39 relation to traditional MOSFET theory I D = W L µ C V eff ox( "V GS T )V DS I = W C V V L µ! ( ) D eff ox GS T low V DS high V DS (long channel) ballistic MOSFET I W C V V % $ F1/ (! F1 # qvds / kbt ) % 1# & F (! ) & 1/ F1 DS = ox ( GS # T )" T ' ( F0(! F1 # qvds / kbt ) ) & 1+ & F (! ) & & * 0 F1 +

40 relation to traditional MOSFET theory q " hd ( E) # I D = ) % & f ( E) $ f ( E) de h!! D ( ) ( ) 1 + ' 1 ( ballistic transport: L! =! = = 1 L ( ) " * x E m /# diffusive transport:! =! = L 1 D eff see: The Ballistic MOSFET, unpublished notes by M.S. Lundstrom, 005

41 Outline I) Traditional MOS theory II) A Bottom-up approach III) The ballistic nanotransistor IV) Discussion V) Summary

42 summary 1) A ballistic, top-of-the barrier model for the MOSFET is easy to formulate. ) The ballistic model provides new insights into the physics of nanoscale MOSFETs. 3) Although not comprehensive, the top-of-the-barrier ballistic model should prove useful in exploring new materials and structures for ultimate CMOS.

43 references the ballistic model: Mark Lundstrom, The Ballistic Nanotransistor, unpublished notes, 005. Anisur Rahman, Jing Guo, Supriyo Datta, and Mark Lundstrom, Theory of Ballistic Nanotransistors, IEEE Trans. Electron. Dev., 50, , 003. scattering in nanotransistors: Mark Lundstrom and Zhibin Ren, Essential Physics of Carrier Transport in Nanoscale MOSFETs, IEEE Trans. Electron Dev., 49, pp , 00.

Physics of Nanoscale Transistors

Physics of Nanoscale Transistors Physics of Nanoscale ransistors Mark Electrical and Computer Engineering University, West Lafayette, IN 1. Introduction 2. he Ballistic MOSFE 3. Scattering heory of the MOSFE 4. Beyond the Si MOSFE 5.