Simple Theory of the Ballistic Nanotransistor

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Transcription:

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

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

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

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

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 )

velocity saturation V DS 1.5V 5 10 4 V/cm L = 60 nm! " velocity cm/s ---> 10 7 " = µe " = " sat 10 4 electric field V/cm --->

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) 0 0.4 0.8 1. 1.4 I D = W C ox V GS "V T ( )# sat I = W C! ( V " V ) D ox sat GS T

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

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

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, 1998. eqn. (3.36) on p. 18 of Fundamentals of Modern VLSI Devices, N A

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

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

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

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

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

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 #

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 1 + 1 + SCF

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

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

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

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

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 "

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

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 )

filling states under bias ε 1 vs. x for V GS = 0.5V f (k x, k y ) ε 1 (ev) ---> Increasing V DS -10-5 0 5 10 X (nm) --->

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) ( )

carrier velocity in a ballistic MOSFET ε 1 vs. x for V GS = 0.5V E C (ev) ---> Increasing V DS Increasing V DS -10-5 0 5 10 X (nm) --->

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) -10-5 0 5 10 X (nm) ---> V DS ---> injection velocity "% T (! ) ( ) kbt $ F % 1/ F1 = * # m & 0! ' ( F F1 )

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)

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

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 (

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, 1994. V DS

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

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 www.nanohub.org I = qn W! " # " [ F ( ) F ( )] D D T 1/ F1 1/ F 6) select new V G, V D, and go to 1

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

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

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

scattering in nano-mosfets measured ballistic Intel 30nm bulk MOSFET Chau et al, IEDM Technical Digest, 000, pp 45-48 MOSFETs operate at 50% of their ballistic limit

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 +

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

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

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.

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, 1853-1864, 003. scattering in nanotransistors: Mark Lundstrom and Zhibin Ren, Essential Physics of Carrier Transport in Nanoscale MOSFETs, IEEE Trans. Electron Dev., 49, pp. 133-141, 00.

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