course230B3

nMOSFET Schematic

Four structural masks: Field, Gate, Contact, Metal., , , Reverse doping polarities for pMOSFET in N-well.

nMOSFET Schematic

polysilicon

gate gate oxide

V_{g oxide Ground potential.}Source terminal:

z g V_ds Ground potential. Gate voltage: V_g Drain voltage: V_ds Substrate bias y n source+ n drain+ 0 L z voltage: V_bs x depletion region inversion (x,y): Band

bending at any point (x,y).

V(y): Quasi Fermi p-type substrate

region

channel

W

V(y): Quasi-Fermi

potential along the channel.

V(y=0) = 0 V(y=L)

-V_bs V(y=0) = 0, V(y=L)

Drain Current Model

Electron concentration:

n x y n N e i q V kT ( , ) ( ) 2

Electric field:

q n q kTN d 2 k /k /k 2 / 2 N_a

C diti

f

f

i

i

kT q e e N n kT q e kTN dx d y x qV kT q kT a i kT q si a _{1 ( 1)} 2 ) , ( / / 2 / 2 E

Condition for surface inversion:

( , )0 y V y( ) 2 _B

Maximum depletion layer width at inversion:

W y V y N dm si B ( ) 2 ( ) 2 qN_a

Gradual Channel Approximation

Assumes that vertical field is stronger than lateral field in the channel region, thus 2-D Poisson’s eq. can be solved in terms of 1 D vertical slices

of 1-D vertical slices.

Current density eq. (both drift and diffusion):

dV y

( )

Integrate in x- and z-directions,

J x y

q n x y

dV y

dy

n

( , )

n

( , )

( )

dV dV

where is the inversion charge/area.

I y W dV dy Q y W dV dy Q V ds( ) eff i( ) eff i( ) Q y_i q xi n x y dx ( ) ( , ) 0

Current continuity requires I_ds independent of y, integration with respect to y from 0 to L yields

I

W

Vds

Q V dV

( )

I

W

L

Q V dV

ds eff i ds

( )

0

Pao-Sah’s Double Integral

g

Change variable from (x,y) to ( ,V),

V ni q V kT ( ) ( ) ( )/ 2 n x y n V N e i a q V kT ( , ) ( , ) ( )/ s B B s d V e N n q d d dx V n q V Q kT V q a i i ) ( ) / ( ) , ( ) ( / ) ( 2 E Substituting into the current expression,

B s d E( ,V) ds s V q V kT a i N e _{d dV} n W q I / ) ( 2 ) / (

where (V) is solved by the gate voltage eq for a

B eff ds d dV V L q I 0 E( , )

where _s(V) is solved by the gate voltage eq. for a vertical slice of the MOSFET:

V V Q C V kTN C q kT n N e g fb s s fb s si a s i q s V kT 2 2 2 1 2 ( )/ / Cox Cox kT Na

Charge Sheet Model

Assumes that all the inversion charges are located at the silicon surface like a sheet of charge and that there is no

t ti l d th i i l

potential drop across the inversion layer.

Depletion charge: Q_d qN_aW_d 2 _siqN_{a s} Inv. charge: Qi Qs Qd Cox(Vgs Vfb s) 2 siqNa s Change variable: sd s s s s s i eff ds d d dV Q L W I , , ) ( q V V C N kT 2 2( )2 Note that: And: kT q kTN V V C n N q kT V s a si s fb gs ox i a s 2 ) ( ln ₂ a si s fb gs ox V V qN C kT dV 2 ( ) 1 2 d s a si s fb gs ox s q C V V qN d 1 2( )2 2

Charge Sheet Model

d s s s s s a si s fb gs ox a si s fb gs ox s a si s fb gs ox eff ds d qN V V C qN V V C q kT qN V V C L W I , , ( ) 2 ) ( 2 2 ) ( 2

Further approximation: for the last term in the bracket.

s a si s fb gs ox V V qN C ( ) 2 Obtain: d s s a si s a si s ox s fb gs ox eff ds _q qN kT qN C q kT V V C L W I , 2 2 3 2 2 1 2 3/2

where _s,s and _s,d are implicit solutions to the V( _s) eq. with

V = 0 and V_ds, respectively. s s q q L , 3 2 V 0 and V_ds, respectively.

The above drain current expression is valid for all regions of MOSFET operation—a continuous model.

Saturation Behavior

3

2 +V

Plot of the V( _s) eq. for different V_gs values:

2 2.5 ( V ) V = 3 V_gs s 2 +V_B 1 1.5 ce Po te nt ial g 2 V 0 0.5 1 Su rf ac ₂ B 1 V 0 0.5 1 1.5 2 2.5 0

Electron Quasi-Fermi Potential, (V)V

sat rates at certain V depending on V

Explicit, Regional Current Model

p

,

g

To obtain explicit expressions of I_ds(V_gsg , V_ds), regional approximations are necessary.

For MOSFETs biased above threshold, but below saturation,

s,s 2 B , and s,d 2 B + Vds. Then I C W L V V V V qN C V ds eff ox g fb B ds ds si a ox B ds B 2 2 2 2 3 2 2 3 2 3 2 ( ) / ( ) /

Note that separate expressions are needed for subthreshold and saturation regions—piecewise models.

Linear Region I-V Characteristics

For V_ds << V_g , I C W L V V qN C V C W L V V V ds eff ox g fb B si a B ds eff ox g t ds 2 4 ( ) 1E+0 0 8

where is the MOSFET V V qN threshold voltage.

C t fb B si a B ox 2 4 L C_ox L 1E-2 1E+0 0.6 0.8 y s ca le ) y scale) 1E-6 1E-4 0.4 g( ) (arbi trar y ear (arb itrar y Ids Ids 0 0 5 1 1 5 2 2 5 3 1E-10 1E-8 0 0.2 Lo g Li ne 0 0.5 1 1.5 2 2.5 3 Gate Voltage, (V)Vg

Saturation Region I-V Characteristics

Keeping the 2nd order terms in V_ds:

where is the body effect coefficient

I C W L V V V m V ds eff ox ( g t) ds ds 2 2 m ₁ siqNa / 4 B ₁ Cdm ₁ 3tox (Vdsat Idsat)

where is the body-effect coefficient. m

C_ox C W dm ox ox dm 1 1 1 Vg4 (Vdsat, Idsat) I_ds I_{dsat eff}C_ox W V( g Vt) 2 Drai n C urrent Vg2 Vg3 when V_ds = V_dsat = (V_g V_t)/m. C L m ds dsat eff ox 2 Vg1 Typically, m 1.2. Drain Voltage y y

The Dimensionless Parameter, m

Incremental change of potential in a MOSFET due to a gate-voltage modulation near or below threshold.

V_{gs s}

C_ox C_{dm Grounding of the body}

anchored the potential on the bulk side of the depletion

3tox Wdm

bulk side of the depletion

region where = 0. Th t ti l d th V_{gs s} (x) x Q Q

The potential drop across the oxide, ( Q/ _ox)t_ox, is equivalent to ( Q/ _si)[( _si/ _ox)t_ox]. ( ) dm ox ox dm s gs W t C C V m 1 1 3

Pinch-off Condition

From inversion charge density point of view, Q V_i( ) C V_ox( _g V_t mV ) while Q V_i( ) I W L Q V dV ds eff i V_ds ( ) 0 At V V (V V )/ Q_i( ) C V V_ox( _{g t}) At V_ds = V_dsat = (V_g V_t)/m, Q_i = 0 and I_ds = max. I_ds g V_d 0 V_ds V_{d t} V 0 V Source Drain V_dsat V V m g t

Pinch-off from Potential Point of View

V y V V m V V m y L V V m V y LV g t g t g t ds ds ( ) 2 2 2

V (y ) At the pinch-off point,

dV/dy Vg Vt m Q /mC Vds L dV/dy Gradual channel approximation breaks Qi/mCox approximation breaks down.

Current is injected into

0

0 L y

V(y)

Vds

j

the bulk depletion region.

Subthreshold Region

S i V V V m ds g t Vds 1E-2 1E-1 1E+0 ary uni ts) Low Drain Bias Saturation region Sub-threshold region _1E-5 1E-4 1E-3 Cu rr en t ( ar bi tr a _Diffusion Component Linear region Vt Vg 0 0.5 1 1.5 2 1E-8 1E-7 1E-6 Dra in _Drift Component Gate Voltage (V)

Subthreshold Currents

2 / 1 / ) ( 2 2 2 q V kT a i s a si s si s e s N n kT q kTN Q _E P i i 1 t t Q 2 d t Q

Power series expansion: 1st term Q_d, 2nd term Q_i,

Q qN kT q n N e i si a s i a q _s V kT 2 2 ( )/ q s a I W L qN kT q n N e e ds eff si a i q _s kT qV_ds kT 2 1 2 2 / / L 2 _s q N_a I C W L m kT q e e ds eff ox q V_g V_t mkT _{qVds kT} ( ₁) ( )/ ₁ / 2 or, L q d(l I ) 1 kT kT C

Inverse subthreshold slope:

S d I dV mkT q kT q C C ds g dm ox (log ) . . 2 3 2 3 1

Body Effect: Dependence of Threshold

S

Voltage on Substrate Bias

If V_bs 0, I C W L V V V V qN C V V V ds eff ox g fb B ds ds si a ox B bs ds B bs 2 2 2 2 3 2 2 3 2 3 2 ( ) / ( ) / 1.6 1.8 ( V ) tox=200 Å Vfb=0 Na=10 16 _cm 3 V V qN V C t fb B si a B bs 2 2 (2 ) 1.2 1.4 old V oltage, Vt Na=3 1015 cm 3 C t fb B ox dV_{t si}qN_a / (2 2 _B V_bs) 0.8 1 Th re sh o dV q C t bs si a B bs ox ( ) 1 m dV_t 0 2 4 6 8 10 0.6 1 m dV_bs

Dependence of Threshold Voltage on

T

t

Temperature

For n+ _{poly gated nMOSFET, V}

fb = (Eg /2q) B V E q qN C t g B si a B ox 2 4 dV 1 dE qN / d 1 dE d dV dT q dE dT qN C d dT q dE dT m d dT t g si a B ox B g B 1 2 1 1 2 2 1 / ( ) dV k N N 3 m 1 dE

From Table 2 1 dE /dT 2 7 10 4 _{eV/K and (N N )}1/2

dV dT m k q N N N m q dE dT t c v a g (2 1) ln 3 2 1

From Table 2.1, dE_g/dT 2.7 10 eV/K and (N_cN_v) 2.4 1019 _cm 3_.

For N_a 1016 _cm 3 _{and m 1.1,}

a ,

MOSFET Channel Mobility

eff n x x

n x dx

n x dx

i i

( )

( )

0

n x dx

( )

0

It was empirically found that when _eff is plotted against an effective normal field E_eff, there exists a “universal

relationship” independent of the substrate bias doping relationship independent of the substrate bias, doping concentration, and gate oxide thickness (Sabnis and Clemens, 1979). Here _E 1 _Q 1 _Q Here _{d i} si eff Q Q 2 E Since Q_d 4 _siqN_{a B} C V_ox( _t V_fb 2 _B)and Q_i C_ox(V_g V_t), ox t g ox B fb t t V V t V V 6 3 2 eff E V V V_t 0 2 Vg Vt V 0.2 eff E

N-channel MOSFET Mobility

Low field region (low

electron density): Limited by electron density): Limited by impurity or Coulomb

scattering (screened at high electron densities).

Intermediate field region: Limited by phonon

scattering,

High field region (> 1

3 / 1

32500 _E

eff

High field region (> 1

MV/cm): Limited by surface roughness scattering (less temp dependence)

Temperature Dependence

of MOSFET Current

P-channel MOSFET Mobility

i d si eff

Q

Q

3

1

1

E

In general, pMOSFET mobility does not exhibit as “universal” behavior as nMOSFET

Intrinsic MOSFET Capacitance

C WL C C WLC g d 1 1 1 Subthreshold region: C_ox C_d

C

_g

WLC

_ox Linear region: Saturation region: V (y ) Q y C V V y L i( ) ox( g t) 1 Saturation region: Vg Vt m Qi/mCox Vds L Q y L i( ) ox( g t) C_g 2WLC_ox 3 0 V(y) Vds 3 0 0 Source Drain y L

Inversion Layer Capacitance

Gate

Vg In the charge-sheet model, Ci =

C g Qs i and Q_i = C_ox(V_g V_t). Cox (inversion)

In reality, inversion layer has a finite thickness and finite capacitance

Ci

(low freq.)

C

d _Q

d Qi

thickness and finite capacitance.

d Q( ) C C 1 p-type substrate n+ channel d Q dV C C C C C C C C i g ox i ox i d ox i ox ( ) / 1 1 1

Inversion Layer Capacitance

1st order approximation, C_i Q_i /(2kT/q) and

Q_i C_ox(V_g V_t), therefore, C_i/C_ox = (V_g V )/(2kT/ ) V_t)/(2kT/q). Q C V V kT q q V V kT i ox g t g t ( ) 2 ln 1 ( ) 2 0.6 0.8 ( µ C /c m ) Q i 2 _{Na=5 10}16 _cm 3 tox=100 Å Note: Linearly extrapolated 0.4 har ge D en si ty , Q Vfb=0 C_ox(V_g V_t) Linearly extrapolated threshold voltage is typically (2-4)kT/q higher than the

0 0.2 Inv ersi on C h Vt

higher than the

threshold voltage V_t at _s(inv.) = 2 _B.

0 0.5 1 1.5 2 2.5 3 3.5

Short-Channel Effect

If L I I C W V( g Vt ) 2 If L , And . But I I C L m ds dsat eff ox 2 C_g 2WLC_ox 33 gate-controlled barrier _1E-4 1E-3 0.8 1 t ( A /µ m) (mA /µ m) Log scale source drain 1E-6 1E-5 0.4 0.6 rai n Cu rren t ain C u rr en t ( scale Slope ~q/kT

Threshold voltage becomes

sensitive to channel length

and drain bias

1E-80 0 2 0 4 0 6 0 8 1 1 2

1E-7 0 0.2 So u rce-D r S our ce -Dr a Linear scale q/kT V_t

and drain bias.

0 0.2 0.4 0.6 0.8 1 1.2

Gate Voltage (V)

Drain-Induced Barrier Lowering

3 0 V Vds= L=0.2 m 1E+0 1E+1 3.0 V 50 mV 1E-3 1E-2 1E-1 1E+0 ent (A /cm ) L=2.0 m tox=100 Å 1E 7 1E-6 1E-5 1E-4 Drai n cu rr e L=0.35 m _{Na=3 10}16_cm 3 -0.5 0 0.5 1 1.5 1E-8 1E-7 Gate voltage (V)

Lateral Field Penetration

2-D Poisson’s Eq.: qN E E si Ex/ x: gate controlled d l ti h si a si y x qN y x E E depletion charge. si Ey/ y: S/D controlled depletion charge depletion charge.

Note that the characteristic Note that the characteristic length of exponential decay is independent of channel length

MOSFET Geometry and Scale Length

2-D Analysis in a Simplified MOSFET Geometry

Gate L Drain Source n+poly N_a t_ox n+ _ox W_{d n}+ n _d n Substrate

A 2-D boundary-value problem

y

p

with Poisson’s equation

Gate + _l

Subthreshold region:

In AFGH ( id ) 2 2 2 2 0 Drain Source H _{G L} y A B C D E F -t_ox n+poly N_a n+ _n+ 0

Subthreshold region:

(oxide), In ABEF 2 2 x y 2 2 qN B C D E W_d In ABEF (silicon), _x2 _y2 qN_a si x Substrate Boundary conditions:

To eliminate the boundary condition

along GH, along AB, along EF, l CD ( 3t_ox, )y V_g V_fb ( , )x 0 _bi ( , )x L _bi V_ds (W ) 0 y

at the Si/oxide interface, the oxide region is replaced by an equivalent

Si region ( _si/ _ox)t_ox 3t_ox thick.

along CD.

General approach to a 2-D boundary value problem

Gate Drain Source H _{G L} y -tox n+poly 0 L A B C D E F Na n+ _n+ 0 W_d x Substrate Let:

( , )

x y

v x y

( , )

u x y

_L

( , )

u x y

_R

( , )

u x y

_B

( , )

v(x,y) is a solution to the inhomogeneous equation and satisfies

the top boundary condition.

u_LL, u_RR, u_BB are solutions to the homogeneous equation such that (x,y) satisfies the other B.C.’s.

Gate D i S t n+poly Drain Source H _{G L} y A B C D E F -t_ox n poly N_a n+ _n+ 0 W_d

For satisfying the

Boundary conditions:

u x y b n L y W t L n x t W t L n d ox ox ( , ) sinh ( ) sin ( ) * 3 3 3 x Substrate n L W t W t d ox d ox n sinh 3 3 1 n y u x y c n y W t n L W t n x t W t R n d ox d ox ox d ox n ( , ) sinh sinh sin ( ) * 3 3 3 3

1

Note that for u=sin(kx),

d

2

_u/dx

2

_=-k

2

_u;

u x y d n x t L n W t n y L B n ox d ox n ( , ) sinh ( ) sinh ( ) sin * 3 3 1

d u/dx

k u;

And that for u=sinh(ky),

d

2

_u/dy

2

_=k

2

_u.

MOSFET Scale Length

V t W V e t ox bi bi ds L W_dm t_ox 24 ₃2 ( ) / 1 f ( V )

m= Define scale length,

W t dm bi ( bi ds) 0.6 0.8 shold roll-of f 1.2 1.3 1.4 W_dm + ( _si/ _ox)t_ox g 0 2 0.4 han ne l t hr es 1.5 To keep short-channel

effect under control,

L_min should be kept

0 1 2 3 4 0 0.2 L / (W 3t ) Sh or t-ch min p

larger than about 2 .

m = V /_{g s}= 1 + 3t /W_{ox dm}

Depletion Width Scaling

W kT N n q N dm si a i 0 2 4 ln( / ) 10 µ m) q N_a 1 et io n W idt h ( µ 0.1 xim um De pl e

1.0E+14 1.0E+15 1.0E+16 1.0E+17 1.0E+18 1.0E+19 0.01

Substrate Doping Concentration (cm )

Ma

x

-3

Generalized Scale Length

Gate

In the one-region model,

the eigenvalues are:

Source

Drain

t_i W_d si i 1 2

the eigenvalues are:

For two regions, assume

ox d n t W n k 3

Body

L D Frank et al EDL 10/98 eigenvalues: n n k

L D. Frank et al., EDL 10/98

1 1 1 ) ( sin ) ( sinh ) , ( n n i n n L t x y L b y x u B.C. at x = 0: u_L1 = u_L2

(du_L1/dy = du_L2/dy)

1 2 2 ) ( sin ) ( sinh ) , ( n n d n n L W x y L b y x u

(du_L1/dy du_L2/dy)

and _idu_L1/dx = _sidu_L2/dx

Generalized Scale Length

si tan( ti / 1) + i tan( Wd / 1) = 0 1 Lowest eigenvalue: t / 0.8 1 ickn ess, 3 1

/

=

i si 1/3 (SiO )₂ t_i / ₁ SCE exp( L/2 1) L_min ~ 2 ₁ 0 4 0.6 n su lat o r T h 10 3 _{Note that:} 1>Wd, and 1>ti 0.2 0.4 ze d G ate I n 1=2Wd =2ti is always a point of symmetry regardless of _i, _si. 0 0.2 0.4 0.6 0.8 1 0

Normalized Si Depletion Depth,

Nor m al i W_d / ₁ If _i= _si, ₁=W_d + t_i

High-k Gate Insulator

High-k gate insulator is an active area of 1 s, i si/

= t /_i is an active area of Si research because it may replace SiO₂ thereby 0.6 0.8 o r T h icknes s 10 3 1 i 1/3 (SiO )₂ thereby circumventing the tunneling problem. 0.4 0.6 G at e Insulat o 10 But ~ W_d + ( _si/ _i)t_i is valid only when

0 0.2 o rmaliz e d G y t_i << . 0 0.2 0.4 0.6 0.8 1 0

Normalized Si Depletion Depth,

N

o

W /_d

Velocity Saturation

Because of velocity

saturation the saturation of saturation, the saturation of drain current in a

short-channel device occurs at a much lower voltage than V much lower voltage than V_dsat = (V_g V_t)/m for long channel devices.

This causes the saturation current, I_dsatdsat, to deviate from the (V_g V_t)2 _{behavior and}

Velocity-Field Relationship

n n eff

v

_1/

1

E

E

c

1

E

At low fields, v = _{effeffE : Ohm’s law.} As E , v = v_sat = _effEc.

Critical Field: _E vsat

Critical Field:

It is commonly believed that:

n = 2 for electrons n = 1 for holes

eff c

E

n = 2 for electrons, n = 1 for holes.

v_sat is independent of _{eff (vertical field), but Ec} depends on _eff.

Analytical Solution for n=1

I WQ v WQ V dV dy v dV dy ds i i eff eff sat ( ) ( / ) ( / )( / ) 1 C t ti it i th t I b t t Current continuity requires that I_ds be a constant, independent of y. I WQ V I v dV dy ds eff i eff ds ( )

Multiplying dy on both sides and integrating from y = 0 to

L and from V = 0 to V_ds, one solves for I_ds:

V v_sat dy I W L Q V dV V v L ds eff i V eff ds sat ds ( / ) ( ) ( / ) 0 1 Charge-sheet model: Therefore Q V_i( ) C V_ox( _g V_t mV ) I effC W L Vox( / ) ( g V Vt) ds ( / )m 2 Vds 2 Therefore, _I V v L ds eff ds sat ( / ) 1

Saturation Drain Voltage and Current

V V V m V V mv L dsat g t eff g t sat 2 1 1 2 ( ) / ( ) / ( )

The saturation voltage, V_dsat, can be found by solving dI_ds/dV_ds = 0: 0 6 I C Wv V V V V mv L V V mv L dsat ox sat g t eff g t sat eff g t sat ( ) ( ) / ( ) ( ) / ( ) 1 2 1 1 2 1

And the saturation current is:

0.5 0.6 (mA /µ m) tox=100Å 0.3 0.4 ration cu rre nt ( L=0 L=2.5 m L=0.5 m (Dashed: long-ch 0.1 0.2 Drai n sa tu r L=10 m (Dashed: long ch. model,

solid: velocity sat. model)

0 1 2 3 4 5

Velocity-Saturation-Limited Current

At the drain end of the channel when V_ds = V_dsat,

d I W Q ( L)

Q y L_i( ) C V_ox( _g V_t mV_dsat)

and I_dsat = Wv_satQ_i(y = L),

i.e., carriers move at the saturation velocity. This implies that dV/dy at the drain.

Therefore, the gradual channel approximation breaks down and the carriers are no longer confined to the surface channel.

I C Wv V V V V mv L V V mv L dsat ox sat g t eff g t sat eff g t sat ( ) ( ) / ( ) ( ) / ( ) 1 2 1 1 2 1

When (V_g V_t) mv_satL/2 _eff,

I C W L V V m dsat eff ox g t ( )2 2 In the limit of L 0, I_dsat C Wv_{ox sat}(V_g V_t) Long channel limit.

L 2m

Velocity Overshoot: Monte Carlo Simulation

Velocity saturation is derived from the drift

d diff i d l

and diffusion model

which assumes that carriers are always in thermal equilibrium thermal equilibrium with the silicon

lattice.

But if the MOSFET is only a few mean free path (~10 nm) long, carriers do not travel enough distance to establish equilibrium

elocit o ershoot i e carrier elocit at the high field velocity overshoot, i.e., carrier velocity at the high field

Velocity Overshoot

Monte-Carlo simulation:

Even in a 30 nm device, nFET/pFET velocity and therefore current ratio is still 2 because of the difference in effective masses.

The velocity at the source does not greatly exceed 107 _cm/s;

therefore c rrent does not greatl e ceed I C W (V V ) therefore, current does not greatly exceed I_dsat C Wv_{ox sat}(V_g V_t)

Distribution Function

f E

_{E E kT}

f

( )

₍

1

_)/

1

Fermi-Dirac distribution under equilibrium:

e

( f)

1

The standard semi-classical transport theory is based on the Boltzmann transport equation (BTE):

where r is the position, k is the momentum, f(r,k,t) is the distribution k k r k k r k r k k k r k r k v f e f S( , )f ( , ,t)1 f ( , ,t) S( , )f ( , ,t)1 f ( , ,t) t f _E p ( )

function, v is the group velocity, E is the electric field, S(k,k’) is the transition probability between the momentum states k and k’.

The summation on the right hand side is the collison term which The summation on the right hand side is the collison term, which accounts for all the scattering events. The terms on the left hand side indicate, respectively, the dependence of the distribution function on time, space (explicitly related to velocity), and , p ( p y y), momentum (explicitly related to electric field).

Velocities at the Source and at the Drain

I = WQ vds i I = WC (V - V )vds ox g t s Source E₀ I = WC (V - V - V )vds ox g t dsat d Drain 1

Inversion charge density at the source is given by C (V V )

Drain 1

r

Inversion charge density at the source is given by C_ox(V_g-V_t). Inversion charge density at the drain is much lower because of the drain bias.

Scattering theory

At high drain bias, T’=0,

TI

I

Source E0 Let r=n_s-_/n s+, the backscattering coefficient.

TI

I

_ds 0

I

/

W

C V

(

V

)

1

r

Then Drain 1 r

Ref. Lundstrom, EDL, p.361, 1997

I

ds

/

W

C V

ox

(

g

V v

t

)

T

₁

_r

Note that r depends on the low-field mobility near the source Note that r depends on the low field mobility near the source. In the ballistic limit, no collisions in the channel, i.e., r = 0, and

Carrier Thermal Injection Velocity

For 2-D nondegenerate carriers,

dE

E

f

E

EN

(

)

(

)

kT

dE

E

f

E

N

dE

E

f

E

EN

E

0 0

)

(

)

(

)

(

)

(

so

m

kT

v

_rms

2

For uni-directional injection,

m kT dv kT mv dv kT mv v v x x x x x T 2 ) 2 / exp( ) 2 / exp( 0 2 0 2

Injection Velocity in the Degenerate Case

At 0 K, all states below the Fermi energy are filled In Fermi energy are filled. In 2-D, define a Fermi circle with velocity v_F. F v y x x

v

dv

dv

v

v

x 0

4

F v y x x

v

dv

dv

v

x

3

0

V

V

C

m

1

(

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1

2

q

V

V

C

n

mv

m

E

E

N

_{F F s} ox

(

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1

)

(

2

1

2 2 Since V V C_ox( _{g t}) 2 4 q m v_T ox g t 3

I-V Curves of a Ballistic MOSFET

Natori, JAP, p. 4879, 1994.

2

(

)

3/2 (T=0 K)

3

4

/

ox _{ox g t} ds

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C

W

I

2/3/2010 53 , , p , Independent of L! ( )

)

(

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ox g t ds

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