Scattering Rates for Confined Carriers

Dragica Vasileska

Professor

Arizona State University

Outline

• General comments on matrix element calculation

• Examples of scattering rates calculation– Acoustic phonon scattering– Interface roughness scattering – dominant

scattering mechanism in nanoscale MOSFETs

Matrix Element Calculation

• Suppose we want to calculate the scattering rate out of state k|| in a subband n.

• For that purpose, we will use Fermi’s Golden Rule result:

2' '|| || || || '

2( , ) ( , ) 'nm nn

S k k M k k E E

Transition rate from a state k|| in a subband n into a state k||’ into a subband n’

Matrix element for scattering betweenstate k|| in a subband n into a state k||’ into a subband n’

'|| ||' 2 *|| ||

1, ( ) ( ) ( )

i k k r

m q nM k k d re dz z H R zA

Acoustic Phonon Scattering

• The matrix element for acoustic phonon scattering in the bulk phonon approximation is:

,

||

( )2

q ( , ), R ( , )

iq R iq Rac q q q

q q

z

H R q e a e a eMN

where q q r z

Restricts to longitudinalmodes only

After integrating over the phonon coordinates, the matrix element for scattering between states k|| and k||’, in subbands n and m becomes:

'|| ||

1/ 2'

|| ||

' *|| ||

22' 2

|| ||

'|| ||

1 1,

2 2 2

( ) ( )

2 1 1, ( )

2 2 2

z

ac qq

iq zm n

nm ac q nm zq

n k m qk

M k k NV

k k q dz z e z

qS k k N I q

V

k k q E E E E

Inm(qz)

• In the elastic and equipartition approximation, the total scattering rate out of state k is of the form:

2

2 ||2||

2 2

1 1

( )

1( ) ( )

B acD n m k

mn s nm

n mnm

k Tg E E E

k v W

dz z zW

2D DOS function

• Effective extent of the interaction in the z-direction

• For infinite well: , L is the well width21 nm

nmW L

Interface-Roughness Scattering

oxide

p-type SC

n+ n+

S D

Gx

y

dy

dF

dxdF yx

L

W

z

Gradual Channel Approximation• This model is due to Shockley.• Assumption: The electric field variation in the direction parallel

to the SC/oxide interface is much smaller than the electric field variation in the direction perpendicular to the interface.

)()( xVVVCxQ TGtotN

VS= 0

EC

EFS

EFD= EFS - VD

VDx=0

V(x)x

dxdV

qndxdn

qDxFqnJ nnnn

negligible

)(

Square-Law Theory• The charge on the gate is completely balanced by QN(x), i.e:

• Total current density in the channel:

• Integrating the current density, we obtain drain current ID:Effective Mobility

( )

0 0

( )

0

( )

( , ) ( , )

( , ) ( , )

( ) ( )

c

c

N eff

y xW

D n

y x

n

Q x

N eff ox eff G T

dVI dz dy qn x y x y

dx

dVW qn x y x y dy

dx

dV dVQ x W C W V V V x

dx dx

Effective electron mobility, in which interface-roughness is taken into account.

Effective electron mobility, in which interface-roughness is taken into account.

Mobility Characterization due to Interface Roughness

High-resolution transmission electron micrograph of the interfacebetween Si and SiO2 (Goodnick et al., Phys. Rev. B 32, p. 8171, 1985)

2.71 Å 3.84 Å

100

200

300

400

1012 1013

experimental datauniformstep-like (low-high)retrograde (Gaussian)M

obili

ty [cm

2 /V-s

]Inversion charge density N

s [cm-2]

(aNs + bN

depl)-1

Ns

-1/3

0

500

1000

1500

1015 1016 1017 1018

Mob

ility

[c

m2 /V

-s]

Doping [cm-3]

Bulk samplesBulk samples Si inversion layersSi inversion layers

Phonon

Coulomb

Interface-roughness

317107 cmN A

Interface Roughness

Mathematical Description of Interface Roughness

• In Monte Carlo device simulations, interface-roughness is treated in real space and approximately 50% of the interactions with the interface are assumed to be specu-lar and 50% to be diffusive

• In k-space treatments of interface roughness, the pertur-bing potential is evaluated from:

( ') '

( ) ( ) ( ) ( )

( ) ( )

V z z z r

VV z V z r V z eE z r

z

H R eE z r

• The matrix element for scattering between states k|| and k||’ is:

'|| ||

'|| || 1 2

' * 2|| ||

2' 2 2 2 2

|| || 1 2 1 22

1, ( ) ( ) ( ) ( )

1, ( ) ( )

i k k r

n mnm

i k k r r

nmnm

M k k e z E z z dz d r r eA

M k k e F d r d r e r rA

Fnm

Random variable that is characterized by its autocovariance function which is obtained by averaging over manySamples – R(r)

'|| || 1 2

2' 2 2 2 2

|| || 1 2 1 22

1, ( ) ( )

i k k r r

nmnm

M k k e F d r d r e r rA

When the random process is stationary, the autocorrelation function depends only upon the difference of the variablesr1 and r2.

If R(r) is the autocorrelation function, then its power spectral density is S(q||) and the transition rate is:

For exponential autocorrelation function we have:

Δ = Roughness correlation lengthL = Ruth Mean Square (r.m.s.) of the roughness

2 ||' 2|| ||

( )2( , ) ( ')nm

S qS k k e F E E

A

2 2

2 /|| 2 21

||2

( )1

r L LR r e S q

L q

Conventional MOSFETs:Scaling MOSFETs Down

When we scale MOSFETs down, we reduce the oxide thickness which in turn leads to increased:

- gate leakage due to direct tunneling- more pronounced influence of remote roughness

No exponential is forever…. But we can delay forever….

Gordon E. Moore